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Electronic structure analysis of challenging open-shell systems: from excited states of copper oxide anions to exchange-coupling in binuclear copper complexes
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Electronic structure analysis of challenging open-shell systems: from excited states of copper oxide anions to exchange-coupling in binuclear copper complexes
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ELECTRONIC STRUCTURE ANALYSIS OF CHALLENGING OPEN-SHELL SYSTEMS: FROM EXCITED STATES OF COPPER OXIDE ANIONS TO EXCHANGE-COUPLING IN BINUCLEAR COPPER COMPLEXES by Natalie F. Orms 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 (CHEMICAL PHYSICS) December 2017 Copyright 2017 Natalie F. Orms Acknowledgements She has pink hair, and she is a cat person. We’re going to get along just fine. This was my first impression of Professor Anna Krylov when I met her in the sum- mer of 2012 as her undergraduate summer research intern. First impressions do not always turn out to be accurate, but five years later, I am pleased to say that this one did. As my research advisor, Anna taught me how to tackle complex scientific problems in careful, methodical, and creative ways. As a mentor, she taught me the value of perse- verance and hard-work, lessons that will serve me for a lifetime. Not many experts in electronic structure theory would look at a student who received a C in college physics and take her seriously when she declares, “I want to study transition metal chemistry from first principles.” Anna gave me that opportunity, never letting me settle for any- thing less than a deep understanding of the science in question. If you are reading this, Boss, I hope you know how grateful I am to you. I hope that when the next first-year graduate student from some little-known college in Kentucky wanders into your office, wanting to push the limits of modern electronic structure theory, you will take a chance on her too. I owe a great debt to my undergraduate research mentor, Professor Jennifer Muzyka, who introduced me to computational chemistry. She was not only a wonderful professor and advisor, but a role model and friend, and will remain the coolest chemist I have ii ever known. I am also indebted to Professor Ed Montgomery, who taught my first college course in quantum mechanics and paved the way for my career at USC. With his contagious passion for electronic structure theory and a simple email introduction to a colleague at Moscow State, he effectively squandered my life-long dream of attending medical school. I cannot thank him enough for ruining my plans. Thanks go to the professors who have educated and advised me during my time at USC, in particular, Professors Curt Wittig and Aiichiro Nakano. Professor Wittig is the kind of professor who shows up in your office with lemon squares when you’re strug- gling through the last week of your qualifying examination. He is the kind of professor who teaches graduate courses on advanced topics simply because his students ask him to. There aren’t enough of him. In my desk, there is a folder of miscellaneous articles labelled “Nakano papers.” They are the culmination of Professor Nakano’s reading rec- ommendations over the years, and they never failed to simultaneously spark new ideas for my projects and distract me from whatever task I was completing at the time. I hope Professor Nakano will continue distracting his students for many years to come. In addition to Professor Nakano and Professor Krylov, special thanks go to the other members of my qualifying exam and thesis committees: Professors Smaranda Mari- nescu, Hanna Reisler, and Oleg Prezhdo. I appreciate how busy your schedules are. I appreciate the time commitment that is required to be a member of these panels. I appreciate you. The work presented in this thesis would be incomplete without the help of my collab- orators: Andreas Dreuw and Dirk Rehn, who contributed ADC(2) results and stimulated great conversations about Chapter 3; Carol Parish, Adam Luxon, and Ren´ e Kanters, who are responsible for much of the energetic data and analysis presented in Chapter 4. From all of you, I have learned much. iii My thesis would also be incomplete without the help of my fellow group mem- bers, past and present: Nadia Korovina, Adele Laurent, Thomas Jagau, Kaushik Nanda, Marc de Wergifosse, Ilya Kaliman, Matthias Schneider, Alex Barrozo, Wojciech Sko- morowski, Marwa Farag, Pavel Pokhilko, Xintian Feng, Sahil Gulania, and Tirthendu Sen, to name a few; Dmitry Zuev and Kirill Khistyaev, who served as my mentors during my first projects in the Krylov group; Shirin Faraji, Samer Gozem, Anastasia Gunina, and Atanu Acharya, who provided invaluable feedback during my qualifying exam. I am grateful to all of you, and wish you nothing but the best in your great deeds to come. When you find yourself in the trenches of graduate school, the first thing you do is look around and discover you aren’t alone. The people next to you become some of your dearest friends. In the trench beside me were, among others, Arman and Anas- tasiia Sadybekov, Bailey Qin, Daniel Kwasniewski, Alexandra Aloia, and Kavita Belli- gund. If I were to list all the ways, big and small, you helped me retain my sanity these last four years, I would need another fifty pages. Thank you for being my “framily.” Thank you for filling my years at USC with laughter, concerts, board games, boba runs, roommates, tubby paste, spontaneous Vegas weekends, all-you-can-eat sushi, Russian pancakes, Ground Zero milkshakes, gingerbread house contests, and more. Thank you. I would also like to thank my family. Janet and Bob Orms, my mother and father, not only provided me with food and shelter for my first twenty-two years of life, but made sure that science was my first love. I would not be who or where I am today without the encouragement they gave me to chase whatever was on the horizon. My sister, Caroline, never failed to give me the good advice (and early-morning FaceTime calls with my niece) that I needed to surmount the obstacles I faced along the way. She will always be the hero I call on when the going gets tough. iv Finally, I’d like to thank my husband and best friend, Ben Slone. Ben has been the most important person in my world for nearly half my life, and proof that the best things—even those you least expect—are owed to science. Thank you, Ben, for sitting next to the skinny girl with braces in our 8 th grade science class, all those years ago. Thank you for loving and supporting her through her last. v Table of Contents Acknowledgements ii List of Tables ix List of Figures xiv Abstract xxiii Chapter 1: Introduction 1 1.1 Multicofigurational Wave Functions: Navigating the Open-Shell Inter- state Highway . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Electronic States of Di- and Triatomic Copper Oxides: Mapping the Low-Energy Spectrum with EOM-CCSD Methods . . . . . . . . . . . 5 1.3 Density-Based Analysis of Benzynes, Xylylenes, and the Bergman Cy- clization Reaction with EOM-SF-CCSD and SF-TDDFT . . . . . . . . 9 1.4 Computing Exchange-Coupling Constants and State Properties in Binu- clear Copper Diradicals by a Spin-Flip Approach . . . . . . . . . . . . 12 Chapter 1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Chapter 2: Modeling photoelectron spectra of CuO, Cu 2 O, and CuO 2 an- ions with equation-of-motion coupled-cluster methods: An ad- venture in Fock space 19 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2 Computational Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.3.1 CuO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.3.2 Cu 2 O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.3.3 CuO 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 2.5 Appendix A: Symmetry labels . . . . . . . . . . . . . . . . . . . . . . 46 2.6 Appendix B: Hartree-Fock molecular orbital diagrams for singlet CuO and additional EOM-CCSD and CIS results . . . . . . . . . . . . . . . 47 vi 2.7 Appendix C: Hartree-Fock molecular orbital diagrams for doublet Cu 2 O and additional EOM-CCSD results . . . . . . . . . . . . . . . . . . . . 51 2.8 Appendix D: Cartesian geometries . . . . . . . . . . . . . . . . . . . . 53 Chapter 2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 Chapter 3: Characterizing bonding patterns in diradicals and triradicals by density-based wave function analysis: A uniform approach 58 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.2 Theoretical methods and computational details . . . . . . . . . . . . . . 65 3.2.1 Computational Details . . . . . . . . . . . . . . . . . . . . . . 67 3.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.3.1 Equilibrium Geometries . . . . . . . . . . . . . . . . . . . . . 69 3.3.2 Head-Gordon’s Indices Along the H 2 Dissociation Curve . . . . 72 3.3.3 EOM-SF-CCSD energies and wave function character in dirad- icals and triradicals . . . . . . . . . . . . . . . . . . . . . . . . 75 3.3.4 Molecular Magnets: Natural Orbitals versus Molecular Orbitals 77 3.3.5 Comparison between EOM-SF-CCSD, SF-TDDFT, and SF-ADC(2)- s wave functions . . . . . . . . . . . . . . . . . . . . . . . . . 79 3.3.6 Adiabatic Singlet-Triplet and Doublet-Quartet Gaps . . . . . . . 81 3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 3.5 Appendix A: Equilibrium structures . . . . . . . . . . . . . . . . . . . 91 3.6 Appendix B: Wave function analysis in meta-benzyne . . . . . . . . . . 92 3.7 Appendix C: H 2 example and comparison of two Head-Gordon’s indices 93 3.8 Appendix D: Comparison of two Head-Gordon’s indices for diradicals and triradicals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 3.9 Appendix E: Results for CUAQAC02 and CITLAT with the B97 func- tional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 3.10 Appendix F: Cartesian geometries . . . . . . . . . . . . . . . . . . . . 98 Chapter 3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 Chapter 4: An ab initio exploration of the Bergman cyclization 115 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 4.2 Computational Details . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 4.2.1 Optimizations and Frequency Calculations . . . . . . . . . . . . 118 4.2.2 Energy Calculations . . . . . . . . . . . . . . . . . . . . . . . . 119 4.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 122 4.3.1 Singlet Pathway . . . . . . . . . . . . . . . . . . . . . . . . . . 122 4.3.2 Triplet Pathway . . . . . . . . . . . . . . . . . . . . . . . . . . 125 4.3.3 Wavefunction Properties and Natural Orbitals . . . . . . . . . . 127 4.3.4 Energetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 4.3.5 Method Comparison . . . . . . . . . . . . . . . . . . . . . . . 129 4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 vii 4.5 Appendix A: Key values: S 0 Pathway . . . . . . . . . . . . . . . . . . 138 4.6 Appendix B: Key Values: T 1 Pathway . . . . . . . . . . . . . . . . . . 140 4.7 Appendix C: Zero Point Energies . . . . . . . . . . . . . . . . . . . . . 142 4.8 Appendix D: Cartesian Geometries . . . . . . . . . . . . . . . . . . . . 143 Chapter 4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 Chapter 5: Singlet-triplet energy gaps and the degree of diradical character in binuclear copper molecular magnets characterized by spin- flip density functional theory 149 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 5.2 Theoretical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 5.2.1 Diradicals and spin-flip approach . . . . . . . . . . . . . . . . . 154 5.2.2 The Heisenberg-Dirac-van-Vleck Hamiltonian for localized, weakly interacting electrons . . . . . . . . . . . . . . . . . . . . . . . . 156 5.2.3 Quantifying radical character and bonding patterns by density- based analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 5.3 Computational Details . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 5.4 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 165 5.4.1 The comparison of singlet-triplet gaps computed by different methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 5.4.2 SF-TDDFT: The effect of ECP and basis sets . . . . . . . . . . 166 5.4.3 SF-TDDFT: Collinear versus non-collinear kernel . . . . . . . 169 5.4.4 The effect of geometry on the computed gaps . . . . . . . . . . 170 5.4.5 Wavefunction analysis . . . . . . . . . . . . . . . . . . . . . . 171 5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 5.6 Appendix A: Derivation of the Heisenberg-Dirac-van-Vleck Hamilto- nian and the Land´ e Interval Rule . . . . . . . . . . . . . . . . . . . . . 184 5.7 Appendix B: ECP Details . . . . . . . . . . . . . . . . . . . . . . . . . 191 5.8 Appendix C: Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . 193 5.9 Appendix D: Optimized structures . . . . . . . . . . . . . . . . . . . . 195 5.10 Appendix E: EOM-SF-CCSD calculations at optimized structures . . . 196 5.11 Appendix F: Singlet-Triplet Gaps with the Collinear TDDFT Kernel . . 197 5.12 Appendix G: Wavefunction Analysis: B97 Frontier natural orbitals of the CUAQAC02 Complex . . . . . . . . . . . . . . . . . . . . . . . . . 198 5.13 Appendix H: Cartesian geometries . . . . . . . . . . . . . . . . . . . . 199 Chapter 5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 Chapter 6: Future Directions 221 Chapter 6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 viii List of Tables 2.1 Six lowest EOM-IP-CCSD/ECP10MDF/aug-cc-pVTZ-PP a detachment energies obtained from the closed-shell singlet CuO reference. ECP applied to Cu only. All-electron aug-cc-pVTZ basis used for oxygen. . 29 2.2 Six lowest excitation energies (from the closed-shell singlet reference) of CuO computed at the EOM-EE-CCSD/ECP10MDF/aug-cc-pVTZ- PP a level of theory. ECP applied to Cu only. All-electron aug-cc-pVTZ basis used for oxygen. . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.3 Six lowest EOM-EE-CCSD/ECP10MDF/aug-cc-pVTZ-PP a excitation energies of closed-shell singlet Cu 2 O. ECP applied to copper only. All- electron basis used for oxygen. . . . . . . . . . . . . . . . . . . . . . . 33 2.4 Lowest vertical electron detachment energy of Cu 2 O calculated by dif- ferent approaches (using the all-electron cc-pVTZ basis, unless other- wise indicated). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.5 12 lowest states of CuO 2 computed by EOM-DIP-CCSD/cc-pVTZ from the closed-shell singlet CuO 3 2 reference. . . . . . . . . . . . . . . . . 39 2.6 Correspondence betweenC 2v andC 1v point groups 54 . . . . . . . . . 46 2.7 Correspondence betweenD 2h andD 1h point groups 54 . . . . . . . . . 46 2.8 Six lowest EOM-IP-CCSD/cc-pVTZ (all-electron) detachment energies obtained from the closed-shell singlet CuO reference. . . . . . . . . . 48 2.9 Six lowest EOM-IP-CCSD/ECP10MDF/cc-pVTZ-PP detachment ener- gies obtained from the closed-shell singlet CuO reference. ECP ap- plied to copper only. All-electron cc-pVTZ basis used for oxygen. . . . 48 2.10 Six lowest excitation energies (from the closed-shell singlet reference) of CuO computed at the EOM-EE-CCSD/cc-pVTZ level of theory. All-electron basis used for copper and oxygen. . . . . . . . . . . . . . 49 2.11 Six lowest excitation energies (from the closed-shell singlet reference) of CuO computed at the EOM-EE-CCSD/ECP10MDFD/cc-pVTZ-PP level of theory. ECP applied to Cu only. All-electron cc-pVTZ basis used for O. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 2.12 Six lowest CIS/cc-pVTZ excitation energies from the closed-shell sin- glet reference of CuO . All-electron basis used for copper and oxygen. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 ix 2.13 Six lowest EOM-EE-CCSD/cc-pVTZ excitation energies of closed-shell singlet Cu 2 O. All-electron basis used for Cu and O. . . . . . . . . . . . 52 2.14 Six lowest EOM-EE-CCSD/ECP10MDF/cc-pVTZ-PP excitation ener- gies of closed-shell singlet Cu 2 O. ECP applied to copper only. All- electron basis used for oxygen. . . . . . . . . . . . . . . . . . . . . . . 52 3.1 EOM-SF-CCSD energy gaps (eV) and wave function properties of low- est two states (singlets andM s = 0 triplets for diradicals,M s = 1/2 dou- blets and quartets for triradicals). . . . . . . . . . . . . . . . . . . . . 77 3.2 SF-TDDFT and ADC(2)-s energy splittings (eV) and wave function properties of lowest singlet and M s = 0 triplet states of diradicals. . . . 82 3.3 SF-TDDFT energy splittings (cm 1 ) and wave function properties of lowest singlet and M s = 0 triplet states of binuclear copper diradicals. . 83 3.4 n u andn u;nl at the SF-TDDFT and ADC(2)-s/cc-pVTZ levels of theory. Values are provided for the lowest singlet and M s = 0 triplet states of each diradical benchmark system. Results for four density functionals are compared. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 3.5 SF-TDDFT/cc-pVTZ energy splittings (eV) and wave function proper- ties of lowest M s = 1/2 doublet and quartet states of triradicals. Results for four density functionals are shown. . . . . . . . . . . . . . . . . . 87 3.6 Adiabatic energy gaps (eV) between the lowest singlet and triplet states of methylene (CH 2 ) and A–C. . . . . . . . . . . . . . . . . . . . . . . 87 3.7 Adiabatic energy gaps (eV) between the lowest doublet and quartet states of F–I. