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Development and application of robust many-body methods for strongly correlated systems: from spin-forbidden chemistry to single-molecule magnets

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Development and application of robust many-body methods for strongly correlated systems: from spin-forbidden chemistry to single-molecule magnets

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Content DEVELOPMENT AND APPLICATION OF ROBUST MANY-BODY METHODS FOR STRONGLY CORRELATED SYSTEMS: FROM SPIN-FORBIDDEN CHEMISTRY TO SINGLE-MOLECULE MAGNETS by Pavel Pokhilko A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (CHEMISTRY) May 2020 Copyright 2020 Pavel Pokhilko Acknowledgements During my last year in Moscow State University, when I was thinking about graduate school, I decided to work with Professor Anna Krylov. I was sure of my decision straight from appli- cation to USC in 2014 and I have never regretted my choice. When I emailed her the first time, I was very surprised how quick her reply was. Ever since, her support has been enormous. Regardless of whether the question is of scientific or personal matters, she is always ready to listen and provide invaluable advice. I am deeply grateful to her. She suggested the research topics that are inline with my scientific interests and guided me through the development and application of electronic structure methods. Anna is a good mentor. Her feedback allowed me to improve my English as well as writing and presenting skills. She encouraged me to visit many conferences that broadened my views on chemistry and physics and sharpened my communica- tion skills. All of these allowed me to grow personally and professionally. Anna definitely is a role model to follow in my future career. I am very lucky to have parents, who encouraged my decision to become a scientist. Seeing my interest in natural sciences, they surrounded me with deep support and care. They bought me books, which was not easy to do after the dissolution of USSR. They have been supporting ii my curiosity to nature from my childhood up to this day. They spent a lot of time and effort educating me and helping me in various aspects of life, which I deeply appreciate. I am very grateful to my previous teachers and advisers, who supported my intentions to build a career path in science and prepared me for further steps. Dr. Tatiana Tzhavoronkova taught me physics and math, which helped me to develop analytical thinking. Diring my first days at Moscow State University, Dr. Artem Kosko taught me calculus in a truly unique way. Without his invaluable course, I would have not been able to gain a solid knowledge of math and develop mathematical aspects of the thesis. Dr. Alexander Abramenkov introduced me to programming in C. Dr. Sergey Petrov taught me classical mechanics, electrodynamics, quan- tum mechanics, and special relativity. His courses sparked my interest in quantum mechan- ics and relativistic physics. Prof. Yulia Novakovskaya and Dr. Vladimir Pupyshev introduced me to electronic structure, spectroscopy, and mathematical foundation of quantum mechanics. Dr. Vladimir Khrustov taught me second quantization and introduced me to the idea of effec- tive Hamiltonians in electronic structure, which further became development within EOM-CC in my thesis. The last but not least, Dr. Dmitri Bezrukov was my diploma thesis adviser at the Moscow State University. He taught me C++ programming language and guided me through my first development in quantum chemistry—adaptation of non-Abelian point-group symmetry in many-body theory. This work is devoted to the development and application of electronic structure methods to strongly correlated systems. The results presented in subsequent chapters are produced with the help of collaborators. Dr. Robin Shannon familiarized me with MESMER—program for iii calculation of rates of chemical processes. Dr. David Glowacki and Dr. Hai Wang provided feedback on the manuscript which later became Chapter 4 of this thesis. Daniil Izmodenov helped me with benchmarks of OSFNO approximation that are reported in Chapter 6. I would especially thank Dr. Evgeny Epifanovsky from Q-Chem Inc., who guided me through Q-Chem’s code, explained to me the structure of the libtensor (the library for tensor contractions), and implemented spin–orbit integrals, contributing to the Chapters 2 and 5. I would like to thank the current and previous members of the group for creating a friendly and supportive environment: Kaushik, Sahil, Tirthendu, Maxim, Sven, Wojtek, Madhubani, Pawel, Goran, Ronit, Sourav, Florian, Ilya, Marwa, Natalie, Samer, Arman, Xintian, Bailey. Thanks for being a family all these years. I gained a lot from working with all of you. In particular, your feedback for my presentations at group meetings allowed me to improve my presenting skills. I am especially grateful to Nastya, who has been supporting and mentoring me throughout my entire graduate life. I was lucky to have good roommates who created a healthy atmosphere in the house: Nastya, Mich, Matteo, Isa, Nicolas, Fabio, and Carolina. It was a lot of fun! Finally, I would like to thank Boson—my special roommate, who has been my source of inspiration for many years a . a No cats were harmed in the writing this thesis. iv Figure 1: Boson, the cat, the source of inspiration of the thesis. v Table of contents Acknowledgements ii List of tables x List of figures xiii Abstract xx Chapter 1: Introduction and overview 1 1.1 Effective theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Patterns of electron correlation . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Spin–orbit interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.4 Chapter 1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Chapter 2: General framework for calculating spin–orbit couplings using spinless one-particle density matrices: Theory and application to the equation- of-motion coupled-cluster wave functions 17 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.2.1 Equation-of-motion coupled-cluster methods with single and double exci- tations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.2.2 Spin properties of operators . . . . . . . . . . . . . . . . . . . . . . . . 26 2.2.3 Evaluation of matrix elements . . . . . . . . . . . . . . . . . . . . . . 30 2.2.4 SOMF approximation for closed- and open-shell references . . . . . . . 33 2.2.5 Averaging scheme for interstate matrix elements within a non-Hermitian framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.3.1 Computational details . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.3.2 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.3.3 ZFS in Fe(II) SMM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.3.4 Impact of violation ofL + =L symmetry and triplet form by canonical SOMF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 vi 2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2.5 Appendix A: Evaluation of Clebsh–Gordan coefficients . . . . . . . . . . . . . 48 2.6 Appendix B: Examples of SOC matrix . . . . . . . . . . . . . . . . . . . . . . 49 2.7 Appendix C: Relevant Cartesian geometries . . . . . . . . . . . . . . . . . . . 50 2.8 Chapter 2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 Chapter 3: Quantitative El-Sayed Rules for Many-Body Wavefunctions from Spin- less Transition Density Matrices 60 3.1 Appendix A: Generalized El-Sayed’s rules . . . . . . . . . . . . . . . . . . . . 73 3.2 Appendix B: Reduced spin–orbit matrix elements . . . . . . . . . . . . . . . . 78 3.3 Appendix C: Normalization of the spinless density matrix . . . . . . . . . . . . 79 3.4 Appendix D: Cartesian geometries . . . . . . . . . . . . . . . . . . . . . . . . 80 3.5 Chapter 3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 Chapter 4: Spin-Forbidden Channels in Reactions of Unsaturated Hydrocarbons with O( 3 P) 84 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.2 Molecular orbital framework . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 4.3 Theoretical methods and computational details . . . . . . . . . . . . . . . . . . 89 4.3.1 Calculation and characterization of MECPs . . . . . . . . . . . . . . . 93 4.3.2 Evaluation of spin–orbit matrix elements . . . . . . . . . . . . . . . . . 96 4.3.3 Calculation of ISC rates . . . . . . . . . . . . . . . . . . . . . . . . . . 98 4.4 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 4.4.1 Relevant structures, potential energy scans, and states ordering . . . . . 98 4.4.2 MECPs and SOCs: Cumulative analysis and implications for ISC rates . 101 4.4.3 Analysis of SOCs and extended El-Sayed’s rules . . . . . . . . . . . . 104 4.4.4 Implications for ISC rates . . . . . . . . . . . . . . . . . . . . . . . . . 105 4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 4.6 Appendix A: Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 4.7 Appendix B: Ordering of low-lying electronic states . . . . . . . . . . . . . . . 111 4.7.1 Ethylene+O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 4.7.2 Acetylene+O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 4.8 Appendix C: Evaluation of spin-orbit matrix elements: Effect of spin-contamination115 4.9 Appendix D: Calculation and characterization of MECPs . . . . . . . . . . . . 116 4.9.1 Reduced-mass elimination . . . . . . . . . . . . . . . . . . . . . . . . 119 4.10 Appendix E: Energy dependence of microcanonical rates . . . . . . . . . . . . 120 4.11 Appendix F: The analysis of SOC matrix elements and natural transition orbitals 120 4.12 Appendix G: Relevant Cartesian geometries . . . . . . . . . . . . . . . . . . . 123 4.12.1 Geometries of triplet minima and transition states . . . . . . . . . . . . 123 4.12.2 Geometries of singlet minima . . . . . . . . . . . . . . . . . . . . . . 129 4.12.3 Geometries of MECP . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 4.13 Chapter 4 references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 vii Chapter 5: Double Precision Is not Needed for Many-Body Calculations: Emergent Conventional Wisdom 138 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 5.2 Algorithms and implementation details . . . . . . . . . . . . . . . . . . . . . . 142 5.3 Computational details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 5.4 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 5.4.1 Accuracy of ground-state energies and properties . . . . . . . . . . . . 147 5.4.2 Accuracy of target-state energies in EOM-CCSD . . . . . . . . . . . . 151 5.4.3 Accuracy of gradient evaluation in single precision . . . . . . . . . . . 151 5.4.4 Accuracy of finite-difference frequencies . . . . . . . . . . . . . . . . 154 5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 5.6 Appendix A: Implementation details . . . . . . . . . . . . . . . . . . . . . . . 156 5.7 Appendix B: Relevant Cartesian geometries . . . . . . . . . . . . . . . . . . . 158 5.8 Appendix C: Frequencies and normal modes for optimized structures computed with single and double precision . . . . . . . . . . . . . . . . . . . . . . . . . 166 5.8.1 Frequencies and normal modes of benzene, computed by finite differ- ence with double precision amplitudes and step size of 0.001 ˚ A . . . . . 166 5.8.2 Frequencies and normal modes of benzene, computed by finite differ- ence with single precision amplitudes and step size of 0.001 ˚ A . . . . . 170 5.8.3 Frequencies and normal modes of benzene, computed by finite differ- ence with single precision amplitudes with cleanup in double precision and step size of 0.001 ˚ A . . . . . . . . . . . . . . . . . . . . . . . . . 174 5.8.4 Frequencies and normal modes of benzene, computed by finite differ- ence with double precision amplitudes and step size of 0.01 ˚ A . . . . . 178 5.8.5 Frequencies and normal modes of benzene, computed by finite differ- ence with single precision amplitudes and step size of 0.01 ˚ A . . . . . . 182 5.8.6 Frequencies and normal modes of benzene, computed by finite differ- ence with single precision amplitudes with cleanup in double precision and step size of 0.01 ˚ A . . . . . . . . . . . . . . . . . . . . . . . . . . 186 5.9 Chapter 5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 Chapter 6: Extension of frozen natural orbital approximation to open-shell ref- erences: Theory, implementation, and application to single-molecule magnets 193 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 6.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 6.2.1 Equation-of-motion coupled-cluster methods . . . . . . . . . . . . . . 200 6.2.2 FNO algorithm for closed- and open-shell references . . . . . . . . . . 202 6.2.3 Computational details . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 6.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 6.3.1 Analysis of the original FNO approximation and comparison with OSFNO208 6.3.2 Benzynes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 viii 6.3.3 Triradicals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 6.3.4 Di-copper SMMs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 6.3.5 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 6.3.6 Compactness of the OSFNO truncated virtual space . . . . . . . . . . . 217 6.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 6.5 Appendix A: Additional FNO and OSFNO results . . . . . . . . . . . . . . . . 218 6.6 Appendix B: Relevant Cartesian geometries . . . . . . . . . . . . . . . . . . . 219 6.7 Chapter 6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 Chapter 7: Effective Hamiltonians derived from equation-of-motion coupled-cluster wave-functions: Theory and application to the Hubbard and Heisen- berg Hamiltonians 231 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 7.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 7.2.1 Equation-of-motion coupled-cluster theory . . . . . . . . . . . . . . . 236 7.2.2 Effective Hamiltonians . . . . . . . . . . . . . . . . . . . . . . . . . . 238 7.3 Numerical examples and discussion . . . . . . . . . . . . . . . . . . . . . . . 244 7.3.1 Computational details . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 7.3.2 Model space selection . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 7.3.3 Extraction of parameters from effective Hamiltonians . . . . . . . . . . 247 7.3.4 Hubbard’s effective Hamiltonian . . . . . . . . . . . . . . . . . . . . . 248 7.3.5 Heisenberg’s effective Hamiltonian . . . . . . . . . . . . . . . . . . . . 251 7.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 7.5 Appendix A: Proof of optimality of des Cloizeaux’ transformation . . . . . . . 254 7.6 Appendix B. Hubbard’s effective Hamiltonians . . . . . . . . . . . . . . . . . 255 7.7 Appendix C. Overlaps between target and model spaces . . . . . . . . . . . . . 259 7.8 Appendix D. Example of input . . . . . . . . . . . . . . . . . . . . . . . . . . 261 7.9 Appendix E. Relevant Cartesian geometries . . . . . . . . . . . . . . . . . . . 262 7.10 Chapter 7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 Chapter 8: Future directions 274 8.1 Method development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 8.1.1 Incorporation of relativistic effects beyond the Breit–Pauli Hamiltonian for heavier elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 8.1.2 Further cost reduction through combination of FNO and localization approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 8.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 8.2.1 Application of EOM-SF to multi-center single-molecule magnets via Mayhall’s extrapolation . . . . . . . . . . . . . . . . . . . . . . . . . . 277 8.3 Chapter 8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 ix List of tables 2.1 Spin–orbit coupling constants (cm 1 ) between the 3 B 1 and 1 A 1 states in selected diradicals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.2 Doublet radicals relevant for laser cooling experiments. SOCCs (cm 1 ) between the lowest degenerate and non-degenerate doublet states are computed with EOM-EA-CCSD/cc-pVTZ from closed-shell cationic references (CaF, CaOCH 3 ) and with EOM-IP-CCSD/cc-pVTZ from neutral closed-shell references (AsN + , GeO + ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.3 Atomic SOCCs (cm 1 ) computed for several ions. EOM-SF-CCSD/cc-pVTZ from the hextet 6 S reference was used for Fe 3+ states; EOM-EA-CCSD/cc- pVTZ with electron attachment was used for isoelectronic Fe 2+ and Mn + states from the 6 S Fe 3+ and Mn 2+ references. . . . . . . . . . . . . . . . . . . 42 2.4 Spin-reversal barrier (spin-splitting gap in the multiplet) computed with SOMF EOM-EA-MP2/cc-pVDZ from the hextet reference. Spin–orbit splittings were computed in state-interaction approach with the indicated number of electronic states. The experimental estimate for the barrier is 158 cm 1 . . . . . . . . . . 45 2.5 Extent of violation ofL + =L -symmetry of SOMF with open-shell references, cm 1 . Only A(lowest energy)!B(higher energy) transition is shown a . . . . . . 47 2.6 Unphysical singlet reduced matrix elements for the considered systems, cm 1 . The hydrides were oriented in the way that thathSjjH Lz jjS 0 i elements are not zero. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.1 Spin–orbit mean-field reduced matrix elements of the considered systems in the selected orientations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 3.2 Spin–orbit reduced matrix elements of the considered systems in the selected orientations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 4.1 Relevant energy differences (kcal/mol) for the lowest triplet states of prototypi- cal Cvetanovi´ c diradicals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 4.2 SOCCs (cm 1 ) at equilibrium triplet geometries and at MECPs. . . . . . . . . 103 4.3 Canonical ISC rates (s 1 ) at 300 K (high-pressure limit) and maximal micro- canonical ISC rates within dissociation energy window a . . . . . . . . . . . . . 107 4.4 Comparison of the uncorrected and corrected SOCC values for the propylene and ethylene-derived intermediates. . . . . . . . . . . . . . . . . . . . . . . . 116 x 4.5 Angle between the gradients in mass-weighted coordinates at the optimized MECPs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 4.6 ’s (weights of triplet states) at MECPs. Energies here were computed by CCSD/cc-pVTZ at the CCSD/6-31G* and EOM-EE-CCSD/CCSD/6-31G* geome- tries for the propyne- and acetylene-derived species. . . . . . . . . . . . . . . 119 5.1 Summary of single and double floating-point IEEE 754 standard. . . . . . . . 140 5.2 Convergence thresholds for CCSD calculations. Convergence for equations is the same as for theT amplitudes. . . . . . . . . . . . . . . . . . . . . . . . 144 5.3 CCSD/cc-pVDZ total energies of water clusters. Differences between single- and double-precision energies are shown in the last column. . . . . . . . . . . 150 5.4 CD-CCSD/cc-pVDZ total energies of water clusters. Cholesky threshold of 10 3 was used. Differences between single and double precision energies are shown in the last column. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 5.5 CCSD total energies (hartree) and dipole moments (, a.u.) of uracil, water dimer, formaldehyde, and ethylene-oxygen adduct in various basis sets a . . . . 152 5.6 Mean average deviation (MAD) and standard deviation (STD), J/mol, from ref- erence double-precision CCSD total energies for the G2 set. . . . . . . . . . . 152 5.7 EOM-EE-CCSD total energies (in a.u.) of excited singlet and triplet states of uracil in various basis sets. In each cell the first number is obtained from double- precision calculation, the second number is obtained from single-precision EOM calculation from double-precision reference, and the third number is single- precision EOM from single-precision reference. The fourth number is the dif- ference between double- and single-precision energies (with double-precision reference). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 5.8 EOM-SF-CCSD total energies (in a.u.) of several electronic states of C 6 H 5 N in various basis sets. Symmetry labels refer to state symmetries. In each cell the first number is from double-precision calculation, the second number is from single-precision EOM calculation, and the third number is the difference between double- and single-precision energies. CCSD equations were solved in double precision in all cases. . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 6.1 Vertical excitation energies relative to the 3 B 1 state in the methylene series com- puted with EOM-SF-CCSD/cc-pVTZ in the full virtual space at the triplet state geometry. UHF and ROHF references were used. . . . . . . . . . . . . . . . . 207 6.2 Vertical singlet–triplet and doublet–quartet gaps (eV) computed with EOM-SF- CCSD/cc-pVTZ in the full virtual space at the triplet/quartet state geometry (negative sign corresponds to the singlet/doublet ground states) using UHF and unrestricted PBE reference orbitals. . . . . . . . . . . . . . . . . . . . . . . . 208 6.3 Singlet–triplet gaps in SMMs (cm 1 ) computed with UHF EOM-SF-CCSD/cc- pVDZ in full virtual spaces (negative sign corresponds to the singlet ground state). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 xi 6.4 hS 2 i values of the reference determinant for different orbital choices in ben- zynes; the cc-pVTZ basis set. . . . . . . . . . . . . . . . . . . . . . . . . . . 213 6.5 Spin–orbit coupling constants (cm 1 ) in selected diradicals, computed with EOM-SF-CCSD/cc-pVTZ in the full virtual orbital space. Couplings of the triplet and two states of closed-shell character are shown. . . . . . . . . . . . . 217 6.6 The number of frozen orbitals in different FNO approaches for the lowest triplet state of formaldehyde. The same freezing criterion was applied: occupation truncation threshold preserving 99% of virtual space population. . . . . . . . . 217 7.1 Exchange coupling constants,J (cm 1 ), extracted from the EOM-SF-CCSD/cc- pVDZ calculations in different ways. . . . . . . . . . . . . . . . . . . . . . . 253 7.2 Molecule 1 and Molecule 2. Individual EOM-SF-CCSD amplitudes for exci- tations between the localized open-shell orbitals (r 1 ) and the squared norm of the single amplitudes (jjR 1 jj 2 ). The energies (E, eV) are relative to the CCSD high-spin reference. The smallest eigenvalue of the intermediate I for Molecule 1 and Molecule 2 are 0.76 and 0.45. . . . . . . . . . . . . . . . . . . . . . . . 259 7.3 Magnet 1 and Magnet 2. Individual EOM-SF-CCSD amplitudes for excitations between the localized open-shell orbitals (r 1 ) and the squared norm of the sin- gle amplitudes (jjR 1 jj 2 ). The energies (E, eV) relative to the CCSD high-spin reference and relative to the lowest state (E, eV) are shown. The smallest eigenvalue of the intermediate I for Magnet 1 and Magnet 2 are 0.91 and 0.93. 260 7.4 Magnet 1 and Magnet 2. Individual EOM-SF-CCSD-S amplitudes (R 2 =0) for excitations between the localized open-shell orbitals (r 1 ) and the squared norm of the single amplitudes (jjR 1 jj 2 ). The energies (E, eV) relative to the CCSD high-spin reference and relative to the lowest state (E, eV) are shown. All states are severely spin-contaminated, as can be seen from the leading determi- nants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 7.5 Effect of different geometries and environment on averagedJ constants (cm 1 ) in Magnet 2. The cc-pVDZ basis set was used in all SF-TDDFT calculations. . 260 xii List of figures 1 Boson, the cat, the source of inspiration of the thesis. . . . . . . . . . . . . . . v 1.1 More detailed, deeper theory can zoom in a physical phenomenon uncovering its origin, while an effective theory provides a bird’s eye view on it. In Bloch’s formalism, these two pictures are connected through a wave operator . . . . . 3 1.2 Landscape of methods in electronic structure. This work holds a direction through the wave-function forest. . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3 Which method should I choose? Among a variety of existing methods in quan- tum chemist’s toolbox, EOM-CC is an attractive choice to nail difficult prob- lems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.4 Different choices of EOM-CC operator ^ R a . . . . . . . . . . . . . . . . . . . . 10 2.1 The workflow of the present algorithm (shown by green arrows), Fedorov’s scheme (red arrows), and the approach implemented in Molcas (blue arrows). Here WE label denotes the application of Wigner–Eckart’s theorem. In our scheme, we start from one transition density matrix, extract the triplet part, con- struct the spinless density, and then form three reduced matrix elements. In the Molcas scheme, the spinless transition density matrix is built directly from spin manifoldsjjISi andjjI 0 S 0 i, also known as “tensor” states. In Fedorov’s scheme, the reduced matrix elements are computed from three matrix elements. 33 2.2 Electronic configurations for reference and target states of the considered sys- tems: (a) Methylene-like diradicals: CH 2 , SiH 2 , NH + 2 , PH + 2 ; (b) Diatomic cations: AsN + , GeO + ; (c) CaF and CaOCH 3 ; (d) and (e) Hextet reference (left), one of the quartet d 5 configurations of Fe 3+ (top), and one of the quintet d 6 con- figurations of Fe 2+ , Mn + , (tpa)Fe (bottom). . . . . . . . . . . . . . . . . . . 39 2.3 Top: Structure of (tpa)Fe (C 15 N 4 H 15 Fe) in the hextet neutral state. Iron is shown in red, nitrogens in blue, carbons in gray, and hydrogens in white. Bottom: The lowest spin-split levels of quintet anionic (tpa)Fe , showing a spin-reversal barrier U (the energy gap between the lowest and the highest spin-split states within the ground state multiplet). . . . . . . . . . . . . . . . . . . . . . . . . 43 xiii 3.1 Left: Structure of (tpa)Fe (C 15 N 4 H 15 Fe). Iron is shown in red, nitrogens in blue, carbons in gray, and hydrogens in white. Right: Frontier MOs and electronic configuration of the d 5 hextet reference and relevant target states. The target states are obtained by the attachment of a -electron to one of the three MOs marked by the dashed box: attachment to the two lowest MOs gives rise to degenerate states 1 and 2 and attachment to the next MO gives rise to state 3. . 69 3.2 Left: Structure of the methylethylgalium (EtMeGa) radical (GaC 3 H 8 ). Gallium is shown in pink, carbons in gray, and hydrogens in white. Right: Electronic configuration of the closed-shell cationic reference and relevant target states. The target states are obtained by electron attachment to the two lowest unoccu- pied orbitals of the cation marked by dashed box. . . . . . . . . . . . . . . . . 69 3.3 Spinless triplet NTOs for the transitions between three lowest quintet states in (tpa)Fe . The states 1 and 2 are degenerate. Red, green, and blue axes indicate X,Y , andZ coordinates axes, respectively. The isovalue of 0.050 was used in all the cases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 3.4 Spinless triplet NTOs for the transitions between the two lowest doublet states of EtMeGa system. Red, green, and blue axes indicateX,Y , andZ coordinates axes, respectively. The isovalue of 0.050 was used in all the cases. . . . . . . . 71 4.1 Total yields of the reaction products via ISC in ethene, propene, acetylene, and propyne. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.2 Top: leading electronic configurations of the relevant electronic states. Lone pairs on oxygen (p(O)) and carbon (p(O)) are singly occupied in the 3 A 00 and 1 A 00 states. Bonding () and antibonding ( ) orbitals are singly occupied in 3 A 0 , but in 1 A 0 their occupations are 1.40 and 0.58, revealing a contribution of a closed-shell configuration. Bottom: Singly occupied natural orbitals for the A 0 ( and ) and A 00 (p(O,a 00 and p(C)) states computed at the 3 A 00 constrained geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 4.3 Natural frontier orbitals and their occupations of the lowest singlet at triplet states at the optimized triplet geometry (C 1 ). . . . . . . . . . . . . . . . . . . 89 4.4 Left: leading electronic configurations of the relevant electronic states of the Z-isomer of the C 2 H 2 O diradical. Right: Natural frontier orbitals of the open- and closed-shell states. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 4.5 Top: Electronic configurations resulting from 4-electrons-in-3-orbitals pattern (onlyM s = 0 determinants are shown). Determinants (1)-(3) are of a closed- shell (CS) type and determinants (4)-(9) are of open-shell (OS) type. In ethy- lene+O, at C s structures, determinants (1)-(5) are of A 0 symmetry, and determi- nants (6)-(9) are A 00 . Bottom: electronic configurations of the reference deter- minants used in DIP, EE, and SF calculations. . . . . . . . . . . . . . . . . . . 92 xiv 4.6 Schematic representation of the MECP position relative to the triplet-state min- imum and crossing coordinates at the MECP. Left panel: In the ethylene and propene adducts, the MECP is structurally and energetically close to the triplet minimum. Right panel: In the acetylene- and propyne-derived diradicals, the structure of MECP is similar to that of the singlet diradical and the crossings is energetically far from the triplet minimum. . . . . . . . . . . . . . . . . . . . 94 4.7 Equilibrium structures of the lowest triplet states of C 2 H 4 O (top) and C 3 H 5 O (bottom) computed with CCSD/cc-pVTZ. Adduct formation energy (relative to RH+O( 3 P)) is shown under each structure (plain text: CCSD/cc-pVTZ; bold: CCSD(dT)/cc-pVTZ; ZPE is not included). Bond lengths are in ˚ A. . . . . . . . 99 4.8 Z and E isomers of the 3 A 00 state of the C 2 H 2 O diradical. Adduct forma- tion energy (relative to RH+O( 3 P)) is shown under each structure (plain text: CCSD/cc-pVTZ; bold: CCSD(dT)/cc-pVTZ; ZPE is not included). Bond lengths are in ˚ A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 4.9 EOM-DIP-CCSD/6-31G* energies of low-lying states of C 2 H 4 O (top) and C 2 H 2 O (bottom) along torsional coordinate. The coordinates for the scans were obtained by constrained optimization of the lowest triplet state (CCSD/6-31G*). Sym- metry labels correspond to the states computed at the symmetric structures (see Figs. 4.2 and 4.4). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 4.10 NTOs and SOCCs between the singlet and triplet states at symmetric C 2 H 4 O geometry computed using EOM-SF-CCSD/6-31G* and EOM-EE-CCSD/6-31G* wave-functions. Only orbitals are shown. NTO analysis was performed using A!B transition matrix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 4.11 NTOs and SOCCs between the lowest triplet and singlet states at the triplet- minimum geometries of ethylene-derived intermediate and TC1a, TC1b, TC2a, TC2b propylene-derived intermediates computed by EOM-SF-CCSD/6-31G*. Only orbitals are shown. NTO analysis was performed usingA! B transi- tion matrix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 4.12 Three geometries of C 2 H 4 O, obtained with CCSD/cc-pVTZ (with core elec- trons frozen) with or without symmetry constraint. Adduct formation energy (relative to C 2 H 4 +O( 3 P)) is shown under each structure (plain text: CCSD/cc- pVTZ; bold: CCSD(dT)/cc-pVTZ; ZPE is not included). The low-symmetry structure is the global minimum on the lowest triplet surface, corresponding to the equilibrium geometry of triplet C 2 H 4 O. The 3 A 00 and 3 A 0 structures are obtained under C s constraint and correspond to the two different occupations of MOs (see Figs. 2 and 3 in the main manuscript); these structures are not PES minima. The 3 A 00 structure is close to 3 A 0 . . . . . . . . . . . . . . . . . . . . . 110 4.13 Vertical state ordering (eV) at the 3 A 00 geometry (optimized by CCSD/cc-pVTZ) computed by different methods with the cc-pVTZ basis. . . . . . . . . . . . . 112 xv 4.14 EOM-DIP-CCSD/6-31G* and EOM-SF-CCSD/6-31G* energies. The coordi- nates for the scans were obtained by constrained optimization of the lowest triplet state (CCSD/6-31G*). Symmetry labels correspond to the states com- puted at the symmetric structures. . . . . . . . . . . . . . . . . . . . . . . . . 113 4.15 Vertical state ordering for the Z and E isomers of the C 2 H 2 O diradical; cc-pVTZ. See Fig. 4 of the main manuscript for MOs and electronic configurations. The bold line marks closed-shell singlet state. SF calculations employ 3 A 00 reference. 114 4.16 Frontier MOs of the E and Z isomers of C 2 H 2 O and allyl radical. D and S denote doubly and singly occupied orbitals. . . . . . . . . . . . . . . . . . . . . . . . 115 4.17 Stationary crossing points in C 2 H 2 O. Left: Out-of-plane mode in the seam of 659.46i cm 1 . Middle: Crossing coordinate at MECP geometry, perpendicular to the seam. Right: Equilibrium singlet geometry. . . . . . . . . . . . . . . . . 118 4.18 MECP crossing coordinates in the diradicals derived from ethylene and propy- lene. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 4.19 Microcanonical rates of triplet-singlet ISC. Left top: ethylene+O diradical; Right top: propylene+O diradicals; Left bottom: acetylene+O diradicals; Right bot- tom: propylene+diradicals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 4.20 NTOs and SOCCs between the singlet and triplet states at symmetric C 2 H 4 O geometry, computed using EOM-SF-CCSD/6-31G* and EOM-EE-CCSD/6-31G* wavefunctions. Only-orbitals are shown. A! B transition matrix was used for singular value decomposition. . . . . . . . . . . . . . . . . . . . . . . . . 122 5.1 Left: Implemented CCSD algorithm. Clean-up iterations in double precision are optional. Right: EOM algorithm. . . . . . . . . . . . . . . . . . . . . . . 145 5.2 Wall time speedup for SP scheme with clean-up iterations for the CCSD/cc- pVDZ energy calculations of water clusters. Theoretical estimate is given by Eq. (5.2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 5.3 Example of the change in the interface. In this code a block-tensor operation over double type (“btod”) is generalized to a block-tensor operation over the template type. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 5.4 Example of the output, produced by CCSD code. The first iteration in dou- ble precision provides a clean-up, which is illustrated by a large net change in amplitudes. Changes of individual amplitudes are small, and the procedure converges at the second iteration. . . . . . . . . . . . . . . . . . . . . . . . . 157 5.5 ATT system a , used for CCSD benchmark. . . . . . . . . . . . . . . . . . . . . 158 xvi 6.1 Frozen natural orbitals (FNOs) are defined as eigenstates of the virtual–virtual block of a correlated state density matrix. In the FNO approach, the occupied space is unchanged, but the virtual orbital space is transformed such that the orbitals can be ordered by their relative significance for the correction energy and the orbitals with the lowest occupations can be frozen. Left and right pan- els highlight the difference between calculations using closed- and open-shell references. Because in the latter the and orbital spaces are different, a spe- cial care is needed to arrange the orbitals by maximum correspondence, so that dropped orbitals do not introduce an imbalance in the singly occupied space (marked by dashed box). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 6.2 In the EOM-CC formalism, different manifolds of target states are described by combining a particular reference state and a particular type of excitation oper- ator (orbitals occupied in the reference state are marked by red dashed boxes). Note that in EOM-IP and EOM-DIP, the principal character of target states is described by the hole operators acting in the occupied orbital space (red boxes), whereas in EOM-EA and EOM-DEA it is described by the particle operators acting in the virtual space. In EOM-EE, the EOM operators act in both spaces. In EOM-SF, the leading electronic configurations of the target states are con- fined to the singly occupied orbital space. . . . . . . . . . . . . . . . . . . . . 198 6.3 Structures of benzynes and dehydro-meta-xylylenes. . . . . . . . . . . . . . . 206 6.4 Structures of SMMs. Color scheme: Bronze (copper), blue (nitrogen), red (oxy- gen), gray (carbon), white (hydrogen). . . . . . . . . . . . . . . . . . . . . . . 207 6.5 Errors in energy differences (E, eV) between the four EOM-SF-CCSD tar- get states (M S = 0 triplet component, open-shell singlet, and two closed-shell singlets) and the reference high-spin triplet CCSD state of CH 2 computed with cc-pVTZ. Two schemes are shown: the original FNO (dashed lines) and the new open-shell variant (solid lines). The errors are computed relative to the full spaces EOM-SF-CCSD values. . . . . . . . . . . . . . . . . . . . . . . . . . 209 6.6 hS 2 i for open-shell singlet and triplet states of CH 2 obtained by the original FNO approximation (left) and OSFNO (right); EOM-SF-CCSD/cc-pVTZ. . . . 209 6.7 Analysis of the imbalance between the and orbital spaces in the FNO EOM- SF-CCSD/cc-pVTZ calculation of the M s =0 triplet state of CH 2 , see text and Eq. (6.13). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 6.8 Errors in energy differences (E, eV) between the four EOM-SF-CCSD tar- get states (M S = 0 triplet component, open-shell singlet, and two closed-shell singlets) and the reference high-spin triplet CCSD state of CH 2 computed with cc-pVTZ. Two schemes are compared: OSFNO and freezing the same fraction of canonical orbitals (CF). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 6.9 Errors in singlet–triplet energy gaps forortho-,meta-, andpara-benzynes com- puted with EOM-SF-CCSD/cc-pVTZ and different orbitals (ROHF, UHF, PBE, and OO-MP2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 xvii 6.10 Errors in doublet–quartet gaps in the two DMX triradicals computed with cc- pVTZ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 6.11 Errors in the singlet–triplet gaps in selected SMMs calculated with cc-pVDZ. . 216 6.12 Errors in spin–orbit coupling constant for transitions between triplet and two closed-shell singlet states of XH 2 , EOM-SF-CCSD/cc-pVTZ. . . . . . . . . . 216 6.13 Statistical characteristics of the OSFNO approximation for EOM-SF-CCSD. The shown values are the errors in energy gaps (eV) between the reference CCSD state and the three target EOM-SF states (M S = 0): triplet (T), closed- shell singlet (CS, S) and open-shell singlet (OS, S) with the cc-pVTZ basis set. Panels (a) and (b) show mean values and standard deviations for CH 2 , SiH 2 , NH + 2 , and PH + 2 . Panels (c) and (d) show the results for ortho-, meta-, and para-benzynes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 6.14 Errors in total energies of the high-spin reference state calculated using the orig- inal FNO and OSFNO schemes for CH 2 , SiH 2 , NH + 2 , and PH + 2 with CCSD/cc- pVTZ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 6.15 Errors in the energy gap between open-shell singlet and triplet states calculated using the original FNO and OSFNO schemes with EOM-SF-CCSD/cc-pVTZ. . 221 7.1 Configurations entering three-electron-in-three-centers model Hamiltonians. Each center (or site) is represented by one localized orbital. The configurations are built from different distributions of three electrons on three localized orbitals (numbered 1, 2, and 3 and colored with yellow, green, and violet circles, respec- tively). Each line represents one configuration. Heisenberg’s model space includes only open-shell configurations in which the orbitals are singly occu- pied. A half-filled (i.e., three-electrons-in-three orbitals) Hubbard’s model space also includes ionic configurations in which localized orbitals can host two elec- trons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 7.2 Target and model spaces are connected through a wave operator . The target space can be the full configurational (i.e., Hilbert) space of the system, while model space is a subspace of the full configurational space. . . . . . . . . . . . 239 7.3 Structures of the considered systems. Color code: Copper (bronze), nitrogen (blue), oxygen (red), carbon (gray), hydrogen (white). In Molecule 1 and Molecule 2, representing organic di/triradicals, the unpaired electrons are local- ized on the odd carbon atoms. In single-molecule magnets (Magnet 1 and Mag- net 2), the unpaired electrons are localized on the copper centers. . . . . . . . 245 7.4 Localized orbitals of Molecule 1 and Bloch’s effective Hamiltonian (eV) con- structed with EOM-SF-CCSD/cc-pVTZ. All energies are shifted to achieve zero trace of open-shell submatrix. CS and OS denote closed-shell ionic determi- nants and open-shell determinants, respectively. Blue color marks the numbers corresponding to Hubbard’s Hamiltonian parameters; red color on antidiagonal marks the numbers that do not enter standard Hubbard’s Hamiltonian. . . . . . 248 xviii 7.5 Localized open-shell orbitals (left), Bloch’s (right, top), and des Cloizeaux’ (right, bottom) effective Hamiltonians for Magnet 1. All matrix elements are in cm 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 7.6 Localized open-shell orbitals (left), Bloch’s (right, top), and des Cloizeaux’ (right, bottom) effective Hamiltonians of Magnet 2. All matrix elements are in cm 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 7.7 Localized orbitals of Molecule 2. . . . . . . . . . . . . . . . . . . . . . . . . 256 8.1 Left: RAHBIM, a wheel-shaped Mn 12 complex. Right: AYUCOM, a wheel- shaped Cr 8 complex. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 xix Abstract Strongly correlated open-shell species are common in atmospheric and combustion chem- istry, catalysis, astrochemistry, molecular magnetism, and quantum information science. Elec- tronic structure of such species can be complex and multiconfigurational, imposing strict require- ments on quantum chemical methods. Some systems of interest, such as single-molecule mag- nets, have hundreds (if not thousands) of electrons, presenting a challenge for current electronic structure methods. Moreover, because non-relativistic calculations often do not capture the physics of the systems, the relativistic effects must be included, which also increase computa- tional cost. This work presents development of robust and efficient many-body methods and their applications to challenging open-shell systems. A typical non-relativistic calculation provides electronic states with only one spin projec- tion. However, correct treatment of spin–orbit interaction requires states with all possible spin projections. In Chapter 2, we show how application of Wigner–Eckart’s theorem to one-particle transition density matrices resolves this problem. We present a general formalism, which relies on only one transition density matrix between electronic states with given spin projections and multiplicities. In this approach, the phases of wave-functions are synchronized through Clebsh– Gordan’s coefficients. This choice of phase synchronization avoids the issue of phase arbitrari- ness, appearing within explicit change of coordinate system (discussed as well in Chapter 4), xx which can be related to the phase-blindless of Born–Oppenheimer’s approximation. We ana- lyze a commonly used reduced-cost spin–orbit mean-field approximation (SOMF) and demon- strate that it leads to violation of physical symmetries when applied to open-shell references. Based on spin-tensor formalism, we propose a modified version of SOMF that preserves the cor- rect physics. We benchmarked the approach on several open-shell species, including high-spin tris(pyrrolylmethyl)amine Fe(II) complex, explaining a large spin-reversal barrier of magneti- zation in this single-molecule magnet. Multiconfigurational wave-functions are difficult to interpret. Moreover, spin-dependent properties require analysis of wave-functions with different spin projections. In Chapter 3, we extend the concept of natural transition orbitals (NTO) to natural orbitals of triplet spinless transition density matrix. This approach allows one to analyze the transitions between many- body wave-functions of different spin projections in a uniform and compact manner. We used this approach to validate, quantify, and extend empirically known El-Sayed’s rules. In Chapter 4, we study electronic structure of Cvetanovi´ c’s diradicals, which are formed in the reactions between atomic oxygen ( 3 P) and unsaturated hydrocarbons. These diradicals show significant variations in intersystem crossing rates, which we explain on the basis of cal- culations. We foudn that minimal energy crossing points (MECP) in ethylene- and propylene- derived species are energetically close to triplet minima, making the spin-forbidden reactions nearly barrierless. MECP barriers in acetylene- and propyne-derived species are higher, corre- lating with experimentally observed slower rates of spin-forbidden transitions. The magnitude of spin–orbit couplings is explained using the NTO analysis. This analysis also highlights the importance of quantification of El-Sayed’s rules: intermediate values of spin–orbit coupling constants are observed for partial rotation ofp-orbitals on oxygen and carbon atoms. In Chapter 5, we reconsider the choice of numerical precision for quantum chemical calcula- tions. Traditionally, double precision is used in most post-Hartree–Fock calculations. We found xxi that single precision can be safely used for both ground and excited-state energies, properties, and geometries. The resulting accuracy of the approach leads to errors that are 100–1000 times smaller than the target thermochemical accuracy and are comparable with the typical thresholds used in calculations. In Chapter 6, we explore virtual space truncation for correlated methods based on open- shell references. Virtual space truncation reduces the cost of calculations, especially for large systems. We show that a typical truncation scheme based on frozen virtual natural orbitals (FNO) gives large errors when applied for equation-of-motion spin-flip coupled-cluster method (EOM-SF-CC). We found that the reason of the large errors lies in the inconsistent freezing of and spin-orbital subspaces. Additional issues appeared due to a partial inclusion of strongly correlated orbitals to the frozen orbital subspace. We developed a new open-shell frozen natural orbital approximation (OSFNO), which separates strongly correlated orbitals based on singular value decomposition (SVD) of the overlap matrix followed by establishing corresponding natural-like orbitals through SVD of one-particle singlet state density matrix in the remaining virtual orbital subspace. OSFNO delives robust performace. This approach can be used to significantly reduce the cost of large systems such as molecular magnets. Model Hamiltonians are widely used to describe strongly correlated systems. Parametriza- tion of these Hamiltonians is often obtained through fitting of experimental observables. How- ever, the connetction to theory is not often simple. For example, the Lande interval rule can be used to extract exchange coupling constants for the systems with two radical centers. More complicated systems, such as molecular magnets with multiple radical centers, present a chal- lenge for interpretation of calculations in terms of Heisenberg’s Hamiltonian. In Chapter 7, we develop an exact mapping between EOM-CC configurations and model spaces. Based on Bloch’s theory, we show how Hubbard’s and Heisenberg’s Hamiltonians can be rigorously xxii parametrized from the EOM-SF-CC wave-functions. This approach allows one to make a rig- orous connection between many-body calculations and experimental observables. xxiii Chapter 1: Introduction and overview Electronic structure is at the botton of many phenomena around us. For example, electronic structure is responsible for optical properties of materials such as absorption and reflectance. Our eyes perceive these properties at certain wavelengths as colors. In section 1.1, we guide the reader toward the idea of microscopic description starting from macroscopic observations. If we want to explain the macroscopic consequences of microscopic phenomena, a quantitative treatment of electronic structure is needed, which requires a proper incorporation of electron correlation. The nature of electron structure is built in quantum chemical methods, target- ing a certain pattern of electron correlation, reviewed in section 1.2. This work is devoted to many-body methods for open-shell systems featuring strong electron correlation. Treatment of such systems requires inclusion of multiple electronic configurations, making these calculations expensive. Description of spin-forbidden processes and zero-field splitting requires inclusion of relativistic effects, which makes the calculations even more expensive. The goal of this work is development of robust many-body methods (combined with treatment of relativistic effects), their application on strongly correlated systems, and the analysis of the underlying electronic structure. The thesis is organized as follows: The rest of the introduction guides through the effective theories, electronic correlation problem, and spin–orbit interaction. 1 Chapter 2 describes a general algorithm for computing spin–orbit couplings within many- body electronic structure methods. An application to simple radicals and single-molecule magnets is presented as well. The results are published in Ref. 1. Chapter 3 presents the analysis of spin–orbit couplings through singular value decom- position of spinless one-electron transition density matrix. This analysis allows one to quantify El-Sayed’s rules. This chapter summarizes the results published in Ref. 2. Chapter 4 is devoted to the analysis of spin-forbidden channels in the reactions between O( 3 P) and unsaturated hydrocarbons. The results presented in this chapter are also pub- lished in Ref. 3. Chapter 5 describes a single-precision setup of EOM-CC calculations to reduce compu- tational cost. Various benchmarks of energies, properties, and optimized geometries are reported. This chapter summarizes the results published in Ref. 4. Chapter 6 contains a derivation of open-shell frozen natural orbital approximation and provides benchmarks on di- and triradicals; these results are published in Ref. 5. Chapter 7 derives effective Hamiltonians from the EOM-CC wave-functions. The cases of Hubbard’s and Heisenberg’s Hamiltonians are presented; these results are published in Ref. 6. Chapter 8 discusses an impact of the thesis on the field and possible directions for future developments. 1.1 Effective theories Effective theory is, perhaps, the most important concept in physics; it runs through nearly all areas of physics, from hydrodynamics to cosmology. A theory is called effective if it can 2 Detailed whisker theory Effective cat theory Ω -1 Ω Figure 1.1: More detailed, deeper theory can zoom in a physical phenomenon uncovering its origin, while an effective theory provides a bird’s eye view on it. In Bloch’s formalism, these two pictures are connected through a wave operator . be derived bottom-up, on the basis of a deeper and more fundamental theory that provides a parametrization of effective interactions. Let us consider a pedagogical example of why the sky is blue a . In 19th century, John Tyndall made a glass tube and observed a white light beam coming through this tube 8 . He noticed that if he fills the tube with a fog, the fog becomes bluish when observed perpendicular to the direction of beam propagation, but reddish when observed directly from the end of the tube. He proposed that the blue light is scattered stronger by fog droplets. On the basis of this observation, he explained that the sky is blue (and the subset is red) due to presence of particles, whose nature was unknown at that time, that scatter the light. This was a phenomenological theory b that explained the presence of color but did not explain its intensity. Later, Lord Rayleigh showed how the intensity and cross-section of elastic light scattering by small objects can be explained from the size and the refractive index of the particle 9–12 . Microscopically, these particles are air molecules (mainly N 2 and O 2 ) and their refractive properties are determined by their polarizabilities 13 . At the same time, most of ultra- violet radiation is absorbed by a thin ozone layer in the stratosphere. To answer the question of a To combat climate change, humankind may dispense small particles in the atmosphere 7 . In this case, future generations would no longer see a blue sky due to a change of a character of light scattering. b Phenomenological theories are theories that are based on empirical data. Phenomenological thermodynamics is an example of such theory. Statistical mechanics, which was developed much later, derived the postulates of thermodynamics from first principles, making the later an effective theory. 3 why molecules have these optical properties, a deeper quantum mechanical consideration is nec- essary. Molecular Schr¨ odinger’s equation can explain spectroscopical observations and provide detailed information about behavior of molecules in the electric fields. However, this equation is effective as well: in its standard form the relativistic effects are not included. Full incorporation of the relativistic effects leads to the Dirac–Coulomb–Breit Hamiltonian. But even this theory is phenomenological 14 , because Dirac’s equation 15, 16 was rigorously derived for one electron only. It is effective, since the electrons and nuclei interact through the Coulomb (or more com- plicated) potential and its solution describes a classical Dirac field. Quantum field theories 17 , such as quantum electrodynamics (QED), quantize both fermionic and bosonic fields. In the QED Lagrangian, the Dirac fermionic field interacts with the bosonic field. The leading order of perturbation of this interaction by charge gives a one-photon exchange interaction between fermions, from which the bosonic degrees of freedom can be integrated out, giving a retarded Coulomb potential 17, 18 . QED diverges at large energies 19 , which makes this theory effective at lower energy scale 17 . In this thesis, the effective theories are used in a various ways. First, we handle cases of strong electron correlation through equation-of-motion coupled-cluster methods (EOM-CC), which are (exact) effective theories (we show their effectiveness in Chapter 7). Then, we incor- porate spin–orbit relativistic effects within the effective spin–orbit mean-field Breit–Pauli treat- ment. Further, we extend natural orbital analysis (NTO) to explain the magnitude of spin-orbit coupling and derive empirical El-Sayed’s rules. NTOs allow us to build an effective picture of interacting electrons in strongly correlated systems. Finally, we apply Bloch’s formalism and derive a number of model Hamiltonians from EOM-CC. 4 1.2 Patterns of electron correlation Generations of students worldwide are taught in general chemistry classes that “electrons occupy orbitals”. This point of view, originating from the pioneers of quantum chemistry 20–25 , is stated in many general chemistry textbooks 26–29 . A deeper quantum mechanical description of electronic structure 30 translates this view onto a single Slater determinant, typically composed of Hartree–Fock (HF) molecular orbitals: 0 = 1 p N! 1 (1) 2 (1) 3 (1) N (1) 1 (2) 2 (2) 3 (2) N (2) . . . . . . . . . . . . . . . 1 (N) 2 (N) 3 (N) N (N) ; (1.1) where i are spin-orbitals,N is a number of electrons. Bold symbols 1; 2;:::; N denote space and spin coordinates of electrons 1; 2;:::;N. The Hartree–Fock orbitals are optimized such that the energyE HF of HF determinant 0 is minimized: E HF =h 0 jH el j 0 i; (1.2) where H el is the electronic Hamiltonian. First-order variations in energy lead to mean-field one-electron equations, in which the interactions with all other electrons are averaged in the form of a mean field. c c Mean-field theories also appear in nuclear physics 31 , electrodynamics 32 , statistical physics 33 , and game the- ory 34 . We will use a similar concept—spin–orbit mean field—in Chapters 2, 3, and 4. 5 Description of an electronic state through the Hartree product H = 1 (1) 2 (2) 3 (3) N (N) or through a Slater determinant (Eq. 1.1) leads to pseudo- independent electrons, meaning that the one-electron density (which is proportional to probability of finding any electron in in the space regionr 1 ::r 1 +dr 1 ) is HF (1) = X i i (1) = X i j i (1)j 2 (1.3) Eq. (1.3) is valid for both the Hartree product and HF determinant. However, the two ans¨ atze yield different two-electron densities. Consider a two-electron system, e.g., a helium atom. Inclusion of fermionic nature of electrons through the HF determinant (Eq. 1.1) gives the expres- sion for probabilities: HF 12 (1; 2) =j 0 j 2 = 1 2 ( 1 (1) 2 (2) + 1 (2) 2 (1)) 1 (1) 2 (2) 1 (2) 1 (2) (1.4) The lack of separability of two-electron probabilities of the HF determinant is called Fermi’s correlation. Other type of correlation—Coulomb correlation—appears in expansions beyond the mean-field HF treatment. An exact solution of electronic Hamiltonian in a finite basis set, given by full configuration interaction (FCI) d , is a linear combination of all possible determi- nants: FCI =C 0 0 + X ia C a i a i + X i<j a<b C ab ij ab ij + ; (1.5) where indices i;j;::: denote occupied set of orbitals, a;b;::: denote virtual set of orbitals, and a i , ab ij , ::: , ab::: ij::: are single, double, ::: , N-tuple excited determinants, C ab::: ij::: are the d FCI is fully orbital invariant—energies and properties do not change with the change of reference orbitals. HF orbitals are a convenient starting point for building FCI expansion, but they are not a requirement. 6 DFT is exact! Wave-function forest Green's function river QMC rose garden FCI top DFT swamp Beware of spikes Figure 1.2: Landscape of methods in electronic structure. This work holds a direction through the wave-function forest. corresponding expansion coefficients. The computational cost of FCI is factorial e , making it feasible for only small molecules. Most electronic structure methods are based on truncation of the expansion in Eq. 1.5. There are two limiting cases of the nature of electron correlation. If the HF solution is qualitatively correct and its contribution to FCI is large (C 0 in Eq. 1.5), the correlation is said to be weak, or dynamical f . If the HF solution does not describe an electronic state at even qualitative level and there are other determinants with comparable or larger FCI coefficients than the HF determinant, e Recent progress in approaches, targeting FCI within a finite threshold 35–37, 37, 38 , can reduce scaling to poly- nomial (although the degree of the polynomial is not always known). Yet, these schemes are still too expensive for practical applications. f The terms “static” and “dynamical” are often defined 39 with respect to the correlation captured by MCSCF method. Today there are many methods correctly describing strong correlation that do not rely on MCSCF. This makes the terms focusing on the nature of electron correlation—“weak” and “strong”—a better terminological choice 7 the correlation is said to be strong, or static f . The nature of electron correlation is reflected in the theoretical chemists’ toolbox of methods. Wave-function methods approximate many- body wave-functions, giving a direct route toward evaluation of energies and properties 30 . In density functional theory 40 (DFT) and conceptually close density matrix functional theory 41 (DMFT g ) and reduced density-matrix methods 42 (RDM), the energy is evaluated from a density or density matrices with a given choice of a functional or a set of constraints. Another view is provided by the methods based on Green’s functions 43 (approximating one- or many-particle Green’s functions). Finally, Quantum Monte-Carlo 44, 45 (QMC) provides an alternative to the deterministic approaches. These branches of methods are not completely distinct and share many similarities. In this work, we focus on deterministic wave-function-based approaches. Traditionally, single-reference methods are developed to capture weak correlation. This set of methods include configuration interaction methods (CISD, CISDT, ::: ), perturbation theory 46 (MP2, MP3, MP4, etc.), coupled-cluster methods 47, 48h (CCSD, CCSDT, etc.), and per- turbative coupled-cluster approximations 49–55 (CCSD(T), CCSD(dT), CCSD(fT), etc.). These methods are designed primarily for the ground state. Excited states can be described by an ansatz comprising excited determinants to describe the target wave-functions (CIS, EOM-CC 56 , ADC 57 ). Multi-reference i methods (MCSCF, DMRG 59 , MRCI, CASPT 60 , NEVPT 61 , etc) use several determinants as a starting point. In these methods, the reference space is usually con- structed over selected active space, and the rest of correlation is included through excitations out of the active space. However, the choice of active spaces is not uniquely defined. Another g Not to be confused with dynamical mean-field theory—a common approximation of Green’s functions in solid-state physics. h Although typically coupled-cluster methods are considered as methods for weak correlation, they can handle some amount of strong correlation. We illustrate this behavior on the intermediate in the reaction between O( 3 P) and ethylene in Chapter 4. i Sometimes, people apply the term “multi-reference” to the strongly correlated systems, but not to the meth- ods. This terminological choice is misleading, since many patterns of strong correlation can be described with single-reference methods. See, for example, an overview in Ref. 58 . 8 BS-DFT EOM-CC ADC ADC Figure 1.3: Which method should I choose? Among a variety of existing methods in quantum chemist’s toolbox, EOM-CC is an attractive choice to nail difficult problems. issue with some multi-reference methods is that they violate size consistency (MRCI, several variants of MRCC). In addition, the growth of the active space size can be computationally very expensive (for example, the cost of CASSCF increases factorially with the size of active space). To circumvent the issues common for multi-reference methods, a number of single-reference methods has been developed to treat strongly correlated systems. A set of approaches replaces the concept of “electrons on orbitals” by correlated orbital pairs—geminals. This type of ans¨ atze (e.g., in the form of the Hartee–Fock–Bogoliubov method) is common in nuclear physics 31 , and is gradually gaining ground in molecular electronic structure 62 . Certain perturbative methods built for Green’s functions (e.g., GF2 63 ) can handle very strong correlation 64 . Finally, EOM-CC methods provide an alternative to multi-reference approaches. The wave-function in EOM-CC 9 IP DIP EA DEA EE SF Figure 1.4: Different choices of EOM-CC operator ^ R a . a Reproduced from Ref. 65 with permission from the PCCP Owner Societies. is parametrized linearly by action of excitation (ionization, electron attachment) operators on the coupled-cluster reference state: EOM = ^ Re ^ T 0 ; (1.6) ^ H ^ R 0 =E exc ^ R 0 ; (1.7) where ^ R is the excitation (ionization, electron attachment operator), ^ T is the coupled-cluster excitation operator, and EOM is the target EOM state. EOM-CC does not rely on active space, yet, it can treat certain types of strong correlation (shown in Figure 1.4). Spin-flip excita- tions 66–68 in ^ R (EOM-SF-CCSD) allow one to treat open- and closed-shell states in diradicals 10 on equal footing starting from a triplet reference. This approach works well for two-electrons- in-two-orbitals or three-electrons-in-three-orbitals strong correlation pattern (appearing, e.g., in single bond breaking), providing a balanced description of strong and weak correlation. Double spin-flip excitations 69 (EOM-DSF-CC(2,3)) can handle four electrons in four orbitals, which is important, for example, in singlet fission 70–72 . EOM-CC is a Fock-space method, meaning that the reference and target states can be in different sectors of Fock space. Ionization 73, 74 (h, EOM-IP-CCSD) and double ionization 75, 76 (2h, EOM-DIP-CCSD) allow one to tackle pattern of one (or two) hole(s) in multiple occupied orbitals. These variants are especially useful when the electronic states are heavily multiconfigurational and the selection of an active space is problematic. In particular, we used EOM-DIP-CCSD for four-electrons-in-three-orbitals strong correlation pattern of C 2 H 4 O in Chapter 4. Similarly, electron-attachment 77 (EOM-EA-CCSD) and double electron-attachment operators provide a balanced description of one electron in many orbitals or two electrons in many orbitals. These situations are common, for example, for Rydberg states of alkali metals or for systems with optical cycling centers, suitable for laser cooling 78 . 1.3 Spin–orbit interaction It is well-known that in classical electrodynamics a system of interacting charges is not fully described by a Lagrange function that depends only on the coordinates and velocities of the particles 79 . A complete set of equations of motion can be obtained from a Lagrangian density that depends on particles’ coordinates and velocities and field’s degrees of freedom. For slowly moving particles, the lowest-order approximation is constructed from non-relativistic kinetic energy and Coulomb interaction. Upon Legendre transform of this Lagrange function and its quantization, the usual non-relativistic molecular Hamiltonian is obtained. However, as in the classical case, it is not exact. If we want to improve it, we may consider relativistic corrections. 11 One of the corrections—spin–orbit interaction—is important in several contexts. For example, it is the driving force of spin-forbidden reactions and also contributes to zero-field splitting, which are considered in the following chapters. The origin of spin–orbit interaction can be understood by using classical arguments. Let us consider the case of a radial electrostatic potential and an electron moving in the potential. It is convenient to consider the rest frame of the electron. Changing electric field generates a magnetic field. This magnetic field interacts with intrinsic magnetic moment (due to spin) of the electron, giving rise to an interaction, which is not present in the standard molecular Hamiltonian. Relativistic kinematic effect (Thomas precession 80, 81 ) can also be included to correct the prefactor of the expression. An alternative derivation 82 is based on the rest frame of the nucleus. Changing an intrinsic magnetic moment of the electron generates an electric dipole. Interaction of this dipole with the electrostatic potential gives rise to spin–orbit interaction. Finally, a strict derivation can be carried out on the basis of many-electron Dirac equation, which is given, for example, in Ref. 14. In subsequent chapters, we consider spin–orbit interaction due to the electron motion in the nuclear potential (one-electron term) as well as the interaction with other electrons (two-electron terms), develop a theoretical framework for its computation, extend El-Sayed’s rule to analyze spin–orbit couplings, and apply the formalism for calculations of spin-forbidden rates and zero-field splittings in single-molecule magnets. 12 1.4 Chapter 1 References 1 P. Pokhilko, E. Epifanovsky, and A. I. Krylov, J. Chem. Phys. 151, 034106 (2019). 2 P. Pokhilko and A. I. Krylov, J. Phys. Chem. Lett. 10, 4857 (2019). 3 P. Pokhilko, R. Shannon, D. Glowacki, H. Wang, and A. I. Krylov, J. Phys. Chem. A 123, 482 (2019). 4 P. Pokhilko, E. Epifanovskii, and A. I. Krylov, J. Chem. Theory Comput. 14, 4088 (2018). 5 P. Pokhilko, D. Izmodenov, and A. I. Krylov, J. Chem. Phys. 152, 034105 (2020). 6 P. Pokhilko and A. I. Krylov, J. Chem. Phys. 152, 094108 (2020). 7 A. Robock, L. Oman, and G. Stenchikov, J. Geophys. Res. Atmos. 113 (2008). 8 J. Tyndall, J. Franklin Inst. 88, 34 (1869). 9 J. W. Strutt, Lond. Edinb. Dublin Philos. Mag. J. Sci. 41, 107 (1871). 10 J. W. Strutt, Lond. Edinb. Dublin Philos. Mag. J. Sci. 41, 274 (1871). 11 J. W. Strutt, Lond. Edinb. Dublin Philos. Mag. J. Sci. 41, 447 (1871). 12 Lord Rayleigh F.R.S., Lond. Edinb. Dublin Philos. Mag. J. Sci. 12, 81 (1881). 13 Lord Rayleigh F.R.S., Lond. Edinb. Dublin Philos. Mag. J. Sci. 47, 375 (1899). 14 S. A. Perera and R. J. Bartlett, A reinvestigation of ramsey’s theory of nmr coupling, in Advances in Quantum Chemistry, volume 48, pages 434–467. 2005. 15 P. A. M. Dirac and R. H. Fowler, Proc. R. Soc. London A 117, 610 (1928). 16 P. A. M. Dirac and R. H. Fowler, Proc. R. Soc. London A 118, 351 (1928). 17 A. Zee, Quantum field theory in a nutshell. Princeton university press, 2 edition, 2010. 18 E. M. Lifshitz, V . B. Berestetskii, and L. P. Pitaevskii, Teoreticheskaya physica (Tom 4. Kvantovaya Electrodimanica). Nauka, Moscow, 3 edition, 1989. 19 L. D. Landau, A. A. Abrikosov, and I. M. Khalatnikov, “asimptoticheskoe vyrazhenie dlya funktsii grina v kvantovoi elektrodinamike”(“asymptotic expression for the green’s function of a photon in quantum electrodynamics”), in Dokl. Akad. Nauk SSSR, volume 95, pages 1177–1180, 1954. 13 20 L. Pauling, The Nature of the Chemical Bond and the Structure of Molecules and Crystals: An Introduction to Modern Structural Chemistry, George Fisher Baker Non-Resident Lecture Series. Cornell University Press, New York, 3 edition, 1960. 21 G. Lewis, Valence and the Structure of Atoms and Molecules, ACS monograph. Chemical Catalog Company, Inc., New York, 1923. 22 J. Slater, Phys. Rev. 37, 481 (1931). 23 J. Slater, Phys. Rev. 38, 1109 (1931). 24 J. Lennard-Jones, Trans. Faraday Soc. 25, 668 (1929). 25 W. Heitler and F. London, Z. Phys. 44, 455 (1927). 26 M. Tamm and Y . Tretyakov, Neorganisheskaya khimiya, volume 1. Izdatelskii tsentr “Academiya”, Moscow, 2004. 27 N. Glinka, Obshaya khimiya. Integral-Press, Moscow, 30 edition, 2004. 28 N. Ahmetov, Obshaya i neorganicheskaya khimiya. Izdatelskii tsentr “Academiya”, Moscow, 4 edition, 2001. 29 S. Zumdahl and S. Zumdahl, Chemistry. Houghton Mifflin, 7 edition, 2007. 30 T. Helgaker, P. Jørgensen, and J. Olsen, Molecular electronic structure theory. Wiley & Sons, 2000. 31 M. Bender, P.-H. Heenen, and P.-G. Reinhard, Rev. Mod. Phys. 75, 121 (2003). 32 K.-H. R¨ adler, Astron. Nachr. 335, 459 (2014). 33 L. Kadanoff, J. Stat. Phys. 137, 777 (2009). 34 J.-M. Lasry and P.-L. Lions, Jpn. J. Math. 2, 229 (2007). 35 J. Eriksen, F. Lipparini, and J. Gauss, J. Phys. Chem. Lett. 8, 4633 (2017). 36 J. J. Ericksen and J. Gauss, J. Chem. Theory Comput. 14, 5180 (2018). 37 J. J. Ericksen and J. Gauss, J. Chem. Theory Comput. 15, 4873 (2019). 38 P. M. Zimmerman, J. Chem. Phys. 146, 104102 (2017). 39 D. Cooper, Valence Bond Theory. Elsevier Science, Amsterdam, 2002. 40 E. Engel and R. Dreizler, Density Functional Theory: An Advanced Course, Theoretical and Mathematical Physics. Springer Berlin Heidelberg, 2011. 14 41 J. Cioslowski, Many-Electron Densities and Reduced Density Matrices, Mathematical and Computational Chemistry. Springer US, 2000. 42 D. Mazziotti and S. Rice, Reduced-Density-Matrix Mechanics: With Application to Many- Electron Atoms and Molecules, Advances in Chemical Physics. Wiley, 2007. 43 J. Linderberg and Y . ¨ Ohrn, Propagators in quantum chemistry. Academic, London, 1973. 44 W. Foulkes, L. Mitas, R. Needs, and G. Rajagopal, Rev. Mod. Phys. 73, 33 (2001). 45 T. Pang, An Introduction to Quantum Monte Carlo Methods, 2053-2571. Morgan & Claypool Publishers, 2016. 46 C. M¨ oller and M. Plesset, Phys. Rev. 46, 618 (1934). 47 J. Noga and R.J. Bartlett, J. Chem. Phys. 86, 7041 (1987). 48 R.J. Bartlett, Coupled-cluster theory: An overview of recent developments, in Modern Elec- tronic Structure Theory, Part II, edited by D. R. Yarkony, pages 1047–1131. World Scientific, Singapore, 1995. 49 K. Raghavachari, G.W . Trucks, J. A. Pople, and M. Head-Gordon, Chem. Phys. Lett. 157, 479 (1989). 50 J. D. Watts, J. Gauss, and R. J. Bartlett, J. Chem. Phys. 98, 8718 (1993). 51 S. R. Gwaltney and M. Head-Gordon, J. Chem. Phys. 115, 2014 (2001). 52 O. Christiansen, H. Koch, and P. Jorgensen, J. Chem. Phys. 105, 1451 (1996). 53 P. Piecuch and M. Włoch, J. Chem. Phys. 123, 224105 (2005). 54 K. Kowalski and P. Piecuch, J. Chem. Phys. 113, 18 (2000). 55 P. U. Manohar and A. I. Krylov, J. Chem. Phys. 129, 194105 (2008). 56 A. I. Krylov, Annu. Rev. Phys. Chem. 59, 433 (2008). 57 A. Dreuw and M. Wormit, WIREs: Comput. Mol. Sci. 5, 82 (2015). 58 R. J. Bartlett, Int. J. Mol. Sci. 3, 579 (2002). 59 Sebastian Wouters and Dimitri Van Neck, Eur. Phys. J. D 68, 272 (2014). 60 K. Andersson, P.- ˚ A. Malmqvist, B. O. Roos, A.J. Sadlej, and K. Wolinski, J. Phys. Chem. 94, 5483 (1990). 61 C. Angeli, S. Borini, and R. Cimiraglia, Theor. Chim. Acta 111, 352 (2004). 15 62 P. Tecmer, K. Boguslawski, P. Johnson, P. Limacher, M. Chan, T. Verstraelen, and P. Ayers, J. Phys. Chem. A 118, 9058 (2014). 63 J. Phillips and D. Zgid, J. Chem. Phys. 140, 241101 (2014). 64 E. M. Lifshitz and L. P. Pitaevskii, Statisticheskaya physica. Chast’ 2. Teoriya konden- sirovannogo sostoyaniya. Fizmatlit, Moscow, 2004. 65 D. Casanova and A. I. Krylov, Phys. Chem. Chem. Phys. (2020), in press; DOI: 10.1039/c9cp06507e. 66 A. I. Krylov, Chem. Phys. Lett. 338, 375 (2001). 67 S. V . Levchenko and A. I. Krylov, J. Chem. Phys. 120, 175 (2004). 68 A. I. Krylov, Acc. Chem. Res. 39, 83 (2006). 69 D. Casanova, L. V . Slipchenko, A. I. Krylov, and M. Head-Gordon, J. Chem. Phys. 130, 044103 (2009). 70 X. Feng, A. V . Luzanov, and A. I. Krylov, J. Phys. Chem. Lett. 4, 3845 (2013). 71 M. H. Farag and A. I. Krylov, J. Phys. Chem. C 122, 25753 (2018). 72 N. V . Korovina, S. Das, Z. Nett, X. Feng, J. Joy, A. I. Krylov, S. E. Bradforth, and M. E. Thompson, J. Am. Chem. Soc. 138, 617 (2016). 73 J. F. Stanton and J. Gauss, J. Chem. Phys. 101, 8938 (1994). 74 P. A. Pieniazek, S. E. Bradforth, and A. I. Krylov, J. Chem. Phys. 129, 074104 (2008). 75 K. W. Sattelmeyer, H. F. Schaefer, and J. F. Stanton, Chem. Phys. Lett. 378, 42 (2003). 76 M. Musiał, A. Perera, and R. J. Bartlett, J. Chem. Phys. 134, 114108 (2011). 77 M. Nooijen and R. J. Bartlett, J. Chem. Phys. 102, 3629 (1995). 78 M. V . Ivanov, F. H. Bangerter, and A. I. Krylov, Phys. Chem. Chem. Phys. 21, 19447 (2019). 79 E. M. Lifshitz and L. P. Pitaevskii, Teoriya polya. Fizmatlit, Moscow, 2003. 80 L. H. Thomas, Nature 117, 514 (1926). 81 L.H. Thomas B.A., Lond. Edinb. Dublin Philos. Mag. J. Sci. 3, 1 (1927). 82 G. Spavieri and M. Mansuripur, Phys. Scr. 90, 085501 (2015). 16 Chapter 2: General framework for calculating spin–orbit couplings using spinless one-particle density matrices: Theory and application to the equation-of-motion coupled-cluster wave functions 2.1 Introduction Relativistic effects are important in chemistry and spectroscopy 1, 2 . Related to the intrinsic magnetic moment of the electron associated with its spin, they give rise to the orbital contrac- tion, an increase in electron binding energies, and spin–orbit interactions. The latter effect, which arises due to the coupling of the angular momentum of an electron and its spin, is of special importance. Although small in magnitude for molecules composed of light atoms, spin–orbit coupling (SOC) leads to the mixing of otherwise non-interacting states, e.g., sin- glets and triplets, and splits electronic degeneracies, e.g., between degenerate or states or 17 between different M s components of spin multiplets. Consequently, SOCs open new reaction channels (via inter-system crossing, ISC) and change the spectroscopic behavior by lighting up dark states via intensity borrowing. SOCs lead to noticeable changes in the wave functions and electronic properties, and are particularly important in open-shell systems. Spin-related properties, such as SOCs and spin–spin interactions, determine magnetic behavior of molecules, which is central for understanding spectroscopic signatures of unpaired electrons (e.g., EPR spectroscopy), their macroscopic magnetic properties (magnetozabilities), and magnetic relaxation times. The ability to model these properties computationally is a key prerequisite for the design of novel materials. For example, single-molecule magnets (SMMs), molecules with several unpaired electrons that have a high-spin ground state in the absence of an applied magnetic field 3–5 , can be used as building blocks to create novel, light-weight, and tunable magnetic materials or as building blocks for quantum information storage and quantum computations 6–9 . The key challenge in realizing the full potential of SMMs is the ability to tune and control their magnetic behavior. Magnetic properties are usually modeled using phenomenological Hamiltonians of vary- ing complexity, parameterized either empirically or by using first-principle calculations 5 . In the context of SMMs uses in information storage, the key quantities needed for the parame- terization of these Hamiltonians are energy levels corresponding to different spin states and properties such as zero-field splittings (ZFS), hyperfine couplings (HFS), g-tensors, and asym- metric Dzyaloshinksii–Moriya (DM) interactions. All these quantities depend on SOCs, either directly or indirectly. For example, SOC significantly contributes to ZFS, which can be estimated perturbatively from the SOC between a selected state with other closely lying states 10–13 . ZFS can be (approx- imately) decomposed into single-ion anisotropy and exchange anisotropy tensors, giving rise 18 to effective Hamiltonians 5 . These quantities ultimately determine the barrier for the reversal of magnetization and magnetic relaxation times. Fully relativistic schemes describe SOC in a natural manner. For light atoms, relativis- tic effects can be treated perturbatively by using the Breit–Pauli (BP) Hamiltonian 14, 15 . Here we consider only such perturbative scheme in which SOCs are computed as matrix ele- ments of the BP Hamiltonian using non-relativistic wave functions 2, 16 . Different implemen- tations of full and approximate SOC treatments have been reported for different types of wave functions 1, 17–31 including those obtained by complete active-space self-consistent field (CASSCF) 16–22 , restricted active space self-consistent field (RASSCF) 29, 30 , multi-reference con- figuration interaction (MRCI) 1, 23 , density-matrix renormalization group (DMRG) 31 , coupled- cluster (CC) response 25 , equation-of-motion CC (EOM-CC) 25, 26, 28 , and multi-reference CC (MRCC) within the Mk-MRCCSD formulation 27, 32 . DFT implementations have also been reported 24, 33–36 . While the working expressions for calculating the matrix elements of the BP Hamiltonian between the pairs of non-relativistic states have been thoroughly described in previous works, one important aspect of the theory deserves further discussion. To construct experimentally relevant quantities, such as rates of ISC crossing, oscillator strengths, or magnetic anisotropies, one needs SOCs computed for all multiplet components. For example, the expression for the SOC constant (SOCC), the quantity that enters the Fermi golden rule expression of the ISC rate, is given by the following expression 2 : SOCC = s X M;M 0 jh SM jH SO j 0 S 0 M 0 ij 2 ; (2.1) 19 where the sum runs over all spin projections of both multiplets. Likewise, calculations of the parameters for effective Hamiltonians, such as magnetic anisotropy or DM tensors, also require matrix elements between all multiplet components. Previously reported implementations of SOC calculations within the EOM-CCSD frame- work 25, 26, 28 enable calculations between the selected target EOM states. In a typical non- relativistic EOM-CC calculation (and in many other excited-state calculations), one computes only one component of the target multiplet because other components are exactly degener- ate. For example, traditionally only M s =0 components of triplet states are computed. This is obviously insufficient for computing the full SOCC matrix. One possible strategy is to explic- itly compute two other components (M s =1), which requires additional coding efforts and increases the cost of the calculations. In addition, a special care needs to be taken to syn- chronize the phases of all computed states a . Alternatively, one can apply coordinate rotations (which is equivalent to changing the quantization axis) and extract the SOC matrix elements in the original coordinate frame from the matrix elements in the rotated frames. However, practi- cal application of this approach is also complicated by the phase problem mentioned above: the phases of the wave functions computed in different coordinate frames are arbitrary. In addition, frame rotations may require disabling point-group symmetry, thus further increasing the cost of calculations. Here we describe how to generate all required SOC matrix elements from a minimal amount of calculations. The theory is formulated in terms of reduced one-particle density matrices, in spin-orbital representation, such that it is ansatz-agnostic and can be applied to any electronic structure method that can furnish required transition densities. We illustrate this approach by using the EOM-CC method with single and double excitations (EOM-CCSD) as an example. a Within Born–Oppenheimer separation of nuclear and electronic degrees of freedom, the phase of the elec- tronic wave function is not defined: it changes at random at different nuclear geometries or even from calculation to calculation at the same geometry. To ensure the consistency of the signs of the matrix elements, the phase should be fixed by an additional condition. 20 The formalism allows one to compute an entire SOC matrix between all multiplet compo- nents at once, without additional calculations in the rotated frames or explicit calculations of the other multiplet components, and without the need to address the phase problem. In essence, the approach is grounded in Wigner–Eckart’s theorem 37–39 that outlines the relationships between the matrix elements involving multiplet components. In the context of SOC, the utility of Wigner–Eckart’s theorem has been recognized in many previous studies 19, 22, 23, 29–31, 33, 36, 40–42 , however, the concrete strategies of its application vary considerably. Reported implementa- tions differ by whether Wigner–Eckart’s theorem is applied to the states or transition densities, which multiplet components are used as generators, and whether spinless or spin-orbital expres- sions are used to construct the working equations. For example, within configuration interac- tion formalism, an algorithm for generating the entire SOC matrix from only three reduced matrix elements has been described by Fedorov 20, 40 . However, as explained below, Fedorov’s approach 20, 40 requires access to several multiplet components (M s =0,1), a feature which is not commonly available in excited-state codes, ensued is also affected by the phase issue unless a special care is taken. A similar strategy has been used in Orca package, where highest-spin multiplet component is used as a generator 12, 33, 36, 43 . An alternative approach, based on using transition densities for the spinless states, was used in the Molcas implementation within the restricted active space state interaction (RAS-SI) framework 29, 30 and in a recent DMRG imple- mentation 31 ; in these studies the required transition densities were obtained from spinless states. Although not always acknowledged, the most general strategy based on the application of Wigner–Eckart’s theorem to transition density has been originally outlined by McWeeny in 1965 44, 45 . Here we follow McWeeny’s proposal and develop second quantized formulation of this approach. The essential components of our formalism are very similar to the Molcas and DMRG implementations 29, 31 . In our approach, one needs to compute only one multiplet compo- nent (for example, the M s =0 state) from which the reduced spinless density matrix is generated. 21 In contrast to Refs. 29, 31, we employ spin-orbital formalism, which allows us to obtain requi- site reduced transition densities for the transitions between any types of open-shell states. This density matrix is then used to compute a reduced matrix element and the entire SOC matrix is obtained by the application of Wigner–Eckart’s theorem. The phase problem is avoided by con- struction, because the density matrix corresponding to a reduced matrix element is computed from only one transition between the pair of target states (or between the reference and the tar- get EOM state). The theory is illustrated by application to the EOM-CCSD wave functions. As numerical examples, we report calculations for a set of small open-shell molecules, considered as candidates for laser cooling experiments, a set of diradicals used as benchmark systems in previous studies 25, 28 , several transition-metal ions, and a Fe(II) SMM with a large spin-reversal barrier (magnetic anisotropy) for which a synthetic analogue has been synthesized 46 . The structure of the paper is as follows. In the next section, we outline essential features of the EOM-CCSD method and the calculation of the state and transition properties within EOM-CC. We then describe spin-tensor formalism and apply it to derive the key equations for calculation of the SOC matrix elements via reduced density matrices and Wigner–Eckart’s theorem. We also discuss the implementation of the algorithm within the EOM-CC suite of methods in the Q-Chem electronic structure package 47, 48 . We discuss additional aspects of the theory in the case of open-shell references. Section 6.3 presents illustrative calculations. 22 2.2 Theory 2.2.1 Equation-of-motion coupled-cluster methods with single and double excitations In the EOM-CC framework 49–53 the target states are parameterized as j R i =Re T j 0 i; (2.2) where 0 is a reference determinant, which defines the many-body vacuum state and, conse- quently, the occupied and virtual orbital spaces b , e T j 0 i is a CC wave function, and R is a general excitation operator c . Explicit form of R depends on the level of correlation included in the model (e.g., it generates up to doubly excited configurations at the EOM-CCSD level) and on the type of the target states. For example, in EOM-CCSD for excitation energies 54 ,R is spin- and particle-conserving, i.e., of 1h1p and 2h2p types at the CCSD level, with an addi- tional constraint that in each operator the number of holes equals the number of particles and the number of holes equals the number of particles. In EOM-CCSD for ionized 55, 56 (EOM-IP-CCSD) or electron-attached 57 (EOM-EA-CCSD) states,R 0 s are not particle conserv- ing. In EOM-IP-CCSD,R is of 1h and 2h1p types and in EOM-EA-CCSD,R is of 1p and 2p1h types. In the spin-flip variant 58, 59 (EOM-SF-CCSD), operatorsR change the spin-projection by flipping the spin of an electron. Other variants include double-ionization potential (DIP) 60, 61 , b Traditionally, orbitals occupied in 0 are denoted byi;j;k;::: and virtual orbitals are denoted bya;b;c;:::. Orbitals that can be either occupied or unoccupied are denoted byp;q;r;:::. Atomic orbitals are denoted by Greek letters:;;;:::. c Excitation/de-excitation operators are defined with respect to the vacuum state, 0 . An excitation operator can only contain creation operators corresponding to the virtual orbitals, referred to as particle (p) operators, and annihilation operators corresponding to the occupied orbitals, referred to as hole (h) operators. EOM operatorsR and CC operatorsT are excitation operators. At the CCSD level, cluster operatorsT are of the 1h1p and 2h2p types. 23 double electron-attachment (DEA), and double spin-flip (DSF) 62 ans¨ atze. R amplitudes are found through the diagonalization of the similarity-transformed Hamiltonian, H = e T He T , in the space of a target set of determinants: HR K =E K R K ; (2.3) L y K H =E K L y K : (2.4) Because H is non-Hermitian, its right and left eigenvectors are not Hermitian conjugates of each other, but form a biorthogonal set: h L I j R J i = IJ : (2.5) The normalization and phases within the right and left sets are arbitrary, but the biorthogonality condition, Eq. (2.5), fixes the relative norms and phases: i.e., if one changes the sign and norm of R K , the sign and norm of L K changes accordingly. In many implementations, the diagonalization is carried out independently for the right and left vectors and the biorthogonality condition, which synchronizes the phases, is applied a posteriori (in the case of degenerate eigenstates, one needs to rotate the degenerate eigenstates such that the overlap matrix between the left and right degenerate manifolds becomes a unit matrix). 24 In many-body theories, the state and transition properties are computed as contractions of the corresponding integrals with reduced density matrices d . One-electron operators require a one-electron density matrix e h I jAj I 0i =Tr [ (I;I 0 )A] = X pq pq A pq : (2.6) and two-electron operators require a two-electron density matrix. (HereA denotes an operator describing a specific property). For the exact states or within the expectation-value formulation of properties 63, 64 , the one-particle density matrix used for property calculation is simply: pq (I;I 0 ) =h I ja y p a q j I 0i: (2.7) Within the response formulation of properties, the relaxed one-particle density matrices are used; they include additional terms describing relaxation of non-variational wave function parameters (e.g., CC amplitudes). Here, we follow the expectation-value approach, as in our previous work 28, 65 . Because of a non-Hermitian nature of EOM-CC, pq (I;I 0 )6= qp (I 0 ;I), which leads to dif- ferent numerical results forh I jAj I 0i andh I 0jAj I i. A common resolution of this problem is to take a geometric average 54 . However, this treatment should be modified for complex-valued tensor properties, in order to preserve phase consistency and to generalize it for a matrix. Below d In the case of non-orthogonal orbitals (e.g., atomic orbitals), it is important to distinguish contravariant and covariant indices. From herein, the integrals are assumed to be contravariant and the densities are assumed to be covariant. Since the orbital spaces have a metric, the overlap matrixS, the relation between the covariant and contravariant density matrices is cov = S 1 contra S 1 . Since the working expressions of EOM-CC calculate the density matrices in the MO basis, covariant densities can be obtained without a need for the inverse overlap: cov (AO) =C (MO)C y . Covariance or contravariance does not change the spin-tensor properties of the density matrix, such as spin projectionM or singlet or triplet type. e ForI = I 0 , is simply one-particle state density matrix and forI6= I 0 , is one-particle transition density matrix. 25 (section 2.2.3) we introduce a spinless density matrix, which enables generation of all phase- consistent SOCs between the selected pairs of states. We also propose a different resolution of the averaging problem. Finally, we would like to comment on spin contamination of EOM-CC states 66 . Although the standard formulation of EOM-CC 50 does not involve explicit spin-adaptation, the CC/EOM- CC states are spin pure when closed-shell reference is used. For example, EOM-IP/EA wave- functions of doublet states are naturally spin-adapted, even when spin symmetry is not explic- itly enforced. For closed-shell references, even in in spin-orbital implementations, the spin- symmetry can be exploited in the same fashion as permutational or point-group symmetries, at the level of tensor library 67 . Only when open-shell references are used in EOM calculations, for example, in spin-flip calculations or in EOM-EE calculations using doublet references, spin contamination becomes an issue 66 . Recently, we discussed the impact of spin contamination on SOCs between the singlet and triplet states of the Cvetanoviˇ c diradicals 68 . By using an approx- imate a posteriori spin-projection technique, we have shown 68 that in these species the impact of spin contamination on SOCCs is small, i.e., less than 1 cm 1 . 2.2.2 Spin properties of operators In wave function approaches spin is usually described as a property of the wave function. However, one can consider properties of a spin operator ^ S alone. It has three Cartesian com- ponents and transforms under rotations as a vector. Operators involving spin may obey other transformation rules. For example, the ^ S 2 operator, formed as a dot product of the two spin 26 operators, is a scalar operator: it has only one component and it does not change upon the rotation of the coordinate frame. The spin–orbit operator of the BP Hamiltonian is H SO BP = 1 2c 2 X i h SO (i) s(i) X i6=j h SOO (i;j) (s(i) + 2s(j) ! ; (2.8) h SO = X K Z K (r i R K ) p i jr i R K j 3 = X K Z k r 3 iK (r iK p i ); (2.9) h SOO (i;j) = (r i r j ) p i jr i r j j 3 = X i6=j 1 r 3 ij (r ij p i ); (2.10) where the coordinates and momenta of electron i are denoted by r i and p i , the charge and coordinates of nucleusK are denoted byZ K and R K , and the relative coordinates of electron i and electron j or nucleus K are denoted by r ij and r iK . Because of the dot product, the spin–orbit operator is a scalar operator. Therefore, its components transform as L S upon the rotation of the coordinate frame. Here we do not consider spatial rotations but focus on the transformations in the spin space. Representation theory tells us that the objects (wave functions or operators) transform according to some representations of the transformation group. These representations are irre- ducible if they cannot be decomposed into other representations. This leads to irreducible ten- sor operators, which are especially useful for second-quantized formulations. It is convenient to define irreducible spin-tensor operators ^ O S;M through commutation relations 64, 69 : h S ; ^ O S;M i = p S(S + 1)M(M 1) ^ O S;M1 ; (2.11) h S z ; ^ O S;M i =M ^ O S;M : (2.12) HereS denotes the spin value andM denotes the spin projection, which goes fromS;S + 1;:::;S1;S. Any set of operators satisfying these relations are called irreducible spin-tensor operators. For example, theS = 0 case corresponds to singlet operators, which commute with 27 S z ,S + , andS . Any spin-independent operator, such as a dipole moment operator, is a singlet operator. Spin–orbit operator depends on spin, and, as shown below, it is composed from triplet spin-tensor operators. In particular, McWeeny writes down both angular momentum and spin in irreducible tensor form for spin–orbit operator in Eqs. (2.1) and (2.2) in Ref. 44 (see also exercise 2.10 in Ref. 64). Because we are interested in second-quantized formulation, it is useful to express spin in second quantization and consider several important examples. If and spin-orbitals have the same spatial part, S =S x iS y ; (2.13) S + = X p a y p a p ; (2.14) S = X p a y p a p ; (2.15) S z = 1 2 X p a y p a p a y p a p : (2.16) One important example is excitation operators. A one-particle singlet excitation operator can be written as T 0;0 pq = 1 p 2 a y p a q +a y p a q ; (2.17) and a set of triplet operators can be written as T 1;1 pq =a y p a q ; (2.18) T 1;0 pq = 1 p 2 a y p a q a y p a q ; (2.19) T 1;1 pq =a y p a q : (2.20) 28 Note thatS andS z are not expressed in the irreducible spin-tensor form. If one wishes to use the irreducible form, these operators should be multiplied by factors from Eqs. (3.13)-(3.15). The spin parts of all terms in the spin–orbit operator are triplet operators. The one-electron part can be written as H 1eSO = 1 2 X pq h 1eSO L + ;pq a y p a q + X pq h 1eSO z;pq a y p a q a y p a q + X pq h 1eSO L ;pq a y p a q ! ; (2.21) where h 1eSO L + =h 1eSO x +ih 1eSO y ; (2.22) h 1eSO L =h 1eSO x ih 1eSO y : (2.23) The one-electron part requires only one-electron transition density matrix for the final SOC matrix elements. The two-electron part requires the spin–orbit two-electron integrals and two- electron transition densities, which makes the full calculation expensive. However, the two- electron contribution can be effectively approximated by considering only the separable part of the two-electron transition density, such that the two-electron spin–orbit integrals are first contracted with the density of the reference determinant (usually Hartree–Fock determinant), and the resulting mean-field-like one-electron operator is then folded into the one-electron part. This spin–orbit mean-field approximation (SOMF) 2, 70 is commonly used in calculations of SOCs 2, 23–26, 28, 71–74 ; it is described in Section 2.2.4. 29 One of the applications of irreducible tensor operators is Wigner–Eckart’s theorem. The theorem allows one to compute matrix elements through a reduced matrix element, which does not depend on spin projection: hS 0 M 0 jO S;M jS 00 M 00 i =hS 00 M 00 ;SMjS 0 M 0 ihS 0 jjO S; jjS 00 i: (2.24) Here the matrix element between the bra state with spinS 0 and spin projectionM 0 and the ket state with spin S 00 and spin projection M 00 is expressed through a Clebsh–Gordan coefficient hS 00 M 00 ;SMjS 0 M 0 i and a reduced matrix elementhS 0 jjO S; jjS 00 i. Here, the superscript S; signifies the reduced matrix element, which does not depend on spin projection. 2.2.3 Evaluation of matrix elements To compute all required matrix elements between the two multiplets, we apply Wigner– Eckart’s theorem in the spin space: hISS 0 z 1jH L + S jI 0 S 0 S 0 z i =hS 0 S 0 z 1 1jSS z 1ihISjjH L + jjL 0 S 0 i; (2.25) hISS 0 z jH LzSz jI 0 S 0 S 0 z i =hS 0 S 0 z 10jSS z ihISjjH Lz jjL 0 S 0 i; (2.26) hISS 0 z + 1jH L S + jI 0 S 0 S 0 z i =hS 0 S 0 z 11jSS 0 z + 1ihISjjH L jjL 0 S 0 i; (2.27) where I and I 0 enumerate multiplets. This strategy has been used, for example, within the CI framework 40 , where the full spin–orbit matrix between I and I 0 spin manifolds was con- structed from three (or two, if symmetry is used) reduced matrix elements. An application of this approach 40 to EOM-CCSD would require calculation of target states of different spin projections (with consistent phases). 30 Rather than applying Wigner–Eckart’s theorem to the spin–orbit operator directly, we apply it to general triplet excitation operators, as was done in Refs. 29, 31, 44, 45: hI 0 S 0 M 0 j ^ T 1;M pq jI 00 S 00 M 00 i =hS 00 M 00 ; 1MjS 0 M 0 ihI 0 S 0 jj ^ T 1; pq jjI 00 S 00 i (2.28) These reduced matrix elements form a new density, which we denote asu: u pq hI 0 S 0 jj ^ T 1; pq jjI 00 S 00 i: (2.29) Being a reduced matrix element,u pq does not depend on the spin projections of the states or the excitation operator. To emphasize this property, we drop the corresponding index in excitation operator and replace it with a dot: T 1; pq . If the transition between the target states (or between the reference and target states) is a spin-flip transition, the spin-tensor form of the one-particle transition density matrix is the transition density matrix up to a sign, see Eq. (3.13) and (3.15). If the transition is spin-conserving, the transition density matrix always can be decomposed into the singlet and triplet components in any spin-adapted basis (i.e., atomic orbitals): Ms=0 = Ms=0 0 0 Ms=0 ! = 1 2 Ms=0 0 0 Ms=0 ! + Ms=0 0 0 Ms=0 !! + 1 2 Ms=0 0 0 Ms=0 ! Ms=0 0 0 Ms=0 !! = Ms=0 singlet + Ms=0 triplet : (2.30) Here the M s = 0 label indicates that the density matrix Ms=0 is computed for the spin- conserving excitations. Importantly, this applies not only to the EOM-EE ansatz, but also to the EOM-SF, EOM-IP, and EOM-EA variants for the transitions between the states with the same spin projections. Non-zero spin blocks are denoted as Ms=0 and Ms=0 , where the 31 spin label indicates the or spin-orbitals in Eq. (2.7). It is easy to see that Ms=0 triplet matrix is constructed through a y p a q a y p a q excitations between the bra and ket states. Although it is a triplet excitation operator, for the consistency with other triplet excitation operators, the Ms=0 triplet matrix should be further divided by p 2, as per definition ofT 1;0 in Eq. (3.14). If the spin projections of bra and ket differ by1, the corresponding excitation operators are spin-flip triplet operators, and no decomposition is needed. However, for spin-tensor form they should be multiplied by the appropriate phase factors from Eqs. (3.13) and (3.15). We note that for these spin-flipping excitations right-to-left and left-to-right density matrices gain opposite phase factors. Once the density matrix between the states with the same spin projection is computed, one can easily calculateu from Eq. (3.20) as u = 1 p 2 Ms=0 Ms=0 =hS 0 M; 10jSMi; (2.31) where is computed for the transition from the state with spin S 0 and spin projection M to the state with spinS and the same spin projectionM. Eq. (2.31) gives a recipe for computing u from M s = 0 transition densities obtained for general wave functions expressed in spin- orbital basis, rather than spinless states 29, 31 (u can be also computed fromM s =1 transition densities). The contraction ofu with the spin–orbit integrals give the spin–orbit reduced matrix elements (see the note 75 ): hISjjH L jjI 0 S 0 i = 1 2 X p;q h 1eSO L ;pq u 1; pq ; (2.32) hISjjH L 0 jjI 0 S 0 i = p 2 2 X pq h 1eSO Lz ;pq u 1; pq ; (2.33) hISjjH L + jjI 0 S 0 i = + 1 2 X pq h 1eSO L + ;pq u 1; pq ; (2.34) 32 and the full spin–orbit matrix is a product of Clebsh–Gordan’s coefficients to these reduced matrix elements. The prefactors in the expressions above arise because the spin part is taken in spin-tensor form whereas the orbital momentum is taken inL + =L ;L z form, which is not a spin-tensor form. The overall workflow of the algorithm is shown in Fig. 5.1. γ pq =hISM|a † p a q |I 0 S 0 M 0 i hISM|H L − |I 0 S 0 M +1i hISM|H L 0 |I 0 S 0 Mi hISM|H L + |I 0 S 0 M−1i hIS||H L − ||I 0 S 0 i hIS||H L 0 ||I 0 S 0 i hIS||H L + ||I 0 S 0 i Entire SOC matrix WE WE hISM|T 1,−1/0/+1 pq |I 0 S 0 M 0 i hIS||T 1,· pq ||I 0 S 0 i hIS|| and||I 0 S 0 i WE Tr[uh] Figure 2.1: The workflow of the present algorithm (shown by green arrows), Fedorov’s scheme (red arrows), and the approach implemented in Molcas (blue arrows). Here WE label denotes the application of Wigner–Eckart’s theorem. In our scheme, we start from one transition density matrix, extract the triplet part, construct the spinless density, and then form three reduced matrix elements. In the Molcas scheme, the spinless transition density matrix is built directly from spin manifoldsjjISi andjjI 0 S 0 i, also known as “ten- sor” states. In Fedorov’s scheme, the reduced matrix elements are computed from three matrix elements. 2.2.4 SOMF approximation for closed- and open-shell references SOMF approximation 2, 70 considerably reduces the cost of the calculation of the two- electron part of SOC while introducing negligible errors. Originally introduced in 1996 70 , SOMF (and some more aggressive approximations) have been used with great success for computing SOCs within wave-function approaches and density functional theory 2, 23–26, 28, 71–74 . 33 Briefly, SOMF entails 28 considering only the contributions from the separable part of the two- particle density matrix. In CC/EOM-CC, the separable part of the two-particle density matrix, ~ , has the following form 76 : ~ pqrs = P + (pr;qs)P (p;q) rp qs = rp qs rq ps + sq pr sp qr ; (2.35) rp = 8 > > < > > : rp r;p2 occupied in 0 0 otherwise (2.36) where the operatorsP + (pr;qs) andP (p;q) generate symmetrized/antisymmetrized expres- sions, respectively. Here is the density of the reference determinant, which is diagonal in case of canonical Hartree–Fock molecular orbitals. When contracted with a two-electron operator A = 1 2 P pqrs A pqrs p y q y sr, the separable part of two-particle density matrix yields: 2e = 1 2 X pq pq " X rs rs (A prqs +A rpsq A prsq A rpqs ) # = X pq pq A SOMF pq ; (2.37) whereA SOMF pq is defined by the above equation. Further simplifications are possible depending on the permutational symmetry of the tensorA. When applied to the two-electron part of the BP Hamiltonian, h SOO (1; 2), the mean-field matrices are h SOMF;2e pq = 1 2 X rs rs [J prqs J prsq J rpqs ]; (2.38) whereJ denotes the two-electron spin–orbit integrals: J prqs =h p (1) r (2)jh soo (1; 2)(s(1) + 2s(2))j q (1) s (2)i: (2.39) 34 A detailed derivation of SOMF, the expressions for the spin–orbit integrals, and the spin- integrated expressions (expressions for h SOMF;2e pq in spatial molecular orbital set) can be found in Ref. 28. An efficient evaluation of h SOMF;2e pq defined by Eq. (2.38) is carried out in atomic orbitals. Since it contains Coulomb-like and two exchange-like terms, the implementation is using the same techniques as in Fock matrix build within the Hartree–Fock procedure. Computation of one- and two-electron spin–orbit integrals is performed with King–Furlani algorithm 18 . As often is the case with mean-field treatments, construction of mean-field effective opera- tors for open-shell references might lead to artifacts. Below we discuss intrinsic issues of SOMF in the case of open-shell references. Note that these problems arise only when the reference, not the target state, have open-shell character. For example, EOM-IP/EA calculations of doublet states using closed-shell references are not affected by this issue, however, EOM-SF calcula- tions using high-spin references are. The analysis is based on considering the second-quantized form of the mean-field spin–orbit operator in a spin-restricted spin-orbital basis (meaning that the spatial parts of spin-orbitals p and p are the same, e.g. AOs, restricted HF orbitals for a closed-shell reference, etc). The two-electron mean-field operator can be written as H SOMF;2e = 1 2 ( X pq h SOMF;2e L + ;pq a y p a q + X pq h SOMF;2e z;pq a y p a q +h SOMF;2e z;pq a y p a q + X pq h SOMF;2e L ;pq a y p a q ); (2.40) where h SOMF;2e L + =h SOMF;2e x; +ih SOMF;2e y; ; (2.41) h SOMF;2e L =h SOMF;2e x; ih SOMF;2e y; : (2.42) 35 The indicesp andq run over spin-restricted orbitals (e.g., AOs) and the spin labels forh SOMF indicate the parts that should be contracted with the corresponding parts of the transition density matrix to yield SOCs (see the note 75 ). In the case of closed-shell references, the SOMF approximation of the two-electron part of SOC can be treated in the same way as the one-electron part. To illustrate the problem of open-shell references, let us first discuss the contributions from the h SOMF;2e z terms. In the case of a closed-shell reference, and parts of the HF density (or, in general, the density of the reference determinant) are the same, which makes h SOMF;2e z;pq equal toh SOMF;2e z;pq . In this case, the SOMF expression reduces to the form of one-electron part, Eq. (2.21). The minus sign comes from the dependence of spin–orbit operators on spin, as per Eq. (2.8), and from the fact that and electrons have the opposite signs of S z . However, in general, h SOMF;2e z;pq 6= h SOMF;2e z;pq for an open-shell reference determinant in which the number of and electrons is different. In this case, and parts of the HF density are not the same; therefore, the mean- field Fock-like matrices from Eq. (2.38) also do not have a proper symmetry of the different spin blocks. Theh SOMF;2e z terms acquire additional unphysical contributions from the singlet part of the transition density matrix, which leads to artifacts illustrated numerically in Section 2.3.4. As shown below, the singlet contribution violates the symmetry of spin–orbit operator. One possible solution, which we adopt in this work, is to antisymmetrize the h SOMF;2e z integrals with respect to the and parts and take a trace triplet transition density matrix for SOCs. This operation eliminates the unphysical contribution from the singlet transition densities. By doing so, one can take theh SOMF;2e z integrals out of parenthesis, making the form of the SOMF expression, Eq. (2.40), the same as for the one-electron part, Eq. (2.21). The contribution of the two remaining terms to SOC in case of the open-shell references require the formation ofh SOMF L not from the fullx andy matrices, as in Eqs. (3.17) and (3.18), 36 but from their parts that will be contracted with transition density matrices ofa y p a q anda y p a q types, respectively (see appendix B in Ref. 28). This leads to the partitioning of the terms as in Eqs. (2.41) and (2.42). Different and parts of the HF density deteriorate the symmetry of the resulting matrix elements as well. The impact of this violation and a possible fix are discussed in Section 2.3.4. 2.2.5 Averaging scheme for interstate matrix elements within a non- Hermitian framework To obtain a Hermitian SOC matrix, the SOC couplings should be averaged in some way. The averaging procedure should satisfy the following requirements: 1. The averaging procedure should be applicable to complex-valued quantities; 2. The averaging procedure should produce meaningful phases; 3. The averaged matrix should transform under the rotations in the same manner as the unaveraged matrices, making the SOCC and splittings invariant with respect to the choice of the coordinate system. A common strategy for computing interstate properties within EOM-CC is to take a geometric average of individual matrix elements 54 . Numerical examples illustrate that, as expected, the phases of A ! B and B ! A matrix elements are not exactly conjugated. This leads to an ambiguity in geometric averaging: one can either average absolute values of the matrix elements and assign phases in some way, or average the Cartesian components of the matrix. We observe that in the absence of point group symmetry geometric averaging of the absolute values violates the rotational invariance of SOCC. In contrast, the arithmetic average satisfies the requirements above while producing the same value regardless of whether the spherical or Cartesian components were averaged. 37 We can consider the transformation properties of the arithmetic average: H 0 ar:av;A!B = H 0 A!B + (H 0 B!A ) y = D y S H A!B D S 0 + (D y S 0 H B!A D S ) y = D y S H A!B D S 0 + (D y S H y B!A D S 0) = D y S H ar:av;A!B D S 0: (2.43) Here D are Wigner’s D-matrices, written in the basis of states with S or S 0 spins, and H denote spin–orbit Hamiltonian matrix, computed between electronic states with spin projec- tionsS;S +1;:::;S1;S andS 0 ;S 0 +1;:::;S 0 1;S 0 . Although weighted arithmetic average is more theoretically justified 77 , a simple arithmetic average is often numerically close to the weighted average 78 . 2.3 Results and discussion 2.3.1 Computational details To illustrate the methodology and to provide reference data for future developments, we considered several groups of molecules with various electronic structure patterns: 1. A set of isoelectronic diradicals, which were investigated in previous studies 28, 79 , CH 2 , NH + 2 , SiH 2 , PH + 2 . 2. A set of doublet radicals, which are of interest in laser cooling experiments: AsN + , GeO + , CaF, and CaOCH 3 . 3. A set of transition-metal ions: Fe 3+ , Fe 2+ , Mn + . 4. A single-center iron-containing SMM derived from the structure from Ref. 46. Figure 6.2 illustrates essential features of the electronic structure patterns of the molecules from the four sets and explains which variants of EOM-CC were used to access the target states. 38 IP DIP EA DEA EE SF Figure 2.2: Electronic configurations for reference and target states of the considered sys- tems: (a) Methylene-like diradicals: CH 2 , SiH 2 , NH + 2 , PH + 2 ; (b) Diatomic cations: AsN + , GeO + ; (c) CaF and CaOCH 3 ; (d) and (e) Hextet reference (left), one of the quartet d 5 con- figurations of Fe 3+ (top), and one of the quintet d 6 configurations of Fe 2+ , Mn + , (tpa)Fe (bottom). The diradical set was treated with the EOM-EE-CCSD and EOM-SF-CCSD methods. The latter provides a balanced treatment of the leading configurations in the target-states wave functions—the determinants shown in Figure 6.2(a) appear at the same level of excitations from the high-spin reference determinant. In set 2, the target states have either one electron or one hole in the two degenerate orbitals ( orE). To achieve a balanced treatment of the corresponding configurations, we used EOM- IP-CCSD and EOM-EA-CCSD. EOM-IP-CCSD removes an electron from the fully occupied 39 degenerate orbitals of the closed-shell neutral reference to describe diatomic cations (Fig- ure 6.2(b)), whereas EOM-EA-CCSD adds an electron to empty degenerate orbitals of the cationic reference to describe neutral calcium derivatives (Figure 6.2(c)). The electronic degeneracies are even more extensive for transition metal ions and the SMM. To treat these systems, we used the EOM-SF-CCSD and EOM-EA-CCSD methods with high- spin hextet references, which afford a balanced treatment of the quartetd 5 and quintetd 6 deter- minants (Figures 6.2(d) and 6.2(e)). We used Dunning’s cc-pVDZ and cc-pVTZ basis sets 80–83 . Core electrons were frozen in all calculations. Unrestricted HF references were used in the calculations using open-shell references. For diradicals, we used the same geometries as in Ref. 28, 79. The calculations for set 2 were carried out at their ground-state doublet geometries. The structures of AsN + and GeO + were optimized with EOM-IP-CCSD/cc-pVDZ from the neutral reference; CaF and CaOCH 3 were optimized with EOM-EA-CCSD/cc-pVDZ from the cationic reference. The structure of a model system representing SMM was derived from the original one 46 by replacing the mesitylene groups in the chelating ligand, tpa Mes , by hydrogens. We refer to the resulting ligand as tris(pyrrolylmethyl)amine (tpa). The ground state of the neutral (tpa)Fe has 5 unpaired electrons, as in Fe 3+ ion. The geometry of the neutral (tpa)Fe complex was optimized with!B97X-D/cc-pVDZ for the hextet state. We used this geometry for the calculations of the anion as well. All relevant Cartesian geometries are given in Appendix C. We used tight convergence criteria for the SOC calculations for all examples except Fe(II) complex: SCF (10 12 ), EOM (10 10 ), CC energy (10 10 ), T and -amplitudes (10 9 ). For (tpa)Fe we used the default Q-Chem’s convergence criteria: SCF (10 8 ), EOM (10 5 ). 40 2.3.2 Numerical examples Tables 2.1, 2.2, and 2.3 show the SOCCs computed with the selected EOM-CC methods. Table 2.1 shows the results for diradicals f . As discussed in previous work 28 , the SOCCs computed with EOM-EE and EOM-SF agree well in the systems with small diradical character (SiH 2 and PH 2 ), but the differences increase when diradical character increases. As one can see, the discrepancy is dominated by the one-electron part; therefore, it is not related to the artifacts due to open-shell references. The results in Table 2.2 show anticipated trends: the magnitude of SOCC increases for heavy elements. Moreover, for these systems the contribution of the two-electron part is smaller than for typical organic molecules. The magnitude of SOCCs between the degenerate pairs of states ( 2 or 2 E) is slightly larger than between - and manifolds, which probably can be rationalized by considering the shapes of the respective MOs and El-Sayed’s rule 84 . Table 2.3 shows atomic SOCCs. For atoms Eq. (2.1) no longer gives a rotationally invariant constant. Therefore, here we summed the matrix elements not only by spin projections, but also by the projections of orbital angular momentum. This also leads to large atomic SOCCs. These values can be compared with experimental data, as explained below. To compare these values with experiment, one should divide the SOCC by p L(L + 1) p S(S + 1). The resulting quan- tity can be compared with Russell–Sauders LS coupling constant () extracted from splittings through the Land´ e interval rule 85 . For example, SOMF gives an estimate for 5 D g levels of Mn + of = 68:29 cm 1 , while the values extracted from experiment are within 58.66–66.99 cm 1 interval 86 . The averaged experimental value of for Fe 2+ is 100 cm 187 , while the SOMF estimate from Table 2.3 is 108.54 cm 1 . f The SOCCs for CH 2 , SiH 2 , PH + 2 , NH + 2 reported in Ref. 28 are off by a factor of p 2. The reason for this unfortunate mistake becomes clear by considering the respective Cartesian geometries listed in the SI of Ref. 28. 41 Table 2.1: Spin–orbit coupling constants (cm 1 ) between the 3 B 1 and 1 A 1 states in selected diradicals. EOM-SF-CCSD/cc-pVTZ EOM-EE-CCSD/cc-pVTZ System 1e SOMF 1e SOMF CH 2 21.79 10.86 26.55 13.22 SiH 2 73.79 56.74 77.86 59.87 NH + 2 31.65 18.26 61.20 35.26 PH + 2 152.26 119.97 160.92 126.78 Table 2.2: Doublet radicals relevant for laser cooling experiments. SOCCs (cm 1 ) between the lowest degenerate and non-degenerate doublet states are computed with EOM-EA- CCSD/cc-pVTZ from closed-shell cationic references (CaF, CaOCH 3 ) and with EOM-IP- CCSD/cc-pVTZ from neutral closed-shell references (AsN + , GeO + ). 2 + = 2 or 2 A 1 = 2 E 2 = 2 or 2 E= 2 E System 1e SOMF 1e SOMF CaF a 28.86 24.95 49.60 39.09 CaOCH b 3 26.57 22.89 43.24 34.42 AsN +a 373.49 349.31 436.81 375.19 GeO +a 97.56 110.07 228.81 166.90 a Transitions between 2 components and 2 + were considered b Transitions between 2 E components and 2 A 1 were considered Table 2.3: Atomic SOCCs (cm 1 ) computed for several ions. EOM-SF-CCSD/cc-pVTZ from the hextet 6 S reference was used for Fe 3+ states; EOM-EA-CCSD/cc-pVTZ with electron attachment was used for isoelectronic Fe 2+ and Mn + states from the 6 S Fe 3+ and Mn 2+ references. System 1e SOMF Fe 3+ , 4 F g 135.90 80.95 Fe 3+ , 4 P g 32.64 19.32 Fe 2+ , 5 D g 2159.21 1189.01 Mn + , 5 D g 1413.78 748.14 2.3.3 ZFS in Fe(II) SMM Iron complexes, which feature rich electronic properties related to the malleability of their spin states, have been extensively studied 88–90 . Here we consider a single-center Fe(II) SMM, which was shown to have a large spin-reversal barrier (magnetic anisotropy) 46 . The structure of the model system representing this SMM is shown in Fig. 2.3. The SMM hasd 6 electronic 42 U E S −2 −1 0 +1 +2 Figure 2.3: Top: Structure of (tpa)Fe (C 15 N 4 H 15 Fe) in the hextet neutral state. Iron is shown in red, nitrogens in blue, carbons in gray, and hydrogens in white. Bottom: The lowest spin-split levels of quintet anionic (tpa)Fe , showing a spin-reversal barrierU (the energy gap between the lowest and the highest spin-split states within the ground state multiplet). configuration, corresponding the overall negative charge 46 . As explained below, this electronic configuration gives rise to electronic degeneracy and Jahn–Teller distortions. EOM-CC methods are capable of tackling Jahn–Teller and pseudo-Jahn–Teller effects very well 56, 66, 91–94 , however, this particular system is different and its description is affected by the non-Hermitian nature of EOM-CC theory 95 . The high-spin hextet neutral (tpa)Fe state has electronic configurationd 5 and, therefore, can be well described by a single Slater determinant, similarly to Fe 3+ and Mn 2+ . This state is not degenerate and it has a geometry with C 3 point group of symmetry, as shown in Fig. 2.3. Although this group is Abelian, it has one- and two-dimensional real irreducible representa- tions (irreps), which can give rise to Jahn–Teller effect for certain electronic configurations of the two-dimensional irrep 96 . For example, the quintet anion (tpa)Fe has doubly degenerate 43 states in C 3 structures. Unlike C 3v group, C 3 group does not have planes of symmetry, which would split a two-dimensional irrep into a symmetric and antisymmetric irrep with respect to the plane. Therefore, the two Jahn–Teller states would not fall into two different irreps, as it happens in the majority of symmetry-imposed degenerate states 56, 66, 91–94 . The only Abelian subgroup of C 3 is C 1 , therefore the two degenerate states belong to the same irrep, giving rise to general conical intersection problem. As documented numerically 97 and theoretically 95 , the description of true conical intersections is problematic in EOM-CC due to the non-Hermitian nature of the theory. In our case this leads to issues with finding left vectors at the symmet- ric geometry. To circumvent this problem, we carried out calculations at slightly asymmetric geometry, coming from DFT optimization (given in Appendix C). Although this geometry is asymmetric only within the magnitude of a symmetry threshold, it leads to a small artificial energy splitting ( 0.01 eV in the EOM-EA calculation from a neutral reference) of the states that should be degenerate. This artifact is small enough to be neglected in typical photochem- ical applications, but it affects the magnitude of spin–orbit splitting. To mitigate this issue, we average the energies of the states that should be degenerate by symmetry. Table 2.4 clearly shows that the major contribution to ZFS of the ground state comes from degenerate state by point group symmetry. ZFS splits not only the spin components: it also splits the degenerate irreps (the symmetry of the system is no longer described by point group symmetry; it is now described by double group symmetry), and the lowest of them looks very similar to the structure shown in Fig. 2.3. We followed the state-interaction procedure in compu- tation of the ZFS splittings: the arithmetically averaged SOC blocks were plugged in the matrix Hamiltonian and the unperturbed state energies were on the diagonal. Then the entire matrix was diagonalized to yield energy-split multiplet states. To study convergence with respect to the number of states included in the calculation, we computed the splittings between 2, 3, and 5 multiplets (10, 15, 25 electronic states with different spin-projections, respectively). This 44 sequence was chosen to include the degenerate irreps of the point group. The values in in Table 2.4 are very close to the experimentally derived result for (tpa Mes )Fe of 158 cm 1 . Table 2.4: Spin-reversal barrier (spin-splitting gap in the multiplet) computed with SOMF EOM-EA-MP2/cc-pVDZ from the hextet reference. Spin–orbit splittings were computed in state-interaction approach with the indicated number of electronic states. The experi- mental estimate for the barrier is 158 cm 1 . # Electronic multiplets Spin-reversal barrier, cm 1 2 173 3 158 5 157 The computed spin-reversal barriers U are related to the axial magnetic anisotropy D through the following relation: U =S 2 jDj (2.44) To estimate a transverse magnetic anisotropy, one should consider Jahn–Teller distortions, allowing mixing ofM s states, which leads to the non-zero probabilities of tunneling under the barrier. 2.3.4 Impact of violation ofL + =L symmetry and triplet form by canon- ical SOMF As explained in Section 2.2.4, the direct application of SOMF in the case of open-shell ref- erences may violateL + =L -symmetry and the triplet structure of spin–orbit operator. L + =L - symmetry becomes evident from the following observation. Eqs. (8), (9), (10) from 40 give hISjjH L + jjI 0 S + 1i =hISjjH L jjI 0 S + 1i : (2.45) 45 Because the requirement of having a Hermitian method does not appear in the proof, the state- ment is applicable to EOM-CC as well. Similarly, Eqs. (11), (12), (13) from 40 lead to the condition for the same multiplicity: hISjjH L + jjI 0 Si =hISjjH L jjI 0 Si : (2.46) Because the part of Hartree–Fock density is not equal to the part for open-shells, the symmetry betweenh SOMF;2e is broken. Table 2.5 shows numerical consequences of such vio- lation. Fortunately, regardless of the size of the system or of the number of unpaired electrons in the reference, the extent of such violation does not exceed 0.05 cm 1 for the reduced matrix elements. To eliminate its influence on SOCC and splittings, we used arithmetic averaging for hSjjH L jjS 0 i andhSjjH L + jjS 0 i, similar to Eq. (2.43). Table 2.6 shows the impact of the singlet component of transition density matrix. The singlet part is small for transitions between states with different multiplicities. It is not zero because of small spin contamination of the electronic states. The magnitude of singlet part can be signif- icant for the transitions between states of the same multiplicities. Using the triplet component of transition density matrix for calculation of matrix elements circumvents this issue. To conclude, the arithmetic averaging betweenhSjjH L jjS 0 i andhSjjH L + jjS 0 i restores the correct symmetry of these elements. Usage of the triplet transition density does not lead to contamination ofhSjjH Lz jjS 0 i matrix element with the unphysical singlet contribution. 2.4 Conclusion We presented a theory for calculating matrix elements of the BP SOC operator based on the application of Wigner–Eckart’s theorem to reduced one-particle density matrices. The key equations are given in the second-quantized spin-orbital form. The algorithm is ansatz-agnostic 46 Table 2.5: Extent of violation of L + =L -symmetry of SOMF with open-shell references, cm 1 . Only A(lowest energy)!B(higher energy) transition is shown a . Element/System CH 2 NH + 2 SiH 2 PH + 2 hSjjH L jjS 0 i -13.241i 22.269i 40.489i -146.113i hSjjH L + jjS 0 i 13.215i -22.264i -40.492i 146.111i (tpa)Fe , 1!2 (tpa)Fe , 1!3 hSjjH L jjS 0 i 0.293-0.096i -32.067+228.232i hSjjH L + jjS 0 i 0.290+0.095i -32.073-228.276i a Same computational setup as in Table 2.1 and Table 2.4. Table 2.6: Unphysical singlet reduced matrix elements for the considered systems, cm 1 . The hydrides were oriented in the way that thathSjjH Lz jjS 0 i elements are not zero. Element/System CH 2 NH + 2 SiH 2 PH + 2 hSjjH singlet Lz jjS 0 i 0.004i -0.001i -0.002i 0.001i (tpa)Fe , 1!2 (tpa)Fe , 1!3 hSjjH singlet Lz jjS 0 i 13.083i 0.004i and can be used with any electronic stricture method for which state and transition density matrices are available. The current implementation is based on the EOM-CCSD suite of meth- ods. The approach allows one to compute the SOC matrix for the entire multiplet from just one transition density matrix, which solves the problem of accessibility of states of different spin projections. It also addresses, by constriction, the phase problem due to Born–Oppenheimer’s separation of the nuclear and electronic degrees of freedom. We also highlighted an important aspect of calculation of SOCCs within a non-Hermitian theory and proposed using arithmetic averaging of the EOM-CC matrix elements, which leads to rotationally invariant SOCCs. The current implementation treats the two-electron part of the BP Hamiltonian via the SOMF approximation. We discuss special aspects of the application of SOMF to open-shell references and propose practical solutions. In particular,L + andL reduced matrix elements are slightly different. This issue can be solved by averaging. Contribution of the (unphysical) singlet part of the transition density can be important. This issue does not occur if only the triplet component is used. 47 2.5 Appendix A: Evaluation of Clebsh–Gordan coefficients The Racah recursions 69 formally give a definite set of Clebsh–Gordan’s coefficients, but their implementation require a certain effort to avoid division by zero and possible phase mis- match. Instead, we follow the Complete Set of Commuting Observables (CSCO) approach 98 in which matrices of commuting operators are constructed and diagonalized. The following algo- rithm produces Clebsh–Gordan’s coefficients in the Condon–Shortley notation. Here indices 1 and 2 refer to the particles 1 and 2, absence of an index refers to the result of addition of angular momentum,j denotes angular momentum quantum number,m denotes the quantum number of the projection of angular momentum. The algorithm proceeds as follows: 1. Takej 1 ,j 2 ,j from the input. 2. Makehj 1 m 1 jJ 1; jj 1 m 0 1 i,hj 2 m 2 jJ 2; jj 2 m 0 2 i. 3. Makehj 1 m 1 ;j 2 m 2 jJ 2 jj 1 m 0 1 ;j 2 m 0 2 i in the tensor space ofjj 1 m 0 1 ;j 2 m 0 2 i. 4. Diagonalizehj 1 m 1 ;j 2 m 2 jJ 2 jj 1 m 0 1 ;j 2 m 0 2 i and save only eigenvectors forj(j + 1) eigen- value. 5. From these eigenvectors, build a projectorP onto their subspace. 6. Diagonalize B = P (J z + C)P , where the shift constant C is large enough to move eigenvalues above zero. 7. Synchronize the phases, multiply the eigenvector vector with the largest eigenvalue ofB by sign ofhj 1 j 1 ;j 2 (jj 2 )jjji. 8. Fix the phases of other eigenvectors recursively: form = j;j 1;:::;j + 1 multiply thejjm 1i vector by sign ofhjm 1jJ jjmi. 48 9. Returnhj 1 m 1 ;j 2 m 2 jjmi coefficients. This algorithm produces a cube of coefficients with them 1 ,m 2 ,m indices, which are needed for calculation of the SOC matrix. 2.6 Appendix B: Examples of SOC matrix Cartesian geometry: $molecule 0 3 C -0.0000000000 0.0000000000 0.1067875138 H -0.9892163971 -0.0000000000 -0.3203625414 H 0.9892163971 0.0000000000 -0.3203625414 $end One-electron SOC, ket state is the triplet, bra state is the singlet: |Sz=-1.00> |Sz=0.00> |Sz=1.00> <Sz=0.00|(-0.00,15.327014)(0.00,-0.00)(-0.00,-15.327014) SOMF(L + =L averaged, without singlet part): |Sz=-1.00> |Sz=0.00> |Sz=1.00> <Sz=0.00|(-0.00,7.638437)(0.00,-0.00)(-0.00,-7.638437) One-electron SOC, bra state is the triplet, ket state is the singlet: |Sz=0.00> <Sz=-1.00|(-0.000000,-15.488633) <Sz=0.00|(0.000000,0.000000) <Sz=1.00|(-0.000000,15.488633) SOMF(L + =L averaged, without singlet part): |Sz=0.00> <Sz=-1.00|(-0.000000,-7.720793) <Sz=0.00|(0.000000,0.000000) <Sz=1.00|(-0.000000,7.720793) 49 Cartesian geometry: $molecule 0 3 C 0.0000000000 0.1067875138 -0.0000000000 H -0.0000000000 -0.3203625414 -0.9892163971 H 0.0000000000 -0.3203625414 0.9892163971 $end One-electron SOC, ket state is the triplet, bra state is the singlet: |Sz=-1.00> |Sz=0.00> |Sz=1.00> <Sz=0.00|(-0.00,0.00)(0.00,21.675671)(-0.00,-0.00) SOMF(L + =L averaged, without singlet part): |Sz=-1.00> |Sz=0.00> |Sz=1.00> <Sz=0.00|(-0.00,0.00)(0.00,10.802382)(-0.00,-0.00) One-electron SOC, bra state is the triplet, ket state is the singlet: |Sz=0.00> <Sz=-1.00|(-0.000000,-0.000000) <Sz=0.00|(0.000000,-21.904235) <Sz=1.00|(-0.000000,0.000000) SOMF(L + =L averaged, without singlet part): |Sz=0.00> <Sz=-1.00|(-0.000000,-0.000000) <Sz=0.00|(0.000000,-10.918850) <Sz=1.00|(-0.000000,0.000000) 2.7 Appendix C: Relevant Cartesian geometries $comment Ethylene+O intermediate, triplet minimum Nuclear Repulsion Energy = 67.6764451387 hartrees Optimized by unrestricted CCSD/cc-pVDZ with frozen core $end $molecule 50 0 3 C -0.1066912528 0.4851410808 -0.0686818253 H -0.1551331936 1.3441018949 0.6100473684 H -0.2678294463 0.9010365297 -1.0850680040 O -1.1884097283 -0.3576957703 0.0928241625 C 1.2002148897 -0.2212534852 0.0159920619 H 1.3003502296 -1.2006527887 -0.4257274024 H 2.0488729957 0.2394985805 0.4969299522 $end $comment CaF Nuclear Repulsion Energy = 43.0525030945 hartrees Optimized by unrestricted EOM-EA-CCSD/cc-pVDZ with frozen core from CaFˆ+ reference $end $molecule Ca 0.0000000000 0.0000000000 -0.6866252069 F 0.0000000000 0.0000000000 1.5258337931 $end $comment CaOCH3 Nuclear Repulsion Energy = 100.9872753173 hartrees Optimized by unrestricted EOM-EA-CCSD/cc-pVDZ with frozen core from CaOCH3ˆ+ reference $end $molecule Ca 0.0000000000 0.0000000000 1.3222507248 O 0.0000000000 0.0000000000 -0.7425868450 C 0.0000000000 0.0000000000 -2.1389498327 H 1.0312218725 0.0000000000 -2.5568735797 H -0.5156109362 0.8930643385 -2.5568735797 H -0.5156109362 -0.8930643385 -2.5568735797 $end $comment AsNˆ+ Nuclear Repulsion Energy = 72.2314383570 hartrees Optimized by unrestricted EOM-IP-CCSD/cc-pVDZ with frozen core from AsN neutral reference $end 51 $molecule As 0.0000000000 0.0000000000 -0.2961589750 N 0.0000000000 0.0000000000 1.3961780250 $end $comment GeOˆ+ Nuclear Repulsion Energy = 81.3966656261 hartrees Optimized by unrestricted EOM-IP-CCSD/cc-pVDZ with frozen core from GeO neutral reference $end $molecule Ge 0.0000000000 0.0000000000 -0.3328622000 O 0.0000000000 0.0000000000 1.3314488000 $end $comment neutral hextet (tpa)Fe geometry, optimized with wB97XD/cc-pVDZ $end $molecule 0 6 N -0.0009594129 -0.0005151132 1.2715236551 N 1.3796505581 -1.3423679276 -0.6796869350 N -1.8531795328 -0.5112691392 -0.6807247131 N 0.4744716301 1.8639100138 -0.6803344605 C 1.1451037230 2.8312819863 -1.3939265612 C 0.4217802546 2.2980622639 0.6398265889 C 1.4983579229 3.8656841193 -0.5571924750 C 1.0421397333 3.5194454229 0.7510412251 C -2.2030430284 -0.7797532699 0.6381174299 C -3.0256606760 -0.4189921773 -1.3959756544 C -4.0980133907 -0.6367821660 -0.5607584307 C -3.5707117890 -0.8596599045 0.7474974740 C 1.8857694100 -2.4045491730 -1.3941612512 C 1.7795278098 -1.5151271547 0.6411864849 C 2.5285507710 -2.6622603345 0.7519965822 C 2.6041637627 -3.2278802178 -0.5569987154 C 1.3583354911 -0.4748939165 1.6338671040 C -0.2702946696 1.4142135906 1.6321594602 C -1.0921896812 -0.9409254395 1.6303855035 H 2.0299585686 4.7692029943 -0.8454370880 H 1.1504775701 4.1046208613 1.6613974936 H -5.1464335392 -0.6320435527 -0.8492763773 52 H -4.1337641022 -1.0619170971 1.6558315805 H 2.9776577184 -3.0528505066 1.6623583170 H 3.1241574763 -4.1380127924 -0.8456881405 H 2.0146733390 0.4089366821 1.5930025488 H 1.3701401594 -0.8641303596 2.6652410494 H -1.3639721681 1.5421160095 1.5900163807 H 0.0583583554 1.6195866609 2.6642430144 H -0.6550743392 -1.9512839337 1.5841045064 H -1.4339350869 -0.7609728844 2.6629427641 H 1.3294318038 2.7092430480 -2.4588421637 H -3.0130140952 -0.1972049798 -2.4606855314 H 1.6918753419 -2.5031502086 -2.4598148605 Fe 0.0009029075 0.0000323988 -0.8532015669 $end 53 2.8 Chapter 2 References 1 D. R. Yarkony, Int. Rev. Phys. Chem. 11, 195 (1992). 2 C. M. Marian, WIREs Comput. Mol. Sci. 2, 187 (2012). 3 P. M. Lahti, editor, Magnetic properties of organic materials. Marcel Dekker, Inc, 1999. 4 M.R. Pederson and T. Baruah, Molecular magnets: Phenomenology and theory, in Handbook of Magnetism and Magnetic Materials, edited by S. Parkin, pages 1–22. J. Wiley and Sons, London, 2007. 5 J.P. Malrieu, R. Caballol, C.J. Calzado, C. de Graaf, and N. Guih´ ery, Chem. Rev. 114, 429 (2013). 6 M. N. Leuenberger and D. Loss, Nature 410, 789 (2001). 7 C. Schlegel, J. van Slageren, M. Manoli, E. K. Brechin, and M. Dressel, Phys. Rev. Lett. 101, 147203 (2008). 8 M. J. Mart´ ınez-P´ erez, S. Cardona-Serra, C. Schlegel, F. Moro, P.J. Alonso, H. Prima-Garc´ ıa, J. M. Clemente-Juan, M. Evangelisti, A. Gaita-Arino, J. Ses´ e, J. van Slageren, E. Coronado, and F. Luis, Phys. Rev. Lett. 108, 247213 (2012). 9 M. D. Jenkins, D. Zueco, O. Roubeau, G. Aromi, J. Majere, and F. Luis, Dalton Trans. 45, 16682 (2016). 10 M.R. Pederson and S.N. Khanna, Phys. Rev. B 60, 9566 (1999). 11 O. Vahtras, O. Loboda, B. Minaev, H. ˚ Argen, and K. Ruud, Chem. Phys. 279, 133 (2002). 12 D. Ganyushin and F. Neese, J. Chem. Phys. 125, 024103 (2006). 13 M. Atanasov, D. Aravena, E. Suturina, E. Bill, D. Maganas, and F. Neese, Coord. Chem. Rev. 289-290, 177 (2015). 14 H. A. Bethe and E. E. Salpeter, Quantum mechanics of one and two electron atoms. Plenum, New York, 1977. 15 P. Schwerdtfeger, editor, Relativistic Electronic Structure Theory. Elsevier, Amsterdam, 2002. 16 D. G. Fedorov, S. Koseki, M. W. Schmidt, and M. S. Gordon, Int. Rev. Phys. Chem. 22, 551 (2003). 54 17 T. R. Furlani and H. F. King, J. Chem. Phys. 82, 5577 (1985). 18 H. F. King and T. R. Furlani, J. Comput. Chem. 9, 771 (1988). 19 O. Vahtras, H. ˚ Agren, P. Jørgensen, H. J. Aa. Jensen, T. Helgaker, and J. Olsen, J. Chem. Phys. 96, 2118 (1992). 20 D. G. Fedorov and M. S. Gordon, J. Chem. Phys. 112, 5611 (2000). 21 D. G. Fedorov and J. P. Finley, Phys. Rev. A 64, 042502 (2001). 22 H. ˚ Agren, O. Vahtras, and B. Minaev, Response theory and calculations of spin-orbit coupling phenomena in molecules, in Advances in Quantum Chemistry, volume 27, pages 71–162. Academic Press, 1996. 23 A. Berning, M. Schweizer, H.-J. Werner, P. Knowles, and P. Palmieri, Mol. Phys. 98, 1823 (2000). 24 F. Neese, J. Chem. Phys. 122, 034107 (2005). 25 O. Christiansen, J. Gauss, and B. Schimmelpfennig, Phys. Chem. Chem. Phys. 2, 965 (2000). 26 K. Klein and J. Gauss, J. Chem. Phys. 129, 194106 (2008). 27 L.A. M¨ uck, Highly Accurate Quantum Chemistry: Spin-Orbit Splittings via Multireference Coupled-Cluster Methods and Applications in Heavy-Atom Main-Group Chemistry, PhD thesis, Johannes-Gutenberg Universit¨ at Mainz, 2013. 28 E. Epifanovsky, K. Klein, S. Stopkowicz, J. Gauss, and A. I. Krylov, J. Chem. Phys. 143, 064102 (2015). 