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 3.8 E tot , n u andn u;nl , and their difference =jn u n u;nl j computed by EOM-SF-CCSD/cc-pVTZ. Values are provided for three states of H 2 at varying internuclear distances,R: the high-spin reference and the lowest singlet and triplet excited states. . . . . . . . . . . . . . . . . . . . . . 93 3.9 E tot ,n u andn u;nl , and their difference =jn u n u;nl j at the B5050LYP/cc- pVTZ level of theory. Values are provided for the lowest singlet and triplet excited states of diatomic H 2 at varying internuclear distances,R. 94 3.10 n u andn u;nl , and their difference =jn u n u;nl j computed by EOM- SF-CCSD/cc-pVTZ. Values are provided for three states of each bench- mark system: the high-spin reference and the lowest high-spin and low- spin states (singlet and triplet states of diradicals, and doublet and quar- tet states of triradicals). . . . . . . . . . . . . . . . . . . . . . . . . . . 95 3.11 n u and n u;nl and their difference =jn u n u;nl j computed by SF- TDDFT/cc-pVTZ with four different functionals. Values are provided for the lowest high-spin and low-spin states of each triradical. . . . . . 96 x 4.1 Vertical singlet-triplet gaps for each structure. Adiabatic gaps for R, TS, P (ZPE corrected). All values are in kcal/mol. A positive value indicates that the singlet is lower in energy. . . . . . . . . . . . . . . . . . . . . 123 4.2 Leading electronic configurations of singlet and triplet states at the re- spective optimized structures. [Core] 30 denotes the first 15 doubly occu- pied lower energy molecular orbitals. . . . . . . . . . . . . . . . . . . 125 4.3 Wavefunction properties of the reactant, transition state, and product in the singlet and triplet pathways of Bergman cyclization. . . . . . . . . 128 4.4 H z and H rxn values for S 0 and T 1 cyclizations alongside available experimental and previous theoretical values. . . . . . . . . . . . . . . 129 4.5 Comparison of key values for S 0 reaction with multiple combinations of method, basis, and spin treatment. Values are from single point calcu- lations (ZPE corrected) on optimized structures in the manner described above. Reference wavefunctions used for each calculation are indicated. When adopting a restricted approach, RHF was used for reactant and ROHF was used for TS and product. This is indicated in the table as RHF/ROHF. Note: Spin flip target state energies are all calculated at the CCSD level even though reference wave functions are optimized at CCSD(T) level. H.S denotes high spin, see Figure 4.3 for details. Total energy of the singlet reactant has units of a.u, all other values are in units of kcal/mol. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 4.6 Comparison of the impact that method, basis set, and spin restrictions have on total energies (a.u) and reaction energetics for the T 1 reaction. Values for triplet H z and H rxn are calculated from ZPE corrected total energies and are in units of kcal/mol. The spin heading refers to the treatment of the reference wavefunction where R = restricted open- shell HF and U = unrestricted HF. . . . . . . . . . . . . . . . . . . . . 141 4.7 Zero Point Energy values for all structures optimized with cc-pVDZ ba- sis set. Calculated with same method as respective geometry optimiza- tion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 4.8 Zero Point Energy values for all structures optimized with cc-pVTZ ba- sis set. Calculated with same method as respective geometry optimiza- tion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 5.1 Experimental exchange-coupling constants for eight binuclear copper diradicals shown in Figure 5.3. . . . . . . . . . . . . . . . . . . . . . . 164 5.2 EOM-SF-CCSD/cc-pVDZ singlet-triplet gaps (computed using the X- ray structures) and experimental exchange-coupling constants, in cm 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 5.3 Singlet-triplet gaps (cm 1 ) for Complexes 1-8 calculated using SF-TDDFT with the LRC-!PBEh method and the non-collinear TDDFT kernel. . . 168 xi 5.4 Singlet-triplet gaps (cm 1 ) for Complexes 1-8 calculated using SF-TDDFT with the PBE0 method and the non-collinear TDDFT kernel. . . . . . . 179 5.5 Singlet-triplet gaps (cm 1 ) for Complexes 1-8 calculated using SF-TDDFT with the PBE50 method. . . . . . . . . . . . . . . . . . . . . . . . . . 180 5.6 Singlet-triplet gaps (cm 1 ) for Complexes 1-8 calculated using SF-TDDFT with the B5050LYP method. . . . . . . . . . . . . . . . . . . . . . . . 181 5.7 Singlet-triplet gaps (cm 1 ) for optimized geometries of Complexes 1, 4 and 8, compared with gaps obtained using the experimentally deter- mined molecular geometries. . . . . . . . . . . . . . . . . . . . . . . . 182 5.8 NC-PBE50/cc-pVDZ energy splittings (cm 1 ) and wavefunction prop- erties of lowest singlet and M s = 0 triplet states of all eight benchmark complexes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 5.9 SF-TDDFT/cc-pVTZ and EOM-SF-CCSD/cc-pVDZ energy splittings (cm 1 ) and wave function properties of lowest singlet and M s = 0 triplet spin states of CUAQAC02. . . . . . . . . . . . . . . . . . . . . . . . . 183 5.10 B5050LYP/cc-pVDZ SF-TDDFT energy splittings (cm 1 ) and wave func- tion properties of lowest singlet and M s = 0 triplet spin states of simpli- fied PATFIA and the full experimental structure, PATFIA + Fe(C 5 H 5 ) 2 . The collinear TDDFT kernel was applied. . . . . . . . . . . . . . . . . 183 5.11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 5.12 ECPs with matching basis sets and core electron approximations for Cu, Cl, C, O, N and H. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 5.13 Mean error ( ME ), mean absolute error ( MAE ), and standard deviation of the error ( STD ) in the calculated values of the exchange coupling constants of AF complexes 1-5 at different levels of theory a . Errors are reported relative to experiment in cm 1 . . . . . . . . . . . . . . . . . . 193 5.14 Mean error ( ME ), mean absolute error ( MAE ), and standard deviation of the error ( STD ) in the calculated values of the singlet-triplet gap of F complexes 6-8 at different levels of theory. Errors are reported relative to experiment in cm 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 5.15 Mean error ( ME ), mean absolute error ( MAE ), and standard deviation of the error ( STD ) in the calculated values of the singlet-triplet gap of all complexes at different levels of theory. Errors are reported relative to experiment in cm 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 5.16 EOM-SF-CCSD/cc-pVDZ singlet-triplet gaps and experimental exchange coupling constants for three binuclear copper diradicals at optimized ge- ometries. Cholesky decomposition with a threshold of 1e-3 was used for two-electron integral calculations. The frozen core approximation was applied. Geometries were optimized at the!B97X-D/cc-pVTZ level of theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 xii 5.17 Singlet-triplet gaps (cm 1 ) for Complexes 1-8 calculated using SF-TDDFT with the LRC-!PBEh method and the collinear TDDFT kernel. Experi- mentally determined molecular geometries were used. . . . . . . . . . 197 5.18 Singlet-triplet gaps (cm 1 ) for Complexes 1-8 calculated using SF-TDDFT with the PBE0 method and the collinear TDDFT kernel. Experimentally determined molecular geometries were used. . . . . . . . . . . . . . . 197 xiii List of Figures 1.1 Traveling across the (N 2) through (N + 1) configurational space of four-electrons-in-three-orbitals, whereN = 4 electrons in the reference state. From references well-described by a single Slater determinant, equation-of-motion coupled-cluster methods and other balanced model chemistries can be applied to access the multiple electronic configura- tions necessary for adequate descriptions of low-lying open and closed- shell states. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Copper-containing compounds relevant to the present work, illustrated as a function of the total number of electrons associated with each sys- tem. Excited states of small molecules, such as di- and triatomic cop- per oxides, can be studied with high-level ab initio methods like EOM- CCSD, where relativistic ECPs may or may not be used to improve the quality of results. For larger systems, such as those complexes on the far right side of the spectrum, alternative methods like TDDFT must be considered. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Photoelectron spectra of CuO at 532, 355, 266, and 193 nm. HB stands for hot band transitions. Reproduced from Ref. 20. The vertical ioniza- tion energies associated with the experimentala,b,X, andY transitions that appear in these and other Cu x O y spectra can be computed directly using the appropriate EOM-CCSD methods and references for each cop- per oxide system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.4 Illustration of the Bergman cyclization reaction. . . . . . . . . . . . . . 12 1.5 DFT functionals are diverse and numerous, and each is parameterized in a different way. While DFT offers an invaluable alternative to higher- level ab initio methods for large systems, extensive benchmarking and a uniform property analysis that facilitates comparisons between wavefunction- based and density functional methods are required. Without robust density- based analysis, important wavefunction details are easily “swept under the rug.” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 xiv 2.1 Different EOM models are defined by choosing the reference ( 0 ) and the form of the operator R. EOM-EE allows access to electronically excited states of closed-shell molecules, EOM-IP and EOM-EA can de- scribe doublet target states, and EOM-SF and EOM-DIP describe mul- ticonfigurational wave function of diradical/triradical character. . . . . 23 2.2 Structures of the three copper oxide anions included in the study: closed- shell singlet CuO (top left), closed-shell doublet Cu 2 O (top right), and triplet CuO 2 (bottom). All structures were optimized at the!B97X- D/cc-pVTZ level of theory. . . . . . . . . . . . . . . . . . . . . . . . . 26 2.3 Six lowest EOM-IP-CCSD/cc-pVTZ electron-detached states obtained from the closed-shell singlet CuO reference. Molecular orbitals asso- ciated with dominant amplitudes in each EOM-IP transition are depicted in red and blue. Dyson orbitals associated with each EOM-IP transition are rendered in magenta and yellow. Correlated DEs are shown (EOM- IP-CCSD values). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.4 Six lowest singlet (left) and triplet (right) EOM-EE-CCSD/cc-pVTZ excited states obtained from the closed-shell singlet CuO reference. Photodetachment threshold indicated by the dotted grey line (the states above the dashed line correspond to autodetaching resonances). . . . . 30 2.5 Comparison of spectral transitions calculated at the EOM-IP-CCSD/cc- pVTZ level of theory (left), spectral transitions calculated at the EOM- IP-CCSD/cc-pVTZ-PP level of theory (middle), and experimental spec- tral transitions (right) for CuO . EOM-IP-CCSD/aug-cc-pVTZ-PP tran- sition energies are reported in the middle panel in parentheses. Energies are in eV . Calculations were performed from the closed-shell singlet ref- erence, with the exception of the 4 B 1 = 4 B 2 ! 2 B 1 = 2 B 2 transition (left), which was calculated at the EOM-SF-CCSD/cc-pVTZ level of theory from the quartet reference. Experimental data is reproduced from Ref. 16. See Table Appendix A for mapping C 2v point group symmetry labels to those of C 1v . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.6 Six lowest singlet (left) and triplet (right) EOM-EE-CCSD/cc-pVTZ ex- cited states obtained from the closed-shell singlet Cu 2 O reference. . . . 34 xv 2.7 Comparison of spectral transitions calculated at the EOM-EE-CCSD/cc- pVTZ level of theory (left), spectral transitions calculated at the EOM- EE-CCSD/cc-pVTZ-PP level of theory (middle), and experimental spec- tral transitions (right) for Cu 2 O . EOM-EE-CCSD/aug-cc-pVTZ-PP transition energies are reported in the middle panel in parentheses. Ex- perimental data is taken from Ref. 17. All transitions were computed with the EOM-EE-CCSD method from the closed-shell (N1)-electron Cu 2 O reference, with the exception of the lowest electron-detachment energies, which were computed as differences in CCSD(T) total energies between the doublet anion and lowest neutral state. The all-electron cc- pVTZ basis (left) and the ECP10MDF/(aug)-cc-pVTZ-PP basis (mid- dle) for copper were used in the E[CCSD(T)] calculations. . . . . . . 36 2.8 Molecular orbitals of the triplet CuO 3 2 closed-shell reference state (bot- tom) and associated low-lying excited states (top). The relative energy separations of the states that are provided in blue were obtained using EOM-DIP-CCSD and the closed-shell CuO 3 2 reference. We consider the ordering of CuO 2 states obtained from this method to be most re- liable, but other relative energy separations using different EOM meth- ods/references are also provided in the diagram when possible. . . . . . 40 2.9 Molecular orbitals of the 4 B 1g CuO 2 reference state (bottom) and its associated excited and electron attached states (top). The relative en- ergy separations andhS 2 i of states that are shown in purple were ob- tained using EOM-SF-CCSD from the quartet CuO 2 reference. The EOM-EA-CCSD result in orange, which connects the neutral and an- ionic state manifolds, provides the energy of the lowest photodetach- ment from CuO 2 (3.53 eV). The electronic configuration boxed in red would require both excitation and photodetachment from the ground- state CuO 2 triplet reference. See Appendix A for mapping C 2v point group symmetry labels to those of D 1h . . . . . . . . . . . . . . . . . . 41 2.10 Electronic configuration of the triplet CuO 2 ground state and closely ly- ing singlet state of the same orbital occupation (left) and of the closed- shell 1 A g states (right). From the 3;1 B 1g configuration, the three lowest- lying Koopmans-allowed detached states (middle) are expected to emerge in the spectrum are indicated by solid green arrows. However, several configurations (middle, dashed red arrows) require excitation as well as removing an electron, making them Koopmans-forbidden from that state. From the 1 A g states, a slightly different set of detached states can be accessed. See Appendix A for mapping C 2v point group symmetry labels to those of D 1h . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 xvi 2.11 Comparison of spectral transitions calculated at the EOM-SF/DIP/EA- CCSD/cc-pVTZ level of theory (left) and experimental spectral transi- tions (right) for CuO 2 . Experimental data is taken from Ref. 16, and spectral notation reflects original experimental assignments. Experi- mental intensities of the spectral transitions are as follows: A> B> X C. Bold lines in the left panel denote several states clustered around the same energy. See Appendix A for mapping C 2v point group symmetry labels to those of D 1h . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.12 Fourteen HOMOs and three LUMOs of singlet CuO . . . . . . . . . . 47 2.13 Fourteen HOMOs and two LUMOs of doublet Cu 2 O . MO surfaces correspond to the closed-shell neutral reference (Cu 2 O). The red box around the 14a 1 LUMO indicates the orbital that becomes occupied in the ground state of the doublet anion species. . . . . . . . . . . . . . . 51 3.1 Wave functions of di- (A) and triradicals (B) that are eigenfunctions of S 2 . In both panels, wave function (i) corresponds to the reference-state wave function: the high-spin M s = 1 triplet state of a diradical or the M s = 3/2 quartet state of a triradical. Wave functions (ii)-(v) in A and (ii)- (x) in B correspond to the low-spin states: M s = 0 singlets and triplets, or M s = 1/2 doublets and quartets. For simplicity, only configurations with positive spin-projection are shown. . . . . . . . . . . . . . . . . . 59 3.2 Structures of the four benzyne diradicals (A-D), methylene (E), and four triradicals (F-I) included in this study. Numbers in italics correspond to the optimized lowest high-spin state (M s = 1 triplets for A through E and M s = 3/2 quartets for F through I), while underlined numbers correspond to the optimized lowest energy singlet (diradicals) or doublet (triradicals) state computed by SF-TDDFT. Additional structural data for compounds D and H is presented in the appendices. Compounds will be referred to by letter designation in the remainder of this work. . 70 3.3 Optimized geometries of the lowest singlet state computed by regular Kohn-Sham DFT (left) and SF-TDDFT (center), and high-spin triplet state of meta-benzyne (B5050LYP/cc-pVTZ). Distance and angle of sep- aration between radical sites are shown. Head-Gordon’s indices for the 3 structures are computed using SF-CCSD wave functions for the lowest singlet state. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 3.4 n u , n u;nl , andjn u - n u;nl j (top), and E B5050LYP - E CCSD (E) and jn B5050LYP u;nl -n CCSD u;nl j (n, bottom) computed for the lowest singlet state of H 2 along the bond-stretching coordinate using EOM-SF-CCSD and SF-TDDFT/B5050LYP with the cc-pVTZ basis set. . . . . . . . . . . . 