29 P. ˚ A. Malmqvist, B. O. Roos, and B. Schimmelphennig, Chem. Phys. Lett. 357, 230 (2002). 30 B. O. Roos and P.- ˚ A. Malmqvist, Phys. Chem. Chem. Phys. 6, 2919 (2004). 31 M. Roemelt, J. Chem. Phys. 143, 044112 (2015). 32 L.A. M¨ uck and J. Gauss, J. Chem. Phys. 136, 111103 (2012). 33 F. Neese, T. Petrenko, D. Ganyushin, and G. Olbrich, Coordination Chemistry Reviews 251, 288 (2007). 34 S.G. Chiodo and N. Russo, Chem. Phys. Lett. 490, 90 (2010). 35 Z. Li, B. Suo, Y . Zhang, Y . Xiao, and W. Liu, Mol. Phys. 111, 3741 (2013). 36 M. Roemelt, D. Maganas, S. DeBeer, and F. Neese, J. Chem. Phys. 138, 204101 (2013). 55 37 C. Eckart, Rev. Mod. Phys. 2, 305 (1930). 38 E. Wigner, Zeits. f. Physik 43, 624 (1927). 39 R.N. Zare, Angular momentum: understanding spatial aspects in chemistry and physics. J. Wiley & Sons, 1988. 40 D. G. Fedorov, Theoretical study of spin-orbit coupling in molecules, PhD thesis, Iowa State University, 1999. 41 S. Schmitt, P. Jost, and C. van W¨ ullen, J. Chem. Phys. 134, 194113 (2011). 42 B. Cahier, R. Maurice, H. Bolvin, T. Mallah, and N. Guih´ ery, Magnetochemistry 2, 31 (2016). 43 D. Ganyushin and F. Neese, J. Chem. Phys. 128, 114117 (2008). 44 R. McWeeny, J. Chem. Phys. 42, 1717 (1965). 45 R. McWeeny, Spins in Chemistry. Academic Press, New York, 1970. 46 D. E. Freedman, W. H. Harman, T. D. Harris, G. J. Long, C. J. Chang, and J. R. Long, J. Am. Chem. Soc. 132, 1224 (2010). 47 Y . Shao, Z. Gan, E. Epifanovsky, A.T.B. Gilbert, M. Wormit, J. Kussmann, A.W. Lange, A. Behn, J. Deng, X. Feng, D. Ghosh, M. Goldey, P.R. Horn, L.D. Jacobson, I. Kaliman, R.Z. Khaliullin, T. Kus, A. Landau, J. Liu, E.I. Proynov, Y .M. Rhee, R.M. Richard, M.A. Rohrdanz, R.P. Steele, E.J. Sundstrom, H.L. Woodcock III, P.M. Zimmerman, D. Zuev, B. Albrecht, E. Alguires, B. Austin, G.J.O. Beran, Y .A. Bernard, E. Berquist, K. Brandhorst, K.B. Bravaya, S.T. Brown, D. Casanova, C.-M. Chang, Y . Chen, S.H. Chien, K.D. Closser, D.L. Crittenden, M. Diedenhofen, R.A. DiStasio Jr., H. Do, A.D. Dutoi, R.G. Edgar, S. Fatehi, L. Fusti-Molnar, A. Ghysels, A. Golubeva-Zadorozhnaya, J. Gomes, M.W.D. Hanson- Heine, P.H.P. Harbach, A.W. Hauser, E.G. Hohenstein, Z.C. Holden, T.-C. Jagau, H. Ji, B. Kaduk, K. Khistyaev, J. Kim, J. Kim, R.A. King, P. Klunzinger, D. Kosenkov, T. Kowal- czyk, C.M. Krauter, K.U. Laog, A. Laurent, K.V . Lawler, S.V . Levchenko, C.Y . Lin, F. Liu, E. Livshits, R.C. Lochan, A. Luenser, P. Manohar, S.F. Manzer, S.-P. Mao, N. Mardirossian, A.V . Marenich, S.A. Maurer, N.J. Mayhall, C.M. Oana, R. Olivares-Amaya, D.P. O’Neill, J.A. Parkhill, T.M. Perrine, R. Peverati, P.A. Pieniazek, A. Prociuk, D.R. Rehn, E. Rosta, N.J. Russ, N. Sergueev, S.M. Sharada, S. Sharmaa, D.W. Small, A. Sodt, T. Stein, D. Stuck, Y .-C. Su, A.J.W. Thom, T. Tsuchimochi, L. V ogt, O. Vydrov, T. Wang, M.A. Watson, J. Wenzel, A. White, C.F. Williams, V . Vanovschi, S. Yeganeh, S.R. Yost, Z.-Q. You, I.Y . Zhang, X. Zhang, Y . Zhou, B.R. Brooks, G.K.L. Chan, D.M. Chipman, C.J. Cramer, W.A. Goddard III, M.S. Gordon, W.J. Hehre, A. Klamt, H.F. Schaefer III, M.W. Schmidt, C.D. Sherrill, D.G. Truhlar, A. Warshel, X. Xu, A. Aspuru-Guzik, R. Baer, A.T. Bell, N.A. Besley, J.-D. Chai, A. Dreuw, B.D. Dunietz, T.R. Furlani, S.R. Gwaltney, C.-P. Hsu, Y . Jung, J. Kong, D.S. Lambrecht, W.Z. Liang, C. Ochsenfeld, V .A. Rassolov, L.V . Slipchenko, J.E. Subotnik, T. Van V oorhis, J.M. Herbert, A.I. Krylov, P.M.W. Gill, and M. Head-Gordon, Mol. Phys. 113, 184 (2015). 56 48 A. I. Krylov and P. M. W. Gill, WIREs: Comput. Mol. Sci. 3, 317 (2013). 49 K. Emrich, Nucl. Phys. A351, 379 (1981). 50 A. I. Krylov, Annu. Rev. Phys. Chem. 59, 433 (2008). 51 R. J. Bartlett, Mol. Phys. 108, 2905 (2010). 52 K. Sneskov and O. Christiansen, WIREs: Comput. Mol. Sci. 2, 566 (2012). 53 R. J. Bartlett, WIREs: Comput. Mol. Sci. 2, 126 (2012). 54 J. F. Stanton and R. J. Bartlett, J. Chem. Phys. 98, 7029 (1993). 55 J. F. Stanton and J. Gauss, J. Chem. Phys. 101, 8938 (1994). 56 P. A. Pieniazek, S. E. Bradforth, and A. I. Krylov, J. Chem. Phys. 129, 074104 (2008). 57 M. Nooijen and R. J. Bartlett, J. Chem. Phys. 102, 3629 (1995). 58 A. I. Krylov, Chem. Phys. Lett. 338, 375 (2001). 59 A. I. Krylov, Acc. Chem. Res. 39, 83 (2006). 60 K. W. Sattelmeyer, H. F. Schaefer, and J. F. Stanton, Chem. Phys. Lett. 378, 42 (2003). 61 M. Musiał, A. Perera, and R. J. Bartlett, J. Chem. Phys. 134, 114108 (2011). 62 D. Casanova, L. V . Slipchenko, A. I. Krylov, and M. Head-Gordon, J. Chem. Phys. 130, 044103 (2009). 63 T. Helgaker, S. Coriani, P. Jørgensen, K. Kristensen, J. Olsen, and K. Ruud, Chem. Rev. 112, 543 (2012). 64 T. Helgaker, P. Jørgensen, and J. Olsen, Molecular electronic structure theory. Wiley & Sons, 2000. 65 K. D. Nanda and A. I. Krylov, J. Chem. Phys. 142, 064118 (2015). 66 A. I. Krylov, The quantum chemistry of open-shell species, in Reviews in Comp. Chem., edited by A. L. Parrill and K. B. Lipkowitz, volume 30, pages 151–224. J. Wiley & Sons, 2017. 67 E. Epifanovsky, M. Wormit, T. Ku´ s, A. Landau, D. Zuev, K. Khistyaev, P. U. Manohar, I. Kali- man, A. Dreuw, and A. I. Krylov, J. Comput. Chem. 34, 2293 (2013). 68 P. Pokhilko, R. Shannon, D. Glowacki, H. Wang, and A. I. Krylov, J. Phys. Chem. A 123, 482 (2019). 57 69 G. Racah, Phys. Rev. 62, 438 (1942). 70 B. A. Hess, C. M. Marian, U. Wahlgren, and O. Gropen, Chem. Phys. Lett. 251, 365 (1996). 71 K. Ruud, B. Schimmelpfennig, and H. ˚ Agren, Chem. Phys. Lett. 310, 215 (1999). 72 M. Kaupp, R. Reviakine, O. L. Malkina, A. Arbuznikov, B. Schimmelpfenning, and V . G. Malkin, J. Comput. Chem. 23, 794 (2001). 73 J. Liu, Y . Shen, A. Asthana, and L. Cheng, J. Chem. Phys. 148, 034106 (2018). 74 J. Liu and L. Cheng, J. Chem. Phys. 148, 144108 (2018). 75 In the case of non-orthogonal orbitals (e.g., atomic orbitals), it is important to distinguish contravariant and covariant indices. From herein, the integrals are assumed to be contravari- ant and the densities are assumed to be covariant. Since the orbital spaces have a metric, the overlap matrix S, the relation between the covariant and contravariant density matrices is cov = S 1 contra S 1 . Since the working expressions of EOM-CC calculate the density matrices in the MO basis, covariant densities can be obtained without a need for the inverse overlap: cov (AO) = C (MO)C y . Covariance or contravariance does not change the spin- tensor properties of the density matrix, such as spin projectionM or singlet or triplet type. 76 S. V . Levchenko, T. Wang, and A. I. Krylov, J. Chem. Phys. 122, 224106 (2005). 77 A. Tajti and P.G. Szalay, J. Chem. Phys. 131, 124104 (2009). 78 S. Faraji, S. Matsika, and A. I. Krylov, J. Chem. Phys. 148, 044103 (2018). 79 Y . A. Bernard, Y . Shao, and A. I. Krylov, J. Chem. Phys. 136, 204103 (2012). 80 T. H. Dunning, Jr., J. Chem. Phys. 90, 1007 (1989). 81 N. B. Balabanov and K. A. Peterson, J. Chem. Phys. 123, 064107 (2005). 82 J. Koput and K. A. Peterson, J. Phys. Chem. A 106, 9595 (2002). 83 A. K. Wilson, D. E. Woon, K. A. Peterson, and T. H. Dunning, J. Chem. Phys. 110, 7667 (1999). 84 M. A. El-Sayed, Acc. Chem. Res. 1, 8 (1968). 85 A. Land´ e, Z. Phys. 15, 189 (1923). 86 A. Kramida, Yu. Ralchenko, J. Reader, and NIST ASD Team, NIST atomic spectra database, v. 5.6.1, https://physics.nist.gov/asd, 2018, National Institute of Standards and Technology, Gaithersburg, MD. DOI: https://doi.org/10.18434/T4W30F. 87 T. M. Dunn, Trans. Faraday Soc. 57, 1441 (1961). 58 88 P. G¨ utlich, Spin crossover in iron(II)-complexes, in Metal Complexes, pages 83–195, Berlin, Heidelberg, 1981. Springer Berlin Heidelberg. 89 P. G¨ utich and H. A. Goodwin, Top. Curr. Chem. 233, 1 (2004). 90 T. J. Penfold, E. Gindensperger, C. Daniel, and C. M. Marian, Chem. Rev. 118, 6975 (2018). 91 V . A. Mozhayskiy, D. Babikov, and A. I. Krylov, J. Chem. Phys. 124, 224309 (2006). 92 V . A. Mozhayskiy and A. I. Krylov, Mol. Phys. 107, 929 (2009). 93 L. V . Slipchenko and A. I. Krylov, J. Chem. Phys. 118, 6874 (2003). 94 L. V . Slipchenko and A. I. Krylov, J. Chem. Phys. 118, 9614 (2003). 95 E.F. Kjønstad and H. Koch, J. Phys. Chem. Lett. 8, 4801 (2017). 96 H.A. Jahn and E. Teller, Proc. R. Soc. London A161, 220 (1937). 97 A. K¨ ohn and A. Tajti, J. Chem. Phys. 127, 044105 (2007). 98 J.-Q. Chen, J. Ping, and F. Wang, Group representation theory for physicists. World Scientific, 2nd edition, 2002. 59 Chapter 3: Quantitative El-Sayed Rules for Many-Body Wavefunctions from Spinless Transition Density Matrices Molecular orbital theory is of central importance in chemistry. Chemists use molecular orbital concepts to explain molecular structure and properties 1 . Orbitals are used to rationalize systematic trends in series of compounds and chemical reactivity in the ground and electron- ically excited states 2, 3 . Molecular orbitals are also commonly invoked to describe electronic transitions and interstate properties, such as transition dipole moments, non-adiabatic and spin- orbit couplings 4–7 . Molecular orbitals are often associated with (pseudo)-non-interacting electrons. In the Hartree–Fock mean-field theory, anN-electron wave function is given by an antisymmetrized product onN molecular orbitals and the differences between two electronic states (e.g., neutral and cation or ground and excited) can be described in terms of individual orbitals. Conse- quently, state and interstate properties can be analyzed in terms of matrix elements between the orbitals. Yet, a quantitative description of anN-electron system entails multi-determinantal ans¨ atze. Even the simplest excited-state theory, configuration interaction singles (CIS), goes beyond the 60 single-determinant representation. One can analyze such wave functions in terms of molecu- lar orbitals by focusing on the dominant configurations, however, such approach is imprecise and quickly becomes impractical. Moreover, the choice of individual orbitals (and, conse- quently, the values of the individual amplitudes in the multi-configurational wave functions) is not unique. For example, energies and all physical observables of the Hartree–Fock and CIS wave functions are invariant with respect to any unitary orbital transformation within the occu- pied and virtual orbital spaces. This is also the case for coupled-cluster (CC) and equation-of- motion coupled-cluster (EOM-CC) theories. Despite these unsettling observations, molecular orbital theory and rigorous description of electron correlation are perfectly compatible with each other. Molecular orbital theory can be extended to correlated many-electron wave func- tions via generalized one-electron quantities such as Dyson 8–10 and natural transition orbitals (NTOs) 11–20 . In this paper, we focus on one-electron transition properties. The key quantity, which allows one to characterize the differences between two states (e.g., F and I ) in terms of one-electron excitations and to compute interstate properties, is one-particle transition density matrix: FI pq h F ja y p a q j I i; (3.1) wherea y p anda q are creation and annihilation operators associated with orbitalsp andq. The interstate matrix element for a one-electron operator ^ A = P pq A pq a y p a q is then a : h F j ^ Aj I i = X pq A pq h F ja y p a q j I i = X pq A pq FI pq =Tr[ A]; (3.2) a If non-orthogonal orbitals are used, a metric tensor should be introduced in the contraction in Eq. (3.2). The equations preserve the same form if contravariant integrals and covariant densities are used, or vice versa. A comprehensive formulation of quantum-chemical methods in non-orthogonal orbital bases is give in Ref. 21. 61 where the sum runs over all orbitals. While the value ofh F j ^ Aj I i evaluated using the above expression is orbital-invariant, the values of the individual integralsA pq =h p j ^ Aj q i and the respective elements of depend on the choice of orbitals. The most compact representation of and, consequently, of Eq. (3.2) is achieved by using diagonal representation of the transition density matrix via singular value decomposition (SVD) 11, 14 : FI = UV y ; (3.3) where is a diagonal matrix with non-negative numbers ( k ) on the diagonal,k th columns of U and V are the pair of right and left singular vectors corresponding tok th singular value k . In this way, one can represent the transition density between statesF andI as a (usually small) number of particle-hole pairs: FI (x p ;x h ) = X k k ~ p k (x p ) ~ h k (x h ); (3.4) wherex denotes spatial and spin coordinates of an electron (xr;s) and ~ p k and ~ h k , which are called particle and hole NTOs, are defined by the singular vectors from Eq. (3.3): ~ p k = X q U qk q ; (3.5) ~ h k = X q V qk q : (3.6) (3.7) FI (x p ;x h ), which sometimes is called exciton wave function 11, 22 , describes the difference between the two many-electron states, F and I , in terms of one-electron (i.e., hole-pair) excitations between the orbitals ~ p k and ~ h k . In this way, NTOs provide a rigorous extension of 62 molecular orbital theory to general correlated wave functions. In the basis of NTOs, Eq. (3.2) becomes h F j ^ Aj I i = X k h ~ p k j ^ Aj ~ h k i k : (3.8) Thus, an interstate property can be represented as a sum of the matrix elements between the hole and particle orbitals and the weight of each term is given by the respective singular value b . Often, only one or two singular values are significant in the expansion, while the contribution of the remaining singular values is small. NTOs are now routinely used to analyze various aspects of bound 16, 19, 20, 23 and metastable 18 excited states, including non-linear optical phenomena 17 . In this contribution, we extend the formalism to the analysis of tensor properties, such as spin–orbit couplings (SOCs), which involve states of different spin projection. Qualitative molecular orbital analysis has been used to rationalize the rates of the spin- forbidden transitions since early days of quantum chemistry 5, 24 . As succinctly explained by El-Sayed 5, 24 , the angular momentum operator of the SOC term induces an orbital torque, which means that large values of the SOC matrix element can be attained only when the transition involve the change of orbital orientation c . In the context of organic photochemistry, this implies that SOCs between the states that have the same orbital character (e.g., triplet and singlet) are small and SOCs between the states that have different orbital character (e.g.,n triplet and singlet) are large. Known as El-Sayed’s rule, this guideline is often invoked to explain many b If the wave functions are normalized to 1, the Frobenius norm of is bounded by 1: jj J I jj 2 = X k 2 k 1: (3.9) , c Because of the denominator in the Breit-Pauli Hamiltonian, the action of the spin–orbit operator is localized. That is why orbital analysis of SOCs as well as El-Sayed’s rules are framed in terms of atomic orbitals. 63 aspects of spin-forbidden processes, such as diradical reactivity 4, 25–27 , reaction of triplet oxygen with unsaturated compounds 26, 27 , and phosphorescent properties of photovoltaic materials 28–33 . Molecular orbital analysis is also invoked to explain the breakdown of El-Sayed’s rules due to vibronic effects 28, 34, 35 — since the molecular vibrations affect the shape of molecular orbitals, they can result in mixing ofn and states, strongly modulating the magnitude of SOCC and the rates of intersystem crossing. Despite a wide-spread use of NTOs and natural orbitals to describe characters of the states involved in the spin-forbidden transitions (as in, for example, Refs. 28, 29, 36), there have been only a couple of studies reporting the NTOs between spin-coupled states 23 or between particular components of a multiplet 27 . In this contribution, we introduce a new type of NTOs suitable for the analysis of SOCs. The definition is based on the spinless density matrices and Wigner– Eckart’s theorem, which are used for the calculation of SOCs for the entire multiplet from just one transition 37, 38 . This approach allows one to treat the transitions between states with arbitrary spin projections in a uniform way and to quantitatively describe the contributions of specific orbital pairs into the overall SOC. We use this tool to rationalize the magnitude of the SOCs computed using EOM-CC wave functions in terms of El-Sayed’s rules 5, 24 . To introduce the key concept, NTOs of the spinless transition density matrix, we first briefly outline the formalism for SOC calculations 38 using Wigner–Eckart’s theorem with the Breit- Pauli Hamiltonian and EOM-CC wave functions. It is convenient to describe spin-dependent operators through spin-tensor operators ^ O S;M . These operators are defined as a set of ^ O S;M , satisfying the following relations 39, 40 : h ^ S ; ^ O S;M i = p S(S + 1)M(M 1) ^ O S;M1 ; (3.10) h ^ S z ; ^ O S;M i =M ^ O S;M ; (3.11) 64 whereS andM label the operator’s spin and spin projection. Spin-independent operators are always singlet operators, because they commute with ^ S z and ^ S . We can split all one-electron excitation operators into the singlet and triplet subsets d : ^ T 0;0 pq = 1 p 2 a y p a q +a y p a q ; (3.12) ^ T 1;1 pq =a y p a q ; (3.13) ^ T 1;0 pq = 1 p 2 a y p a q a y p a q ; (3.14) ^ T 1;1 pq =a y p a q : (3.15) ^ T 0;0 , ^ T 1;1 , ^ T 1;0 , and ^ T 1;1 form a basis in the space of one-particle excitation operators. Because of this partitioning, the transition density matrix FI can be broken down into the singlet and triplet components. The singlet density matrices have identical and parts, singlet = singlet ; M = 0 triplet density matrices have the opposite signs of and parts triplet, M = 0 = triplet, M = 0 . Because of this permutational property, the singlet and triplet den- sity matrices are orthogonal to each other e . To evaluate the transition matrix element, the oper- ators of the singlet type require only singlet part of the transition density matrix from Eq. (3.2), while the triplet operators require only the triplet part. Since different parts of the density matrix give rise to different physical observables, one should analyze the part that is relevant for the investigated property. That is, for rationalizing transition dipole moments or non-adiabatic cou- plings between spin-coupled states 41 , one should consider the singlet part of , whereas for computing SOC between the non-relativistic zero-order states one should consider the triplet part of . The latter is the key quantity in our analysis. d Here we assume a spin-restricted set of orbitals that have the same spatial parts for and spins. e In the sense of the inner product of two operators defined ash ^ A; ^ Bi = P pq ^ A pq ^ B pq =Tr[A y B] 65 One-electron and mean-field Breit–Pauli spin–orbit operators have the following form 38f : ^ H SO = 1 2 X pq h SO L + ;pq a y p a q + X pq h SO z;pq a y p a q a y p a q + X pq h SO L ;pq a y p a q ! = 1 2 X pq h SO L + ;pq ^ T 1;1 pq + p 2 X pq h SO z;pq ^ T 1;0 X pq h SO L ;pq ^ T 1;1 pq ! ; (3.16) whereL + andL parts are spherical components: h 1eSO L + =h 1eSO x +ih 1eSO y ; (3.17) h 1eSO L =h 1eSO x ih 1eSO y : (3.18) The spherical components for the spin–orbit mean-field (SOMF) operator are defined in a sim- ilar way 38 . In practical calculations 37, 38 of SOCs, it is convenient to apply Wigner–Eckart’s theorem for matrix elements with spin-tensor operators ^ O S;M : hS 0 M 0 j ^ O S;M jS 00 M 00 i =hS 00 M 00 ;SMjS 0 M 0 ihS 0 jj ^ O S; jjS 00 i; (3.19) wherehS 00 M 00 ;SMjS 0 M 0 i is a Clebsh–Gordan coefficient,hS 0 jj ^ O S; jjS 00 i is reduced matrix ele- ment, which does not depend on spin projections. Applying Wigner–Eckart’s theorem to the triplet density matrices, we obtain a spinless triplet transition density matrixu 1; : u 1; pq =hJSjj ^ T 1; pq jjIS 0 i =hJSMj ^ T 1;m jIS 0 M 0 i=hS 0 M 0 ; 1mjSMi (3.20) f We assume that the artificial singlet part of SOMF has been already removed, as in Ref. 38 66 Doing the same for the spin–orbit operator, we obtain the following relations: hJSjj ^ H L jjIS 0 i = 1 2 X p;q h SO L ;pq u 1; pq ; (3.21) hJSjj ^ H L 0 jjIS 0 i = p 2 2 X pq h SO Lz ;pq u 1; pq ; (3.22) hJSjj ^ H L + jjIS 0 i = + 1 2 X pq h SO L + ;pq u 1; pq : (3.23) As one can see, all relevant matrix elements are computed fromu 1; . Thus, to analyze the entire spin–orbit matrix between the two multiplets, we perform SVD of u 1; and obtain an analogue of Eq. (3.2). The reduced spin–orbit matrix elements can be then written as hJSjj ^ H L jjIS 0 i = 1 2 X k h p k jh SO L j h k i! k ; (3.24) hJSjj ^ H L 0 jjIS 0 i = p 2 2 X k h p k jh SO Lz j h k i! k ; (3.25) hJSjj ^ H L + jjIS 0 i = + 1 2 X k h p k jh SO L + j h k i! k ; (3.26) where h;p k (r) and ! k are the new spinless NTOs and the respective weights (singular values ofu 1; ). Because all reduced spin–orbit matrix elements are expressed through the spin–orbit integrals over the spinless NTO pairs and the respective weights! k are the same for all three reduced spin–orbit matrix elements, one can analyze different components of the spin–orbit matrix in a uniform way. We note that, in contrast to the parent , the norm ofu 1; is not bounded by one because of the division by the Clebsh-Gordan coefficient, as per Eq. (3.20). Consequently, the magnitude of the respective singular values (! k ) and their sum can exceed one. However, the extent of the 67 collectivity of the transition (i.e., the number of significant NTO pairs) can be characterized by the participation ratio computed using the same expression as for regular NTOs 20 : PR NTO = ( P k ! 2 k ) 2 P k ! 4 k (3.27) For example, for a transition dominated by a single NTO pair, PR NTO =1, for a transition described by two NTO pairs with equal weights,PR NTO =2, and so on. Below we illustrate the utility of the spinless NTOs by considering several examples of open-shell species described using EOM-CC wave functions. EOM-CC target electronic states are parameterized using linear excitation (ionization, electron attachment) operators acting on the exponential coupled-cluster ansatz describing the reference state: j EOM i = ^ Re ^ T j 0 i; (3.28) h EOM j =h 0 j ^ Le ^ T ; (3.29) where ^ T is a coupled-cluster excitation operator, ^ R and ^ L are the EOM right excitation and left de-excitation operators. In the singles and doubles variant 42 (EOM-CCSD), ^ T is truncated after single and double excitations. In the electron excitation (EOM-EE-CCSD) and spin-flip 43, 44 (EOM-SF-CCSD) variants, ^ R comprises 1h1p (one-hole-one-particle) and 2h2p (two-holes- two-particles) operators. In the ionization 45, 46 (EOM-IP-CCSD) and electron-attached 47 (EOM- EA-CCSD) variants, ^ R comprises 1h, 2h1p and 1p, 1h2p operators, respectively. To illustrate the analysis, we consider tris(pyrrolylmethyl)amine Fe(II) complex (which we hereafter denote as (tpa)Fe ), exhibiting a large spin-reversal barrier 48 , and methylethylgalium (EtMeGa) radical; their structures are shown in Figures 3.1 and 3.2, respectively. To describe quintet states of (tpa)Fe , we used EOM-EA-MP2/cc-pVDZ fromd 5 hextet reference (Fig. 3.1, right panel). In this calculation, we used the same geometry and followed the same protocol as 68 8 statesofmethylethylgaliumweredescribedbyEOM-EA-CCSDfromtheclosed-shellcationic reference; these calculations were carried out at the doublet state geometry optimized with !B97X-D (given in SI). (1) (2) (3) FIG. 1: Top, left: schematic picture of the hextet reference of (tpa)Fe. Electron attachment is shown by dashed lines. The attachments to the orbitals (1) and (2) generate the leading configura- tions of the doubly degenerate quintet ground state of the anion, (tpa)Fe . These electronic states are denoted as State 1 and State 2 later. The attachment to the orbital (3) gives the leading config- uration of the non-degenerate State 3. Top, right: schematic picture of electron attachment to the EtMeGa + closed-shell reference, generating the leading configurations of the two lowest doublets of the neutral EtMeGa molecule. Bottom: considered geometries of the considered systems. We consider the three lowest quintet states of (tpa)Fe . The generation of these states is depictedschematicallyinFigure1. Thegroundstatequintetisdoublydegeneratebecauseof point-group symmetry (C3); these states are denoted as State 1 and 2. The non-degenerate quintetisdenotedasState3. Wehaveshown[37]recentlythattakingintoaccountonlythese three states leads to the spin-reversal barrier, close to the experimental value [43]. Figure 2 shows the leading spinless NTO pair of the reduced triplet transition density matrix u 1,· for (tpa)Fe . We show the transitions between the manifolds of the degenerate and non- degeneratestates. ThesingularvaluesoftheremainingNTOsarebytwoordersofmagnitude smaller than of those shown in Figure 2. TableIillustrateshowwelltheleadingNTOpairrepresentsthereducedspin–orbitmatrix elements (full EOM-EA-MP2 values). As one can see, the truncation of the sums from Eqs. (23), (24), and (25), by retaining only the first leading term recovers 99% of the respective full value CHECK. Sometimes is is 97%. For small values it is worse. In other terms, only one spinless NTO pair is needed to describe the physics of these systems and to 8 statesofmethylethylgaliumweredescribedbyEOM-EA-CCSDfromtheclosed-shellcationic reference; these calculations were carried out at the doublet state geometry optimized with !B97X-D (given in SI). (1) (2) (3) FIG. 1: Top, left: schematic picture of the hextet reference of (tpa)Fe. Electron attachment is shown by dashed lines. The attachments to the orbitals (1) and (2) generate the leading configura- tions of the doubly degenerate quintet ground state of the anion, (tpa)Fe . These electronic states are denoted as State 1 and State 2 later. The attachment to the orbital (3) gives the leading config- uration of the non-degenerate State 3. Top, right: schematic picture of electron attachment to the EtMeGa + closed-shell reference, generating the leading configurations of the two lowest doublets of the neutral EtMeGa molecule. Bottom: considered geometries of the considered systems. We consider the three lowest quintet states of (tpa)Fe . The generation of these states is depictedschematicallyinFigure1. Thegroundstatequintetisdoublydegeneratebecauseof point-group symmetry (C3); these states are denoted as State 1 and 2. The non-degenerate quintetisdenotedasState3. Wehaveshown[37]recentlythattakingintoaccountonlythese three states leads to the spin-reversal barrier, close to the experimental value [43]. Figure 2 shows the leading spinless NTO pair of the reduced triplet transition density matrix u 1,· for (tpa)Fe . We show the transitions between the manifolds of the degenerate and non- degeneratestates. ThesingularvaluesoftheremainingNTOsarebytwoordersofmagnitude smaller than of those shown in Figure 2. TableIillustrateshowwelltheleadingNTOpairrepresentsthereducedspin–orbitmatrix elements (full EOM-EA-MP2 values). As one can see, the truncation of the sums from Eqs. (23), (24), and (25), by retaining only the first leading term recovers 99% of the respective full value CHECK. Sometimes is is 97%. For small values it is worse. In other terms, only one spinless NTO pair is needed to describe the physics of these systems and to 8 statesofmethylethylgaliumweredescribedbyEOM-EA-CCSDfromtheclosed-shellcationic reference; these calculations were carried out at the doublet state geometry optimized with !B97X-D (given in SI). (1) (2) (3) FIG. 1: Top, left: schematic picture of the hextet reference of (tpa)Fe. Electron attachment is shown by dashed lines. The attachments to the orbitals (1) and (2) generate the leading configura- tions of the doubly degenerate quintet ground state of the anion, (tpa)Fe . These electronic states are denoted as State 1 and State 2 later. The attachment to the orbital (3) gives the leading config- uration of the non-degenerate State 3. Top, right: schematic picture of electron attachment to the EtMeGa + closed-shell reference, generating the leading configurations of the two lowest doublets of the neutral EtMeGa molecule. Bottom: considered geometries of the considered systems. We consider the three lowest quintet states of (tpa)Fe . The generation of these states is depictedschematicallyinFigure1. Thegroundstatequintetisdoublydegeneratebecauseof point-group symmetry (C3); these states are denoted as State 1 and 2. The non-degenerate quintetisdenotedasState3. Wehaveshown[37]recentlythattakingintoaccountonlythese three states leads to the spin-reversal barrier, close to the experimental value [43]. Figure 2 shows the leading spinless NTO pair of the reduced triplet transition density matrix u 1,· for (tpa)Fe . We show the transitions between the manifolds of the degenerate and non- degeneratestates. ThesingularvaluesoftheremainingNTOsarebytwoordersofmagnitude smaller than of those shown in Figure 2. TableIillustrateshowwelltheleadingNTOpairrepresentsthereducedspin–orbitmatrix elements (full EOM-EA-MP2 values). As one can see, the truncation of the sums from Eqs. (23), (24), and (25), by retaining only the first leading term recovers 99% of the respective full value CHECK. Sometimes is is 97%. For small values it is worse. In other terms, only one spinless NTO pair is needed to describe the physics of these systems and to 8 statesofmethylethylgaliumweredescribedbyEOM-EA-CCSDfromtheclosed-shellcationic reference; these calculations were carried out at the doublet state geometry optimized with !B97X-D (given in SI). (1) (2) (3) FIG. 1: Top, left: schematic picture of the hextet reference of (tpa)Fe. Electron attachment is shown by dashed lines. The attachments to the orbitals (1) and (2) generate the leading configura- tions of the doubly degenerate quintet ground state of the anion, (tpa)Fe . These electronic states are denoted as State 1 and State 2 later. The attachment to the orbital (3) gives the leading config- uration of the non-degenerate State 3. Top, right: schematic picture of electron attachment to the EtMeGa + closed-shell reference, generating the leading configurations of the two lowest doublets of the neutral EtMeGa molecule. Bottom: considered geometries of the considered systems. We consider the three lowest quintet states of (tpa)Fe . The generation of these states is depictedschematicallyinFigure1. Thegroundstatequintetisdoublydegeneratebecauseof point-group symmetry (C3); these states are denoted as State 1 and 2. The non-degenerate quintetisdenotedasState3. Wehaveshown[37]recentlythattakingintoaccountonlythese three states leads to the spin-reversal barrier, close to the experimental value [43]. Figure 2 shows the leading spinless NTO pair of the reduced triplet transition density matrix u 1,· for (tpa)Fe . We show the transitions between the manifolds of the degenerate and non- degeneratestates. ThesingularvaluesoftheremainingNTOsarebytwoordersofmagnitude smaller than of those shown in Figure 2. TableIillustrateshowwelltheleadingNTOpairrepresentsthereducedspin–orbitmatrix elements (full EOM-EA-MP2 values). As one can see, the truncation of the sums from Eqs. (23), (24), and (25), by retaining only the first leading term recovers 99% of the respective full value CHECK. Sometimes is is 97%. For small values it is worse. In other terms, only one spinless NTO pair is needed to describe the physics of these systems and to State 1 State 2 Reference Reference 8 statesofmethylethylgaliumweredescribedbyEOM-EA-CCSDfromtheclosed-shellcationic reference; these calculations were carried out at the doublet state geometry optimized with !B97X-D (given in SI). (1) (2) (3) FIG. 1: Top, left: schematic picture of the hextet reference of (tpa)Fe. Electron attachment is shown by dashed lines. The attachments to the orbitals (1) and (2) generate the leading configura- tions of the doubly degenerate quintet ground state of the anion, (tpa)Fe . These electronic states are denoted as State 1 and State 2 later. The attachment to the orbital (3) gives the leading config- uration of the non-degenerate State 3. Top, right: schematic picture of electron attachment to the EtMeGa + closed-shell reference, generating the leading configurations of the two lowest doublets of the neutral EtMeGa molecule. Bottom: considered geometries of the considered systems. We consider the three lowest quintet states of (tpa)Fe . The generation of these states is depictedschematicallyinFigure1. Thegroundstatequintetisdoublydegeneratebecauseof point-group symmetry (C3); these states are denoted as State 1 and 2. The non-degenerate quintetisdenotedasState3. Wehaveshown[37]recentlythattakingintoaccountonlythese three states leads to the spin-reversal barrier, close to the experimental value [43]. Figure 2 shows the leading spinless NTO pair of the reduced triplet transition density matrix u 1,· for (tpa)Fe . We show the transitions between the manifolds of the degenerate and non- degeneratestates. ThesingularvaluesoftheremainingNTOsarebytwoordersofmagnitude smaller than of those shown in Figure 2. TableIillustrateshowwelltheleadingNTOpairrepresentsthereducedspin–orbitmatrix elements (full EOM-EA-MP2 values). As one can see, the truncation of the sums from Eqs. (23), (24), and (25), by retaining only the first leading term recovers 99% of the respective full value CHECK. Sometimes is is 97%. For small values it is worse. In other terms, only one spinless NTO pair is needed to describe the physics of these systems and to 8 statesofmethylethylgaliumweredescribedbyEOM-EA-CCSDfromtheclosed-shellcationic reference; these calculations were carried out at the doublet state geometry optimized with !B97X-D (given in SI). (1) (2) (3) FIG. 1: Top, left: schematic picture of the hextet reference of (tpa)Fe. Electron attachment is shown by dashed lines. The attachments to the orbitals (1) and (2) generate the leading configura- tions of the doubly degenerate quintet ground state of the anion, (tpa)Fe . These electronic states are denoted as State 1 and State 2 later. The attachment to the orbital (3) gives the leading config- uration of the non-degenerate State 3. Top, right: schematic picture of electron attachment to the EtMeGa + closed-shell reference, generating the leading configurations of the two lowest doublets of the neutral EtMeGa molecule. Bottom: considered geometries of the considered systems. We consider the three lowest quintet states of (tpa)Fe . The generation of these states is depictedschematicallyinFigure1. Thegroundstatequintetisdoublydegeneratebecauseof point-group symmetry (C3); these states are denoted as State 1 and 2. The non-degenerate quintetisdenotedasState3. Wehaveshown[37]recentlythattakingintoaccountonlythese three states leads to the spin-reversal barrier, close to the experimental value [43]. Figure 2 shows the leading spinless NTO pair of the reduced triplet transition density matrix u 1,· for (tpa)Fe . We show the transitions between the manifolds of the degenerate and non- degeneratestates. ThesingularvaluesoftheremainingNTOsarebytwoordersofmagnitude smaller than of those shown in Figure 2. TableIillustrateshowwelltheleadingNTOpairrepresentsthereducedspin–orbitmatrix elements (full EOM-EA-MP2 values). As one can see, the truncation of the sums from Eqs. (23), (24), and (25), by retaining only the first leading term recovers 99% of the respective full value CHECK. Sometimes is is 97%. For small values it is worse. In other terms, only one spinless NTO pair is needed to describe the physics of these systems and to 8 statesofmethylethylgaliumweredescribedbyEOM-EA-CCSDfromtheclosed-shellcationic reference; these calculations were carried out at the doublet state geometry optimized with !B97X-D (given in SI). (1) (2) (3) FIG. 1: Top, left: schematic picture of the hextet reference of (tpa)Fe. Electron attachment is shown by dashed lines. The attachments to the orbitals (1) and (2) generate the leading configura- tions of the doubly degenerate quintet ground state of the anion, (tpa)Fe . These electronic states are denoted as State 1 and State 2 later. The attachment to the orbital (3) gives the leading config- uration of the non-degenerate State 3. Top, right: schematic picture of electron attachment to the EtMeGa + closed-shell reference, generating the leading configurations of the two lowest doublets of the neutral EtMeGa molecule. Bottom: considered geometries of the considered systems. We consider the three lowest quintet states of (tpa)Fe . The generation of these states is depictedschematicallyinFigure1. Thegroundstatequintetisdoublydegeneratebecauseof point-group symmetry (C3); these states are denoted as State 1 and 2. The non-degenerate quintetisdenotedasState3. Wehaveshown[37]recentlythattakingintoaccountonlythese three states leads to the spin-reversal barrier, close to the experimental value [43]. Figure 2 shows the leading spinless NTO pair of the reduced triplet transition density matrix u 1,· for (tpa)Fe . We show the transitions between the manifolds of the degenerate and non- degeneratestates. ThesingularvaluesoftheremainingNTOsarebytwoordersofmagnitude smaller than of those shown in Figure 2. TableIillustrateshowwelltheleadingNTOpairrepresentsthereducedspin–orbitmatrix elements (full EOM-EA-MP2 values). As one can see, the truncation of the sums from Eqs. (23), (24), and (25), by retaining only the first leading term recovers 99% of the respective full value CHECK. Sometimes is is 97%. For small values it is worse. In other terms, only one spinless NTO pair is needed to describe the physics of these systems and to State 1 State 2 State 3 Figure 3.1: Left: Structure of (tpa)Fe (C 15 N 4 H 15 Fe). Iron is shown in red, nitrogens in blue, carbons in gray, and hydrogens in white. Right: Frontier MOs and electronic config- uration of thed 5 hextet reference and relevant target states. The target states are obtained by the attachment of a -electron to one of the three MOs marked by the dashed box: attachment to the two lowest MOs gives rise to degenerate states 1 and 2 and attachment to the next MO gives rise to state 3. 8 statesofmethylethylgaliumweredescribedbyEOM-EA-CCSDfromtheclosed-shellcationic reference; these calculations were carried out at the doublet state geometry optimized with !B97X-D (given in SI). (1) (2) (3) FIG. 1: Top, left: schematic picture of the hextet reference of (tpa)Fe. Electron attachment is shown by dashed lines. The attachments to the orbitals (1) and (2) generate the leading configura- tions of the doubly degenerate quintet ground state of the anion, (tpa)Fe . These electronic states are denoted as State 1 and State 2 later. The attachment to the orbital (3) gives the leading config- uration of the non-degenerate State 3. Top, right: schematic picture of electron attachment to the EtMeGa + closed-shell reference, generating the leading configurations of the two lowest doublets of the neutral EtMeGa molecule. Bottom: considered geometries of the considered systems. We consider the three lowest quintet states of (tpa)Fe . The generation of these states is depictedschematicallyinFigure1. Thegroundstatequintetisdoublydegeneratebecauseof point-group symmetry (C 3 ); these states are denoted as State 1 and 2. The non-degenerate quintetisdenotedasState3. Wehaveshown[37]recentlythattakingintoaccountonlythese three states leads to the spin-reversal barrier, close to the experimental value [43]. Figure 2 shows the leading spinless NTO pair of the reduced triplet transition density matrix u 1,· for (tpa)Fe . We show the transitions between the manifolds of the degenerate and non- degeneratestates. ThesingularvaluesoftheremainingNTOsarebytwoordersofmagnitude smaller than of those shown in Figure 2. TableIillustrateshowwelltheleadingNTOpairrepresentsthereducedspin–orbitmatrix elements (full EOM-EA-MP2 values). As one can see, the truncation of the sums from Eqs. (23), (24), and (25), by retaining only the first leading term recovers 99% of the respective full value CHECK. Sometimes is is 97%. For small values it is worse. In other terms, only one spinless NTO pair is needed to describe the physics of these systems and to 8 statesofmethylethylgaliumweredescribedbyEOM-EA-CCSDfromtheclosed-shellcationic reference; these calculations were carried out at the doublet state geometry optimized with !B97X-D (given in SI). (1) (2) (3) FIG. 1: Top, left: schematic picture of the hextet reference of (tpa)Fe. Electron attachment is shown by dashed lines. The attachments to the orbitals (1) and (2) generate the leading configura- tions of the doubly degenerate quintet ground state of the anion, (tpa)Fe . These electronic states are denoted as State 1 and State 2 later. The attachment to the orbital (3) gives the leading config- uration of the non-degenerate State 3. Top, right: schematic picture of electron attachment to the EtMeGa + closed-shell reference, generating the leading configurations of the two lowest doublets of the neutral EtMeGa molecule. Bottom: considered geometries of the considered systems. We consider the three lowest quintet states of (tpa)Fe . The generation of these states is depictedschematicallyinFigure1. Thegroundstatequintetisdoublydegeneratebecauseof point-group symmetry (C 3 ); these states are denoted as State 1 and 2. The non-degenerate quintetisdenotedasState3. Wehaveshown[37]recentlythattakingintoaccountonlythese three states leads to the spin-reversal barrier, close to the experimental value [43]. Figure 2 shows the leading spinless NTO pair of the reduced triplet transition density matrix u 1,· for (tpa)Fe . We show the transitions between the manifolds of the degenerate and non- degeneratestates. ThesingularvaluesoftheremainingNTOsarebytwoordersofmagnitude smaller than of those shown in Figure 2. TableIillustrateshowwelltheleadingNTOpairrepresentsthereducedspin–orbitmatrix elements (full EOM-EA-MP2 values). As one can see, the truncation of the sums from Eqs. (23), (24), and (25), by retaining only the first leading term recovers 99% of the respective full value CHECK. Sometimes is is 97%. For small values it is worse. In other terms, only one spinless NTO pair is needed to describe the physics of these systems and to 8 statesofmethylethylgaliumweredescribedbyEOM-EA-CCSDfromtheclosed-shellcationic reference; these calculations were carried out at the doublet state geometry optimized with !B97X-D (given in SI). (1) (2) (3) FIG. 1: Top, left: schematic picture of the hextet reference of (tpa)Fe. Electron attachment is shown by dashed lines. The attachments to the orbitals (1) and (2) generate the leading configura- tions of the doubly degenerate quintet ground state of the anion, (tpa)Fe . These electronic states are denoted as State 1 and State 2 later. The attachment to the orbital (3) gives the leading config- uration of the non-degenerate State 3. Top, right: schematic picture of electron attachment to the EtMeGa + closed-shell reference, generating the leading configurations of the two lowest doublets of the neutral EtMeGa molecule. Bottom: considered geometries of the considered systems. We consider the three lowest quintet states of (tpa)Fe . The generation of these states is depictedschematicallyinFigure1. Thegroundstatequintetisdoublydegeneratebecauseof point-group symmetry (C 3 ); these states are denoted as State 1 and 2. The non-degenerate quintetisdenotedasState3. Wehaveshown[37]recentlythattakingintoaccountonlythese three states leads to the spin-reversal barrier, close to the experimental value [43]. Figure 2 shows the leading spinless NTO pair of the reduced triplet transition density matrix u 1,· for (tpa)Fe . We show the transitions between the manifolds of the degenerate and non- degeneratestates. ThesingularvaluesoftheremainingNTOsarebytwoordersofmagnitude smaller than of those shown in Figure 2. TableIillustrateshowwelltheleadingNTOpairrepresentsthereducedspin–orbitmatrix elements (full EOM-EA-MP2 values). As one can see, the truncation of the sums from Eqs. (23), (24), and (25), by retaining only the first leading term recovers 99% of the respective full value CHECK. Sometimes is is 97%. For small values it is worse. In other terms, only one spinless NTO pair is needed to describe the physics of these systems and to 8 statesofmethylethylgaliumweredescribedbyEOM-EA-CCSDfromtheclosed-shellcationic reference; these calculations were carried out at the doublet state geometry optimized with !B97X-D (given in SI). (1) (2) (3) FIG. 1: Top, left: schematic picture of the hextet reference of (tpa)Fe. Electron attachment is shown by dashed lines. The attachments to the orbitals (1) and (2) generate the leading configura- tions of the doubly degenerate quintet ground state of the anion, (tpa)Fe . These electronic states are denoted as State 1 and State 2 later. The attachment to the orbital (3) gives the leading config- uration of the non-degenerate State 3. Top, right: schematic picture of electron attachment to the EtMeGa + closed-shell reference, generating the leading configurations of the two lowest doublets of the neutral EtMeGa molecule. Bottom: considered geometries of the considered systems. We consider the three lowest quintet states of (tpa)Fe . The generation of these states is depictedschematicallyinFigure1. Thegroundstatequintetisdoublydegeneratebecauseof point-group symmetry (C 3 ); these states are denoted as State 1 and 2. The non-degenerate quintetisdenotedasState3. Wehaveshown[37]recentlythattakingintoaccountonlythese three states leads to the spin-reversal barrier, close to the experimental value [43]. Figure 2 shows the leading spinless NTO pair of the reduced triplet transition density matrix u 1,· for (tpa)Fe . We show the transitions between the manifolds of the degenerate and non- degeneratestates. ThesingularvaluesoftheremainingNTOsarebytwoordersofmagnitude smaller than of those shown in Figure 2. TableIillustrateshowwelltheleadingNTOpairrepresentsthereducedspin–orbitmatrix elements (full EOM-EA-MP2 values). As one can see, the truncation of the sums from Eqs. (23), (24), and (25), by retaining only the first leading term recovers 99% of the respective full value CHECK. Sometimes is is 97%. For small values it is worse. In other terms, only one spinless NTO pair is needed to describe the physics of these systems and to State 1 State 2 Reference Reference 8 statesofmethylethylgaliumweredescribedbyEOM-EA-CCSDfromtheclosed-shellcationic reference; these calculations were carried out at the doublet state geometry optimized with !B97X-D (given in SI). (1) (2) (3) FIG. 1: Top, left: schematic picture of the hextet reference of (tpa)Fe. Electron attachment is shown by dashed lines. The attachments to the orbitals (1) and (2) generate the leading configura- tions of the doubly degenerate quintet ground state of the anion, (tpa)Fe . These electronic states are denoted as State 1 and State 2 later. The attachment to the orbital (3) gives the leading config- uration of the non-degenerate State 3. Top, right: schematic picture of electron attachment to the EtMeGa + closed-shell reference, generating the leading configurations of the two lowest doublets of the neutral EtMeGa molecule. Bottom: considered geometries of the considered systems. We consider the three lowest quintet states of (tpa)Fe . The generation of these states is depictedschematicallyinFigure1. Thegroundstatequintetisdoublydegeneratebecauseof point-group symmetry (C 3 ); these states are denoted as State 1 and 2. The non-degenerate quintetisdenotedasState3. Wehaveshown[37]recentlythattakingintoaccountonlythese three states leads to the spin-reversal barrier, close to the experimental value [43]. Figure 2 shows the leading spinless NTO pair of the reduced triplet transition density matrix u 1,· for (tpa)Fe . We show the transitions between the manifolds of the degenerate and non- degeneratestates. ThesingularvaluesoftheremainingNTOsarebytwoordersofmagnitude smaller than of those shown in Figure 2. TableIillustrateshowwelltheleadingNTOpairrepresentsthereducedspin–orbitmatrix elements (full EOM-EA-MP2 values). As one can see, the truncation of the sums from Eqs. (23), (24), and (25), by retaining only the first leading term recovers 99% of the respective full value CHECK. Sometimes is is 97%. For small values it is worse. In other terms, only one spinless NTO pair is needed to describe the physics of these systems and to 8 statesofmethylethylgaliumweredescribedbyEOM-EA-CCSDfromtheclosed-shellcationic reference; these calculations were carried out at the doublet state geometry optimized with !B97X-D (given in SI). (1) (2) (3) FIG. 1: Top, left: schematic picture of the hextet reference of (tpa)Fe. Electron attachment is shown by dashed lines. The attachments to the orbitals (1) and (2) generate the leading configura- tions of the doubly degenerate quintet ground state of the anion, (tpa)Fe . These electronic states are denoted as State 1 and State 2 later. The attachment to the orbital (3) gives the leading config- uration of the non-degenerate State 3. Top, right: schematic picture of electron attachment to the EtMeGa + closed-shell reference, generating the leading configurations of the two lowest doublets of the neutral EtMeGa molecule. Bottom: considered geometries of the considered systems. We consider the three lowest quintet states of (tpa)Fe . The generation of these states is depictedschematicallyinFigure1. Thegroundstatequintetisdoublydegeneratebecauseof point-group symmetry (C 3 ); these states are denoted as State 1 and 2. The non-degenerate quintetisdenotedasState3. Wehaveshown[37]recentlythattakingintoaccountonlythese three states leads to the spin-reversal barrier, close to the experimental value [43]. Figure 2 shows the leading spinless NTO pair of the reduced triplet transition density matrix u 1,· for (tpa)Fe . We show the transitions between the manifolds of the degenerate and non- degeneratestates. ThesingularvaluesoftheremainingNTOsarebytwoordersofmagnitude smaller than of those shown in Figure 2. TableIillustrateshowwelltheleadingNTOpairrepresentsthereducedspin–orbitmatrix elements (full EOM-EA-MP2 values). As one can see, the truncation of the sums from Eqs. (23), (24), and (25), by retaining only the first leading term recovers 99% of the respective full value CHECK. Sometimes is is 97%. For small values it is worse. In other terms, only one spinless NTO pair is needed to describe the physics of these systems and to 8 statesofmethylethylgaliumweredescribedbyEOM-EA-CCSDfromtheclosed-shellcationic reference; these calculations were carried out at the doublet state geometry optimized with !B97X-D (given in SI). (1) (2) (3) FIG. 1: Top, left: schematic picture of the hextet reference of (tpa)Fe. Electron attachment is shown by dashed lines. The attachments to the orbitals (1) and (2) generate the leading configura- tions of the doubly degenerate quintet ground state of the anion, (tpa)Fe . These electronic states are denoted as State 1 and State 2 later. The attachment to the orbital (3) gives the leading config- uration of the non-degenerate State 3. Top, right: schematic picture of electron attachment to the EtMeGa + closed-shell reference, generating the leading configurations of the two lowest doublets of the neutral EtMeGa molecule. Bottom: considered geometries of the considered systems. We consider the three lowest quintet states of (tpa)Fe . The generation of these states is depictedschematicallyinFigure1. Thegroundstatequintetisdoublydegeneratebecauseof point-group symmetry (C 3 ); these states are denoted as State 1 and 2. The non-degenerate quintetisdenotedasState3. Wehaveshown[37]recentlythattakingintoaccountonlythese three states leads to the spin-reversal barrier, close to the experimental value [43]. Figure 2 shows the leading spinless NTO pair of the reduced triplet transition density matrix u 1,· for (tpa)Fe . We show the transitions between the manifolds of the degenerate and non- degeneratestates. ThesingularvaluesoftheremainingNTOsarebytwoordersofmagnitude smaller than of those shown in Figure 2. TableIillustrateshowwelltheleadingNTOpairrepresentsthereducedspin–orbitmatrix elements (full EOM-EA-MP2 values). As one can see, the truncation of the sums from Eqs. (23), (24), and (25), by retaining only the first leading term recovers 99% of the respective full value CHECK. Sometimes is is 97%. For small values it is worse. In other terms, only one spinless NTO pair is needed to describe the physics of these systems and to State 1 State 2 State 3 Figure 3.2: Left: Structure of the methylethylgalium (EtMeGa) radical (GaC 3 H 8 ). Gal- lium is shown in pink, carbons in gray, and hydrogens in white. Right: Electronic config- uration of the closed-shell cationic reference and relevant target states. The target states are obtained by electron attachment to the two lowest unoccupied orbitals of the cation marked by dashed box. in Ref. 38. The relevant states of methylethylgalium were described by EOM-EA-CCSD from the closed-shell cationic reference (Fig. 3.2, right panel); these calculations were carried out at the doublet state geometry optimized with!B97X-D (given in the Appendix D). We consider the three lowest quintet states of (tpa)Fe . The character of these states is depicted schematically in Figure 3.1. The ground state state is a doubly degenerate quintet because of point-group symmetry (C 3 ); we refer to these states as State 1 and 2. The next quintet state, State 3, is non-degenerate. As shown in a recent paper 38 , taking into account only these three lowest states is sufficient for accurate description of the spin-reversal barrier 48 . Figure 3.3 shows the leading spinless NTO pair of the reduced triplet transition density matrix u 1; for (tpa)Fe . The singular values of the remaining NTOs are by two orders of magnitude smaller than of those shown in Figure 3.3 and the participation ratio for these transitions are 69 1.002 and 1.001. As one can see, the transitions between all three states can be described as transition between nearly perfect d-orbitals, d yz ! d xz and d yz ! d z 2. The respective value of SOC (given in Table 3.2) are large, which can be explained by generalized El-Sayed’s rules, as discussed below. The NTOs corresponding to the transitions between the two doublet states in EtMeGa a pair of flippedp-like orbitals giving rise to a substantial value of SOC. The participation ratio for this transition is 1.008. As per Eq. (3.20), the normalization ofu 1; is different from the normalization of regular one-particle transition densities ( ). Consequently, in contrast to the singular values of , the singular values of u 1; (! k ) may exceed 1, as it happens in EtMeGa. u 1; connects the spin- flip ( SF ) and spin-preserving triplet ( EE triplet ) transition density matrices via Clebsh–Gordan coefficients. The numeric consequences of this relationship are analyzed in the Appendix C by using the electronic states of EtMeGa as an example. Table 3.2 also illustrates how well the leading NTO pair represents the reduced spin–orbit matrix elements (full EOM-EA values). As one can see, the truncation of the sums from Eqs. (3.24), (3.25), and (3.26) by retaining only the first leading term recovers the full value with good accuracy. In other terms, only one spinless NTO pair is needed to describe the physics of these systems and to recover the full SOC matrix with high accuracy in both one-electron and mean-field cases, which is consistent with the respective participation ratios. We conclude our analysis by discussing the computed NTOs and SOCs in terms of El- Sayed’s rules 5, 24 . Originally introduced on the basis of the selection rules for p-orbitals, El- Sayed’s rule 5 explains why the change in orbital character (e.g.,n!) is needed for attaining large SOCs. In the context of transition-metal photochemistry, following El-Sayed’s reasoning explains why mixing of metal-to-ligand charge-transfer configurations facilitates spin-forbidden transitions on the metal and that large values of SOC can be obtained when the respective sin- glet and triplet states involve different components of thed-orbital manifold 30 . Here we follow 70 State 1, hole (d yz ) State 1, hole (d yz ) State 2, particle (d xz ) State 3, particle (d z 2 ) ω=0.88 ω=0.87 Figure 3.3: Spinless triplet NTOs for the transitions between three lowest quintet states in (tpa)Fe . The states 1 and 2 are degenerate. Red, green, and blue axes indicateX,Y , and Z coordinates axes, respectively. The isovalue of 0.050 was used in all the cases. ω=1.19 State 1, hole (p xy ) State 2, hole (p z ) Figure 3.4: Spinless triplet NTOs for the transitions between the two lowest doublet states of EtMeGa system. Red, green, and blue axes indicate X, Y , and Z coordinates axes, respectively. The isovalue of 0.050 was used in all the cases. this strategy and generalize El-Sayed’s rules to the case of arbitrary orbitals involved in a local- ized transition. The generalization is based on the representations of the angular momentum operators in real spherical harmonics. The derivation and the key expressions are given in the Appendix A. This generalization provides selection rules, which can be used to rationalize the magnitude of SOC for a particular transition and to also explain the effect of molecular vibra- tions on the SOCs. It also explains relative values of different components of the SOC. Figure 3.4 shows the leading spinless NTO pair of the type of p xy ! p z , where p xy is a linear combination of p x and p y orbitals in the shown coordinate system. According to the 71 Table 3.1: Spin–orbit mean-field reduced matrix elements of the considered systems in the selected orientations. (tpa)Fe , 1! 2 h p jh SO j h i! full EOM-EA-MP2 hSjjH SO L jjSi 0:30 + 0:09i 0:29 + 0:10i hSjjH SO L 0 jjSi 209:04 211:34i hSjjH SO L + jjSi 0:30 0:09i 0:29 0:10i (tpa)Fe , 1! 3 h p jh SO j h i! full EOM-EA-MP2 hSjjH SO L jjSi 31:79 + 224:48i 32:07 + 228:23i hSjjH SO L 0 jjSi 0:04i 0:07i hSjjH SO L + jjSi 31:79 224:44i 32:07 228:23i EtMeGa, 1! 2 h p jh SO j h i! full EOM-EA-CCSD hSjjH SO L jjSi 135:87 + 210:65i 132:69 + 205:38i hSjjH SO L 0 jjSi 1:68i 1:80i hSjjH SO L + jjSi 135:87 210:65i 132:69 205:38i matrix representation of the angular momentum in the basis ofp-orbitals, thehp x j ^ L jp z i matrix elements are non-zero, which leads to largehSjj ^ H SO L jjSi andhSjj ^ H SO L + jjSi components of spin– orbit coupling, whilehSjj ^ H SO Lz jjSi is close to zero. The generalized El-Sayed’s rules predict not only the transitions with large SOC, but also estimate the relative values of SOC for different transitions in the same molecule. For example, the transitions in Figure 3.3 correspond to thed yz ! d xz andd yz ! d z 2 cases. The case ofd yz ! d xz is allowed through ^ L z operator with a relative magnitude ofX. The main contribution to the SOC from this transition in this orientation indeed comes fromhSjj ^ H SO L 0 jjSi (Table 3.2). Thed yz !d z 2 transition is allowed through ^ L + and ^ L with a relative magnitude ofX p 3. The computed spin–orbit matrix elements confirm this prediction as well: the main contribution to the couplings comes from the imaginary parts ofhSjj ^ H SO L jjSi andhSjj ^ H SO L + jjSi. There is also an increase in the magnitude relative to the first case. The increase is not exactly p 3, which is likely due to a small mismatch of the coordinate system (real parts ofhSjj ^ H SO L jjSi andhSjj ^ H SO L + jjSi) and deviation of the shape of the NTOs from ideal d-orbitals, which is especially clear ford z 2-like orbital. 72 To conclude, we introduced a new type of NTOs, which are suitable for describing spin- forbidden transitions. The new NTOs are obtained by the SVD procedure of the reduced spin- less transition density matrix, the quantity determining the SOC values for the entire multiplet by virtue of Wigner–Eckart’s theorem. These spinless NTOs describe the transitions between states with arbitrary spin projections in a uniform way. In addition to pictorial representation of the transition, the analysis also yields quantitative contributions of hole-particle pairs into the overall many-body matrix elements and helps to rationalize the magnitude of computed SOCs in terms of El-Sayed’s rules. By providing a clear orbital picture of the transitions at different geometries, this analysis can also be used to explain vibronic effects on the rate of spin-forbidden transitions. We hope that these tools will be helpful in deriving insight from high-level electronic structure calculations of phenomena facilitated by SOCs. 3.1 Appendix A: Generalized El-Sayed’s rules In this section, the angular momentum operators are represented in real harmonics, giving the expressions for the integrals between the orbitals of different types. These integrals are used in the derivation of the conventional El-Sayed’s rules as well as their extensions. Starting from spherical harmonicsY m l , which are the eigen-functions of ^ L 2 and ^ L z , we can define real harmonics as (see, for example, book 49 ) Y lm = 8 > > > > > > < > > > > > > : i p 2 Y m l (1) m Y m l ; ifm< 0 Y m l ; ifm = 0 1 p 2 Y m l + (1) m Y m l ; ifm> 0 (3.30) 73 Spherical harmonics are related to real harmonics as Y m l = 8 > > > > > > < > > > > > > : 1 p 2 (Y l;m iY l;m ); ifm< 0 Y lm ; ifm = 0 (1) m p 2 (Y l;m +iY l;m ); ifm> 0 (3.31) Now the matrix elements are hY lm 0j ^ L z jY lm i = 8 > > > > > > < > > > > > > : i p 2 mhY lm 0jY m l i (1) m (m)hY lm 0jY m l i ; ifm< 0 0; ifm = 0 1 p 2 mhY lm 0jY m l i + (1) m mhY lm 0jY m l i ; ifm> 0 (3.32) and hY lm 0j ^ L jY lm i = 8 > > > > > > < > > > > > > : i p 2 A l;m hY lm 0jY m1 l i (1) m (A l;m hY lm 0jY m1 l i ; ifm< 0 A l;0 hY lm 0jY 1 l i; ifm = 0 1 p 2 A l;m hY lm 0jY m1 l i + (1) m A l;m hY lm 0jY m1 l i ; ifm> 0 (3.33) where A l;m = p l(l + 1)m(m 1) (3.34) 74 In the case ofl = 1,Y 1;1 =p y ,Y 1;0 =p z ,Y 1;1 =p x . L z = 0 B B B B B @ jY 1;1 i jY 1;0 i jY 1;+1 i hY 1;1 j 0 0 i hY 1;0 j 0 0 0 hY 1;+1 j i 0 0 1 C C C C C A (3.35) L + = 0 B B B B B @ jY 1;1 i jY 1;0 i jY 1;+1 i hY 1;1 j 0 i 0 hY 1;0 j i 0 1 hY 1;+1 j 0 1 0 1 C C C C C A (3.36) L = 0 B B B B B @ jY 1;1 i jY 1;0 i jY 1;+1 i hY 1;1 j 0 i 0 hY 1;0 j i 0 1 hY 1;+1 j 0 1 0 1 C C C C C A (3.37) L x = 0 B B B B B @ jY 1;1 i jY 1;0 i jY 1;+1 i hY 1;1 j 0 i 0 hY 1;0 j i 0 0 hY 1;+1 j 0 0 0 1 C C C C C A (3.38) 75 L y = 0 B B B B B @ jY 1;1 i jY 1;0 i jY 1;+1 i hY 1;1 j 0 0 0 hY 1;0 j 0 0 i hY 1;+1 j 0 i 0 1 C C C C C A (3.39) In the case ofl = 2,Y 2;2 =d xy ,Y 2;1 =d yz ,Y 2;0 =d z 2,Y 2;1 =d xz ,Y 2;2 =d x 2 y 2 L z = 0 B B B B B B B B B B B B B @ jY 2;2 i jY 2;1 i jY 2;0 i jY 2;1 i jY 2;2 i hY 2;2 j 0 0 0 0 2i hY 2;1 j 0 0 0 i 0 hY 2;0 j 0 0 0 0 0 hY 2;1 j 0 i 0 0 0 hY 2;2 j 2i 0 0 0 0 1 C C C C C C C C C C C C C A (3.40) L + = 0 B B B B B B B B B B B B B @ jY 2;2 i jY 2;1 i jY 2;0 i jY 2;1 i jY 2;2 i hY 2;2 j 0 1 0 i 0 hY 2;1 j 1 0 p 3i 0 i hY 2;0 j 0 p 3i 0 p 3 0 hY 2;1 j i 0 p 3 0 1 hY 2;2 j 0 i 0 1 0 1 C C C C C C C C C C C C C A (3.41) 76 L = 0 B B B B B B B B B B B B B @ jY 2;2 i jY 2;1 i jY 2;0 i jY 2;1 i jY 2;2 i hY 2;2 j 0 1 0 i 0 hY 2;1 j 1 0 p 3i 0 i hY 2;0 j 0 i p 3 0 p 3 0 hY 2;1 j i 0 p 3 0 1 hY 2;2 j 0 i 0 1 0 1 C C C C C C C C C C C C C A (3.42) L x = 0 B B B B B B B B B B B B B @ jY 2;2 i jY 2;1 i jY 2;0 i jY 2;1 i jY 2;2 i hY 2;2 j 0 0 0 i 0 hY 2;1 j 0 0 p 3i 0 i hY 2;0 j 0 p 3i 0 0 0 hY 2;1 j i 0 0 0 0 hY 2;2 j 0 i 0 0 0 1 C C C C C C C C C C C C C A (3.43) L y = 0 B B B B B B B B B B B B B @ jY 2;2 i jY 2;1 i jY 2;0 i jY 2;1 i jY 2;2 i hY 2;2 j 0 i 0 0 0 hY 2;1 j i 0 0 0 0 hY 2;0 j 0 0 0 i p 3 0 hY 2;1 j 0 0 i p 3 0 i hY 2;2 j 0 0 0 i 0 1 C C C C C C C C C C C C C A (3.44) 77 3.2 Appendix B: Reduced spin–orbit matrix elements Table 3.2: Spin–orbit reduced matrix elements of the considered systems in the selected orientations. (tpa)Fe , 1! 2 h p jh SO j h i! 1e SOMF hSjjH SO L jjSi 0:54 + 0:17i 0:30 + 0:09i hSjjH SO L 0 jjSi 382:78i 209:04 hSjjH SO L + jjSi 0:54 0:17i 0:30 0:09i (tpa)Fe , 1! 3 h p jh SO j h i! 1e SOMF hSjjH SO L jjSi 56:82 + 409:06 31:79 + 224:48i hSjjH SO L 0 jjSi 0:06i 0:04i hSjjH SO L + jjSi 56:82 409:06 31:79 224:44i EtMeGa, 1! 2 h p jh SO j h i! 1e SOMF hSjjH SO L jjSi 149:57 + 231:86i 135:87 + 210:65i hSjjH SO L 0 jjSi 1:87i 1:68i hSjjH SO L + jjSi 149:57 231:86i 135:87 210:65i (tpa)Fe , 1! 2 full EOM-EA-MP2 1e SOMF hSjjH SO L jjSi 0:54 + 0:18i 0:29 + 0:10i hSjjH SO L 0 jjSi 385:57i 211:34i hSjjH SO L + jjSi 0:54 0:18i 0:29 0:10i (tpa)Fe , 1! 3 full EOM-EA-MP2 1e SOMF hSjjH SO L jjSi 57:20 + 414:88i 32:07 + 228:23i hSjjH SO L 0 jjSi 0:12i 0:07i hSjjH SO L + jjSi 57:20 414:88i 32:07 228:23i EtMeGa, 1! 2 full EOM-EA-CCSD 1e SOMF hSjjH SO L jjSi 146:08 + 226:07i 132:69 + 205:38i hSjjH SO L 0 jjSi 2:01i 1:80i hSjjH SO L + jjSi 146:08 226:07i 132:69 205:38i 78 3.3 Appendix C: Normalization of the spinless density matrix In this section, we explain the consequences of different normalization ofu 1; and the rela- tionship between the singular values ofu 1; and of . As explained in the main text,u 1; connects the spin-flip ( SF ) and spin-preserving triplet ( EE triplet ) transition density matrices via Clebsh– Gordan coefficients. Specifically, in the case of EtMeGa: SF =h1=2; 1=2; 1;1j1=2;1=2iu 1; = r 2 3 u 1; ; (3.45) jju 1; jj = s X k ! 2 k = 1:19; (3.46) jj SF jj = s X k ! 2 k 2 3 = 0:975; (3.47) jj EE triplet jj =jh1=2; 1=2; 1; 0j1=2; 1=2ijjju 1; jj = r 1 3 jju 1; jj: (3.48) Expressed in spin-orbitals, the singlet and triplet components of EE are orthogonal to each other. Their norms are jj EE triplet jj = 0:689 (3.49) and jj EE singlet jj = q jj EE jj 2 jj EE triplet jj 2 = 0:698: (3.50) In a model system with only one -electron (e.g., hydrogen atom), the spin-preserving one- particle density matrix has only one non-zero block,, leading to the equal EE singlet and EE triplet components. In EtMeGa, the norms of EE singlet and EE triplet are very close, which corresponds to an effective one-electron character of the transition. 79 3.4 Appendix D: Cartesian geometries $comment Methylethylgallum, doublet minimum Nuclear Repulsion Energy = 244.6096981601 hartrees Optimized with wB97XD $end $molecule 0 2 Ga 0.5438707666 -1.0736702386 0.4833465235 C 1.7543032024 0.4731216565 0.0780379946 H 1.4694411892 1.3192325020 0.7281328449 H 1.5974303623 0.7870753610 -0.9689496550 H 2.8197675072 0.2526158165 0.2309158586 C -1.4312317653 -0.8301334526 0.1263697681 H -1.9948816119 -1.2380131401 0.9823006497 H -1.6797193720 -1.4830281640 -0.7300666936 C -1.8232912915 0.6234059339 -0.1585738352 H -2.9033215249 0.7237104058 -0.3623612862 H -1.2886819645 1.0273543114 -1.0332959018 H -1.5929865490 1.2848498369 0.6924112591 $end 80 3.5 Chapter 3 References 1 C. J. Cramer, Essentials of Computational Chemistry: Theories and models. Wiley & Sons, New York, 2002. 2 R.B. Woodward and R. Hoffmann, The conservation of orbital symmetry. VCH, New York, 1970. 3 J. Michl and V . Bonaˇ ci´ c-Kouteck´ y, Electronic Aspects of Organic Photochemistry. Wiley, New York, 1990. 4 L. Salem and C. Rowland, Angew. Chem., Int. Ed. 11, 92 (1972). 5 M. A. El-Sayed, Acc. Chem. Res. 1, 8 (1968). 6 J.C Johnson, A.J. Nozik, and J. Michl, Acc. Chem. Res. 46, 1290 (2013). 7 M. V . Ivanov, S. A. Reid, and R. Rathore, J. Phys. Chem. Lett. 9, 3978 (2018). 8 J. Linderberg and Y . ¨ Ohrn, Propagators in quantum chemistry. Academic, London, 1973. 9 J. V . Ortiz, Adv. Quantum Chem. 35, 33 (1999). 10 C. M. Oana and A. I. Krylov, J. Chem. Phys. 127, 234106 (2007). 11 A. V . Luzanov, A. A. Sukhorukov, and V . E. Umanskii, Theor. Exp. Chem. 10, 354 (1976). 12 M. Head-Gordon, A. M. Grana, D. Maurice, and C. A. White, J. Phys. Chem. 99, 14261 (1995). 13 R. L. Martin, J. Phys. Chem. A 118, 4775 (2003). 14 A. V . Luzanov and O. A. Zhikol, Excited state structural analysis: TDDFT and related mod- els, in Practical aspects of computational chemistry I: An overview of the last two decades and current trends, edited by J. Leszczynski and M.K. Shukla, pages 415–449. Springer, 2012. 15 F. Plasser, M. Wormit, and A. Dreuw, J. Chem. Phys. 141, 024106 (2014). 16 F. Plasser, S. A. B¨ appler, M. Wormit, and A. Dreuw, J. Chem. Phys. 141, 024107 (2014). 17 K. Nanda and A. I. Krylov, J. Phys. Chem. Lett. 8, 3256 (2017). 18 W. Skomorowski and A. I. Krylov, J. Phys. Chem. Lett. 9, 4101 (2018). 81 19 S. Mewes, F. Plasser, A. I. Krylov, and A. Dreuw, J. Chem. Theory Comput. 14, 710 (2018). 20 S. A. Mewes and A. Dreuw, Phys. Chem. Chem. Phys. 21, 2843 (2019). 21 M. Head-Gordon, P.E. Maslen, and C.A. White, J. Chem. Phys. 108, 616 (1998). 22 S. A. B¨ appler, F. Plasser, M. Wormit, and A. Dreuw, Phys. Rev. A 90, 052521 (2014). 23 S. Mai, F. Plasser, J. Dorn, M. Fumanal, C. Daniel, and L. Gonzi´ alez, Coord. Chem. Rev. 361, 74 (2018). 24 M. A. El-Sayed, J. Chem. Phys. 38, 2834 (1963). 25 J. Michl, J. Am. Chem. Soc. 118, 3568 (1996). 26 A. Bergantini, M. J. Abplanalp, P. Pokhilko, A. I. Krylov, C. N. Shingledecker, E. Herbst, and R. I. Kaiser, Astrophys. J. 860, 108 (2018). 27 P. Pokhilko, R. Shannon, D. Glowacki, H. Wang, and A. I. Krylov, J. Phys. Chem. A 123, 482 (2019). 28 C. M. Marian, WIREs Comput. Mol. Sci. 2, 187 (2012). 29 Q. Ou and J. E. Subotnik, J. Phys. Chem. C 117, 9839 (2013). 30 E. Y .-T. Li, T.-Y . Jiang, Y . Chi, and P.-T. Chou, Phys. Chem. Chem. Phys. 16, 26184 (2014). 31 P. K. Samanta, D. Kim, V . Coropceanu, and J.-L. Br´ edas, J. Am. Chem. Soc. 139, 4042 (2017). 32 I. Lyskov and C. M. Marian, J. Phys. Chem. C 121, 21145 (2017). 33 H. Ma, Q. Peng, Z. An, W. Huang, and Z. Shuai, J. Am. Chem. Soc. 141, 1010 (2019). 34 L. V . Poluyanov and W. Domcke, Spin–orbit vibronic coupling in Jahn–Teller and Renner systems, in The Jahn–Teller Effect: Fundamentals and Implications for Physics and Chem- istry, edited by H. K¨ oppel, D. R. Yarkony, and H. Barentzen, pages 77–97. Springer Berlin Heidelberg, Berlin, Heidelberg, 2009. 35 T. J. Penfold, E. Gindensperger, C. Daniel, and C. M. Marian, Chem. Rev. 118, 6975 (2018). 36 T. Zeng, D. G. Fedorov, M. W. Schmidt, and M. Klobukowski, J. Chem. Theory Comput. 7, 2864 (2011). 37 P. ˚ A. Malmqvist, B. O. Roos, and B. Schimmelphennig, Chem. Phys. Lett. 357, 230 (2002). 38 P. Pokhilko, E. Epifanovsky, and A. I. Krylov, J. Chem. Phys. 151, 034106 (2019). 82 39 T. Helgaker, P. Jørgensen, and J. Olsen, Molecular electronic structure theory. Wiley & Sons, 2000. 40 G. Racah, Phys. Rev. 62, 438 (1942). 41 S. Matsika, X. Feng, A. V . Luzanov, and A. I. Krylov, J. Phys. Chem. A 118, 11943 (2014). 42 J. F. Stanton and R. J. Bartlett, J. Chem. Phys. 98, 7029 (1993). 43 A. I. Krylov, Chem. Phys. Lett. 338, 375 (2001). 44 A. I. Krylov, Acc. Chem. Res. 39, 83 (2006). 45 J. F. Stanton and J. Gauss, J. Chem. Phys. 101, 8938 (1994). 46 P. A. Pieniazek, S. E. Bradforth, and A. I. Krylov, J. Chem. Phys. 129, 074104 (2008). 47 M. Nooijen and R. J. Bartlett, J. Chem. Phys. 102, 3629 (1995). 48 D. E. Freedman, W. H. Harman, T. D. Harris, G. J. Long, C. J. Chang, and J. R. Long, J. Am. Chem. Soc. 132, 1224 (2010). 49 E. T. Whittaker and G. N. Watson, A course of modern analysis. Cambridge University Press, 4th edition, 1996. 83 Chapter 4: Spin-Forbidden Channels in Reactions of Unsaturated Hydrocarbons with O( 3 P) 4.1 Introduction Among various reaction products, triplet oxygen atoms reacting with unsaturated hydrocar- bons (RH) can produce Cvetanovi´ c diradicals 1, 2 , species in which O( 3 P) is added to a double or triple bond. Because oxygen is a triplet, the diradical is formed in the triplet state. It can then undergo series of reactions on the triplet potential energy surface (PES) as well as intersystem crossing (ISC) to the singlet manifold. The reactions of Cvetanovi´ c diradicals are important in hydrocarbon combustion, and, to an extent, in atmospheric and interstellar chemistry 1–5 . For example, the reaction of ethylene with O( 3 P) can lead to effective secondary chain-branching and thus an enhanced fuel oxidation rate. Although numerous studies have investigated the role of Cvetanovi´ c diradicals and ISC in the reactions of O( 3 P) with unsaturated hydrocar- bons 3, 4, 6–15 , the key details of ISC are still unresolved. One particularly interesting question is what aspects of the electronic structure of the Cvetanovi´ c diradicals derived from different unsaturated hydrocarbons control the rate of ISC. Fig. 4.1 shows the yields of singlet reaction products in the reaction of O( 3 P) with ethylene 16 , acetylene 17 , propene 18 , and propyne 19 . In 84 these four unsaturated hydrocarbon species, the branching ratios between the singlet and triplet pathways vary from 10% to 84%. The efficiency of ISC in alkenes and alkynes is quite differ- ent: while in ethylene the branching ratio for the singlet products is above 50%, in acetylene it is less than 10%. Most intriguing are the variations due to substituting one of the hydrogens by the methyl group: in alkenes, this substitution results in a moderate drop in the yield of singlet products, whereas in alkynes the branching ratio increases eightfold. The goal of the present study is to characterize the electronic structure of the prototypical Cvetanovi´ c diradicals derived by O( 3 P) addition to ethene, propene, acetylene, and propyne, with an emphasis on fac- tors relevant to ISC. While the quantitative determination of the yield of the reaction products formed via ISC to the singlet state requires full multi-well modeling of the reaction 20, 21 at the conditions that match the available experiments, the rate of the actual ISC step is determined by two quantities 15, 22–25 —the spin–orbit coupling constant (SOCC) and the location of minimal- energy crossing point (MECP). Generally, larger SOCs and lower MECPs lead to faster ISC rates, however, the interplay between these two parameters in specific molecules can lead to distinctly different mechanisms. Consequently, the origin of the variations in the observed branching ratios in homologically similar compounds could be attributed to different factors. For example, in species with very low MECPs, the variations in ISC rates are due to SOCCs, whereas in systems with high MECPs, the variations in SOCC values become less important. Concluding'Remarks' Acetylene (<10%) a Ethene (>50%) b Propene (40%) c Allene (90%) d Propyne (81%) a G.*Capozza,*E.*Segoloni,*F.*Leonori,*G.G.*Volpi,*and*P.*Casavecchia*(2004);* b P.*Casavecchia,*G.*Capozza,*E.*Segoloni,*F.*Leonori,*N.*Balucani,*and*G.G.*Volpi*(2005);* c J.D.* Savee,*O.*Welz,*C.A.*Taatjes,*and*D.L.*Osborn*(2012);* d F.*Leonori,*A.*Occhiogrosso,*N.*Balucani,*A.*Bucci,*R.*Petrucci,*and*P.*Casavecchia*(2012).** Concluding'Remarks' Acetylene (<10%) a Ethene (>50%) b Propene (40%) c Allene (90%) d Propyne (81%) a G.*Capozza,*E.*Segoloni,*F.*Leonori,*G.G.*Volpi,*and*P.*Casavecchia*(2004);* b P.*Casavecchia,*G.*Capozza,*E.*Segoloni,*F.*Leonori,*N.*Balucani,*and*G.G.*Volpi*(2005);* c J.D.* Savee,*O.*Welz,*C.A.*Taatjes,*and*D.L.*Osborn*(2012);* d F.*Leonori,*A.*Occhiogrosso,*N.*Balucani,*A.*Bucci,*R.*Petrucci,*and*P.*Casavecchia*(2012).** Concluding'Remarks' Acetylene (<10%) a Ethene (>50%) b Propene (40%) c Allene (90%) d Propyne (81%) a G.*Capozza,*E.*Segoloni,*F.*Leonori,*G.G.*Volpi,*and*P.*Casavecchia*(2004);* b P.*Casavecchia,*G.*Capozza,*E.*Segoloni,*F.*Leonori,*N.*Balucani,*and*G.G.*Volpi*(2005);* c J.D.* Savee,*O.*Welz,*C.A.*Taatjes,*and*D.L.*Osborn*(2012);* d F.*Leonori,*A.*Occhiogrosso,*N.*Balucani,*A.*Bucci,*R.*Petrucci,*and*P.*Casavecchia*(2012).** Concluding'Remarks' Acetylene (<10%) a Ethene (>50%) b Propene (40%) c Allene (90%) d Propyne (81%) a G.*Capozza,*E.*Segoloni,*F.*Leonori,*G.G.*Volpi,*and*P.*Casavecchia*(2004);* b P.*Casavecchia,*G.*Capozza,*E.*Segoloni,*F.*Leonori,*N.*Balucani,*and*G.G.*Volpi*(2005);* c J.D.* Savee,*O.*Welz,*C.A.*Taatjes,*and*D.L.*Osborn*(2012);* d F.*Leonori,*A.*Occhiogrosso,*N.*Balucani,*A.*Bucci,*R.*Petrucci,*and*P.*Casavecchia*(2012).** Ethene (>50%) Propene (40%) Acetylene (<10%) Propyne (84%) Figure 4.1: Total yields of the reaction products via ISC in ethene, propene, acetylene, and propyne. 85 The electronic structure of C 2 H 4 O has been extensively studied 4, 9, 11, 14, 15, 26 . ISC has been studied by Bowman and co-workers, by surface-hopping 9, 10 , and by Klippenstein and co- workers, by non-adiabatic transition-state theory (NA-TST) and dynamical simulations 4, 15 . While most these ISC calculations were carried out for only a single triplet surface, Ref. 15 considered the contribution to the rate from the two closely lying triplet states. The simulations revealed that ISC in Cvetanovi´ c diradicals can indeed be very effective and showed that differ- ent treatments of ISC yield different kinetic predictions at high temperature. In this work we explore the qualitative differences between the electronic structure of several Cvetanovi´ c dirad- icals and evaluate important trends and features relevant to ISC. For this purpose, we assume ergodicity, i.e., treating vibrational energy redistribution to be fast in our rate treatment. This treatment allows us to focus on general trends rather than on quantitative rate modeling. For O + C 2 H 4 reaction, a more detailed treatment of the branching ratio following ISC with a classical trajectory study is given in Ref. 15. The paper’s outline is as follows. In the next section, we describe the molecular orbital (MO) framework and the nature of relevant electronic states. Sec- tion 4.3 outlines theoretical methods employed and computational details. Results of electronic structure calculations as well as their implication for kinetics are presented in Section 4.4. 4.2 Molecular orbital framework The electronic structure of Cvetanovi´ c diradicals features a manifold of closely lying elec- tronic states derived by distributing 4 electrons over 3 orbitals. To simplify the description of the electronic states of the diradical, we begin by considering a symmetric geometry (with the CCO symmetry plane constraint) of the lowest triplet state of the ethylene adduct, as in previous studies 7, 26 . Although this structure is not a true minimum, the analysis of the electronic configu- rations at this geometry is a good starting point for understanding the electronic structure of the diradical at low-symmetry geometries. The structures and relevant energetics are shown in Fig. 86 4.12 in the Appendix A. Figure 4.2 shows frontier MOs of C 2 H 4 O at the optimized geometry of the 3 A 00 state and the leading electronic configurations of the relevant states. Although the exact shape of orbitals depends on a method, their characters can be reliably described as an out-of-plane lone pair on oxygen, p(O), and a bonding-antibonding pair formed by an in-plane p-orbital of oxygen and ap-orbital of carbon. Different electronic distributions of 4 electrons on these 3 orbitals give rise to several closely lying electronic states: 4 singlets and 2 triplets. The leading electronic configurations of the relevant electronic states are shown in the mid- dle panel of Fig. 4.2. At symmetric (C s ) structures, one can generate 2 different triplet states ( 3 A 0 and 3 A 00 ) and two open-shell singlets with the same orbital occupations. At low-symmetry structures, all electronic configurations of the same multiplicity can mix, giving rise to heavily multi-configurational wave-functions. This multi-configurational character and multiple closely lying electronic states pose a challenge for theory. It is easy to rationalize why there are two nearly degenerate triplet states in C 2 H 4 O by con- sidering the correlation between the MOs of the adduct and the reactants. At the dissociation limit, the O( 3 P) term comprises 9 triplet states (3 p-states for each S z ). From these, one p- orbital (call it p z (O)) can form a bonding combination with the -orbitals of ethylene, while others yield non-bonding combinations. In the triplet state with doubly occupiedp z (O), an anti- bonding orbital is occupied, leading to a repulsive state, while in two other triplets, differing in occupation of non-bonding lone pairs, there is bonding interaction between O and C. The singlet A 0 states can also contain closed-shell configurations. The weight of the closed-shell configuration in the lowest singlet is comparable with the weights of open-shell configurations. This closed-shell configuration imparts bonding character to the two interacting radical cen- ters. The weight of this configuration and, consequently, the bonding interactions increase at smaller CCO angles, leading to important products of the reaction of oxygen with unsaturated hydrocarbons: ethylene and propylene oxides. 87 Fig. 4.3 shows frontier MOs and their occupations at the optimized geometry of the lowest triplet state. The displacement leading to this lower-symmetry structure, which can be described as a torsional motion of CH 2 , upsets the bonding and antibonding interactions, and the MOs can be described as almost non-interacting lone pairs on carbon and oxygen. +/- +/- p(O,a”) p(O,a”) p(C,a’) p(O,a’) σ = p(O,a')+p(C) σ* = p(O,a')-p(C) p(O,a'') p(C) Figure 4.2: Top: leading electronic configurations of the relevant electronic states. Lone pairs on oxygen (p(O)) and carbon (p(O)) are singly occupied in the 3 A 00 and 1 A 00 states. Bonding () and antibonding ( ) orbitals are singly occupied in 3 A 0 , but in 1 A 0 their occupations are 1.40 and 0.58, revealing a contribution of a closed-shell configuration. Bottom: Singly occupied natural orbitals for the A 0 ( and ) and A 00 (p(O,a 00 and p(C)) states computed at the 3 A 00 constrained geometry. Fig. 4.4 shows relevant MOs in the acetylene-derived diradical, C 2 H 2 O. The electronic structure of diradicals derived from triple-bond compounds is different from that of C 2 H 4 O. In the lowest triplet state, there are two planar isomers, Z andE. The symmetry plane sepa- rates the orbitals into(a 00 ) and(a 0 ) manifolds. The-system and the orbital occupations are remarkably similar to the-system of the allyl radical (see Fig. 4.16 in Appendix B), giving rise to the CC and CO bond orders of1.5 and and an increased barrier to rotation. The-system 88 1.00 0.98 0.72 1.20 A 3 A 1 Figure 4.3: Natural frontier orbitals and their occupations of the lowest singlet at triplet states at the optimized triplet geometry (C 1 ). comprises lone-pair type orbitals on oxygen and carbon. While these non-bonding orbitals do not contribute towards bonding in the triplet state, they are important for the analysis of excited states and transition properties because they are involved in the A 00 transitions. lp(O) (a') π (a'') Figure 4.4: Left: leading electronic configurations of the relevant electronic states of the Z-isomer of the C 2 H 2 O diradical. Right: Natural frontier orbitals of the open- and closed- shell states. 4.3 Theoretical methods and computational details Electronic degeneracies and the multi-configurational character of the low-lying states of the Cvetanovi´ c diradicals make electronic structure calculations rather involved. Figure 4.5 shows all M s = 0 electronic configurations that can be generated by distributing 4 electrons in 89 3 orbitals. There are five A 0 determinants that can give rise to closed- and open-shell 1 A 0 con- figurations. Of these, determinants (4) and (5) give rise to the 3 A 0 state. Determinants (6)-(9) give rise to open-shell 1 A 00 and 3 A 00 states. In contrast to many previous studies, which have employed multi-reference methods based on the complete-active-space self-consistent field (CASSCF) method augmented by various treatments of dynamic correlation, here we explore an alternative approach: equation-of-motion coupled-cluster methods with single and double exci- tations (EOM-CCSD) 27–29 . By combining deliberately chosen references with various types of EOM operators, EOM-CC can describe a wide range of multi-configurational wave-functions within a single-reference formalism 30 . Particularly attractive features of EOM-CC methods are (i) simultaneous account of dynamic and non-dynamic correlation; (ii) their black-box nature (no active space selection or state averaging is involved); and (iii) their ability to describe mul- tiple interacting states in the same calculation. The last feature simplifies the calculation of interstate properties, such as spin–orbit and non-adiabatic couplings 31, 32 , and the analysis of transition properties 33, 34 . However, despite its flexibility, some electronic degeneracy patterns are beyond the reach of currently available EOM-CC variants 35 . Here we employ the following variants of EOM-CC: EOM-CCSD for excitation energies (EOM-EE) 36 ; in this approach the target states are described as spin-conserving 1h1p and 2h2p excitations from a closed-shell reference; EOM-CCSD with spin-flip (EOM-SF) 37, 38 ; in this approach the excitation operators are particle-conserving but flip the spin of an electron, so that the low-spin target states are generated from a high-spin (M s =1) triplet reference; EOM-CCSD with double ionization (EOM-DIP) 39–41 ; in this approach a closed-shell ref- erence state with 2 extra electrons is used, and the target states are generated by 2h and 3h1p excitation operators. 90 Hereh andp refer to the hole and particle operators defined with respect to the reference deter- minant. The bottom panel of Fig. 4.5 shows electronic configurations, which are used as ref- erences in different variants of EOM-CC. As one can see, only the DIP configurational space includes all 9 determinants on an equal footing, thus providing the most balanced description of the electronic states in these species. In contrast, in EE calculations, determinant (1) represents the reference and determinants (4)-(7) are single excitations, and the rest are double excita- tions. The analysis of the singlet CCSD wave-function reveals that the closed-shell reference is ill-behaved near the lowest triplet minimum, i.e., the norms of the cluster amplitudes exceed unity. Consequently, EOM-EE-CCSD leads to an unbalanced treatment of configurations and an incorrect state ordering. Moreover, the results of (T) calculations are quite wrong; how- ever, (dT) and f(T) corrections do give consistent and reasonable results (see Appendix C). In EOM-SF, the set of target determinants depends on which triplet reference is used. We assess the validity of different EOM models by comparing them with each other. As expected, EOM-SF and EOM-DIP yields similar results, while EOM-EE produces a different state ordering. EOM-SF treats the determinants of the target 1 A 0 states in a balanced manner (in this case the occupation of orbitals is approximately the same as in the reference triplet). How- ever, in the target 1 A 00 states the treatment of leading determinants is unbalanced because from 3 A 00 reference determinant (4) is generated by a single excitation whereas its spin-complete partner (5) cannot be obtained by a single excitation; the same is true for the 1 A 0 reference and determinants (6) and (7). The wave-functions of the resulting A 00 states are severely spin- contaminated, withhS 2 i close to 1. The computed PES (discussed in Section 4.4.1 and in Appendix B) of this state lies in between singlet and triplet states from the EOM-DIP calcula- tion, giving an averaged description. 91 (1) (2) (3) (4) (5) (6) (7) (8) (9) CS OS DIP EE SF (1) (2) (3) (4) (5) (6) (7) (8) (9) CS OS DIP EE SF Figure 4.5: Top: Electronic configurations resulting from 4-electrons-in-3-orbitals pattern (only M s = 0 determinants are shown). Determinants (1)-(3) are of a closed-shell (CS) type and determinants (4)-(9) are of open-shell (OS) type. In ethylene+O, at C s structures, determinants (1)-(5) are of A 0 symmetry, and determinants (6)-(9) are A 00 . Bottom: elec- tronic configurations of the reference determinants used in DIP, EE, and SF calculations. To analyze the electronic wave-functions, we employed the libwfa library 33 to compute natural orbitals (NOs) and natural transition orbitals (NTOs). NOs, defined as eigenvectors of a state one-particle density matrix, pq h jp y qj i; (4.1) provide a compact representation of the wave-function, which facilitates the analysis of elec- tronic states 42 . NTOs are a generalization of this concept to electronic transitions; they are defined as singular vectors of a one-particle transition density matrix : FI pq h F jp y qj I i; (4.2) 92 where I and F denote the initial and final states. Corresponding left and right singular vectors form hole and particle pairs. The transition density matrix is diagonal in the NTO basis, thus, one-particle transition properties can be represented as matrix elements between the NTOs: h F jAj I i =Tr[A FI ] = X K K h p K jAj h K i; (4.3) where h K and p K are a hole and a particle pair corresponding to the singular value K . Very often, only a few singular values are non-vanishing and the sum can be approximated by a short truncated expansion. The sum of the squares of singular values gives the squared norm of the transition density matrix,jj jj 2 . We computed relevant stationary points on the lowest triplet state PES with CCSD/6-31G* and CCSD/cc-pVTZ. For acetylene- and propyne-derived species, the MECPs were computed 43 with EOM-EE-CCSD/6-31G*. For ethylene and propene, the MECPs were computed by EOM- DIP-CCSD/6-31G*. We used such a small basis set to avoid problems due to an unstable dianionic reference 30, 44 . The calculations of MECPs are described in Section 4.3.1 below. All electronic structure calculations were carried out using the Q-Chem package 45, 46 . 4.3.1 Calculation and characterization of MECPs In NA-TST, MECPs play a role similar to those of transition states in adiabatic kinet- ics 23–25, 47 . We carried out MECP optimizations using the Lagrangian formalism defining the generalized gradient as the gradient on one of the states including the condition of the degen- eracy (the description of the algorithm can be found in Ref. 43; additional details are given in Appendix D). To validate the MECPs, we analyzed the components of the gradients and the Hessians along the seam and perpendicular to it. 93 The MECP calculations revealed important differences between the double- and triple-bond adducts. First, nuclear motions leading to MECP are different in ethylene- and acetylene- derived diradicals, as illustrated in Fig. 4.6. In ethylene-derived adducts, the crossing coordinate at MECP is the bending of the OCC angle (see also Fig. 4.18), whereas in acetylene-derived adducts, the participation of O is small and the distortion can be described as HCCH deforma- tion (as shown in Figs. 4.6 and Fig. 4.17). Second, as schematically illustrated in Fig. 4.6, in the ethylene-derived diradicals, the MECP is structurally and energetically close to the triplet diradical minimum, whereas in the acetylene-derived ones it is close to the singlet state. Con- sequently, the MECP height is larger in the acetylene and propyne diradicals. The behavior of the MECP optimization algorithm was also very different in the two families. T S Figure6: SchematicrepresentationoftheMECPpositionrelativetothetriplet-stateminimum. In the acetylene- and propyne-derived diradicals, the structure of MECP is similar to that of the singlet diradical and the crossings is energetically far from the triplet minimum (left panel). In the ethylene and propene adducts, the MECP is structurally and energetically close to the triplet minimum. In the acetylene- and propyne-derived diradicals, we were able to locate MECP by simply optimizing the seam with and without constraint of the symmetry plane of the triplet geome- tries. To verify that the found stationary crossing points are minima, we performed frequency analysis along the seam (this procedure 48 is analogous to the verification of minima versus saddle points in geometry optimization). We found that the symmetric structures have an out-of-plane imaginary mode of the e↵ ective Hessian (see Fig. S6 in SI). The gradients at the optimized MECP geometries are almost collinear, as they should be for a valid MECP (see Sec- tion 4 in SI). All angles are close to 180 , corresponding to the crossing of two shifted parabolas with the crossing point between the respective minima. The deviation of the angles between thegradientsislargerforthepropyne-deriveddiradicalthanfortheacetylene-deriveddiradical, likely because of a larger number of degrees of freedom. For acetylene- and propyne-derived systems, the MECP geometries are close to the equilibrium geometry of the singlet diradical. Consequently, the MECP and the singlet minima are also close energetically (see Table S2 in the SI and Fig. 6). Another observation is that the contribution of singlet Hessian dominates the e↵ ective Hessian and frequencies of MECP are singlet-like (Table S3 in SI). In contrast, in the ethylene- and propene-derived diradicals, the MECPs are very close to the triplet state minima, both structurally and energetically (see Fig. 6). For these systems, optimization of MECP was non-trivial because of a very shallow PES and the lack of ana- lytic gradients for EOM-DIP-CCSD (which for this system is the only EOM model capable of describing the relevant states around the MECP in a balanced way). We used the conjugate gradient method and steepest descent starting from the optimized triplet geometries. Often the optimization procedure converged to a stationary point with one or two imaginary frequency along the seam. In these cases, we located the true MECPs by applying displacements from these stationary points and rerunning the optimization. 8 T S Figure6: SchematicrepresentationoftheMECPpositionrelativetothetriplet-stateminimum. In the acetylene- and propyne-derived diradicals, the structure of MECP is similar to that of the singlet diradical and the crossings is energetically far from the triplet minimum (left panel). In the ethylene and propene adducts, the MECP is structurally and energetically close to the triplet minimum. In the acetylene- and propyne-derived diradicals, we were able to locate MECP by simply optimizing the seam with and without constraint of the symmetry plane of the triplet geome- tries. To verify that the found stationary crossing points are minima, we performed frequency analysis along the seam (this procedure 48 is analogous to the verification of minima versus saddle points in geometry optimization). We found that the symmetric structures have an out-of-plane imaginary mode of the e↵ ective Hessian (see Fig. S6 in SI). The gradients at the optimized MECP geometries are almost collinear, as they should be for a valid MECP (see Sec- tion 4 in SI). All angles are close to 180 , corresponding to the crossing of two shifted parabolas with the crossing point between the respective minima. The deviation of the angles between thegradientsislargerforthepropyne-deriveddiradicalthanfortheacetylene-deriveddiradical, likely because of a larger number of degrees of freedom. For acetylene- and propyne-derived systems, the MECP geometries are close to the equilibrium geometry of the singlet diradical. Consequently, the MECP and the singlet minima are also close energetically (see Table S2 in the SI and Fig. 6). Another observation is that the contribution of singlet Hessian dominates the e↵ ective Hessian and frequencies of MECP are singlet-like (Table S3 in SI). In contrast, in the ethylene- and propene-derived diradicals, the MECPs are very close to the triplet state minima, both structurally and energetically (see Fig. 6). For these systems, optimization of MECP was non-trivial because of a very shallow PES and the lack of ana- lytic gradients for EOM-DIP-CCSD (which for this system is the only EOM model capable of describing the relevant states around the MECP in a balanced way). We used the conjugate gradient method and steepest descent starting from the optimized triplet geometries. Often the optimization procedure converged to a stationary point with one or two imaginary frequency along the seam. In these cases, we located the true MECPs by applying displacements from these stationary points and rerunning the optimization. 8 should be positive. Here P is a projector into the space spanned by the translation and rotation vectors as well as on the crossing coordinate along the gradients. The necessary and su cient conditions can be written as: g 1 + (g 2 g 1 )=0, (5) = (g 1 ,g 1 g 2 ) ||g 1 || · ||g 1 g 2 || ||g 1 || ||g 1 g 2 || , (6) where g 1 , g 2 are the gradients of surfaces 1 and 2, is a Lagrange multiplier, (g1,g1 g2) ||g1 ||·||g1 g2 || is a cosine between g 1 and the gradient di↵ erence, which ideally should be ±1. This approach has been used before 4 with the assumption of a good collinearity of gradient vectors. Here we use the angle between the gradients (and the cosine of this angle) as a simple metric of the quality of MECP. Another important quantity used in NA-TST approaches is reduced mass. Here we used the mass-weighted coordinates for the gradients, which allowed us to eliminate the reduced mass (see derivation below). Figure S6: Stationary crossing points in C 2 H 2 O. Left: Out-of-plane mode in the seam of 659.46i cm 1 . Middle: Crossing coordinate at MECP geometry, perpendicular to the seam. Right: Equilibrium singlet geometry. Figure S7: MECP crossing coordinates in the diradicals derived from ethylene and propylene. 4.1 Reduced-mass elimination In NA-TST it is assumed that there is a reaction coordinate q, and the conjugated momentum is included in only one term in kinetic energy. All other terms do not have p q . Around MECP S7 should be positive. Here P is a projector into the space spanned by the translation and rotation vectors as well as on the crossing coordinate along the gradients. The necessary and su cient conditions can be written as: g1 + (g2 g1)=0, (5) = (g1,g1 g2) ||g1|| · ||g1 g2|| ||g1|| ||g1 g2|| , (6) where g1, g2 are the gradients of surfaces 1 and 2, is a Lagrange multiplier, (g1,g1 g2) ||g1 ||·||g1 g2 || is a cosine between g1 and the gradient di↵ erence, which ideally should be ±1. This approach has been used before 4 with the assumption of a good collinearity of gradient vectors. Here we use the angle between the gradients (and the cosine of this angle) as a simple metric of the quality of MECP. Another important quantity used in NA-TST approaches is reduced mass. Here we used the mass-weighted coordinates for the gradients, which allowed us to eliminate the reduced mass (see derivation below). Figure S6: Stationary crossing points in C2H2O. Left: Out-of-plane mode in the seam of 659.46i cm 1 . Middle: Crossing coordinate at MECP geometry, perpendicular to the seam. Right: Equilibrium singlet geometry. Figure S7: MECP crossing coordinates in the diradicals derived from ethylene and propylene. 4.1 Reduced-mass elimination In NA-TST it is assumed that there is a reaction coordinate q, and the conjugated momentum is included in only one term in kinetic energy. All other terms do not have p q. Around MECP S7 Figure 4.6: Schematic representation of the MECP position relative to the triplet-state minimum and crossing coordinates at the MECP. Left panel: In the ethylene and propene adducts, the MECP is structurally and energetically close to the triplet minimum. Right panel: In the acetylene- and propyne-derived diradicals, the structure of MECP is sim- ilar to that of the singlet diradical and the crossings is energetically far from the triplet minimum. In the acetylene- and propyne-derived diradicals, we were able to locate MECP by simply optimizing the seam with and without constraint of the symmetry plane of the triplet geometries. 94 To verify that the found stationary crossing points are minima, we performed frequency anal- ysis along the seam (this procedure 48 is analogous to the verification of minima versus saddle points in geometry optimization). We found that the symmetric structures have an out-of-plane imaginary mode of the effective Hessian (see Fig. 4.17 in Appendix D). The gradients at the optimized MECP geometries are almost collinear, as they should be for a valid MECP (see Appendix D). All angles are close to 180 , corresponding to the crossing of two shifted parabo- las with the crossing point between the respective minima. The deviation of the angles between the gradients is larger for the propyne-derived diradical than for the acetylene-derived diradical, likely because of a larger number of degrees of freedom. For acetylene- and propyne-derived systems, the MECP geometries are close to the equilibrium geometry of the singlet diradical. Consequently, the MECP and the singlet minima are also close energetically (see Table 4.5 in the Appendix D and Fig. 4.6). Another observation is that the contribution of singlet Hes- sian dominates the effective Hessian and frequencies of MECP are singlet-like (Table 4.6 in Appendix D). In contrast, in the ethylene- and propene-derived diradicals, the MECPs are very close to the triplet state minima, both structurally and energetically (see Fig. 4.6). For these systems, optimization of MECP was non-trivial because of a very shallow PES and the lack of ana- lytic gradients for EOM-DIP-CCSD (which for this system is the only EOM model capable of describing the relevant states around the MECP in a balanced way). We used the conjugate gradient method and steepest descent starting from the optimized triplet geometries. Often the optimization procedure converged to a stationary point with one or two imaginary frequency along the seam. In these cases, we located the true MECPs by applying displacements from these stationary points and rerunning the optimization. 95 4.3.2 Evaluation of spin–orbit matrix elements The quantity relevant for the ISC rates calculations is the SOC constant (SOCC) 49, 50 , which has the following form in spherical coordinates: jSOCC(s;s 0 )j 2 s X ms=s s 0 X m 0 s =s 0 h h (s;m s )jH so L j 0 (s 0 ;m 0 s )i 2 + h (s;m s )jH so L 0 j 0 (s 0 ;m 0 s )i 2 + h (s;m s )jH so L + j 0 (s 0 ;m 0 s )i 2 i = s X ms=s s 0 X m 0 s =s 0 jh (s;m s )jH so j 0 (s 0 ;m 0 s )ij 2 (4.4) wheres;s 0 are spins andm s ;m 0 s are spin projections for the two states. We computed SOCCs using the EOM-CCSD wave-functions 31 . Calculations of SOCCs are complicated by several issues: 1. For non-spin-adapted wave-functions (i.e., EOM-SF) spin contamination may affect the SOCC values. 2. Within the adiabatic approximation, the phases of electronic wave-functions are not defined. Therefore, the phase changes randomly at different geometries or molecular orientations. This should be correctly taken care of when computing interstate matrix elements. 3. The phase consistency between the states of different spin projections in the same mul- tiplet is important, because it defines the action of the ladder operators on the states and the transformation properties of the states. In the second quantization, the same applies to the components of tensor operators, such as the transition density matrix. We tackled these issues as follows. The symmetry of one- and two-electron parts of the spin–orbit operator is the same as the symmetry of ^ ~ L ^ ~ S. Because dot product is rotationally 96 invariant, we compute matrix elements in different molecular orientations and trace how the states change. Using properties of Wigner’sD-matrices and the resolution-of-the-identity, one can establish the following relation: hSMjH so jS 0 M 0 i = SMjR 1 RH so R 1 RjS 0 M 0 = SMjR 1 H so RjS 0 M 0 = X S;S 0 z hSS z jH so jS 0 S 0 z i D (S) Sz;M y D (S 0 ) S 0 z ;M 0 ; (4.5) whereR is the rotation operator,M denotes spin projections in the original coordinate system, andS z is a spin projection in the rotated coordinate system. In the case of singlet–triplet SOCC, D 0 0;0 =1. One can compute matrix elements 00jH SO j10 by EOM-EE (as the reference–EOM matrix elements) and EOM-SF (as the EOM–EOM matrix elements) or 00jH SO j11 by EOM-SF (reference–EOM) at three different molecular orientations and solve the system of linear equa- tions. By randomizing the signs of the SOC elements, we established that phase inconsistency affects the final result of SOCC when the latter strategy is used. The same test yields identical values of SOCCs if 00jH SO j10 is computed. We estimated the effect of spin contamination on the EOM-SF SOCCs for ethylene and propylene adducts by using a procedure described in Appendix C. We found that the effect is small and the corrected SOCC values differ from the raw SOCC values by 1 cm 1 or less (Table 4.4 in Appendix C). 97 4.3.3 Calculation of ISC rates In NA-TST, the probability of the ISC step is evaluated using the Landau–Zener formula, which gives the probability of staying on the initial diabatic state (i.e., triplet PES): P =exp 2SOCC 2 vjs 1 s 2 j ; (4.6) wherev is a velocity ands 1 ands 2 are the slopes of the two potential energy curves. Tunneling can be included through a one-dimensional Wentzel–Kramers–Brillouin (WKB) model, which gives an approximate local overlap of the nuclear wave-functions. Details can be found in Ref. 51. We modeled the kinetics of ISC by applying NA-TST 23, 47 using the Landau–Zener formula, Eq. (4.6). We carried out these calculations using the open-source master equation codeMES- MER, which has been described in detail previously 24 . For the sake of simplicity, here we focus on the canonical formalism; the dependence of microcanonical rates on energy is discussed in Appendix E. Rovibrational densities of states were computed using the rigid-rotor harmonic- oscillator approximation. 4.4 Results and discussion 4.4.1 Relevant structures, potential energy scans, and states ordering The first question is the energy ordering of the relevant states and their structures. The triplet states can be well described by the CCSD method. Therefore, we used CCSD/cc-pVTZ for geometry optimizations. Fig. 4.7 shows optimized structures of the lowest triplet state of C 2 H 4 O and C 3 H 5 O. The equilibrium structures are C 1 . For C 2 H 4 O, the structures optimized under C s constraint are very close to the C 1 one, both geometrically and energetically (these 98 1.482 1.375 1.380 1.484 1.484 1.375 1.484 1.484 1.380 1.520 1.487 1.547 1.483 1.376 21.33 kcal/mol 21.54 kcal/mol TC1a: 23.78 kcal/mol TC1b: 23.27 kcal/mol TC2a: 23.83 kcal/mol TC2b: 23.86 kcal/mol Figure 4.7: Equilibrium structures of the lowest triplet states of C 2 H 4 O (top) and C 3 H 5 O (bottom) computed with CCSD/cc-pVTZ. Adduct formation energy (relative to RH+O( 3 P)) is shown under each structure (plain text: CCSD/cc-pVTZ; bold: CCSD(dT)/cc-pVTZ; ZPE is not included). Bond lengths are in ˚ A. structures are shown in Appendix A). In the propene-derived diradical, oxygen can attach to either the terminal carbon or to the middle carbon, giving rise to nearly isoenergetic TC1 and TC2 isomers. For each structure, there are two conformers separated by relatively small barriers (about 1 kcal/mol). Fig. 4.8 shows two lowest-energy structures of the C 2 H 2 O diradical ( 3 A 00 ). Both structures are true minima. The barrier for rotation is about 7 kcal/mol. It is unclear if there are stable minima on the singlet PES of the diradicals derived from ethy- lene and propylene. Our attempts to optimize the singlet structure by CCSD was unsuccessful. EOM-SF-CCSD and EOM-DIP-CCSD optimizations yield structures that might correspond to a very shallow minimum; tiny displacements lead to hydrogen shifts toward aldehyde or ketone products. 99 1.421 1.421 1.219 TZ TE 1.224 48.39 kcal/mol 49.07 kcal/mol 49.55 kcal/mol 50.57 kcal/mol T1Z T1E 49.47 kcal/mol 51.36 kcal/mol 1.473 1.424 1.224 1.230 1.426 1.473 T2Z T2E 48.25 kcal/mol 50.40 kcal/mol 1.519 1.222 1.443 1.226 1.443 1.506 Figure 4.8: Z and E isomers of the 3 A 00 state of the C 2 H 2 O diradical. Adduct formation energy (relative to RH+O( 3 P)) is shown under each structure (plain text: CCSD/cc-pVTZ; bold: CCSD(dT)/cc-pVTZ; ZPE is not included). Bond lengths are in ˚ A. Fig. 4.9 shows potential energy profiles of the lowest electronic states of C 2 H 4 O and C 2 H 2 O along torsional motion of the CH 2 group. In C 2 H 4 O the triplet surface exhibits a rather small barrier (about 1 kcal/mol) for internal rotation of the CH 2 group. The rotation in the triplet C 2 H 4 O is nearly free because there is no additional CC bonding (CC bond length is close to the single bond in ethane). Due to the Pauli principle, the two unpaired electrons in the triplet state cannot occupy the same bonding MO, leading to non-bonding interactions between the 2 radical centers. The rotation leads to the crossing with the lowest singlet surface, giving an upper bound of the MECP energy of 0.05 eV . The singlet PESs show much larger variations 100 along the torsional coordinate. Importantly, at large dihedral angles the two singlets are scram- bled and one cannot correlate the individual states with the 1 A 0 or 1 A 00 states; that is, almost all configurations from Fig. 4.5 are present in the respective wave-functions. At the symmetric geometry, closed-shell configuration (1) from Fig. 4.5, which contributes to bonding between the radical centers, has a weight comparable to the open-shell determinants. Along the displace- ment leading to a cyclic (epoxide) structure, the energy drops and the weight of determinant (1) becomes dominant. We also note that the gap between the two triplets remains nearly constant (0.2 eV). The vertical state orderings at the selected geometries are given in Appendix B. The triplet PES of the acetylene-derived diradical (Fig. 4.9, bottom) is very different: the barrier is an order of magnitude higher than in C 2 H 4 O. This difference can be explained by the orbital pattern: the -subsystem of the triplet state resembles that of the allyl radical, giving rise to an increased CC bond order (which is also manifested by a shorter CC distance). The gap between the triplet and the lowest singlet is also larger than in C 2 H 4 O (see Appendix B for the state orderings at selected geometries). The character of the singlet state depends on geometry: near the triplet 3 A 00 minimum this state has an open-shell 1 A 00 -like character, but in the structures with rotated CH, it acquires a closed-shell character. The scans (Fig. 4.9) suggest a higher-lying singlet–triplet MECP, which is confirmed by the calculations. 4.4.2 MECPs and SOCs: Cumulative analysis and implications for ISC rates Table 4.1 and 4.2 show relevant energy gaps and SOCCs. As explained in Section 4.3.1, the calculations of MECPs in these species were far from trivial. The triplet propylene adducts are nearly isoenergetic, differing by less than 1 kcal/mol. At the optimized triplet geometries, the singlet is 2.3 kcal/mol above the triplet state. The MECP heights relative to the triplet minima are very small, i.e., less than 1 kcal/mol. 101 −0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0 10 20 30 40 50 60 70 80 90 T1 3 A 00 1 A 00 , 1 A 0 3 A 0 E , e V 6 OCCH −0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 0 20 40 60 80 100 120 140 160 180 3 A 00 1 A 00 1 A 0 Z E E , e V 6 OCCH Figure 4.9: EOM-DIP-CCSD/6-31G* energies of low-lying states of C 2 H 4 O (top) and C 2 H 2 O (bottom) along torsional coordinate. The coordinates for the scans were obtained by constrained optimization of the lowest triplet state (CCSD/6-31G*). Symmetry labels correspond to the states computed at the symmetric structures (see Figs. 4.2 and 4.4). In the propylene adduct, the arithmetic mean of SOCC is 12.03 cm 1 and 15.00 cm 1 for the triplet geometries and the MECP respectively. These values are smaller than for the ethy- lene adduct, indicative of less efficient ISC, which qualitatively agrees with the experimentally observed smaller yield of singlet products in propylene. For the adducts derived from acetylene and propyne, the difference in isomer energetics is slightly larger, up to 3 kcal/mol. As discussed above, the electronic structure of these adducts is very different from ethylene-derived species and vertical singlet–triplet energy gaps are greater 102 Table 4.1: Relevant energy differences (kcal/mol) for the lowest triplet states of prototypi- cal Cvetanovi´ c diradicals. Molecule E a f E b E c ST MECP d Ethylene T 20.43 0.0 1.36 0.06 Propene TC1a 23.40 0.08 1.84 0.24 TC1b 22.89 0.59 0.78 0.05 TC2a 22.81 0.0 1.90 0.35 TC2b 22.79 0.02 2.32 0.29 Acetylene TZ 47.85 0 23.74(A 00 ); 23.82(A 0 ) 10.39 TE 49.03 1.18 23.03(A 00 ); 32.01(A 0 ) 11.57 Propyne T1E 49.40 1.91 23.92(A 00 ); 31.81(A 0 ) 7.76 T1Z 51.31 0 25.01(A 00 ); 28.88(A 0 ) 9.67 T2E 48.20 3.11 28.20(A 00 ); 32.12(A 0 ) 7.52 T2Z 50.37 0.94 28.29(A 00 ); 27.33(A 0 ) 9.68 a Energy relative to the RH+O( 3 P) asymptote, CCSD/cc-pVTZ. b Energy relative to the lowest isomer for each species. c Vertical singlet triplet energy gap (E T E S ), EOM-DIP-CCSD/6-31G*. d MECP location relative to the equilibrium structure of each isomer. Table 4.2: SOCCs (cm 1 ) at equilibrium triplet geometries and at MECPs. Molecule SOCC at T SOCC at MECP a Ethylene T 17.14 14.28 Propene TC1a 13.68 17.37 TC1b 13.33 10.54 TC2a 16.56 18.27 TC2b 4.55 13.81 Acetylene TZ 0.27(A 00 ); 10.52(A 0 ) 13.53 TE 0.32(A 00 ); 7.13(A 0 ) 13.53 Propyne T1E 0.24(A 00 ); 1.43(A 0 ) 15.07 T1Z 0.17(A 00 ); 8.17(A 0 ) 15.07 T2E 0.26(A 00 ); 7.81(A 0 ) 16.67 T2Z 0.36(A 00 ); 7.17(A 0 ) 16.67 a Coupling with the lowest singlet. than 20 kcal/mol. There is a significant difference between the MECP heights in acetylene- derived and propyne-derived intermediates, sometimes up to a factor of two. This result is also 103 consistent with the experimental finding: the branching ratio indicates much smaller yield of the singlet products in C 2 H 2 O than in C 2 H 4 O. The difference between the branching ratios in the propyne+O and acetylene+O reactions can be explained by the MECP heights as well: smaller yields of singlet products in the acetylene+O reaction can be attributed to higher MECPs relative to propyne. 4.4.3 Analysis of SOCs and extended El-Sayed’s rules El-Sayed’s rules 52 predict different magnitude of SOCs for the two triplet states in C 2 H 4 O: the coupling between 3 A 00 and 1 A 0 is expected to be much larger than that between 3 A 0 and 1 A 0 or between 3 A 00 and 1 A 00 because a rotation of the p(O) orbital is needed for significant cou- pling. The calculations show a two orders of magnitude difference between SOCC( 3 A 00 , 1 A 0 ) and SOCC( 3 A 00 , 1 A 00 ), whereas SOCC( 3 A 0 , 1 A 0 ) has an intermediate value. To rationalize these couplings in terms of El-Sayed’s rules, we use NTOs. Figs. 4.20 and 4.11 show the NTOs, norms of one-particle transition density matrix, and SOCCs for several states in ethylene- derived diradical and for the lowest triplet and singlet states in various isomers and conformers of the propylene-derived diradicals. Fig. 4.20 shows that for the 3 A 00 , 1 A 00 transition, there is no rotation of hole and particle orbitals, which gives rise to a weak coupling. Note a similarity between the holes and particles: there is no change in orbital shape, and these two states can be represented well by spin-adapted configurations of two open-shell determinants constructed from the respective NTOs. There is always a sign change in these combinations, coming from the Clebsh-Gordon coefficients, which can be observed in one of the NTO pairs for ( 3 A 00 , 1 A 00 ), because singular value decom- position preserves phases. 1 A 0 , 3 A 00 transition leads to 90 rotation ofp(O), giving rise to large 104 coupling. The 3 A 0 , 1 A 0 transition leads to partial rotation ofp(O) andp(C) in the plane of sym- metry. The angle of this rotation is about 30 . In Appendix F, we extend this analysis and illustrate that an intermediate value of SOCC corresponds to partial rotation of NTOs. 0.24 SOCC = 0.39 cm -1 ||γ|| 2 = 0.96 3 A’’ 1 A’ (SF) 0.24 SOCC = 10.11 cm -1 ||γ|| 2 = 0.97 0.35 0.14 3 A’ 1 A’ (SF) 0.28 SOCC = 17.66 cm -1 ||γ|| 2 = 0.65 1 A’ 3 A’ (EE) SOCC = 58.15 cm -1 ||γ|| 2 = 0.47 0.28 1 A’ 3 A’’ (EE) Figure 4.10: NTOs and SOCCs between the singlet and triplet states at symmetric C 2 H 4 O geometry computed using EOM-SF-CCSD/6-31G* and EOM-EE-CCSD/6-31G* wave- functions. Only orbitals are shown. NTO analysis was performed usingA!B transi- tion matrix. 4.4.4 Implications for ISC rates Although full rate and branching ratios require complicated multi-well calculations, some qualitative predictions can be made by simply considering the Landau–Zener formula, Eq. (4.6). This expression indicates that the non-adiabatic (here, triplet–singlet) transitions are suppressed 105 0.29 0.20 SOCC = 17.14 cm -1 ||γ|| 2 = 0.93 3 A 1 A (SF) 0.28 0.20 3 A 1 A (SF) SOCC = 13.68 cm -1 ||γ|| 2 = 0.93 0.29 0.19 3 A 1 A (SF) SOCC = 13.33 cm -1 ||γ|| 2 = 0.93 0.25 0.23 3 A 1 A (SF) SOCC = 4.55 cm -1 ||γ|| 2 = 0.96 0.29 0.20 3 A 1 A (SF) SOCC = 16.56 cm -1 ||γ|| 2 = 0.91 Figure 4.11: NTOs and SOCCs between the lowest triplet and singlet states at the triplet- minimum geometries of ethylene-derived intermediate and TC1a, TC1b, TC2a, TC2b propylene-derived intermediates computed by EOM-SF-CCSD/6-31G*. Only orbitals are shown. NTO analysis was performed usingA!B transition matrix. when the crossing velocity is high. In NA-TST treatment the probabilities of transition are com- puted in an averaged way, and the velocity is computed from an energy excess of a given micro- canonical state. Taking probabilities and statistical factors together, one can show that asymp- totically the microcanonical ISC rate behaves as k(E) E 1=2 (this trend was derived for 106 Table 4.3: Canonical ISC rates (s 1 ) at 300 K (high-pressure limit) and maximal micro- canonical ISC rates within dissociation energy window a . Molecule Canonical LZ rate Canonical WKB rate Max microcanonical LZ rate Ethylene T 1:2 10 10 5:67 10 9 1:50 10 10 Propene 7:2 10 10 5:7 10 10 1:79 10 11 TC1a 3:4 10 10 2:7 10 10 1:02 10 11 TC1b 2:0 10 10 1:2 10 10 4:08 10 10 TC2a 1:0 10 10 9:9 10 9 1:99 10 10 TC2b 8:4 10 9 8:2 10 9 1:62 10 10 Acetylene 1:9 10 4 2:0 10 4 1:14 10 10 TZ 2:1 10 3 3:5 10 3 5:44 10 9 TE 1:7 10 4 1:7 10 4 5:99 10 9 Propyne 10:0 10 4 2:5 10 5 1:44 10 10 T1E 4:1 10 4 6:3 10 4 2:28 10 9 T1Z 9:7 10 2 1:5 10 3 5:17 10 9 T2E 5:7 10 4 1:8 10 5 2:62 10 9 T2Z 1:4 10 3 4:4 10 3 4:36 10 9 a The rates were computed using harmonic approximation for all modes. The total rate of ISC (shown in the same row as acetylene, propyne, etc) is a sum of the rates of from all intermedi- ates. an N-dimensional harmonic oscillator in the original NA-TST paper 47 ). However, this regime is observed only for energies comparable to or higher than dissociation limit, and for propy- lene, acetylene, and propyne derivatives it increases monotonically up to a dissociation limit, as shown in Fig. 4.19 in Appendix E. As described in Section 4.3.3 we modeled the kinetics of ISC in these systems by applying NA-TST 23, 47 in its microcanonical form. By using the MESMER software, we calculated high- pressure limiting ISC rates for each triplet species; these are given in Table 4.3. These rates agree qualitatively with the experimental branching ratios 16–19 shown in Figure 4.1 and illus- trate the relative ISC rates between the different systems. For example, the computed rates are similar in the ethene and propene-derived species, whereas in the triple-bond adducts the methyl group leads to a 5-10-fold rate increase, consistently with a significant increase in the yield of singlet products in propyne relative to acetylene. These high-pressure conditions correspond to 107 ultrafast vibrational relaxation, often observed in condensed phase. Interestingly, in the experi- ments studying oxygen addition to double-bond species 2 in liquid nitrogen and in solid films the fragmentation products have not been observed. We note that under the combustion conditions this thermal, high-pressure-limit picture can be misleading. In flames, the Cvetanovi´ c diradicals are formed with excess internal energy and the ISC (and other reactive channels on the triple surface) compete with collisional energy transfer with the bath. Since the difference between microcanonical and canonical rates is not large for double-bond species, a weak dependence of ISC on pressure is expected. In contrast, the difference between microcanonical and canonical rates is 5-6 orders of magnitude in the triple-bond species, which we attribute to the relatively high MECPs. 4.5 Conclusion We characterized four prototypical Cvetanovi´ c diradicals, C 2 H 4 O, C 3 H 6 O, C 2 H 2 O, and C 3 H 4 O, with an emphasis on the electronic structure aspects relevant to the ISC rates. We found that methylation has a relatively small effect on the electronic structure. In contrast, the diradicals derived from double- and triple-bond unsaturated hydrocarbons are very different. In ethylene-derived diradicals, there are four closely lying electronic states differing by the occupation of oxygen’s p-orbitals. The lowest singlet state is multi-configurational and the weights of different configurations depend on the geometry. While its diradical structure was described long time ago 2 , the exact shape and orientation of the orbitals can be established only by accurate correlated calculations. The NTO analysis using SF wave-functions confirms the open-shell nature of the singlet states and serves as a basis for the extension of El-Sayed’s rules. The lowest triplet state features a very shallow PES profile along the torsional coordinate, with a low-energy (1 kcal/mol) MECP located close to the equilibrium triplet structure. There is no stable minimum corresponding to the singlet diradical. 108 In contrast to ethylene, in the acetylene-derived adducts the barriers for the internal rotation and singlet–triplet MECPs are much higher. Moreover, these species form a stable closed- shell singlet diradical (its structure is close to the MECP). Because the singlet geometry is MECP-like, the difference in the MECP heights in the acetylene- and propyne-derived species correlates with the adiabatic singlet–triplet gaps. The magnitude of SOCCs depends strongly on the molecular structure, which can be explained by El-Sayed’s rules. In the acetylene/propyne adducts, the magnitude of SOCCs at MECP is almost 10 times larger than at the equilibrium structures. The calculated MECPs heights and SOCCs reveal that the different effect of the methyl group on the ISC rates in the double- and triple-bond compounds can be attributed to different mechanisms. In the former, the MECP is very low and the rate is controlled by the variations of SOCC, whereas in the latter, the MECP is high and the main effect of methyl substitutions is explained by the variations in MECP heights. Our study also highlights the great complexity of these small species, drawing attention to the methodological challenges and possible pitfalls in theoretical modeling of their reactivity. 4.6 Appendix A: Structures 109 1.488 1.381 20.46 kcal/mol 20.66 kcal/mol Equilibrium 3 A 1.392 1.489 1.395 1.504 19.76 kcal/mol 19.98 kcal/mol Constrained 3 A" 17.26 kcal/mol 17.40 kcal/mol Constrained 3 A' Figure 4.12: Three geometries of C 2 H 4 O, obtained with CCSD/cc-pVTZ (with core elec- trons frozen) with or without symmetry constraint. Adduct formation energy (rela- tive to C 2 H 4 +O( 3 P)) is shown under each structure (plain text: CCSD/cc-pVTZ; bold: CCSD(dT)/cc-pVTZ; ZPE is not included). The low-symmetry structure is the global min- imum on the lowest triplet surface, corresponding to the equilibrium geometry of triplet C 2 H 4 O. The 3 A 00 and 3 A 0 structures are obtained under C s constraint and correspond to the two different occupations of MOs (see Figs. 2 and 3 in the main manuscript); these structures are not PES minima. The 3 A 00 structure is close to 3 A 0 . 110 4.7 Appendix B: Ordering of low-lying electronic states 4.7.1 Ethylene+O Figure 4.13 shows vertical state ordering for C 2 H 4 O at the 3 A 00 geometry computed by CCSD, CCSD(dT), and selected EOM methods (EOM-EE, DSF, and DIP). Since this structure is very close to the 3 A minimum, this state ordering is representative of the state ordering at the minimum-energy structure. The DIP method provides the most balanced description of all electronic states. DIP-CCSD predicts that 3 A 00 is the lowest triplet (lying 0.20 eV below 3 A 0 ). It also shows that there is an open-shell singlet, 1 A 00 , which is nearly degenerate with the closed- shell singlet ( 1 A 0 ). The vertical energy gap between 1 A 0 and 3 A 00 is 0.1 eV . EOM-EE gives an incorrect ordering of the two triplets, probably because configurations (8) and (9) from Fig. 5 in the main manuscript are doubly excited with respect to the closed-shell EOM-EE reference. The most reliable energetics is given by CCSD(dT) method, which shows that 3 A 00 is the lowest triplet state and that the gap between the lowest triplet and 1 A 0 is 0.30 eV (note that CCSD(T) reverses ordering of the two triplets). To put these energy differences in context, recall that the 3 A 00 structure is 0.88 eV below the entrance channel, so all these states are energetically accessible. The analysis of the wavefunctions reveals their complex nature. The DIP wavefunctions show that “closed-shell” singlet state is, in fact, a combination of closed-shell and open- shell diradical configurations, i.e., determinants (1), (2), (4) and (5) from Fig. 5 in the main manuscript. This is probably why this state is so poorly described both by CCSD and EOM-SF, although it is not clear whether large weights of these OS configurations in DIP states arise due to intrinsic state character or due to large orbital relaxation effects. The 3 A 00 state is a lin- ear combination of determinants (6)-(9), with (6) and (7) dominating. We note that the CCSD 111 description of the closed-shell singlet state surface is in remarkably good agreement with DIP, despite alarmingly largeT 1 amplitudes. 3 A 0 3 A 00 3 A Structure 3 A” 3 A’ 1 A’ 0.16 0.17 0.17 0.71 0.13 0.18 CCSD CCSD(dT) CCSD(fT) 3 A” 3 A’ 1 A’ EE-CCSD 0.49 0.03 1 A” 3 A” 1 A’ 0.08 0.01 DIP-CCSD 3 A’ 0.10 1 A” 0.24 Figure 4.13: Vertical state ordering (eV) at the 3 A 00 geometry (optimized by CCSD/cc- pVTZ) computed by different methods with the cc-pVTZ basis. 112 Additional PES scans −0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0 10 20 30 40 50 60 70 80 90 T1 3 A 00 1 A 00 , 1 A 0 3 A 0 E , e V 6 OCCH −0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0 10 20 30 40 50 60 70 80 90 3 A 00 1 A 00 ? A 0 E , e V 6 OCCH CCSD, triplet Figure 4.14: EOM-DIP-CCSD/6-31G* and EOM-SF-CCSD/6-31G* energies. The coor- dinates for the scans were obtained by constrained optimization of the lowest triplet state (CCSD/6-31G*). Symmetry labels correspond to the states computed at the symmetric structures. 4.7.2 Acetylene+O In the C 2 H 2 O diradical, the energy gaps between different electronic states are larger (see Fig. 4 of the main manuscript for frontier MOs and electronic configurations of the relevant states). Importantly, all methods (EE, DIP, SF, etc) predict that 3 A 00 state is the lowest-energy 113 state and is separated from the closed-shell singlet by1 eV . Figure 4.15 shows vertical state ordering computed at the equilibrium geometries of the two isomers. EE-CCSD 1 A’ 3 A” 1 A” 0.92 0.03 DIP-CCSD 1 A’ 0.60 Acetylene+O, Z izomer 3 A’ 0.24 1 A’ 3 A” 1 A” 1.17 0.30 1 A’ 0.82 3 A’ 1.27 Z isomer SF-CCSD 1 A’ 3 A” 1 A” 1.36 0.01 1 A’ 0.60 3 A’ 1.27 EE-CCSD 1 A’ 3 A” 1 A” 0.92 0.03 DIP-CCSD 1 A’ 0.60 Acetylene+O, Z izomer 3 A’ 0.24 1 A’ 3 A” 1 A” 1.17 0.30 1 A’ 0.82 3 A’ 1.27 Z isomer SF-CCSD 1 A’ 3 A” 1 A” 1.36 0.01 1 A’ 0.60 3 A’ 1.27 Figure 4.15: Vertical state ordering for the Z and E isomers of the C 2 H 2 O diradical; cc- pVTZ. See Fig. 4 of the main manuscript for MOs and electronic configurations. The bold line marks closed-shell singlet state. SF calculations employ 3 A 00 reference. 114 HOMO-1 (D) HOMO (S) LUMO Allyl radical E-isomer Z-isomer HOMO (S) LUMO HOMO (S) LUMO HOMO-3 (D) HOMO-3 (D) (D) (S) Figure 4.16: Frontier MOs of the E and Z isomers of C 2 H 2 O and allyl radical. D and S denote doubly and singly occupied orbitals. 4.8 Appendix C: Evaluation of spin-orbit matrix elements: Effect of spin-contamination To estimate the effect of spin contamination on the EOM-SF SOCCs for the ethylene and propylene diradicals, we used the following procedure. We note that the triplet reference state and theM s =0 EOM triplet states are almost spin-pure (hS 2 i2), whereas spin contamination is larger for the singlet state. Fig. 5 in main manuscript suggests that the majority of non-singlet components in this wavefunction are triplet configurations. Becauseh10jH SO j10i = 0, spin 115 contamination affects only the normalization of the wavefunction. Thus, the combined weight of contaminating triplet configurations can be computed fromhS 2 i as follows: = singlet + triplet ; (4.7) hS 2 i = 2 2 1 + 2 ; (4.8) Scaling = 1 + 2 ; (4.9) and one can correct the SOCC for spin contamination by multiplying the computed value by the above scaling coefficient. The SOCC values are given in Table 4.4; they differ from the raw SOCC values by 1 cm 1 or less. Table 4.4: Comparison of the uncorrected and corrected SOCC values for the propylene and ethylene-derived intermediates. Molecule SOCC, uncorrected SOCC, corrected Ethylene T 17.14 18.19 Propene TC1a 13.68 14.22 TC1b 13.33 13.84 TC2a 16.56 17.58 TC2b 4.55 4.58 4.9 Appendix D: Calculation and characterization of MECPs MECPs can be found by optimization of a crossing seam using analytical gradient tech- niques 43 . Despite their conceptual similarity, the mathematical description of MECPs is differ- ent from that of TS. For example, the reaction coordinate of interstate transition via MECP is not associated with imaginary frequency. In addition to the obvious degeneracy requirement, a valid MECP is characterized by: (i) the two gradients being collinear with each other and perpendicular to the seam and (ii) the norms of the gradients define an effective Hessian, which 116 has nonnegative frequencies on the seam. Condition (i), which is the necessary condition, can be easily seen from the Lagrange multipliers 53 . Condition (ii) is the sufficient condition; it comes from a bordered Hessian—at MECP, any small displacement in the tangent space should increase the energy 54 . Therefore, 3N 7 frequencies of effective Hessian,H eff , in the seam H eff = (1P )(H 1 +(H 2 H 1 ))(1P ); (4.10) should be positive. HereP is a projector into the space spanned by the translation and rotation vectors as well as on the crossing coordinate along the gradients. The necessary and sufficient conditions can be written as: g 1 +(g 2 g 1 ) = 0; (4.11) = (g 1 ;g 1 g 2 ) jjg 1 jjjjg 1 g 2 jj jjg 1 jj jjg 1 g 2 jj ; (4.12) whereg 1 ,g 2 are the gradients of surfaces 1 and 2, is a Lagrange multiplier, (g 1 ;g 1 g 2 ) jjg 1 jjjjg 1 g 2 jj is a cosine betweeng 1 and the gradient difference, which ideally should be1. This approach has been used before 48 with the assumption of a good collinearity of gradient vectors. Here we use the angle between the gradients (and the cosine of this angle) as a simple metric of the quality of MECP. Another important quantity used in NA-TST approaches is reduced mass. Here we used the mass-weighted coordinates for the gradients, which allowed us to eliminate the reduced mass (see derivation below). 117 Figure 4.17: Stationary crossing points in C 2 H 2 O. Left: Out-of-plane mode in the seam of 659.46i cm 1 . Middle: Crossing coordinate at MECP geometry, perpendicular to the seam. Right: Equilibrium singlet geometry. Figure 4.18: MECP crossing coordinates in the diradicals derived from ethylene and propylene. Table 4.5: Angle between the gradients in mass-weighted coordinates at the optimized MECPs. Molecule Angle, mwjjg T jj, mwjjg S jj, mwjjg S g T jj, degrees a.u./(bohr p amu) a.u./(bohr p amu) a.u./(bohr p amu) Ethylene 160.2 0.0009173 0.0091759 0.0100439 Acetylene 180.0 0.0326737 0.00659588 0.0392696 Propylene, TC1a 165.5 0.0028755 0.00788105 0.0106893 Propylene, TC1b 169.2 0.0020593 0.00756541 0.009596 Propylene, TC2a 164.4 0.0052458 0.00528123 0.0104294 Propylene, TC2b 168.9 0.0033739 0.00653306 0.00986535 Propyne, 1 170.8 0.0153565 0.005559 0.0208629 Propyne, 2 180.0 0.024858 0.00956675 0.0344246 118 Table 4.6: ’s (weights of triplet states) at MECPs. Energies here were computed by CCSD/cc-pVTZ at the CCSD/6-31G* and EOM-EE-CCSD/CCSD/6-31G* geometries for the propyne- and acetylene-derived species. Molecule E MECP E S , kcal/mol Ethylene 0.913577 Acetylene 0.167964 0.66 Propylene, TC1a 0.737282 Propylene, TC1b 0.788392 Propylene, TC2a 0.506377 Propylene, TC2b 0.662223 Propyne, 1 0.266453 0.90 Propyne, 2 0.277901 2.30 4.9.1 Reduced-mass elimination In NA-TST it is assumed that there is a reaction coordinateq, and the conjugated momentum is included in only one term in kinetic energy. All other terms do not havep q . Around MECP we can write q = (g 1 x 1 +g 2 x 2 + +g n x n )=jgj; whereg i is thei th component of the gradient vector alongx i . Neglecting the curvature of the path and the seam and the cross-terms in kinetic energy we obtain: @ 2 @x 2 = (q 0 x ) 2 @ 2 @q 2 ; X i 1 m i @ 2 @x 2 i = X i g 2 i m i jgj 2 @ 2 @q 2 : Therefore, reduced mass can be found from 1 = X i g 2 i m i jgj 2 : 119 In a special case of two atoms, after separating the movement of the center-of-mass, the reduced mass is given by the familiar expression: 1 = 1 m 1 + 1 m 2 : In another special case of mass-weighted coordinatesx i ,m i = 1 and = 1. 4.10 Appendix E: Energy dependence of microcanonical rates Fig. 4.19 shows pressure dependence of the computed ISC rates. To understand the main trends, we considered a simple model: hydrocarbon+O( 3 P) limit, the triplet diradical intermedi- ate (all isomers), the singlet diradical intermediate (all isomers), and the MECPs connecting the singlet and triplet intermediates. For the triple-bond adducts, the MECP geometries connecting the triplet Z and E isomers with the corresponding singlet minima were so close that we treated those MECPs as single species, i.e., as one MECP in the acetylene-derived diradical and as two MECPs in the propyne-derived diradicals. 4.11 Appendix F: The analysis of SOC matrix elements and natural transition orbitals To illustrate that intermediate coupling corresponds to partial rotation of NTOs, consider two EOM-EE transitions shown in Fig. 10 of the main manuscript. From the 1 A 0 to 3 A 00 tran- sition one can extract the value of spin-orbit integral, assuming zero overlap between the p- orbitals on O and C (this assumption for spin-orbit integral is justified because of the local 120 0 2e+09 4e+09 6e+09 8e+09 1e+10 1.2e+10 1.4e+10 1.6e+10 0 5000 10000 15000 20000 25000 30000 35000 kISC , s −1 Energy ab o v e singlet minim um, cm −1 Disso ciation limit 0 2e+10 4e+10 6e+10 8e+10 1e+11 1.2e+11 0 5000 10000 15000 20000 25000 30000 kISC , s −1 Energy ab o v e TC2b, cm −1 Disso ciation limit TC1a TC1b TC2a TC2b 0 1e+09 2e+09 3e+09 4e+09 5e+09 6e+09 7e+09 8e+09 0 5000 10000 15000 20000 25000 kISC , s −1 Energy ab o v e TZ, cm −1 Disso ciation limit TZ TE 0 1e+09 2e+09 3e+09 4e+09 5e+09 6e+09 7e+09 8e+09 0 5000 10000 15000 20000 25000 kISC , s −1 Energy ab o v e T1Z, cm −1 Disso ciation limit T1Z T1E T2Z T2E Figure 4.19: Microcanonical rates of triplet-singlet ISC. Left top: ethylene+O diradical; Right top: propylene+O diradicals; Left bottom: acetylene+O diradicals; Right bottom: propylene+diradicals. nature of the spin-orbit operator). Assuming that the weights ofp-orbitals on O and C are the same, the absolute value of atomic spin-orbit integral can be computed as: hp(O)jhjp(O) +p(C)i p p(O) +p(C)jp(O) +p(C) = 1 p 2 hp(O)jhjp(O)i; (4.13) hp(O)jhjp(O)i = p 2 SOCC 1e ( 1 A 0 ; 3 A 00 ) ( 1 A 0 ; 3 A 00 ) : (4.14) Since one-electron SOC scales as Z 4 , where Z is the nuclear charge, hp(O)jhjp(O)i=hp(C)jhjp(C)i = (6=8) 4 . After simple algebra, the ratio of SOCCs becomes SOCC 1e ( 1 A 0 ; 3 A 00 ) SOCC 1e ( 1 A 0 ; 3 A 0 ) = ( 1 A 0 ; 3 A 00 ) ( 1 A 0 ; 3 A 0 ) sin 30 p 2 2 1 + 6 8 4 ! : (4.15) Substituting all values, the left part is equal 0.39 (computed only fromA!B density matrix), and the right part is equal 0.43; the two values agree well within the approximations used. 121 The phases of integrals are the same, because p-orbital on O is rotated on 30 clockwise (in the chosen phases of electronic states), andp-orbital on C is rotated on 150 , while sin 30 = sin 150 = 1 2 . σ 2 = 0.24 σ 2 = 0.24 P σ 2 iα = 0.48 ||γ|| 2 = 0.96 SOCC( 3 A 00 , 1 A 00 ): 0.39 cm −1 SF σ 2 = 0.35 σ 2 = 0.14 P σ 2 iα = 0.49 ||γ|| 2 = 0.97 3 A 0 → 1 A 0 SOCC: 10.11 cm −1 SF σ 2 = 0.28 ||γ|| 2 = 0.65 1 A 0 → 3 A 0 SOCC: 17.66 cm −1 EE σ 2 = 0.23 ||γ|| 2 = 0.47 1 A 0 → 3 A 00 SOCC: 58.15 cm −1 EE Figure 4.20: NTOs and SOCCs between the singlet and triplet states at symmetric C 2 H 4 O geometry, computed using EOM-SF-CCSD/6-31G* and EOM-EE-CCSD/6-31G* wave- functions. Only -orbitals are shown. A! B transition matrix was used for singular value decomposition. The leading NTO pairs for the 3 A 0 - 1 A 0 transition are very similar for EE and SF. To explain the difference in the SOCC values computed using the EOM-EE-CCSD and EOM-SF-CCSD wavefunctions, it is sufficient to consider theh 3 A 0 ;M s = 0jH SO j 1 A 0 i matrix element, because the SOCC can be reconstructed from zero-spin components by rotating the coordinate system and applying Wigner’sD-matrices. This matrix element is imaginary 55 (unless a special time- reversal-invariant basis is invoked 56 ). Since the hole and particle orbitals in the second NTO 122 pair of ( 3 A 0 , 1 A 0 ) from the SF calculation are almost identical to the particle and hole in the first NTO pair, we can assign an effective singular value eff = 1 2 =0.2165 for these two NTO pairs (the minus sign arises due to the swapping hole and particle orbitals, leading to complex conjugation of the respective integral). Now, if the effective NTO pair for SF and the leading NTO pair for EE are the same, they yield the same one-electron integrals. Therefore, the ratio of SOCCs should be equal the ratio of singular values. Indeed, this simple reasoning is confirmed by the calculated values: eff (SF )=(EE) = 2.4 and the ratio of one-electron SOCCs SOCC 1e (SF)/SOCC 1e (EE) = 1.8. A small discrepancy between the two ratios can be explained by a non-Hermitian nature of EOM-CC: onlyA! B transition density matrix was used for the NTO analysis, while the properties were computed through geometric average 31 . While for EOM-SF theA!B andB!A norms of transition density are similar, they are very different for EOM-EE: for the 1 A 0 to 3 A 0 transitionjj 2 jj=0.6470 and 1.4115 respectively. The same reasoning is valid for the full SOCC with both one- and two-electron parts: the mean- field approximation for SOCCs works extremely well; since the NTO pairs are the same for both methods, all the integrals are the same, and the ratio is still controlled by the leading singular value in the case of good separation of singular values. 