73 xvii 3.5 EOMSF-SF-CCSD/cc-pVTZ natural orbitals of lowest singlet/doublet and triplet/quartet (low-spin) states of A-I. n =jn +n j, with n = jn n j provided in parentheses. n s and n t correspond to n values obtained from the occupancies of the singlet and triplet natural orbitals, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 3.6 Spin-difference densities, unrestricted singly occupied (SO) molecular and natural frontier orbitals of high-spin triplet states of CUAQAC02 (left) and CITLAT (right) computed by B5050LYP/cc-pVTZ. . . . . . 79 3.7 ADC(2)s/cc-pVTZ natural orbitals of lowest singlet and triplet (low- spin) states of CH 2 and A-D. n = jn +n j, with n = jn n j provided in parentheses. n s and n t correspond to n values obtained from the occupancies of the singlet and triplet natural orbitals, respectively. . 81 3.8 Frontier natural orbitals of lowest singlet and M s = 0 triplet states of A-E with four DFT functionals. n =jn +n j, with n =jn n j provided in parentheses. n s and n t correspond to n values obtained from the occupancies of the singlet and triplet natural orbitals, respectively. . 85 3.9 Natural orbitals of lowest singlet andM s = 0 triplet states of CUAQAC02 and CITLAT with the PBE50, B5050LYP-collinear, and LDA DFT functionals. n =jn +n j, with n =jn n j provided in paren- theses. n s and n t correspond to n values obtained from the occupancies of the singlet and triplet natural orbitals, respectively. . . . . . . . . . . 86 3.10 Natural orbitals of lowest doublet and M s = 1/2 quartet states of F-I with four density functionals. n =jn +n j, with n =jn n j provided in parentheses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 3.11 Structures 40, 41 of the two binuclear copper diradicals included the SF- TDDFT benchmark set. . . . . . . . . . . . . . . . . . . . . . . . . . 91 3.12 Expanded structural data for D and H. Numbers in plain font correspond to the singlet (for D) or doublet (for H) state optimized by Kohn-Sham DFT. Italics denotes the optimized lowest high-spin state (triplet for D and quartet for H) optimized by Kohn-Sham DFT. Underlined numbers correspond to the optimized lowest SF-TDDFT state (triplet for D and quartet for H). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 3.13 EOM-SF-CCSD/cc-pVTZ frontier natural orbitals of lowest singlet and triplet states of m-benzyne at three geometries. Red corresponds to the structure optimized by Kohn-Sham DFT for closed-shell singlet singlet configuration. Pink corresponds to the SF-DFT optimized structure of the lowest singlet state. Blue corresponds to the structure optimized by Kohn-Sham DFT for the triplet state. Orbital occupancy of each natural orbital is shown below orbital pictures. At each geometry, wave function properties computed for the lowest singlet state are presented in purple, while properties computed for the triplet state are presented in green. . 92 xviii 3.14 SF-TDDFT/cc-pVTZ frontier orbitals of CUAQAC02 and CITLAT with the B97 functional. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 4.1 Bergman cyclization of (Z)-hex-3-ene-1,5-diyne. . . . . . . . . . . . . 115 4.2 Various ways to distribute two electrons in two nearly degenerate or- bitals for P. All fourM s = 0 determinants, pictured on the right, can be obtained from the singleM s = 1 Slater determinant with just one spin flipping excitation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 4.3 Determination of E a and E rxn for singlet reaction pathway. All geome- tries were optimized using the respective ground state singlet wave func- tion. Single-point calculations of triplet states at the singlet geometries (TS HS , P HS ) were performed using CCSD(T)/ROHF/cc-pVTZ. Singlet triplet vertical gaps were calculated using EOM-SF-CCSD/ROHF/cc- pVTZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 4.4 Stationary points along singlet and triplet pathways. 1 R optimized with CCSD/cc-pVTZ. 1 TS and 1 P optimized with EOM-SF-CCSD/cc-pVTZ. 3 R, 3 TS, and 3 P optimized with CCSD/cc-pVTZ. Images generated with Jmol. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 4.5 Transformation of frontier molecular orbitals along the singlet reaction coordinate. Orbitals were obtained using the HF/cc-pVTZ triplet refer- ence state at the UCCSD/cc-pVDZ optimized structures. . . . . . . . . 132 4.6 Transformation of frontier molecular orbitals along the triplet reaction coordinate. Orbitals were obtained using the HF/cc-pVTZ triplet refer- ence state at the UCCSD/cc-pVDZ optimized structures. . . . . . . . . 133 4.7 Unpaired spin densities for triplet optimized structures. From right to left: 3 R, 3 TS, 3 P. Isovalue of 0.075. IQmol used for visualization. . . . 134 4.8 EOM-SF-CCSD/cc-pVDZ frontier natural orbitals of lowest singlet states of the reactant (left), transition state (middle), and product (right) in the singlet pathway of Bergman cyclization. -orbitals are shown. n = jn +n j, with n =jn n j provided in parentheses. . . . . . . . . 135 4.9 CCSD/cc-pVDZ frontier natural orbitals of high-spin triplet states of the reactant (left), transition state (middle), and product (right) in the triplet pathway of Bergman cyclization. -orbitals are shown. n =jn +n j, with n =jn n j provided in parentheses. . . . . . . . . . . . . . . 136 4.10 Energetic diagram along S 0 and T 1 pathways. Relative electronic ener- gies are shown and ZPE corrected energies are in parenthesis. All values are in kcal/mol. The 3 R energy shown is the R adiabatic E ST . The 3 TS and 3 P energies relative to 1 R were calculated by adding the TS and P adiabatic E ST;a , to the energy of 1 TS and 1 P, respectively. . . . . . . 137 xix 5.1 Wave functions of diradicals that are eigenfunctions of S 2 (only con- figurations with positive spin projections are shown). Wave function (i) corresponds to the high-spin M s = 1 triplet state. Wave functions (ii)-(v) correspond to the low-spin states: M s = 0 singlets and triplets. Note that all M s = 0 configurations can be formally generated by a spin-flipping excitation of one electron from the high-spin M s = 1 configuration. . . 150 5.2 Spin difference densities, unrestricted singly occupied (SO) molecular and natural frontier orbitals of triplet CUAQAC02 (left) and CITLAT (right) at the B5050LYP/cc-pVTZ level of theory. CUAQAC02 has 202 electrons while CITLAT has 278, making their low-lying states and as- sociated orbital surfaces numerous and complex. Reproduced with per- mission from Ref. 11. . . . . . . . . . . . . . . . . . . . . . . . . . . 159 5.3 angle=90 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 5.4 Mean absolute error (MAE) in the singlet-triplet gap for the eight cop- per benchmark systems. Error relative to experimental values of the exchange-coupling constant, J, are presented for fifteen density func- tionals and EOM-SF-CCSD. The all-electron cc-pVDZ basis set was used for all atoms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 5.5 Theoretical singlet-triplet gaps computed with the PBE0, LRC-!-PBEh, B3LYP, and B5050LYP functionals (y-axis) versus experimentally de- rived values of the exchange-coupling parameter (x-axis). The inset in the bottom right zooms into the region spanning the range of -50 to 10 cm 1 . Dashed diagonal line marks perfect agreement between the- oretical and experimental values. The all-electron cc-pVDZ basis was applied to all atoms. The collinear SF-TDDFT kernel was used with the B5050LYP functional, and non-collinear SF-TDDFT kernel with all other functionals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 5.6 Singlet-triplet gaps for Complexes 1-8 calculated using the PBE0 (A), LRC-!PBEh (B), PBE50 (C), and B5050LYP (D) density functionals. The performances of the non-collinear and collinear TDDFT kernels are compared in (C) and (D). The ECP10MDF pseudopotential and match- ing cc-pVDZ-PP basis set were applied to Cu atoms. The cc-pVDZ basis was used for all non-Cu atoms. . . . . . . . . . . . . . . . . . . . . . . 170 5.7 Frontier natural orbitals of lowest singlet and M s = 0 triplet states of cop- per diradicals at the PBE50/cc-pVDZ level of theory. With the exception of YAFZOU, for other systems only orbitals of the triplet states are shown, since spatial extent of relevant paired and natural orbitals— and the appearance of frontier natural orbitals associated with the lowest singlet and triplet states—does not differ. n =jn +n j, withjn n j provided in parentheses. n s and n t correspond to n values obtained from the occupancies of the singlet and triplet natural orbitals, respectively. . 173 xx 5.8 Frontier natural orbitals of lowest singlet and M s = 0 triplet states of CUAQAC02 at the SF-TDDFT/cc-pVTZ and EOM-SF-CCSD/cc- pVDZ levels of theory. Natural orbital surfaces obtained with three dif- ferent density functionals are compared. The collinear TDDFT kernel was used with the B5050LYP functional. n =jn +n j, withjn n j provided in parentheses. n s and n t correspond to n values obtained from the occupancies of the singlet and triplet natural orbitals, respectively. . 175 5.9 Frontier natural orbitals of lowest singlet and M s = 0 triplet states of simplified PATFIA and the full experimental structure, PATFIA + Fe(C 5 H 5 ) 2 . Natural orbitals were obtained at the B5050LYP/cc-pVDZ level of theory with the collinear TDDFT kernel applied. n =jn +n j, withjn n j provided in parentheses. n s and n t correspond to n val- ues obtained from the occupancies of the singlet and triplet natural or- bitals, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 5.10 Optimized triplet geometries of BISDOW (top left), PATFIA (middle), and CITLAT (top right). Interatomic Cu-Cu distances in parentheses correspond to experimental geometries. All three complexes were opti- mized at the!B97X-D/cc-pVTZ level of theory. . . . . . . . . . . . . 195 5.11 SF-TDDFT/cc-pVTZ frontier orbitals of CUAQAC02 with the B97 func- tional (non-collinear TDDFT kernel). . . . . . . . . . . . . . . . . . . 198 6.1 CuO 3 (left) and two structural isomers of CuO 4 : the Cu(O 2 ) 2 complex (middle) and the O 2 -solvated OCuO structure (right). Copper atoms are shown in gold, oxygen atoms in red. The left and middle structures undergo photodisocciation to yield O 2 at 355 nm. The structure on the right gives rise to a photodetachment spectrum similar to CuO 2 , with the solvating O 2 slightly perturbing the transitions. 3 . . . . . . . . . . . . . 222 6.2 Binuclear copper diradicals not considered in the present work. Experi- mental structures 4–7 are labeled with their Cambridge Crystal Structure Database reference codes. Counter ions are shown, but would be re- moved in the final analysis. . . . . . . . . . . . . . . . . . . . . . . . . 223 6.3 Two examples of trinuclear copper triradicals. Experimental structures 11 are labeled with their Cambridge Crystal Structure Database reference codes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 6.4 Experimental structure of [Mn 6 O 2 (sao) 6 (O 2 CPh) 2 (EtOH) 4 ], where “sao” is short for salicylaldoxime. Mn atoms depicted in red, oxygen in green, and nitrogen in blue. J exp = 1.6 cm 1 . 10 The shaded rhombus shows the plane of symmetry that divides the two triangular groups of three Mn(III) spin centers. . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 xxi 6.5 There will likely always be a place in industry for traditional magnetic magnetic materials, which have been a part of human culture for thou- sands of years. Nevertheless, studies like those presented here bring us ever closer to the day when molecular magnetic devices can compete with their conventional predecessors. . . . . . . . . . . . . . . . . . . 226 xxii Abstract Modeling the electronic structure of open-shell species is both essential to compu- tational studies of excited-state processes and difficult due to the multi-configurational nature of the states involved. Choosing a reliable methodology becomes even more dif- ficult when the systems one wishes to study have a large number of electrons, or when inclusion of relativistic effects is necessary for accurate description. The work pre- sented here provides several diverse examples of challenging open-shell systems. For each system, we offer a detailed account of how the challenges inherent in their study were overcome for accurate description of the relevant chemical and physical properties, ranging from photodetachment energies to Heisenberg exchange-coupling interactions. The experimental photoelectron spectra of di- and triatomic copper oxide anions have been reported previously. In Chapter 2, we present an analysis of the experimen- tal spectra of the CuO , Cu 2 O , and CuO 2 anions using equation-of-motion coupled- cluster (EOM-CC) methods. The open-shell electronic structure of each cluster demands a unique combination of EOM-CC methods to achieve an accurate and balanced rep- resentation of the multi-configurational anionic and neutral state manifolds. Analysis of the Dyson orbitals associated with photodetachment from CuO reveals the strong non-Koopmans character of the CuO states. A perturbative triples correction to the xxiii coupled-cluster singles and doubles ansatz is required for accurate description of elec- tron detachment from Cu 2 O . Use of a relativistic pseudopotential and matching basis set improves quality of results in most cases. EOM-DIP-CCSD analysis of the low-lying states of CuO 2 reveals multiple singlet and triplet anionic states near the triplet ground state, adding a extra layer of complexity to the interpretation of the experimental CuO 2 photoelectron spectrum. In Chapter 3, we present an analysis of the wavefunction properties of classic open- shell systems. Density-based wave function analysis enables unambiguous comparisons of electronic structure computed by different methods and removes ambiguity of orbital choices. We use this tool to investigate the performance of different spin-flip methods for several prototypical diradicals and triradicals. In contrast to previous calibration studies that focused on energy gaps between high and low spin-states, we focus on the properties of the underlying wave functions, such as the number of effectively unpaired electrons. Comparison of different density functional and wave function theory results provides insight into the performance of the different methods when applied to strongly correlated systems such as polyradicals. We show that canonical molecular orbitals for species like large copper-containing diradicals fail to correctly represent the underlying electronic structure due to highly non-Koopmans character, while density-based analysis of the same wave function delivers a clear picture of bonding pattern. Chapter 4 extends the density-based wavefunction analysis presented in Chapter 3 to stationary points in the Bergman cyclization, an important reaction in which an enediyne cyclizes to produce a highly reactive diradical species, p-benzyne. Enediyne motifs are found in natural antitumor antibiotic compounds, such as calicheammicin and dyne- micin. Understanding of the energetics of the cyclization is required in order to better control the initiation of the cyclization, which induces cell death. The singlet and triplet xxiv potential energy surfaces for the thermochemically induced Bergman cyclization of (Z)- hex-3-ene-1,5-diyne have been computed by the CCSD and EOM-SF-CCSD methods. The triplet enediyne and transition state were found to have C 2 symmetry, which con- trasts with the singlet reactant and transition state that possess C 2 v symmetry. The fron- tier orbitals of both cyclization pathways were analyzed to explain the large energetic barrier of the triplet cyclization. Reaction energies were calculated using CCSD(T)/cc- pVTZ single point calculations on structures optimized with CCSD/cc-pVDZ. The sin- glet reaction was found to be slightly endothermic (H rxn = 13.76 kcal/mol) and the triplet reaction was found to be highly exothermic (H rxn = -33.29 kcal/mol). The adi- abatic singlet-triplet gap of p-benzyne, computed with EOM-SF-CCSD/cc-pVTZ, was found to be 3.56 kcal/mol, indicating a ground state singlet. Molecular magnets, defined here as organic polyradicals, can be used as building blocks in the fabrication of novel and structurally diverse magnetic light-weight mate- rials. In Chapter 5, we present a theoretical investigation of the lowest spin states of several binuclear copper diradicals. In contrast to previous studies, we consider not only the energetics of the low-lying states (which are related to the exchange-coupling parameter within the Heisenberg-Dirac-van-Vleck model), but also the character of the diradical states themselves. We use natural orbitals, their occupations, and the num- ber of effectively unpaired electrons to quantify bonding patterns in these systems. We compare the performance of spin-flip time-dependent density functional theory (SF- TDDFT) using various functionals and effective core potentials against the wave func- tion based approach, equation-of-motion spin-flip coupled-cluster method with single and double substitutions (EOM-SF-CCSD). We find that SF-TDDFT paired with the PBE50 and B5050LYP functionals performs comparably to EOM-SF-CCSD, with re- spect to both singlet-triplet gaps and states’ characters. Visualization of frontier natural xxv orbitals shows that the unpaired electrons are localized on copper centers, in some cases exhibiting slight through-bond interaction via copperd-orbitals andp-orbitals of neigh- boring ligand atoms. The analysis reveals considerable interactions between the for- mally unpaired electrons in the antiferromagnetic diradicaloids, meaning that they are poorly described by the Heisenberg-Dirac-van-Vleck model. Thus, for these systems the experimentally derived exchange-coupling parameters are not directly comparable with the singlet-triplet gaps. This explains systematic discrepancies between the com- puted singlet-triplet energy gaps and the exchange-coupling parameters extracted from experiment. xxvi Chapter 1: Introduction 1.1 Multicofigurational Wave Functions: Navigating the Open-Shell Interstate Highway To study excited-state chemistry from the theoretical perspective, one must choose methodology suited for the description of open-shell systems. While this statement is easy enough to make, developing and correctly applying open-shell electronic structure methods to understand increasingly larger and more complex systems is an active area of research, 1–4 and the focus of the work presented herein. Consider the electronic configurations on the blackboard in Figure 1.1. From its inception, computational quantum chemistry has been most readily applicable in the analysis of electronic states well-represented by a single Slater determinant (e.g., states arising from the closed-shell configurations in Figure 1.1). 5–14 Ignoring a priori the full-suite of non-black-box multi-reference approaches, 15 one requires a method which facilitates electron migration through Fock space via one (or two, or three,...) electronic excitations from a well-described single reference. Moreover, the configurational space accessible through the chosen method must comprise all configurations important in the description of dynamical and static correlation (and, preferably, do so in a spin- complete fashion). When these criteria are met, one is free to explore the rich landscape of ionized, excited, and electron attached states of a molecular system without facing 1 "I thought you signed us up for driver's ed..." Interstate Travel EOM-EA EOM-IP EOM-DIP EOM-IP EOM-EE EOM-SF EOM-SF ©2017 Figure 1.1: Traveling across the (N 2) through (N + 1) configurational space of four-electrons-in-three-orbitals, where N = 4 electrons in the reference state. From references well-described by a single Slater determinant, equation-of-motion coupled-cluster methods and other balanced model chemistries can be applied to access the multiple electronic configurations necessary for adequate descriptions of low-lying open and closed-shell states. 2 the limitations of multi-reference wave functions, which are limited to configurations spanning a user-defined subset of Hilbert space. The problem of choosing an appropriate model chemistry is compounded when the system(s) one wishes to study are large or contain heavy atoms, such as transition met- als. The presence of heavy atoms calls for the use of relativistic basis sets and/or ef- fective core potentials (ECPs), both to capture the essential physics of the system and to reduce computational cost by decreasing the number of explicitly treated electrons. While excited-state methodologies based on a coupled-cluster singles and doubles ref- erence, such as equation-of-motion (EOM-CCSD), 16 would be suited for systems with as many as 250 electrons,N 6 scaling makes the application of coupled-cluster methods to systems with 350 electrons or more a challenge. Examples relevant to the present work are shown in Figure 1.2. Methods based on time-dependent density functional theory (TDDFT), which do not scale as steeply with the total number of electrons, are an attractive alternative for larger systems, but careful benchmarking is required when extending them to the description of systems or physical properties for which there is little or no precedent for their usage. If one must embark on a benchmark study, then depending on the physical questions that need answering, small errors in state energies (relative to experiment and/or high- level ab initio results) may be insufficient to conclude that a lower-level methodology can be employed with minimal sacrifice in the quality of results. With regard to DFT methods, parameterization of functionals is typically based upon structural data and energy differences, with little attention given to underlying state character or properties. Relative energies of well-defined states are often the only information needed to answer meaningful physical questions. But for some processes, such as those encountered in bond-breaking and magnetochemistry, more detailed information regarding extent and 3 # of e - 0 -100 150-200 250-300 350+ 1 Figure 1.2: Copper-containing compounds relevant to the present work, illustrated as a function of the total number of electrons associated with each system. Excited states of small molecules, such as di- and triatomic copper oxides, can be studied with high-level ab initio methods like EOM-CCSD, where relativistic ECPs may or may not be used to improve the quality of results. For larger systems, such as those complexes on the far right side of the spectrum, alternative methods like TDDFT must be considered. localization of unpaired electron density in open-shell states is required. Therefore, to conduct a holistic benchmark study and to better understand the nature of the open- shell states themselves, a uniform state analysis must be performed in such a way that cross-methodological comparisons can be made. These and related challenges are confronted in the work reported in the following chapters. The remaining sections in Chapter 1 provide a brief overview of the systems under investigation, physical quantities of interest, and molecular electronic structure approaches that will be discussed in detail in Chapters 2 through 5. We begin by intro- ducing an EOM-CCSD computational approach to modeling the photoelectron spectrum of small, single-molecule copper oxides in Chapter 2. Copper atoms return to compli- cate the electronic structure picture in Chapter 5, this time emerging as spin centers of 4 large, bimetallic coordinate-covalent complexes with the potential for use in magnetic materials. In Chapter 5, we focus on accurate computing of the small energy difference between low-lying spin states as well as state character. For the analysis of bonding patterns, we introduce an approach based on the one-particle state density matrix in Chapter 3 and illustrate its application to classic di- and triradical benchmark systems. In Chapter 4, we apply the same state analysis tool to study singlet and triplet species along the Bergman cyclization reaction pathway. 1.2 Electronic States of Di- and Triatomic Copper Ox- ides: Mapping the Low-Energy Spectrum with EOM-CCSD Methods Copper oxides serve as components of semiconductors, 17 pigments in glazes for ceramics, 18 and have even been assembled in nanoparticles for antimicrobial applica- tions. 19 Photoelectron spectra of di- and triatomic copper oxides were reported in the late 1990’s (example spectrum for CuO in Figure 1.3). 20 We provide an EOM-CCSD anal- ysis of the photodetachment spectra of CuO , Cu 2 O , and CuO 2 in Chapter 2. 21 These three copper oxide anions possess a singlet, closed-shell doublet, and triplet ground state, respectively, 20, 22 each requiring a different variant(s) of EOM-CCSD for the most accurate description of their photodetached states. 16 The motivation for the study was to provide high-level ab initio results that could be used to justify the application of lower- level electronic structure methods—including Kohn-Sham DFT or its Green’s function- based extension, GW 23 —to analysis of photoelectron spectra of Cu x O y clusters. For closed-shell singlet CuO , the straightforward application of the EOM-IP- CCSD method to the singlet ground state describes the desired manifold of neutral states 5 photon energies, each revealing more transitions (Figure 2). There are two major features labeled as X and Y, two weak features labeled as a and b, and a broad band starting near 4 eV BE labeled as Z. The X band at 532 nm shows clear hot band transitions. The Y band at 355 nm is vibrationally resolved. The broad feature labeled Z is partially observed at 266 nm. It is fully observed at 193 nm although the spectrum becomes rather noisy at the high BE side. The CuO 2 - spectra areshownattwophotonenergiesinFigure3. At266nm,four bands are observed and are labeled as X, A, B, and C. The A bandcontainsclearvibrationalstructures. At193nm,twonew features are observed and are labeled as D and E. ForCuO 3 - toCuO 5 - ,eachisstudiedatthreephotonenergies (Figures4-6). ThespectraofCuO 6 - (Figure7)areonlytaken attwophotonenergieswithrelativelylowcountingratebecause the CuO 6 - anion mass abundance is rather weak. This makes it very difficult to measure the PES spectrum at 193 nm due to the severe noise problem. For CuO 3 - (Figure 4), the 355 nm spectrum shows surprisingly features due to CuO - besides the features labeled as X and A due to CuO 3 - . This is compared to the spectrum of CuO - in Figure 8 on the same BE energy scale, where the peak labeled A in Figure 4 is not shown. And the feature near 3.2 eV that is labeled X in Figure 4 clearly overlaps with the feature labeled as b in the CuO - spectrum. Additionally, the CuO - spectrum from the CuO 3 - beam shows significant hot band transitions and an extra feature labeled as a′ in Figure 8. At 266 nm, more features are observed and are labeledasB,C,andD(Figure4). However,theCuO - features that show up in the 355 nm disappear in the 266 nm spectrum. A close examination reveals that there is a very weak feature near 1.23 eV BE that can be attributed to Cu - . Both the Cu - and CuO - features must be due to photodissociations of the parent CuO 3 - anions and subsequent detachment of the anion product by a second photon from the same detachment laser pulse (see below). The photodissociations appear to be wavelength dependent. At 193 nm, no signals that can be attributed to either Cu - or CuO - are observed, and a weak feature at high BE near 5.8 eV, labeled as E, is observed. Figure 2. Photoelectron spectra of CuO - at 532, 355, 266, and 193 nm. HB stands for hot band transitions. Figure 3. Photoelectron spectra of CuO2 - at 266 and 193 nm. Figure 4. Photoelectron spectra of CuO3 - at 355, 266, and 193 nm. Figure 5. Photoelectron spectra of CuO4 - at 355, 266, and 193 nm. Chemical Bonding between Cu and Oxygen J. Phys. Chem. A, Vol. 101, No. 11, 1997 2105 Figure 1.3: Photoelectron spectra of CuO at 532, 355, 266, and 193 nm. HB stands for hot band transitions. Reproduced from Ref. 20. The vertical ionization energies associated with the experimentala,b,X, andY transitions that appear in these and other Cu x O y spectra can be computed directly using the appropriate EOM-CCSD methods and references for each copper oxide system. in a balanced way. In every application of EOM-CCSD, the target states are obtained from a reference described by the the CCSD ansatz: j CCSD i =e T j 0 i; (1.1) wherej 0 i defines the Hartree-Fock vacuum. The cluster operator,T , includes all singly and doubly-excited determinants. ^ T =T 1 +T 2 = X i X a t i a ^ a a ^ a i + 1 4 X i;j X a;b t ij ab ^ a a ^ a b ^ a j ^ a i ; (1.2) 6 wherea a anda b are creation operators,a i anda j are annihilation operators, andt i a and t ij ab are amplitudes. Cluster amplitudes, T , define the similarity transformed Hamilto- nian: H =e T He T : (1.3) It is the form of the linear excitation operator, R, which acts on the CCSD reference, that distinguishes one variant of EOM-CCSD from another. Diagonalization yields the EOM states and their associated eigenvalues: HR =ER: (1.4) As a general rule,R is truncated in the same manner asT (i.e.,R only generates singly and doubly excited amplitudes in EOM-CCSD). In the case of EOM-IP-CCSD,R re- moves an electron from the (N + 1)-electron reference wavefunction: (N) = ^ R(1) 0 (N + 1): (1.5) The doublet ground state of Cu 2 O requires the use of EOM-EA-CCSD and EOM- EE-CCSD. The former utilizes the (N 1)-electron reference that is well-described by a single Slater determinant—a closed-shell singlet state of Cu 2 O. The EOM operator, R, adds an electron to a virtual orbital: (N) = ^ R(+1) 0 (N 1): (1.6) EOM-EA can be used to obtain what is formally the electron detachment energy associ- ated with the radical electron in the Cu 2 O ground state. All other detachment energies 7 can be described relative to the (N1)-electron closed-shell singlet reference by EOM- EE, which generates singly excited states that preserve the number of and electrons: (M s = 0) = ^ R(M s = 0) 0 (M s = 0): (1.7) CuO 2 possesses a triplet ground state. The manifold of low-lying singlets and triplets associated with the anion are well described by EOM-DIP-CCSD, in which two electrons are removed from the closed-shell singlet CuO 3 2 : (N) = ^ R(2) 0 (N + 2): (1.8) The lowest energy CuO 2 state, a quartet, can be mapped to the ground state CuO 2 triplet by application of EOM-EA-CCSD, thereby connecting the neutral and anionic state manifolds. EOM-SF-CCSD, a method that underpins much of the work presented throughout this manuscript, is an EOM variant that generates target states that have a different spin projection than the reference. In Chapter 2, it is used to map the energy landscape of doublet and quartet CuO 2 states from the high-spin quartet reference. The spin-flip ansatz can be described as (M s = 0) = ^ R(M s =1) 0 (M s = +1): (1.9) Other variants of the spin-flip approach exist, including one approach based on time- dependent DFT (SF-TDDFT) 24, 25 and one based on the algebraic-diagrammatic con- struction (ADC) scheme for the polarization propagator 26 (SF-ADC). 27, 28 These variants will be discussed in the following section and applied in Chapters 3 and 5. 8 As revealed through analysis of Dyson orbitals 29 associated with EOM-CCSD states, use of electronic structure methods restricted to a single Slater determinant may be jus- tified for the description of electron detachment energies of CuO , where the ground state is relatively well described by a single electronic configuration. However, caution is advised for these strongly correlated systems. The lowest electron detached state of CuO is not associated with removal of an electron from the highest occupied Hartree- Fock molecular orbital. The same can be said for Cu 2 O , where triples corrections must be incorporated in the description of the ground state to capture dynamical correlation. CuO 2 , with its numerous high spin states that are close in energy to the triplet ground state—EOM-CCSD methods, which better account for the multi-configurational nature of the anionic and neutral states, are required. 1.3 Density-Based Analysis of Benzynes, Xylylenes, and the Bergman Cyclization Reaction with EOM-SF- CCSD and SF-TDDFT For species with intermediate radical character, known as radicaloids, classification of low-lying states as open or closed-shell type is non-trivial. When wave functions comprise a large number of electron configurations (e.g. when many amplitudes con- tribute to an EOM-CCSD or TDDFT state, including some arising from excitations of doubly occupied orbitals), understanding the true nature of the radical(oid) state be- comes even more difficult. The Kohn-Sham DFT and Hartree-Fock molecular orbitals are themselves non-unique, to the extent that any rotation of the orbitals which pre- serves the orthonormality condition and does not mix occupied and virtual subspaces is 9 allowed. What is required, then, is a method of radical(oid) state analysis that is 1) com- pact in its representation of the wavefunction, 2) uniformly applicable across electronic structure methods, and 3) yields results that are unique to the state/system of interest. Chapter 3 details the solution to this requirement: one-particle state density analy- sis, as developed and implemented by Wormit et al, 30, 31 which exploits the concept of natural orbitals (NOs). 30–35 Density-based analysis can be applied within the framework of any electronic structure method, without impacting scaling or overall computational cost. From the one-particle state density matrix—which can be obtained for any state, arising from any electronic configuration(s), by any electronic structure method—one can arrive at the NO representation of the wavefunction. NOs afford the most compact representation of a state 36 and, as visualization of the surfaces associated with large sys- tems reveals, greatly simplify the orbital picture relative to their Hartree-Fock or DFT canonical molecular orbital counterparts. Occupancies of the natural orbitals can be used to obtain the total number of unpaired electrons associated with a state, indicative of the degree of radical(oid) character. We demonstrate the utility of this approach using classic di- and triradical examples. The work presented in Chapter 3 was done in collaboration with Andreas Drew and Dirk Rehn of the University of Heidelberg. 