4.12 Appendix G: Relevant Cartesian geometries 4.12.1 Geometries of triplet minima and transition states $comment Ethylene+O intermediate, triplet minimum Nuclear Repulsion Energy = 67.3565911086 hartrees Optimized by unrestricted CCSD/6-31G * $end $molecule 0 3 C -0.1020917254 0.4901591223 -0.0668443892 123 H -0.1576090430 1.3433908147 0.6297996702 H -0.2665843795 0.9248573043 -1.0815071723 O -1.1925759333 -0.3613638787 0.0847921047 C 1.2016662910 -0.2249951364 0.0205561341 H 1.2966532303 -1.2120920557 -0.4198051998 H 2.0707002656 0.2437710514 0.4709053947 $end $comment Ethylene+O intermediate, triplet saddle point (symmetric geometry) Nuclear Repulsion Energy = 67.0486114153 hartrees Optimized by unrestricted CCSD/6-31G * $end $molecule 0 3 C 0.1102536036 0.4943869475 -0.0000000015 H 0.2515135105 1.1468858499 0.8793151540 H 0.2515135105 1.1468858499 -0.8793151540 O 1.1899882580 -0.4040049195 0.0000000025 C -1.2276070738 -0.1633624873 0.0000000001 H -1.6594061308 -0.5239395524 -0.9292662805 H -1.6594061308 -0.5239395524 0.9292662805 $end $comment Propylene+O intermediate, triplet minimum, TC1a structure (conf1) Nuclear Repulsion Energy = 115.1925838489 hartrees Optimized by unrestricted CCSD/6-31G * $end $molecule 0 3 C -0.8777648067 -0.4768053058 0.1786368007 H -1.5114902438 -1.3361416051 -0.0923878246 H -1.0385138217 -0.3059846614 1.2686043823 O -1.3655073262 0.7072536210 -0.3792062417 C 0.5675598439 -0.7136637902 -0.1074474007 H 0.8567829816 -1.6125809337 -0.6476192030 C 1.5551565224 0.3869512834 0.0966630777 H 1.3334509243 0.9497493484 1.0133723366 H 2.5770395786 -0.0019316416 0.1685701470 H 1.5300791311 1.1120521050 -0.7310796407 $end 124 $comment Propylene+O intermediate, triplet minimum, TC1b structure (conf3) Nuclear Repulsion Energy = 113.0693912113 hartrees Optimized by unrestricted CCSD/6-31G * $end $molecule 0 3 C 0.7127803075 0.4388068407 -0.1697020848 H 0.5627433938 1.3930335726 0.3680962814 H 0.9045775250 0.7327961225 -1.2313221069 O 1.9018604685 -0.1735864713 0.2061755135 C -0.4710758144 -0.4606958091 -0.0448868428 H -0.3173221131 -1.5099273794 -0.2876999119 C -1.8517232192 0.0867958228 0.1067755612 H -1.8622843552 0.9591503193 0.7741218149 H -2.5339164505 -0.6650483033 0.5192767611 H -2.2774250009 0.4158810290 -0.8566977755 $end $comment Propylene+O intermediate, triplet minimum, TC2b structure (conf1) Nuclear Repulsion Energy = 117.7863052538 hartrees Optimized by unrestricted CCSD/6-31G * $end $molecule 0 3 C -0.0785997086 0.0883585818 0.3743862233 H -0.0500136468 0.1095369579 1.4777900598 O 0.1169957620 1.3666986154 -0.1443543823 C -1.3600225648 -0.5012083002 -0.1109625302 H -1.9130588901 -1.2107316480 0.4973162418 H -1.6760013055 -0.3150651866 -1.1329680384 C 1.1633125177 -0.7082729960 -0.1097503507 H 1.1697214056 -0.7508937310 -1.2028989302 H 1.1100631959 -1.7253761701 0.2916501033 H 2.0812433602 -0.2260612109 0.2414985825 $end $comment Propylene+O intermediate, triplet minimum, TC2a structure (conf2) Nuclear Repulsion Energy = 117.3415476201 hartrees Optimized by unrestricted CCSD/6-31G * $end 125 $molecule 0 3 C -0.0587443845 0.0813798968 0.2982604152 H -0.0960306144 0.2592315441 1.4005423144 O -0.3717112254 1.3494417182 -0.1976351509 C -1.1179421922 -0.9049223211 -0.0772702958 H -2.1613105990 -0.6108461191 -0.0299579990 H -0.8626657499 -1.9179666010 -0.3728423345 C 1.3572119380 -0.3552748584 -0.0751387661 H 2.0776066870 0.4079632395 0.2346462343 H 1.4322136226 -0.4861908489 -1.1603413861 H 1.6137975271 -1.3051650041 0.4085634578 $end $comment Acetylene+O intermediate, triplet A" minimum, E-isomer Nuclear Repulsion Energy = 56.8483304025 hartrees Optimized by unrestricted CCSD/6-31G * $end $molecule 0 3 C 0.0560218409 -0.3523098317 0.0000914604 H 0.0143617053 -1.4613954323 0.0006271993 O 1.1420923476 0.2323199030 0.0004912433 C -1.2160043755 0.3028508182 -0.0010643873 H -2.2284355221 -0.0895061937 -0.0019908528 $end $comment Acetylene+O intermediate, triplet A" minimum, Z-isomer Nuclear Repulsion Energy = 57.1998483089 hartrees Optimized by unrestricted CCSD/6-31G * $end $molecule 0 3 C 0.0838484021 -0.4380127045 0.0000077072 H 0.2250213718 -1.5307078814 -0.0000190175 O 1.0513715372 0.3344149625 -0.0000018864 C -1.2550922590 0.0651904427 -0.0000015328 H -1.6085305280 1.0923217525 -0.0000029379 $end 126 $comment Propyne+O intermediate, triplet A" minimum, 1E-isomer Nuclear Repulsion Energy = 99.3730744713 hartrees Optimized by unrestricted CCSD/6-31G * $end $molecule 0 3 C 0.7811814860 0.3259449682 0.0000000000 H 0.6718333508 1.4330858818 0.0000000000 O 1.9057930434 -0.1887354375 0.0000000000 C -0.4481092073 -0.4011445424 0.0000000000 C -1.8651140250 0.0212132461 0.0000000000 H -2.3907859423 -0.3589985590 -0.8858354584 H -2.3907859423 -0.3589985590 0.8858354584 H -1.9455894580 1.1194279342 0.0000000000 $end $comment Propyne+O intermediate, triplet A" minimum, 1E-isomer Nuclear Repulsion Energy = 99.3730744713 hartrees Optimized by unrestricted CCSD/6-31G * $end $molecule 0 3 C 0.7811814860 0.3259449682 0.0000000000 H 0.6718333508 1.4330858818 0.0000000000 O 1.9057930434 -0.1887354375 0.0000000000 C -0.4481092073 -0.4011445424 0.0000000000 C -1.8651140250 0.0212132461 0.0000000000 H -2.3907859423 -0.3589985590 -0.8858354584 H -2.3907859423 -0.3589985590 0.8858354584 H -1.9455894580 1.1194279342 0.0000000000 $end $comment Propyne+O intermediate, triplet A" minimum, 1Z-isomer Nuclear Repulsion Energy = 101.6571638040 hartrees Optimized by unrestricted CCSD/6-31G * $end $molecule 0 3 C 0.9233531922 0.4806957755 0.0000000000 127 H 1.5763033036 1.3692474335 0.0000000000 O 1.4044728413 -0.6643110749 0.0000000000 C -0.4899958187 0.6964293895 0.0000000000 C -1.6156461894 -0.2622991077 0.0000000000 H -2.2492052878 -0.1283149243 -0.8863796290 H -2.2492052878 -0.1283149243 0.8863796290 H -1.2199425629 -1.2870853297 0.0000000000 $end $comment Propyne+O intermediate, triplet A" minimum, 2E-isomer Nuclear Repulsion Energy = 103.8752888376 hartrees Optimized by unrestricted CCSD/6-31G * $end $molecule 0 3 C 0.1724502722 0.0760732438 0.0000000000 O 0.6943475177 1.1945681589 0.0000000000 C 0.9751934735 -1.1262712332 0.0000000000 H 0.7218077684 -2.1830333954 0.0000000000 C -1.3386843238 -0.1046427374 0.0000000000 H -1.8064351441 0.8835773597 0.0000000000 H -1.6619535211 -0.6640012374 -0.8865117128 H -1.6619535211 -0.6640012374 0.8865117128 $end $comment Propyne+O intermediate, triplet A" minimum, 2Z-isomer Nuclear Repulsion Energy = 104.0372232415 hartrees Optimized by unrestricted CCSD/6-31G * $end $molecule 0 3 C 0.1061426424 0.0448408022 0.0000000000 O 0.5685726158 1.1924270891 0.0000000000 C 1.0316484730 -1.0653509703 0.0000000000 H 2.1183789490 -1.0383343492 0.0000000000 C -1.3775613048 -0.2389660430 0.0000000000 H -1.9247589873 0.7074179023 0.0000000000 H -1.6517586359 -0.8258125578 -0.8850540698 H -1.6517586359 -0.8258125578 0.8850540698 $end 128 4.12.2 Geometries of singlet minima $comment Ethylene+O intermediate, singlet Nuclear Repulsion Energy = 67.5140556288 hartrees Optimized by unrestricted EOM-SF-CCSD/6-31G * from triplet reference $end $molecule 0 1 C 0.0878430121 -0.4417006760 0.0087899372 C -1.2435014090 0.2123388142 -0.0421209411 O 1.2217618006 0.3201687133 0.0223164096 H 0.1610684623 -1.0152398597 0.9580246148 H 0.1766129227 -1.2224618972 -0.7719578390 H -1.3302240858 1.2934524577 -0.0041098398 H -2.1396857240 -0.3936129103 0.0503008323 $end $comment Ethylene+O intermediate, singlet Nuclear Repulsion Energy = 67.9191649772 hartrees Optimized by restricted EOM-DIP-CCSD/6-31G * from the singlet dianion reference $end $molecule 0 1 C 0.1064888717 0.4162358738 0.0667003419 H 0.1496949964 1.2614312305 0.7702026124 H 0.1510980091 0.9060552055 -0.9390919421 O 1.2557585287 -0.2884118883 0.0152034083 C -1.2242206722 -0.2235832436 0.0546191920 H -1.3213012555 -1.2805256780 -0.1491257186 H -2.0981428626 0.4069819100 -0.0128851269 $end 4.12.3 Geometries of MECP $comment Ethylene+O intermediate Nuclear Repulsion Energy = 67.3754726819 hartrees Geometry is taken from relaxed CCSD/6-31G * coordinate 129 and EOM-DIP-CCSD/6-31G * scan E(triplet) = -153.222140654 hartree E(singlet) = -153.222098265 hartree $end $molecule 0 1 C -0.0992764967 0.4926323492 -0.0253595296 H -0.1594701669 1.2748658741 0.7539739635 H -0.2544814988 1.0425445924 -0.9839169079 O -1.1951185036 -0.3573854754 0.0419624125 C 1.2040861224 -0.2326998331 -0.0044007617 H 1.2165768323 -1.3078062785 -0.1453620956 H 2.1294651075 0.2898845191 0.2181674877 $end $comment Ethylene+O intermediate Nuclear Repulsion Energy = 67.3363541056 hartrees Geometry is optimized by EOM-DIP-CCSD/6-31G * E(triplet) = -153.22253399 hartree E(singlet) = -153.22253410 hartree $end $molecule 0 1 C -0.0980090864 0.4840931760 -0.0422803843 H -0.1664875032 1.2887872330 0.6999155519 H -0.2495089684 0.9849574261 -1.0219270457 O -1.2051215095 -0.3521209228 0.0570430396 C 1.2124060630 -0.2303538933 0.0162736796 H 1.2540339501 -1.2735473485 -0.2577424376 H 2.1165527379 0.2943343757 0.2794498431 $end $comment Acetylene+O pseudo-MECP with imaginary mode in the seam Nuclear Repulsion Energy = 57.4436807979 hartrees Geometry is optimized by MECP optimization by EOM-EE-CCSD between CCSD reference and the lowest triplet state lambda = 0.342082 (weight of triplet) $end $molecule 0 1 130 C 0.0426587516 -0.1985354252 0.0000000000 H -0.2547137818 -1.3064126203 0.0000000000 O 1.2165183426 0.1406143480 0.0000000000 C -1.2421849991 0.2375732834 0.0000000000 H -2.2802944065 -0.0527249646 0.0000000000 $end $comment Acetylene+O MECP Nuclear Repulsion Energy = 56.8210285225 hartrees Geometry is optimized by MECP optimization by EOM-EE-CCSD between CCSD reference and the lowest triplet state lambda = 0.167964 (weight of triplet) $end $molecule 0 1 C -0.0303943366 -0.3149506151 -0.0436827067 H -0.1031279784 -1.4353675705 -0.1647400816 O 1.0178175761 0.2350702146 0.2517598501 C -1.3252768442 0.1905302054 -0.4205132671 H -2.0767072171 0.1461814657 0.3842573052 $end $comment Propyne+O MECP, 1 Nuclear Repulsion Energy = 102.0357814877 hartrees Geometry is optimized by MECP optimization by EOM-EE-CCSD between CCSD reference and the lowest triplet state lambda = 0.266453 (weight of triplet) $end $molecule 0 1 C -0.0944466128 0.1321138133 0.1159666008 H 0.4966754719 0.8179339984 0.7555458779 O 0.3217885971 -0.9827454041 -0.2040291337 C -1.3205936023 0.6418037133 -0.4642549265 C -2.6081841023 0.0097114307 -0.1068507869 H -3.4348108451 0.3163213561 -0.7539000278 H -2.8654168744 0.2553888722 0.9375369568 H -2.4818920901 -1.0852441254 -0.1454338387 $end $comment 131 Propyne+O MECP, 2 Nuclear Repulsion Energy = 104.3412122911 hartrees Geometry is optimized by MECP optimization by EOM-EE-CCSD between CCSD reference and the lowest triplet state lambda = 0.278104 (weight of triplet) $end $molecule 0 1 C 0.1202265444 0.0614643238 -0.1280326435 O 0.5724397422 1.2035407039 -0.2638260692 C 1.1250743852 -0.9638082137 -0.3070741576 H 1.8550909586 -1.0875345311 0.5026400273 C -1.3428371471 -0.2556736271 0.0267213727 H -1.8811458514 0.6651688261 0.2687516922 H -1.7285708001 -0.6693003710 -0.9129737389 H -1.5007296241 -1.0028849234 0.8140209169 $end $comment Propylene+O MECP, TC1a Nuclear Repulsion Energy = 115.3760477800 hartrees Geometry is optimized with EOM-DIP-CCSD/6-31G * $end $molecule 0 1 C -0.8709234216 -0.5036567334 0.1683443696 H -1.5033283487 -1.2947522195 -0.2493056638 H -1.0397591053 -0.5428664196 1.2667396841 O -1.3902860603 0.7435426736 -0.1629537308 C 0.5814606756 -0.7024496761 -0.1364697146 H 0.8943158637 -1.6136155244 -0.6288147952 C 1.5316229985 0.4242888908 0.0621745728 H 1.3464747516 0.9187322149 1.0241801779 H 2.5670222083 0.0806763176 0.0375135951 H 1.4046015975 1.1943893542 -0.7109785185 $end $comment Propylene+O MECP, TC1b Nuclear Repulsion Energy = 113.1039551913 hartrees Geometry is optimized with EOM-DIP-CCSD/6-31G * $end 132 $molecule 0 1 C 0.7059967779 0.4192729869 -0.0929118163 H 0.5960908508 1.2822449987 0.5822652979 H 0.8380583311 0.8781561538 -1.0996958054 O 1.9275561729 -0.1981543094 0.1253759111 C -0.4890731167 -0.4777305725 -0.0483739076 H -0.3338488728 -1.5196375700 -0.3017624719 C -1.8578954635 0.0914298235 0.0640363241 H -1.8823713991 0.9165309278 0.7867477882 H -2.5745430288 -0.6668135185 0.3865865285 H -2.2180044500 0.4969200564 -0.8936522271 $end $comment Propylene+O MECP, TC2a Nuclear Repulsion Energy = 117.4958438305 hartrees Geometry is optimized with EOM-DIP-CCSD/6-31G * $end $molecule 0 1 C -0.0682097273 0.0914895725 0.2844347190 H -0.0921700028 0.2423620346 1.3880595130 O -0.1433511309 1.4029293951 -0.1620172291 C -1.2437125516 -0.7667794225 -0.0727560624 H -2.2291724332 -0.3254923051 -0.1158948229 H -1.1216963911 -1.8186439732 -0.2836203387 C 1.2768440362 -0.5345743520 -0.0761695755 H 2.0895486273 0.1065261542 0.2681102156 H 1.3483290011 -0.6491955472 -1.1612115921 H 1.3624397018 -1.5198063120 0.3876403712 $end $comment Propylene+O MECP, TC2b Nuclear Repulsion Energy = 117.6542013992 hartrees Geometry is optimized with EOM-DIP-CCSD/6-31G * $end $molecule 0 1 C -0.0706766448 0.0850856078 0.3466578249 H -0.0296411525 0.1194944860 1.4478179975 O 0.1443107331 1.3819887070 -0.1069785128 133 C -1.3702634662 -0.5119914066 -0.1013087371 H -1.7801004792 -1.3752314074 0.4012221964 H -1.8999713635 -0.0734453240 -0.9346021060 C 1.1507983339 -0.7318526660 -0.1311862479 H 1.1725601164 -0.7378779608 -1.2236139679 H 1.0571413162 -1.7571910225 0.2302114204 H 2.0663763607 -0.2791076390 0.2498155228 $end 134 4.13 Chapter 4 references 1 R. J. Cvetanovi´ c, J. Phys. Chem. Ref. Data 16, 261 (1987). 2 R. J. Cvetanovi´ c and D. L. Singleton, Rev. Chem. Int. 5, 183 (1984). 3 C. A. Taatjes, D. L. Osborn, T. M. Selby, G. Meloni, A. J. Trevitt, E. Epifanovsky, A. I. Krylov, B. Sirjean, E. Dames, and H. Wang, J. Phys. Chem. A 114, 3355 (2010). 4 H. Sabbah, L. Biennier, I.A. Sims, Y . Georgievskii, S.J. Klippenstein, and I.W.M. Smith, Science 317, 102 (2007). 5 A. Bergantini, M. J. Abplanalp, P. Pokhilko, A. I. Krylov, C. N. Shingledecker, E. Herbst, and R. I. Kaiser, Astrophys. J. 860, 108 (2018). 6 B.F. Minaev, V .V . Kukueva, and A.S. Bykov, Colloid. Czech. Chem. Commun (2004). 7 M. Dupuis, J.J. Wendoloski, T. Takada, and W.A. Lester, J. Chem. Phys. 76, 481 (1982). 8 S. Zhao, W. Wu, H. Zhao, H. Wang, C. Yang, K. Liu, and H. Su, J. Phys. Chem. A 113, 23 (2009). 9 B. Fu, Y .-C. Han, J.M. Bowman, F. Leonori, N. Balucani, L. Angelucci, A. Occhiogrosso, R. Petrucci, and P. Casavecchia, J. Chem. Phys. 137, 22A532 (2012). 10 B. Fu, Y .-C. Han, J.M. Bowman, L. Angelucci, N. Balucani, F. Leonori, and P. Casavecchia, Proc. Nat. Acad. Sci. 109, 9733 (2012). 11 T.L. Nguyen, L. Vereecken, X.J. Hou, M.T. Nguyen, and J. Peeters, J. Phys. Chem. A 109 (2005). 12 D. Hodgson, H.-Y . Zhang, M.R. Nimlos, and J.T. McKinnon, J. Phys. Chem. A 105, 4316 (2001). 13 T. L. Nguyen, J. Peeters, and L. Vereecken, J. Phys. Chem. A 111, 3836 (2007). 14 A.C. West, J.D. Lynch, B. Sellner, H. Lischka, W.L. Hase, and T.L. Windus, Theor. Chim. Acta 131, 1123 (2012). 15 X. Li, A. W. Jasper, J. Z´ ador, J. A. Miller, and S. J. Klippenstein, Proceedings of the Com- bustion Institute 36, 219 (2016). 16 P. Casavecchia, G. Capozza, E. Segoloni, F. Leonori, N. Balucani, and G. G. V olpi, J. Phys. Chem. A 109, 3527 (2005). 135 17 G. Capozza, E. Segoloni, F. Leonori, G. G. V olpi, and P. Casavecchia, J. Chem. Phys. 120, 4557 (2004). 18 J.D. Savee, O. Welz, C.A. Taatjes, and D.L. Osborn, J. Chem. Phys. 14, 10410 (2012). 19 J. D. Savee, S. Borkar, O. Welz, B. Sztar’a, C. A. Taatjes, and D. L. Osborn, J. Phys. Chem. A 119, 7388 (2015). 20 D. L. Osborn, Annu. Rev. Phys. Chem. 68, 233 (2017). 21 L. Vereecken, D. R. Glowacki, and M. J. Pilling, Chem. Rev. 115, 4063 (2015). 22 C. M. Marian, WIREs Comput. Mol. Sci. 2, 187 (2012). 23 J. N. Harvey, Phys. Chem. Chem. Phys. 2007, 331 (2006). 24 D. R. Glowacki, C.-H. Liang, C. Morley, M. J. Pilling, and S. H. Robertson, J. Phys. Chem. A 116, 9545 (2012). 25 A. O. Lykhin, D. S. Kaliakin, G. E. dePolo, A. A. Kuzubov, and S. A. Varganov, Int. J. Quant. Chem. 116, 750 (2016). 26 A.C. West, J.S. Kretchmer, B. Sellner, K. Park, W.L. Hase, H. Lischka, and T.L. Windus, J. Phys. Chem. A 113, 12663 (2009). 27 A. I. Krylov, Annu. Rev. Phys. Chem. 59, 433 (2008). 28 K. Sneskov and O. Christiansen, WIREs: Comput. Mol. Sci. 2, 566 (2012). 29 R. J. Bartlett, WIREs: Comput. Mol. Sci. 2, 126 (2012). 30 S. Gulania, T.-C. Jagau, and A. I. Krylov, Faraday Discuss. 217, 514 (2019). 31 E. Epifanovsky, K. Klein, S. Stopkowicz, J. Gauss, and A. I. Krylov, J. Chem. Phys. 143, 064102 (2015). 32 S. Faraji, S. Matsika, and A. I. Krylov, J. Chem. Phys. 148, 044103 (2018). 33 F. Plasser, M. Wormit, and A. Dreuw, J. Chem. Phys. 141, 024106 (2014). 34 S. Mewes, F. Plasser, A. I. Krylov, and A. Dreuw, J. Chem. Theory Comput. 14, 710 (2018). 35 N. Orms and A. I. Krylov, J. Phys. Chem. A 122, 3653 (2018). 36 J. F. Stanton and R. J. Bartlett, J. Chem. Phys. 98, 7029 (1993). 37 A. I. Krylov, Chem. Phys. Lett. 338, 375 (2001). 38 A. I. Krylov, Acc. Chem. Res. 39, 83 (2006). 136 39 M. Wladyslawski and M. Nooijen, The photoelectron spectrum of the NO 3 radical revisited: A theoretical investigation of potential energy surfaces and conical intersections, in ACS Symposium Series, volume 828, pages 65–92, 2002. 40 K. W. Sattelmeyer, H. F. Schaefer, and J. F. Stanton, Chem. Phys. Lett. 378, 42 (2003). 41 M. Musiał, A. Perera, and R. J. Bartlett, J. Chem. Phys. 134, 114108 (2011). 42 N. Orms, D. R. Rehn, A. Dreuw, and A. I. Krylov, J. Chem. Theory Comput. 14, 638 (2018). 43 E. Epifanovsky and A. I. Krylov, Mol. Phys. 105, 2515 (2007). 44 T. Ku´ s and A. I. Krylov, J. Chem. Phys. 135, 084109 (2011). 45 Shao, Y .; Gan, Z.; Epifanovsky, E.; Gilbert, A.T.B.; Wormit, M.; Kussmann, J.; Lange, A.W.; Behn, A.; Deng, J.; Feng, X., et al., Mol. Phys. 113, 184 (2015). 46 A. I. Krylov and P. M. W. Gill, WIREs: Comput. Mol. Sci. 3, 317 (2013). 47 J. C. Lorquet and B. Leyh-Nihant, J. Phys. Chem. 92, 4778 (1988). 48 J. N.Harvey and M. Aschi, Phys. Chem. Chem. Phys. 1, 5555 (1999). 49 D. G. Fedorov and M. S. Gordon, J. Chem. Phys. 112, 5611 (2000). 50 D. G. Fedorov, S. Koseki, M. W. Schmidt, and M. S. Gordon, Int. Rev. Phys. Chem. 22, 551 (2003). 51 J. N.Harvey and M. Aschi, Faraday Discuss. 124, 129 (2003). 52 M. A. El-Sayed, Acc. Chem. Res. 1, 8 (1968). 53 N. Koga and K. Morokuma, Chem. Phys. Lett. 119, 371 (1985). 54 R. Fletcher, Practical methods of optimization. J. Wiley and Sons, second edition, 1987. 55 D. G. Fedorov, Theoretical study of spin-orbit coupling in molecules, PhD thesis, Iowa State University, 1999. 56 E.P. Wigner, Group Theory. New York: Academic Press, 1959. 137 Chapter 5: Double Precision Is not Needed for Many-Body Calculations: Emergent Conventional Wisdom 5.1 Introduction Quantum chemistry is one of the most demanding fields in terms of computational resources. Standard formulations of many-body theories result in large amounts of data (wave-function parameters) and steep computational scaling. For example, storage and floating point opera- tions requirements of the coupled-cluster method with single and double substitutions (CCSD) scale asN 4 andN 6 with the system size, respectively 1 . This steep scaling limits the applica- bility of these highly reliable methods. Large memory footprint, inherent to correlated theories, also creates a hurdle for efficient parallelization and utilization of accelerators (such as graphic processing units, GPUs). In this paper, we present a production-level implementation of CCSD and EOM-CCSD (equation-of-motion CCSD) methods 2–8 and investigate the impact of using reduced precision on the computational efficiency and accuracy of the results. Chemical accuracy, the minimal accuracy for quantum chemical calculations of thermo- chemical data, is defined as 1 kcal/mol (which equals 4.2 kJ/mol or 1.59310 3 hartree) 9 . How- ever, today’s standards for chemical kinetics call for a sub-kJ/mol accuracy 10, 11 . Typical desired 138 accuracy for excitation and ionization energies is 0.01-0.1 eV . The errors in the calculations arise due to the intrinsic errors of the methods because of approximations made in many-electron and one-electron bases 1 , as well as finite convergence thresholds in employed iterative algorithms. Typically, convergence thresholds are much tighter than the methods’ error bars. The IEEE 754 standard 12, 13 defines single and double precision floating-point arithmetic, which are almost universally used to represent numbers on most modern CPUs and GPUs. The numbers in this format are represented in the scientific notation (1) s b 0 :b 1 b 2 b 3 :::b p1 2 E ; (5.1) where s is the sign bit, p is the precision of significand, E is the exponent, and bits b i can take values 0 or 1 a . The exponent can take positive and negative values. The format specifies a biased exponent e = E +bias, allowing to encode exponentE by a non-negative value of e. Although the second revision of the standard 13 generalizes base, or radix, here we consider only binary formats b . The attributes of single- and double-precision floating-point numbers are summarized in Table 5.1, which shows that a single precision number takes half the space of a double precision number and for all practical purposes gives 7 significant figures, whereas double gives 15. Typical implementations of most ab initio methods use double precision arithmetic. The single-precision format uses half the number of bits of double precision, thereby allowing to store twice as many values. Another benefit is proportionally faster memory access and disk input/output (in terms of the number of elements per second). This is important in practice because memory speed improves at a slower rate than CPUs. Furthermore, CPU caches can a Althoughb 0 value is implicit and encoded by the exponent, one should consider both implicitly and explicitly stored bits when discussing decimal precision shown in Table 5.1 b The number is called normal ifb 0 is 1 and subnormal otherwise. Subnormal numbers fill the gap between the smallest positive normal number and zero. 139 Table 5.1: Summary of single and double floating-point IEEE 754 standard. Single Double Total size, bits 32 64 Exponent size, bits 8 11 Exponent bias 127 1023 Sign, bits 1 1 Significant (explicit), bits 23 52 Decimal precision log 10 (2 24 ) 7 log 10 (2 53 ) 16 Smallest normal number 2 126 2 1022 accommodate twice as many floating point numbers potentially leading to less frequent cache misses. In terms of computational time, single precision gives a twofold speedup on CPUs for most modern architectures and the gains on GPUs can be much larger. Thus, single-precision implementation of quantum-chemistry methods can extend the scope of systems amenable to these treatments, decrease time-to-solution, and reduce energy footprint. Using too much energy and power per calculation is recognized as one of the biggest challenges in high perfor- mance computing and using reduced precision or even entirely different representation of real numbers have been advocated 14 . Although early electronic-structure codes, which were very mindful of effective resource usage, utilized mixed precision, today double precision is a de facto standard in quantum chem- istry and other scientific calculations. However, many algorithms can be re-designed to work in mixed precision, as was recently done in such diverse areas as lattice quantum chromody- namics 15 , molecular dynamics 16 , and general linear algebra algorithms with iterative refine- ments 17 . The renewed interest in single precision computations has been largely driven by potential benefits offered by GPUs 18–24 . In quantum chemistry mixed-precision algorithms have been explored in the context of integral calculations 25, 26 within Hartree–Fock (HF) and density functional theory (DFT) 18–20, 27 . The main conclusion from these studies was that pure single precision is not sufficient for integral and HF/DFT calculations and one should only deploy it in a mixed-precision fashion, i.e., such that some operations are performed in single precision 140 and others in double. The utility of single precision has not yet been thoroughly investigated in post-HF calculations. Previous studies 28–30 focused on non-iterative methods, such as MP2 and triples correction for CCSD. Single-precision MP2 was tested within resolution-of-the-identity (RI) 28 and Cholesky decomposition (CD) 29 schemes. These studies have shown that commonly used RI bases and CD thresholds (10 2 –10 3 ) yield much larger errors in total energies than errors due to using single-precision arithmetic. Within coupled-cluster theory, single-precision algorithms were analyzed within CCD 21 and perturbative triples, (T), correction 30 calculations. GPU implementation of CCD 21 in single precision showed that the numerical error is 10 5 – 10 6 hartree. Numerical analysis of the (T) correction for CCSD 30 has shown that the result is stable with respect to numerical noise, justifying a single-precision implementation. Despite these encouraging findings, the extent of applicability of single precision in many- body theories is not fully understood. There are several open questions: 1. Does numerical error accumulate in iterative procedures such as those used to solve CCSD and EOM equations? 2. What is the impact of using single precision on molecular properties and excited states? 3. Can one reliably compute analytic nuclear gradients and optimize structures within pure single precision? In this paper, we show that using pure single-precision implementation of post-HF methods is sufficient for most quantum-chemistry applications. Thus, one can reap the benefits of reduced costs without invoking more sophisticated mixed-precision algorithms, which can be reserved for more exotic situations when much higher numeric accuracy is desired. 141 As a standard practice in quantum chemistry, typical numerical convergence thresholds are tight enough to not affect the resulting accuracy. For example, Q-Chem’s default CCSD conver- gence criteria are 10 6 hartree for energies and 10 4 for amplitudes c ; Molpro uses 10 6 hartree for energies and 10 5 for amplitudes d . Some packages use a single threshold, for example, GAMESS uses 10 7 for amplitudes e and ORCA uses 10 5 –10 6 hartree for energies. Thus, it appears that 7 decimal digits is sufficient for correlation energy (we note that 10 7 hartree is three orders of magnitude tighter that chemical accuracy). In this paper, we describe a general implementation of CC/EOM-CC methods that allows users to perform calculations in either single or double precision. We implemented both the standard variant and one using CD and RI representation of electron-repulsion integrals. In addition, our implementation allows one to follow up a single-precision calculation with clean- up iterations, in which the full double-precision accuracy can be recovered 21 . The code is based on the libtensor 31 and libxm 32 libraries for many-body electronic structure calculations. The production-level code is implemented in the Q-Chem electronic structure package 33, 34 . 5.2 Algorithms and implementation details Libtensor 31 was developed to provide a high-level interface for tensor operations and to deliver efficient performance. The library has been used to implement a large number of c The metric for the amplitude convergence is the norm of the difference between the amplitudes from the successive iterations, i.e.,jjT (i) T i1 jj. d The actual metric used in Molpro is the square of the norm of the difference between the amplitudes from the successive iterations, with the respective default threshold of 10 10 . e The metric for the amplitude convergence in GAMESS is the absolute value of the largest element of the difference between the amplitudes from the successive iterations, i.e., maxjT (i) T i1 j. 142 CC 2–4 , EOM-CC 6–8, 35 , and ADC (algebraic diagrammatic construction) 36 methods for calcu- lating energies, properties, and nuclear gradients in Q-Chem. Libtensor supports tensor sym- metries, block-sparsity, several contraction algorithms, different BLAS implementations, and several computational backends 32, 37 . We extended the libtensor library to incorporate single- precision operations by generalizing all tensor operations for the general element type (Fig. 5.3 gives an example of the code). The code is available in the original repository on GitHub f . The numerical tests presented in this paper were performed with a developer version of Q-Chem and the modified libtensor, compiled with the GCC-6.4.0 compiler using the ’-O3’ optimization flag and linked against the Intel MKL library (2017.0 version). Fig. 5.1 shows an overview of the CCSD and EOM-CCSD workflows. In our imple- mentation, we compute all integrals and solve HF equations in double precision; the integral transformation step is also performed in double precision. We then convert the tensors (inte- grals) to single precision and perform all tensor operations (contractions, additions, etc.) in the CCSD/EOM-CCSD calculations using single precision. In the RI/CD variants, we also employ the single-precision algorithm at the CCSD step. Once the CCSD equations converge in sin- gle precision, the procedure can switch to double precision to perform clean-up iterations to recover the double-precision result. We follow the same strategy for -amplitudes. We also implemented single-precision calculations of various density matrices needed for property and nuclear gradient calculations. The EOM workflow entails solving the CCSD equations and evaluating the similarity- transformed Hamiltonian ( H) intermediates followed by the computation of EOM energies and amplitudes by iterative diagonalization of the similarity-transformed Hamiltonian using f The templated libtensor code enabling both single- and double-precision calculations is available in the sp branch at https://github.com/epifanovsky/libtensor. 143 the Davidson algorithm. We extended the routines 38 that compute the intermediates and - vectors (products of the similarity-transformed Hamiltonian acting on EOM trial states) for EOM-IP/EA/SF/EE-CCSD to support calculations in single precision. It is expected that using single precision provides a speedup factor of two on CPUs (this estimate does not take into account reduced I/O, which can speed the calculation even further if single-precision tensors fit in RAM, while double-precision tensors do not). If the single- precision calculation is followed up by clean-up iterations to recover double-precision accuracy, the theoretical estimate for the overall speedup is: Theor. speedup = N dp 0:5N sp +N cleanup ; (5.2) whereN dp andN sp are the number of iterations in double and single precisions,N cleanup is the number of clean-up iterations in double precision. 5.3 Computational details Table 5.2: Convergence thresholds for CCSD calculations. Convergence for equations is the same as for theT amplitudes. System E, sp, hartree T , sp E, dp, hartree T , dp Water clusters 10 5 10 3 10 6 10 4 Water dimer (dipole moment run) 10 6 10 4 10 6 10 4 Uracil 10 6 10 4 10 6 10 4 CH 2 O 10 6 10 4 10 6 10 4 C 2 H 4 O 10 6 10 4 10 6 10 4 ATT 10 6 10 4 10 6 10 4 G2 set, pure sp or dp 10 6 10 4 10 6 10 4 G2 set, sp with clean up 10 5 10 3 10 6 10 4 G2 set, sp with energy in dp 10 5 10 3 — — G2 set, pure sp or dp, optimization 10 7 10 5 10 7 10 5 C 6 H 5 N 10 6 10 4 10 6 10 4 benzene 10 10 10 9 10 7 10 5 144 HF, in tegral calculations and trans- formations in double precision Con v ersion of in tegrals and F o c k matrix to single precision CCSD in single precision Con v ersion of amplitudes (and optionally in- tegrals and F o c k matrix) to double precision CCSD in double precision T arget: canonical CCSD in double precision CCSD in single or double precision If not done b efore, con v ert in tegrals and amplitudes to single precision In termediates in single precision EOM in single precision T arget: EOM in single precision Figure 5.1: Left: Implemented CCSD algorithm. Clean-up iterations in double precision are optional. Right: EOM algorithm. Benchmark calculations of water clusters were performed on a Dell server with four 8-core Intel Xeon E5-4640 processors using 4 threads. We used 4 threads to avoid bias for small jobs, which cannot be effectively parallelized using too large a number of threads. All other calcu- lations were performed on Dell servers with four 8-core Intel Xeon E5-4640, two 8-core Intel Xeon E5-2690, or two 10-core Intel Xeon E5-2689 v4 processors with the best settings, i.e., utilizing all cores and using maximum available memory. The benchmark set comprises diverse types of electronic structure, including water clusters of increasing size, the uracil molecule, the 145 formaldehyde molecule, oxygen-ethylene adduct (C 2 H 4 O), a nucleobase trimer (ATT, adenine- thymine-thymine), an aromatic diradical (C 6 H 5 N), the benzene molecule, and the G2 set 39 con- taining 148 molecules. To asses numerical errors in very large systems, we used the taxol molecule (110 atoms, 446 electrons). We tested basis sets of the double-, triple-, quadruple-, and pentuple-zeta quality, with and without diffuse functions, and including polarized-core variants. Frozen core was used for all calculations, except water dimer and formaldehyde calculations with cc-pCVXZ basis sets. All Cartesian geometries are given in Appendix B. The structure of taxol is from Ref. 29. Convergence thresholds for CCSD equations are summarized in Table 5.2. Since single pre- cision provides7 decimal digits of precision, the tightest threshold of convergence for energy in single precision is 10 7 hartree (assuming correlation energies of1 hartree). By numer- ical experimentation we found that single precision iterations converge smoothly with energy threshold of 10 5 –10 6 hartree; therefore, we used this threshold for the single precision part of the calculation in most cases. In the double-precision calculations, we used default convergence except for the benzene benchmark. In properties calculations, we used the same convergence criteria forT and amplitudes. Unrelaxed one-particle density matrices used in dipole moment calculations were computed in double and single precision from respective double and single precision intermediates and amplitudes. In all EOM calculations a 10 5 convergence threshold for EOM amplitudes g was used in the Davidson procedure. For geometry optimization (ben- zene) and finite-difference frequency analysis, much tighter convergence criteria were used: 10 10 for energies, 10 9 forT and amplitudes in double precision; 10 7 for energies, 10 5 for T and amplitudes in single precision. The criteria of convergence in geometry opti- mization were the same for the double and single precision calculations: 2 10 5 a.u. for the maximum component of the gradient 5 10 5 a.u. for the maximum atomic displacement, and g The metric for the EOM amplitude convergence is the norm of the residual vector,jj HR k ! k R k jj, where R k and! k are EOM amplitudes and energies at a given iteration. 146 1 10 7 a.u. for energy. Such tight criteria would reveal how different the “best” geometry from single-precision calculation is from the geometry from the “best” double-precision calculation. For geometry optimization of the systems from G2 set, we used default Q-Chem conver- gence criteria on gradient, displacement and energy change; 10 7 a.u. for energies, 10 5 forT and amplitudes for both precisions, which are the default for double precision. 5.4 Results and discussion 5.4.1 Accuracy of ground-state energies and properties We found that the CC amplitude convergence rate is not affected by using single precision and that the number of CCSD iterations in calculations with single-precision CCSD followed by clean-up iterations is the same as in reference double-precision calculations. The typical output of a single-precision calculation with clean-up iterations is shown in (Fig. 5.4). Table 5.3 shows the results for water clusters. Pure single precision calculations (without clean-up iterations) do not introduce significant numerical errors in total energies, i.e. the typical difference between single and double precision energies is only several J/mol, which is three orders of magnitude below chemical accuracy. Moreover, double-precision numerical accuracy is fully recovered when clean-up iterations are performed. The single-precision calculation is twice as fast than the double-precision one. In calculations with clean-up iterations, observed speedup quickly approaches the theoretical maximum (about 1.6, for these parameters) with the increasing number of water molecules (see Fig. 5.2). For all water clusters only two clean-up iterations were needed to converge calculations within the convergence criteria (Table 5.2). At the first clean-up iteration the change in energy is small, but the net change of amplitudes is large (this is illustrated in Fig. 5.4). If only energy criterion of convergence is used, only one clean- up iteration is necessary in most cases. The explanation of the net change in the amplitudes is 147 simple: switching to double precision changes all amplitudes by a small amount (that is why it is “clean-up”) but because the number of amplitudes is large, the net change is large (this is also confirmed by the increase of this net change with the size of the cluster). The second clean-up iteration does not include change of precision and the calculation quickly converges. We observed similar performance and accuracy for RI/CD-CCSD calculations of these clusters. 0 0.5 1 1.5 2 2 3 4 5 6 7 W all time sp eedup, times W ater cluster size sp+dp theor Figure 5.2: Wall time speedup for SP scheme with clean-up iterations for the CCSD/cc- pVDZ energy calculations of water clusters. Theoretical estimate is given by Eq. (5.2). To test whether numerical errors increase with system size, we compared single and double precision CCSD for an adenine-thymine-thymine system (ATT) and found that the difference between single and double precision total energies is 15 J/mol (again, full double precision accu- racy can be recovered with one or two clean-up iterations). As an example of an even larger system, we carried out RI-MP2/cc-pVDZ calculations of taxol with double-precision ampli- tudes and with the amplitudes converted to single precision. The resulting single- and double- precision energies are -2918.93641306 and -2918.93643602 hartree, respectively. Although the error due to single precision is larger (60 J/mol), it is still well below 1 kJ/mol. The MP2 corre- lation energy is -8.9646559 10 hartree, thus, the tightest convergence criterion by energy for a corresponding single-precision CCSD calculation would be 10 6 hartree. 148 We investigated basis set effects on the error due to single precision by using uracil, formaldehyde, and the water dimer. Table 5.5 presents total energies and dipole moments for all species. The results show that the errors in energies are negligible and that the differences in dipole moments computed in single and double precision are less than the number of digits printed in the output. Increasing the basis set from cc-pVDZ to cc-pVTZ and aug-cc-pVTZ does not increase the errors. The results for formaldehyde and water dimer and for which larger set of basis sets were used (up to cc-pCV5Z) follow the same trend and the magnitude of errors due to single precision does not increase. In order to meaningfully compare the differences between single and double precision, one should keep in mind that in these calculations the energy threshold for CCSD convergence was 10 6 hartree2.6 J/mol. Within this threshold, the error of the calculations with the clean-up iterations is the same as in the single-precision calculation. Interestingly, calculation of double- precision energies from single-precision amplitudes gives larger errors, which can be explained by the convergence criteria used: the amplitudes are underoptimized in comparison with the amplitudes, used for other calculations (shown in Table 5.2). To test the behavior of the single-precision algorithm in strongly correlated systems, we considered ethylene-oxygen adduct, an important intermediate in the reaction of atomic oxy- gen with ethylene. This molecule can be described as an oxyrane ring with one broken C–O bond, which results in strong diradical character. This is a somewhat artificial example, because standard CCSD is expected to perform poorly in strongly correlated systems. These types of electronic structure should be described by, for example, spin-flip 40, 41 or double ionization potential 42 variants of EOM-CC. We performed CCSD calculations for the singlet and triplet states. The triplet solution is well-behaved, e.g.,jjT 1 jj 2 =0.0115 andjjT 2 jj 2 = 0.1724 in the cc- pVQZ basis, and for either precision CCSD converges in 10, 11, 11 iterations with cc-pVDZ, 149 cc-pVTZ, and cc-pVQZ, respectively. As expected, the description of the singlet state is prob- lematic (jjT 1 jj 2 =1.1296 andjjT 2 jj 2 =0.5336 in cc-pVQZ). Such largeT 1 amplitudes are due to open-shell character of the state, which renders a closed-shell determinant to be a poor quality zero-order wave-function. This is a likely cause of the observed slow convergence of CCSD (31-35 iterations forT and 51-56 iterations for ), yet the number of iterations in single and double precision were the same, except for cc-pVQZ where the number of iterations in single and double precision differed by one. Importantly, the errors of pure single precision CCSD total energies are of the same magnitude as in the previous examples. To confirm that these observations hold for other systems, we compared the CCSD total energies computed in single and double precision for the G2 set 39 using the the 6-31G(d), cc- pVDZ, cc-pVTZ, and cc-pVQZ basis sets. In addition, we compared single-precision results with those obtained with clean-up iterations, as well as a calculation in which energy is com- puted in double precision using double-precision integrals and single-precision amplitudes. The results are summarized in Table 5.6. As one can see, the MAD (mean absolute deviation) in single-precision calculations is less than 4 J/mol. The MADs, STDs (standard deviation, unbi- ased estimate), and maximum errors in cc-pVQZ basis are only slightly larger than in cc-pVDZ. Table 5.3: CCSD/cc-pVDZ total energies of water clusters. Differences between single- and double-precision energies are shown in the last column. Cluster size E sp , a.u. E sp+dp , a.u. E dp , a.u. dp;sp , J/mol 2 -152.48710832 -152.48710752 -152.48710767 1.7 3 -228.74911206 -228.74911017 -228.74911059 3.9 4 -305.00868540 -305.00868416 -305.00868472 1.8 5 -381.26858576 -381.26858599 -381.26858585 -0.2 6 -457.51949116 -457.51949130 -457.51949145 -0.8 7 -533.78980571 -533.78980623 -533.78980640 -1.8 150 Table 5.4: CD-CCSD/cc-pVDZ total energies of water clusters. Cholesky threshold of 10 3 was used. Differences between single and double precision energies are shown in the last column. Cluster size E sp , a.u. E sp+dp , a.u. E dp , a.u. dp;sp , J/mol 2 -152.48713650 -152.48713571 -152.48713586 1.7 3 -228.74916915 -228.74916737 -228.74916779 3.6 4 -305.00875903 -305.00875662 -305.00875717 4.9 5 -381.26864546 -381.26864554 -381.26864540 0.2 6 -457.51960841 -457.51960843 -457.51960858 -0.4 7 -533.78994271 -533.78994226 -533.78994243 0.7 5.4.2 Accuracy of target-state energies in EOM-CCSD To test the accuracy of single-precision calculations of excited states, we carried out EOM- EE-CCSD calculations for uracil (Table 5.7) and EOM-SF-CCSD calculations for C 6 H 5 N (Table 5.8). We considered different types of states: singlets and triplets, states with differ- ent symmetry (symmetric and antisymmetric with respect to the symmetry plane), closed-shell and open-shell types, valence and Rydberg states. In all cases, the difference between total energies computed in double and single precision does not exceed 3 J/mol (3 10 5 eV), which is much smaller than the intrinsic error bars of EOM-CCSD (0.1-0.3 eV). This result is compa- rable to the differences in the CCSD total energies. The respective excitation energies are the same within at least 4 decimal places. 5.4.3 Accuracy of gradient evaluation in single precision To test how the choice of precision affects the accuracy of the nuclear gradient and the result- ing optimized structures, we first optimized the benzene molecule at the CCSD/cc-pVDZ level of theory using tight convergence criteria. In these calculations, density matrices were com- puted in the respective precisions; orbital response was computed in double precision. Geome- try optimization was performed by gradient optimization (gradient was evaluated analytically). 151 Table 5.5: CCSD total energies (hartree) and dipole moments (, a.u.) of uracil, water dimer, formaldehyde, and ethylene-oxygen adduct in various basis sets a . Uracil, Basis set E sp , a.u. E dp , a.u. dp;sp , J/mol , sp b , dp c cc-pVDZ -413.72139806 -413.72139812 -0.2 1.550917 1.550917 cc-pVTZ -414.10170402 -414.10170403 0.0 1.643366 1.643366 aug-cc-pVTZ -414.13153216 -414.13153108 2.8 1.689405 1.689405 cc-pVQZ -414.22039427 -414.22039569 -3.7 1.683603 1.683603 (H 2 O) 2 , Basis set d E sp , a.u. E dp , a.u. dp;sp , J/mol , sp b , dp c cc-pCVTZ -152.77124280 -152.77124301 -0.6 0.758536 0.758536 cc-pCVQZ -152.83005083 -152.83005083 0 0.746618 0.746618 cc-pCV5Z -152.84874263 -152.84874291 -0.7 0.740770 0.740770 CH 2 O, Basis set d E sp , a.u. E dp , a.u. dp;sp , J/mol , sp b , dp c cc-pCVTZ -114.42374047 -114.42374037 0.3 0.912793 0.912793 cc-pCVQZ -114.46445277 -114.46445284 -0.2 0.946294 0.946294 cc-pCV5Z -114.47693774 -114.47693771 0.1 0.961034 0.961035 C 2 H 4 O (T 1 ), Basis set E sp , a.u. E dp , a.u. dp;sp , J/mol , sp b , dp c cc-pVDZ -153.28012439 -153.28012442 -0.1 0.713475 0.713475 cc-pVTZ -153.42725145 -153.42725167 -0.6 0.752959 0.752959 cc-pVQZ -153.47046520 -153.47046539 -0.5 0.778434 0.778434 C 2 H 4 O (S 0 ), Basis set E sp , a.u. E dp , a.u. dp;sp , J/mol , sp b , dp c cc-pVDZ -153.24187533 -153.24187486 1.2 0.930752 0.930749 cc-pVTZ -153.39033087 -153.39033115 -0.7 1.155325 1.155318 cc-pVQZ -153.43514375 -153.43514491 -3.0 1.272352 1.272339 a Convergence thresholds for single and double precision are the same (see Table 5.2). b Equations for T and equations are solved in single precision; intermediates and unrelaxed density matrices are evaluated in single precision. c Equations for T and equations are solved in double precision; intermediates and unrelaxed density matrices are evaluated in double precision. d All electron were active in the cc-pCVTZ, cc-pCVQZ, and cc-pCV5Z calculations. Table 5.6: Mean average deviation (MAD) and standard deviation (STD), J/mol, from reference double-precision CCSD total energies for the G2 set. Basis sp sp with dp energy sp with cleanup MAD STD MAD STD MAD STD 6-31G(d) 0.12 0.2 3.7 6.6 1.5 1.4 cc-pVDZ 0.15 0.2 3.6 6.7 1.4 1.4 cc-pVTZ 0.30 0.5 3.4 5.3 1.5 1.4 cc-pVQZ 0.68 0.9 3.7 5.0 1.6 1.5 152 Table 5.7: EOM-EE-CCSD total energies (in a.u.) of excited singlet and triplet states of uracil in various basis sets. In each cell the first number is obtained from double-precision calculation, the second number is obtained from single-precision EOM calculation from double-precision reference, and the third number is single-precision EOM from single- precision reference. The fourth number is the difference between double- and single- precision energies (with double-precision reference). Singlets Basis set 1A 0 2A 0 1A 00 2A 00 cc-pVDZ -413.50624322 -413.50624315 -413.50624328 7 10 8 -413.46584035 -413.46584026 -413.46584041 9 10 8 -413.52849406 -413.52849409 -413.52849406 3 10 8 -413.47629005 -413.47628997 -413.47629005 8 10 8 cc-pVTZ -413.89150482 -413.89150498 -413.89150563 1:6 10 7 -413.84840838 -413.84840792 -413.84840854 4:6 10 7 -413.90876266 -413.90876251 -413.90876290 1:5 10 7 -413.85792039 -413.85792058 -413.85792110 1:9 10 7 aug-cc- pVTZ -413.92599740 -413.92599743 -413.92599918 3 10 8 -413.88373275 -413.88373272 -413.88373397 3 10 8 -413.94000534 -413.94000543 -413.94000635 9 10 8 -413.90496469 -413.90496414 -413.90496581 5:5 10 7 Triplets Basis set 1A 0 2A 0 1A 00 2A 00 cc-pVDZ -413.57714822 -413.57714817 -413.57714827 5 10 8 -413.51667861 -413.51667858 -413.51667865 3 10 8 -413.53936723 -413.53936718 -413.53936732 5 10 8 -413.48574474 -413.48574485 -413.48574489 1:1 10 7 cc-pVTZ -413.95932894 -413.95932869 -413.95932945 2:5 10 7 -413.89779284 -413.89779281 -413.89779339 3 10 8 -413.91902233 -413.91902181 -413.91902247 5:2 10 7 -413.86657196 -413.86657206 -413.86657255 1:0 10 7 aug-cc- pVTZ -413.99008713 -413.99008666 -413.99008779 4:7 10 7 -413.92935022 -413.92935024 -413.92935158 2 10 8 -413.94969790 -413.94969820 -413.94969912 3:0 10 7 -413.90754309 -413.90754252 -413.90754422 5:7 10 7 Starting from the same initial geometry (MP2/cc-pVDZ optimized structure), both optimiza- tions converged in 6 iterations (single precision run converged by gradient and displacement). The resulting geometries (given in Appendix B) are nearly identical, with mean absolute error (computed for non-zero Cartesian coordinates) of 1:9 10 9 ˚ A. 153 Table 5.8: EOM-SF-CCSD total energies (in a.u.) of several electronic states of C 6 H 5 N in various basis sets. Symmetry labels refer to state symmetries. In each cell the first number is from double-precision calculation, the second number is from single-precision EOM calculation, and the third number is the difference between double- and single-precision energies. CCSD equations were solved in double precision in all cases. Basis set 1A 2 2A 2 1A 1 2A 1 cc-pVDZ -285.46570954 -285.46570956 2 10 8 -285.42658952 -285.42658954 2 10 8 -285.40655223 -285.40655224 1 10 8 -285.37250140 -285.37250139 1 10 9 cc-pVTZ -285.79837285 -285.79837303 1:8 10 7 -285.76114321 -285.76114343 2:2 10 7 -285.74329035 -285.74328950 8:5 10 7 -285.70728705 -285.70728629 7:6 10 7 To confirm this result for a larger set of systems, we carried out geometry optimizations the G2 set with cc-pVTZ in both precisions. The set includes 712 bonds. The bond lengths were extracted from the Cartesian geometries using the Openbabel package 43 . The errors with respect to double precision are small: MAD is 6:7 10 5 ˚ A, STD is 2:1 10 4 ˚ A, and maximal absolute deviation is 2:5 10 3 ˚ A. Although the standard deviation is larger in comparison with the benzene result, we note that the shape of statistical distribution is very narrow with a sharp spike at zero. An error interval (-10 7 ˚ A, 10 7 ˚ A) contains about a half of the total number of bonds, explaining discrepancy with standard deviation. To put these differences in the context, we note that the CCSD/cc-pVTZ MAD relative to the experimental bond lengths is about 0.64 pm=0.006 ˚ A 1 , which is by two orders of magnitude larger than the MAD due to using single precision. 5.4.4 Accuracy of finite-difference frequencies To investigate whether finite-difference calculations of frequencies (using analytic gradi- ents) can be carried out in single precision, we computed the frequencies and normal modes for benzene at the respective optimized geometries with the same convergence criteria as used for geometry optimizations with the step size of 0.001 ˚ A. The resulting frequencies (given in 154 Appendix C) are very different: real and imaginary frequencies of15,000 cm 1 occur along with several imaginary frequencies. Thus, not surprisingly, single precision can cause problems in finite-difference calculations. However, finite-difference calculations using single-precision calculation with the double-precision clean-up iterations fully recovers double precision fre- quencies, while affording1.3 speedup of the calculation. We note that finite difference derivative evaluation is always a balance between the size of the step and the accuracy due to finite numerical precision. The default step size of 0.001 ˚ A requires energies converged to 10 9 or so. Thus, an erratic behavior of single-precision finite- difference calculations can be remedied by using a larger step size. The calculation with 0.01 ˚ A displacements yielded almost identical frequencies in double and single precision with maxi- mal difference in frequencies of 0.64 cm 1 . The MAD between double precision frequencies computed with 0.001 and 0.01 ˚ A steps is 0.21 cm 1 . 5.5 Conclusion In this contribution, we report single- and mixed-precision implementation of the CCSD and EOM-CCSD energies, analytic gradients, and properties. Using single precision results in reduced memory footprint, considerable computation speed-up, and reduced energy-to-solution. That is, single precision calculations use half the memory (or disk) space and produce a speedup factor of 2 on CPUs. The main focus of this paper was on assessing the impact on accuracy of the resulting energies and properties in post-HF calculations, in particular in iterative schemes. The results show that the rate of error accumulation in single precision is sufficiently slow, and overall single precision introduces negligible errors for total energies, excitation energies, forces, and properties. Moreover, single precision can be used in geometry optimizations with analytical gradients. Thus, we conclude that for most type of calculations, straight single- precision implementation of post-HF methods can be used and one can reap the benefits of 155 reduced costs without invoking more sophisticated mixed-precision algorithms, which can be reserved for more exotic situations when much higher numeric accuracy is desired. If tight convergence is desired (e.g. in finite-difference calculations), single precision can be used to speedup iterations in the beginning, converging to the single-precision result first and continuing in double precision. In most cases the total number of iterations is not affected and the speedup is close to the theoretical estimate. 5.6 Appendix A: Implementation details template<size_t N, typename T> class bto_copy : public additive_gen_bto<N, typename bto_traits<T>::bti_traits>, public noncopyable { // .... // Declaration of the templated class } template<size_t N> using btod_copy = bto_copy<N, double>; Figure 5.3: Example of the change in the interface. In this code a block-tensor operation over double type (“btod”) is generalized to a block-tensor operation over the template type. 156 Running a single precision version CCSD T amplitudes will be solved using DIIS. Start Size MaxIter EConv TConv 3 7 100 1.00e-05 1.00e-03 ----------------------------------------------------------------- Energy (a.u.) Ediff Tdiff Comment ----------------------------------------------------------------- -533.73456142 1 -533.75538383 2.08e-02 1.05e+00 2 -533.78203182 2.66e-02 1.12e-01 3 -533.78610147 4.07e-03 3.66e-02 4 -533.78905248 2.95e-03 1.32e-02 Switched to DIIS 5 -533.78979316 7.41e-04 6.75e-03 6 -533.78980959 1.64e-05 9.56e-04 7 -533.78980583 3.76e-06 3.78e-04 ----------------------------------------------------------------- -533.78980583 CCSD T converged. Running a double precision version (after sp) CCSD T amplitudes will be solved using DIIS. Start Size MaxIter EConv TConv 3 7 100 1.00e-06 1.00e-04 ----------------------------------------------------------------- Energy (a.u.) Ediff Tdiff Comment ----------------------------------------------------------------- -533.78980618 1 -533.78980625 6.86e-08 1.28e+00 2 -533.78980623 1.93e-08 4.31e-05 ----------------------------------------------------------------- -533.78980623 CCSD T converged. Figure 5.4: Example of the output, produced by CCSD code. The first iteration in double precision provides a clean-up, which is illustrated by a large net change in amplitudes. Changes of individual amplitudes are small, and the procedure converges at the second iteration. 157 5.7 Appendix B: Relevant Cartesian geometries The Cartesian geometries (in ˚ A) used in the benchmark calculations are given below. Figure 5.5: ATT system a , used for CCSD benchmark. a The structure was obtained by removing one adenine molecule from the AATT structure from Ref. 44 $comment Water dimer. Nuclear Repulsion Energy = 36.5381789890 hartrees Optimized with MP2/6-31G * This structure was also used for calculations with cc-pCXVZ basis sets $end $molecule 0 1 O -1.5167088799 -0.0875022822 0.0744338901 H -0.5688047242 0.0676402012 -0.0936613229 H -1.9654552961 0.5753254158 -0.4692384530 O 1.3898685804 0.0960995460 -0.0761488482 H 1.5926924704 -0.8335878302 -0.2679884752 H 1.5164596797 0.1745974125 0.8831816344 $end $comment Water trimer. Nuclear Repulsion Energy = 84.7431618876 hartrees Optimized with MP2/6-31G * $end 158 $molecule 0 1 O -1.4765014766 -0.6332885087 0.0898827346 H -1.9838499390 -0.7470663319 -0.7281466521 H -1.1474120728 0.2937933586 0.0499611499 O 1.3009060824 -0.9725326095 0.1123300788 H 1.6231813194 -1.4263451978 -0.6812347553 H 0.3383752926 -1.1745452783 0.1460364674 O 0.2066022512 1.5796259613 -0.1194070925 H 0.8505520801 0.8353634930 -0.0913819530 H 0.3749571446 2.0763034073 0.6962042911 $end $comment Water tetramer. Nuclear Repulsion Energy = 140.8533786467 hartrees Optimized with MP2/6-31G * $end $molecule 0 1 O -1.9292712102 -0.0645569159 0.1117679206 H -2.4914119096 0.0073852098 -0.6743533999 H -1.3279797920 0.7200450587 0.0572351830 O 0.0556775311 -1.9834041949 0.1251954700 H -0.7452932853 -1.4024421984 0.1534116762 H -0.0578054438 -2.5263517257 -0.6694026790 O -0.0284527051 1.9145766189 -0.1269877741 H 0.0826356714 2.4643236809 0.6632720821 H 2.4967666426 -0.0808647657 0.7228603336 O 1.9591510378 -0.0026698519 -0.0796304864 H 0.7727195176 1.3336367194 -0.1473249006 H 1.3542775597 -0.7856500486 -0.0462427112 $end $comment Cluster of 5 water molecules. Nuclear Repulsion Energy = 220.0747025347 hartrees Optimized with MP2/6-31G * $end $molecule 0 1 O -0.2876645445 -1.7012437583 0.2029164243 H -0.5715171913 -2.6278936014 0.2177698265 159 H -1.1211952210 -1.1567502352 0.1410434902 O 1.4494811981 -0.1832356135 -1.3664308071 H 0.8495877249 -0.8784465376 -1.0148018507 H 1.7885868060 -0.5222367118 -2.2083896644 O -2.3074119608 0.0600951514 -0.0549593403 H -2.8203525262 0.2892204097 0.7353401875 H 0.2626613561 1.4965603247 0.5922725167 O -0.2424726136 1.8126064878 -0.1909073151 H -1.6640497911 0.8139612893 -0.1634430169 H 0.2878907915 1.4335334111 -0.9188689235 O 1.3813546468 0.2415962603 1.5136093974 H 0.8356415445 -0.5611886184 1.3988328403 H 1.9779314005 0.1892600736 0.7452520171 $end $comment Cluster of 6 water molecules. Nuclear Repulsion Energy = 285.4680301551 hartrees Optimized with MP2/6-31G * $end $molecule 0 1 O -1.0056893157 -0.1043674637 1.7352314342 H -1.0664276447 -0.3762387797 2.6634016204 H -1.5071027077 -0.7913788396 1.2185432381 O 1.1470564813 0.3117796882 -0.0477367962 H 0.6103606068 0.0433421321 0.7275681882 H 2.0791424333 0.1211528401 0.1794662548 O -2.2063795158 -1.8574899369 0.0516378330 H -3.1585603577 -1.7418528352 -0.0901912407 H -1.2703988826 0.5556485786 -1.4822595530 O -0.9686544539 -0.3213451857 -1.8092615790 H -1.7829183163 -1.4200976013 -0.7381133801 H -0.0506196244 -0.3301023162 -1.4666019324 O -1.1414038621 2.0193691143 -0.2156011398 H -1.3619510638 1.5449391020 0.6091355621 H -0.1726333256 1.9018225581 -0.2491925658 O 3.9130618957 -0.1477904028 0.5773094099 H 4.3921274685 0.6778964502 0.4038875571 H 4.3084563274 -0.7931325208 -0.0300166098 $end $comment Cluster of 7 water molecules. 160 Nuclear Repulsion Energy = 396.5486015625 hartrees Optimized with MP2/6-31G * $end $molecule 0 1 O -0.2578815727 0.8589987101 1.0903178485 H 0.5724321179 0.9957545630 1.6134194631 H -0.3379575048 -0.1213199214 1.0257127358 O 0.7053437993 1.0245432710 -1.4943936709 H 0.3698578944 1.1189598601 -0.5656693305 H 1.0352513822 1.9002918475 -1.7483588311 O -0.3458650799 -1.7904919129 0.2521395252 H -0.8067013063 -2.5793300144 0.5772833215 H -2.2204485019 0.0514045678 -1.2281883065 O -1.6470360380 -0.5567512801 -1.7481190861 H -0.8635215557 -1.4793338112 -0.5471909111 H -0.9032735713 0.0137100383 -2.0286082118 O -2.8270840011 1.2361544476 0.0947724694 H -2.0207774964 1.2912094835 0.6512402118 H -3.0198162448 2.1508831339 -0.1596488039 O 2.3155522081 0.8259479989 1.9468048976 H 2.6243054144 0.3877453957 2.7538641108 H 2.4860312413 0.1745463055 1.2179450536 O 2.3555214451 -0.9273649757 -0.1787945084 H 1.5622606691 -1.4797809021 -0.0212223033 H 2.0415014395 -0.3067984167 -0.8674027785 $end $comment C$_6$H$_5$N diradical. Nuclear Repulsion Energy = 245.9819800369 hartrees $end $molecule 0 3 C 0.0000000007 0.0000000000 1.1804212019 C 1.2065229753 0.0000000000 0.3467052071 C 1.1981069765 0.0000000000 -1.1011558332 C -0.0000000015 -0.0000000000 -1.8337538347 C -1.1981069770 -0.0000000000 -1.1011558292 C -1.2065229768 -0.0000000000 0.3467052112 H -2.1662140349 -0.0000000000 0.8762281317 H -2.1574219857 -0.0000000000 -1.6342787851 H 0.0000000000 0.0000000000 -2.9251948347 161 H 2.1574219856 0.0000000000 -1.6342787885 H 2.1662140349 0.0000000000 0.8762281317 N 0.0000000000 0.0000000000 2.4878142019 $end $comment Uracil. Nuclear Repulsion Energy = 356.7964376322 hartrees The structure of uracil molecule was optimized with wB97X-D/6-31G * . $end $molecule 0 1 C 1.1965439001 1.1059975965 0.0000000000 C -0.0106406622 1.6974586473 0.0000000000 N -1.1759657685 0.9706327770 0.0000000000 C 1.2905520182 -0.3511811123 0.0000000000 N 0.0394261039 -0.9922273002 0.0000000000 C -1.2164849902 -0.4175249683 0.0000000000 O -2.2534478773 -1.0446838400 0.0000000000 O 2.3153668717 -1.0016569295 0.0000000000 H 2.1145824324 1.6760396163 0.0000000000 H -0.1369773173 2.7740930054 0.0000000000 H -2.0769371453 1.4242304202 0.0000000000 H 0.0555272212 -2.0045027192 0.0000000000 $end $comment ATT complex. Nuclear Repulsion Energy = 2683.0524561513 hartrees $end $molecule 0 1 N 4.648954 0.062237 -2.370046 C 5.222661 1.044161 -1.595124 N 4.354894 1.743856 -0.892815 C 3.124382 1.191641 -1.234691 C 1.796538 1.473710 -0.820053 N 1.477058 2.412334 0.063341 N 0.802690 0.730580 -1.347517 C 1.118897 -0.233683 -2.222372 N 2.315056 -0.597666 -2.682467 C 3.287489 0.159535 -2.145551 N -4.119880 1.175240 -1.668947 162 C -2.788993 0.852210 -1.845154 O -2.385787 0.134394 -2.748531 N -1.937862 1.401121 -0.916872 C -2.275607 2.205332 0.157303 O -1.389232 2.644961 0.899530 C -3.691851 2.470201 0.309209 C -4.156175 3.291113 1.475440 C -4.531960 1.955432 -0.607897 N -3.093123 -2.936903 -1.004080 C -1.789262 -2.710900 -0.630457 O -0.840457 -3.167912 -1.243704 N -1.636359 -1.922963 0.479694 C -2.636971 -1.336214 1.225506 O -2.346142 -0.645842 2.195441 C -3.987755 -1.598884 0.766076 C -5.137909 -0.995328 1.511021 C -4.143924 -2.378097 -0.316323 H 6.291440 1.207255 -1.586397 H 2.216641 2.881413 0.561742 H 0.519215 2.485870 0.406134 H 0.269720 -0.796087 -2.601331 H -0.930607 1.148159 -1.047409 H -5.240856 3.424662 1.455065 H -3.680063 4.275568 1.470584 H -3.878536 2.804296 2.415249 H -5.602410 2.125849 -0.555843 H -0.639080 -1.781127 0.770889 H -6.090057 -1.247239 1.035981 H -5.040838 0.094030 1.551165 H -5.161876 -1.348358 2.546381 H -5.126930 -2.611972 -0.712652 H 5.101053 -0.601273 -2.978111 H -4.769876 0.803247 -2.342853 H -3.229023 -3.459136 -1.855586 $end $comment Formaldehyde. Optimized with MP2/cc-pVDZ Nuclear Repulsion Energy = 30.9973671953 hartrees $end $molecule 0 1 C -0.2581670178 0.0631333004 0.0000000000 163 O 0.9398801946 -0.1450331544 0.0000000000 H -0.8478178295 0.1654099608 -0.9426389633 H -0.8478178295 0.1654099608 0.9426389633 $end $comment C2H4O system Triplet is optimized with CCSD/6-31G * Nuclear Repulsion Energy = 67.3565977393 hartrees $end $molecule 0 3 C 0.1068645036 0.4873547877 -0.0717777121 H 0.1745652108 1.3412829765 0.6229328603 H 0.2634473223 0.9195625491 -1.0887529280 O 1.1938322951 -0.3703483305 0.0699496107 C -1.2001021873 -0.2199777358 0.0301457864 H -1.3054161399 -1.2071535294 -0.4076817671 H -2.0616826952 0.2545694374 0.4886865543 $end $comment Benzene structure optimized in double-precision CCSD calculation. Nuclear Repulsion Energy = 201.7916204645 hartrees $end $molecule 0 1 C 1.4059535336 0.0000000000 0.0000000000 C 0.7029767668 1.2175914766 0.0000000000 C 0.7029767668 -1.2175914766 0.0000000000 C -0.7029767668 1.2175914766 0.0000000000 C -0.7029767668 -1.2175914766 0.0000000000 C -1.4059535336 0.0000000000 0.0000000000 H 2.5018761059 0.0000000000 0.0000000000 H 1.2509380530 2.1666882648 0.0000000000 H 1.2509380530 -2.1666882648 0.0000000000 H -1.2509380530 2.1666882648 0.0000000000 H -1.2509380530 -2.1666882648 0.0000000000 H -2.5018761059 0.0000000000 0.0000000000 $end $comment Benzene structure optimized in single-precision CCSD calculation. 