37 Di- and tridehydrobenzynes, methylene, and triradical xylylenes are examined by density-based analysis using spin-flip variants of EOM-CCSD, TDDFT, and ADC(2). By the TDDFT method, spin-flipped excited states are obtained as non-spin-conserving poles of the first-order linear response equations applied to the high-spin reference state Kohn-Sham density. 24, 25 Similar to the SF- CIS(D 1 ), 38 single and double excitations are included in the SF-ADC(2) scheme, with the latter emerging as zeroth order corrections from perturbation theory. 10 Examining state properties obtained from SF-TDDFT methods relative to those ob- tained from EOM-SF-CCSD and SF-ADC(2), we show that for most benchmark sys- tems, the wavefunction-based methods and SF-TDDFT paired with hybrid functionals agree upon radical character and relative energies of low-lying spin states. Incorporated in the SF-TDDFT benchmark set are two copper-containing diradicals. Density-based analysis and the NO surfaces obtained as a result greatly simplify the orbital picture of low-lying spin states in these systems, facilitating more informed predictions of their potential magnetic properties. This is a topic we revisit in Chapter 5. Chapter 4 is devoted to the application of density-based analysis to the reaction pathways of Bergman cyclization (Figure 1.4). The product of the Bergman cyclization process is the highly reactive para-benzyne diradical examined in the benchmark study presented in Chapter 3, a species that plays a role in DNA scission, 39 the development of aromatic species in the interstellar medium, 40 antitumor therapeutics, 41–43 and the forma- tion of biomolecules in marine organisms. 44 This work, performed in collaboration with Adam Luxon, Carol Parish, and Rene Kanters of the University of Richmond, 45 consti- tutes the first uniform theoretical analysis of the multiconfigurational states and energy barriers along the singlet and triplet cyclization pathways. We utilize EOM-CCSD and density-based analysis to illustrate the degree of open-shell character associated with reactants, intermediates, and products, and for visualization of the frontier orbitals that play a role in bond breaking and forming along each trajectory. 11 enediyene transition state para-benzyne ∆ Figure 1.4: Illustration of the Bergman cyclization reaction. 1.4 Computing Exchange-Coupling Constants and State Properties in Binuclear Copper Diradicals by a Spin-Flip Approach In the quest for light-weight magnetic devices, identification of molecular building blocks with stable high-spin ground states is essential. In Chapter 5, we investigate the energies and properties of low-lying spin states of eight binuclear copper diradi- cals with EOM-SF-CCSD and SF-TDDFT methods. 46 Fifteen density functionals are included in the benchmark study, covering hybrid, GGA, and dispersion-corrected fam- ilies of functionals. Different quality basis sets, ECPs, and equilibrium geometries are compared. Errors in singlet-triplet energy gaps are reported relative to experiment. Un- paired electron density is visualized through NO analysis, clarifying understanding of the wavefunctions associated with each state. NO occupation analysis reveals that cer- tain complexes with singlet ground states have a significant amount of ionic character due to their imperfect diradical character. 12 The latter point is of particular significance, as the imperfect diradical nature of cer- tain antiferromagnetic complexes is missed entirely in previous experimental and theo- retical treatments of these systems. 47–57 For the magnetic exchange-coupling parameters extracted from EPR, magnetic susceptibility, and neutron scattering experiments to be held in 1:1 correspondence with the energy difference between lowest-lying singlet and triplet states, the spin states in question must arise from spatially separated, weakly cou- pled electrons. This relation is known as the Land´ e interval rule, 58 which can be derived from the Heisenberg-Dirac-van-Vleck Hamiltonian. 59–61 Diradicaloids and singlet diradicals that include ionic configurations violate the ten- ants of this model. When the performance of density functional methods are poorly benchmarked with only relative energy errors in mind, or when symmetry-breaking methods are applied for the description of antiferromagnetic and ferromagnetic states, probability of identifying complexes that violate the Heisenberg physics of weakly in- teracting spin moments is greatly diminished. Consistent and expanded density-based analysis is paramount to future theoretical investigations of polyradical molecular mag- nets, an argument that is explored in detail in the final chapter of this work. 13 Quote of the Day: "If at first you don't succeed, try a different functional." Meow! catvids.com B3LYP LDA MN15 PBE0 B5050LYP B97 PW91 PBE50 "It'll be okay... no one ever checks under here..." Figure 1.5: DFT functionals are diverse and numerous, and each is parameter- ized in a different way. 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Clarendon Press, Oxford, 1932. 18 Chapter 2: Modeling photoelectron spectra of CuO, Cu 2 O, and CuO 2 anions with equation-of-motion coupled-cluster methods: An adventure in Fock space 2.1 Introduction Photoelectron spectroscopy (PES) allows one to characterize multiple electronic states of the excited or photodetached species in a single experimental setup. 1–4 Photodetach- ment from anions, which can be mass-selected, is often used to characterize transient open-shell neutral states. 5–7 A particularly attractive feature of PES is that, owing to dif- ferent selection rules, it can characterize energy differences between electronic states from different spin manifolds. In the present work, we apply state-of-the-art quan- tum chemistry methods based on the equation-of-motion coupled-cluster (EOM-CC) approach 8–10 to model photoelectron spectra in di- and triatomic copper oxide anions. 19 Copper oxides serve as a component of semiconductors, 11 a pigment in glazes for ceram- ics, 12 and have even been assembled in nanoparticles for antimicrobial applications. 13 Complexes of copper with oxygen are also involved in catalytic oxidation and oxygen transport in biomolecules 14 and in catalysis. 15 The structure and low-lying electronic states of transition metal oxide clusters are important in the study of corrosion pro- cesses and the reaction of oxygen with metal surfaces. In addition to their relevance to applications, these simple molecules are also popular because of their fundamental importance — they illustrate basic bonding concepts of transition metal compounds. Several experimental studies investigated neutral and anionic copper oxide clusters by using PES. 4, 16, 17 Different aspects of their electronic structure have been also interro- gated theoretically. The bulk of previous theoretical studies have been limited to meth- ods based on density functional theory (DFT) 18–36 with BLYP, 22, 24, 32, 35 BPW91, 22, 32 B3LYP, 22, 26–28, 32–36 and PBE 25 functionals. Time-dependent DFT has been used to compute excited states. 24 Previous treatments that utilize wavefunction methods include multi-configurational self-consistent field methods, 18, 30 configuration interaction-based methods, 18–21, 23, 29 perturbation theory, 23, 27 and coupled-cluster-based methods. 23, 27, 34 In this contribution, we report a systematic study of low-lying electron-detached and excited states of the CuO , Cu 2 O , and CuO 2 anions using a uniform single-reference approach capable of treating dynamical and non-dynamical correlation, equation-of- motion coupled-cluster method with single and double excitations (EOM-CCSD). We describe the excitation and electron detachment transitions that give rise to the low- energy photoelectron spectra of these species. On the basis of the analysis of the under- lying wave functions, we show that extreme open-shell character of these species calls 20 for a careful choice of methodology. In EOM-CCSD, the target states are describes using the following ansatz: = ( ^ R 1 + ^ R 2 )e ^ T 1 + ^ T 2 0 ; (2.1) wheree ^ T 1 + ^ T 2 0 is the reference CCSD wave function and operator ^ R is a general ex- citation operator. By a deliberate combination of an appropriate reference state and a particular form of the EOM operator, 8 the EOM-CC family of methods allows one to access different types of states probed in these experiments. Here, we utilize the fol- lowing EOM-CCSD methods: EE (excitation energies), SF (spin-flip), IP (ionization potentials), EA (electron attachment), and DIP (double IP). Figure 2.1 illustrates types of target-state manifolds accessed via these methods. Operator ^ R 1 describes the singly excited part of the target wave function, i.e., in EOM-EE it generates all hole-particle configurations from the reference determinant 0 , whereas in EOM-IP it generates all singly ionized Koopmans-like determinants (hole configurations). The relative mag- nitude of ^ R 1 and ^ R 2 , which reflects the correlation effects, can be quantified by their norms: jjR EE 1 jj 2 = X ia (r a i ) 2 jjR IP 1 jj 2 = X i (r i ) 2 ::: (2.2) The states that are predominantly singly excited with respect to the reference have jjR EE 1 jj 2 1. In addition to the magnitude of R 1 , it is also sometimes instructive to look at the leading amplitude, to distinguish whether the target state is dominated by one or several configurations. The latter situation is described as non-dynamical correlation. 21 The role ofR 2 is to describe dynamical correlation. The states withjjR EE 1 jj 2 1 can be described by configuration interaction singles (CIS) or TDDFT ansatz. Likewise, singly ionized states for which there is one leadingr i amplitude andjjR IP 1 jj 2 1 can, in prin- ciple be described by Koopmans approximation. The overall singly ionized character (jjR IP 1 jj 2 ) is related to the norm of the Dyson orbital, 37 which determines the magnitude of the photoionization/photodetachment cross section. 38, 39 Detached states that can be generated by removing one electron from the reference have norms of Dyson orbitals close to 1 and result in principal peaks in PES, whereas the states that have largejjR IP 2 jj 2 have small norms of Dyson orbitals and show up as satellite shake-up excitations. Our analysis of the excited and ionized states in neutral and anionic copper oxides can serve as a basis for calibrating other electronic structure methods and as a tutorial into practical application of EOM-CC methods. To include the effect of higher exci- tations, for selected states, we also performed CCSD(T) calculations. One of the aims of our analysis is to outline the domain of applicability of more approximate methods, such as Kohn-Sham DFT 40–42 and its extensions, 43 for modeling PES in copper oxide anions and similar systems. For states dominated by a single electronic configuration, these methods are justifiable. 44 However, as we show below, the three di- and triatomic copper oxide species considered here posses low-lying anionic and neutral states aris- ing from multiple near-degenerate electronic configurations. For highly-correlated sys- tems such as these, Koopmans-like behavior is rarely observed, and more sophisticated methodology, like EOM-CCSD, is required for description of photoelectron spectra. The paper is organized as follows: In the next section, we briefly outline the com- putational details and methodologies applied. Where applicable, we refer the reader to previous work, wherein the relevant methods are explained in detail. 8, 45–47 We then present our results for the Cu x O y anions in the following order: CuO , Cu 2 O , and 22 EOM-EE: Ψ(N) =R(Ms=0)Ψ 0 (N) 0 a i EOM-IP: Ψ(N) =R(-1)Ψ 0 (N+1) EOM-EA: Ψ(N) =R(+1)Ψ 0 (N-1) 0 a i 0 EOM-SF: Ψ(Ms=0) =R(Ms=-1)Ψ 0 (Ms=1) 0 a i EOM-DIP: Ψ(N-2) =R(-2)Ψ 0 (N) 0 ij Figure 2.1: Different EOM models are defined by choosing the reference ( 0 ) and the form of the operator R. EOM-EE allows access to electronically excited states of closed-shell molecules, EOM-IP and EOM-EA can describe doublet target states, and EOM-SF and EOM-DIP describe multiconfigurational wave function of diradical/triradical character. CuO 2 . We conclude with a brief commentary on the electronic structure of each sys- tem, including the extent of open-shell and multi-configurational character. 2.2 Computational Methods The Q-Chem electronic structure package was used for all electronic structure calcula- tions. 48, 49 We optimized the structures of the ground states of CuO (closed-shell sin- glet), Cu 2 O (closed-shell doublet), and CuO 2 (triplet) using the!B97X-D method 50 23 and all-electron cc-pVTZ basis; relevant Cartesian geometries are provided in the appen- dices. We report only vertical electronic energy separations, computed at the equilib- rium structures of the anionic ground states. Zero-point energies are not included. Since the available experimental spectra 16 are broad and not vibrationally resolved, vertical detachment energies (DE) are sufficient for an adequate comparison. All EOM–EE/IP/DIP/SF/EA–CCSD and CCSD(T) calculations were performed us- ing an all-electron cc-pVTZ basis, or the Stuttgart-Cologne ECP10MDF pseudopoten- tial 51 paired with a matching correlation-consistent basis for Cu (indicated in results tables). ECP10MDF has been shown to perform well in the EOM-CCSD analysis of small copper compounds. 52 User-defined basis sets for Cu were obtained from Ref. 53. For linear molecules (CuO and CuO 2 ), we report symmetry labels corresponding to the largest Abelian subgroup. Tables 2.6 and 2.7 in the Appendix A show the map- ping betweenC 2v andC 1v , andD 2h andD 1h groups, 54 which can be used to obtain proper spectroscopic term labels. We note that Q-Chem does not follow the standard Mulliken convention 55 for molecular orientation, consequently, some symmetry labels (in particular, for C 2v symmetry) differ from the standard notation. Core electrons (1s on oxygen and 1s, 2s, 2p, 3s, and 3p on copper) were frozen in the all-electron EOM calculations. For each system, the specific choice of an EOM-CCSD method (or a combination of several EOM-CC approaches) depends on an open-shell pattern exhibited by the re- spective ground-state anions and target states accessible via one-electron detachment processes. We will describe the approaches used for CuO , Cu 2 O , and CuO 2 on a case-by-case basis in the following section. In selected cases, we also computed Dyson orbitals: 37 d (1) = Z N (1; 2;:::;N) N1 (2; 3;:::;N)d2:::dN (2.3) 24 Dyson orbitals are related to photoionization/photodetachment cross sections: 38, 39 their norms quantify the extent of differential electron correlation in the initial (N-electron) and final (N 1 electron) states. For example, for Hartree-Fock (or Kohn-Sham) wave functions and within Koopmans approximation, the Dyson orbitals are just canonical SCF orbitals and their norms equal one. We note that even when the ionized states have predominantly Koopmans character, i.e., when the respective correlated wave functions have one leading ionized determinant, the quantitative accuracy of Koopmans theorem can be rather poor. For example, it is not uncommon that the order of ionized states changes relative to that predicted by the Koopmans theorem, when correlation is in- cluded. 2.3 Results and discussion Figure 2.2 shows the structures of the three copper oxide anions studied. Although they are superficially similar, these species feature distinct electronic structure patterns, calling for different electronic structure approaches. CuO , which has the singlet ground state, is the simplest of the three oxides. In this system, most of the target neutral states have clear Koopmans character. The ground states of Cu 2 O and CuO 2 are a doublet and triplet, respectively, which results in more complicated target-state patterns. In all three oxides, the attached electron is rather strongly bound, with the detachment energies exceeding 1 eV . 2.3.1 CuO The Hartree-Fock molecular orbitals (MOs) of closed-shell singlet CuO are depicted in Figure 2.12 in the Appendix B. As a singlet closed-shell anion, CuO is the easiest 25 180.0°& 1.737&Å& Cu& O& O& 88.1°% Cu% O% Cu% 1.672&Å& Cu& O Figure 2.2: Structures of the three copper oxide anions included in the study: closed-shell singlet CuO (top left), closed-shell doublet Cu 2 O (top right), and triplet CuO 2 (bottom). All structures were optimized at the !B97X-D/cc-pVTZ level of theory. to model: the majority of transitions that appear in the experimental PES of CuO can be computed via EOM-IP-CCSD from the closed-shell singlet reference. Most of the target states have well-defined Koopmans character, as evidenced by the norms of the respective Dyson orbitals and the values of the leading EOM-IP amplitudes. We note that CuO features low-lying excited states; these states can be computed by a straightforward application of EOM-EE-CCSD. Importantly, the lowest triplet state lies only 0.5 eV (vertically) above the singlet. The triplet and other excited states can be populated at the experimental conditions, 16 giving rise to additional transitions. For most target states, we computed the energies of these hot bands by combining EOM-EE excitation energies (of the closed-shell anion) with EOM-IP detachment energies of the singlet anion. In order to compute the energy corresponding to an open-shell quartet state ( 4 B 1 /B 2 ), which cannot be accessed by detachment from the singlet, we performed EOM-SF calculations from the quartet reference. 56 This scheme allows us to accurately estimate the energy gap between the quartet and open-shell doublet states, so that the energy of this state can be computed relative to other detached states. 26 IE HOMO 10a 1 HOMO-4 8a 1 HOMO-3 9a 1 HOMO-2 4b 1 HOMO-1 4b 2 HOMO-5 1a 2 1.628 eV 2.167 eV 4.996 eV 4.913 eV Figure 2.3: Six lowest EOM-IP-CCSD/cc-pVTZ electron-detached states obtained from the closed-shell singlet CuO reference. Molecular orbitals associated with dominant amplitudes in each EOM-IP transition are depicted in red and blue. Dyson orbitals associated with each EOM-IP transition are rendered in magenta and yellow. Correlated DEs are shown (EOM-IP-CCSD values). Table 2.1 and Figure 2.3 summarize the results of the EOM-IP calculations (the results with other basis sets are collected in Appendix B). Table 2.2 and Figure 2.4 summarize the results of excited-state calculations (in Appendix B, we also provide EOM-EE-CCSD excitation energies computed with smaller basis sets and CIS/cc-pVTZ excitation energies for comparison). Finally, Figure 2.5 compares the computed energy differences with the experimentally derived values. 16 To quantify the effect of electron correlation, we analyze the amplitudes of the EOM- CCSD wave functions. The norm ofR 1 gives the weight of one-electron configurations (1h in EOM-IP and 1h1p in EOM-EE) in the EOM-CCSD wave functions. The devia- tion ofR 2 1 from one quantifies the weight of 2h1p/2h2p configurations, which describe dynamic correlation of the target states. The absolute value of the leadingR 1 amplitude shows how well this state can be described by Koopmans approximation, e.g., several amplitudes with comparable values signify mixing of several Koopmans configurations in the target states, which can be described in terms of non-dynamical correlation. The 27 most experimentally relevant quantity is the norm of the Dyson orbitals, which deter- mine the magnitude of the photodetachment cross sections: e.g., transitions which can- not be described by one-electron ejection, the norms of Dyson orbitals give rise to small norms and, consequently, low cross sections. jjR 1 jj 2 and leading amplitudes (leadingr i ) of the EOM-IP-CCSD transitions indicate that the states arising from photodetachment of CuO have one-electron character and are dominated by a single electronic configuration. Yet, there is notable dissimilarity between the correlated Dyson orbitals and the canonical Hartree-Fock MOs associated with the leading amplitudes of the lowest three EOM-IP-CCSD/cc-pVTZ transitions (Figure 2.3). Moreover, the lowest EOM-IP-CCSD detached state obtained from the closed-shell CuO reference does not arise from detachment of an electron occupying the HOMO, and instead arises from detachment from the HOMO-1/HOMO-2. These results highlight the quantitative limitations of Koopmans’ approximation in describing the detached states of CuO . Norms of the Dyson orbitals (Table 2.1, and Tables 2.8 and 2.9 in Appendix B) are all greater than 0.9 and approximately equal, indicating that the corresponding exper- imental peaks should have similar intensities. With the exception of the quartet state, most transitions have dominant Koopmans character and, therefore, can be modeled by Kohn-Sham DFT and GW type of approaches. The EOM-IP-CCSD/aug-cc-pVTZ level of theory where the ECP10MDF pseudopotential is used for Cu yields EOM-IP-CCSD transitions from the CuO reference that are in closest agreement with experimentally obtained values, relative to other choices of basis sets. The EOM-EE-CCSD/cc-pVTZ level of theory best reproduces the energy gaps between the closed-shell singlet CuO ground state and the two lowest-lying triplet states. The energy of the second-lowest 28 EOM-IP-CCSD detached state is more sensitive to choice of basis set than the lowest detached state. Table 2.1: Six lowest EOM-IP-CCSD/ECP10MDF/aug-cc-pVTZ-PP a detachment energies obtained from the closed-shell singlet CuO reference. ECP applied to Cu only. All-electron aug-cc-pVTZ basis used for oxygen. Orbital b State DE (eV) jjR 1 jj 2 Leadingr i Dyson norm c 3b 1 /3b 2 2 B 1 / 2 B 2 1.889 0.901 0.942 0.930 7a 1 2 A 1 2.556 0.928 0.959 0.956 5a 1 2 A 1 4.829 0.907 0.953 0.941 1a 2 2 A 2 4.829 0.907 0.953 0.941 6a 1 2 A 1 5.114 0.868 0.922 0.911 2b 1 /2b 2 2 B 1 / 2 B 2 5.571 0.898 0.941 0.934 a Augmented by even-tempered diffuse 6s, 6p, 3d functions. b See Appendix B for the symmetries and shapes of Hartree-Fock orbitals. c Norm of the right Dyson orbital associated with the EOM-IP transition. Table 2.2: Six lowest excitation energies (from the closed-shell singlet reference) of CuO computed at the EOM-EE-CCSD/ECP10MDF/aug-cc-pVTZ-PP a level of theory. ECP applied to Cu only. All-electron aug-cc-pVTZ basis used for oxygen. Transition b State E ex (eV) jjR 1 jj 2 Leadingr a i 7a 1 ! 24a 1 3 A 1 0.925 0.956 0.358 3b 1 /3b 2 ! 4b 1 /4b 2 1 A 1 , 1 A 2 1.901 0.901 0.393 3 A 1 , 3 A 2 1.902 0.901 0.31 3b 1 /3b 2 ! 24a 1 1 B 1 , 1 B 2 0.861 0.907 0.334 3 B 1 , 3 B 2 0.442 0.919 0.325 3b 1 /3b 2 ! 9a 1 1 B 1 , 1 B 2 1.894 0.901 0.411 3 B 1 , 3 B 2 0.375 a Augmented by even-tempered diffuse 6s, 6p, 3d functions. b See Appendix B for the symmetries and shapes of Hartree-Fock orbitals. 29 1 A 1 # 1 A 1# 10a 1############# 11a 1 # 1 A 1# 8a 1############# 11a 1 # 1 A 1# 9a 1############# 11a 1 # 1 B 1# 4b 1############# 11a 1 # 1 B 2# 4b 2############# 11a 1 # 1 A 2# 1a 2############## 11a 1 # 0.919%eV% 2.095%eV% 2.886%eV% 4.128%eV% IE# 1 A 1 # 3 A 1$ 10a 1$$$$$$$$$$$$$ 11a 1 $ 3 A 1$ 8a 1$$$$$$$$$$$$$ 11a 1 $ 3 A 1$ 9a 1$$$$$$$$$$$$$ 11a 1 $ 3 B 1$ 4b 1$$$$$$$$$$$$$ 11a 1 $ 3 B 2$ 4b 2$$$$$$$$$$$$$ 11a 1 $ 3 A 2$ 1a 2$$$$$$$$$$$$$$ 11a 1 $ 0.488%eV% 0.873%eV% 2.629%eV% 3.019%eV% IE$ Figure 2.4: Six lowest singlet (left) and triplet (right) EOM-EE-CCSD/cc-pVTZ excited states obtained from the closed-shell singlet CuO reference. Photodetach- ment threshold indicated by the dotted grey line (the states above the dashed line correspond to autodetaching resonances). 30 1 A 1 CS CuO - 3 B 1 / 3 B 2 2 B 1 / 2 B 2 CuO 0 0.49 0.87 0 0.53 3.28 1.63 2.16 5.00 3 A 1 2 A 1 2 A 1 / 2 A 2 4.91 1 A 1 CS 3 B 1 / 3 B 2 2 B 1 / 2 B 2 0 0.44 0.93 0 0.67 2.94 1.67 (1.89) 2.33 (2.56) 4.87 (5.11) 3 A 1 2 A 1 2 A 1 / 2 A 2 4.58 (4.83) 3.37 2 A 1 3.22 2 A 1 4 B 1 / 4 B 2 1.83 1.83 1 Σ + 3d 10 2pσ 2 2pπ 4 3d 10 2pσ 2 2pπ 3 4sσ 1 3 Π 2 3 Π 0 X 2 Π Y 2 Σ + b 4 Σ - Z 3d 10 2pσ 2 2pπ 3 3d 10 2pσ 1 2pπ 4 3d 10 2pσ 2 2pπ 2 4sσ 1 0 0.51 0.75 0 0.97 1.91 ~2.2 X, 1.78 Y, 2.75 Z, ~4 a, 1.27 b, 3.18 a’, 1.03 CuO CuO - Figure 2.5: Comparison of spectral transitions calculated at the EOM-IP- CCSD/cc-pVTZ level of theory (left), spectral transitions calculated at the EOM- IP-CCSD/cc-pVTZ-PP level of theory (middle), and experimental spectral tran- sitions (right) for CuO . EOM-IP-CCSD/aug-cc-pVTZ-PP transition energies are reported in the middle panel in parentheses. Energies are in eV . Calcula- tions were performed from the closed-shell singlet reference, with the exception of the 4 B 1 = 4 B 2 ! 2 B 1 = 2 B 2 transition (left), which was calculated at the EOM-SF- CCSD/cc-pVTZ level of theory from the quartet reference. Experimental data is reproduced from Ref. 16. See Table Appendix A for mapping C 2v point group symmetry labels to those of C 1v . 31 2.3.2 Cu 2 O Figure 2.13 in the Appendix C shows the Hartree-Fock MOs of closed-shell singlet Cu 2 O; the ground-state structure of the anion is shown in Figure 2.2. Because the ground state of Cu 2 O is a doublet, the majority of photodetached states that appear in the experimental PES of Cu 2 O can be computed relative to each other via EOM-EE-CCSD from the closed-shell singlet (N 1)-electron reference. Table 2.3 and Figure 2.6 show the results (see also Appendix C). The lowest photodetachment energy—i.e., the energy gap between the closed-shell singlet reference and the lowest energy (doublet) anion— can be computed by a number of approaches, which are compared in Table 2.4. Our best results are summarized and compared with experiment in Figure 2.7. jjR 1 jj 2 of the EOM-EE-CCSD states show that these transitions have one-electron character (Table 2.3 and S8/S9). The lowest EOM-EE-CCSD excited state obtained from the closed-shell Cu 2 O reference does not arise from a HOMO! LUMO excita- tion, and instead arises from a HOMO-2! LUMO excitation. This result indicate that Koopmans’ approximation based on Hartree-Fock MOs is insufficient for modeling the PES in this system featuring strong correlation. Of the model chemistries used to calculate the energy separation between the dou- blet anion and the lowest photodetached state (presented in Table 2.4), our best results were obtained when the onset of electron detachment was calculated as a difference in CCSD(T)/aug-cc-pVTZ total energies of the two states. The inclusion of perturbative triples recovers 0.2 eV , and the inclusion of diffuse functions in the basis recovers 0.2 eV , relative to CCSD/cc-pVTZ. Our best estimate for this lowest energy transition (X-band) is 1.00 eV is within 0.1 eV from the experimental peak position. Combining the results of the EOM-EE-CCSD and CCSD(T) calculations to obtain higher electron detached states and comparing with experiment (Figure 2.7), we find 32 that the use of the ECP10MDF pseudopotential generally improves the quality of re- sults. The inclusion of diffuse functions in the basis set improves the quality of lower energy transitions while a non-augmented set better reproduces the energy of high-lying photodetached states. Table 2.3: Six lowest EOM-EE-CCSD/ECP10MDF/aug-cc-pVTZ-PP a excitation energies of closed-shell singlet Cu 2 O. ECP applied to copper only. All-electron basis used for oxygen. Orbital b State E ex (eV) jjR 1 jj 2 Leadingr a i 4b 2 ! 10a 1 1 B 2 1.620 0.920 0.451 3 B 2 1.180 0.928 0.432 7b 1 ! 10a 1 1 B 1 1.676 0.925 0.450 3 B 1 1.228 0.937 0.432 4b 2 ! 8b 1 1 A 2 2.085 0.926 0.296 2a 2 ! 10a 1 3 A 2 1.670 0.932 0.280 7b 1 ! 8b 1 1 A 1 2.189 0.928 0.276 3 A 1 1.672 0.936 0.252 6b 1 ! 10a 1 1 B 1 2.329 0.930 0.333 3 B 2 1.823 0.940 0.350 9a 1 ! 10a 1 1 A 1 2.375 0.930 0.323 3 A 1 1.581 0.947 0.337 a No diffuse f or g functions. b See Appendix C for the symmetries and shapes of Hartree-Fock orbitals. 33 1 A 1 # 1 A 2$ 5b 2$$$$$$$$$$$$$ 12b 1 $ 1 A 1$ 11b 1$$$$$$$$$$$$$ 12b 1 $ 1 A 1$ 13a 1$$$$$$$$$$$$$ 14a 1 $ 1 B 2$ 5b 2$$$$$$$$$$$$$ 14a 1 $ 1.696%eV% 2.274%eV% 2.374%eV% 2.517%eV% 1 B 1$ 11b 1$$$$$$$$$$$$$ 14a 1 $ 1.774%eV% 1 B 1$ 10b 1$$$$$$$$$$$$$ 14a 1 $ 1 A 1 # 3 A 2$ 5b 2$$$$$$$$$$$$$ 12b 1 $ 3 A 1$ 11b 1$$$$$$$$$$$$$ 12b 1 $ 3 B 2$ 5b 2$$$$$$$$$$$$$ 14a 1 $ 1.283&eV& 1.896&eV& 1.905&eV& 2.088&eV& 3 B 1$ 11b 1$$$$$$$$$$$$$ 14a 1 $ 1.379&eV& 3 B 1$ 10b 1$$$$$$$$$$$$$ 14a 1 $ 3 A 1$ 13a 1$$$$$$$$$$$$$ 14a 1 $ 1.675&eV& Figure 2.6: Six lowest singlet (left) and triplet (right) EOM-EE-CCSD/cc-pVTZ excited states obtained from the closed-shell singlet Cu 2 O reference. 34 Table 2.4: Lowest vertical electron detachment energy of Cu 2 O calculated by different approaches (using the all-electron cc-pVTZ basis, unless otherwise indi- cated). Method Reference state DE (eV) jjR 1 jj 2 Leadingr 1 Experiment – 1.10 – – CCSD[Cu 2 O ]- doublet Cu 2 O and 0.584 – – CCSD[Cu 2 O] closed-shell singlet Cu 2 O CCSD(T)[Cu 2 O] - doublet Cu 2 O and 0.751 – – CCSD(T)[Cu 2 O ] closed-shell singlet Cu 2 O CCSD(T)[Cu 2 O] - a doublet Cu 2 O and 0.811 – – CCSD(T)[Cu 2 O ] a closed-shell singlet Cu 2 O CCSD(T)[Cu 2 O] - doublet Cu 2 O and 0.996 – – CCSD(T)[Cu 2 O ] b closed-shell singlet Cu 2 O EOM-EA-CCSD closed-shell singlet Cu 2 O 0.353 0.955 0.968 EOM-EA-CCSD b closed-shell singlet Cu 2 O 0.578 0.957 0.945 EOM-IP-CCSD doublet Cu 2 O 0.574 0.970 0.956 EOM-IP-CCSD[Cu 2 O 2 ]- closed-shell singlet Cu 2 O 2 0.579 0.958, 0.941, EOM-DIP-CCSD[Cu 2 O 2 ] 0.954 0.946 a Calculated using the ECP10MDF pseudopotential with a cc-pVTZ-PP basis for Cu. The cc-pVTZ set was used for O. b Calculated using the ECP10MDF pseudopotential with a partial aug-cc-pVTZ-PP basis for Cu (diffuse f and g functions were omitted). The aug-cc-pVTZ set was used for O. 35 Cu 2 O - 1 A 1 CS 1 B 2 Cu 2 O 0 0 1.70 1.77 2.37 1.70 1.77 2.52 1 B 1 1 A 1 2.37 2.52 1 A 1 / 1 B 1 1 A 2 2.27 2 A 1 2.27 0.75 Cu 2 O 1 A 1 CS 1 B 2 1.60 (1.62) 1.69 (1.68) 2.32 (2.33) 1 B 1 1 A 1 2.18 (2.19) 1 B 1 1 A 2 2 A 1 2.07 (2.09) 0.81 (1.00) 0 1.60 1.69 2.18 2.32 2.07 0 Cu 2 O - X A B 0 0 1.43 1.56 X, 1.10 A, 2.53 B, 2.66 Cu 2 O C 1.75 C, 2.85 D 1.85 D, 2.95 Cu 2 O Figure 2.7: Comparison of spectral transitions calculated at the EOM-EE- CCSD/cc-pVTZ level of theory (left), spectral transitions calculated at the EOM- EE-CCSD/cc-pVTZ-PP level of theory (middle), and experimental spectral tran- sitions (right) for Cu 2 O . EOM-EE-CCSD/aug-cc-pVTZ-PP transition energies are reported in the middle panel in parentheses. Experimental data is taken from Ref. 17. All transitions were computed with the EOM-EE-CCSD method from the closed-shell (N 1)-electron Cu 2 O reference, with the exception of the lowest electron-detachment energies, which were computed as differences in CCSD(T) to- tal energies between the doublet anion and lowest neutral state. The all-electron cc- pVTZ basis (left) and the ECP10MDF/(aug)-cc-pVTZ-PP basis (middle) for copper were used in the E[CCSD(T)] calculations. 36 2.3.3 CuO 2 Of the three molecules studied here, CuO 2 presents the most challenging electronic structure pattern. Its ground-state equilibrium geometry is shown in Figure 2.2. Al- though earlier studies argued in favor of bent molecular structure, the most recent ex- perimental study 4 of photodetachment from CuO 2 has presented conclusive evidence for the linear structure being predominant in the gas phase. The difficulties in choosing an appropriate reference for describing relevant elec- tronic states of the neutral CuO 2 and CuO 2 become apparent upon examining frontier MOs and possible electronic configurations. Let us begin with the anion. Formally, it has an even number of electrons, so one might attempt to employ a closed-shell singlet reference and to describe low-lying open-shell singlets and triplets by EOM-EE-CCSD, as we did in CuO . This strategy is perfectly legitimate provided that all pairs of for- mally degenerate orbitals ( orbitals) are either fully occupied or empty and that all relevant low-lying states can be described as single electron excitations from such refer- ence. Unfortunately, the degeneracy pattern of the frontier orbitals in CuO 2 precludes us from using this approach and the only way to correctly describe the manifold of low- lying states is to use the reference with 2 additional electrons and employ EOM-DIP- CCSD. Figure 2.8 shows frontier Hartree-Fock MOs of the closed-shell singlet CuO 3 2 state and the energies and electronic configurations of the low-lying states of CuO 2 . The results of EOM-DIP-CCSD/cc-pVTZ calculation are collected in Table 2.5. As one can see, the lowest electronic state of CuO 2 is triplet ( 3 B 1g or 3 g ), in agreement with theoretical result by Zhu and coworkers. 31 Within 0.2 eV , there are 4 singlet states, 1 B 1g ( 1 g ), a doubly degenerate closed-shell like singlet ( 1 g ), and nearly degenerate singlet 37 and triplet pair ( 1;3 A u , corresponding to 1;3 g ). All these states (and possibly higher ly- ing ones too) can be populated at the experimental conditions, giving rise to different electron-detached states. To describe target states that can be generated by detaching an electron from the lowest state of the anion ( 3 B 1g or 3 g ), we use a quartet reference state ( 4 B 1g ) in which each degenerate-orbital is singly occupied and describe the three lowest doublet states (of which two are a degenerate pair) and M s =1/2 component of the quartet state by EOM-SF-CCSD. The frontier MOs and electronic configuration of the 4 B 1g state and the resulting doublet states are shown in Figure 2.9. As one see, these calculations predict the ground state of CuO 2 to be a quartet, but the doublet states are very close (within 0.3 eV). The two manifolds (EOM-DIP manifold of the anionic sates and EOM-SF manifold of the neutral states) can be connected by EOM-EA-CCSD calculation using a neutral quartet reference (in which each degenerate -orbital is singly occupied) producing a target 3 B 1g state. This energy (3.53 eV) corresponds to the high-intensityA-peak in the experimental spectra (using labels by Wuetal: 16 ), which has been later assigned to the lowest detachment from the ground-state ( 3 g ) of the anion. 31 The presence of multiple low-lying states in CuO 2 , each of them giving rise to dif- ferent target states, makes the assignment of spectral transitions difficult. Figure 2.10 shows electronic configurations of the lowest states of CuO 2 and low-lying states of the neutral CuO 2 and shows which pairs of states are connected by one-electron transi- tion (i.e., are Koopmans-allowed) and which ones are not. Finally, in Figure 2.11, we present an energy level diagram for comparison with experimental transitions. 