164 Nuclear Repulsion Energy = 201.7916207506 hartrees $end $molecule 0 1 C 1.4059535336 0.0000000000 0.0000000000 C 0.7029767633 1.2175914744 0.0000000000 C 0.7029767633 -1.2175914744 0.0000000000 C -0.7029767633 1.2175914744 0.0000000000 C -0.7029767633 -1.2175914744 0.0000000000 C -1.4059535336 0.0000000000 0.0000000000 H 2.5018761033 0.0000000000 0.0000000000 H 1.2509380519 2.1666882634 0.0000000000 H 1.2509380519 -2.1666882634 0.0000000000 H -1.2509380519 2.1666882634 0.0000000000 H -1.2509380519 -2.1666882634 0.0000000000 H -2.5018761033 0.0000000000 0.0000000000 $end 165 5.8 Appendix C: Frequencies and normal modes for opti- mized structures computed with single and double pre- cision 5.8.1 Frequencies and normal modes of benzene, computed by finite dif- ference with double precision amplitudes and step size of 0.001 ˚ A Mode: 1 2 3 Frequency: 404.62 404.62 611.93 Force Cnst: 0.2891 0.2891 1.3307 Red. Mass: 2.9967 2.9967 6.0316 IR Active: YES YES YES IR Intens: 0.000 0.000 0.000 Raman Active: YES YES YES X Y Z X Y Z X Y Z C 0.000 -0.000 0.246 -0.000 -0.000 -0.000 -0.360 0.000 -0.000 C -0.000 0.000 -0.123 -0.000 -0.000 -0.212 -0.024 0.222 0.000 C 0.000 0.000 -0.122 0.000 -0.000 0.213 -0.024 -0.221 0.000 C 0.000 -0.000 -0.122 -0.000 0.000 0.213 0.024 0.221 -0.000 C -0.000 0.000 -0.123 0.000 -0.000 -0.212 0.024 -0.222 -0.000 C -0.000 -0.000 0.246 0.000 0.000 -0.000 0.360 -0.000 0.000 H 0.000 -0.000 0.523 -0.000 -0.000 -0.001 -0.357 -0.000 -0.000 H -0.000 0.000 -0.262 -0.000 -0.000 -0.452 0.263 0.055 0.000 H -0.000 0.000 -0.260 -0.000 -0.000 0.453 0.263 -0.054 0.000 H 0.000 -0.000 -0.260 -0.000 0.000 0.453 -0.263 0.054 -0.000 H -0.000 -0.000 -0.262 0.000 -0.000 -0.452 -0.263 -0.055 -0.000 H -0.000 -0.000 0.523 0.000 -0.000 -0.001 0.357 0.000 -0.000 TransDip -0.000 -0.000 -0.000 -0.000 -0.000 0.000 -0.000 0.000 -0.000 Mode: 4 5 6 Frequency: 611.94 654.18 691.23 Force Cnst: 1.3307 1.4174 0.3054 Red. Mass: 6.0315 5.6214 1.0848 IR Active: YES YES YES IR Intens: 0.000 0.000 88.372 Raman Active: YES YES YES X Y Z X Y Z X Y Z C 0.000 0.152 -0.000 -0.000 -0.000 0.264 -0.000 0.000 -0.034 C 0.221 0.232 -0.000 -0.000 0.000 -0.264 0.000 -0.000 -0.034 C -0.221 0.232 0.000 0.000 -0.000 -0.264 -0.000 -0.000 -0.034 C 0.221 -0.232 0.000 -0.000 0.000 0.264 -0.000 0.000 -0.034 C -0.221 -0.232 -0.000 0.000 -0.000 0.264 0.000 0.000 -0.034 C -0.000 -0.152 -0.000 0.000 0.000 -0.264 0.000 0.000 -0.034 H 0.000 -0.231 0.000 -0.000 0.000 0.311 -0.000 0.000 0.407 H 0.054 0.326 0.000 0.000 -0.000 -0.311 0.000 -0.000 0.407 H -0.055 0.326 0.000 0.000 -0.000 -0.311 0.000 -0.000 0.407 H 0.055 -0.326 0.000 -0.000 0.000 0.311 -0.000 0.000 0.407 H -0.054 -0.326 0.000 -0.000 -0.000 0.311 0.000 0.000 0.407 H -0.000 0.231 0.000 0.000 -0.000 -0.311 0.000 -0.000 0.407 TransDip 0.000 0.000 0.000 -0.000 -0.000 -0.000 0.000 0.000 0.301 166 Mode: 7 8 9 Frequency: 867.75 867.75 973.24 Force Cnst: 0.5537 0.5537 0.7523 Red. Mass: 1.2480 1.2480 1.3479 IR Active: YES YES YES IR Intens: 0.000 0.000 0.000 Raman Active: YES YES YES X Y Z X Y Z X Y Z C -0.000 -0.000 0.000 -0.000 -0.000 0.085 0.000 -0.000 0.000 C 0.000 0.000 -0.074 0.000 0.000 0.043 -0.000 -0.000 0.088 C -0.000 0.000 0.074 0.000 0.000 0.043 -0.000 0.000 -0.088 C 0.000 -0.000 -0.074 -0.000 -0.000 -0.043 -0.000 0.000 -0.088 C 0.000 -0.000 0.074 -0.000 -0.000 -0.043 -0.000 -0.000 0.088 C 0.000 -0.000 -0.000 0.000 0.000 -0.085 0.000 0.000 0.000 H -0.000 -0.000 -0.001 -0.000 -0.000 -0.571 0.000 -0.000 -0.000 H 0.000 0.000 0.494 0.000 0.000 -0.286 -0.000 -0.000 -0.492 H -0.000 0.000 -0.495 0.000 0.000 -0.285 -0.000 0.000 0.492 H -0.000 -0.000 0.495 -0.000 -0.000 0.285 -0.000 0.000 0.492 H -0.000 0.000 -0.494 -0.000 -0.000 0.286 -0.000 -0.000 -0.492 H 0.000 0.000 0.001 0.000 0.000 0.571 0.000 0.000 -0.000 TransDip -0.000 -0.000 -0.000 0.000 0.000 0.000 -0.000 -0.000 0.000 Mode: 10 11 12 Frequency: 973.24 977.87 1007.88 Force Cnst: 0.7523 0.6276 3.9446 Red. Mass: 1.3479 1.1140 6.5907 IR Active: YES YES YES IR Intens: 0.000 0.000 0.000 Raman Active: YES YES YES X Y Z X Y Z X Y Z C -0.000 -0.000 -0.102 0.000 -0.000 0.040 0.291 0.000 -0.000 C -0.000 0.000 0.051 -0.000 0.000 -0.040 -0.145 -0.252 -0.000 C 0.000 0.000 0.051 0.000 -0.000 -0.040 -0.145 0.252 0.000 C 0.000 0.000 0.051 0.000 -0.000 0.040 -0.145 0.252 0.000 C -0.000 0.000 0.051 0.000 -0.000 0.040 -0.145 -0.252 -0.000 C -0.000 -0.000 -0.102 -0.000 0.000 -0.040 0.291 0.000 -0.000 H -0.000 -0.000 0.568 0.000 -0.000 -0.406 0.286 0.000 0.000 H -0.000 0.000 -0.285 -0.000 0.000 0.406 -0.143 -0.248 0.000 H 0.000 0.000 -0.284 0.000 0.000 0.406 -0.143 0.248 -0.000 H 0.000 0.000 -0.284 0.000 0.000 -0.406 -0.143 0.248 -0.000 H -0.000 0.000 -0.285 0.000 -0.000 -0.406 -0.143 -0.248 0.000 H -0.000 -0.000 0.568 -0.000 0.000 0.406 0.286 0.000 0.000 TransDip -0.000 -0.000 -0.000 -0.000 0.000 0.000 0.000 0.000 -0.000 Mode: 13 14 15 Frequency: 1022.99 1065.79 1065.80 Force Cnst: 3.6743 1.0983 1.0983 Red. Mass: 5.9591 1.6411 1.6411 IR Active: YES YES YES IR Intens: 0.000 3.561 3.566 Raman Active: YES YES YES X Y Z X Y Z X Y Z C -0.274 -0.000 0.000 0.114 -0.000 0.000 0.001 0.079 0.000 C -0.137 -0.237 0.000 -0.030 0.084 -0.000 -0.084 -0.065 0.000 C -0.137 0.237 -0.000 -0.031 -0.083 -0.000 0.083 -0.066 -0.000 C 0.137 -0.237 0.000 -0.031 -0.083 0.000 0.083 -0.066 -0.000 C 0.137 0.237 0.000 -0.030 0.084 -0.000 -0.084 -0.065 0.000 C 0.274 -0.000 -0.000 0.114 -0.000 -0.000 0.001 0.079 0.000 H -0.303 0.000 -0.000 0.125 -0.003 -0.000 0.001 0.546 -0.000 H -0.151 -0.262 -0.000 -0.377 0.290 0.000 -0.292 0.045 -0.000 H -0.151 0.262 0.000 -0.380 -0.291 0.000 0.289 0.042 0.000 H 0.151 -0.262 -0.000 -0.380 -0.291 -0.000 0.289 0.042 0.000 H 0.151 0.262 -0.000 -0.377 0.290 -0.000 -0.292 0.045 -0.000 167 H 0.303 -0.000 0.000 0.125 -0.003 0.000 0.001 0.546 -0.000 TransDip -0.000 -0.000 -0.000 -0.060 -0.000 -0.000 -0.000 0.060 0.000 Mode: 16 17 18 Frequency: 1157.22 1195.22 1195.23 Force Cnst: 0.9203 0.9466 0.9466 Red. Mass: 1.1664 1.1247 1.1247 IR Active: YES YES YES IR Intens: 0.000 0.000 0.000 Raman Active: YES YES YES X Y Z X Y Z X Y Z C -0.000 0.049 -0.000 -0.006 0.000 0.000 -0.000 -0.059 -0.000 C 0.042 -0.025 0.000 -0.043 0.028 -0.000 -0.028 0.011 -0.000 C -0.042 -0.025 0.000 -0.043 -0.028 0.000 0.028 0.011 0.000 C -0.042 -0.025 0.000 0.043 0.028 -0.000 -0.028 -0.011 0.000 C 0.042 -0.025 0.000 0.043 -0.028 -0.000 0.028 -0.011 -0.000 C -0.000 0.049 -0.000 0.006 -0.000 -0.000 0.000 0.059 0.000 H -0.000 0.405 0.000 -0.005 0.001 0.000 -0.000 -0.574 0.000 H 0.351 -0.203 -0.000 -0.429 0.250 0.000 -0.251 0.140 0.000 H -0.351 -0.203 -0.000 -0.430 -0.251 -0.000 0.250 0.140 -0.000 H -0.351 -0.203 -0.000 0.430 0.251 0.000 -0.250 -0.140 -0.000 H 0.351 -0.203 -0.000 0.429 -0.250 0.000 0.251 -0.140 0.000 H -0.000 0.405 0.000 0.005 -0.001 -0.000 0.000 0.574 -0.000 TransDip 0.000 -0.000 -0.000 -0.000 0.000 0.000 0.000 -0.000 0.000 Mode: 19 20 21 Frequency: 1318.66 1371.46 1521.78 Force Cnst: 4.6952 1.3830 2.8642 Red. Mass: 4.5829 1.2480 2.0992 IR Active: YES YES YES IR Intens: 0.000 0.000 7.303 Raman Active: YES YES YES X Y Z X Y Z X Y Z C 0.000 0.233 -0.000 -0.000 0.060 0.000 0.098 0.000 0.000 C 0.202 -0.116 0.000 -0.052 0.030 0.000 -0.090 0.109 0.000 C -0.202 -0.116 0.000 0.052 0.030 -0.000 -0.090 -0.109 0.000 C -0.202 -0.116 0.000 -0.052 -0.030 0.000 -0.090 -0.109 -0.000 C 0.202 -0.116 0.000 0.052 -0.030 0.000 -0.090 0.109 -0.000 C 0.000 0.233 -0.000 0.000 -0.060 -0.000 0.098 0.000 0.000 H 0.000 -0.335 0.000 -0.000 -0.404 -0.000 0.122 -0.000 -0.000 H -0.290 0.168 -0.000 0.350 -0.202 -0.000 0.431 -0.179 -0.000 H 0.290 0.168 -0.000 -0.350 -0.202 -0.000 0.431 0.178 -0.000 H 0.290 0.168 -0.000 0.350 0.202 0.000 0.431 0.178 0.000 H -0.290 0.168 -0.000 -0.350 0.202 0.000 0.431 -0.179 0.000 H 0.000 -0.335 0.000 0.000 0.404 0.000 0.122 -0.000 -0.000 TransDip -0.000 -0.000 -0.000 -0.000 -0.000 0.000 0.087 -0.000 0.000 Mode: 22 23 24 Frequency: 1521.79 1672.03 1672.04 Force Cnst: 2.8642 9.2999 9.3002 Red. Mass: 2.0992 5.6459 5.6461 IR Active: YES YES YES IR Intens: 7.293 0.000 0.000 Raman Active: YES YES YES X Y Z X Y Z X Y Z C 0.000 -0.153 0.000 0.000 -0.343 -0.000 0.152 0.000 0.000 C 0.109 0.035 0.000 -0.083 0.200 -0.000 -0.295 0.083 -0.000 C -0.109 0.035 -0.000 0.083 0.200 -0.000 -0.295 -0.083 0.000 C -0.109 0.035 0.000 -0.083 -0.200 0.000 0.295 0.083 -0.000 C 0.109 0.035 -0.000 0.083 -0.200 -0.000 0.295 -0.083 -0.000 C 0.000 -0.153 0.000 -0.000 0.343 0.000 -0.152 -0.000 -0.000 H 0.000 0.534 -0.000 0.000 0.402 0.000 0.175 -0.000 0.000 H -0.178 0.225 -0.000 0.250 0.031 0.000 0.258 -0.250 0.000 168 H 0.179 0.225 0.000 -0.250 0.031 -0.000 0.258 0.250 -0.000 H 0.179 0.225 -0.000 0.250 -0.031 -0.000 -0.258 -0.250 0.000 H -0.178 0.225 0.000 -0.250 -0.031 0.000 -0.258 0.250 0.000 H 0.000 0.534 -0.000 -0.000 -0.402 0.000 -0.175 0.000 0.000 TransDip 0.000 0.086 -0.000 0.000 -0.000 0.000 -0.000 0.000 0.000 Mode: 25 26 27 Frequency: 3195.41 3206.15 3206.15 Force Cnst: 6.5119 6.5896 6.5896 Red. Mass: 1.0824 1.0880 1.0880 IR Active: YES YES YES IR Intens: 0.000 0.000 0.000 Raman Active: YES YES YES X Y Z X Y Z X Y Z C -0.034 0.000 0.000 0.049 0.000 -0.000 0.000 -0.003 -0.000 C 0.017 0.029 0.000 -0.014 -0.020 -0.000 0.020 0.038 -0.000 C 0.017 -0.029 0.000 -0.014 0.020 -0.000 -0.020 0.038 0.000 C 0.017 -0.029 -0.000 0.014 -0.020 0.000 0.020 -0.038 -0.000 C 0.017 0.029 0.000 0.014 0.020 0.000 -0.020 -0.038 0.000 C -0.034 -0.000 -0.000 -0.049 -0.000 0.000 -0.000 0.003 0.000 H 0.407 -0.000 0.000 -0.575 -0.000 -0.000 -0.000 0.002 0.000 H -0.203 -0.352 -0.000 0.146 0.248 0.000 -0.248 -0.432 -0.000 H -0.203 0.352 -0.000 0.145 -0.248 0.000 0.248 -0.432 -0.000 H -0.203 0.352 0.000 -0.145 0.248 0.000 -0.248 0.432 0.000 H -0.203 -0.352 -0.000 -0.146 -0.248 -0.000 0.248 0.432 -0.000 H 0.407 -0.000 0.000 0.575 0.000 -0.000 0.000 -0.002 -0.000 TransDip 0.000 -0.000 -0.000 -0.000 -0.000 -0.000 0.000 0.000 -0.000 Mode: 28 29 30 Frequency: 3223.73 3223.74 3234.77 Force Cnst: 6.7165 6.7166 6.7916 Red. Mass: 1.0969 1.0969 1.1016 IR Active: YES YES YES IR Intens: 36.855 36.896 0.000 Raman Active: YES YES YES X Y Z X Y Z X Y Z C 0.000 0.004 -0.000 -0.052 0.000 0.000 -0.038 -0.000 -0.000 C -0.024 -0.038 0.000 -0.010 -0.024 0.000 -0.019 -0.033 0.000 C 0.024 -0.038 -0.000 -0.010 0.024 0.000 -0.019 0.033 0.000 C 0.024 -0.038 0.000 -0.010 0.024 -0.000 0.019 -0.033 0.000 C -0.024 -0.038 -0.000 -0.010 -0.024 -0.000 0.019 0.033 -0.000 C 0.000 0.004 -0.000 -0.052 0.000 -0.000 0.038 -0.000 0.000 H -0.003 -0.004 -0.000 0.575 -0.000 0.000 0.407 0.000 -0.000 H 0.250 0.429 0.000 0.142 0.253 -0.000 0.203 0.352 -0.000 H -0.252 0.431 0.000 0.139 -0.249 -0.000 0.203 -0.352 0.000 H -0.252 0.431 -0.000 0.139 -0.249 -0.000 -0.203 0.352 0.000 H 0.250 0.429 0.000 0.142 0.253 0.000 -0.203 -0.352 0.000 H -0.003 -0.004 -0.000 0.575 -0.000 0.000 -0.407 -0.000 -0.000 TransDip 0.001 -0.194 -0.000 -0.195 -0.001 0.000 -0.000 0.000 0.000 169 5.8.2 Frequencies and normal modes of benzene, computed by finite dif- ference with single precision amplitudes and step size of 0.001 ˚ A Mode: 1 2 3 Frequency: ******** -1944.03 -1064.99 Force Cnst: 243.5195 3.6738 1.0206 Red. Mass: 1.9748 1.6499 1.5272 IR Active: YES YES YES IR Intens: 14.753 2.614 12.108 Raman Active: YES YES YES X Y Z X Y Z X Y Z C -0.001 0.163 0.003 0.020 -0.006 -0.009 -0.089 0.011 -0.050 C 0.060 -0.059 -0.032 0.077 -0.084 0.032 0.059 0.062 0.003 C -0.059 -0.060 0.015 0.104 0.044 -0.043 0.003 0.024 -0.035 C 0.059 0.061 -0.055 -0.104 -0.056 0.017 -0.018 0.001 0.046 C -0.060 0.059 -0.007 -0.068 0.091 -0.031 -0.083 -0.083 -0.043 C 0.001 -0.163 -0.042 -0.033 -0.003 -0.007 0.097 -0.003 -0.009 H 0.008 -0.001 0.020 -0.064 0.070 -0.005 0.038 -0.142 -0.018 H -0.008 -0.016 0.248 0.167 -0.004 0.327 0.108 -0.114 0.063 H 0.007 -0.007 -0.051 0.046 0.033 -0.066 0.254 0.164 0.888 H -0.008 0.007 0.796 -0.017 -0.028 -0.490 -0.061 -0.007 0.041 H -0.001 0.004 0.458 -0.078 0.133 0.723 -0.043 -0.067 0.063 H 0.000 -0.001 -0.059 -0.025 -0.032 0.002 0.078 0.024 0.015 TransDip -0.000 0.001 0.123 0.002 -0.007 0.051 0.012 0.002 0.111 Mode: 4 5 6 Frequency: -94.51 468.39 628.64 Force Cnst: 0.0125 0.3546 0.9312 Red. Mass: 2.3661 2.7431 3.9995 IR Active: YES YES YES IR Intens: 0.064 11.563 0.682 Raman Active: YES YES YES X Y Z X Y Z X Y Z C -0.097 -0.016 -0.107 -0.022 0.004 0.178 0.040 0.004 -0.199 C -0.062 -0.032 0.077 -0.013 -0.011 -0.086 -0.005 -0.033 0.040 C 0.055 -0.122 -0.030 0.006 -0.023 -0.170 -0.008 0.060 0.323 C -0.067 0.135 -0.075 -0.023 0.036 -0.190 -0.001 -0.017 -0.105 C 0.053 0.022 0.171 0.003 -0.013 -0.063 -0.013 0.001 -0.275 C 0.099 0.016 -0.046 0.039 0.010 0.212 -0.042 -0.008 0.178 H -0.056 0.200 -0.144 0.001 0.055 0.604 0.087 0.037 -0.189 H 0.168 -0.133 0.770 0.069 -0.059 0.278 0.027 -0.050 0.601 H -0.097 -0.137 0.007 -0.009 -0.009 0.066 -0.175 0.015 0.074 H 0.049 0.129 -0.107 0.007 0.032 -0.032 0.234 0.066 -0.046 H 0.075 -0.023 -0.279 0.021 -0.034 -0.102 0.207 -0.155 -0.278 H 0.092 -0.072 -0.131 0.036 -0.010 0.607 -0.045 -0.000 0.275 TransDip 0.006 -0.002 0.006 0.003 -0.000 0.109 0.008 -0.001 0.025 Mode: 7 8 9 Frequency: 669.91 856.64 906.78 Force Cnst: 1.6173 0.8430 0.6592 Red. Mass: 6.1164 1.9497 1.3608 IR Active: YES YES YES IR Intens: 0.696 0.774 12.114 Raman Active: YES YES YES X Y Z X Y Z X Y Z C 0.063 0.008 -0.107 0.071 0.026 -0.032 0.009 0.001 -0.018 C 0.034 0.084 0.463 0.057 0.010 -0.066 0.007 0.004 0.018 C -0.061 0.101 -0.128 -0.068 0.152 -0.064 -0.010 0.022 0.019 C 0.082 -0.115 -0.381 0.034 -0.101 0.025 0.009 -0.017 0.024 C -0.076 -0.051 0.126 -0.055 -0.058 0.125 -0.004 -0.003 0.038 170 C -0.018 -0.032 0.073 -0.069 -0.019 -0.014 -0.015 -0.005 -0.166 H 0.036 -0.187 -0.359 0.106 -0.075 0.343 0.013 -0.012 0.021 H -0.134 0.171 -0.444 0.006 0.057 0.531 -0.008 0.015 0.064 H 0.036 0.120 -0.018 -0.256 0.095 0.015 -0.054 0.003 -0.006 H -0.085 -0.173 0.049 0.258 -0.022 -0.055 0.047 -0.002 -0.006 H -0.132 0.001 0.181 0.316 -0.298 -0.231 0.062 -0.044 -0.031 H -0.014 0.127 0.058 -0.075 0.121 -0.295 -0.016 0.017 0.975 TransDip -0.006 0.001 -0.026 0.013 -0.005 0.025 0.002 -0.001 0.111 Mode: 10 11 12 Frequency: 913.18 983.96 1023.00 Force Cnst: 0.6558 2.5544 3.6065 Red. Mass: 1.3349 4.4781 5.8490 IR Active: YES YES YES IR Intens: 7.819 0.101 0.021 Raman Active: YES YES YES X Y Z X Y Z X Y Z C -0.007 -0.008 -0.153 0.302 -0.027 -0.013 -0.220 0.004 0.001 C -0.013 0.008 0.044 -0.152 -0.276 0.034 -0.164 -0.259 -0.002 C 0.016 -0.039 0.026 -0.035 0.143 -0.028 -0.155 0.246 0.002 C 0.003 0.006 0.019 -0.144 0.243 0.009 0.123 -0.209 0.009 C 0.015 0.028 0.004 -0.073 -0.089 0.013 0.114 0.203 -0.006 C 0.000 0.002 -0.006 0.084 0.015 -0.024 0.316 0.006 -0.002 H -0.019 -0.007 0.945 0.337 -0.187 0.052 -0.253 0.061 0.027 H -0.054 0.027 -0.192 -0.318 -0.247 -0.253 -0.212 -0.264 -0.060 H 0.053 -0.036 -0.018 0.137 0.209 0.209 -0.176 0.259 -0.010 H -0.071 -0.019 0.024 0.038 0.379 0.096 0.077 -0.265 0.004 H -0.092 0.098 0.072 -0.041 -0.093 0.024 0.058 0.272 0.029 H 0.002 -0.026 -0.042 0.071 -0.175 -0.013 0.345 0.037 -0.012 TransDip -0.008 0.002 0.089 0.006 -0.000 0.008 -0.004 0.002 -0.001 Mode: 13 14 15 Frequency: 1065.67 1068.47 1141.80 Force Cnst: 1.1172 1.1720 0.9520 Red. Mass: 1.6696 1.7424 1.2394 IR Active: YES YES YES IR Intens: 3.472 2.630 0.357 Raman Active: YES YES YES X Y Z X Y Z X Y Z C -0.048 0.068 -0.001 0.112 0.053 -0.003 -0.009 0.026 -0.006 C -0.069 -0.095 0.009 -0.033 0.050 -0.030 0.033 -0.041 0.024 C 0.096 -0.039 -0.009 -0.021 -0.054 0.024 -0.063 0.030 0.007 C 0.098 -0.034 -0.002 -0.018 -0.069 0.017 -0.015 0.040 0.003 C -0.047 -0.067 0.007 -0.106 -0.065 -0.006 0.053 -0.079 -0.014 C -0.077 0.073 -0.004 0.146 0.033 -0.003 -0.018 0.026 -0.004 H -0.045 0.463 0.008 0.106 0.416 -0.011 -0.045 0.379 0.046 H -0.152 -0.072 -0.041 -0.199 0.188 0.174 0.558 -0.329 -0.177 H 0.440 0.150 0.029 -0.303 -0.209 -0.086 -0.114 -0.013 -0.078 H 0.449 0.166 0.015 -0.273 -0.245 -0.064 -0.388 -0.157 -0.011 H -0.070 -0.063 0.004 -0.451 0.127 -0.009 0.245 -0.180 0.116 H -0.082 0.481 -0.004 0.154 0.344 0.013 -0.019 0.281 -0.007 TransDip 0.025 0.054 0.001 -0.045 0.026 0.001 0.015 -0.004 -0.011 Mode: 16 17 18 Frequency: 1234.28 1313.07 1347.04 Force Cnst: 1.2435 3.8848 1.5446 Red. Mass: 1.3854 3.8242 1.4448 IR Active: YES YES YES IR Intens: 0.618 0.052 1.181 Raman Active: YES YES YES X Y Z X Y Z X Y Z C 0.043 0.028 -0.004 0.008 0.203 -0.013 0.065 -0.058 -0.001 C -0.015 -0.043 0.003 0.184 -0.095 0.031 -0.040 0.011 0.008 171 C 0.010 -0.090 -0.017 -0.180 -0.120 -0.013 -0.013 -0.046 -0.033 C -0.055 -0.042 -0.003 -0.168 -0.129 -0.010 0.073 -0.030 0.004 C 0.045 0.102 -0.001 0.155 -0.072 -0.001 -0.069 0.117 -0.002 C -0.026 0.056 0.004 -0.005 0.222 -0.002 -0.037 0.028 -0.000 H 0.103 -0.386 -0.049 0.040 0.174 -0.022 0.148 0.604 -0.035 H -0.276 0.043 0.039 -0.235 0.099 -0.103 -0.074 -0.066 -0.048 H 0.068 -0.063 0.243 -0.048 -0.056 0.162 -0.005 -0.053 0.345 H -0.312 -0.187 0.040 0.402 0.213 0.043 -0.127 -0.138 0.060 H 0.423 -0.107 -0.047 -0.083 0.075 0.015 0.356 -0.110 -0.015 H -0.029 0.574 0.003 -0.006 -0.619 0.003 -0.041 -0.509 -0.009 TransDip -0.004 -0.001 0.025 0.002 -0.002 0.007 0.010 -0.007 0.033 Mode: 19 20 21 Frequency: 1395.98 1399.64 1522.60 Force Cnst: 1.6164 1.3569 2.8841 Red. Mass: 1.4078 1.1756 2.1115 IR Active: YES YES YES IR Intens: 0.523 0.469 7.244 Raman Active: YES YES YES X Y Z X Y Z X Y Z C 0.088 0.055 -0.007 -0.007 -0.032 -0.006 -0.060 0.128 -0.002 C -0.089 0.012 -0.017 0.063 -0.034 -0.024 -0.032 -0.089 -0.000 C -0.008 -0.016 -0.007 -0.054 0.006 0.029 0.153 0.030 0.001 C 0.014 -0.080 -0.005 0.031 0.029 -0.011 0.128 0.024 -0.000 C 0.042 0.052 0.024 -0.009 -0.022 0.022 -0.064 -0.085 0.001 C -0.070 0.007 -0.002 -0.010 0.039 0.001 -0.039 0.132 -0.000 H 0.164 -0.200 -0.032 -0.039 0.017 -0.002 -0.077 -0.431 -0.003 H 0.507 -0.401 0.149 -0.159 0.162 0.259 -0.080 -0.079 0.022 H 0.177 0.102 0.159 0.574 0.403 -0.242 -0.394 -0.298 -0.009 H -0.028 -0.124 0.070 -0.433 -0.273 -0.046 -0.382 -0.286 -0.010 H -0.457 0.328 -0.197 -0.104 0.003 -0.126 -0.036 -0.131 0.003 H -0.077 -0.071 0.009 -0.010 -0.152 0.019 -0.050 -0.450 0.001 TransDip 0.012 -0.012 0.016 -0.009 0.005 -0.019 -0.045 -0.074 0.000 Mode: 22 23 24 Frequency: 1592.24 2532.65 3202.62 Force Cnst: 2.5188 8.4084 6.5636 Red. Mass: 1.6863 2.2249 1.0861 IR Active: YES YES YES IR Intens: 9.132 4.289 2.171 Raman Active: YES YES YES X Y Z X Y Z X Y Z C -0.028 -0.060 -0.011 0.048 0.008 -0.012 -0.002 -0.000 -0.000 C 0.083 -0.067 -0.002 -0.148 0.058 0.013 -0.005 0.005 0.002 C 0.025 0.076 -0.031 -0.144 -0.063 -0.042 0.016 -0.037 -0.003 C 0.024 0.058 -0.002 0.150 0.070 0.036 0.024 -0.031 0.003 C 0.130 -0.005 0.004 0.126 -0.093 -0.050 0.022 0.026 -0.003 C -0.117 -0.061 0.005 -0.025 0.005 -0.003 -0.053 0.002 -0.001 H 0.075 0.302 -0.048 -0.098 -0.151 -0.024 0.052 -0.005 -0.000 H -0.381 0.071 0.095 -0.079 0.240 0.322 -0.019 -0.023 0.005 H -0.327 -0.101 0.424 -0.054 0.074 0.171 -0.240 0.413 0.014 H -0.208 -0.055 0.069 0.098 -0.050 -0.429 -0.222 0.387 -0.021 H -0.409 0.320 -0.107 0.107 0.090 0.657 -0.207 -0.364 0.043 H -0.149 0.169 0.008 -0.052 -0.023 -0.006 0.620 -0.001 -0.001 TransDip -0.074 0.045 0.043 -0.003 -0.025 0.061 0.001 -0.047 0.005 Mode: 25 26 27 Frequency: 3220.06 3224.44 3228.47 Force Cnst: 6.6746 6.7012 6.7275 Red. Mass: 1.0926 1.0939 1.0955 IR Active: YES YES YES IR Intens: 17.747 22.996 21.122 Raman Active: YES YES YES 172 X Y Z X Y Z X Y Z C -0.003 0.002 -0.002 -0.002 0.006 -0.001 -0.067 -0.000 0.003 C -0.000 -0.005 -0.004 -0.000 -0.006 -0.005 -0.025 -0.048 0.004 C -0.029 0.049 0.000 0.009 -0.011 0.005 0.011 -0.007 0.001 C 0.006 0.000 -0.002 0.031 -0.055 -0.004 -0.007 0.011 0.001 C -0.010 -0.024 0.004 -0.033 -0.044 0.006 -0.005 -0.009 -0.002 C -0.061 -0.003 0.001 0.019 -0.000 0.001 -0.004 0.003 -0.001 H 0.019 0.001 -0.004 0.016 -0.002 -0.001 0.744 -0.002 0.006 H 0.019 0.041 0.058 0.026 0.033 0.041 0.308 0.528 -0.073 H 0.330 -0.571 0.008 -0.066 0.122 -0.038 -0.076 0.122 -0.019 H -0.030 0.044 -0.008 -0.368 0.630 0.005 0.078 -0.139 0.019 H 0.146 0.268 -0.022 0.312 0.542 -0.046 0.055 0.093 0.006 H 0.674 -0.001 0.003 -0.207 -0.005 0.002 0.046 -0.000 -0.003 TransDip -0.132 0.025 0.004 0.033 -0.150 -0.005 -0.130 -0.069 -0.007 Mode: 28 29 30 Frequency: 3238.32 3344.51 14577.28 Force Cnst: 6.8147 7.2219 249.0864 Red. Mass: 1.1030 1.0958 1.9895 IR Active: YES YES YES IR Intens: 2.574 6.365 14.565 Raman Active: YES YES YES X Y Z X Y Z X Y Z C 0.006 0.001 0.003 -0.055 -0.000 0.005 0.001 -0.165 0.003 C 0.019 0.020 0.001 0.021 0.060 0.003 -0.060 0.060 -0.032 C 0.030 -0.038 0.004 -0.019 0.007 0.005 0.059 0.061 0.015 C -0.027 0.035 -0.002 0.013 0.002 0.001 -0.059 -0.062 -0.055 C -0.031 -0.042 0.002 0.009 -0.009 -0.003 0.060 -0.061 -0.007 C -0.028 0.001 -0.000 0.012 -0.000 -0.002 -0.001 0.164 -0.042 H -0.069 0.000 0.005 0.623 -0.022 0.010 -0.009 0.003 0.020 H -0.138 -0.253 -0.050 -0.358 -0.591 0.061 0.008 0.017 0.247 H -0.258 0.447 -0.042 0.073 -0.104 -0.217 -0.008 0.006 -0.052 H 0.234 -0.409 0.024 -0.029 0.066 -0.129 0.008 -0.007 0.795 H 0.279 0.482 -0.023 -0.015 -0.056 0.166 0.001 -0.002 0.458 H 0.316 0.000 -0.002 -0.076 0.000 -0.005 0.000 -0.001 -0.060 TransDip -0.040 -0.031 -0.011 -0.022 0.077 -0.014 0.000 -0.001 0.122 173 5.8.3 Frequencies and normal modes of benzene, computed by finite dif- ference with single precision amplitudes with cleanup in double pre- cision and step size of 0.001 ˚ A Mode: 1 2 3 Frequency: 397.20 404.63 607.86 Force Cnst: 0.2724 0.2891 1.1235 Red. Mass: 2.9305 2.9967 5.1608 IR Active: YES YES YES IR Intens: 0.029 0.000 3.408 Raman Active: YES YES YES X Y Z X Y Z X Y Z C -0.000 0.022 -0.093 0.000 -0.000 0.227 0.001 -0.140 -0.007 C 0.019 0.011 -0.148 -0.000 -0.000 -0.195 -0.200 -0.207 -0.003 C -0.019 0.011 0.237 0.000 -0.000 -0.032 0.200 -0.206 0.055 C 0.019 -0.011 0.236 -0.000 0.000 -0.032 -0.200 0.206 0.064 C -0.019 -0.011 -0.146 -0.000 0.000 -0.195 0.200 0.207 -0.023 C 0.000 -0.022 -0.093 0.000 -0.000 0.227 -0.001 0.140 0.001 H -0.000 -0.003 -0.189 0.000 0.000 0.482 0.001 0.203 -0.179 H 0.008 0.017 -0.301 0.000 -0.000 -0.415 -0.051 -0.291 -0.214 H -0.008 0.017 0.524 0.000 -0.000 -0.068 0.050 -0.290 -0.112 H 0.008 -0.017 0.539 -0.000 0.000 -0.068 -0.050 0.290 -0.228 H -0.008 -0.017 -0.311 -0.000 0.000 -0.415 0.051 0.291 -0.140 H 0.000 0.003 -0.189 0.000 -0.000 0.483 -0.001 -0.203 -0.173 TransDip 0.000 0.000 0.005 0.000 -0.000 0.000 0.000 -0.000 -0.059 Mode: 4 5 6 Frequency: 611.84 654.18 691.27 Force Cnst: 1.3300 1.4175 0.3157 Red. Mass: 6.0302 5.6216 1.1212 IR Active: YES YES YES IR Intens: 0.000 0.001 84.640 Raman Active: YES YES YES X Y Z X Y Z X Y Z C -0.359 -0.000 0.001 0.001 -0.001 0.265 -0.000 -0.006 -0.034 C -0.024 0.221 -0.001 -0.003 -0.004 -0.264 -0.017 -0.023 -0.034 C -0.024 -0.222 -0.001 0.003 -0.003 -0.264 0.017 -0.024 -0.034 C 0.024 0.222 0.001 -0.003 0.003 0.265 -0.017 0.024 -0.034 C 0.024 -0.221 0.001 0.003 0.004 0.264 0.017 0.023 -0.034 C 0.360 0.000 -0.001 -0.001 0.001 -0.264 0.000 0.006 -0.034 H -0.357 0.000 0.000 0.001 0.004 0.309 -0.000 0.031 0.405 H 0.263 0.054 -0.001 -0.001 -0.005 -0.314 -0.001 -0.032 0.405 H 0.263 -0.055 -0.003 -0.000 -0.005 -0.313 0.001 -0.033 0.405 H -0.263 0.055 0.001 0.000 0.005 0.308 -0.001 0.033 0.405 H -0.263 -0.054 -0.001 0.001 0.005 0.309 0.001 0.032 0.405 H 0.357 -0.000 -0.002 -0.001 -0.004 -0.313 0.000 -0.031 0.405 TransDip -0.000 0.000 -0.000 0.000 0.000 -0.001 -0.000 0.000 0.295 Mode: 7 8 9 Frequency: 866.06 867.75 966.36 Force Cnst: 0.5534 0.5537 0.7558 Red. Mass: 1.2522 1.2480 1.3736 IR Active: YES YES YES IR Intens: 0.014 0.000 0.028 Raman Active: YES YES YES X Y Z X Y Z X Y Z C 0.000 0.005 0.009 0.000 0.000 0.085 0.000 0.014 0.038 C -0.002 -0.009 -0.067 0.000 -0.000 0.049 -0.000 -0.015 0.068 174 C 0.002 -0.009 0.074 0.000 -0.000 0.036 0.000 -0.015 -0.103 C -0.002 0.009 -0.081 -0.000 0.000 -0.036 -0.000 0.015 -0.100 C 0.002 0.009 0.072 -0.000 -0.000 -0.049 0.000 0.015 0.059 C -0.000 -0.005 -0.006 -0.000 0.000 -0.085 -0.000 -0.014 0.042 H 0.000 0.022 -0.062 0.000 0.000 -0.569 0.000 0.053 -0.202 H 0.005 -0.013 0.451 0.000 -0.000 -0.329 0.016 -0.025 -0.389 H -0.005 -0.013 -0.510 0.000 0.000 -0.239 -0.017 -0.025 0.553 H 0.005 0.014 0.527 -0.000 0.000 0.239 0.017 0.025 0.549 H -0.005 0.014 -0.479 -0.000 -0.000 0.329 -0.016 0.024 -0.315 H -0.000 -0.022 0.041 -0.000 -0.000 0.569 -0.000 -0.053 -0.244 TransDip -0.000 0.000 -0.004 0.000 0.000 -0.000 0.000 -0.000 -0.005 Mode: 10 11 12 Frequency: 973.24 977.85 1007.88 Force Cnst: 0.7523 0.6278 3.9445 Red. Mass: 1.3479 1.1144 6.5906 IR Active: YES YES YES IR Intens: 0.000 0.000 0.000 Raman Active: YES YES YES X Y Z X Y Z X Y Z C -0.000 0.000 -0.094 -0.000 -0.000 0.042 0.291 -0.000 0.000 C 0.000 -0.000 0.080 -0.000 0.000 -0.038 -0.145 -0.252 -0.000 C 0.000 0.000 0.013 -0.000 0.000 -0.044 -0.145 0.252 -0.000 C -0.000 -0.000 0.013 0.000 -0.001 0.036 -0.145 0.252 0.000 C -0.000 0.000 0.081 0.000 -0.000 0.042 -0.145 -0.252 -0.000 C -0.000 -0.000 -0.094 0.000 0.000 -0.038 0.291 0.000 -0.000 H -0.000 -0.000 0.525 -0.000 -0.002 -0.415 0.286 -0.000 -0.001 H 0.000 -0.000 -0.450 -0.001 0.001 0.394 -0.143 -0.248 0.002 H 0.000 0.000 -0.074 0.000 0.001 0.428 -0.143 0.248 0.002 H -0.000 -0.000 -0.076 -0.000 -0.001 -0.383 -0.143 0.248 -0.003 H -0.000 0.000 -0.451 0.001 -0.000 -0.419 -0.143 -0.248 -0.001 H -0.000 0.000 0.526 0.000 0.001 0.397 0.286 0.000 0.002 TransDip 0.000 -0.000 0.000 -0.000 -0.000 0.000 0.000 0.000 -0.000 Mode: 13 14 15 Frequency: 1022.99 1065.79 1065.80 Force Cnst: 3.6743 1.0983 1.0983 Red. Mass: 5.9590 1.6411 1.6411 IR Active: YES YES YES IR Intens: 0.000 3.561 3.566 Raman Active: YES YES YES X Y Z X Y Z X Y Z C 0.274 -0.000 -0.000 0.114 -0.000 -0.000 0.000 0.079 -0.000 C 0.137 0.237 -0.000 -0.030 0.084 0.000 -0.084 -0.066 0.000 C 0.137 -0.237 -0.000 -0.031 -0.083 0.000 0.083 -0.066 -0.000 C -0.137 0.237 0.000 -0.031 -0.083 -0.000 0.083 -0.066 -0.000 C -0.137 -0.237 0.000 -0.030 0.084 -0.000 -0.084 -0.066 0.000 C -0.274 0.000 -0.000 0.114 -0.000 -0.000 0.000 0.079 -0.000 H 0.303 -0.000 -0.001 0.125 -0.002 0.000 0.001 0.546 -0.000 H 0.151 0.262 0.002 -0.378 0.290 -0.000 -0.292 0.044 0.000 H 0.151 -0.262 0.001 -0.380 -0.291 0.000 0.289 0.042 0.000 H -0.151 0.262 -0.003 -0.380 -0.291 -0.000 0.289 0.042 -0.000 H -0.151 -0.262 -0.001 -0.377 0.290 0.000 -0.292 0.044 -0.000 H -0.303 0.000 0.002 0.125 -0.002 0.000 0.001 0.546 0.000 TransDip -0.000 0.000 -0.000 -0.060 -0.000 -0.000 -0.000 0.060 -0.000 Mode: 16 17 18 Frequency: 1157.22 1195.45 1198.37 Force Cnst: 0.9203 0.9467 0.9449 Red. Mass: 1.1664 1.1244 1.1167 IR Active: YES YES YES IR Intens: 0.000 0.000 0.080 Raman Active: YES YES YES 175 X Y Z X Y Z X Y Z C 0.000 0.049 0.000 -0.006 0.000 -0.000 -0.000 -0.057 0.001 C 0.042 -0.025 -0.000 -0.043 0.028 0.000 -0.028 0.009 0.000 C -0.043 -0.025 0.000 -0.043 -0.028 0.000 0.028 0.009 -0.004 C -0.042 -0.024 -0.000 0.043 0.028 0.000 -0.028 -0.009 -0.009 C 0.042 -0.025 0.000 0.043 -0.028 -0.000 0.028 -0.009 0.005 C -0.000 0.049 0.000 0.006 -0.000 -0.000 0.000 0.057 0.001 H 0.000 0.406 -0.000 -0.005 0.001 0.000 -0.000 -0.571 -0.008 H 0.351 -0.203 0.000 -0.429 0.250 -0.000 -0.251 0.138 -0.010 H -0.351 -0.203 -0.000 -0.430 -0.251 -0.001 0.249 0.137 0.049 H -0.351 -0.202 0.001 0.430 0.251 -0.001 -0.249 -0.138 0.091 H 0.351 -0.203 -0.000 0.429 -0.250 -0.000 0.251 -0.138 -0.036 H -0.000 0.405 -0.000 0.005 -0.001 0.000 0.000 0.571 -0.011 TransDip 0.000 -0.000 0.000 0.000 -0.000 -0.000 -0.000 -0.000 0.009 Mode: 19 20 21 Frequency: 1318.66 1371.46 1521.78 Force Cnst: 4.6950 1.3830 2.8642 Red. Mass: 4.5827 1.2480 2.0992 IR Active: YES YES YES IR Intens: 0.000 0.000 7.303 Raman Active: YES YES YES X Y Z X Y Z X Y Z C -0.000 0.233 0.000 0.000 0.060 -0.000 0.098 -0.000 -0.000 C 0.202 -0.116 -0.000 -0.052 0.030 0.000 -0.090 0.109 0.000 C -0.202 -0.116 0.000 0.052 0.030 -0.000 -0.090 -0.109 0.000 C -0.202 -0.116 0.000 -0.052 -0.030 -0.000 -0.091 -0.109 -0.000 C 0.202 -0.116 -0.000 0.052 -0.030 -0.000 -0.090 0.109 -0.000 C 0.000 0.233 -0.000 -0.000 -0.060 0.000 0.098 -0.000 -0.000 H -0.000 -0.335 0.000 0.000 -0.404 -0.000 0.122 0.000 -0.000 H -0.290 0.168 0.000 0.350 -0.202 0.000 0.431 -0.178 0.000 H 0.290 0.168 -0.000 -0.350 -0.202 0.000 0.431 0.179 -0.000 H 0.291 0.168 -0.000 0.350 0.202 0.000 0.431 0.179 0.000 H -0.290 0.168 -0.000 -0.350 0.202 -0.000 0.431 -0.179 0.000 H 0.000 -0.335 0.000 -0.000 0.404 0.000 0.122 0.000 -0.000 TransDip -0.000 -0.000 -0.000 0.000 0.000 0.000 0.087 0.000 0.000 Mode: 22 23 24 Frequency: 1521.78 1671.93 1677.89 Force Cnst: 2.8642 9.3132 9.1134 Red. Mass: 2.0992 5.6547 5.4942 IR Active: YES YES YES IR Intens: 7.293 0.000 0.120 Raman Active: YES YES YES X Y Z X Y Z X Y Z C -0.000 -0.153 0.000 -0.152 0.003 0.000 -0.001 -0.338 -0.001 C 0.109 0.035 -0.000 0.296 -0.084 -0.000 -0.080 0.195 -0.005 C -0.109 0.035 -0.000 0.295 0.081 -0.000 0.084 0.196 -0.001 C -0.109 0.035 -0.000 -0.295 -0.081 0.000 -0.084 -0.196 -0.013 C 0.109 0.035 -0.000 -0.296 0.084 0.000 0.080 -0.195 0.005 C -0.000 -0.153 0.000 0.152 -0.003 -0.000 0.001 0.338 -0.002 H -0.000 0.534 -0.000 -0.176 -0.003 0.000 -0.001 0.389 -0.007 H -0.179 0.225 -0.000 -0.260 0.250 0.001 0.241 0.033 -0.004 H 0.178 0.225 0.000 -0.257 -0.251 -0.000 -0.245 0.030 0.099 H 0.178 0.225 0.000 0.255 0.250 -0.001 0.245 -0.030 0.188 H -0.179 0.225 -0.000 0.259 -0.249 0.001 -0.241 -0.033 -0.063 H -0.000 0.534 -0.000 0.175 0.003 -0.000 0.001 -0.389 -0.010 TransDip -0.000 0.086 0.000 -0.000 -0.000 0.000 -0.000 -0.000 0.011 Mode: 25 26 27 Frequency: 3195.41 3206.14 3206.16 Force Cnst: 6.5119 6.5895 6.5897 Red. Mass: 1.0824 1.0880 1.0880 176 IR Active: YES YES YES IR Intens: 0.000 0.000 0.000 Raman Active: YES YES YES X Y Z X Y Z X Y Z C -0.034 0.000 -0.000 -0.049 -0.000 -0.000 -0.002 0.003 0.000 C 0.017 0.029 0.000 0.015 0.021 -0.000 -0.020 -0.037 0.000 C 0.017 -0.029 -0.000 0.014 -0.019 -0.000 0.021 -0.038 -0.000 C 0.017 -0.029 0.000 -0.014 0.019 -0.000 -0.021 0.038 0.000 C 0.017 0.029 -0.000 -0.015 -0.021 -0.000 0.020 0.037 -0.000 C -0.034 -0.000 0.000 0.049 0.000 -0.000 0.002 -0.003 0.000 H 0.407 -0.000 -0.000 0.575 0.000 -0.000 0.021 -0.002 0.000 H -0.203 -0.352 0.000 -0.154 -0.263 0.000 0.243 0.423 -0.000 H -0.203 0.352 0.000 -0.137 0.233 0.000 -0.253 0.441 -0.001 H -0.203 0.352 -0.000 0.137 -0.233 0.000 0.253 -0.441 -0.001 H -0.203 -0.352 0.000 0.154 0.263 0.000 -0.243 -0.423 0.000 H 0.407 0.000 0.000 -0.575 -0.000 -0.000 -0.020 0.002 0.000 TransDip 0.000 0.000 -0.000 -0.000 -0.000 0.000 -0.000 0.000 -0.000 Mode: 28 29 30 Frequency: 3223.73 3223.74 3234.77 Force Cnst: 6.7165 6.7166 6.7916 Red. Mass: 1.0969 1.0969 1.1016 IR Active: YES YES YES IR Intens: 36.857 36.897 0.000 Raman Active: YES YES YES X Y Z X Y Z X Y Z C -0.000 0.004 -0.000 -0.052 -0.000 0.000 0.038 0.000 0.000 C -0.024 -0.038 -0.000 -0.010 -0.024 -0.000 0.019 0.033 -0.000 C 0.024 -0.038 -0.000 -0.010 0.024 0.000 0.019 -0.033 0.000 C 0.024 -0.038 -0.000 -0.010 0.024 -0.000 -0.019 0.033 -0.000 C -0.024 -0.038 -0.000 -0.010 -0.024 -0.000 -0.019 -0.033 0.000 C -0.000 0.004 -0.000 -0.052 -0.000 -0.000 -0.038 -0.000 -0.000 H 0.001 -0.004 -0.000 0.575 0.000 0.000 -0.407 -0.000 0.000 H 0.251 0.431 0.000 0.140 0.250 0.000 -0.203 -0.352 -0.000 H -0.251 0.430 0.000 0.141 -0.252 0.000 -0.203 0.352 -0.000 H -0.251 0.430 0.000 0.141 -0.252 0.000 0.203 -0.352 0.000 H 0.251 0.430 -0.000 0.140 0.250 0.000 0.203 0.352 0.000 H 0.001 -0.004 -0.000 0.575 0.000 -0.000 0.407 0.000 -0.000 TransDip -0.000 -0.194 0.000 -0.195 0.000 0.000 -0.000 -0.000 0.000 177 5.8.4 Frequencies and normal modes of benzene, computed by finite dif- ference with double precision amplitudes and step size of 0.01 ˚ A Mode: 1 2 3 Frequency: 404.72 404.72 611.79 Force Cnst: 0.2892 0.2892 1.3295 Red. Mass: 2.9970 2.9970 6.0288 IR Active: YES YES YES IR Intens: 0.000 0.000 0.000 Raman Active: YES YES YES X Y Z X Y Z X Y Z C 0.000 0.000 0.246 -0.000 -0.000 -0.000 0.360 0.000 -0.000 C 0.000 -0.000 -0.123 -0.000 -0.000 0.213 0.024 -0.221 0.000 C -0.000 -0.000 -0.123 0.000 -0.000 -0.213 0.024 0.221 0.000 C -0.000 -0.000 -0.123 -0.000 0.000 -0.213 -0.024 -0.221 0.000 C 0.000 -0.000 -0.123 0.000 0.000 0.213 -0.024 0.221 0.000 C -0.000 0.000 0.246 -0.000 0.000 -0.000 -0.360 -0.000 0.000 H 0.000 0.000 0.523 -0.000 0.000 -0.000 0.357 -0.000 -0.000 H 0.000 -0.000 -0.261 0.000 -0.000 0.453 -0.263 -0.054 -0.000 H -0.000 -0.000 -0.261 0.000 -0.000 -0.452 -0.263 0.054 -0.000 H -0.000 -0.000 -0.261 0.000 0.000 -0.452 0.263 -0.054 -0.000 H 0.000 -0.000 -0.261 0.000 0.000 0.453 0.263 0.054 -0.000 H -0.000 0.000 0.523 -0.000 -0.000 -0.000 -0.357 0.000 -0.000 TransDip -0.000 0.000 0.000 -0.000 -0.000 -0.000 0.000 -0.000 -0.000 Mode: 4 5 6 Frequency: 611.90 654.40 691.47 Force Cnst: 1.3305 1.4171 0.3056 Red. Mass: 6.0312 5.6166 1.0848 IR Active: YES YES YES IR Intens: 0.000 0.000 88.291 Raman Active: YES YES YES X Y Z X Y Z X Y Z C 0.000 -0.152 -0.000 -0.000 0.000 -0.264 0.000 -0.000 -0.034 C -0.221 -0.232 -0.000 0.000 0.000 0.264 -0.000 -0.000 -0.034 C 0.221 -0.232 0.000 -0.000 -0.000 0.264 0.000 0.000 -0.034 C -0.221 0.232 0.000 0.000 -0.000 -0.264 0.000 -0.000 -0.034 C 0.221 0.232 -0.000 -0.000 0.000 -0.264 -0.000 0.000 -0.034 C -0.000 0.152 0.000 -0.000 -0.000 0.264 -0.000 -0.000 -0.034 H 0.000 0.231 -0.000 -0.000 -0.000 -0.311 0.000 -0.000 0.407 H -0.054 -0.326 0.000 0.000 0.000 0.311 -0.000 -0.000 0.407 H 0.054 -0.326 0.000 0.000 -0.000 0.311 -0.000 0.000 0.407 H -0.054 0.326 -0.000 0.000 -0.000 -0.311 0.000 0.000 0.407 H 0.054 0.326 -0.000 -0.000 0.000 -0.311 0.000 0.000 0.407 H -0.000 -0.231 0.000 -0.000 0.000 0.311 -0.000 -0.000 0.407 TransDip -0.000 -0.000 -0.000 -0.000 -0.000 -0.000 -0.000 -0.000 0.301 Mode: 7 8 9 Frequency: 867.93 867.93 973.41 Force Cnst: 0.5539 0.5539 0.7525 Red. Mass: 1.2480 1.2480 1.3479 IR Active: YES YES YES IR Intens: 0.000 0.000 0.000 Raman Active: YES YES YES X Y Z X Y Z X Y Z C -0.000 0.000 -0.085 -0.000 0.000 0.000 0.000 -0.000 0.000 C 0.000 0.000 -0.043 -0.000 -0.000 -0.074 -0.000 -0.000 0.088 C 0.000 -0.000 -0.043 -0.000 0.000 0.074 -0.000 0.000 -0.088 C 0.000 -0.000 0.043 0.000 -0.000 -0.074 -0.000 0.000 -0.088 C 0.000 0.000 0.043 0.000 0.000 0.074 -0.000 -0.000 0.088 178 C -0.000 -0.000 0.085 0.000 0.000 -0.000 0.000 0.000 0.000 H -0.000 -0.000 0.571 -0.000 -0.000 -0.000 0.000 0.000 -0.001 H 0.000 0.000 0.286 -0.000 -0.000 0.494 -0.000 -0.000 -0.492 H 0.000 -0.000 0.285 -0.000 0.000 -0.495 -0.000 0.000 0.493 H 0.000 -0.000 -0.285 0.000 -0.000 0.495 -0.000 0.000 0.493 H 0.000 0.000 -0.286 0.000 0.000 -0.494 -0.000 -0.000 -0.492 H -0.000 0.000 -0.571 0.000 -0.000 0.000 0.000 -0.000 -0.001 TransDip -0.000 -0.000 -0.000 -0.000 0.000 -0.000 0.000 0.000 -0.000 Mode: 10 11 12 Frequency: 973.41 978.07 1007.82 Force Cnst: 0.7525 0.6280 3.9440 Red. Mass: 1.3479 1.1142 6.5906 IR Active: YES YES YES IR Intens: 0.000 0.000 0.000 Raman Active: YES YES YES X Y Z X Y Z X Y Z C 0.000 0.000 -0.102 0.000 -0.000 -0.040 0.291 0.000 0.000 C 0.000 0.000 0.051 -0.000 -0.000 0.040 -0.146 -0.252 -0.000 C -0.000 -0.000 0.051 -0.000 0.000 0.040 -0.146 0.252 -0.000 C -0.000 0.000 0.051 -0.000 0.000 -0.040 -0.146 0.252 0.000 C 0.000 -0.000 0.051 -0.000 -0.000 -0.040 -0.146 -0.252 0.000 C -0.000 0.000 -0.102 0.000 0.000 0.040 0.291 0.000 -0.000 H 0.000 0.000 0.568 0.000 0.000 0.406 0.287 0.000 -0.000 H 0.000 0.000 -0.285 -0.000 -0.000 -0.406 -0.144 -0.248 0.000 H -0.000 -0.000 -0.283 -0.000 0.000 -0.406 -0.144 0.248 0.000 H -0.000 -0.000 -0.283 -0.000 0.000 0.406 -0.144 0.248 -0.000 H 0.000 -0.000 -0.285 -0.000 -0.000 0.406 -0.144 -0.248 -0.000 H -0.000 0.000 0.568 0.000 0.000 -0.406 0.287 0.000 0.000 TransDip 0.000 0.000 0.000 -0.000 0.000 -0.000 -0.000 0.000 0.000 Mode: 13 14 15 Frequency: 1023.12 1065.65 1065.85 Force Cnst: 3.6752 1.0973 1.0980 Red. Mass: 5.9591 1.6400 1.6405 IR Active: YES YES YES IR Intens: 0.000 3.538 3.546 Raman Active: YES YES YES X Y Z X Y Z X Y Z C -0.274 -0.000 -0.000 0.114 -0.000 -0.000 0.000 0.079 -0.000 C -0.137 -0.237 0.000 -0.030 0.084 0.000 -0.083 -0.066 0.000 C -0.137 0.237 -0.000 -0.030 -0.083 0.000 0.083 -0.066 0.000 C 0.137 -0.237 0.000 -0.030 -0.083 -0.000 0.083 -0.066 0.000 C 0.137 0.237 -0.000 -0.030 0.084 -0.000 -0.083 -0.066 0.000 C 0.274 0.000 -0.000 0.114 -0.000 0.000 0.000 0.079 -0.000 H -0.303 -0.000 0.000 0.124 -0.000 0.000 0.000 0.546 0.000 H -0.151 -0.262 -0.000 -0.379 0.291 -0.000 -0.291 0.044 -0.000 H -0.151 0.262 -0.000 -0.379 -0.291 -0.000 0.291 0.044 -0.000 H 0.151 -0.262 -0.000 -0.379 -0.291 0.000 0.291 0.044 -0.000 H 0.151 0.262 0.000 -0.379 0.291 0.000 -0.291 0.044 -0.000 H 0.303 -0.000 0.000 0.124 -0.000 -0.000 0.000 0.546 0.000 TransDip -0.000 0.000 -0.000 -0.060 -0.000 0.000 -0.000 0.060 -0.000 Mode: 16 17 18 Frequency: 1157.13 1195.00 1195.21 Force Cnst: 0.9190 0.9461 0.9465 Red. Mass: 1.1650 1.1244 1.1246 IR Active: YES YES YES IR Intens: 0.000 0.000 0.000 Raman Active: YES YES YES X Y Z X Y Z X Y Z C -0.000 0.049 0.000 0.006 0.000 0.000 0.000 -0.059 -0.000 C 0.042 -0.024 -0.000 0.043 -0.028 -0.000 -0.028 0.011 -0.000 179 C -0.042 -0.024 -0.000 0.043 0.028 0.000 0.028 0.011 -0.000 C -0.042 -0.024 -0.000 -0.043 -0.028 -0.000 -0.028 -0.011 -0.000 C 0.042 -0.024 -0.000 -0.043 0.028 0.000 0.028 -0.011 0.000 C -0.000 0.049 0.000 -0.006 -0.000 0.000 -0.000 0.059 0.000 H -0.000 0.406 -0.000 0.005 0.000 -0.000 0.000 -0.574 -0.000 H 0.351 -0.203 0.000 0.430 -0.251 0.000 -0.251 0.140 0.000 H -0.351 -0.203 0.000 0.430 0.251 -0.000 0.251 0.140 0.000 H -0.351 -0.203 0.000 -0.430 -0.251 0.000 -0.251 -0.140 0.000 H 0.351 -0.203 0.000 -0.430 0.251 -0.000 0.251 -0.140 -0.000 H -0.000 0.406 -0.000 -0.005 -0.000 -0.000 -0.000 0.574 0.000 TransDip -0.000 0.000 0.000 0.000 0.000 -0.000 0.000 0.000 0.000 Mode: 19 20 21 Frequency: 1318.87 1371.23 1521.59 Force Cnst: 4.7193 1.3826 2.8658 Red. Mass: 4.6049 1.2480 2.1009 IR Active: YES YES YES IR Intens: 0.000 0.000 7.261 Raman Active: YES YES YES X Y Z X Y Z X Y Z C -0.000 0.233 0.000 0.000 -0.060 0.000 0.098 0.000 0.000 C 0.202 -0.117 -0.000 0.052 -0.030 0.000 -0.090 0.109 0.000 C -0.202 -0.117 -0.000 -0.052 -0.030 -0.000 -0.090 -0.109 0.000 C -0.202 -0.117 -0.000 0.052 0.030 -0.000 -0.090 -0.109 -0.000 C 0.202 -0.117 -0.000 -0.052 0.030 -0.000 -0.090 0.109 -0.000 C -0.000 0.233 0.000 0.000 0.060 -0.000 0.098 0.000 -0.000 H -0.000 -0.335 -0.000 0.000 0.404 0.000 0.122 -0.000 0.000 H -0.290 0.167 0.000 -0.349 0.202 -0.000 0.431 -0.178 -0.000 H 0.290 0.167 0.000 0.349 0.202 0.000 0.431 0.178 -0.000 H 0.290 0.167 0.000 -0.349 -0.202 0.000 0.431 0.178 0.000 H -0.290 0.167 0.000 0.349 -0.202 -0.000 0.431 -0.178 0.000 H -0.000 -0.335 -0.000 0.000 -0.404 -0.000 0.122 -0.000 -0.000 TransDip 0.000 -0.000 0.000 -0.000 -0.000 -0.000 0.086 -0.000 -0.000 Mode: 22 23 24 Frequency: 1521.83 1672.13 1672.16 Force Cnst: 2.8657 9.3165 9.3077 Red. Mass: 2.1001 5.6554 5.6499 IR Active: YES YES YES IR Intens: 7.283 0.000 0.000 Raman Active: YES YES YES X Y Z X Y Z X Y Z C 0.000 -0.153 -0.000 -0.152 -0.000 -0.000 0.000 -0.343 -0.000 C 0.109 0.035 -0.000 0.295 -0.083 -0.000 -0.083 0.200 -0.000 C -0.109 0.035 -0.000 0.295 0.083 0.000 0.083 0.200 -0.000 C -0.109 0.035 -0.000 -0.295 -0.083 -0.000 -0.083 -0.200 0.000 C 0.109 0.035 -0.000 -0.295 0.083 0.000 0.083 -0.200 0.000 C 0.000 -0.153 -0.000 0.152 0.000 0.000 -0.000 0.343 0.000 H 0.000 0.534 -0.000 -0.175 0.000 0.000 0.000 0.403 -0.000 H -0.178 0.225 0.000 -0.258 0.250 0.000 0.250 0.031 0.000 H 0.178 0.225 0.000 -0.258 -0.250 -0.000 -0.250 0.031 0.000 H 0.178 0.225 0.000 0.258 0.250 -0.000 0.250 -0.031 -0.000 H -0.178 0.225 0.000 0.258 -0.250 -0.000 -0.250 -0.031 0.000 H 0.000 0.534 -0.000 0.175 -0.000 0.000 -0.000 -0.403 0.000 TransDip 0.000 0.086 0.000 -0.000 -0.000 0.000 -0.000 -0.000 0.000 Mode: 25 26 27 Frequency: 3195.89 3206.51 3206.82 Force Cnst: 6.5139 6.5911 6.5923 Red. Mass: 1.0824 1.0880 1.0880 IR Active: YES YES YES IR Intens: 0.002 0.000 0.000 Raman Active: YES YES YES 180 X Y Z X Y Z X Y Z C -0.033 0.000 0.000 -0.000 0.003 0.000 -0.049 -0.000 0.000 C 0.017 0.029 0.000 -0.020 -0.038 0.000 0.014 0.020 0.000 C 0.017 -0.029 0.000 0.020 -0.038 0.000 0.014 -0.020 -0.000 C 0.017 -0.029 -0.000 -0.020 0.038 -0.000 -0.014 0.020 0.000 C 0.017 0.029 -0.000 0.020 0.038 -0.000 -0.014 -0.020 -0.000 C -0.033 0.000 -0.000 0.000 -0.003 -0.000 0.049 0.000 0.000 H 0.402 0.000 0.000 0.000 -0.002 0.000 0.572 0.000 0.000 H -0.204 -0.354 -0.000 0.248 0.432 -0.000 -0.147 -0.251 -0.000 H -0.204 0.354 -0.000 -0.248 0.432 0.000 -0.147 0.251 -0.000 H -0.204 0.354 0.000 0.248 -0.432 0.000 0.147 -0.251 -0.000 H -0.204 -0.354 0.000 -0.248 -0.432 -0.000 0.147 0.251 -0.000 H 0.402 0.000 -0.000 -0.000 0.002 -0.000 -0.572 -0.000 0.000 TransDip 0.001 0.000 0.000 0.000 -0.000 0.000 0.000 0.000 0.000 Mode: 28 29 30 Frequency: 3224.02 3224.34 3235.23 Force Cnst: 6.7178 6.7191 6.7934 Red. Mass: 1.0969 1.0969 1.1016 IR Active: YES YES YES IR Intens: 36.867 36.821 0.000 Raman Active: YES YES YES X Y Z X Y Z X Y Z C -0.000 0.004 0.000 -0.052 -0.000 -0.000 -0.038 -0.000 0.000 C -0.024 -0.038 0.000 -0.010 -0.024 -0.000 -0.019 -0.032 0.000 C 0.024 -0.038 0.000 -0.010 0.024 -0.000 -0.019 0.032 -0.000 C 0.024 -0.038 0.000 -0.010 0.024 0.000 0.019 -0.032 0.000 C -0.024 -0.038 0.000 -0.010 -0.024 0.000 0.019 0.032 -0.000 C -0.000 0.004 0.000 -0.052 -0.000 0.000 0.038 -0.000 0.000 H 0.000 -0.004 0.000 0.578 0.000 -0.000 0.411 0.000 0.000 H 0.251 0.430 -0.000 0.139 0.248 0.000 0.202 0.350 -0.000 H -0.251 0.430 -0.000 0.139 -0.248 0.000 0.202 -0.350 0.000 H -0.251 0.430 -0.000 0.139 -0.248 -0.000 -0.202 0.350 -0.000 H 0.251 0.430 -0.000 0.139 0.248 -0.000 -0.202 -0.350 0.000 H 0.000 -0.004 0.000 0.578 0.000 0.000 -0.411 0.000 0.000 TransDip -0.000 -0.194 0.000 -0.194 0.000 -0.000 -0.000 0.000 0.000 181 5.8.5 Frequencies and normal modes of benzene, computed by finite dif- ference with single precision amplitudes and step size of 0.01 ˚ A Mode: 1 2 3 Frequency: 404.72 404.72 611.15 Force Cnst: 0.2892 0.2892 1.3253 Red. Mass: 2.9970 2.9970 6.0225 IR Active: YES YES YES IR Intens: 0.000 0.000 0.000 Raman Active: YES YES YES X Y Z X Y Z X Y Z C 0.000 -0.000 -0.050 0.000 -0.000 0.241 -0.359 -0.006 0.000 C -0.000 -0.000 0.233 0.000 -0.000 -0.077 -0.033 0.211 -0.000 C 0.000 0.000 -0.183 0.000 0.000 -0.163 -0.015 -0.231 -0.000 C -0.000 -0.000 -0.183 -0.000 -0.000 -0.163 0.015 0.231 -0.000 C 0.000 0.000 0.233 -0.000 0.000 -0.077 0.034 -0.211 -0.000 C -0.000 0.000 -0.050 -0.000 0.000 0.241 0.359 0.006 -0.000 H 0.000 -0.000 -0.106 0.000 0.000 0.512 -0.356 0.009 0.000 H -0.000 -0.000 0.496 -0.000 -0.000 -0.164 0.261 0.040 0.000 H -0.000 -0.000 -0.390 0.000 0.000 -0.347 0.265 -0.069 0.000 H 0.000 0.000 -0.390 0.000 -0.000 -0.347 -0.266 0.068 0.000 H 0.000 0.000 0.496 0.000 0.000 -0.164 -0.261 -0.040 0.000 H -0.000 0.000 -0.106 -0.000 -0.000 0.512 0.356 -0.011 0.000 TransDip 0.000 -0.000 0.000 0.000 0.000 0.000 -0.000 -0.000 0.000 Mode: 4 5 6 Frequency: 612.12 654.40 691.47 Force Cnst: 1.3310 1.4171 0.3056 Red. Mass: 6.0289 5.6166 1.0848 IR Active: YES YES YES IR Intens: 0.000 0.000 88.291 Raman Active: YES YES YES X Y Z X Y Z X Y Z C 0.015 -0.152 0.000 -0.000 -0.000 -0.264 0.000 -0.000 -0.034 C -0.220 -0.241 -0.000 -0.000 -0.000 0.264 -0.000 -0.000 -0.034 C 0.222 -0.222 0.000 0.000 -0.000 0.264 -0.000 0.000 -0.034 C -0.222 0.222 0.000 -0.000 0.000 -0.264 0.000 -0.000 -0.034 C 0.220 0.241 -0.000 0.000 -0.000 -0.264 0.000 0.000 -0.034 C -0.015 0.151 -0.000 0.000 0.000 0.264 -0.000 0.000 -0.034 H 0.015 0.231 -0.000 -0.000 0.000 -0.311 0.000 0.000 0.407 H -0.066 -0.328 -0.000 0.000 -0.000 0.311 -0.000 -0.000 0.407 H 0.043 -0.323 -0.000 0.000 -0.000 0.311 -0.000 -0.000 0.407 H -0.043 0.323 -0.000 -0.000 0.000 -0.311 0.000 0.000 0.407 H 0.065 0.328 -0.000 -0.000 0.000 -0.311 0.000 0.000 0.407 H -0.015 -0.232 -0.000 0.000 -0.000 0.311 -0.000 -0.000 0.407 TransDip 0.000 -0.000 -0.000 0.000 0.000 -0.000 -0.000 -0.000 0.301 Mode: 7 8 9 Frequency: 867.93 867.93 973.41 Force Cnst: 0.5539 0.5539 0.7525 Red. Mass: 1.2480 1.2480 1.3479 IR Active: YES YES YES IR Intens: 0.000 0.000 0.000 Raman Active: YES YES YES X Y Z X Y Z X Y Z C 0.000 -0.000 -0.003 -0.000 -0.000 -0.085 -0.000 -0.000 0.012 C 0.000 0.000 0.072 0.000 -0.000 -0.046 0.000 0.000 -0.093 C 0.000 0.000 -0.076 0.000 -0.000 -0.040 0.000 0.000 0.081 C -0.000 -0.000 0.076 -0.000 -0.000 0.040 -0.000 -0.000 0.081 C -0.000 -0.000 -0.072 -0.000 0.000 0.046 0.000 -0.000 -0.093 182 C -0.000 0.000 0.003 0.000 0.000 0.085 0.000 0.000 0.012 H 0.000 -0.000 0.023 -0.000 0.000 0.571 -0.000 -0.000 -0.069 H -0.000 0.000 -0.483 0.000 -0.000 0.305 -0.000 0.000 0.523 H -0.000 -0.000 0.506 0.000 0.000 0.265 0.000 0.000 -0.454 H 0.000 0.000 -0.506 -0.000 -0.000 -0.265 -0.000 -0.000 -0.454 H 0.000 -0.000 0.483 -0.000 0.000 -0.305 0.000 -0.000 0.523 H -0.000 0.000 -0.023 0.000 0.000 -0.571 0.000 0.000 -0.069 TransDip -0.000 -0.000 0.000 -0.000 0.000 -0.000 0.000 -0.000 -0.000 Mode: 10 11 12 Frequency: 973.41 978.07 1007.82 Force Cnst: 0.7525 0.6280 3.9440 Red. Mass: 1.3479 1.1142 6.5906 IR Active: YES YES YES IR Intens: 0.000 0.000 0.000 Raman Active: YES YES YES X Y Z X Y Z X Y Z C -0.000 -0.000 -0.101 -0.000 -0.000 0.040 0.291 0.000 0.000 C 0.000 0.000 0.040 -0.000 -0.000 -0.040 -0.146 -0.252 -0.000 C 0.000 0.000 0.061 0.000 0.000 -0.040 -0.146 0.252 0.000 C -0.000 -0.000 0.061 -0.000 -0.000 0.040 -0.146 0.252 -0.000 C -0.000 -0.000 0.040 -0.000 0.000 0.040 -0.146 -0.252 -0.000 C 0.000 0.000 -0.101 0.000 0.000 -0.040 0.291 -0.000 0.000 H -0.000 -0.000 0.564 -0.000 0.000 -0.406 0.287 0.000 0.000 H 0.000 -0.000 -0.222 0.000 -0.000 0.406 -0.144 -0.248 0.000 H 0.000 0.000 -0.342 0.000 0.000 0.406 -0.144 0.248 -0.000 H -0.000 -0.000 -0.342 -0.000 -0.000 -0.406 -0.144 0.248 0.000 H -0.000 0.000 -0.222 -0.000 0.000 -0.406 -0.144 -0.247 0.000 H 0.000 0.000 0.564 0.000 0.000 0.406 0.287 -0.000 -0.000 TransDip 0.000 0.000 -0.000 -0.000 0.000 0.000 -0.000 -0.000 0.000 Mode: 13 14 15 Frequency: 1023.12 1065.65 1065.85 Force Cnst: 3.6752 1.0973 1.0980 Red. Mass: 5.9591 1.6400 1.6405 IR Active: YES YES YES IR Intens: 0.000 3.538 3.547 Raman Active: YES YES YES X Y Z X Y Z X Y Z C 0.274 0.000 0.000 0.114 -0.000 -0.000 0.000 0.079 0.000 C 0.137 0.237 -0.000 -0.030 0.084 -0.000 -0.083 -0.066 -0.000 C 0.137 -0.237 -0.000 -0.031 -0.083 0.000 0.083 -0.066 0.000 C -0.137 0.237 0.000 -0.031 -0.083 0.000 0.083 -0.066 -0.000 C -0.137 -0.237 0.000 -0.030 0.084 -0.000 -0.084 -0.065 -0.000 C -0.274 -0.000 0.000 0.114 -0.000 -0.000 0.001 0.079 0.000 H 0.303 0.000 -0.000 0.124 -0.002 0.000 0.000 0.546 0.000 H 0.150 0.263 0.000 -0.378 0.291 0.000 -0.292 0.044 0.000 H 0.150 -0.263 0.000 -0.381 -0.291 -0.000 0.290 0.043 -0.000 H -0.150 0.263 -0.000 -0.379 -0.290 -0.000 0.289 0.042 0.000 H -0.150 -0.263 -0.000 -0.377 0.290 0.000 -0.294 0.045 0.000 H -0.303 -0.000 0.000 0.124 -0.002 0.000 0.001 0.546 -0.000 TransDip -0.000 -0.000 0.000 -0.060 -0.000 -0.000 -0.000 0.060 0.000 Mode: 16 17 18 Frequency: 1157.13 1195.21 1196.24 Force Cnst: 0.9191 0.9467 0.9466 Red. Mass: 1.1651 1.1248 1.1227 IR Active: YES YES YES IR Intens: 0.000 0.000 0.000 Raman Active: YES YES YES X Y Z X Y Z X Y Z C -0.000 0.049 0.000 0.001 -0.059 0.000 -0.006 -0.007 -0.000 C 0.042 -0.025 0.000 -0.023 0.007 0.000 -0.046 0.029 0.000 183 C -0.042 -0.025 0.000 0.033 0.014 -0.000 -0.039 -0.026 0.000 C -0.042 -0.024 0.000 -0.033 -0.014 -0.000 0.039 0.026 0.000 C 0.042 -0.024 -0.000 0.023 -0.007 0.000 0.046 -0.029 0.000 C 0.000 0.049 -0.000 -0.001 0.059 0.000 0.006 0.007 -0.000 H -0.000 0.407 0.000 0.001 -0.569 -0.000 -0.005 -0.066 0.000 H 0.351 -0.203 -0.000 -0.199 0.110 -0.000 -0.455 0.265 -0.000 H -0.351 -0.203 -0.000 0.299 0.168 0.000 -0.397 -0.233 -0.000 H -0.350 -0.202 0.000 -0.300 -0.169 0.000 0.398 0.233 -0.000 H 0.350 -0.202 0.000 0.200 -0.111 -0.000 0.456 -0.266 -0.000 H 0.000 0.405 -0.000 -0.001 0.570 -0.000 0.005 0.067 0.000 TransDip 0.000 0.000 -0.000 -0.000 -0.000 0.000 0.000 0.000 -0.000 Mode: 19 20 21 Frequency: 1318.83 1371.23 1521.59 Force Cnst: 4.7174 1.3826 2.8658 Red. Mass: 4.6034 1.2480 2.1009 IR Active: YES YES YES IR Intens: 0.000 0.000 7.261 Raman Active: YES YES YES X Y Z X Y Z X Y Z C 0.000 0.233 0.000 -0.000 -0.060 -0.000 0.098 -0.000 -0.000 C 0.202 -0.117 -0.000 0.052 -0.030 -0.000 -0.090 0.109 0.000 C -0.202 -0.117 -0.000 -0.052 -0.030 0.000 -0.090 -0.109 -0.000 C -0.202 -0.117 0.000 0.052 0.030 -0.000 -0.091 -0.109 0.000 C 0.202 -0.117 0.000 -0.052 0.030 0.000 -0.091 0.109 -0.000 C -0.000 0.233 0.000 0.000 0.060 0.000 0.098 -0.000 0.000 H 0.000 -0.335 -0.000 -0.000 0.405 -0.000 0.122 0.001 0.000 H -0.290 0.167 0.000 -0.349 0.202 0.000 0.430 -0.178 -0.000 H 0.290 0.167 0.000 0.349 0.201 0.000 0.431 0.178 0.000 H 0.290 0.168 0.000 -0.349 -0.201 0.000 0.432 0.179 -0.000 H -0.290 0.168 -0.000 0.349 -0.202 -0.000 0.431 -0.178 0.000 H -0.000 -0.336 -0.000 0.000 -0.405 0.000 0.122 0.001 0.000 TransDip -0.000 -0.000 0.000 -0.000 -0.000 -0.000 0.086 0.000 0.000 Mode: 22 23 24 Frequency: 1521.83 1671.61 1671.99 Force Cnst: 2.8657 9.3932 9.2974 Red. Mass: 2.1001 5.7055 5.6448 IR Active: YES YES YES IR Intens: 7.283 0.000 0.000 Raman Active: YES YES YES X Y Z X Y Z X Y Z C -0.000 -0.153 0.000 -0.150 0.061 -0.000 -0.027 -0.337 0.000 C 0.109 0.035 0.000 0.307 -0.117 -0.000 -0.030 0.182 0.000 C -0.108 0.036 0.000 0.279 0.047 -0.000 0.133 0.211 -0.000 C -0.109 0.035 0.000 -0.278 -0.046 -0.000 -0.133 -0.211 -0.000 C 0.109 0.035 -0.000 -0.307 0.117 -0.000 0.030 -0.182 0.000 C 0.000 -0.153 0.000 0.150 -0.060 -0.000 0.026 0.338 -0.000 H -0.000 0.535 0.000 -0.174 -0.073 -0.000 -0.031 0.395 -0.000 H -0.179 0.226 -0.000 -0.297 0.239 0.000 0.202 0.074 -0.000 H 0.177 0.224 -0.000 -0.211 -0.253 0.000 -0.293 -0.013 0.000 H 0.178 0.225 -0.000 0.206 0.249 0.000 0.290 0.012 0.000 H -0.179 0.225 0.000 0.295 -0.239 0.000 -0.202 -0.074 -0.000 H 0.000 0.534 -0.000 0.173 0.068 -0.000 0.031 -0.396 -0.000 TransDip -0.000 0.086 0.000 -0.000 -0.000 0.000 -0.000 -0.000 0.000 Mode: 25 26 27 Frequency: 3195.89 3206.53 3206.78 Force Cnst: 6.5139 6.5915 6.5917 Red. Mass: 1.0824 1.0881 1.0879 IR Active: YES YES YES IR Intens: 0.002 0.000 0.000 Raman Active: YES YES YES 184 X Y Z X Y Z X Y Z C -0.033 -0.000 -0.000 -0.000 -0.003 0.000 -0.049 0.000 -0.000 C 0.017 0.029 -0.000 0.020 0.038 -0.000 0.014 0.020 -0.000 C 0.017 -0.029 0.000 -0.020 0.038 0.000 0.014 -0.021 -0.000 C 0.017 -0.029 -0.000 0.020 -0.038 -0.000 -0.014 0.021 -0.000 C 0.017 0.029 0.000 -0.020 -0.038 0.000 -0.014 -0.020 -0.000 C -0.033 0.000 -0.000 0.000 0.003 0.000 0.049 -0.000 -0.000 H 0.402 0.000 0.000 0.001 0.002 0.000 0.572 -0.000 0.000 H -0.204 -0.354 0.000 -0.248 -0.433 -0.000 -0.147 -0.250 0.000 H -0.204 0.354 -0.000 0.248 -0.431 0.000 -0.148 0.252 0.000 H -0.205 0.354 0.000 -0.248 0.431 0.000 0.148 -0.252 0.000 H -0.204 -0.354 0.000 0.248 0.433 -0.000 0.147 0.250 0.000 H 0.402 0.000 -0.000 -0.001 -0.002 0.000 -0.572 0.000 -0.000 TransDip 0.001 0.000 0.000 0.000 -0.000 -0.000 -0.000 0.000 0.000 Mode: 28 29 30 Frequency: 3224.02 3224.34 3235.23 Force Cnst: 6.7178 6.7191 6.7935 Red. Mass: 1.0969 1.0969 1.1016 IR Active: YES YES YES IR Intens: 36.868 36.822 0.000 Raman Active: YES YES YES X Y Z X Y Z X Y Z C -0.000 0.004 -0.000 -0.052 -0.000 -0.000 0.038 0.000 0.000 C -0.024 -0.038 0.000 -0.010 -0.024 -0.000 0.019 0.033 0.000 C 0.024 -0.038 -0.000 -0.010 0.024 0.000 0.019 -0.033 -0.000 C 0.024 -0.038 0.000 -0.010 0.024 0.000 -0.019 0.033 -0.000 C -0.024 -0.038 -0.000 -0.010 -0.024 0.000 -0.019 -0.033 -0.000 C -0.000 0.004 0.000 -0.052 -0.000 0.000 -0.038 -0.000 -0.000 H 0.000 -0.004 -0.000 0.578 0.000 0.000 -0.412 -0.000 -0.000 H 0.251 0.430 -0.000 0.139 0.248 0.000 -0.202 -0.350 0.000 H -0.251 0.430 0.000 0.139 -0.248 -0.000 -0.202 0.350 0.000 H -0.251 0.430 -0.000 0.139 -0.248 -0.000 0.202 -0.350 0.000 H 0.251 0.430 -0.000 0.139 0.248 0.000 0.202 0.350 0.000 H 0.000 -0.004 0.000 0.578 0.000 0.000 0.412 0.000 -0.000 TransDip -0.000 -0.194 -0.000 -0.194 0.000 -0.000 -0.000 0.000 0.000 185 5.8.6 Frequencies and normal modes of benzene, computed by finite dif- ference with single precision amplitudes with cleanup in double pre- cision and step size of 0.01 ˚ A Mode: 1 2 3 Frequency: 404.72 404.72 611.79 Force Cnst: 0.2892 0.2892 1.3295 Red. Mass: 2.9970 2.9970 6.0288 IR Active: YES YES YES IR Intens: 0.000 0.000 0.000 Raman Active: YES YES YES X Y Z X Y Z X Y Z C -0.000 0.000 0.246 -0.000 -0.000 -0.001 0.360 -0.000 0.000 C 0.000 -0.000 -0.124 -0.000 0.000 -0.212 0.024 -0.222 -0.000 C -0.000 -0.000 -0.122 -0.000 0.000 0.213 0.024 0.221 -0.000 C -0.000 0.000 -0.122 0.000 -0.000 0.213 -0.024 -0.221 -0.000 C 0.000 -0.000 -0.124 -0.000 -0.000 -0.212 -0.024 0.222 -0.000 C 0.000 0.000 0.246 0.000 0.000 -0.001 -0.360 0.000 -0.000 H -0.000 0.000 0.522 -0.000 -0.000 -0.003 0.357 0.000 0.000 H 0.000 -0.000 -0.264 -0.000 0.000 -0.451 -0.263 -0.055 0.000 H -0.000 -0.000 -0.259 -0.000 0.000 0.454 -0.263 0.054 0.000 H -0.000 0.000 -0.259 -0.000 -0.000 0.454 0.263 -0.054 0.000 H 0.000 -0.000 -0.264 -0.000 -0.000 -0.451 0.263 0.055 0.000 H 0.000 0.000 0.522 0.000 0.000 -0.003 -0.357 -0.000 0.000 TransDip 0.000 0.000 0.000 0.000 -0.000 -0.000 -0.000 0.000 0.000 Mode: 4 5 6 Frequency: 611.90 654.40 691.47 Force Cnst: 1.3305 1.4171 0.3056 Red. Mass: 6.0312 5.6166 1.0848 IR Active: YES YES YES IR Intens: 0.000 0.000 88.291 Raman Active: YES YES YES X Y Z X Y Z X Y Z C -0.000 -0.152 -0.000 -0.000 0.000 -0.264 -0.000 -0.000 -0.034 C -0.221 -0.231 0.000 0.000 0.000 0.264 -0.000 0.000 -0.034 C 0.221 -0.232 0.000 -0.000 -0.000 0.264 0.000 -0.000 -0.034 C -0.221 0.232 0.000 0.000 -0.000 -0.264 0.000 0.000 -0.034 C 0.221 0.231 0.000 -0.000 0.000 -0.264 -0.000 0.000 -0.034 C 0.000 0.152 0.000 -0.000 -0.000 0.264 0.000 -0.000 -0.034 H -0.000 0.231 -0.000 -0.000 0.000 -0.311 -0.000 0.000 0.407 H -0.054 -0.326 -0.000 -0.000 0.000 0.311 0.000 -0.000 0.407 H 0.055 -0.326 -0.000 -0.000 -0.000 0.311 0.000 -0.000 0.407 H -0.055 0.326 -0.000 0.000 -0.000 -0.311 -0.000 0.000 0.407 H 0.054 0.326 -0.000 0.000 0.000 -0.311 -0.000 0.000 0.407 H 0.000 -0.231 -0.000 -0.000 0.000 0.311 0.000 -0.000 0.407 TransDip -0.000 -0.000 -0.000 -0.000 0.000 -0.000 -0.000 -0.000 0.301 Mode: 7 8 9 Frequency: 867.93 867.93 973.41 Force Cnst: 0.5539 0.5539 0.7525 Red. Mass: 1.2480 1.2480 1.3479 IR Active: YES YES YES IR Intens: 0.000 0.000 0.000 Raman Active: YES YES YES X Y Z X Y Z X Y Z C -0.000 0.000 -0.085 -0.000 0.000 0.000 0.000 0.000 -0.102 C 0.000 0.000 -0.043 -0.000 -0.000 -0.074 0.000 -0.000 0.051 186 C 0.000 -0.000 -0.043 -0.000 0.000 0.074 -0.000 -0.000 0.050 C 0.000 -0.000 0.043 0.000 -0.000 -0.074 -0.000 -0.000 0.050 C 0.000 0.000 0.043 -0.000 0.000 0.074 0.000 -0.000 0.051 C -0.000 -0.000 0.085 0.000 0.000 -0.000 -0.000 0.000 -0.102 H -0.000 0.000 0.571 -0.000 0.000 -0.000 0.000 0.000 0.568 H -0.000 0.000 0.286 -0.000 -0.000 0.494 0.000 0.000 -0.286 H 0.000 -0.000 0.285 0.000 0.000 -0.495 -0.000 -0.000 -0.282 H 0.000 -0.000 -0.285 0.000 -0.000 0.495 -0.000 -0.000 -0.282 H 0.000 0.000 -0.286 -0.000 0.000 -0.494 0.000 -0.000 -0.286 H -0.000 -0.000 -0.571 0.000 0.000 0.000 -0.000 0.000 0.568 TransDip -0.000 0.000 0.000 -0.000 0.000 -0.000 0.000 0.000 0.000 Mode: 10 11 12 Frequency: 973.41 978.07 1007.82 Force Cnst: 0.7525 0.6280 3.9440 Red. Mass: 1.3479 1.1142 6.5906 IR Active: YES YES YES IR Intens: 0.000 0.000 0.000 Raman Active: YES YES YES X Y Z X Y Z X Y Z C 0.000 -0.000 0.000 -0.000 0.000 0.040 0.291 0.000 0.000 C -0.000 -0.000 0.088 0.000 0.000 -0.040 -0.146 -0.252 -0.000 C -0.000 0.000 -0.088 0.000 -0.000 -0.040 -0.146 0.252 -0.000 C -0.000 0.000 -0.088 0.000 -0.000 0.040 -0.146 0.252 0.000 C -0.000 -0.000 0.088 0.000 0.000 0.040 -0.146 -0.252 0.000 C 0.000 0.000 0.000 -0.000 -0.000 -0.040 0.291 0.000 -0.000 H 0.000 -0.000 -0.003 -0.000 0.000 -0.406 0.287 0.000 -0.000 H -0.000 -0.000 -0.491 0.000 0.000 0.406 -0.144 -0.248 0.000 H -0.000 0.000 0.493 0.000 -0.000 0.406 -0.144 0.248 0.000 H -0.000 0.000 0.493 0.000 -0.000 -0.406 -0.144 0.248 -0.000 H -0.000 -0.000 -0.491 0.000 0.000 -0.406 -0.144 -0.248 -0.000 H 0.000 0.000 -0.003 -0.000 0.000 0.406 0.287 0.000 0.000 TransDip 0.000 0.000 -0.000 0.000 0.000 -0.000 -0.000 0.000 0.000 Mode: 13 14 15 Frequency: 1023.12 1065.65 1065.85 Force Cnst: 3.6752 1.0973 1.0980 Red. Mass: 5.9591 1.6400 1.6405 IR Active: YES YES YES IR Intens: 0.000 3.538 3.546 Raman Active: YES YES YES X Y Z X Y Z X Y Z C -0.274 0.000 -0.000 0.114 0.000 -0.000 -0.000 0.079 -0.000 C -0.137 -0.237 0.000 -0.030 0.083 -0.000 -0.083 -0.066 0.000 C -0.137 0.237 0.000 -0.030 -0.083 0.000 0.083 -0.066 0.000 C 0.137 -0.237 0.000 -0.030 -0.083 0.000 0.083 -0.066 0.000 C 0.137 0.237 -0.000 -0.030 0.083 -0.000 -0.083 -0.066 -0.000 C 0.274 -0.000 -0.000 0.114 0.000 0.000 -0.000 0.079 -0.000 H -0.303 0.000 0.000 0.124 0.000 0.000 -0.000 0.546 0.000 H -0.151 -0.262 -0.000 -0.379 0.291 -0.000 -0.291 0.044 -0.000 H -0.151 0.262 -0.000 -0.379 -0.291 -0.000 0.291 0.044 -0.000 H 0.151 -0.262 -0.000 -0.379 -0.291 0.000 0.291 0.044 -0.000 H 0.151 0.262 0.000 -0.379 0.291 0.000 -0.291 0.044 0.000 H 0.303 -0.000 -0.000 0.124 0.000 -0.000 -0.000 0.546 -0.000 TransDip -0.000 -0.000 -0.000 -0.060 0.000 0.000 0.000 0.060 -0.000 Mode: 16 17 18 Frequency: 1157.13 1195.00 1195.21 Force Cnst: 0.9190 0.9461 0.9465 Red. Mass: 1.1650 1.1244 1.1246 IR Active: YES YES YES IR Intens: 0.000 0.000 0.000 Raman Active: YES YES YES 187 X Y Z X Y Z X Y Z C 0.000 0.049 0.000 0.006 0.000 -0.000 -0.000 0.059 0.000 C 0.042 -0.024 -0.000 0.043 -0.028 0.000 0.028 -0.011 0.000 C -0.042 -0.024 -0.000 0.043 0.028 -0.000 -0.028 -0.011 0.000 C -0.042 -0.024 -0.000 -0.043 -0.028 0.000 0.028 0.011 -0.000 C 0.042 -0.024 -0.000 -0.043 0.028 0.000 -0.028 0.011 0.000 C -0.000 0.049 0.000 -0.006 -0.000 -0.000 0.000 -0.059 -0.000 H 0.000 0.406 -0.000 0.005 0.001 0.000 -0.000 0.574 0.000 H 0.351 -0.203 0.000 0.430 -0.251 -0.000 0.250 -0.140 -0.000 H -0.351 -0.203 0.000 0.429 0.250 0.000 -0.252 -0.141 -0.000 H -0.351 -0.203 0.000 -0.429 -0.250 -0.000 0.252 0.141 0.000 H 0.351 -0.203 0.000 -0.430 0.251 -0.000 -0.250 0.140 -0.000 H -0.000 0.406 -0.000 -0.005 -0.001 0.000 0.000 -0.574 -0.000 TransDip -0.000 0.000 0.000 0.000 -0.000 -0.000 -0.000 -0.000 -0.000 Mode: 19 20 21 Frequency: 1318.87 1371.23 1521.59 Force Cnst: 4.7193 1.3826 2.8658 Red. Mass: 4.6049 1.2480 2.1009 IR Active: YES YES YES IR Intens: 0.000 0.000 7.261 Raman Active: YES YES YES X Y Z X Y Z X Y Z C -0.000 0.233 0.000 0.000 -0.060 0.000 0.098 -0.000 0.000 C 0.202 -0.117 -0.000 0.052 -0.030 -0.000 -0.090 0.109 0.000 C -0.202 -0.117 -0.000 -0.052 -0.030 -0.000 -0.090 -0.109 0.000 C -0.202 -0.117 -0.000 0.052 0.030 -0.000 -0.090 -0.109 0.000 C 0.202 -0.117 -0.000 -0.052 0.030 -0.000 -0.090 0.109 -0.000 C 0.000 0.233 0.000 -0.000 0.060 -0.000 0.098 0.000 -0.000 H -0.000 -0.335 -0.000 0.000 0.404 -0.000 0.122 0.000 0.000 H -0.290 0.167 0.000 -0.349 0.202 -0.000 0.431 -0.178 -0.000 H 0.290 0.167 0.000 0.349 0.202 0.000 0.431 0.178 -0.000 H 0.290 0.167 0.000 -0.349 -0.202 0.000 0.431 0.178 -0.000 H -0.290 0.167 0.000 0.349 -0.202 0.000 0.431 -0.178 0.000 H 0.000 -0.335 -0.000 -0.000 -0.404 -0.000 0.122 -0.000 -0.000 TransDip 0.000 -0.000 0.000 -0.000 -0.000 0.000 0.086 0.000 -0.000 Mode: 22 23 24 Frequency: 1521.83 1672.13 1672.16 Force Cnst: 2.8657 9.3165 9.3077 Red. Mass: 2.1001 5.6554 5.6499 IR Active: YES YES YES IR Intens: 7.283 0.000 0.000 Raman Active: YES YES YES X Y Z X Y Z X Y Z C -0.000 -0.153 -0.000 -0.152 -0.004 -0.000 -0.002 0.343 0.000 C 0.109 0.035 -0.000 0.294 -0.080 0.000 0.086 -0.201 0.000 C -0.109 0.035 0.000 0.296 0.085 -0.000 -0.079 -0.199 0.000 C -0.109 0.035 -0.000 -0.296 -0.085 0.000 0.079 0.199 -0.000 C 0.109 0.035 0.000 -0.294 0.080 0.000 -0.086 0.201 0.000 C 0.000 -0.153 0.000 0.152 0.004 0.000 0.002 -0.343 -0.000 H -0.000 0.534 -0.000 -0.175 0.005 0.000 -0.002 -0.402 0.000 H -0.178 0.225 0.000 -0.255 0.250 -0.000 -0.253 -0.028 -0.000 H 0.178 0.225 0.000 -0.261 -0.250 0.000 0.247 -0.034 -0.000 H 0.178 0.225 0.000 0.261 0.250 -0.000 -0.247 0.034 0.000 H -0.178 0.225 -0.000 0.255 -0.250 -0.000 0.253 0.028 -0.000 H 0.000 0.534 0.000 0.175 -0.005 0.000 0.002 0.402 -0.000 TransDip -0.000 0.086 0.000 0.000 -0.000 -0.000 0.000 -0.000 -0.000 Mode: 25 26 27 Frequency: 3195.89 3206.51 3206.82 Force Cnst: 6.5139 6.5911 6.5923 Red. Mass: 1.0824 1.0880 1.0880 188 IR Active: YES YES YES IR Intens: 0.002 0.000 0.000 Raman Active: YES YES YES X Y Z X Y Z X Y Z C -0.033 0.000 0.000 -0.000 -0.003 -0.000 0.049 0.000 -0.000 C 0.017 0.029 0.000 0.020 0.038 -0.000 -0.014 -0.020 -0.000 C 0.017 -0.029 0.000 -0.020 0.038 0.000 -0.014 0.020 0.000 C 0.017 -0.029 -0.000 0.020 -0.038 0.000 0.014 -0.020 -0.000 C 0.017 0.029 -0.000 -0.020 -0.038 0.000 0.014 0.020 -0.000 C -0.033 -0.000 -0.000 0.000 0.003 0.000 -0.049 -0.000 -0.000 H 0.402 0.000 0.000 0.000 0.002 -0.000 -0.572 0.000 -0.000 H -0.204 -0.354 -0.000 -0.248 -0.432 0.000 0.147 0.251 0.000 H -0.204 0.354 -0.000 0.248 -0.432 -0.000 0.147 -0.251 -0.000 H -0.204 0.354 0.000 -0.248 0.432 -0.000 -0.147 0.251 0.000 H -0.204 -0.354 0.000 0.248 0.432 -0.000 -0.147 -0.251 0.000 H 0.402 0.000 -0.000 -0.000 -0.002 0.000 0.572 -0.000 -0.000 TransDip 0.001 0.000 0.000 -0.000 0.000 -0.000 -0.000 -0.000 -0.000 Mode: 28 29 30 Frequency: 3224.02 3224.34 3235.23 Force Cnst: 6.7178 6.7191 6.7934 Red. Mass: 1.0969 1.0969 1.1016 IR Active: YES YES YES IR Intens: 36.867 36.821 0.000 Raman Active: YES YES YES X Y Z X Y Z X Y Z C -0.000 0.004 0.000 -0.052 -0.000 -0.000 -0.038 0.000 0.000 C -0.024 -0.038 0.000 -0.010 -0.024 -0.000 -0.019 -0.032 0.000 C 0.024 -0.038 0.000 -0.010 0.024 -0.000 -0.019 0.032 -0.000 C 0.024 -0.038 0.000 -0.010 0.024 0.000 0.019 -0.032 0.000 C -0.024 -0.038 0.000 -0.010 -0.024 0.000 0.019 0.032 -0.000 C -0.000 0.004 0.000 -0.052 -0.000 0.000 0.038 -0.000 -0.000 H 0.000 -0.004 0.000 0.578 0.000 -0.000 0.411 -0.000 0.000 H 0.251 0.430 -0.000 0.139 0.248 0.000 0.202 0.350 -0.000 H -0.251 0.430 -0.000 0.139 -0.248 0.000 0.202 -0.350 -0.000 H -0.251 0.430 -0.000 0.139 -0.248 -0.000 -0.202 0.350 -0.000 H 0.251 0.430 0.000 0.139 0.248 -0.000 -0.202 -0.350 0.000 H 0.000 -0.004 0.000 0.578 0.000 0.000 -0.411 0.000 -0.000 TransDip -0.000 -0.194 0.000 -0.194 0.000 -0.000 -0.000 0.000 0.000 189 5.9 Chapter 5 References 1 T. Helgaker, P. Jørgensen, and J. Olsen, Molecular electronic structure theory. Wiley & Sons, 2000. 2 R. J. Bartlett and I. Shavitt, Many-Body Methods in Chemistry and Physics: MBPT and Coupled-Cluster Theory. Cambridge University Press, 2009. 3 R. J. Bartlett, Mol. Phys. 108, 2905 (2010). 4 R. J. Bartlett, How and why coupled-cluster theory became the preeminent method in ab initio quantum chemistry, in Theory and Applications of computational chemistry, edited by C. Dykstra, G. Frenking, and G. Scuseria. Elsevier, 2005. 5 R. J. Bartlett, Int. J. Mol. Sci. 3, 579 (2002). 6 A. I. Krylov, Annu. Rev. Phys. Chem. 59, 433 (2008). 7 K. Sneskov and O. Christiansen, WIREs: Comput. Mol. Sci. 2, 566 (2012). 8 R. J. Bartlett, WIREs: Comput. Mol. Sci. 2, 126 (2012). 9 J.A. Pople, Rev. Mod. Phys. 71, 1267 (1999). 10 B. Ruscic, R. E Pinzon, G. von Laszewski, D. Kodeboyina, A. Burcat, D. Leahy, D. Montoy, and A. F. Wagner, J. Phys. Conf. Ser. 16, 561 (2005). 11 A. Tajti, P. G. Szalay, A. G. Cs´ asz´ ar, M. K´ allay, J. Gauss, E. F. Valeev, B. A. Flowers, J. V´ azquez, and J. F. Stanton, J. Chem. Phys. 121, 11599 (2004). 12 IEEE Standard for Binary Floating-Point Arithmetic. ANSI/IEEE Std 754-1985, 1985. 13 IEEE Standard for Floating-Point Arithmetic - Redline. IEEE Std 754-2008 - Redline, 1985. 14 G. J. Gustafson, Right-sizing precision: Unleashed computing: The need to right-size precision to save energy, bandwidth, storage, and electrical power, https://web.archive.org/web/20160606203112/http://www.johngustafson.net/presentations/Right- SizingPrecision1.pdf, 2015, Presentation from web archive. 15 M.A. Clark, R. Babich, K. Barros, R.C. Brower, and C. Rebbi, Comp. Phys. Comm. 181, 1517 (2010). 16 S. L. Grand, A. W. G¨ otz, and R. C. Walker, Comp. Phys. Comm. 184, 374 (2013). 190 17 M. Baboulin, A. Buttari, J. Dongarra, J. Kurzak, J. Langou, J. Langou, P. Luszczek, and S. Tomov, Comp. Phys. Comm. 180, 2526 (2009), 40 YEARS OF CPC: A celebratory issue focused on quality software for high performance, grid and novel computing architectures. 18 I. S. Ufimtsev and T. J. Mart´ ınez, J. Chem. Theory Comput. 4, 222 (2008). 19 I. S. Ufimtsev and T. J. Mart´ ınez, J. Chem. Theory Comput. 5, 1004 (2009). 20 I. S. Ufimtsev and T. J. Mart´ ınez, J. Chem. Theory Comput. 5, 2619 (2009). 21 A. E. DePrince and J. R. Hammond, J. Chem. Theory Comput. 7, 1287 (2011). 22 A. E. DePrince, M. R. Kennedy, B. G. Sumpter, and C. D. Sherrill, Mol. Phys. 112, 844 (2014). 23 J. J. Eriksen, Chapter 12 - incrementally accelerating the RI-MP2 correlated method of elec- tronic structure theory using OpenACC compiler directives, in Parallel Programming with OpenACC, edited by R. Farber, pages 241–265. Morgan Kaufmann, 2017. 24 J. J. Eriksen, Mol. Phys. 115, 2086 (2017). 25 A. Asadchev and M. S. Gordon, Comp. Phys. Comm. 183, 1563 (2012). 26 ´ A. R´ ak and G. Cserey, Chem. Phys. Lett. 622, 92 (2015). 27 N. Luehr, I. S. Ufimtsev, and T. J. Mart´ ınez, J. Chem. Theory Comput. 7, 949 (2011). 28 L. V ogt, R. Olivares-Amaya, S. Kermes, Y . Shao, C. Amador-Bedolla, and A. Aspuru-Guzik, J. Phys. Chem. A 112, 2049 (2008). 29 V . P. Vysotskiy and L. S. Cederbaum, J. Chem. Theory Comput. 7, 320 (2011). 30 G. Knizia, W. Li, S. Simon, and H.-J. Werner, J. Chem. Theory Comput. 7, 2387 (2011). 31 E. Epifanovsky, M. Wormit, T. Ku´ s, A. Landau, D. Zuev, K. Khistyaev, P. U. Manohar, I. Kali- man, A. Dreuw, and A. I. Krylov, J. Comput. Chem. 34, 2293 (2013). 32 I. Kaliman and A. I. Krylov, J. Comput. Chem. 38, 842 (2017). 33 A. I. Krylov and P. M. W. Gill, WIREs: Comput. Mol. Sci. 3, 317 (2013). 34 Y . Shao, Z. Gan, E. Epifanovsky, A.T.B. Gilbert, M. Wormit, J. Kussmann, A.W. Lange, A. Behn, J. Deng, X. Feng, D. Ghosh, M. Goldey, P.R. Horn, L.D. Jacobson, I. Kaliman, R.Z. Khaliullin, T. Kus, A. Landau, J. Liu, E.I. Proynov, Y .M. Rhee, R.M. Richard, M.A. Rohrdanz, R.P. Steele, E.J. Sundstrom, H.L. Woodcock III, P.M. Zimmerman, D. Zuev, B. Albrecht, E. Alguires, B. Austin, G.J.O. Beran, Y .A. Bernard, E. Berquist, K. Brandhorst, K.B. Bravaya, S.T. Brown, D. Casanova, C.-M. Chang, Y . Chen, S.H. Chien, K.D. Closser, 191 D.L. Crittenden, M. Diedenhofen, R.A. DiStasio Jr., H. Do, A.D. Dutoi, R.G. Edgar, S. Fatehi, L. Fusti-Molnar, A. Ghysels, A. Golubeva-Zadorozhnaya, J. Gomes, M.W.D. Hanson- Heine, P.H.P. Harbach, A.W. Hauser, E.G. Hohenstein, Z.C. Holden, T.-C. Jagau, H. Ji, B. Kaduk, K. Khistyaev, J. Kim, J. Kim, R.A. King, P. Klunzinger, D. Kosenkov, T. Kowal- czyk, C.M. Krauter, K.U. Laog, A. Laurent, K.V . Lawler, S.V . Levchenko, C.Y . Lin, F. Liu, E. Livshits, R.C. Lochan, A. Luenser, P. Manohar, S.F. Manzer, S.-P. Mao, N. Mardirossian, A.V . Marenich, S.A. Maurer, N.J. Mayhall, C.M. Oana, R. Olivares-Amaya, D.P. O’Neill, J.A. Parkhill, T.M. Perrine, R. Peverati, P.A. Pieniazek, A. Prociuk, D.R. Rehn, E. Rosta, N.J. Russ, N. Sergueev, S.M. Sharada, S. Sharmaa, D.W. Small, A. Sodt, T. Stein, D. Stuck, Y .-C. Su, A.J.W. Thom, T. Tsuchimochi, L. V ogt, O. Vydrov, T. Wang, M.A. Watson, J. Wenzel, A. White, C.F. Williams, V . Vanovschi, S. Yeganeh, S.R. Yost, Z.-Q. You, I.Y . Zhang, X. Zhang, Y . Zhou, B.R. Brooks, G.K.L. Chan, D.M. Chipman, C.J. Cramer, W.A. Goddard III, M.S. Gordon, W.J. Hehre, A. Klamt, H.F. Schaefer III, M.W. Schmidt, C.D. Sherrill, D.G. Truhlar, A. Warshel, X. Xu, A. Aspuru-Guzik, R. Baer, A.T. Bell, N.A. Besley, J.-D. Chai, A. Dreuw, B.D. Dunietz, T.R. Furlani, S.R. Gwaltney, C.-P. Hsu, Y . Jung, J. Kong, D.S. Lambrecht, W.Z. Liang, C. Ochsenfeld, V .A. Rassolov, L.V . Slipchenko, J.E. Subotnik, T. Van V oorhis, J.M. Herbert, A.I. Krylov, P.M.W. Gill, and M. Head-Gordon, Mol. Phys. 113, 184 (2015). 35 T.-C. Jagau, K. B. Bravaya, and A. I. Krylov, Annu. Rev. Phys. Chem. 68, 525 (2017). 36 A. Dreuw and M. Wormit, WIREs: Comput. Mol. Sci. 5, 82 (2015). 37 K. Z. Ibrahim, E. Epifanovsky, S. Williams, and A. I. Krylov, J. Parallel Distrib. Comput. 106, 92 (2017). 38 S. V . Levchenko and A. I. Krylov, J. Chem. Phys. 120, 175 (2004). 39 L. A. Curtiss, K. Raghavachari, and J. A. Pople, J. Chem. Phys. 106, 1063 (1997). 40 A. I. Krylov, Acc. Chem. Res. 39, 83 (2006). 41 L. V . Slipchenko and A. I. Krylov, J. Chem. Phys. 117, 4694 (2002). 42 K. W. Sattelmeyer, H. F. Schaefer, and J. F. Stanton, Chem. Phys. Lett. 378, 42 (2003). 43 N. M. O’Boyle, M. Banck, C. A. James, C. Morley, T. Vandermeersch, and G. R. Hutchison, J. Cheminform. 3, 33 (2011). 44 K. B. Bravaya, E. Epifanovsky, and A. I. Krylov, J. Phys. Chem. Lett. 3, 2726 (2012). 192 Chapter 6: Extension of frozen natural orbital approximation to open-shell references: Theory, implementation, and application to single-molecule magnets 6.1 Introduction The scope of applicability of correlated quantum chemical methods is limited by their steep scaling with respect to the system size 1 . To formulate the algorithms in terms of basic linear algebra operations such as matrix multiplication, many-body wave-functions are represented by Slater determinants composed of molecular orbitals, which are, in turn, represented as an expansion over one-electron basis sets (i.e., atomic orbitals). Consequently, the ultimate scaling of a many-body method is determined not only by the number of electrons, but also by the number of basis functions. For example, the overallN 6 scaling of the coupled-cluster method with single and double excitations (CCSD) comes from the computational step involvingO 2 V 4 operations, whereO andV denote the number of occupied and virtual orbitals, respectively 2 . Whereas the former is determined by the number of electrons, the latter depends on the number of basis functions. Full inclusion of triple excitations, as in CCSDT 3 , leads toN 8 complexity, 193 while perturbative treatments of triples, such as in (T) 4–6 , ( ~ T) 7 , (2) 8 , (dT) 9 , (fT) 9 , and related approaches 10–12 , give rise toN 7 scaling. There are various strategies of reducing computational costs by using more compact rep- resentations of many-body wave-functions and the Hamiltonian. The sparsity of two-electron repulsion integrals (ERI) can be utilized through the approximate representation using auxiliary basis sets obtained by density fitting, as in the resolution-of-identity (RI) methods 13–21 , or by the Cholesky decomposition of ERIs 22–31 , or using alternative schemes such as pseudospectral decomposition 32–34 , chain-of-spheres exchange 35–37 , and tensor hypercontraction 38–42 . Decom- position of amplitudes can also be used to reduce computational costs 43–46 . A number of strategies exploit the physical decay of Coulomb interaction with a distance by using localized orbitals 47–50 . Based on orbital localization and truncation of non-interacting orbital subsets, reduced-scaling versions of MP2 51, 52 , CCSD 53, 54 , (T) correction 55–57 and beyond 58 have been developed. More compact representations of correlated wave-functions can be achieved by using var- ious flavors of natural orbitals (NOs), which has been exploited since early days of quantum chemistry. Natural orbitals, as introduced by L¨ owdin 59 , are defined as eigenvectors of a state one-particle density matrix: pq =h jp y qj i; (6.1) wherep y /q denote creation and annihilation operators corresponding to spin orbitals p = q , and indices p, q run over all spin-orbitals. The trace of equals the number of electrons and its eigenvalues are non-negative numbers between 0 and 1. The magnitude of these eigenvalues, called occupation numbers, reflects the relative weights of configurations in which the respec- tive NO is occupied. For example, for a single Slater determinant composed of a subset 194 off p g, pq is a diagonal matrix with pp = 1 for the orbitals occupied in and zero oth- erwise. Thus, molecular orbitals from a Hartree–Fock calculation are also natural orbitals of the underlying Hartree–Fock wave-function. Moreover, because all occupied orbitals have unit occupancy and all virtual orbitals have zero occupancy, any unitary transformation within the occupied or virtual space yields a set of Hartree–Fock natural orbitals. For multideterminantal wave-functions, the occupation numbers become fractional. Because the occupations are pro- portional to the relative weights of the configurations in which a particular orbital is occupied, one can use natural orbitals to compress the orbital space by dropping the orbitals with low occupation numbers. This utility of natural orbitals to compress correlated wave-functions 59 has been exploited in a variety of configuration interaction methods 60–68 . More recently, hybrid approaches using both localization techniques and natural or pair-natural orbitals have led to the development of very effective computer implementations of modern many-body theories 69–77 , enabling, for example, CC calculations on a protein (crambin) consisting of more than 600 atoms and with more than 6000 basis functions 73 . Here we are concerned with a very simple strategy to reduce computational cost of single- reference correlated calculations, that is, truncating the virtual orbital space while leaving the occupied space untouched, such that the exact correlation energy of a given method is smoothly recovered as the fraction of frozen virtual orbitals approaches zero. By using perturbation the- ory arguments, one can, of course, justify simply freezing high-lying canonical Hartree–Fock orbitals. However, natural occupations provide a much better gauge of the relative importance of orbitals in terms of their contributions into the total correlation energy. In order to preserve the definition of the vacuum (which is determined by choosing a particular reference determi- nant), the concept of Frozen Natural Orbitals (FNOs) was introduced 78 . FNOs are eigenstates of the virtual–virtual block of the one-particle density matrix; thus, they can be used to trans- form only the virtual space, without changing the reference determinant. Of course, to compute 195 FNOs one needs to know the full correlated wave-function. A typical strategy of using FNOs as a mean to reduce computational costs is to use a density matrix computed at a lower level of theory (say, MP2, which scales asN 5 ) to reduce the cost of higher-level calculations, such as CCSD or CCSD(T). This simple yet effective idea has been pioneered by Bartlett and co- workers 79–81 in the context of the ground-state CC calculations. The benchmarks have shown that significant computational gains can be achieved while introducing insignificant errors in the optimized geometries, relative energies, barrier heights 80, 81 , as well as in noncovalent inter- actions 82 . However, calculations of response properties have proven to be more challenging, requiring modification of the approach 83, 84 . Frozen NOs Active virtual MOs Singly occupied MOs Doubly occupied MOs Figure 6.1: Frozen natural orbitals (FNOs) are defined as eigenstates of the virtual–virtual block of a correlated state density matrix. In the FNO approach, the occupied space is unchanged, but the virtual orbital space is transformed such that the orbitals can be ordered by their relative significance for the correction energy and the orbitals with the lowest occupations can be frozen. Left and right panels highlight the difference between calculations using closed- and open-shell references. Because in the latter the and orbital spaces are different, a special care is needed to arrange the orbitals by maximum correspondence, so that dropped orbitals do not introduce an imbalance in the singly occu- pied space (marked by dashed box). By definition, natural orbitals are state-specific, reflecting the state-specific nature of corre- lation. Consequently, exploiting natural orbitals within multi-state calculations is not straight- forward. The difficulty of applying natural orbitals to excited states stems from the fact that a 196 one-size-fits-all truncation of orbital space is likely to introduce an imbalance in the excited- states description. One can even imagine an extreme situation when the truncation based on correlation for one state would exclude the orbitals that are needed to describe the princi- pal character of another state. For example, choosing orbitals based on their contributions to the ground-state correlation energy is likely to provide a reasonable virtual space for low- lying valence states (e.g., HOMO-LUMO excitations); however, it would exclude orbitals needed for the description of Rydberg states because diffuse orbitals are not very important for ground-state correlation. Nevertheless, several ideas have been explored in the context of single-reference excited-state calculations, including using state-specific or averaged NOs 85 , pair-natural orbitals 86 , and natural transition orbitals 87, 88 . In contrast to general excited-state calculations, virtual space truncation schemes based on a single-state density matrix are admissible in selected multi-state Fock-space methods, such as equation-of-motion coupled-cluster (EOM-CC) 89–92 . For example, multiple ionized states can be described in a balanced way by using a virtual space truncation scheme based on the density matrix computed for the closed-shell reference state 93 because the principal character of ionized (i.e., hole) states is confined to the occupied orbital space. This idea 93 , originally explored within FNOs 93 , has been recently extended to the domain local pair-natural orbitals 77 . Here we extend the idea of using reference-state FNOs to other EOM-CC methods. The EOM-CC family of methods 89–92 is based on the Fock-space parameterization of the wave-function. EOM-CC is a multi-state approach in which different manifolds of target states are described by choosing a particular combination of a reference state and a general excitation operator. General excitation operators in single-reference theories are defined with respect to the reference state, which defines the separation of the occupied and virtual orbital spaces; they can include only creation (i.e., particle) operators corresponding to the virtual orbitals and annihila- tion (i.e., hole) operators corresponding to the occupied orbitals. The reference state is described 197 IP DIP EA DEA EE SF Figure 6.2: In the EOM-CC formalism, different manifolds of target states are described by combining a particular reference state and a particular type of excitation operator (orbitals occupied in the reference state are marked by red dashed boxes). Note that in EOM-IP and EOM-DIP, the principal character of target states is described by the hole operators acting in the occupied orbital space (red boxes), whereas in EOM-EA and EOM- DEA it is described by the particle operators acting in the virtual space. In EOM-EE, the EOM operators act in both spaces. In EOM-SF, the leading electronic configurations of the target states are confined to the singly occupied orbital space. by the CC ansatz, which incorporates large amount of correlation and endows the theory with such important property as size-extensivity. Different flavors of EOM-CC are illustrated in Fig- ure 6.2. Electronically excited states are described by particle- and spin-conserving excitations from (usually) a closed-shell reference corresponding to the ground state, giving rise to the EOM-EE-CC method. Cationic or neutral doublet states can be described by ionizing opera- tors such as 1-hole (1h) and 2-hole-1-particle (2p1p), acting on a closed-shell reference, giving rise to EOM-IP 94, 95 . Doubly ionizing operators (2h and 3h1p) can be used to access diradical- type or doubly ionized states (EOM-DIP). In a similar fashion, electron attaching operators (1p, 1h2p, etc) acting on a closed-shell reference provide access to anionic or neutral doublet states (EOM-EA) 96 . Double electron attaching operators (2p and 3p1h) provide access to diradical states or to a subset of excited states derived from excitation from HOMO (EOM-DEA). By 198 using high-spin references and spin-flipping operators 97–99 , other types of multi-configurational states can be described (EOM-SF, EOM-DSF). Figure 6.2 clearly distinguishes between three groups of EOM-CC methods: one in which the principal character of target states is described by annihilation operators (EOM-IP, EOM- DIP), one in which the principal character is described by creation operators (EOM-EA, EOM- DEA), and one in which the principal character is described by both creation and annihilation operators (EOM-EE, EOM-SF, EOM-DSF). For the first group of methods, FNOs derived from the reference state density matrix should provide a reasonable recipe for the virtual space trun- cation, since the virtual space is primarily responsible for describing correlation of the target states. However, for the second group, truncation of the virtual space affects the description of the leading configurations of the target states, thus suggesting that the FNOs derived from the reference-state density matrix may not provide an optimal and balanced truncation scheme. The methods from the third group present an interesting case. Whereas the quality of the EOM- EE states can be adversely affected by the virtual space truncation based on the ground state correlation (as in the hypothetical example of Rydberg and valence states discussed above), the quality of the EOM-SF states should not be compromised by the virtual space truncation based on the reference-state correlation because the virtual orbitals needed to describe the target spin- flip states should have the same character as the occupied orbitals hosting the unpaired electrons in the high-spin reference. In other words, because the singly occupied orbital space (see Figure 6.1) is well defined by the choice of the high-spin reference, one can develop an effective trun- cation scheme of the rest of the virtual space based on the FNOs defined by the reference-state density matrix. Here we develop this idea into a practical algorithm of virtual space trunca- tion by using singular value decomposition (SVD) procedure and illustrate the performance of the resulting method by calculations of multiple electronic states and interstate properties in selected diradicals and triradicals. We note that although EOM-SF-CC is a single-reference 199 method and does not employ active spaces, the singly occupied orbital space defined by our procedure is conceptually similar to an active space of strongly correlated orbitals used within multi-reference formalisms 67, 100, 101 . The utility of natural orbitals computed for high-spin states in multi-reference calculations of ground and low-lying excited states has been successfully exploited 67, 68 by Lu and Matsika. 6.2 Theory 6.2.1 Equation-of-motion coupled-cluster methods The wave-function in EOM-CC methods is parameterized as j I i = ^ R I e ^ T j 0 i; (6.2) where ^ T is a coupled-cluster excitation operator, 0 is a reference determinant, and ^ R is an excitation 102 (EOM-EE-CCSD), spin-flip 97–99 (EOM-SF-CCSD, EOM-DSF-CCSD), ion- ization 94, 95 (EOM-IP-CCSD, EOM-DIP-CCSD), or electron-attachment 96 (EOM-EA-CCSD, EOM-DEA-CCSD) operator. EOM amplitudes R are obtained by diagonalization of the similarity-transformed Hamiltonian H in the basis of the determinants from the correspond- ing sector of the Fock space: H =e T He T ; (6.3) HR I =E I R I ; (6.4) L I H =L I E I ; (6.5) 200 Because H is non-Hermitian, it has distinct right and left eigenvectors, which can be chosen to form a biorthogonal set: L I R J = IJ : (6.6) Once the left and right eigenstates of H are computed, properties are evaluated as con- traction of the corresponding integrals with appropriate density matrices. Evaluation of the expectation value of a one-electron operator ^ A requires one-electron density matrix : ^ A = X pq h p jAj q ip y q; (6.7) h I j ^ Aj J i = X pq h p jAj q ih I jp y qj J i = X pq h p jAj q i I!J pq ; (6.8) where the labelsI andJ enumerate electronic states. Because of the non-Hermitian nature of EOM-CC,I!J andJ!I transition densities are different, resulting in different numerical values of the A IJ and A JI matrix elements. Geometric 102 or arithmetic average 103–105 of the matrix elements can be used to handle this disrepancy. When ^ R includes all possible excitations, EOM-CC is equivalent to full configuration inter- action (FCI). Practical approximate methods are based on truncation of ^ R (and ^ T ) at some excitation level, i.e., in EOM-SF-CCSD, ^ R is truncated at single and double excitations: R SF = X ia r a i a y i + 1 4 X ijab r ab ij a y b y ji; (6.9) where i and a denote occupied and virtual (with respect to 0 ) spin-orbitals. ^ T is usually truncated at the same excitation level and has the same general form as Eq. (6.9), except that ^ T is an M S = 0 operator (i.e., only the and blocks are non-zero in T 1 ) whereas R SF 201 is anM S =1 operator (only the block is non-zero inR SF 1 ). The truncation at the level of double excitations results in theN 6 scaling of the method. The approximations used within the EOM-CC framework preserve a number of important properties, such as orbital invariance with respect to the rotations in the occupied or virtual spaces and size consistency (or size- intesivity). The accuracy of EOM-SF can be systematically improved (up to the FCI limit) by including higher excitations. The benchmarks illustrate that even the lowest-level of EOM-SF methods yield accurate energy gaps due to the built-in balanced treatment of multiple electronic states. For example, when applied for calculating singlet and triplet states in diradicals, EOM- SF-CCSD provides excitation energies with the error bar of 0.03–0.05 eV 9, 106 ; the benchmark study on dicopper single-molecule magnets (SMMs) 107 have shown that even very small energy gaps of several hundreds or tens of wave numbers can be resolved. 6.2.2 FNO algorithm for closed- and open-shell references The original closed-shell FNO approximation is described by the following algorithm 93 : 1. Start from canonicalized orbitals. 2. Compute MP2T 2 amplitudes. 3. Compute theVV part of the MP2 density matrix for the reference state, vv . 4. Diagonalize vv to obtain NOs, sort them according to their occupation numbers in the descending order. 5. Freeze the NOs with the lowest occupation numbers according to a given criterion (either a fixed fraction of the virtual space or a fraction of the virtual space needed to recover the specified population threshold 93 ). 202 6. Semi-canonicalize active orbitals (this step is optional; it is recommended for faster con- vergence of iterative eigensolvers). 7. Carry out the transformation of the required integral blocks into the new orbital basis (only the blocks involving virtual orbitals need to be transformed). 8. Execute CCSD, CCSD(T), EOM-IP-CCSD, EOM-DIP-CCSD, etc in the truncated orbital space. As illustrated below by numerical examples, the application of this procedure to an open- shell reference results in an erratic behavior, which can be attributed to an unbalanced truncation of the and virtual orbital spaces. To solve this problem, we developed an algorithm for separating the virtual space into a subspace matching the orbitals which are singly occupied in the high-spin reference and the rest. This procedure (which we named OSFNO) effectively determines the singly occupied subspace and uses the same strategy as proposed in Ref. 108. We note that the issue of unbalanced truncation of and spaces does not appear in fully spin-adapted formulations, such as one used by Lu and Matsika 67, 68 . The correspondence between two sets of orbitals can be established by means of SVD of the overlap matrix: its singular values reveal the orbitals matching exactly, whereas smaller val- ues would correspond to partial overlap and zeros would correspond to orthogonal subspaces. Thus, SVD of the overlap between the occupied and virtual orbitals allows us to identify the orbitals from the singly occupied subspace. In all considered systems, two (for triplet ref- erences) or three (for quartet references) singular values of the overlap matrix are very close to one, clearly identifying the open-shell electrons, while all other singular values are much smaller. Assuming that the reference is a high-spin state in which the number of electrons is larger than the number of electrons, the OSFNO algorithm proceeds as follows: 203 1. Compute the overlap matrix between the occupied and virtual orbitals: S ov = (C MO ;o ) y S AO C MO ;v . 2. Perform SVD of the overlap matrix:S ov =U SVD (V SVD ) y . 3. Save singular vectorsU SVD andV SVD . 4. Compute newC-matrices: C SVDMO ;o =C MO ;o U SVD , C SVDMO ;v =C MO ;v V SVD . 5. Using near-unity singular values as a guide, eliminate open-shell subspacev; to form a virtual subspace ~ v from which the open-shell orbitals are excluded. 6. Compute MP2T 2 amplitudes in the basis of the original MOs. 7. Compute the VV part of the MP2 state density matrix vv in the basis of the original MOs. 8. Carry out MO!AO transformation (without multiplication by S 1 ) of vv , make AO and AO . 9. Extract the singlet density AO;S = 1 p 2 AO + AO . 10. Transform AO;S to the new ~ v orbitals: ~ v~ v;S =C y ;~ v S AO AO;S S AO C ;~ v : 11. Perform SVD of ~ v~ v;S . 12. Sort density singular values, freeze pairs according to a given criterion (the criteria are the same as in the original FNO scheme 93 ). 204 13. Semi-canonicalize the active orbitals (optional). 14. Transform the required integral blocks to the new orbital basis. 15. Carry out CCSD and EOM-SF-CCSD in the truncated orbital space. Steps 1–5 separate the open-shell space from the rest of the orbitals. The singlet part of the density is taken because it has the same and parts and it is consistent with the closed- shell case. The correspondence between the and NOs is established through SVD of the density matrix between the and orbitals. The selection criterion of the pairs is the respective singular values. We implemented this algorithm in the Q-Chem electronic structure package 109, 110 . 6.2.3 Computational details We investigated the performance of the FNO approximation within the EOM-SF-CCSD method for the set of prototypical diradicals and triradicals: 1. Methylene (CH 2 ) and isolectronic species: NH + 2 , SiH 2 , PH + 2 . We used geometries of 3 B 1 states from Ref. 111. Dunning’s cc-pVTZ basis set was used. 2. Benzynes: ortho-, meta-, andpara-benzynes (Figure 6.3). We used geometries of the 3 B 2 states from Ref. 111 (optimized with NC-SF-TDDFT/LDA/cc-pVTZ). We used the cc-pVTZ basis set. 3. 2- and 5-dehydro-meta-xylylene (DMX) triradicals (Figure 6.3). The geometries were optimized with the CCSD/cc-pVDZ for the quartet states are given in Appendix B. Single- point EOM-SF-CCSD calculations were performed with the cc-pVTZ basis set. 4. Two-center copper SMMs: PATFIA (without ferrocene group, see Figure 7.3), CIT- LAT, and BISDOW. The geometries (optimized with!B97X-D/cc-pVTZ for the triplet 205 state) were taken from Ref. 107. Triplet’s!B97X-D/cc-pVTZ geometry of CUAQAC02 is given in Appendix B. The EOM-SF-CCSD calculations of SMMs were carried out using single precision for CCSD, the intermediates, and EOM iterations 112 , and with the Cholesky decomposition of ERIs 29 (threshold of 10 2 ). Most of EOM-SF-CCSD calcu- lations were performed with the cc-pVDZ basis set; one single point calculation was also evaluated with the cc-pVTZ basis set for PATFIA. The reported symmetry labels of the electronic states and MOs correspond to Mulliken’s con- vention 113a . To compare the compactness of virtual spaces obtained with the FNO and OSFNO schemes with the results of spin-adapted FNO-CISD calculations by Lu and Matsika 67 , we considered the lowest triplet state of formaldehyde at the ground-state CCSD/cc-pVDZ geometry; the Carte- sian coordinates given in Appendix B. Core electrons were frozen in correlated calculations. All calculations were carried out with the Q-Chem package 109, 110 . ortho meta para 2-DMX 5-DMX Figure 6.3: Structures of benzynes and dehydro-meta-xylylenes. a Depending on molecular orientation, symmetry labels corresponding to the same orbital or vibrational mode may be different. Q-Chem’s standard molecular orientation is different from that of Mulliken 113 . For example, Q-Chem places water molecule in thexz-plane instead ofyz. Consequently, for C 2v symmetry,b 1 andb 2 labels are flipped. More details can be found at http://iopenshell.usc.edu/resources/howto/symmetry 206 PATFIA CUAQAC02 CITLAT BISDOW Figure 6.4: Structures of SMMs. Color scheme: Bronze (copper), blue (nitrogen), red (oxygen), gray (carbon), white (hydrogen). 6.3 Results and discussion Table 6.1: Vertical excitation energies relative to the 3 B 1 state in the methylene series com- puted with EOM-SF-CCSD/cc-pVTZ in the full virtual space at the triplet state geometry. UHF and ROHF references were used. Systems UHF ROHF ~ b 1 B 1 ~ a 1 A 1 ~ c 1 A 1 ~ b 1 B 1 ~ a 1 A 1 ~ c 1 A 1 CH 2 1.52 0.94 3.29 1.52 0.94 3.29 NH + 2 1.94 1.82 3.57 1.94 1.82 3.57 SiH 2 1.13 -0.42 3.40 1.13 -0.42 3.40 PH + 2 1.27 -0.17 3.72 1.27 -0.18 3.72 Tables 6.1, 6.2, and 6.3 summarize relevant energy gaps computed with EOM-SF-CCSD/cc- pVXZ for the diradicals and triradicals used as a benchmark in this work. The structures of the molecules are shown in Figures 6.3 and 7.3. Detailed discussion of their underlying electronic structure can be found in Refs. 106, 114, 115, and 107. The singlet–triplet gaps in copper diradicals are given in Table 6.3. For the methylene series, we consider the manifold of all 4 diradical states (2 closed-shell singlets, open-shell singlet and triplet), whereas for the rest of the systems we focus on the two lowest states and consider only the singlet-triplet and doublet- quartet gaps. 207 Table 6.2: Vertical singlet–triplet and doublet–quartet gaps (eV) computed with EOM- SF-CCSD/cc-pVTZ in the full virtual space at the triplet/quartet state geometry (negative sign corresponds to the singlet/doublet ground states) using UHF and unrestricted PBE reference orbitals. Reference S T o-C 6 H 3 m-C 6 H 3 p-C 6 H a 3 UHF 1 A 1 3 B 2 -0.99 -0.51 -0.14 PBE 1 A 1 3 B 2 -1.00 -0.52 -0.14 Reference D Q 2-DMX 5-DMX UHF 2 B 2 4 B 2 0.45 -0.53 PBE 2 B 2 4 B 2 0.42 -0.20 a For para-benzyne the symmetries of electronic states in the full point-group symmetry are 1 A g and 3 B 1u . Table 6.3: Singlet–triplet gaps in SMMs (cm 1 ) computed with UHF EOM-SF-CCSD/cc- pVDZ in full virtual spaces (negative sign corresponds to the singlet ground state). Molecule E PATFIA -85 CITLAT 121 CUAQAC02 -191 BISDOW -225 6.3.1 Analysis of the original FNO approximation and comparison with OSFNO As the first example, we consider the methylene diradical. Direct application of the original FNO scheme to the CCSD calculation of the high-spin triplet state yields errors of a similar magnitude as in the closed-shell FNO calculations 93 , as illustrated in Figure 6.14. However, errors in the EOM-SF-CCSD energies are considerably larger, as shown in Figure 6.5. Already at a small truncation of the total population (a few percent), the errors reach a magnitude of one electron-volt. Such a rapid growth of the errors can be explained in terms of contributions of the physically important orbitals responsible for strong correlation (i.e., those from the singly occupied subspace) to the frozen virtual orbital space. The singlet–triplet gap between the two lowest EOM states also shows large errors (Figure 6.15). The origin of the problematic behavior is revealed by thehS 2 i calculations: as shown in the left panel of Figure 6.6, the trends in spin 208 0 1 2 3 4 5 6 7 65 70 75 80 85 90 95 100 0.01 0.05 99 100 ΔΔE , e V T otal p opulation, % OSFNO, 3 B 1 OSFNO, 1 B 1 OSFNO, ˜ a 1 A 1 OSFNO, ˜ c 1 A 1 FNO, 3 B 1 FNO, 1 B 1 FNO, ˜ a 1 A 1 FNO, ˜ c 1 A 1 0.01 0.05 99 100 Figure 6.5: Errors in energy differences (E, eV) between the four EOM-SF-CCSD tar- get states (M S = 0 triplet component, open-shell singlet, and two closed-shell singlets) and the reference high-spin triplet CCSD state of CH 2 computed with cc-pVTZ. Two schemes are shown: the original FNO (dashed lines) and the new open-shell variant (solid lines). The errors are computed relative to the full spaces EOM-SF-CCSD values. 0 0.5 1 1.5 2 2.5 65 70 75 80 85 90 95 100 hS 2 i T otal p opulation, % 0.001 0.002 0.003 0.004 65 70 75 80 85 90 95 100 2.001 2.002 2.003 2.004 2.005 hS 2 i T otal p opulation, % Figure 6.6:hS 2 i for open-shell singlet and triplet states of CH 2 obtained by the original FNO approximation (left) and OSFNO (right); EOM-SF-CCSD/cc-pVTZ. contamination of the open-shell states follow each other in a symmetric manner, suggesting that 209 these two states are mixed. We rationalize this trend inhS 2 i by considering the following model for open-shell states: T = 1 p 1 + 2 ( T + S ) (6.10) h T jS 2 j T i = 2 1 + 2 ; (6.11) whereT andS denote the triplet and the singlet, respectively. Let us denote the states computed with the full virtual orbital space as T and S and the states computed with the truncated virtual space as T and S . The mixing parameter connectshS 2 i and the occupancies of frontier natural orbitals: n n = 2 1 + 2 ; (6.12) (n n ) 2 =h T jS 2 j T i 2 + 2h T jS 2 j T i; (6.13) wheren andn are the occupancies of the corresponding, frontier natural orbitals of the triplet state. For spin-pure open-shell states with zero spin projection, n = n . Figure 6.7 shows (n n ) 2 versushS 2 i, both computed for the EOM-SF-CCSD wave-functions of CH 2 as a function of the FNO threshold. A close correlation between the two quantities confirms that the above model of state mixing indeed captures the essence of the problem. Because the and blocks of the density matrix were diagonalized separately, the there is no guarantee of direct correspondence between the and NOs. Moreover, the numerical values of the and occupancies are different. Sometimes the and orbitals are ordered in different manner. This introduces an imbalance between the frozen and orbital subspaces, causing spin contamination. 210 0.2 0.4 0.6 0.8 1 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 (n α −n β ) 2 hS 2 T i Figure 6.7: Analysis of the imbalance between the and orbital spaces in the FNO EOM-SF-CCSD/cc-pVTZ calculation of the M s =0 triplet state of CH 2 , see text and Eq. (6.13). As described in Section 6.2.2, the OSFNO algorithm allows one to identify the open-shell orbitals and to match the virtual natural orbitals in a consistent manner, without introducing an imbalance between the and orbital spaces. Consequently, the spin contamination of the resulting states is minimal, as illustrated in the right panel of Fig. 6.6. The errors in energy gaps are also greatly reduced (Fig. 6.5). We observe consistent robust performance of OSFNO for all test cases. We note that the CCSD energies appear to be much more forgiving to the orbital space truncation than the EOM- CCSD ones: the errors in FNO-CCSD relative to the full-space CCSD are always relatively small and behave smoothly with respect to the truncation threshold. This observation is con- sistent with small errors reported by Neese and co-workers 76 for their DLPNO scheme. Conse- quently, OSFNO and FNO results for the high-spin triplet reference states are very similar for all considered systems, as shown in Figure 6.14. For the target EOM-SF states, OSFNO leads to consistent improvement over the original FNO scheme: the position of the EOM-SF-CCSD states relative to the reference as well as the energy gaps between the target EOM states are improved dramatically (Figures 6.5, 6.13, and 6.15). Figure 6.8 compares errors in energy differences between the four EOM-SF-CCSD target states computed with the same number of frozen virtual orbitals using the OSFNO scheme and 211 −0.005 0 0.005 0.01 0.015 0.02 0.025 0.03 65 70 75 80 85 90 95 100 ΔΔE , e V Num b er of activ e virtual orbitals, % OSFNO, 3 B 1 OSFNO, 1 B 1 OSFNO, ˜ a 1 A 1 OSFNO, ˜ c 1 A 1 CF, 3 B 1 CF, 1 B 1 CF, ˜ a 1 A 1 CF, ˜ c 1 A 1 Figure 6.8: Errors in energy differences (E, eV) between the four EOM-SF-CCSD tar- get states (M S = 0 triplet component, open-shell singlet, and two closed-shell singlets) and the reference high-spin triplet CCSD state of CH 2 computed with cc-pVTZ. Two schemes are compared: OSFNO and freezing the same fraction of canonical orbitals (CF). canonical Hartree–Fock orbitals. Just as in the closed-shell case 93 , OSFNO performs better than freezing the same number of the canonical orbitals. Interestingly, frozen canonical orbitals leads to erratic behavior when very small number of orbitals is frozen (10%), likely also because of the mismatch between the canonical and virtual spaces. Overall, for the methylene-like diradicals XH 2 , mean errors of OSFNO for a typical total population of 99% are 0.02–0.03 eV (130 cm 1 for open-shell singlet–triplet gap, 200 cm 1 for a gap between closed-shell singlet and the triplet states). 6.3.2 Benzynes An excellent performance of the SF methods in calculations of singlet–triplet gaps in ben- zynes and similar aromatic di- and tri-radicals has been illustrated in several benchmark stud- ies 9, 106, 115–120 . An important prerequisite for accuracy is using not too spin-contaminated ref- erence. In the case of large spin contamination of the UHF reference, ROHF orbitals, orbitals 212 optimized for a correlated ansatz (i.e., approximate Brueckner’s orbitals b ), such as OO-CCD 122 or OO-MP2 123, 124 , or even Kohn-Sham DFT orbitals 125–127 can be used. As shown below, the spin contamination of the reference state also affects the performance of the OSFNO approxi- mation. In theorto! meta! para benzynes series, the computed singlet–triplet energy gaps (-0.99, -0.51, -0.14 eV) reflect an increased diradical character. In calculations with the UHF orbitals we observe the largest error due to OSFNO in the singlet–triplet gap for the meta isomer, up to 200 cm 1 at 99% of the total unfrozen population (Figure 6.9, top right). The large errors for meta-benzyne can be attributed to the significant spin contamination of the reference determinant. To mitigate the effect of spin contamination, we explored using the ROHF orbitals, PBE orbitals, and OO-MP2 orbitals (computed with the RI approximation for the MP2 part). Table 6.4 compares the degree of spin contamination with different orbital choices. For benzynes, PBE and OO-MP2 provide a comparable reduction of spin contamination of the reference determinant, reducing the error in singlet–triplet gap at 99% of total population to 54 and 60 cm 1 , respectively. The errors in singlet–triplet gaps with the ROHF orbitals formeta- and para-isomers are similar to those with the PBE and OO-MP2 orbitals: 66 and 8 cm 1 , respectively. We could not converge a ROHF calculation forortho-benzyne. Table 6.4:hS 2 i values of the reference determinant for different orbital choices in ben- zynes; the cc-pVTZ basis set. Isomer UHF PBE OO-MP2 Ortho 2.206 2.007 2.011 Meta 2.681 2.015 2.025 Para 2.022 2.007 2.010 b The effect of orbital optimization on spin contamination was studied, for example, in Ref. 121. 213 −20 −10 0 10 20 30 40 50 60 99 99.2 99.4 99.6 99.8 100 PBE 0 50 100 150 200 250 99 99.2 99.4 99.6 99.8 100 UHF 0 10 20 30 40 50 60 70 99 99.2 99.4 99.6 99.8 100 ROHF −10 0 10 20 30 40 50 60 99 99.2 99.4 99.6 99.8 100 OOMP2 ΔE ST , cm −1 T otal p opulation, % ΔE ST , cm −1 T otal p opulation, % o-b enzyne m-b enzyne p-b enzyne ΔE ST , cm −1 T otal p opulation, % ΔE ST , cm −1 T otal p opulation, % Figure 6.9: Errors in singlet–triplet energy gaps for ortho-, meta-, and para-benzynes computed with EOM-SF-CCSD/cc-pVTZ and different orbitals (ROHF, UHF, PBE, and OO-MP2). 6.3.3 Triradicals To investigate the applicability of the OSFNO algorithm beyond diradicals, we considered two isomers of the dehydro-meta-xylylene (DMX) triradicals 116, 128 , 2- and 5-DMX. Previous studies 116, 128 have shown that the ground state of 2-DMX is quartet 4 B 2 and the ground state of 5-DMX is doublet 2 B 2 . The computed doublet–quartet energy gaps are: 0.45 and -0.53 eV for 2- DMX and 5-DMX, respectively. Figure 6.10 shows the errors in doublet–quartet gaps in 2- and 5-DMX. As in the case of benzynes, the UHF reference determinant of DMX is significantly 214 spin-contaminated: hS 2 i is 4.45 for 2-DMX and 4.29 for 5-DMX. The spin contamination of the reference determinant is eliminated almost entirely using the PBE orbitals, which also significantly reduces the errors of OSFNO for doublet–quartet gaps. −30 −20 −10 0 10 20 30 40 50 60 99 99.2 99.4 99.6 99.8 100 PBE −120 −100 −80 −60 −40 −20 0 20 40 60 80 100 99 99.2 99.4 99.6 99.8 100 UHF ΔEDQ , cm −1 T otal p opulation, % ΔEDQ , cm −1 T otal p opulation, % 2-DMX 5-DMX Figure 6.10: Errors in doublet–quartet gaps in the two DMX triradicals computed with cc-pVTZ. 6.3.4 Di-copper SMMs The lowest singlet and triplet states in the di-copper SMMs have nearly perfect diradical character 107, 115 . The interaction between the two radical centers is rather weak, owing to the large spatial separation between them and a relatively compact size of the d-orbitals. Con- sequently, the dynamic correlation in the singlet and triplet states is very similar, leading to rather small errors in the OSFNO calculations of the singlet–triplet gaps not exceeding 18 cm 1 (Figure 6.11). Encouraged by small errors introduced by OSFNO, we carried out the EOM- SF-CCSD calculation for PATFIA with cc-pVTZ (1,038 orbitals in total). Using the threshold 99% of total population corresponds to freezing of 429 virtual orbitals. The resulting exchange constant differs by less than 1 cm 1 from the cc-pVDZ result, thus validating the computational protocol used in Ref. 107. 215 −10 −5 0 5 10 15 20 99 99.2 99.4 99.6 99.8 100 ΔE ST , cm −1 T otal population, % P A TFIA CITLA T BISDO W CUA QA C02 Figure 6.11: Errors in the singlet–triplet gaps in selected SMMs calculated with cc-pVDZ. 6.3.5 Properties −2.5 −2 −1.5 −1 −0.5 0 0.5 99 99.2 99.4 99.6 99.8 100 ΔSOCC , cm −1 T otal p opulation, % CH 2 3 B 1 /˜ a 1 A 1 CH 2 3 B 1 /˜ c 1 A 1 SiH 2 3 B 1 /˜ a 1 A 1 SiH 2 3 B 1 /˜ c 1 A 1 NH + 2 3 B 1 /˜ a 1 A 1 NH + 2 3 B 1 /˜ c 1 A 1 PH + 2 3 B 1 /˜ a 1 A 1 PH + 2 3 B 1 /˜ c 1 A 1 Figure 6.12: Errors in spin–orbit coupling constant for transitions between triplet and two closed-shell singlet states of XH 2 , EOM-SF-CCSD/cc-pVTZ. To benchmark OSFNO beyond energy gaps, we computed spin–orbit coupling constants (SOCC) in the methylene series between the triplet and closed-shell EOM-SF states within mean-field spin-orbit approximation using the algorithm described in Ref. 105. In all cases, we observed that the errors in SOCC due to OSFNO are less than 2.5 cm 1 , which is comparable 216 Table 6.5: Spin–orbit coupling constants (cm 1 ) in selected diradicals, computed with EOM-SF-CCSD/cc-pVTZ in the full virtual orbital space. Couplings of the triplet and two states of closed-shell character are shown. System 3 B 1 /~ a 1 A 1 3 B 1 /~ c 1 A 1 CH 2 10.9 20.7 SiH 2 56.7 86.0 NH + 2 18.3 68.4 PH + 2 119.9 175.6 with the accuracy of the mean-field approximation. The good performance of OSFNO for properties indicates that the truncation does not compromise the quality of the wave-functions. 6.3.6 Compactness of the OSFNO truncated virtual space We conclude by comparing the compactness of the truncated virtual space obtained with the OSFNO and FNO schemes. Table 6.6 shows the number of frozen orbitals in the calculation of the lowest triplet state of formaldehyde for the population threshold of 99%. The OSFNO leads to smaller numbers of frozen orbitals than FNO. However, the differences between the two schemes are very small and the overall compactness of the resulting active virtual spaces is similar. We also compare our results with the FNO-CISD results of Lu and Matsika obtained within spin-adapted framework and observe similar behavior. Table 6.6: The number of frozen orbitals in different FNO approaches for the lowest triplet state of formaldehyde. The same freezing criterion was applied: occupation trun- cation threshold preserving 99% of virtual space population. Basis FNO-CISD a FNO OSFNO cc-pVDZ 4 6 5 cc-pVTZ 30 32 30 cc-pVQZ 97 96 92 a from Ref. 67. 217 6.4 Conclusion Using natural orbitals computed as the eigenstates of the virtual-virtual block of the state density matrix instead of the canonical Hartree–Fock molecular orbitals results in smaller errors when the same fraction of virtual orbitals is frozen 93 . Moreover, the errors due to the virtual space truncation decrease monotonically when the FNO scheme is used. However, the direct application of the FNO strategy to open-shell references, as needed in EOM-SF calculations, leads to much larger errors than in the case of closed-shell references. We analyzed the appli- cability of the FNO approximation for open-shell CCSD and investigated the reasons of its breakdown in open-shell systems within the EOM-SF-CCSD framework. The main cause of the breakdown is the mismatch between the and FNOs and the consequent imbalance between the and active orbital spaces. A similar mismatch happens when canonical Hartree–Fock orbitals are frozen, leading to large errors in the computed energies. To address these issues, we developed a new scheme, called OSFNO. Benchmark calculations illustrated that OSFNO delivers robust performance, similarly to the original FNO scheme in the case of closed-shell references. A typical OSFNO population cutoff of 99% leads to the errors in singlet–triplet gaps of 130 cm 1 for same-center diradicals,10–60 cm 1 for benzynes, 20–50 cm 1 for dehydro- meta-xylylenes, and <18 cm 1 for dicopper SMMs. For SMMs, this approach reduces the virtual space by22% for cc-pVDZ and by46% for cc-pVTZ. Using OSFNO has enabled the EOM-SF-CCSD calculations with more than 1,000 basis functions on a single node. Finally, this approximation has a negligible impact on the spin–orbit coupling constants, enabling cal- culations of magnetic properties in large systems. 6.5 Appendix A: Additional FNO and OSFNO results 218 0 0.1 0.2 0.3 0.4 0.5 0.6 65 70 75 80 85 90 95 100 a 0 0.05 0.1 0.15 0.2 0.25 65 70 75 80 85 90 95 100 b 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 65 70 75 80 85 90 95 100 c 0 0.02 0.04 0.06 0.08 0.1 0.12 65 70 75 80 85 90 95 100 d 0.005 0.02 99 100 0.005 0.01 0.015 99 100 0.01 0.04 0.08 99 100 0.005 0.01 0.015 99 100 T CS, S OS, S 0.005 0.02 99 100 0.005 0.01 0.015 99 100 0.01 0.04 0.08 99 100 0.005 0.01 0.015 99 100 Figure 6.13: Statistical characteristics of the OSFNO approximation for EOM-SF-CCSD. The shown values are the errors in energy gaps (eV) between the reference CCSD state and the three target EOM-SF states (M S = 0): triplet (T), closed-shell singlet (CS, S) and open-shell singlet (OS, S) with the cc-pVTZ basis set. Panels (a) and (b) show mean values and standard deviations for CH 2 , SiH 2 , NH + 2 , and PH + 2 . Panels (c) and (d) show the results forortho-,meta-, andpara-benzynes. 6.6 Appendix B: Relevant Cartesian geometries $comment 2-DMX The quartet state is optimized with CCSD/cc-pVDZ Nuclear Repulsion Energy = 306.66084559 hartrees $end $molecule 0 4 H 2.1779993229 1.6126219346 0.0220094335 C 1.2316383235 1.0591102020 0.0170509023 C 0.0067407068 1.7449750260 0.0232715697 C -1.2218068005 1.0657498072 0.0172414394 H -2.1651940217 1.6243154403 0.0223406129 219 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 99.1 99.4 99.7 100 CH 2 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 99.1 99.4 99.7 100 SiH 2 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 99.1 99.4 99.7 100 NH + 2 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 99.1 99.4 99.7 100 PH + 2 E, e V T otal p opulation, % OSFNO FNO E, e V T otal p opulation, % E, e V T otal p opulation, % E, e V T otal p opulation, % Figure 6.14: Errors in total energies of the high-spin reference state calculated using the original FNO and OSFNO schemes for CH 2 , SiH 2 , NH + 2 , and PH + 2 with CCSD/cc-pVTZ. C -1.2697547499 -0.3837381005 0.0041875309 C -0.0007117314 -1.0178133387 -0.0015490490 C 1.2717398995 -0.3905766458 0.0039764389 C 2.4834586177 -1.1052465217 -0.0026169423 C -2.4853377419 -1.0917894515 -0.0022303563 H 2.4896680064 -2.1991429278 -0.0125192653 H 3.4421181549 -0.5764388687 0.0020686734 H -2.4975169877 -2.1856368798 -0.0121519881 H -3.4411032659 -0.5577713316 0.0025807846 H 0.0097060682 2.8409134546 0.0330871775 $end $comment 5-DMX The quartet state is optimized with CCSD/cc-pVDZ Nuclear Repulsion Energy = 308.26134251 hartrees $end 220 0 200 400 600 800 1000 1200 1400 1600 1800 99 99.2 99.4 99.6 99.8 100 ΔE ST , cm −1 T otal p opulation, % CH 2 , FNO SiH 2 , FNO NH + 2 , FNO PH + 2 , FNO CH 2 , OSFNO SiH 2 , OSFNO NH + 2 , OSFNO PH + 2 , OSFNO Figure 6.15: Errors in the energy gap between open-shell singlet and triplet states calcu- lated using the original FNO and OSFNO schemes with EOM-SF-CCSD/cc-pVTZ. $molecule 0 4 H 2.1787597750 1.7307689406 0.0023968634 C 1.2380017709 1.1699498565 0.0019847041 C 0.0000546892 1.7926615226 0.0038293650 C -1.2379475829 1.1700256439 0.0033293323 H -2.1786628337 1.7309114321 0.0048874326 C -1.2551488653 -0.2801695418 0.0006282192 C -0.0000414472 -0.9630803594 -0.0012762989 H -0.0000720323 -2.0597600167 -0.0033445618 C 1.2551272622 -0.2802577900 -0.0005802981 C 2.4763594548 -0.9889377656 -0.0022116137 C -2.4764396979 -0.9887718906 -0.0001269957 H 2.4898330646 -2.0837945681 -0.0041277531 H 3.4345798928 -0.4596458546 -0.0016942084 H -3.4346164231 -0.4594059385 0.0014919777 H -2.4899798840 -2.0836264692 -0.0023117943 $end $comment CUAQAC02 The triplet state is optimized with wB97X-D/cc-pVTZ Nuclear Repulsion Energy = 3174.87801338 hartrees $end 221 $molecule 0 3 Cu -0.4475643610 -0.0120963053 -1.2187131839 C -1.5516738597 -1.9174459710 0.6216855123 C -2.4404422951 -3.0767823055 0.9891436674 C 1.7884683766 -1.6766996168 -0.6862923408 C 2.8741200254 -2.6422066929 -1.0840583784 H -1.8293382510 -3.9790509196 1.0248530293 H -3.2258364073 -3.2142752931 0.2520287447 H -2.8627428887 -2.9234745771 1.9793067222 H 2.9887046283 -3.4087941897 -0.3208559893 H 2.6596867698 -3.0896360878 -2.0494344473 H 3.8148426588 -2.0943168142 -1.1445126369 H -2.3601778555 0.6855733244 -2.7932158752 H -2.2881709015 -0.8355308500 -2.8370271162 O -1.6039546736 -1.4940498512 -0.5633853188 O 0.8033864073 1.4688139814 -1.5261581319 O 0.9814838837 -1.3115042001 -1.5765454220 O -1.7814477378 1.2892257086 -0.5128288755 O -1.7940678183 -0.0428536234 -3.0630837058 Cu 0.4475643610 0.0120963053 1.2187131839 O -0.8033864073 -1.4688139814 1.5261581319 C 1.5516738597 1.9174459710 -0.6216855123 O 1.7814477378 -1.2892257086 0.5128288755 C -1.7884683766 1.6766996168 0.6862923408 O 1.6039546736 1.4940498512 0.5633853188 O -0.9814838837 1.3115042001 1.5765454220 O 1.7940678183 0.0428536234 3.0630837058 C 2.4404422951 3.0767823055 -0.9891436674 C -2.8741200254 2.6422066929 1.0840583784 H 2.3601778555 -0.6855733244 2.7932158752 H 2.2881709015 0.8355308500 2.8370271162 H 1.8293382510 3.9790509196 -1.0248530293 H 3.2258364073 3.2142752931 -0.2520287447 H 2.8627428887 2.9234745771 -1.9793067222 H -2.9887046283 3.4087941897 0.3208559893 H -2.6596867698 3.0896360878 2.0494344473 H -3.8148426588 2.0943168142 1.1445126369 $end $comment Ground-state formaldehyde geometry, optimized with CCSD/cc-pVDZ Nuclear Repulsion Energy = 31.09707922 hartrees 222 $end $molecule 0 1 C 0.0000000000 0.0000000000 0.5115557582 O 0.0000000000 0.0000000000 -0.6987691962 H 0.9441753015 0.0000000000 1.1099901293 H -0.9441753015 0.0000000000 1.1099901293 $end 223 6.7 Chapter 6 References 1 T. Helgaker, P. Jørgensen, and J. Olsen, Molecular electronic structure theory. Wiley & Sons, 2000. 2 G. D. Purvis and R. J. Bartlett, J. Chem. Phys. 76, 1910 (1982). 3 J. Noga and R.J. Bartlett, J. Chem. Phys. 86, 7041 (1987). 4 K. Raghavachari, G.W . Trucks, J. A. Pople, and M. Head-Gordon, Chem. Phys. Lett. 157, 479 (1989). 5 J. D. Watts and R. J. Bartlett, Chem. Phys. Lett. 233, 81 (1995). 6 J. D. Watts, J. Gauss, and R. J. Bartlett, J. Chem. Phys. 98, 8718 (1993). 7 J. D. Watts and R. J. Bartlett, Chem. Phys. Lett. 258, 581 (1996). 8 S. R. Gwaltney and M. Head-Gordon, J. Chem. Phys. 115, 2014 (2001). 9 P. U. Manohar and A. I. Krylov, J. Chem. Phys. 129, 194105 (2008). 10 K. Kowalski and P. Piecuch, J. Chem. Phys. 115, 2966 (2001). 11 K. Kowalski and P. Piecuch, J. Chem. Phys. 120, 1715 (2004). 12 M. Włoch, M. D. Lodriguito, P. Piecuch, and J. R. Gour, Mol. Phys. 104, 2149 (2006). 13 J. L. Whitten, J. Chem. Phys. 58, 4496 (1973). 14 B. I. Dunlap, J. W. D. Connolly, and J. R. Sabin, J. Chem. Phys. 71, 4993 (1979). 15 K. Eichkorn, O. Treutler, H. ¨ Ohm, M. H¨ aser, and R. Ahlrichs, Chem. Phys. Lett. 240, 283 (1995). 16 A. E. DePrince, M. R. Kennedy, B. G. Sumpter, and C. D. Sherrill, Mol. Phys. 112, 844 (2014). 17 D. Kumar, A. K. Dutta, and P. U. Manohar, J. Chem. Sci. 129, 1611 (2017). 18 U. Bozkaya and C. D. Sherrill, J. Chem. Phys. 144, 174103 (2016). 19 U. Bozkaya and C. D. Sherrill, J. Chem. Phys. 147, 044104 (2017). 224 20 X. Wang, A. Y . Sokolov, J. M. Turney, and H. F. Schaefer, J. Chem. Theory Comput. 12, 4833 (2016). 21 T. Shen, Z. Zhu, I. Y . Zhang, and M. Scheffler, J. Chem. Theory Comput. 15, 4721 (2019). 22 N. H. F. Beebe and J. Linderberg, Int. J. Quant. Chem. 12, 683 (1977). 23 H. Koch, A. S. de Mer´ as, and T. B. Pedersen, J. Chem. Phys. 118, 9481 (2003). 24 F. Aquilante, R. Lindh, and T. B. Pedersen, J. Chem. Phys. 127, 114107 (2007). 25 L. Boman, H. Koch, and A. Sanchez de Meras, J. Chem. Phys. 129, 134107 (2008). 26 F. Aquilante, T. B. Pedersen, and R. Lindh, Theor. Chem. Acc. 124, 1 (2009). 27 F. Aquilante, L. Gagliardi, T. B. Pedersen, and R. Lindh, J. Chem. Phys. 130, 154107 (2009). 28 F. Aquilante, L. Boman, J. Bostr¨ om, H. Koch, R. Lindh, A. S. de Mer´ as, and T. B. Pedersen, Cholesky decomposition techniques in electronic structure theory, in Linear-Scaling Tech- niques in Computational Chemistry and Physics, edited by R. Zale´ sny, M.G. Papadopoulos, P.G. Mezey, and J. Leszczynski, Challenges and advances in computational chemistry and physics, pages 301–343. Springer, 2011. 29 E. Epifanovsky, D. Zuev, X. Feng, K. Khistyaev, Y . Shao, and A. I. Krylov, J. Chem. Phys. 139, 134105 (2013). 30 S. D. Folkestad, E. F. Kjønstad, and H. Koch, J. Chem. Phys. 150, 194112 (2019). 31 X. Feng, E. Epifanovsky, J. Gauss, and A. I. Krylov, J. Chem. Phys. 151, 014110 (2019). 32 T. J. Mart´ ınez, A. Mehta, and E. A. Carter, J. Chem. Phys. 97, 1876 (1992). 33 T. J. Mart´ ınez and E. A. Carter, J. Chem. Phys. 100, 3631 (1994). 34 T. J. Mart´ ınez and E. A. Carter, J. Chem. Phys. 102, 7564 (1995). 35 S. Kossmann and F. Neese, J. Chem. Theory Comput. 6, 2325 (2010). 36 F. Neese, F. Wennmohs, A. Hansen, and U. Becker, Chem. Phys. 356, 98 (2009). 37 A. K. Dutta, F. Neese, and R. Izs´ ak, J. Chem. Phys. 144, 034102 (2016). 38 E. G. Hohenstein, R. M. Parrish, and T. J. Mart´ ınez, J. Chem. Phys. 137, 044103 (2012). 39 E. G. Hohenstein, R. M. Parrish, C. D. Sherrill, and T. J. Mart´ ınez, J. Chem. Phys. 137, 221101 (2012). 40 R. M. Parrish, E. G. Hohenstein, T. J. Mart´ ınez, and C. D. Sherrill, J. Chem. Phys. 137, 224106 (2012). 225 41 S. I. L. Kokkila Schumacher, E. G. Hohenstein, R. M. Parrish, L.-P. Wang, and T. J. Mart´ ınez, J. Chem. Theory Comput. 11, 3042 (2015). 42 E. G. Hohenstein, S. I. L. Kokkila, R. M. Parrish, and T. J. Mart´ ınez, J. Phys. Chem. B 117, 12972 (2013). 43 J. Brabec, C. Yang, E. Epifanovsky, A. I. Krylov, and E. Ng, J. Comput. Chem. 37, 1059 (2016). 44 R. M. Parrish, Y . Zhao, E. G. Hohenstein, and T. J. Mart´ ınez, J. Chem. Phys. 150, 164118 (2019). 45 N. K. Madsen, I. H. Godtliebsen, S. A. Losilla, and O. Christiansen, J. Chem. Phys. 148, 024103 (2018). 46 N. J. Mayhall, J. Chem. Theory Comput. 13, 4818 (2017). 47 P. Pulay, Chem. Phys. Lett. 100, 151 (1983). 48 P. Pulay and S. Saebø, Theor. Chim. Acta 69, 357 (1986). 49 S. Saebø and P. Pulay, J. Chem. Phys. 86, 914 (1987). 50 S. Saebø and P. Pulay, Chem. Phys. Lett. 113, 13 (1985). 51 M. Sch¨ utz, G. Hetzer, and H.-J. Werner, J. Chem. Phys. 111, 5691 (1999). 52 G. Hetzer, M. Sch¨ utz, H. Stoll, and H.-J. Werner, J. Chem. Phys. 113, 9443 (2000). 53 H.-J. Werner and M. Sch¨ utz, J. Chem. Phys. 135, 144116 (2011). 54 M. Sch¨ utz and H.-J. Werner, J. Chem. Phys. 114, 661 (2001). 55 M. Sch¨ utz and H.-J. Werner, Chem. Phys. Lett. 318, 370 (2000). 56 M. Sch¨ utz, J. Chem. Phys. 113, 9986 (2000). 57 M. Sch¨ utz, O. Masur, and D. Usvyat, J. Chem. Phys. 140, 244107 (2014). 58 M. Sch¨ utz, J. Chem. Phys. 116, 8772 (2002). 59 P.-O. L¨ owdin, Phys. Rev. 97, 1474 (1955). 60 C. F. Bender and E. R. Davidson, J. Chem. Phys. 47, 4972 (1967). 61 R. J. Buenker and S. D. Peyerimhoff, Theor. Chim. Acta 35, 33 (1974). 62 K. H. Thunemann, J. R¨ omelt, S. D. Peyerimhoff, and R. J. Buenker, Int. J. Quant. Chem. XI, 743 (1977). 226 63 R. J. Cave, S. S. Xantheas, and D. Feller, Theor. Chim. Acta 83, 31 (1992). 64 C.D. Sherrill and H.F. Schaefer III, Adv. Quantum Chem. 34, 143 (1999). 65 J. Ivanic and K. Ruedenberg, Theor. Chim. Acta 107, 220 (2002). 66 M. L. Abrams and C. D. Sherrill, J. Chem. Phys. 118, 1604 (2003). 67 Z. Lu and S. Matsika, J. Chem. Theory Comput. 8, 509 (2011). 68 Z. Lu and S. Matsika, J. Phys. Chem. A 117, 7421 (2013). 69 F. Neese, F. Wennmohs, and A. Hansen, J. Chem. Phys. 130, 114108 (2009). 70 P. R. Nagy, G. Samu, and M. K´ allay, J. Chem. Theory Comput. 14, 4193 (2018). 71 C. Riplinger and F. Neese, J. Chem. Phys. 138, 034106 (2013). 72 F. Neese, A. Hansen, and D. G. Liakos, J. Chem. Phys. 131, 064103 (2009). 73 C. Riplinger, B. Sandhoefer, A. Hansen, and F. Neese, J. Chem. Phys. 139, 134101 (2013). 74 P. Pinski, C. Riplinger, E. F. Valeev, and F. Neese, J. Chem. Phys. 143, 034108 (2015). 75 C. Riplinger, P. Pinski, U. Becker, E. F. Valeev, and F. Neese, J. Chem. Phys. 144, 024109 (2016). 76 M. Saitow, U. Becker, C. Riplinger, E. F. Valeev, and F. Neese, J. Chem. Phys. 146, 164105 (2017). 77 A. K. Dutta, M. Saitow, C. Riplinger, F. Neese, and R. Izs´ ak, J. Chem. Phys. 148, 244101 (2018). 78 T. L. Barr and E. R. Davidson, Phys. Rev. A 1, 644 (1970). 79 C. Sosa, J. Geertsen, G. W. Trucks, and R. J. Bartlett, Chem. Phys. Lett. 159, 148 (1989). 80 A. G. Taube and R. J. Bartlett, Collect. Czech. Chem. Commun 70, 837 (2005). 81 A. G. Taube and R. J. Bartlett, J. Chem. Phys. 128, 164101 (2008). 82 A. E. DePrince and C. D. Sherrill, J. Chem. Theory Comput. 9, 293 (2013). 83 A. Kumar and T. D. Crawford, J. Phys. Chem. A 121, 708 (2017). 84 T. D. Crawford, A. Kumar, A. P. Bazant´ e, and R. Di Remigio, Wiley Interdiscip. Rev.: Comput. Mol. Sci. 9, e1406 (2019). 85 D. Mester, P. R. Nagy, and M. K´ allay, J. Chem. Phys. 146, 194102 (2017). 227 86 C. Peng, M. C. Clement, and E. F. Valeev, J. Chem. Theory Comput. 14, 5597 (2018). 87 S. H¨ ofener and W. Klopper, Chem. Phys. Lett. 679, 52 (2017). 88 P. Baudin and K. Kristensen, J. Chem. Phys. 146, 214114 (2017). 89 A. I. Krylov, Annu. Rev. Phys. Chem. 59, 433 (2008). 90 R. J. Bartlett, Mol. Phys. 108, 2905 (2010). 91 K. Sneskov and O. Christiansen, WIREs: Comput. Mol. Sci. 2, 566 (2012). 92 R. J. Bartlett, WIREs: Comput. Mol. Sci. 2, 126 (2012). 93 A. Landau, K. Khistyaev, S. Dolgikh, and A. I. Krylov, J. Chem. Phys. 132, 014109 (2010). 94 J. F. Stanton and J. Gauss, J. Chem. Phys. 101, 8938 (1994). 95 P. A. Pieniazek, S. E. Bradforth, and A. I. Krylov, J. Chem. Phys. 129, 074104 (2008). 96 M. Nooijen and R. J. Bartlett, J. Chem. Phys. 102, 3629 (1995). 97 A. I. Krylov, Chem. Phys. Lett. 338, 375 (2001). 98 D. Casanova, L. V . Slipchenko, A. I. Krylov, and M. Head-Gordon, J. Chem. Phys. 130, 044103 (2009). 99 A. I. Krylov, Acc. Chem. Res. 39, 83 (2006). 100 F. Aquilante, T. K. Todorova, L. Gagliardi, T. B. Pedersen, and B. O. Roos, J. Chem. Phys. 131, 034113 (2009). 101 J. Segarra-Mart´ ı, M. Garavelli, and F. Aquilante, J. Chem. Theory Comput. 11, 3772 (2015). 102 J. F. Stanton and R. J. Bartlett, J. Chem. Phys. 98, 7029 (1993). 103 A. Tajti and P.G. Szalay, J. Chem. Phys. 131, 124104 (2009). 104 S. Faraji, S. Matsika, and A. I. Krylov, J. Chem. Phys. 148, 044103 (2018). 105 P. Pokhilko, E. Epifanovsky, and A. I. Krylov, J. Chem. Phys. 151, 034106 (2019). 106 L. V . Slipchenko and A. I. Krylov, J. Chem. Phys. 117, 4694 (2002). 107 N. Orms and A. I. Krylov, Phys. Chem. Chem. Phys. 20, 13127 (2018). 108 N. J. Mayhall and M. Head-Gordon, J. Phys. Chem. Lett. 6, 1982 (2015). 228 109 Y . Shao, Z. Gan, E. Epifanovsky, A.T.B. Gilbert, M. Wormit, J. Kussmann, A.W. Lange, A. Behn, J. Deng, X. Feng, D. Ghosh, M. Goldey, P.R. Horn, L.D. Jacobson, I. Kaliman, R.Z. Khaliullin, T. Kus, A. Landau, J. Liu, E.I. Proynov, Y .M. Rhee, R.M. Richard, M.A. Rohrdanz, R.P. Steele, E.J. Sundstrom, H.L. Woodcock III, P.M. Zimmerman, D. Zuev, B. Albrecht, E. Alguires, B. Austin, G.J.O. Beran, Y .A. Bernard, E. Berquist, K. Brand- horst, K.B. Bravaya, S.T. Brown, D. Casanova, C.-M. Chang, Y . Chen, S.H. Chien, K.D. Closser, D.L. Crittenden, M. Diedenhofen, R.A. DiStasio Jr., H. Do, A.D. Dutoi, R.G. Edgar, S. Fatehi, L. Fusti-Molnar, A. Ghysels, A. Golubeva-Zadorozhnaya, J. Gomes, M.W.D. Hanson-Heine, P.H.P. Harbach, A.W. Hauser, E.G. Hohenstein, Z.C. Holden, T.-C. Jagau, H. Ji, B. Kaduk, K. Khistyaev, J. Kim, J. Kim, R.A. King, P. Klunzinger, D. Kosenkov, T. Kowalczyk, C.M. Krauter, K.U. Laog, A. Laurent, K.V . Lawler, S.V . Levchenko, C.Y . Lin, F. Liu, E. Livshits, R.C. Lochan, A. Luenser, P. Manohar, S.F. Manzer, S.-P. Mao, N. Mardirossian, A.V . Marenich, S.A. Maurer, N.J. Mayhall, C.M. Oana, R. Olivares-Amaya, D.P. O’Neill, J.A. Parkhill, T.M. Perrine, R. Peverati, P.A. Pieniazek, A. Prociuk, D.R. Rehn, E. Rosta, N.J. Russ, N. Sergueev, S.M. Sharada, S. Sharmaa, D.W. Small, A. Sodt, T. Stein, D. Stuck, Y .-C. Su, A.J.W. Thom, T. Tsuchimochi, L. V ogt, O. Vydrov, T. Wang, M.A. Wat- son, J. Wenzel, A. White, C.F. Williams, V . Vanovschi, S. Yeganeh, S.R. Yost, Z.-Q. You, I.Y . Zhang, X. Zhang, Y . Zhou, B.R. Brooks, G.K.L. Chan, D.M. Chipman, C.J. Cramer, W.A. Goddard III, M.S. Gordon, W.J. Hehre, A. Klamt, H.F. Schaefer III, M.W. Schmidt, C.D. Sherrill, D.G. Truhlar, A. Warshel, X. Xu, A. Aspuru-Guzik, R. Baer, A.T. Bell, N.A. Besley, J.-D. Chai, A. Dreuw, B.D. Dunietz, T.R. Furlani, S.R. Gwaltney, C.-P. Hsu, Y . Jung, J. Kong, D.S. Lambrecht, W.Z. Liang, C. Ochsenfeld, V .A. Rassolov, L.V . Slipchenko, J.E. Subotnik, T. Van V oorhis, J.M. Herbert, A.I. Krylov, P.M.W. Gill, and M. Head-Gordon, Mol. Phys. 113, 184 (2015). 110 A. I. Krylov and P. M. W. Gill, WIREs: Comput. Mol. Sci. 3, 317 (2013). 111 Y . A. Bernard, Y . Shao, and A. I. Krylov, J. Chem. Phys. 136, 204103 (2012). 112 P. Pokhilko, E. Epifanovskii, and A. I. Krylov, J. Chem. Theory Comput. 14, 4088 (2018). 113 R. S. Mulliken, J. Chem. Phys. 23, 1997 (1955). 114 A. I. Krylov, J. Phys. Chem. A 109, 10638 (2005). 115 N. Orms, D. R. Rehn, A. Dreuw, and A. I. Krylov, J. Chem. Theory Comput. 14, 638 (2018). 116 T. Wang and A. I. Krylov, Chem. Phys. Lett. 425, 196 (2006). 117 T. Wang and A. I. Krylov, J. Chem. Phys. 123, 104304 (2005). 118 L. V . Slipchenko and A. I. Krylov, J. Chem. Phys. 118, 9614 (2003). 119 A. M. C. Cristian, Y . Shao, and A. I. Krylov, J. Phys. Chem. A 108, 6581 (2004). 229 120 P. U. Manohar, L. Koziol, and A. I. Krylov, J. Phys. Chem. A 113, 2519 (2009). 121 A. I. Krylov, J. Chem. Phys. 113, 6052 (2000). 122 C. D. Sherrill, A. I. Krylov, E. F. C. Byrd, and M. Head-Gordon, J. Chem. Phys. 109, 4171 (1998). 123 J. Lee and M. Head-Gordon, J. Chem. Theory Comput. 14, 5203 (2018). 124 J. Lee and M. Head-Gordon, Phys. Chem. Chem. Phys. 21, 4763 (2019). 125 P. G. Wenthold, J. Org. Chem. 77, 208 (2012). 126 E. Hossain and P. G. Wenthold, Comp. Theo. Chem. 1020, 180 (2013). 127 E. Hossain, S. M. Deng, S. Gozem, A. I. Krylov, X.-B. Wang, and P. G. Wenthold, J. Am. Chem. Soc. 139, 11138 (2017). 128 L. V . Slipchenko, T. E. Munsch, P. G. Wenthold, and A. I. Krylov, Angew. Chem., Int. Ed. 43, 742 (2004). 230 Chapter 7: Effective Hamiltonians derived from equation-of-motion coupled-cluster wave-functions: Theory and application to the Hubbard and Heisenberg Hamiltonians 7.1 Introduction Coarse graining is commonly used in computational chemistry and physics. It can be deployed in the real space, when a group of atoms is replaced by an effective particle, or in the Hilbert space, when a group of many-body electronic states is represented by a set of states expressed in a compact basis of model states a . Coarse-graining is exploited in a number of classic models serving as a foundation of modern solid-state physics: tight binding 1, 2 , Drude– Sommerfeld’s model 3–5 , Hubbard’s 6 and Heisenberg’s 7–9 Hamiltonians. These models explain macroscopic properties of materials through effective interactions whose strengths are treated a In condensed matter community, such downfolding of the Hilbert space is often referred to as “low-energy models”. 231 as model parameters. The values of these parameters are determined either from more sophis- ticated theoretical models or by fitting to experimental observables. For example, in Hubbard’s 1 2 3 Heisenberg Hubbard Figure 7.1: Configurations entering three-electron-in-three-centers model Hamiltonians. Each center (or site) is represented by one localized orbital. The configurations are built from different distributions of three electrons on three localized orbitals (numbered 1, 2, and 3 and colored with yellow, green, and violet circles, respectively). Each line represents one configuration. Heisenberg’s model space includes only open-shell configurations in which the orbitals are singly occupied. A half-filled (i.e., three-electrons-in-three orbitals) Hubbard’s model space also includes ionic configurations in which localized orbitals can host two electrons. model, the model space comprises configurations obtained by distributingN electrons overM sites and each site is represented by just one localized orbital, as illustrated in Figure 7.1 for for the three-electrons-in-three-centers case. Hubbard’s Hamiltonian includes electron-hopping 232 terms (one-electron transitions facilitated by the couplings between neutral and ionic configu- rations) described by the parametert and the site energies (energies of the ionic configurations) described by the parameterU: H Hub = M X i6=j X t ij a y i a j +U M X i n i n i ; (7.1) where indicesi;j denote localized orbitals b and denotes spin part or, operatorsa y i and a i are the creation and annihilation operators of an electron on a localized orbitali with spin , andn i is a particle number operator:n i =a y i a i . While the configurational space in Hub- bard’s model is the same as in the complete-active space wave-function (using the same orbital space), the two models are not the same: Eq. (7.1) does not admit two-electron interactions, such as, for example, direct exchange terms that couple first three configurations from Fig. 7.1. Although Hubbard’s model parameterizes only one-particle interactions between the configura- tions, it includes the Coulomb repulsionU between the conducting (active) electrons, making it sufficiently flexible to describe non-trivial physical phenomena, such as Mott–Hubbard’s phase transition 10 . Heisenberg’s model 7–9 describes the interaction between open-shell configurations in terms of local spin (an effective quantity which we discuss below). This model is most commonly used to describe magnetic properties of solids 10 . It can be written as H Heis = X A<B J AB S A S B ; (7.2) whereA;B enumerate radical (or magnetic) centers,J AB is an effective exchange constant, S I is an effective localized spin operator associated with centerI. Heisenberg’s Hamiltonian can b Most often, Hubbard’s models include only the hops between the nearest neighbors, but in this work we waive this limitation and consider all possible one-electrons hops. 233 be derived from Hubbard’s Hamiltonian through degenerate 11 and canonical perturbation theo- ries 12 ; thus, it can be considered an effective theory with respect to Hubbard’s model. Although Heisenberg’s model space does not admit ionic configurations, their effect is folded into the effective exchange parameters. Effective Hamiltonian theory provides a powerful framework for a rigorous construction of effective Hamiltonians from multiconfigurational many-electron wave functions computed ab initio. Pioneering works of Kato 13 , ˆ Okubo 14 , Bloch 15 , des Cloizeaux 16 have established the foundations of the operator effective Hamiltonian theory. Further development came from L¨ owdin’s partitioning 17 technique, Feshbach’s formalism 18 , and generalizations 19–22 . The his- tory and recent developments of effective Hamiltonian theories are summarized in comprehen- sive reviews 23, 24 . Effective theories fulfill a dual role. On one hand, they can be used to develop new elec- tronic structure methods by systematically improvable description of effective Hamiltonians. For example, perturbative construction of the wave operator and effective Hamiltonians have been exploited in the development of various multireference perturbative 25–27 and and coupled- cluster 28–30 methods. On another hand, effective theories can be used as an interpretation tool by providing an essential description of complex electronic structure. The distinction between using effective Hamiltonians for analysis of electronic structure and method development is not always binary. For example, introduction of Hubbard’s repulsion term U into a density functional expression allowed Anisimov and co-workers to develop a widely used DFT+U method 31–33 . Mayhall and Head-Gordon noticed that effective exchange couplings J from a single spin-flip (SF) calculation agree with the couplings extracted from more computationally expensive n-SF calculations 34 , which led to the development of an effec- tive computational scheme for strongly correlated systems exploiting a coarse-graining idea 35 . 234 In Mayhall’s protocol, a single spin-flip calculation is performed first. Then, under the assump- tion of Heisenberg’s physics, the exchange couplings are extracted, and an effective Hamil- tonian is constructed to compute the spin states that are not accessible in the single spin-flip calculation. The parameters of effective Hamiltonians also can be obtained indirectly. Experimentally, exchange couplings can be extracted from the temperature dependence of magnetic suscepti- bility, electron paramagnetic resonance, and neutron scattering experiments 36–42 . Theoretically, if there are known relations between the states of interest, such as the Land´ e interval rule 43 , advanced electronic structure methods can be easily applied 44 . Even more indirectly, exchange couplings can be extracted from contaminated solutions 45 obtained in broken-symmetry density functional theory (BS-DFT) or broken-symmetry coupled-cluster 46 calculations. Numerous studies 47–50 have used Bloch’s and des Cloizeaux’ formalisms to build Heisen- berg’s and Hubbard’s effective Hamiltonians from configuration interaction (CI) wave func- tions. In this contribution, we apply Bloch’s formalism to the equation-of-motion coupled- cluster (EOM-CC) wave functions to rigorously derive effective Hamiltonians in the Bloch’s and des Cloizeaux’ forms. We report the key equations and illustrate the theory by examples of systems with several unpaired electrons, giving rise to electronic states of covalent and ionic characters. Our goal is to establish a rigorous mapping between the EOM-CC solutions and effective Hamiltonians and to provide a theoretical basis for the extraction of effective param- eters from the EOM-CC calculations. These parameters can be directly compared with the parameters extracted from experimental measurements, facilitating unambiguous comparison between the theory and experiment. In addition, such a mapping provides physical insights into complex electronic structure of strongly correlated systems. Finally, the theoretical devel- opments presented here serve as a stepping stone toward developing coarse-grained models of electron correlation in large systems, following Mayhall’s and Head-Gordon’s ideas 34, 35 . 235 The structure of the paper is as follows. In the next section, we provide an overview of the EOM-CC theory and Bloch’s formalism. We then apply Bloch’s formalism to the EOM-CC wave functions and derive the working expressions. Next, we consider several molecules for which we discuss Hubbard’s and Heisenberg’s Hamiltonians rigorously constructed from the EOM-SF-CCSD solutions. Whenever possible, we compare the results from extraction through the Land´ e rule and from the effective Hamiltonians. 