4, 17 Our assignment of the low part of the spectrum differs from earlier ones. 4, 17, 31 While we 38 agree with Zhu and coworkers, 31 who assigned the high-intensityA-peak to the detach- ment from the ground-state anion ( 3 g ) and the low intensityX-peak to the detachment from an electronically excited anion, we point out that the ground state of the neutral is quartet ( 4 g ). Our calculations also suggest that more than one electronic state of the anion contributes to the electronic hot band (X peak), which is probably responsible for its relatively high intensity. The experimental value of 3.47 eV (ofX-peak) can be compared to our theoretical value of 3.33 eV . We note that the computed splitting betweenA andX peak (0.2 eV) is in good agreement with the experimental value 17 of 0.3 eV . Table 2.5: 12 lowest states of CuO 2 computed by EOM-DIP-CCSD/cc-pVTZ from the closed-shell singlet CuO 3 2 reference. Orbitals a State Configuration E ex (eV) jjR 2 jj 2 Leadingr ij 2b 2g + 2b 3g 3 B 1g triplet 0.000 0.838 0.543 1 B 1g open-shell singlet 0.106 0.841 0.521 2b 2g /2b 3g 1 A g closed-shell singlet 0.163 0.843 0.509 (3b 2u + 2b 2g ), 1 A u open-shell singlet 0.165 0.845 0.458 (3b 3u + 2b 3g ) 3 A u triplet 0.193 0.845 0.458 (3b 3u + 2b 2g ), 3 B 1u triplet 0.211 0.845 0.458 (3b 2u + 2b 3g ) (8a g + 2b 2g )/ (8a g + 2b 3g ) 1 B 2g / 1 B 3g open-shell singlet 0.421 0.845 0.6078 3 B 2g / 3 B 3g triplet 0.370 0.842 0.613 (8a g + 3b 2u )/ (8a g + 3b 3u ) 1 B 2u / 1 B 3u open-shell singlet 1.246 0.850 0.483 3 B 2u / 3 B 3u triplet 1.082 0.852 0.493 8a g + 5b 1u 3 B 1u triplet 1.546 0.851 0.631 8a g 1 A g closed-shell singlet 1.686 0.877 0.800 a See Fig. 2.8 for the symmetries and shapes of Hartree-Fock orbitals. 39 3 B 1u 1 A g (refA) 1 B 1g 3 B 1 g 1 A g Blue= EOM-DIP-CCSD/cc-pVTZ from CuO 2 3- Purple = EOM-EE-CCSD/cc-pVTZ From CuO 2 - ref A Yellow= EOM-EE-CCSD/cc-pVTZ From CuO 2 - ref B 0 0.454;0.211;-0.632 0.570;0.106 KEY 0.639;0.163 1.686 1 A g 0.163 8a g 9a g 2b 3g 2b 2 g 3b 2u 3b 3 u 5b 1u 5b 1u 8a g 2b 3g 2b 2g 3b 2u 3b 3u 9a g 2b 3g 3b 3u 2b 2g 8a g 3b 2u 2b 3g 3b 3u 2b 2g 8a g 3b 2u 2b 3g 3b 3u 2b 2g 8a g 3b 2u 2b 3g 3b 3u 2b 2g 8a g 3b 2u 3b 3u 2b 2g 8a g 2b 3g 3b 2u 1 A g (refB) > 1.686 3b 3u 2b 2g 8a g 2b 3g 3b 2u 1 A g 3b 3u 2b 2g 8a g 2b 3g 3b 2u 1 A u 0.165 2b 3g 3b 3u 2b 2g 8a g 3b 2u 3 A u 0.193 > 1.686 Figure 2.8: Molecular orbitals of the triplet CuO 3 2 closed-shell reference state (bottom) and associated low-lying excited states (top). The relative energy separa- tions of the states that are provided in blue were obtained using EOM-DIP-CCSD and the closed-shell CuO 3 2 reference. We consider the ordering of CuO 2 states ob- tained from this method to be most reliable, but other relative energy separations using different EOM methods/references are also provided in the diagram when possible. 40 4 B 1g (Reference) 2 B 2g 2 A g 3 B 1g (CuO 2 - ) Orange = EOM-EA-CCSD/cc-pVTZ Purple = EOM-SF-CCSD/cc-pVTZ 0 0.29 KEY 0.33 2 B 3g 0.33 8a g 9a g 2b 3g 2b 2g 3b 2u 3b 3u 5b 1u 5b 1u 8a g 2b 3g 2b 2g 3b 2u 3b 3u 9a g 2b 3g 3b 3u 2b 2g 8a g 3b 2u 2b 3g 3b 3u 2b 2g 8a g 3b 2u 2b 3g 3b 3u 2b 2g 8a g 3b 2u 3.53 2b 3g 3b 3u 2b 2g 8a g 3b 2u 0 <S 2 > = 1.78 <S 2 > = 1.10 <S 2 > = 1.10 2b 3g 3b 3u 2b 2g 8a g 3b 2u Other states > 1 eV Figure 2.9: Molecular orbitals of the 4 B 1g CuO 2 reference state (bottom) and its as- sociated excited and electron attached states (top). The relative energy separations andhS 2 i of states that are shown in purple were obtained using EOM-SF-CCSD from the quartet CuO 2 reference. The EOM-EA-CCSD result in orange, which connects the neutral and anionic state manifolds, provides the energy of the lowest photodetachment from CuO 2 (3.53 eV). The electronic configuration boxed in red would require both excitation and photodetachment from the ground-state CuO 2 triplet reference. See Appendix A for mapping C 2v point group symmetry labels to those of D 1h . 41 2b 3g 3b 3u 2b 2g 8a g 3b 2u 3 B 1g / 1 B 1g 3b 3u 2b 2g 8a g 3b 3u 2b 2g 8a g 3b 3u 2b 2g 8a g B 2u / B 3u B 2g / B 3g B 1g 3b 3u 2b 2g 8a g 3b 3u 2b 2g 8a g 3b 3u 2b 2g 8a g B 2u / B 3u B 2g / B 3g A g 2b 3g 3b 2u 2b 3g 3b 2u 2b 3g 3b 2u 2b 3g 3b 2u 2b 3g 3b 2u 2b 3g 3b 2u CuO 2 - 2b 3g 3b 3u 2b 2g 8a g 3b 2u 1 A g CuO 2 - Figure 2.10: Electronic configuration of the triplet CuO 2 ground state and closely lying singlet state of the same orbital occupation (left) and of the closed-shell 1 A g states (right). From the 3;1 B 1g configuration, the three lowest-lying Koopmans- allowed detached states (middle) are expected to emerge in the spectrum are in- dicated by solid green arrows. However, several configurations (middle, dashed red arrows) require excitation as well as removing an electron, making them Koopmans-forbidden from that state. From the 1 A g states, a slightly different set of detached states can be accessed. See Appendix A for mapping C 2v point group symmetry labels to those of D 1h . 42 3 B 1g 0 A, 3.53 0 4 B 1g 1 Σ + OCuO - X 2 Π g A B 0 0 0.32 0.63 X, 3.47 A, 3.79 B, 4.10 OCuO C 0.81 C, 4.28 D 1.20 D, 4.67 E 1.69 E, 5.16 ~ 0.3 2 B 2g , 2 B 3g , 2 A g B, 3.83 3.63 ~ 0.2 1 A g , 3,1 A u , 1 B 1g , 3 B 1u X, 3.33 Figure 2.11: Comparison of spectral transitions calculated at the EOM- SF/DIP/EA-CCSD/cc-pVTZ level of theory (left) and experimental spectral transi- tions (right) for CuO 2 . Experimental data is taken from Ref. 16, and spectral no- tation reflects original experimental assignments. Experimental intensities of the spectral transitions are as follows: A> B> X C. Bold lines in the left panel de- note several states clustered around the same energy. See Appendix A for mapping C 2v point group symmetry labels to those of D 1h . 43 2.4 Conclusions In this contribution, we show that variants of the EOM-CCSD family of methods can be combined and applied to describe the photoelectron spectra of CuO , Cu 2 O , and CuO 2 . For the former two systems, the calculations agree well with the experimen- tally derived values. We note improvement in quality of results when the ECP10MDF pseudopotential is applied to Cu, suggesting the relativistic corrections afforded by the use of ECP10MDF (and its associated highly-contracted, correlation-consistent basis) might be necessary for accurate description of valence photodetachment. Dyson or- bitals and analysis of dominant amplitudes in the EOM-CCSD wave functions reveal the non-Koopmans character and multi-configurational nature of the anionic and neutral states that give rise to the CuO and Cu 2 O PES. For Cu 2 O , the inclusion of pertur- bative triples correction, CCSD(T), was required for an accurate description of the first photodetachment energy of the doublet anion. Due to the highly correlated nature of the CuO and Cu 2 O systems, careful benchmarking and density-based state analysis 57, 58 are recommended when lower-level correlation methods, like DFT and its extensions, are applied to model photodetachment. For CuO 2 , we note good agreement with experiment in the low-energy part of the spectrum. However, adequate description of higher-lying photodetached states remains elusive. Our results indicate that the experimental PES of CuO 2 may arise from multiple neutral states clustered around specific energies, which makes an accurate and balanced description of all of the photodetached states that contribute to the low-energy PES chal- lenging. A variety of EOM-CCSD methods—IP, EE, DIP, SF—from many neutral and 44 anionic CuO 2 references were required to capture this phenomenon. By highlighting ex- treme open-shell character of CuO 2 , our study contributes towards fundamental under- standing of bonding patterns and electronic structure of transition metal oxides, which are important in many areas of chemistry including catalysis and materials. The anal- ysis of electronic structure patterns in these three small molecules provides a practical tutorial for using EOM-CC methods to describe various open-shell systems. 47 Given the high level of theory employed, the reported results can serve as a valuable benchmark for future method development. 45 2.5 Appendix A: Symmetry labels Table 2.6: Correspondence betweenC 2v andC 1v point groups 54 C 1v C 2v + A 1 A 2 E 1 = B 1 =B 2 E 2 = A 1 =A 2 Table 2.7: Correspondence betweenD 2h andD 1h point groups 54 D 1h D 2h + g A g g B 1g g B 2g =B 3g g A g =B 1g + u B 1u u A u u B 2u =B 3u u A u =B 1u 46 2.6 Appendix B: Hartree-Fock molecular orbital dia- grams for singlet CuO and additional EOM-CCSD and CIS results MO#6# %0.648# Energy'' 4a 1 # 5a 1 # 6a 1 # 2b 1 # 2b 2 # 3b 1 # 3b 2 # 7a 1 # 1a 2 # 8a 1 # 9a 1 # 4b 2 # 4b 1 # 10a 1 # 11a 1 # 11a 1 # 10a 1 # 9a 1 # 7a 1 # #2b 2 # 6a 1 # 5a 1 # 4a 1 # 3b 2 # 1a 2 # 4b 2 # #2b 1 # 3b 1 # 8a 1 # 4b 1 # 5b 2 # 5b 1 # 5b 1 # 5b 2 # Figure 2.12: Fourteen HOMOs and three LUMOs of singlet CuO . 47 Table 2.8: Six lowest EOM-IP-CCSD/cc-pVTZ (all-electron) detachment energies obtained from the closed-shell singlet CuO reference. Orbital a State DE (eV) jjR 1 jj 2 b Leadingr i Dyson norm b 4b 1 /4b 2 2 B 1 / 2 B 2 1.628 0.898 0.944 0.927 10a 1 2 A 1 2.167 0.923 0.958 0.947 1a 2 2 A 2 4.913 0.905 0.951 0.939 8a 1 2 A 1 4.913 0.905 0.951 0.939 9a 1 2 A 1 4.996 0.853 0.920 0.904 3b 1 /3b 2 2 B 1 / 2 B 2 5.561 0.895 0.943 0.931 a See Fig. 2.12 for the symmetries and shapes of Hartree-Fock orbitals. b Norm of the right Dyson orbital associated with the EOM-IP transition. Table 2.9: Six lowest EOM-IP-CCSD/ECP10MDF/cc-pVTZ-PP detachment ener- gies obtained from the closed-shell singlet CuO reference. ECP applied to copper only. All-electron cc-pVTZ basis used for oxygen. Orbital a State DE (eV) jjR 1 jj 2 Leadingr i Dyson norm b 3b 1 /3b 2 2 B 1 / 2 B 2 1.674 0.901 0.942 0.932 7a 1 2 A 1 2.332 0.931 0.960 0.954 1a 2 2 A 2 4.578 0.909 0.953 0.940 5a 1 2 A 1 4.578 0.909 0.953 0.940 6a 1 2 A 1 4.869 0.864 0.919 0.908 2b 1 /2b 2 2 B 1 / 2 B 2 5.339 0.899 0.942 0.933 a See Fig. 2.12 for the symmetries and shapes of Hartree-Fock orbitals. b Norm of the right Dyson orbital associated with the EOM-IP transition. 48 Table 2.10: Six lowest excitation energies (from the closed-shell singlet reference) of CuO computed at the EOM-EE-CCSD/cc-pVTZ level of theory. All-electron basis used for copper and oxygen. Transition a State E ex (eV) jjR 1 jj 2 Leadingr a i 4b 1 /4b 2 ! 11a 1 1 B 1 , 1 B 2 0.919 0.898 0.594 3 B 1 , 3 B 2 0.488 0.915 0.578 10a 1 ! 11a 1 1 A 1 2.095 0.910 0.522 3 A 1 0.873 0.957 0.596 8a 1 ! 11a 1 1 A 1 2.886 0.914 0.579 3 A 1 2.629 0.923 0.525 1a 2 ! 11a 1 1 A 2 2.886 0.914 0.579 3 A 2 2.629 0.923 0.525 9a 1 ! 11a 1 1 A 1 4.128 0.860 0.458 3 A 1 3.019 0.923 0.496 a See Fig. 2.12 for the symmetries and shapes of Hartree-Fock orbitals. Table 2.11: Six lowest excitation energies (from the closed-shell singlet reference) of CuO computed at the EOM-EE-CCSD/ECP10MDFD/cc-pVTZ-PP level of the- ory. ECP applied to Cu only. All-electron cc-pVTZ basis used for O. Transition a State E ex (eV) jjR 1 jj 2 Leadingr a i 4b 1 /4b 2 ! 8a 1 1 B 1 , 1 B 2 0.933 0.906 0.575 3 B 1 , 3 B 2 0.457 0.920 0.556 7a 1 ! 8a 1 1 A 1 2.140 0.916 0.513 3 A 1 0.941 0.958 0.582 1a 2 ! 8a 1 1 A 2 2.537 0.919 0.590 3 A 2 2.239 0.928 0.576 5a 1 ! 8a 1 3 A 1 2.240 0.928 0.576 6a 1 ! 8a 1 3 A 1 2.913 0.927 0.488 2b 1 /2b 2 ! 8a 1 1 B 1 , 1 B 2 3.422 0.895 0.535 3 B 1 , 3 B 2 3.219 0.910 0.525 a See Fig. 2.12 for the symmetries and shapes of Hartree-Fock orbitals. 49 Table 2.12: Six lowest CIS/cc-pVTZ excitation energies from the closed-shell sin- glet reference of CuO . All-electron basis used for copper and oxygen. Transition State E (eV) 4b 1 /4b 2 ! 11a 1 1 B 1 , 1 B 2 1.208 3 B 1 , 3 B 2 0.539 10a 1 ! 11a 1 1 A 1 1.842 3 A 1 -0.371 8a 1 ! 11a 1 1 A 1 3.884 3 A 1 3.375 1a 2 ! 11a 1 1 A 2 3.884 3 A 2 3.375 9a 1 ! 11a 1 1 A 1 4.494 3 A 1 2.229 a See Fig. 2.12 for the symmetries and shapes of Hartree-Fock orbitals. 50 2.7 Appendix C: Hartree-Fock molecular orbital dia- grams for doublet Cu 2 O and additional EOM- CCSD results MO#6# %0.648# Energy'' 9a 1 # 10a 1 # 3b 2 # 8b 1 # 11a 1 # 3a 2 # 4a 2 # 4b 2 # 9b 1 # 12a 1 # 10b 1 # 5b 2 # 11b 1 # 13a 1 # 14a 1 # 14a 1 $ 13a 1 $ 12a 1 $ 4b 2 $ $8b 1 $ 3b 2 $ 10a 1 $ 9a 1 $ 3a 2 $ 9b 1 $ 11b 1 $ $11a 1 $ 4a 2 $ 10b 1 $ 5b 2 $ 12b 1 $ 12b 1 # Figure 2.13: Fourteen HOMOs and two LUMOs of doublet Cu 2 O . MO surfaces correspond to the closed-shell neutral reference (Cu 2 O). The red box around the 14a 1 LUMO indicates the orbital that becomes occupied in the ground state of the doublet anion species. 51 Table 2.13: Six lowest EOM-EE-CCSD/cc-pVTZ excitation energies of closed-shell singlet Cu 2 O. All-electron basis used for Cu and O. Orbital a State E ex (eV) jjR 1 jj 2 Leadingr a i 5b 2 ! 14a 1 1 B 2 1.696 0.920 0.552 3 B 2 1.283 0.928 0.537 11b 1 ! 14a 1 1 B 1 1.774 0.926 0.546 3 B 1 1.379 0.938 0.528 5b 2 ! 12b 1 1 A 2 2.274 0.926 0.449 3 A 2 1.905 0.932 0.427 11b 1 ! 12b 1 1 A 1 2.374 0.928 0.415 3 A 1 1.896 0.937 0.418 13a 1 ! 14a 1 1 A 1 2.517 0.932 0.407 3 A 1 1.675 0.947 0.481 10b 1 ! 14a 1 1 B 1 2.517 0.932 0.388 3 B 1 2.088 0.941 0.410 a See Fig. 2.13 for the symmetries and shapes of Hartree-Fock orbitals. Table 2.14: Six lowest EOM-EE-CCSD/ECP10MDF/cc-pVTZ-PP excitation ener- gies of closed-shell singlet Cu 2 O. ECP applied to copper only. All-electron basis used for oxygen. Orbital a State E ex (eV) jjR 1 jj 2 Leadingr a i 4b 2 ! 10a 1 1 B 2 1.604 0.920 0.537 3 B 2 1.158 0.928 0.522 7b 1 ! 10a 1 1 B 1 1.678 0.925 0.536 3 B 1 1.225 0.938 0.522 4b 2 ! 8b 1 1 A 2 2.069 0.926 0.423 3 A 2 1.656 0.932 0.402 7b 1 ! 8b 1 1 A 1 2.182 0.927 0.391 3 A 1 1.662 0.936 0.363 6b 1 ! 10a 1 1 B 1 2.323 0.930 0.409 3 B 1 1.821 0.940 0.434 9a 1 ! 10a 1 1 A 1 2.377 0.930 0.384 3 A 1 1.571 0.948 0.407 a See Fig. 2.13 for the symmetries and shapes of Hartree-Fock orbitals. 52 2.8 Appendix D: Cartesian geometries Closed-shell singlet geometry of CuO- anion optimized at the wB97X-D/cc-pVTZ level of theory. 1 Cu 0.0000000000 0.0000000000 -0.3616121081 2 O 0.0000000000 0.0000000000 1.3108438919 Nuclear Repulsion Energy: 73.4065 hartrees Closed-shell doublet geometry of the (Cu2O)- anion optimized at the wB97X-D/cc-pVTZ level of theory. 1 Cu 1.2492124637 0.0000000000 0.1565252450 2 O 0.0000000000 0.0000000000 -1.1348080259 3 Cu -1.2492124637 0.0000000000 0.1565252450 Nuclear Repulsion Energy: 314.7893 hartrees Triplet geometry of (CuO2)- anion optimized at the wB97X-D/cc-pVTZ level of theory. 1 Cu 0.0000000000 0.0000000000 0.0000000000 2 O 0.0000000000 0.0000000000 1.7367260000 3 O 0.0000000000 0.0000000000 -1.7367260000 53 Nuclear Repulsion Energy: 151.1303 hartrees 54 Chapter 2 References [1] A. 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Phys. 141, 024107 (2014). 57 Chapter 3: Characterizing bonding patterns in diradicals and triradicals by density-based wave function analysis: A uniform approach 3.1 Introduction Chemists define diradicals and triradicals as species with two and three unpaired elec- trons. 1–4 This bonding pattern arises when two (or three) electrons are distributed in two (three) nearly degenerate molecular orbitals. For same-spin electrons, there are only two possible arrangements: M s 1 and M s 3=2. For the states with lower spin-projections, more configurations can be generated, as illustrated in Fig. 3.1. The energy gaps, rela- tive state ordering, and relative weights of the Slater determinants (i.e., coefficients in Fig. 3.1) depend on the nature and energy separation of the frontier molecular orbitals (MOs). The character of the MOs also determines the character of the wave function, e.g., whether it has predominantly covalent (i.e., two electrons residing on different parts of a molecule) or ionic character. 58 + + - - 2 - -!’ + !’ -!” + !” + !’’’ -!’’’ (i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (ix) (x) ! + + - (i) (ii) (iii) (iv) (A) (B) -! (v) Figure 3.1: Wave functions of di- (A) and triradicals (B) that are eigenfunctions of S 2 . In both panels, wave function (i) corresponds to the reference-state wave function: the high-spin M s = 1 triplet state of a diradical or the M s = 3/2 quartet state of a triradical. Wave functions (ii)-(v) in A and (ii)-(x) in B correspond to the low-spin states: M s = 0 singlets and triplets, or M s = 1/2 doublets and quartets. For simplicity, only configurations with positive spin-projection are shown. To illustrate this point, consider a simple diradical, such as a molecule with a broken bond (i.e., H 2 at dissociation limit or twisted ethylene). In this case, the two MOs are a bonding and antibonding pair (such as and orbitals in stretched H 2 or and in twisted ethylene). When the two MOs are exactly degenerate,=1 in wave functions (iv) and (v) in Fig. 3.1A. Wave function (iii) corresponds to the covalent diradical singlet state (H H ), which is exactly degenerate with the triplet state, whereas (v) represents a purely ionic (charge-resonance) state (i.e., H H + H + H ). This situation is often de- scribed as a perfect diradical. If the gap between bonding and antibonding orbitals is large, becomes small and the ground state can be represented by a single Slater de- terminant; the wave function in this case is a mixture of covalent and charge-resonance 59 configurations and the molecule can be described as a closed-shell species. In between there is a continuum of intermediate bonding patterns, which are sometimes described as diradicaloids. 