7.2 Theory 7.2.1 Equation-of-motion coupled-cluster theory Similarly to CI, EOM-CC methods 51–54 utilize linear parameterization of the wave function: j R I i = ^ R I j 0 i; (7.3) h L I j =h 0 j ^ L I y ; (7.4) where I is a EOM target state, ^ R and ^ L are general excitation operators, and 0 is the reference determinant, which defines the separation between occupied and virtual orbital spaces. The equations for the amplitudes of the EOM operators are derived variationally, leading to a CI- like eigenproblem 55 . In contrast to CI, the EOM theory employs a (non-Hermitian) similarity- transformed Hamiltonian: ^ H =e ^ T ^ He ^ T ; (7.5) where ^ T is an excitation operator. Because ^ H has the same spectrum as the bare Hamiltonian regardless of the choice of ^ T , solving the EOM eigen problem in the full configurational space, 236 i.e., when ^ R includes all possible excitations, recovers the exact (full CI, FCI) limit. In practical calculations, these operators are truncated, most often to single and double excitations, giving rise to EOM-CCSD ansatz, and the choice of the operator ^ T becomes important. For example, choosing ^ T = ^ 1 leads to plain CISD, but taking ^ T from the coupled-cluster equations for the reference state amounts to including correlation effects and ensures size-intensivity. Most often, ^ T is truncated at the same level as ^ R, e.g., in EOM-CCSD one uses ^ T = X ia t a i a y i + 1 4 X ijab t ab ij a y b y ji +:::; (7.6) wherei;j;::: anda;b;::: denote occupied and virtual (with respect to 0 ) orbitals. When ^ T satisfies the CC equations, then the reference determinant 0 is also an eigen-state of H. Because ^ H is non-Hermitian, the left and right EOM eigenstates are not Hermitian conju- gates of each other: ^ H ^ R I j 0 i =E I ^ R I j 0 i; (7.7) h 0 j ^ L I y ^ H =h 0 j ^ L I y E I ; (7.8) but are often chosen to form a biorthogonal set: h 0 j ^ L y I ^ R J j 0 i = IJ : (7.9) Because the excitation operators ^ T and ^ R commute, one can also write the EOM states as j R I i = ^ R I j CC i; (7.10) 237 where j CC i =e ^ T j 0 i: (7.11) This form shows clearly that the EOM-CC states include higher excitations than the respective CI states by virtue of the wave operator e ^ T and that the EOM states are excited states with respect to the reference-state CC wave function. Different choices of ^ R allow access to different manifolds of target states, giving rise to a variety of EOM-CC methods 51–54 . Common choices include excitation 56 , spin-flip 57–59 , ioniza- tion 60, 61 or electron-attachment 62 operators. For example, spin-flip operators are ^ R SF = X ia r a i a y i + 1 4 X ijab r ab ij a y b y ji +:::; (7.12) ^ L SF = X ia l a i a y i + 1 4 X ijab l ab ij a y b y ji +:::; (7.13) where the number of creation operators corresponding to orbitals is not equal to the num- ber of annihilation operators corresponding to orbitals, giving rise to a non-zeroM S (spin- projection) of ^ R (in contrast, the EOM-EE operators are of theM S =0 type). 7.2.2 Effective Hamiltonians General formalism Effective theories operate with target (large) and model (small) spaces, as illustrated in Figure 7.2. Both spaces describe the same set of eigenstates, but using different sets of config- urations. The target space contains long multiconfigurational expansions, i.e., it can be a full Hilbert space of a many-body system. The model space contains expansions over a small set of configurations. 238 Target space,P + +c 1 +c 2 Ω −1 Model space,P 0 + Figure 7.2: Target and model spaces are connected through a wave operator . The target space can be the full configurational (i.e., Hilbert) space of the system, while model space is a subspace of the full configurational space. Bloch’s theory usually starts from target states, which are eigenvectors of the Hermitian Hamiltonian: Hj i =E j i; (7.14) P = n X j iN 2 h j: (7.15) Here the eigenstates are not necessarily normalized to one. P denotes a projector onto the target space,N is the normalization factor of the eigenstate , andE is the energy of ,n is the dimension of the target space. The selection of the eigenstates in the target space is deter- mined by the problem at hand— we include only those eigenstates that are relevant to a specific physics. The number of states included in the model space (i.e., the dimension of the model space) is the same as that in the corresponding target space. Effective Hamiltonians are defined 239 in the model space such that, by construction, their eigenvalues reproduce the eigenvalues of the target-space Hamiltonian: H eff j ~ R i =E j ~ R i; (7.16) P 0 = n X j ~ R ih ~ L j; (7.17) wherej ~ R i andh ~ L j are right and left eigenstates of the effective Hamiltonian,P 0 is the pro- jector onto the model space. In other words, the idea behind the effective Hamiltonian entails the transformation of the interaction between the model vectors to reproduce the energies in the target space. The model space is connected to the target space through a wave operator : j i = j ~ i: (7.18) The relations (7.14–7.18) lead to generalized Bloch’s equation, establishing a connection between the wave operator and the effective Hamiltonian: H P 0 = P 0 H eff P 0 : (7.19) Effective Hamiltonians in Eq. (7.16) are defined up to a similarity transformation. There- fore, a constraint is needed for a unique definition. Bloch’s constraint (which we denote through a subscript “B”), also known as intermediate normalization, is P 0 B P 0 =P 0 (7.20) 240 This constraint leads to the following expressions for the wave operator and effective Hamilto- nian 19 : B =P (P 0 PP 0 ) 1 (only in subspaces); (7.21) H eff B =P 0 H eff B P 0 =P 0 H B P 0 : (7.22) Eqs. (7.21) and (7.22) mean that if the exact eigen-states and eigen-energies are known, one can select a model space, compute the projectors (and, consequently, wave operator), and construct H eff B , which, by construction, will yield the same eigen-energies asH. As an interesting example of an effective theory, one can consider coupled-cluster theory. The exponential operatore ^ T with untruncated ^ T has the meaning of the wave operator connect- ing the model space (a single Slater determinant) with the exact (FCI) space. Coupled-cluster equations require that the reference determinant 0 is the eigenfunction of the effective Hamil- tonian ^ H: j 0 i !j CC i = 0 : (7.23) Thus,e ^ T maps the exact many-body ground state into a single Slater determinant. Bloch’s effective Hamiltonian is not Hermitian, which may not be convenient for the analy- sis. Des Cloizeaux gave a recipe for producing a Hermitian effective Hamiltonian 16 by means of 241 symmetric orthogonalization of Bloch’s eigenvectors. An arbitrary Hermitian effective Hamil- tonian can be obtained from Bloch’s effective Hamiltonian through a similarity transformation A. For any Bloch’s eigenprojector, this can be written as A 1 j ~ R B; iE h ~ L B; jA =j ~ H; iE h ~ H; j; (7.24) A 1 j ~ R B; i =j ~ H; i; (7.25) h ~ L B; jA =h ~ H; j; (7.26) j ~ R B; i =AA y j ~ L B; i: (7.27) Since Bloch’s eigenvectors are biorthogonal, AA y = X j ~ R B; ih ~ R B; j; (7.28) AA y 1 = X j ~ L B; ih ~ L B; j; (7.29) A is defined up to a unitary transformation. Polar decomposition of a matrix gives unique positive-semidefinite Hermitian matrixP and a unitary partU: A =PU; P = (AA y ) 1=2 : (7.30) To make the Hermitian effective Hamiltonian physically close to Bloch’s effective Hamiltonian, one has to minimize the degree of unitary rotations, i.e., consider only the square root (AA y ) 1=2 . It is easy to show (see Appendix A) that, similarly to L¨ owdin’s symmetric orthogonalization, this transformation changes the eigenvectors to a minimal possible extent. 242 Effective Hamiltonians from EOM-CC Here we treat the EOM-CC space as the target space:j R i = ^ R j 0 i andh L j =h 0 j ^ L y . Our model space depends on a problem and will be defined for each specific case; it will contain a subset of single EOM-CC amplitudes. The interaction between the configurations in this target space is described through ^ H. After substitution into Eq. (7.22) and cancellation of the normalization prefactors, the final expressions are: I 0 ; 00 = X h 0j R ih L j 00i; (7.31) H eff B = X ;; 0 ; 00 j ih j R iE h L j 0i I 1 0 ; 00 h 00j: (7.32) Here we assumed that the model functions i are orthonormal. Eigenfunctions of Bloch’s Hamiltonian are j ~ R B; i = X j ih j R i; (7.33) h ~ L B; j = X 0 ; 00 h L j 0i I 1 0 ; 00 h 00j: (7.34) Assuming thath j R i andh L j 0i are invertible, one can get biorthogonality of Bloch’s eigenstates: L B y R B = L EOM y I 1 R EOM = 1; (7.35) where columns of R B and L B are Bloch’s eigenvectors in the basis of model functions, and columns of R EOM and L EOM are overlaps of the EOM vectors with model functionsh j R i andh j L i. This relation reveals the physical meaning of I: it is a metric tensor for the EOM vectors projected onto the model space; thus, it quantifies how well the model states can 243 represent the target states. This observation suggest using the overlaps (specifically, the lowest absolute value of the eigenvalues of I) as a diagnostic of whether or not a particular model space is capable of accurate representation of the given target space. Des Cloizeaux’ transformation is obtained in a similar way: AA y = R EOM R EOM y ; (7.36) A C = (AA y ) 1=2 ; (7.37) where A denotes similarity transformationA in a basis of orthonormal model functions. 7.3 Numerical examples and discussion 7.3.1 Computational details We implemented the tools for constructing the effective Hamiltonians from the EOM-CC wavefunctions within the Q-Chem package 63, 64 . The illustrative examples in this work use EOM-SF-CCSD 55, 59 , but the same workflow can be executed with other EOM-CC methods. Briefly, the calculation proceeds as follows: 1. Perform an SCF calculation for a high-spin reference state. 2. Determine single occupied orbital space by computing corresponding and orbitals by means of singular value decomposition (SVD) of the block of the overlap matrix. 3. Localize orbitals within singly occupied space. 4. (Optional) Apply open-shell frozen natural orbital approximation (OSFNO) 65 in the vir- tual space. 244 5. Construct the model space using an appropriate subset of EOM-CC amplitudes see Sec- tions 7.3.4 and 7.3.5 for specific examples. 6. Build effective Hamiltonians in Bloch’s and des Cloizeaux’ forms from the EOM-CC energies and relevant amplitudes, as described above. The validity of the selected model space can be assessed by considering the overlaps between the target and model spaces. Once the effective Hamiltonian is computed, one can extract parameters of Hubbard’s or Heisenberg Hamiltonians and assess the validity of these models. Here we considered non-relativistic EOM-CC calculations. The algorithm can be gen- eralized to include perturbative treatment of spin–orbit couplings 66 , to enable the extraction of other magnetic properties, such as magnetic anisotropies. Additional details of the algorithm are given below; an example of an input is provided in the Appendix D. Magnet 1 Magnet 2 Molecule 1 Molecule 2 Figure 7.3: Structures of the considered systems. Color code: Copper (bronze), nitrogen (blue), oxygen (red), carbon (gray), hydrogen (white). In Molecule 1 and Molecule 2, representing organic di/triradicals, the unpaired electrons are localized on the odd carbon atoms. In single-molecule magnets (Magnet 1 and Magnet 2), the unpaired electrons are localized on the copper centers. To illustrate the theory, we constructed effective Hamiltonians for four representative molecules (structures are shown in Fig. 7.3) with two and three strongly correlated electrons 245 (i.e., diradicals 67 and triradicals 68 ): Hubbard’s model: propane-1,3-diyl (Molecule 1) and pentane-1,3,5-triyl (Molecule 2). These molecules are examples of organic di- and triradicals. Molecule 1 (di-methilene- methane) was used to illustrate construction of effective Hamiltonians within the CI framework 49 . We also consider Heisenberg’s model for these systems. The geometries were optimized with CCSD/cc-pVTZ and are given in the Appendix E. Heisenberg’s Hamiltonian: PATFIA (Magnet 1, Ref. 69) and tris-(3-acetylamino-1,2,4- triazolate) motif of HUKDOG (Magnet 2, Ref. 70). These systems are examples of single molecule magnets (SMMs)— they can also be described as di- and tri-radicals, but with metallic radical centers. The geometry of Magnet 1 optimized with !B97X-D/cc-pVTZ for the triplet state was taken from Ref. 44. As in Ref. 44, the ferrocene group was not included in the calculations. We also excluded water molecules, sulfate groups, and perchlorate anions in the structure of Magnet 2. The geometry of Magnet 2 was optimized with!B97X-D/cc-pVTZ and is given in the Appendix E. We optimized the geometry of Magnet 2 with finite thresholds: 10 4 tolerance on the maximal gradient component (a.u.) and 10 6 a.u. on the energy change; these thresholds are tighter that Q-Chem’s default criteria for geometry optimization. The resulting geometry is slightly asymmetric within these criteria (for example, Cu–Cu distances are 3.416, 3.420, and 3.418 ˚ A). Tighter thresholds lead to a symmetric geometry. C 3 symmetry point group has degenerate irreducible representations, which leads to problems with degenerate EOM solutions, described in Ref. 66. To circumvent these problems, we used the lower symmetry geometry, which we also provide in the Appendix E. In this non-symmetric geometry, the doublet states are split by only 2.5 cm 1 . 246 In the SMM calculations, we used EOM-SF-CCSD with OSFNO truncation of the virtual space 65 with the total population threshold of 99%. We used single precision 71 capability of libxm tensor contraction library 72 in calculations of copper SMMs. Unrestricted Hartree-Fock references were used in all correlated calculations. Core electrons were frozen. All calculations were performed with the Q-Chem software 63, 64 . 7.3.2 Model space selection Starting from a high-spin reference determinant, we define the open-shell orbitals through SVD of the overlap matrix of occupied and virtual spin-orbitals, as in the OSFNO scheme 65 and in Ref. 35. The singular vector pairs corresponding to open-shell and spin-orbitals have near unity singular values. All other singular values are zero (ROHF) or small (UHF), enabling the separation of the open-shell subspace. This is followed by the localization of open-shell orbitals. Here we used Foster–Boys’ localization criterion 73 and a gradient-based algorithm 74–76 with the BFGS optimizer. The model spaces were generated by one-electron spin-flipping excitations acting on the reference determinant and constrained to open-shell sub- spaces. For Heisenberg’s model space we considered only open-shell determinants, while for Hubbard’s model space we also included ionic determinants (shown in Figure 7.1). 7.3.3 Extraction of parameters from effective Hamiltonians To construct Heisenberg’s Hamiltonian, one should first compute the effective Hamiltonian using the model space with open-shell configurations only (see Fig. 7.1). Then, the parame- ters for Heisenberg’s model (J AB ) are given by the sum of matrix element pairs between the respective configurations. For Hubbard’s Hamiltonian, the model space should include both open-shell and ionic con- figurations. Once the effective Hamiltonian is constructed, the extraction of the parameters is 247 also rather straightforward. A simple relation between the matrix elements and effective param- eters is established by substitution of Hubbard’s Hamiltonian definition and second quantization algebra. LetA andB subscripts denote radical centersA andB, CS and OS denote closed-shell and open-shell determinants, and REF denotes a high-spin reference state (in this example, it is a high-spin triplet state). The expressions for the diagonal and off-diagonal matrix elements are: hCS A jH Hub jCS A i =hCS A jU X i n i n i jCS A i =UhCS A j(1 + 0)jCS A i =U; (7.38) hCS A jH Hub jOS A i =hCS A j X i6=j X t ij a y i a j jOS A i = hREFj(a y A; a B; ) y X i6=j X t ij a y i a j ! a y A; a A; jREFi = hREFj(a y A; a B; ) y t BA a y A a B a y A; a A; jREFi =t BA : (7.39) 7.3.4 Hubbard’s effective Hamiltonian CS OS OS CS CS 4.424 0.088 -0.089 -0.130 OS 0.056 0.000 -0.042 -0.144 OS -0.144 -0.042 0.000 0.056 CS -0.130 -0.089 0.088 4.424 Figure 7.4: Localized orbitals of Molecule 1 and Bloch’s effective Hamiltonian (eV) con- structed with EOM-SF-CCSD/cc-pVTZ. All energies are shifted to achieve zero trace of open-shell submatrix. CS and OS denote closed-shell ionic determinants and open-shell determinants, respectively. Blue color marks the numbers corresponding to Hubbard’s Hamiltonian parameters; red color on antidiagonal marks the numbers that do not enter standard Hubbard’s Hamiltonian. 248 Localized orbitals (shown in Figures 7.4 and 7.7) enable unequivocal identification of cova- lent or ionic character of the electronic configurations constructed from these orbitals. Bloch’s and des Cloizeaux’ effective Hamiltonians for Molecules 1 and 2 built for covalent and ionic configurations from EOM-SF-CCSD/cc-pVTZ are shown in Figure 7.4 and Eqs. 7.41, 7.42, 7.43 in Appendix B. We note that des Cloizeaux’ transformation either does not change the matrix elements of effective Hamiltonian on the diagonal (Molecule 1) or alter them in a minimal way (Molecule 2). Des Cloizeaux’ transformation averages the off-diagonal matrix elements yield- ing a Hermitian effective Hamiltonian. The validity of the selected model spaces is reflected by the weights of the leading configurations shown in Table 7.2 in the Appendix C and by the smallest eigenvalue of the metric tensor I. For Molecule 1, it is 0.76, confirming that the selected model space adequately represents the chosen set of the target states. For Molecule 2, we obtain a smaller value (0.45), suggesting that some of the selected states are represented not as well as the states in Molecule 1. The obtained matrix elements of effective Hamiltonians directly correspond to the coupling parameters in Hubbard’s Hamiltonian, as explained in Section 7.3.3. In particular, the ele- ments corresponding to hopping couplingt and energies of ionic configurationsU, see Eqns. (7.39), are colored in blue in Figure 7.4. However, Hubbard’s model does not include two- electron couplings, such as an effective direct exchange between the open-shell determinants and effective interaction between the ionic determinants. The singlet–triplet gap (EOM-SF- CCSD/cc-pVTZ) between the two lowest states in Molecule 1 is 0.077 eV; by construction, it is reproduced exactly by Bloch’s and des Cloizeaux’ effective Hamiltonians. The contribution of the ionic configurations and the validity of Hubbard’s model are revealed by the magnitudes of the matrix elements of effective Hamiltonians. The truncated (inexact) effective Hamilto- nian, including only open-shell determinants (Figure 7.4), gives the singlet–triplet gap of 0.084 eV . This observation can be also rationalized from perturbative arguments: the effective direct 249 exchange splits the open-shell states in the first order of degenerate perturbation theory. The indirect exchange from the ionic configurations comes only from the second-order perturba- tion, having a magnitude of 0:1 2 =4 = 2:5 10 3 eV , which is by one order of magnitude smaller that the direct exchange. This analysis reveals the nature of interaction of open-shell configura- tions in this system: the singlet–triplet gap primarily comes from the effective direct exchange rather from the interaction with the ionic configurations. The standard Hubbard’s hamiltonian (blue numbers in Fig. 7.4) does not include these interactions; consequently, its diagonaliza- tion yields a poor value of the singlet–triplet gap (-0.008 eV—a wrong magnitude and sign). To accommodate this physics, Hubbard’s Hamiltonian can be extended to include direct exchange contribution, as was done, for example, in Ref. 50. It is instructive to compare the open-shell subblock of this effective Hamiltonian with the Heisenberg’s Hamiltonian constructed considering only open-shell configurations (shown in the Appendix B, Eqns. (7.44), (7.45). By construction, this Hamiltonian yields exactly the same singlet–triplet gap as the one constructed using Hubbard’s model space. The coupling matrix elements give the values of effective exchange constantsJ, which include all many-body effects present in the EOM-SF-CCSD wavefunctions. These effects, for example, include the contri- butions of the ionic configurations, which are treated explicitly in the case Hubbard’s model space. The difference between the OS-OS couplings from Heisenberg’s Hamiltonian (0.038 eV) and the matrix elements between the OS configurations from the hamiltonian constructed using Hubbard’s space (0.042 eV) quantify the contribution of the ionic configurations into the effective J values. Finally, the lowest absolute eigenvalue of I is 0.96, meaning that Heisen- berg’s model captures low-energy physics in Molecule 1 very well. The considered triradical—Molecule 2—exhibits similar trends and similar effective inter- action constants shown in Eq. (7.42), (7.43), and Figure 7.7. Although all the energies in Bloch’s 250 and des Cloizeaux’ effective Hamiltonians has been shifted to produce a zero trace in the open- shell submatrix, the individual energies of the open-shell determinants are not zero. This is likely due to some degree of spin contamination of the EOM-SF-CCSD states used to build the effective Hamiltonians. Spin contamination is especially large in the highest ionic states: the corresponding< S 2 > values are 1.06 and 1.17. This degree of spin contamination can dete- riorate the quality of energy, resulting in somewhat higher energies of the corresponding ionic configurations: 6.5 and 6.3 eV . These configurations are not well represented by the model space, resulting in a relatively small lowest eigenvalue of I of 0.45, which is smaller than the corresponding value (0.76) of the Molecule 1. We note that there are several spin-contaminated electronic states with lower energies that are not represented well by the model space; these states were not included in the effective Hamiltonian. Interestingly, Heisenberg’s model space for this system (see Eqns. (7.46), (7.47)) shows much better overlap, with the lowest eigenvalue of I of 0.96, indicating that open-shell states are described well by this effective Hamiltonian. 7.3.5 Heisenberg’s effective Hamiltonian OS OS OS -1.48 35.32 OS 35.08 1.48 OS OS OS -1.48 35.20 OS 35.20 1.48 B C Figure 7.5: Localized open-shell orbitals (left), Bloch’s (right, top), and des Cloizeaux’ (right, bottom) effective Hamiltonians for Magnet 1. All matrix elements are in cm 1 . 251 OS OS OS OS 0.91 48.21 46.95 OS 48.51 -2.59 48.57 OS 46.66 48.94 1.68 OS OS OS OS 0.90 48.36 46.82 OS 48.36 -2.58 48.75 OS 46.82 48.75 1.67 B C Figure 7.6: Localized open-shell orbitals (left), Bloch’s (right, top), and des Cloizeaux’ (right, bottom) effective Hamiltonians of Magnet 2. All matrix elements are in cm 1 . Figures 7.5 and 7.6 show localized open-shell orbitals and the effective Hamiltonians con- structed from the open-shell determinants expressed using these orbitals. The complex char- acter of the wave functions has been illustrated before 44 , yet, the SVD-transformed open-shell orbitals yield a relatively simple expansion of wave functions over this small set of configura- tions. Open-shell configurations in these orbitals are the dominant configurations contributing more than 90% of the total amplitude sum (Table 7.3), which confirms the validity of Heisen- berg’s model in representing these systems. Our proposed diagnostic, the lowest absolute eigen- values of I, equal 0.91 and 0.93, which means that the model space represents the target wave functions very well. We note that the quality of this compact model space by no means mean that the effect of other electronic configurations in the full Hilbert space (e.g., double EOM-SF-CCSD ampli- tudes) can be neglected. To illustrate this point, we computed the eigenvalues of H in the space of single excitations only, as in EOM-SF-CCSD-S 77 ; the results are shown in Table 7.4 in the SI. 252 The energy gaps between the target states are reproduced reasonably well (albeit not exactly), however, the resulting states are severely spin-contaminated. Table 7.1: Exchange coupling constants,J (cm 1 ), extracted from the EOM-SF-CCSD/cc- pVDZ calculations in different ways. Magnet 1 Magnet 2 Bloch -70.40 -96.73, -93.61, -97.51 Des Cloizeaux -70.40 -96.72, -93.64, -97.50 Land´ e rule -70.47 -95.96 a a Here we used a three-center generalization of the Land´ e rule 78 :E(Q)E(D) = 3 2 J. Energies of the two doublet states were averaged. The obtained effective Hamiltonians shown in Figures 7.5 and 7.6, were shifted to achieve a zero trace, as in the case of Hubbard’s Hamiltonian. Although the diagonal elements are small, they deviate from zero by 0.9–2.6 cm 1 . This artifact is a violation of time-reversal symmetry, which can be explained by a small spin contamination of EOM-SF-CCSD states entering the effective Hamiltonian construction. The effective exchange coupling constants J (shown in Table 7.1) were extracted from the off-diagonal elements of the effective Hamiltonians and through the Land´ e interval rule 43 . Bloch’s and des Cloizeaux’ forms of effective Hamiltonians predict nearly identical values ofJ, slightly different from the Land´ e rule. This can be attributed to small non-zero diagonal terms contributing to eigenvalues and entering the energy gaps used in the Land´ e rule. 7.4 Conclusion We generalized Bloch’s formalism to a non-Hermitian Hamiltonian ( H) and presented the application of the resulting theory to the EOM-CC wave functions. We compared Bloch’s and Cloizeaux’ versions of the effective Hamiltonian approach and demonstrated that they perform similarly. The constructed Bloch’s and des Cloizeaux’ effective Hamiltonians provide a direct way to interpret the EOM-CC wave functions in terms of effective theories. The theory is 253 formulated using model spaces built from localized open-shell orbitals computed via SVD of the overlap between the and HF orbitals (such as in OSFNO). The localization of open- shell orbitals allows one to quantify the ionic and covalent characters of the wave functions. The effective Hamiltonians, which are constructed in a rigorous manner, supply the interac- tion constants for the model Hamiltonians, such as the Heisenberg and Hubbard Hamiltonians. The physical quality of the selected models can be assessed by considering the eigenvalues of I, the matrix which quantifies the overlap between the model and target spaces. We pro- pose to use the lowest absolute eigenvalue of I as a diagnostic in construction of the effective Hamiltonians. We observe that Hubbard’s model space gives smaller values of the diagnostic than Heisenberg’s model space. This can be attributed to the effect of ionic configurations: the states with ionic character are known to have large dynamic correlation (due to Coulomb repulsion), which leads to more multiconfigurational wave-functions and, therefore, reduces the overlap with model spaces. The quality of electronic structure calculations, such as the degree of spin contamination, enters the constructed effective Hamiltonians and affects the respective parameters. In particular, spin contamination leads to a small difference in magnetic exchange couplings extracted in different ways. This work provides a foundation for direct comparison of the results of many-body calculations with the experimentally derived effective parameters for magnetic systems and for further development of coarse-grained approaches to strong cor- relation. 7.5 Appendix A: Proof of optimality of des Cloizeaux’ trans- formation Let us find a similarity transformation A of Bloch’s effective Hamiltonian such that the eigenvectors of the resulting Hermitian effective Hamiltonian differ from Bloch’s eigenvectors 254 to a minimal extent. For this purpose, we formulate the problem as the overlap maximization with right and left Bloch’s eigenvectors: X h ~ L B; j ~ H; i +h ~ H; j ~ R B; i = X h ~ H; jA 1 j ~ H; i +h ~ H; jAj ~ H; i ! max: (7.40) This problem can be viewed as a trace maximization: Find such a unitary matrixU for a given A that maximizes Tr(AU) + Tr(U 1 A 1 ). Based on singular value decompositionA =W V y and Cauchy–Schwartz inequality, one can prove 79 that the maximum of both Tr(AU) and Tr(U 1 A 1 ) is achieved atU opt = VW y . Therefore,AU opt = W V y VW y = W W y , which is the polar part ofA. 7.6 Appendix B. Hubbard’s effective Hamiltonians Des Cloizeaux’ effective Hamiltonian for Molecule 1: H C (eV ) = 0 B B B B B B B B B @ CS OS OS CS CS 4:424 0:073 0:115 0:130 OS 0:073 0:000 0:042 0:115 OS 0:115 0:042 0:000 0:073 CS 0:130 0:115 0:073 4:424 1 C C C C C C C C C A (7.41) 255 Figure 7.7: Localized orbitals of Molecule 2. Effective Hamiltonians of Molecule 2: H B (eV ) = 0 B B B B B B B B B B B B B B B B B B B B B B B B B B B B @ CS CS CS OS OS OS CS CS CS CS 4:027 0:100 0:065 0:108 0:105 0:001 0:113 0:008 0:027 CS 0:093 4:560 0:057 0:118 0:122 0:002 0:055 0:023 0:001 CS 0:002 0:034 6:451 0:032 0:001 0:030 0:003 0:008 0:147 OS 0:156 0:106 0:042 0:016 0:038 0:000 0:059 0:003 0:003 OS 0:091 0:172 0:019 0:037 0:029 0:040 0:001 0:098 0:008 OS 0:002 0:006 0:054 0:000 0:041 0:013 0:041 0:015 0:095 CS 0:131 0:007 0:001 0:030 0:002 0:032 6:335 0:062 0:003 CS 0:009 0:021 0:019 0:001 0:051 0:050 0:047 4:563 0:116 CS 0:031 0:001 0:025 0:000 0:048 0:049 0:011 0:130 4:064 1 C C C C C C C C C C C C C C C C C C C C C C C C C C C C A (7.42) 256 H C (eV ) = 0 B B B B B B B B B B B B B B B B B B B B B B B B B B B B @ CS CS CS OS OS OS CS CS CS CS 4:027 0:100 0:065 0:108 0:105 0:001 0:113 0:008 0:027 CS 0:093 4:560 0:057 0:118 0:122 0:002 0:055 0:023 0:001 CS 0:002 0:034 6:451 0:032 0:002 0:030 0:003 0:008 0:147 OS 0:156 0:106 0:042 0:016 0:038 0:000 0:059 0:003 0:003 OS 0:091 0:172 0:019 0:037 0:029 0:040 0:001 0:098 0:008 OS 0:002 0:006 0:054 0:000 0:041 0:013 0:041 0:015 0:095 CS 0:131 0:007 0:001 0:030 0:002 0:032 6:335 0:062 0:003 CS 0:009 0:021 0:019 0:001 0:051 0:050 0:047 4:563 0:116 CS 0:031 0:001 0:025 0:000 0:048 0:049 0:011 0:130 4:064 1 C C C C C C C C C C C C C C C C C C C C C C C C C C C C A (7.43) Heisenberg’s effective Hamiltonian for Molecule 1: H B (eV ) = 0 B @ OS OS OS 0:000 0:038 OS 0:038 0:000 1 C A (7.44) H C (eV ) = 0 B @ OS OS OS 0:000 0:038 OS 0:038 0:000 1 C A (7.45) 257 Heisenberg’s effective Hamiltonian for Molecule 2: H B (eV ) = 0 B B B B B @ OS OS OS OS 0:018 0:031 0:000 OS 0:030 0:027 0:039 OS 0:000 0:040 0:009 1 C C C C C A (7.46) H C (eV ) = 0 B B B B B @ OS OS OS OS 0:018 0:031 0:000 OS 0:031 0:027 0:039 OS 0:000 0:039 0:009 1 C C C C C A (7.47) 258 7.7 Appendix C. Overlaps between target and model spaces Table 7.2: Molecule 1 and Molecule 2. Individual EOM-SF-CCSD amplitudes for exci- tations between the localized open-shell orbitals (r 1 ) and the squared norm of the single amplitudes (jjR 1 jj 2 ). The energies (E, eV) are relative to the CCSD high-spin reference. The smallest eigenvalue of the intermediate I for Molecule 1 and Molecule 2 are 0.76 and 0.45. Molecule 1 E, eV jjR 1 jj 2 r 1 Triplet 0.9713 0.0301 0.6945, 0.6945, 0.0000, 0.0000 OS singlet 0.9705 0.1068 0.6940, 0.6940, -0.0272, 0.0272 CS singlet 1 0.8997 4.3666 0.0126, 0.0126, -0.6244, -0.6244 CS singlet 2 0.8985 4.6343 -0.0282, 0.0282, -0.6360, -0.6360 Molecule 2 E, eV jjR 1 jj 2 r 1 Quartet 0:0318 0:9704 0:5844, 0:0008, 0:0001,0:0012,0:5463, 0:0004,0:0001,0:0002,0:5687 OS doublet 1 0:0639 0:9708 0:7226, 0:0202, 0:0065, 0:0211, 0:0876, 0:0065,0:0063, 0:0060, 0:6585 OS doublet 2 0:1391 0:9684 0:3175, 0:0297, 0:0001, 0:0293, 0:8054, 0:0149,0:0019, 0:0138, 0:4589 CS doublet 3 4:0760 0:8990 0:0255,0:1235, 0:0280,0:7586,0:0117, 0:4563, 0:0398, 0:0736, 0:0100 CS doublet 4 4:1342 0:8992 0:0149, 0:0983, 0:0438, 0:4600, 0:0089, 0:7359,0:0324, 0:1815, 0:0159 CS doublet 5 4:6536 0:8984 0:0196, 0:6450,0:0267,0:1462,0:0149, 0:1533,0:0058, 0:5658,0:0039 CS doublet 6 4:6921 0:9011 0:0161,0:5648,0:0041, 0:0838, 0:0373, 0:1528,0:0316, 0:6492,0:0065 doublet 5:9429 0:9327 not included due to small overlaps doublet 6:3660 0:9400 not included due to small overlaps CS doublet 7 6:4282 0:8993 0:0084, 0:0254, 0:0144, 0:0373, 0:0018, 0:0041,0:7444,0:0196,0:0047 doublet 6:5223 0:9300 not included dues to small overlaps CS doublet 8 6:5360 0:9050 0:0036, 0:0228,0:7101,0:0206, 0:0011, 0:0079,0:0194,0:0083, 0:0060 259 Table 7.3: Magnet 1 and Magnet 2. Individual EOM-SF-CCSD amplitudes for excitations between the localized open-shell orbitals (r 1 ) and the squared norm of the single ampli- tudes (jjR 1 jj 2 ). The energies (E, eV) relative to the CCSD high-spin reference and relative to the lowest state (E, eV) are shown. The smallest eigenvalue of the intermediate I for Magnet 1 and Magnet 2 are 0.91 and 0.93. Magnet 1 E E jjR 1 jj 2 r 1 Singlet 0.2181 0.0 0.9662 0.6915, -0.6610 Triplet 0.2268 0.0087 0.9666 0.6904, 0.6643 Magnet 2 E E jjR 1 jj 2 r 1 Doublet 1 0.2173 0.0 0.9696 0.7895,-0.4282,-0.3498 Doublet 2 0.2179 0.0006 0.9696 -0.7003, 0.6608, 0.0455 Quartet 0.2354 0.0181 0.9704 0.5631, 0.5584, 0.5524 Table 7.4: Magnet 1 and Magnet 2. Individual EOM-SF-CCSD-S amplitudes (R 2 =0) for excitations between the localized open-shell orbitals (r 1 ) and the squared norm of the sin- gle amplitudes (jjR 1 jj 2 ). The energies (E, eV) relative to the CCSD high-spin reference and relative to the lowest state (E, eV) are shown. All states are severely spin-contaminated, as can be seen from the leading determinants. Magnet 1 E E R 2 1 r 1 State 1 2.8626 0.0 1.0000 -0.9984,-0.0139 State 2 2.8739 0.0113 1.0000 -0.0144, 0.9984 Magnet 2 E E R 2 1 r 1 State 1 2.7673 0.0 1.0000 0.9907, -0.1187, -0.0361 State 2 2.7679 0.0006 1.0000 -0.1031, -0.9918, 0.0502 State 3 2.7707 0.0034 1.0000 -0.0416, -0.0417, -0.9967 Table 7.5: Effect of different geometries and environment on averagedJ constants (cm 1 ) in Magnet 2. The cc-pVDZ basis set was used in all SF-TDDFT calculations. Magnet 2 No ligands, opt 2H 2 O H 2 O H 2 O and SO 2 4 b 2H 2 O and SO 2 4 b PBE50 -132 -144 -87 -417 -751 NC-PBE50 -81 -86 -141 -603 -696 B5050LYP -127 -138 -137 -379 -751 NC-B5050LYP -83 -88 -90 -1889 -820 a The experimental value is 185 cm 1 , taken from Ref. 70. b When SO 2 4 is included in the calculation, all SF states become severally spin-contaminated. 260 7.8 Appendix D. Example of input Example of an input section controlling construction of effective Hamiltonians: $eff_ham state_list sf_states 1 1 sf_states 1 2 sf_states 1 3 end_list eff_ham heisenberg $end Here three EOM-SF states in irrep 1 were used to construct an effective Hamiltonian for Mag- net 2. 261 7.9 Appendix E. Relevant Cartesian geometries $comment 1,3-didehydropropane The triplet state is optimized with CCSD/cc-pVTZ Nuclear Repulsion Energy = 69.51653737 hartrees $end $molecule 0 3 C 0.0559165541 -0.2758342919 1.2464551621 C -0.0667896209 0.5345764922 0.0000000000 C 0.0559165541 -0.2758342919 -1.2464551621 H 0.2438914109 0.1999693240 2.1973120933 H -0.1874920523 -1.3283233639 1.2406211626 H 0.6750737956 1.3401092145 0.0000000000 H -1.0377500799 1.0655164781 0.0000000000 H 0.2438914109 0.1999693240 -2.1973120933 H -0.1874920523 -1.3283233639 -1.2406211626 $end 1,3,5-tridehydropentane The triplet state is optimized with CCSD/cc-pVTZ Nuclear Repulsion Energy = 159.18736737 hartrees $molecule 0 4 C 0.0001480271 -0.2469316289 0.1009807613 C -1.2994218656 0.4852965694 0.0234382720 C -2.4844262500 -0.4141018744 -0.0817901094 H 0.0055528133 -1.2574173227 0.4896891488 H -1.2644954649 1.1781394971 -0.8334936376 H -1.4198666326 1.1466819006 0.8959214398 H -3.4738390259 -0.0550000449 0.1596973160 H -2.3916510948 -1.3818483731 -0.5531240662 C 1.2932987451 0.4908165118 -0.0075353813 C 2.4879589324 -0.3978758530 -0.0725109653 H 1.2700058296 1.1564463570 -0.8790150739 H 1.3901219510 1.1831497441 0.8525925549 H 2.4701345251 -1.3711582632 0.3961579091 H 3.4307205364 -0.0302843048 -0.4490637924 $end tris-(3-acetylamino-1,2,4-triazolate) motif of HUKDOG, Magnet 2. 262 The triplet state is optimized with wB97X-D/cc-pVTZ. This geometry is not symmetrized---see the main text for details. This geometry was used for EOM-CC calculations. Nuclear Repulsion Energy = 5414.92994620 hartrees $molecule 2 4 Cu 1.8781988232 0.6005044909 -0.2340318339 Cu -1.4642747543 1.3223461629 -0.2025260179 Cu -0.4155736253 -1.9311493273 -0.2039772115 O -0.0014320472 -0.0040167873 -0.7690269628 H -0.0146763978 -0.0041832477 -1.7324740033 N 2.3437079919 -1.2387974493 -0.0711943629 N 1.4824301409 -2.2970609219 -0.0905656962 N 3.5350341262 -3.0917663227 0.2010243617 N -2.7279986093 -0.1401112324 -0.0840467249 N -2.2434817071 -1.4157531656 -0.0527153163 N -4.4462380897 -1.5167977125 0.2018618698 N -0.0987614643 2.6476246591 -0.0882756361 N 1.2482768995 2.4292378933 -0.1258616001 N 0.9169659490 4.6068114792 0.1522044084 N 4.7008773002 -1.0137575019 0.1923022413 H 5.5476883069 -1.5477578448 0.3193483651 N -3.2327886776 -3.5667571136 0.2119444585 H -4.1203480336 -4.0326013152 0.3292078083 N -1.4640492069 4.5823800354 0.1812193012 H -1.4206356088 5.5846884634 0.2912397059 C 6.1200222258 0.9513492273 0.3037245745 H 6.9286979945 0.2260385923 0.3184958789 H 6.1280049968 1.5155456975 1.2367030245 H 6.2767028918 1.6586951934 -0.5084785048 C 4.7761889100 0.3259002484 0.1486486557 C 3.5419622111 -1.7717349196 0.1065054768 C 2.2382081564 -3.3754877952 0.0762777395 H 1.8222247504 -4.3685575430 0.1060654104 C -2.2412220380 -5.7806538296 0.3045489494 H -3.2725966369 -6.1116466471 0.3885341182 H -1.6854930712 -6.1017162106 1.1850480392 H -1.7792372953 -6.2472839720 -0.5643475660 C -2.1103905237 -4.3018364984 0.1739325220 C -3.3066019502 -2.1845359599 0.1205678649 C -4.0412601388 -0.2531162962 0.0731221608 H -4.6931035233 0.6040734633 0.0929846351 C -3.8740344640 4.8351983113 0.3149322830 263 H -3.6485451372 5.8979550323 0.2986888138 H -4.3570287316 4.5848265917 1.2598158045 H -4.5732215466 4.5965193631 -0.4842003562 C -2.6629892206 3.9794435129 0.1661094928 C -0.2315574924 3.9535309181 0.0813686311 C 1.8080416500 3.6235822353 0.0220119366 H 2.8763841244 3.7606237312 0.0342268115 O 3.7740236320 1.0544921041 -0.0015306117 O -0.9775924969 -3.7963622735 0.0360749284 O -2.7955155090 2.7445703827 0.0428549052 $end tris-(3-acetylamino-1,2,4-triazolate) motif of HUKDOG, Magnet 2. The triplet state is optimized with wB97X-D/cc-pVTZ with tight thresholds. This geometry is symmetrized---see the main text for details. Nuclear Repulsion Energy = 5414.95410199 hartrees $molecule 2 4 Cu 1.9731954866 -0.0001770889 -0.1938397208 Cu -0.9864443798 1.7089259625 -0.1938397208 Cu -0.9867511068 -1.7087488736 -0.1938397208 O 0.0000000000 0.0000000000 -0.7504040200 H 0.0000000000 0.0000000000 -1.7139381614 N 1.8521683296 -1.8952825858 -0.0425468126 N 0.7073499470 -2.6383090691 -0.0683477551 N 2.4157704814 -4.0253033145 0.2263331750 N -2.6385176504 0.7065715111 -0.0683477551 N -2.5674470314 -0.6563835326 -0.0425468126 N -4.6939001690 -0.0794669494 0.2263331750 N 0.7152787018 2.5516661184 -0.0425468126 N 1.9311677034 1.9317375581 -0.0683477551 N 2.2781296876 4.1047702639 0.2263331750 N 4.1639731331 -2.4076594006 0.2245870273 H 4.8050790859 -3.1770569898 0.3493235188 N -4.1670807711 -2.4022768136 0.2245870273 H -5.1539516054 -2.5727920607 0.3493235188 N 0.0031076380 4.8099362142 0.2245870273 H 0.3488725195 5.7498490505 0.3493235188 C 6.1222216774 -0.9761472198 0.3046866133 H 6.6613105829 -1.9163936956 0.3794010966 H 6.3203505133 -0.3733206072 1.1905162745 H 6.4813545347 -0.4190721451 -0.5594177875 264 C 4.6484521273 -1.1568827267 0.1755121276 C 2.8280247531 -2.7711265849 0.1363670516 C 1.0948909074 -3.8970049051 0.0966281715 H 0.3967054691 -4.7167757813 0.1217658401 C -3.9064791289 -4.8139258903 0.3046866133 H -4.9903009155 -4.8106673395 0.3794010966 H -3.4834803863 -5.2869238017 1.1905162745 H -3.6036043910 -5.4034816054 -0.5594177875 C -3.3261158942 -3.4472362671 0.1755121276 C -3.8138783962 -1.0635779863 0.1363670516 C -3.9223507002 1.0002991124 0.0966281715 H -4.2832003851 2.0148308766 0.1217658401 C -2.2157425485 5.7900731101 0.3046866133 H -1.6710096674 6.7270610351 0.3794010966 H -2.8368701270 5.6602444089 1.1905162745 H -2.8777501437 5.8225537505 -0.5594177875 C -1.3223362331 4.6041189939 0.1755121276 C 0.9858536431 3.8347045712 0.1363670516 C 2.8274597928 2.8967057928 0.0966281715 H 3.8864949160 2.7019449047 0.1217658401 O 3.9178652572 -0.1554121852 0.0307824456 O -2.0935235291 -3.3152647488 0.0307824456 O -1.8243417282 3.4706769340 0.0307824456 $end tris-(3-acetylamino-1,2,4-triazolate) motif of HUKDOG, with 2H2O and sulfate anion Magnet 2. The geometry was taken from crystal structure of HUKDOG. Nuclear Repulsion Energy = 7987.96037149 hartrees $molecule 0 4 Cu -5.2442800 -0.8903900 1.0561400 Cu -2.4947400 -2.4717500 0.0184500 Cu -4.8777500 -4.1896300 1.6013100 S -3.5278000 -3.3516200 4.6613800 O -3.9828100 -2.4268800 1.3432400 H -3.5486400 -2.2365500 2.2016000 N -6.7021700 -1.9969900 1.6325700 N -6.6158000 -3.3365300 1.8150100 N -8.7584900 -2.7626100 1.9995100 N -2.2819700 -4.4006500 0.2501700 N -3.3401300 -5.0521900 0.8476100 N -1.9576700 -6.6148100 0.0713100 265 N -2.7151300 -0.5868000 -0.1719900 N -3.7958000 0.1305200 0.2653200 N -2.4968500 1.5403400 -0.9006400 N -8.4609200 -0.4142400 1.6053500 H -9.3076900 -0.3059900 1.7044400 N -3.9642100 -7.3163800 1.1725900 H -3.7080600 -8.1348500 1.1024600 N -0.7916100 -0.0437100 -1.4943600 H -0.3202500 0.5896700 -1.8346900 C -8.4775500 1.9581100 1.3433800 H -9.4076600 1.7902900 1.5092600 H -8.3789800 2.3908100 0.4897700 H -8.1253500 2.5263100 2.0307700 C -7.7378800 0.6758500 1.3315000 C -7.9781200 -1.6986700 1.7445700 C -7.8578100 -3.7323300 2.0240200 H -8.0818600 -4.6208800 2.1771900 C -5.9323500 -8.2977600 2.1798500 H -5.4376400 -9.0931000 1.9724500 H -6.7799600 -8.3209200 1.7292400 H -6.0756700 -8.2474700 3.1280100 C -5.1604400 -7.0990800 1.7284300 C -3.1053500 -6.3492900 0.7017400 C -1.5022900 -5.3801600 -0.1768800 H -0.6973100 -5.2245600 -0.6165000 C 0.9698500 -1.3981600 -2.3864400 H 1.2470700 -0.5286800 -2.6831100 H 0.8279700 -1.9661900 -3.1472300 H 1.6522200 -1.7784200 -1.8269300 C -0.3307000 -1.2757600 -1.5924600 C -2.0034700 0.3023800 -0.8695800 C -3.6351000 1.3516400 -0.2027900 H -4.2567000 2.0289000 -0.0629300 O -6.5150200 0.6509500 1.1377300 O -5.6388200 -5.9718500 1.9204200 O -0.8540100 -2.2980100 -1.1280700 $end tris-(3-acetylamino-1,2,4-triazolate) motif of HUKDOG, with H2O and sulfate anion Magnet 2. The geometry was taken from crystal structure of HUKDOG. Nuclear Repulsion Energy = 7572.14770622 hartrees $molecule 266 0 4 H -10.4966300 -3.3279400 4.0020200 Cu -10.3420000 -2.3055700 5.7929600 Cu -13.0915500 -0.7242100 6.8306400 Cu -10.7085300 0.9936700 5.2477800 O -11.6034700 -0.7690800 5.5058500 H -12.0376500 -0.9594100 4.6474900 N -8.8841100 -1.1989700 5.2165200 N -8.9704800 0.1405700 5.0340800 N -6.8277900 -0.4333500 4.8495900 N -13.3043100 1.2046900 6.5989300 N -12.2461500 1.8562300 6.0014800 N -13.6286100 3.4188500 6.7777800 N -12.8711500 -2.6091700 7.0210900 N -11.7904800 -3.3264800 6.5837700 N -13.0894300 -4.7363000 7.7497400 N -7.1253600 -2.7817200 5.2437400 H -6.2785900 -2.8899600 5.1446600 N -11.6220700 4.1204200 5.6765000 H -11.8782200 4.9388900 5.7466400 N -14.7946700 -3.1522400 8.3434500 H -15.2660300 -3.7856300 8.6837800 C -7.1087300 -5.1540700 5.5057200 H -6.1786200 -4.9862500 5.3398300 H -7.2073000 -5.5867700 6.3593200 H -7.4609300 -5.7222700 4.8183300 C -7.8484000 -3.8718100 5.5176000 C -7.6081600 -1.4972900 5.1045200 C -7.7284700 0.5363700 4.8250800 H -7.5044200 1.4249200 4.6719000 C -9.6539300 5.1018000 4.6692400 H -10.1486400 5.8971400 4.8766400 H -8.8063200 5.1249600 5.1198500 H -9.5106100 5.0515100 3.7210800 C -10.4258400 3.9031200 5.1206600 C -12.4809300 3.1533300 6.1473500 C -14.0839900 2.1842100 7.0259700 H -14.8889700 2.0286000 7.4655900 C -16.5561400 -1.7978000 9.2355300 H -16.8333500 -2.6672800 9.5322100 H -16.4142500 -1.2297700 9.9963200 H -17.2385000 -1.4175400 8.6760300 C -15.2555800 -1.9202000 8.4415500 C -13.5828100 -3.4983300 7.7186800 267 C -11.9511800 -4.5476000 7.0518800 H -11.3295800 -5.2248600 6.9120300 O -9.0712600 -3.8469100 5.7113600 O -9.9474600 2.7758900 4.9286700 O -14.7322700 -0.8979500 7.9771600 $end 268 7.10 Chapter 7 References 1 A. W. Maue, Z. Phys. 94, 717 (1935). 2 E. T. Goodwin, Math. Proc. Camb. Philos. Soc. 35, 221 (1939). 3 P. Drude, Ann. Phys. 306, 566 (1900). 4 P. Drude, Ann. Phys. 308, 369 (1900). 5 A. Sommerfeld, Z. Phys. 47, 1 (1928). 6 J. Hubbard and B. H. Flowers, Proc. R. Soc. London A 276, 238 (1963). 7 W. Heisenberg, Z. Phys. 49, 619 (1928). 8 P. A. M. Dirac, Proc. R. Soc. London A 112, 661 (1926). 9 J. H. van Vleck, The Theory of Electric and Magnetic Susceptibilities. Clarendon Press, Oxford, 1932. 10 S. H. Simon, The Oxford solid state basics. Oxford University Press, 2013. 11 C. L. Cleveland and R. Medina A., Am. J. Phys. 44, 44 (1976). 12 K. A. Chao, J. Spalek, and A. M. Oles, J. Phys. C: Solid State Phys. 10, L271 (1977). 13 T. Kato, Prog. Theor. Phys. 4, 514 (1949). 14 S. ˆ Okubo, Prog. Theor. Phys. 12, 603 (1954). 15 C. Bloch, Nucl. Phys. 6, 329 (1958). 16 J. des Cloizeaux, Nucl. Phys. 20, 321 (1960). 17 P.-O. L¨ owdin, J. Math. Phys. 3, 969 (1962). 18 H. Feshbach, Ann. Phys. (N.Y .) 19, 287 (1962). 19 P. Durand, Phys. Rev. A 28, 3184 (1983). 20 C. E. Soliverez, Am. J. Phys. 35, 624 (1967). 21 C. E. Soliverez, J. Phys. C: Solid State Phys. 2, 2161 (1969). 22 C. E. Soliverez, Phys. Rev. A 24, 4 (1981). 269 23 J. P. Killingbeck and G. Jolicard, J. Phys. A: Math. Gen. 36, R105 (2003). 24 G. Jolicard and J. P. Killingbeck, Journal of Physics A: Mathematical and General 36, R411 (2003). 25 J. Finley, P.- ˚ A. Malmqvist, B. O. Roos, and L. Serrano-Andr´ es, Chem. Phys. Lett. 288, 299 (1998). 26 A. Zaitsevskii and J.-P. Malrieu, Chem. Phys. Lett. 233, 597 (1995). 27 A.A. Granovsky, J. Chem. Phys. 134, 214113 (2011). 28 B. Jeziorski and J. Paldus, J. Chem. Phys. 88, 5673 (1988). 29 N. Oliphant and L. Adamowicz, J. Chem. Phys. 94, 1229 (1991). 30 C. M. L. Rittby and R. J. Bartlett, Theor. Chim. Acta 80, 469 (1991). 31 V . I. Anisimov, J. Zaanen, and O. K. Andersen, Phys. Rev. B 44, 943 (1991). 32 V . I. Anisimov, I. V . Solovyev, M. A. Korotin, M. T. Czy˙ zyk, and G. A. Sawatzky, Phys. Rev. B 48, 16929 (1993). 33 V . I. Anisimov, F. Aryasetiawan, and A. I. Lichtenstein, J. Phys.: Condens. Matter 9, 767 (1997). 34 N. J. Mayhall and M. Head-Gordon, J. Chem. Phys. 141, 134111 (2014). 35 N. J. Mayhall and M. Head-Gordon, J. Phys. Chem. Lett. 6, 1982 (2015). 36 B. Bleaney and K. D. Bowers, Proc. R. Soc. London A 214, 451 (1952). 37 S. A. Al’tshuler; B. M. Kozyrev, Electron Paramagnetic Resonance. Academic Press, New York, 1964. 38 B. Sieklucka and D. Pinkowicz, Molecular Magnetic Materials: Concepts and Applications. Wiley, 2017. 39 I. Mirabeau; M. Hennion; H. Casalta; H. Andres; H. U. G¨ udel; A. V . Irodova; A. Caneschi, Phys. Rev. Lett. 83, 628 (1999). 40 A. Michels, J. Phys.: Condens. Matter 26, 383201 (2014). 41 S. M¨ uhlbauer, D. Honecker, E. A. P´ erigo, F. Bergner, S. Disch, A. Heinemann, S. Erokhin, D. Berkov, C. Leighton, M. R. Eskildsen, and A. Michels, Rev. Mod. Phys. 91, 015004 (2019). 270 42 G. Amoretti, R. Caciuffo, J. Combet, A. Murani, and A. Caneschi, Phys. Rev. B 62, 3022 (2000). 43 A. Land´ e, Z. Phys. 15, 189 (1923). 44 N. Orms and A. I. Krylov, Phys. Chem. Chem. Phys. 20, 13127 (2018). 45 L. Noodleman, J. Chem. Phys. 74, 5737 (1981). 46 H. F. Schurkus, D.-T. Chen, H.-P. Cheng, G. K.-L. Chan, and John F. Stanton, arXiv , https://arxiv.org/abs/2001.03226 (2020). 47 C. J. Calzado, J. Cabrero, J. P. Malrieu, and R. Caballol, J. Chem. Phys. 116, 3985 (2002). 48 R. Maurice, R. Bastardis, C. de Graaf, N. Suaud, T. Mallah, and N. Guih` ery, J. Chem. Theory Comput. 5, 2977 (2009). 49 A. Monari, D. Maynau, and J.-P. Malrieu, J. Chem. Phys. 133, 044106 (2010). 50 J.P. Malrieu, R. Caballol, C.J. Calzado, C. de Graaf, and N. Guih´ ery, Chem. Rev. 114, 429 (2013). 51 A. I. Krylov, Annu. Rev. Phys. Chem. 59, 433 (2008). 52 R. J. Bartlett, Mol. Phys. 108, 2905 (2010). 53 K. Sneskov and O. Christiansen, WIREs: Comput. Mol. Sci. 2, 566 (2012). 54 R. J. Bartlett, WIREs: Comput. Mol. Sci. 2, 126 (2012). 55 S. V . Levchenko and A. I. Krylov, J. Chem. Phys. 120, 175 (2004). 56 J. F. Stanton and R. J. Bartlett, J. Chem. Phys. 98, 7029 (1993). 57 A. I. Krylov, Chem. Phys. Lett. 338, 375 (2001). 58 D. Casanova, L. V . Slipchenko, A. I. Krylov, and M. Head-Gordon, J. Chem. Phys. 130, 044103 (2009). 59 A. I. Krylov, Acc. Chem. Res. 39, 83 (2006). 60 J. F. Stanton and J. Gauss, J. Chem. Phys. 101, 8938 (1994). 61 P. A. Pieniazek, S. E. Bradforth, and A. I. Krylov, J. Chem. Phys. 129, 074104 (2008). 62 M. Nooijen and R. J. Bartlett, J. Chem. Phys. 102, 3629 (1995). 271 63 Y . Shao, Z. Gan, E. Epifanovsky, A.T.B. Gilbert, M. Wormit, J. Kussmann, A.W. Lange, A. Behn, J. Deng, X. Feng, D. Ghosh, M. Goldey, P.R. Horn, L.D. Jacobson, I. Kaliman, R.Z. Khaliullin, T. Kus, A. Landau, J. Liu, E.I. Proynov, Y .M. Rhee, R.M. Richard, M.A. Rohrdanz, R.P. Steele, E.J. Sundstrom, H.L. Woodcock III, P.M. Zimmerman, D. Zuev, B. Albrecht, E. Alguires, B. Austin, G.J.O. Beran, Y .A. Bernard, E. Berquist, K. Brandhorst, K.B. Bravaya, S.T. Brown, D. Casanova, C.-M. Chang, Y . Chen, S.H. Chien, K.D. Closser, D.L. Crittenden, M. Diedenhofen, R.A. DiStasio Jr., H. Do, A.D. Dutoi, R.G. Edgar, S. Fatehi, L. Fusti-Molnar, A. Ghysels, A. Golubeva-Zadorozhnaya, J. Gomes, M.W.D. Hanson- Heine, P.H.P. Harbach, A.W. Hauser, E.G. Hohenstein, Z.C. Holden, T.-C. Jagau, H. Ji, B. Kaduk, K. Khistyaev, J. Kim, J. Kim, R.A. King, P. Klunzinger, D. Kosenkov, T. Kowal- czyk, C.M. Krauter, K.U. Laog, A. Laurent, K.V . Lawler, S.V . Levchenko, C.Y . Lin, F. Liu, E. Livshits, R.C. Lochan, A. Luenser, P. Manohar, S.F. Manzer, S.-P. Mao, N. Mardirossian, A.V . Marenich, S.A. Maurer, N.J. Mayhall, C.M. Oana, R. Olivares-Amaya, D.P. O’Neill, J.A. Parkhill, T.M. Perrine, R. Peverati, P.A. Pieniazek, A. Prociuk, D.R. Rehn, E. Rosta, N.J. Russ, N. Sergueev, S.M. Sharada, S. Sharmaa, D.W. Small, A. Sodt, T. Stein, D. Stuck, Y .-C. Su, A.J.W. Thom, T. Tsuchimochi, L. V ogt, O. Vydrov, T. Wang, M.A. Watson, J. Wenzel, A. White, C.F. Williams, V . Vanovschi, S. Yeganeh, S.R. Yost, Z.-Q. You, I.Y . Zhang, X. Zhang, Y . Zhou, B.R. Brooks, G.K.L. Chan, D.M. Chipman, C.J. Cramer, W.A. Goddard III, M.S. Gordon, W.J. Hehre, A. Klamt, H.F. Schaefer III, M.W. Schmidt, C.D. Sherrill, D.G. Truhlar, A. Warshel, X. Xu, A. Aspuru-Guzik, R. Baer, A.T. Bell, N.A. Besley, J.-D. Chai, A. Dreuw, B.D. Dunietz, T.R. Furlani, S.R. Gwaltney, C.-P. Hsu, Y . Jung, J. Kong, D.S. Lambrecht, W.Z. Liang, C. Ochsenfeld, V .A. Rassolov, L.V . Slipchenko, J.E. Subotnik, T. Van V oorhis, J.M. Herbert, A.I. Krylov, P.M.W. Gill, and M. Head-Gordon, Mol. Phys. 113, 184 (2015). 64 A. I. Krylov and P. M. W. Gill, WIREs: Comput. Mol. Sci. 3, 317 (2013). 65 P. Pokhilko, D. Izmodenov, and A. I. Krylov, J. Chem. Phys. 152, 034105 (2020). 66 P. Pokhilko, E. Epifanovsky, and A. I. Krylov, J. Chem. Phys. 151, 034106 (2019). 67 L. Salem and C. Rowland, Angew. Chem., Int. Ed. 11, 92 (1972). 68 A. I. Krylov, J. Phys. Chem. A 109, 10638 (2005). 69 C. L´ opez, R. Costa, F. Illas, C. de Graaf, M. M. Turnbull, C. P. Landee, E. Espinosa, I. Mata, and E. Molins, Dalton Trans. 13, 2322 (2005). 70 S. Ferrer, F. Lloret, I. Bertomeu, G. Alzuet, J. Borr´ as, S. Garc´ ıa-Granda, M. Liu-Gonz´ alez, and J. G. Haasnoot, Inorg. Chem. 41, 5821 (2002). 71 P. Pokhilko, E. Epifanovskii, and A. I. Krylov, J. Chem. Theory Comput. 14, 4088 (2018). 72 I. Kaliman and A. I. Krylov, J. Comput. Chem. 38, 842 (2017). 73 S.F. Boys, Rev. Mod. Phys. 32, 296 (1960). 272 74 T. E. Abrudan, J. Eriksson, and V . Koivunen, IEEE Transactions on Signal Processing 56, 1134 (2008). 75 T. Abrudan, J. Eriksson, and V . Koivunen, Signal Processing 89, 1704 (2009). 76 S. Lehtola and H. J´ onsson, J. Chem. Theory Comput. 9, 5365 (2013). 77 A. Sadybekov and A. I. Krylov, J. Chem. Phys. 147, 014107 (2017). 78 B. S. Tsukerblat, B. Ya. Kuyavskaya, M. I. Belinskii, A. V . Ablov, V . M. Novotortsev, and V . T. Kalinnikov, Theor. Chim. Acta 38, 131 (1975). 79 See https://web.archive.org/web/20160321110525/https://math. stackexchange.com/questions/754301/maximizing-the-trace. 273 Chapter 8: Future directions In the previous chapters, I presented the development of reduced-cost electronic structure methods, incorporation of spin–orbit effects, analysis of many-body wave-function multiplets, construction of exact effective Hamiltonians from the EOM-CC wave functions, and applica- tions of these approaches to studies of challenging systems: reactive diradicals in combustion reactions and single-molecule magnets. Our work is a step forward in understanding strongly correlated systems, molecular magnetism, and relativistic effects. We already see the impact in the scientific community. For example, recently Nick Mayhall’s group applied the general algorithm for evaluating spin–orbit matrix elements, presented in Chapter 2 and Ref. 1, to RAS- nSF-IP/EA-CI methods 2 . Several other groups are incorporating this algorithm in other RAS-CI frameworks and CVS-EOM-CC methods targeting core-ionized and core-excited states. Our work on single and mixed precision in correlated calculations (Chapter 5 and Ref. 3) served as a motivation for error analysis of lossy compression used for checkpointing in NWChem 4 . Right now, single-precision setup is de facto a standard way of performing computations in our group. The work presented in the previous chapters of the thesis can be further developed in the directions outlined below. 274 8.1 Method development 8.1.1 Incorporation of relativistic effects beyond the Breit–Pauli Hamil- tonian for heavier elements Traditionally, Breit–Pauli’s Hamiltonian is thought to be applicable only for light atoms, because of the incorrect treatment of particles with high momentum (due to the perturbative structure of the approximation), relevant for the low-lying orbitals in heavy elements. An exten- sion of the Breit–Pauli approach for heavy elements is possible thourgh effective core potentials (ECP), which can involve static relativistic effects directly into effective potential 5 . Although this approach is commonly used in practice, it has several disadvantages: 1. Usage of ECPs requires modification of operators describing interaction of electrons and nuclei. For example, Coulomb and one-particle spin–orbit operators should be replaced by their effective forms 6, 7 . The implementation of the integrals involving these effective operators is very difficult. 2. The ECPs are designed matching basis sets. Currently, systematic improvement of heavy element calculations with respect to basis sets is troublesome a , which complicates quan- titative predictions for heavy elements. 3. Two-electron spin–orbit contribution is typically neglected in this treatment. 4. ECPs are optimized only for valence states. They may lead to worse errors in the cases with significant changes in the valence structure, e.g., for addition or removal of several electrons. Moreover, ECP cannot treat core-excited or core-ionized states, which are crucial for interpreting X-ray spectroscopic experiments. a This is an ongoing topic of research. To name a few, systematically convergent basis sets for 3d 8 , 4d 9 , and 5d 10 elements have been developed. 275 The schemes that do not rely on parametrization of the effective potential can, inprinciple, pro- vide more accurate results. Exploration of such schemes within advanced ab initio methods, such as EOM-CC, is an interesting direction for future research. In particular, the Douglas– Kroll–Hess (DKH) Hamiltonian 11, 12 has been used for the treatment of heavy elements for decades. Recent exact-two component family of approximations (X2C) 13–15 provide an alterna- tive framework for relativistic calculations. One can also combine X2C and DKH Hamiltonians together 16, 17 . 8.1.2 Further cost reduction through combination of FNO and localiza- tion approaches Although the FNO scheme was originally developed for the ground state 18–21 , it can be used for multiple ionized states as well 22 . In Chapter 6 and Ref. 23, we showed how this scheme can be extended for covalent and ionic states of di- and triradicals within EOM-SF-CCSD, leading to the OSFNO approximation. OSFNO can reduce the cost of the EOM-SF-CCSD calculations by more than an order of magnitude, which allowed us to perform the largest EOM-SF-CCSD calculation on only one computational node (1038 atomic orbitals, 47 atoms, 196 electrons). The OSFNO and FNO approximations reduce the size of the virtual space by utilizing its spar- sity. This also can be described as a compression of the wave function in the virtual subspace. This compression has limits: if the basis set is compact, the virtual space is not sparse enough to provide large speedups. Such situations require other strategies to reduce computational cost. As discussed in the introduction, weak electron correlation arises from Coulomb interaction that has a local nature. Configurations containing electrons on distant localized orbitals can be efficiently eliminated. Because this strategy is orthogonal to the strategy based on natu- ral orbitals, one can combine them in a very computationally efficient approximation. These hybrid approaches 24–32 allow coupled-cluster calculations of whole proteins containing several thousands of basis functions. The OSFNO scheme employs the separation of strongly and 276 weakly interacting orbitals through singular value decomposition, which would also work for these hybrid schemes. 8.2 Applications 8.2.1 Application of EOM-SF to multi-center single-molecule magnets via Mayhall’s extrapolation Figure 8.1: Left: RAHBIM, a wheel-shaped Mn 12 complex. Right: AYUCOM, a wheel- shaped Cr 8 complex. Treatment of single-molecule magnets with multiple radical centers requires nSF excita- tions acting on a high-spin reference. The cost of such approaches grows factorially with the number of flips (n). Nick Mayhall and Martin Head-Gordon have introduced a novel computa- tional protocol, which allows the evaluation of eigenenergies for arbitrary spin state manifolds based on a single RAS-SF calculation. In this protocol, a Heisenberg effective Hamiltonian is built for the RAS-SF states first; then, the extracted effective exchange couplings are used to construct the Heisenbseg Hamiltonian so that the states that are not available in a single SF calculation can be computed. Therefore, it can be used for ab initio predictions of magnetic properties of multi-center metal complexes such as the wheel-shaped RAHBIM and AYUCOM magnets 33, 34 shown in Fig. 8.1. In our work, we prepared the main ingredients for cooking this 277 magnetic dish within EOM-CC—reduced-cost strategies such as the usage of single precision and OSFNO approximation (Chapters 5 and 6) can make these systems tractable with EOM- SF-CCSD, while our construction of effective Hamiltonians from the EOM-CC wave functions provide (Chapter 7) a tool for the extraction of the effective parameters. The last but not least is the general algorithm for the evaluation of spin–orbit matrix elements and the extension of NTO analysis for spin–orbit couplings (Chapters 2 and 3), which can elucidate magnetic anisotropy in metal complexes. 278 8.3 Chapter 8 References 1 P. Pokhilko, E. Epifanovsky, and A. I. Krylov, J. Chem. Phys. 151, 034106 (2019). 2 O. Meitei, S. Houck, and N. Mayhall, (2020). 3 P. Pokhilko, E. Epifanovskii, and A. I. Krylov, J. Chem. Theory Comput. 14, 4088 (2018). 4 T. Reza, J. Calhoun, K. Keipert, S. Di, and F. Cappello, Analyzing the performance and accuracy of lossy checkpointing on sub-iteration of NWChem, in 2019 IEEE/ACM 5th Inter- national Workshop on Data Analysis and Reduction for Big Scientific Data (DRBSD-5), pages 23–27, 2019. 5 O. Gropen, The Relativistic Effective Core Potential Method, pages 109–135. Springer US, Boston, MA, 1988. 6 R. B. Ross, J. M. Powers, T. Atashroo, W. C. Ermler, L. A. LaJohn, and P. A. Christiansen, J. Chem. Phys. 93, 6654 (1990). 7 R. B. Ross, S. Gayen, and W. C. Ermler, J. Chem. Phys. 100, 8145 (1994). 8 N. B. Balabanov and K. A. Peterson, J. Chem. Phys. 123, 064107 (2005). 9 K. A. Peterson, D. Figgen, M. Dolg, and H. Stoll, J. Chem. Phys. 126, 124101 (2007). 10 D. Figgen, K. A. Peterson, M. Dolg, and H. Stoll, J. Chem. Phys. 130, 164108 (2009). 11 G. Jansen and B. A. Hess, Phys. Rev. A 39, 6016 (1989). 12 M. Reiher, Theor. Chim. Acta 116, 241 (2006). 13 K. G. Dyall, J. Chem. Phys. 115, 9136 (2001). 14 W. Liu and D. Peng, J. Chem. Phys. 131 (2009). 15 L. Cheng and J. Gauss, J. Chem. Phys. 141, 164107 (2014). 16 Z. Li, X. Xiao, and W. Liu, J. Chem. Phys. 137, 154114 (2012). 17 L. Cheng, F. Wang, J. F. Stanton, and J. Gauss, J. Chem. Phys. 148, 044108 (2018). 18 T. L. Barr and E. R. Davidson, Phys. Rev. A 1, 644 (1970). 19 C. Sosa, J. Geertsen, G. W. Trucks, and R. J. Bartlett, Chem. Phys. Lett. 159, 148 (1989). 20 A. G. Taube and R. J. Bartlett, Collect. Czech. Chem. Commun 70, 837 (2005). 279 21 A. G. Taube and R. J. Bartlett, J. Chem. Phys. 128, 164101 (2008). 22 A. Landau, K. Khistyaev, S. Dolgikh, and A. I. Krylov, J. Chem. Phys. 132, 014109 (2010). 23 P. Pokhilko, D. Izmodenov, and A. I. Krylov, J. Chem. Phys. 152, 034105 (2020). 24 F. Neese, F. Wennmohs, and A. Hansen, J. Chem. Phys. 130, 114108 (2009). 25 P. R. Nagy, G. Samu, and M. K´ allay, J. Chem. Theory Comput. 14, 4193 (2018). 26 C. Riplinger and F. Neese, J. Chem. Phys. 138, 034106 (2013). 27 F. Neese, A. Hansen, and D. G. Liakos, J. Chem. Phys. 131, 064103 (2009). 28 C. Riplinger, B. Sandhoefer, A. Hansen, and F. Neese, J. Chem. Phys. 139, 134101 (2013). 29 P. Pinski, C. Riplinger, E. F. Valeev, and F. Neese, J. Chem. Phys. 143, 034108 (2015). 30 C. Riplinger, P. Pinski, U. Becker, E. F. Valeev, and F. Neese, J. Chem. Phys. 144, 024109 (2016). 31 M. Saitow, U. Becker, C. Riplinger, E. F. Valeev, and F. Neese, J. Chem. Phys. 146, 164105 (2017). 32 A. K. Dutta, M. Saitow, C. Riplinger, F. Neese, and R. Izs´ ak, J. Chem. Phys. 148, 244101 (2018). 33 E. M. Rumberger, L. N. Zakharov, A. L. Rheingold, and D. N. Hendrickson, 43, 6531 (2004). 34 P. Christian, G. Rajaraman, A. Harrison, J. J. W. McDouall, J. T. Raftery, and R. E. P. Win- penny, Dalton Trans. , 1511 (2004). 280

Abstract (if available)

Abstract Strongly correlated open-shell species are common in atmospheric and combustion chemistry, catalysis, astrochemistry, molecular magnetism, and quantum information science. Electronic structure of such species can be complex and multiconfigurational, imposing strict requirements on quantum chemical methods. Some systems of interest, such as single-molecule magnets, have hundreds (if not thousands) of electrons, presenting a challenge for current electronic structure methods. Moreover, because non-relativistic calculations often do not capture the physics of the systems, the relativistic effects must be included, which also increase computational cost. This work presents development of robust and efficient many-body methods and their applications to challenging open-shell systems. ❧ A typical non-relativistic calculation provides electronic states with only one spin projection. However, correct treatment of spin–orbit interaction requires states with all possible spin projections. In Chapter 2, we show how application of Wigner–Eckart’s theorem to one-particle transition density matrices resolves this problem. We present a general formalism, which relies on only one transition density matrix between electronic states with given spin projections and multiplicities. In this approach, the phases of wave-functions are synchronized through Clebsh–Gordan’s coefficients. This choice of phase synchronization avoids the issue of phase arbitrariness, appearing within explicit change of coordinate system (discussed as well in Chapter 4), which can be related to the phase-blindless of Born–Oppenheimer’s approximation. We analyze a commonly used reduced-cost spin–orbit mean-field approximation (SOMF) and demonstrate that it leads to violation of physical symmetries when applied to open-shell references. Based on spin-tensor formalism, we propose a modified version of SOMF that preserves the correct physics. We benchmarked the approach on several open-shell species, including high-spin tris(pyrrolylmethyl)amine Fe(II) complex, explaining a large spin-reversal barrier of magnetization in this single-molecule magnet. ❧ Multiconfigurational wave-functions are difficult to interpret. Moreover, spin-dependent properties require analysis of wave-functions with different spin projections. In Chapter 3, we extend the concept of natural transition orbitals (NTO) to natural orbitals of triplet spinless transition density matrix. This approach allows one to analyze the transitions between many-body wave-functions of different spin projections in a uniform and compact manner. We used this approach to validate, quantify, and extend empirically known El-Sayed’s rules. ❧ In Chapter 4, we study electronic structure of Cvetanović’s diradicals, which are formed in the reactions between atomic oxygen (³P) and unsaturated hydrocarbons. These diradicals show significant variations in intersystem crossing rates, which we explain on the basis of calculations. We found that minimal energy crossing points (MECP) in ethylene- and propylene-derived species are energetically close to triplet minima, making the spin-forbidden reactions nearly barrierless. MECP barriers in acetylene- and propylene-derived species are higher, correlating with experimentally observed slower rates of spin-forbidden transitions. The magnitude of spin–orbit couplings is explained using the NTO analysis. This analysis also highlights the importance of quantification of El-Sayed’s rules: intermediate values of spin–orbit coupling constants are observed for partial rotation of p-orbitals on oxygen and carbon atoms. ❧ In Chapter 5, we reconsider the choice of numerical precision for quantum chemical calculations. Traditionally, double precision is used in most post-Hartree–Fock calculations. We found that single precision can be safely used for both ground and excited-state energies, properties, and geometries. The resulting accuracy of the approach leads to errors that are 100–1000 times smaller than the target thermochemical accuracy and are comparable with the typical thresholds used in calculations. ❧ In Chapter 6, we explore virtual space truncation for correlated methods based on open-shell references. Virtual space truncation reduces the cost of calculations, especially for large systems. We show that a typical truncation scheme based on frozen virtual natural orbitals (FNO) gives large errors when applied for equation-of-motion spin-flip coupled-cluster method (EOM-SF-CC). We found that the reason of the large errors lies in the inconsistent freezing of α and β spin-orbital subspaces. Additional issues appeared due to a partial inclusion of strongly correlated orbitals to the frozen orbital subspace. We developed a new open-shell frozen natural orbital approximation (OSFNO), which separates strongly correlated orbitals based on singular value decomposition (SVD) of the overlap matrix followed by establishing corresponding natural-like orbitals through SVD of one-particle singlet state density matrix in the remaining virtual orbital subspace. OSFNO delivers robust performance. This approach can be used to significantly reduce the cost of large systems such as molecular magnets. ❧ Model Hamiltonians are widely used to describe strongly correlated systems. Parametrization of these Hamiltonians is often obtained through fitting of experimental observables. However, the connection to theory is not often simple. For example, the Lande interval rule can be used to extract exchange coupling constants for the systems with two radical centers. More complicated systems, such as molecular magnets with multiple radical centers, present a challenge for interpretation of calculations in terms of Heisenberg’s Hamiltonian. In Chapter 7, we develop an exact mapping between EOM-CC configurations and model spaces. Based on Bloch’s theory, we show how Hubbard’s and Heisenberg’s Hamiltonians can be rigorously parametrized from the EOM-SF-CC wave-functions. This approach allows one to make a rigorous connection between many-body calculations and experimental observables.

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Creator Pokhilko, Pavel (author)

Core Title Development and application of robust many-body methods for strongly correlated systems: from spin-forbidden chemistry to single-molecule magnets

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Chemistry

Publication Date 04/21/2020

Defense Date 03/19/2020

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag combustion chemistry,coupled cluster,effective Hamiltonians,EOM-CC,equation of motion coupled cluster,OAI-PMH Harvest,single-molecule magnet,spin-flip,spin-orbit coupling,spin-orbit interaction,strong correlation

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Krylov, Anna (committee chair), Nakano, Aiichiro (committee member), Prezhdo, Oleg (committee member)

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Permanent Link (DOI) https://doi.org/10.25549/usctheses-c89-283829

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