2 Detailed analysis of different types of diradical electronic structure can be found in classic papers; 1, 2 for a review of triradicals, one can consult Ref. 3. The wave functions composed by Slater determinants in which two electrons occupy the same MO, e.g., (iv) and (v) in (A) and (v)-(x) in (B) are commonly referred to as of “closed-shell” type (although they can correspond to completely unpaired electrons, as in the case of A(v) with=1), in contrast to “open-shell” wave functions (i)-(iii) in (A) and (i)-(iv) in (B) (which can correspond to a purely ionic and, therefore, closed- shell pattern). Here we follow this accepted terminology for consistency with other studies. 4 An important difference between the two types of open-shell wave functions is that in A(v) the coefficients depend on the orbital degeneracy whereas in A(ii)- (iii) the relative weights of the contributing Slater determinants are determined by spin symmetry alone. 1 In model pedagogical examples of two-electrons-in-two orbitals or three-electrons- in-three-orbitals, the number of unpaired electrons can be unambiguously determined from the wave functions. For example, in diradicals, (i) and (ii) correspond to the two unpaired electrons, whereas the relative weight of purely covalent configuration in (v) is determined by, i.e.,=1 corresponds to the 2 unpaired electrons. However, this simple picture does not apply for realistic many-body wave functions. Even in the simplest case of two electrons, the dynamic correlation and the arbitrariness in orbital choice result in multi-configurational wave functions whose character cannot be easily assigned on the basis of just 2 leading configurations. In many-electron systems the contributions of electrons occupying lower MOs give rise to through-bond interactions between the radical centers, further complicating the wave function analysis. Although the number 60 of effectively unpaired electrons is not an observable physical property, it is related to the bonding pattern, which, in turn, manifests itself in concrete structural, spectroscopic, and thermochemical quantities. 3–5 Thus, the ability to assign an effective number of un- paired electrons to a particular electronic wave function is valuable for chemical insight, akin to other methods of wave function analysis. 6, 7 Several solutions towards this goal have been proposed. 8–10 As with many other wave function analysis tools, 6, 7, 11–14 they exploit the concept of natural orbitals. 15 Nat- ural orbitals ( i ) are eigenstates of the one-particle reduced density matrix: i (x) = X j c ji j (x); (3.1) c =cn; (3.2) pq h jp y qj i (3.3) where ^ p y and ^ q are the creation and annihilation operators corresponding to p and q MOs. The respective eigenvalues (n i ) are non-negative, add up to the total number of electrons, and can be interpreted as orbital occupations. Natural orbitals afford the most compact representation of the electron density (x) = X pq pq p (x) q (x) = X i n i 2 i (x) (3.4) and reflect multi-configurational character of wave functions. For example, for a single Slater determinant, natural occupations are two (for occupied orbitals) and zero (for virtuals). For a two-determinantal wave function (v) with=1, natural occupations are 1. For multi-configurational wave functions, the partial occupations of natural orbitals can be used to derive an effective number of the unpaired electrons. For example, for the perfectly diradical wave functions, such as A(i)-(iii) and (iv) with=1 in Fig. 3.1, 61 n 1 =n 2 =1 and the number of unpaired electrons is 2, whereas for the closed-shell case, A(iv) with=0,n 1 =2,n 2 =0 and the number of unpaired electrons is 0. Several ways to compute an effective number of unpaired electrons from the one- particle density matrices have been proposed. 8–10 In this work, we make use of the two indices,n u andn u;nl , proposed by Head-Gordon 9 as an extension of work by Yamaguchi et al.: 8 n u = X i min( n i ; 2 n i ) (3.5) n u;nl = X i n 2 i (2 n i ) 2 (3.6) In both equations, the sum runs over all natural orbitals and the contributions of the doubly occupied and unoccupied orbitals are exactly zero. As one can easily verify, both formulas yield correct answers for the model examples from Fig. 3.1, i.e., 2 for A(i)-(iii) and (iv) with=1, 0 for A(iv) with=0, and so on. The two expressions differ by how they account for partially occupied orbitals. Numerical experimentation has shown that for many-electron wave functions, Eq. (5.8) consistently gives physically meaningful values, whereasn u often produces the number of unpaired electrons which is too high (as compared to chemical intuition). This happens because then u index does not suppress dynamic correlation contributions (which come from a large number of small natural occupation numbers) to the total number of unpaired electrons, whereas n u;nl emphasizes radical character atn i values near one and closed-shell character for n i values close to zero and two. Below we refer ton u;nl as Head-Gordon’s index. For the two-electrons-in-two-orbitals case, the coefficient andn u;nl are related by the following expression: 5 n u;nl = 32 4 (1 + 2 ) 4 : (3.7) 62 As illustrated by the numerical examples in Ref. 5,n u;nl provides a much more robust and reliable measure of the diradical character than . Furthermore, unlike ’s, n u;nl does not depend on orbitals used in the correlated calculation (unrestricted or restricted open-shell Hartree-Fock, Kohn-Sham orbitals, etc). In addition to enabling calculations of the number of effectively unpaired electrons, natural orbitals afford visualization of the frontier orbitals that are associated with corre- lated many-electron wave functions, thus departing from canonical Hartree-Fock MOs. Comparing natural occupations of the frontier orbitals with the computed number of effectively unpaired electrons informs us of how well the frontier orbital picture (e.g., two-electrons-in-two-orbitals representation of diradicalas) represents the reality. In the idealized diradical case, only two frontier natural orbitals contribute to Eq. (5.8). Because the one-particle density matrix is a reduced quantity, which can be com- puted for any wave function (or for Kohn-Sham DFT and TD-DFT states), the analyses based on density matrices afford comparisons of bonding patterns computed by different methods and are also orbital-invariant. Here we apply this tool to analyze the electronic structure of several prototypical diradicals and triradicals. Polyradicals play important roles in fields as diverse as photochemistry, 16 atmo- spheric chemistry, 17 and molecular magnetism. 18 Depending on the type of frontier MOs, they can be described as all- or all-, , or spatially separated atom- centric. The degree of interaction and (nominal) bonding between unpaired electrons differs for each type and each molecular species. We consider the following dirad- icals: methylene (CH 2 , same-center diradical); ortho-, meta-, and para-benzyne (); 1-(2-dehydroisopropyl)-4-dehydrobenzyne (), wherein radical electrons are localized on the tertiary carbon of the propane substituent and the para position of the benzene 63 ring; binuclear copper complexes CUAQAC02 and CITLAT (spatially separated atom- centric). Model triradicals include 1,3,5- and 1,2,4-tridehydrobenzyne (all-), and 5- dehydro- and 2-dehydro-meta-xylylene (). Together, these molecules constitute a diverse benchmark set, representative of various bonding patterns one encounters in open-shell species. In this chapter we employ several variants of spin-flip (SF) methods. 19–23 The SF approach affords robust and accurate description of diradicals and triradicals within a black-box single-reference formalism. The performance of different variants of the SF method has been extensively benchmarked, focusing on the energy gaps between the electronic states and sometimes their structures. 21–31 The most insightful comparisons are based on photoelectron spectra, which provide information about the electronic en- ergies and the structures. 28, 29 These studies illustrated that the SF-CCSD method pro- vides reliable energy gaps, with errors close to 1 kcal/mol, as well as reliable photo- electron spectra. 28, 29 Subchemical accuracy can be achieved by including the effect of higher excitations perturbatively. 32 Within SF-DFT, the best performance is delivered by B5050LYP (in collinear formulation, 21 where a large fraction of exact exchange is needed), and by PBE50 (within non-collinear formulation). 22 However, no analysis of the underlying wave functions has been performed. One interesting question is how the level of correlation (or the functional, in SF-DFT methods) affects the effective num- ber of unpaired electrons. This question is addressed here, by means of the analysis of the density matrices of states obtained by the different post-Hartree-Fock and TDDFT methods. The structure of the chapter is as follows. In the next section, we describe theoretical methods and outline computational protocols. We then present our results, beginning 64 with a comparison of the equilibrium geometries computed by Kohn-Sham and SF- TDDFT. We usemeta-benzyne to illustrate the failure of standard Kohn-Sham DFT to describe singlet-state structures of strongly correlated systems. We proceed to discuss wave function properties—in particular, n u andn u;nl —using the example of H 2 along the dissociation coordinate. In Section 3.3.3, we analyze the character of EOM-SF- CCSD wave functions of methylene, the di- and tridehydrobenzynes, and the triradical xylylenes. We then compare these results with the analysis of SF-TDDFT and SF- ADC(2) wave functions. Included in the SF-TDDFT and SF-ADC(2) results section are the CUAQAC02 and CITLAT copper complexes. We conclude by summarizing relative performance of the different approaches. 3.2 Theoretical methods and computational details As illustrated in Fig. 3.1, diradical character results in multi-configurational wave func- tions. This multi-configurational character arises due to electronic near-degeneracies of the frontier molecular orbitals. Standard electronic structure methods, 33 which are based on the hierarchical improvements of a single-determinantal Hartree-Fock wave function, fail in situations where more than one Slater determinant has a large contribu- tion. The Kohn-Sham DFT also breaks down in this case. Such strongly correlated sys- tems are sometimes treated by multi-reference methods based on a multi-configurational SCF ansatz and separate description of static and dynamic correlation. Here we em- ploy an alternative approach, the SF method, 19, 34 which allows one to describe multi- configurational wave functions of the diradical and triradical types in a simple single- reference framework. 65 The SF approach is based on the observation that high-spin states, such as M s =1 triplet or M s =3/2 quartet, can be well described by a single determinant and that the low- spin states (singlets and doublets) are formally single excitations from the respective high-spin reference states. Thus, in the SF approach, a high-spin state is used as the reference state and the target manifold of states is generated by applying a linear spin- flipping excitation operator to the reference state: s;t Ms=0 = ^ R Ms=1 t Ms=1 ; (3.8) d;q Ms=1=2 = ^ R Ms=1 q Ms=3=2 : (3.9) For diradicals, a triplet reference (M s =1) is used, and a quartet reference (M s =3/2) is used for triradicals. Using different approaches to describe correlation in the reference state gives rise to different SF methods. For example, applying this strategy to a Kohn- Sham determinant leads to SF-TDDFT 21, 22 (in this case, the operator ^ R includes only single excitations that flip the spin of an electron). In wave-function methods, one can use an uncorrelated reference (Hartree-Fock), which gives rise to SF-CIS, 19, 35 or a corre- lated one, such as MP2 or CCSD. The accuracy of SF calculations can be systematically improved (up to the exact result) by increasing the level of correlation. In this chapter, we consider two wave function approaches: one based on coupled-cluster theory (EOM- SF-CCSD or SF-CCSD) 20, 24 and one based on the algebraic-diagrammatic construction (ADC) scheme for the polarization propagator 36 (SF-ADC). 23, 30 In EOM-SF-CCSD, the operator ^ R includes single and double excitations that flip the spin of an electron. In the SF-ADC(2) method, 23 both single and double excitations are included, but the doubly excited determinants are treated only in zeroth order of perturbation theory. The math- ematical structure of the 2 nd order correction is similar to the one of the SF-CIS(D 1 ) method, 37 however, in CIS(D 1 ) it is evaluated perturbatively only once using CIS wave 66 functions, whereas in ADC(2) it is included already within the iterative solution for the ADC vectors, similarly to the CC2 method. 38 For selected examples, we also present results for SF-TDDFT. 21, 22 As one can see, all configurations that appear in electronic states shown in Figure 3.1 are treated on an equal footing within the SF formalism. In diradicals, the target-state manifold comprises the singlets and the M s =0 component of the triplet state. Likewise, in triradicals the target-state manifold comprises the doublets and the low-spin (M s =1/2) component of the quartet state. Because of the balanced description of all target states, the energy gaps between the target SF states are more accurate than energy gaps be- tween the reference and the target states. SF methods can be employed to optimize the geometries of the target states using analytic gradients. 20–23 SF wave functions can be used to construct one-particle density matrices that can then be analyzed using natural orbitals and wave function analysis tools. 3.2.1 Computational Details We performed geometry optimizations of the high-spin triplet and quartet states by DFT/B5050LYP/cc-pVTZ. To obtain the structures of the lowest closed-shell singlet or doublet states, we used SF-TDDFT/B5050LYP/cc-pVTZ. For methylene, we used the FCI/TZ2P structures. 39 These structures are shown in Fig. 3.2. In calculations of the CUAQAC02 and CITLAT binuclear copper diradicals, we used X-ray crystal struc- tures from the Cambridge Crystal Structure Database. 40, 41 The perchlorate counterion from the crystal structure of CITLAT was omitted. These structures are shown in the appendices. 67 To assess the effect of the method on the structure, for selected systems, we per- formed additional geometry optimizations of the singlet and triplet (diradicals) or dou- blet and quartet (triradicals) Kohn-Sham references. Relevant Cartesian coordinates are provided in the appendices. For all benchmark systems, we present vertical energy gaps, computed as energy dif- ferences at the ground-state equilibrium geometry (optimized by SF-TDDFT). Energy gaps are defined as follows: E =E lowspin E highspin ; (3.10) i.e., negative E corresponds to the low-spin ground states (singlets or doublets). Nat- ural orbitals and Head-Gordon’s indices of different states are also computed at the respective equilibrium geometries. We used the cc-pVTZ basis in all calculations. For selected examples, we compare the following levels of theory: EOM-SF-CCSD, SF-ADC(2)-s, and SF-TDDFT. In SF-TDDFT we employed the following function- als: PBE50 (50% PBE 42 and 50% Hartree-Fock exchange with 100% PBE correlation), B5050LYP (50% Hartree-Fock + 8% Slater + 42% Becke 43 for exchange, with 19% VWN + 81% LYP 44 for correlation), 21 B97, 45 and LDA (Slater exchange with VWN correlation) functionals. We used the collinear formulation with B5050LYP 21 and non- collinear formulation 22, 46, 47 with all other functionals. We used thelibwfa module 12, 13 of Q-Chem to compute and visualize natural orbitals and Head-Gordon’s indices. We reportn u;nul , Eq. (5.8), and natural occupations of the frontier orbitals. Because the present implementation of the SF methods are not spin- adapted, the SF states show some (usually small) spin-contamination even if ROHF references are employed. Consequently, and frontier orbitals (and the respective 68 occupations) are slightly different. Below we report spin-average natural occupation, n i : n =jn +n j: (3.11) We also report the difference between the and natural occupations ( =jn n j), which provides an additional measure of spin-contamination. In all figures, we show only orbitals of the triplet states of diradicals, because the shapes of frontier natural orbitals for singlet and triplet states are indistinguishable for all systems considered in this study. We only show-orbitals, as the shapes of paired and natural orbitals are the same. For triradicals, we show orbitals for the doublet and quartet states. The Q-Chem electronic structure package was used for all calculations, 48, 49 and orbitals were rendered using Jmol. 3.3 Results and discussion 3.3.1 Equilibrium Geometries The extent of open-shell character has concrete structural implications, 3, 4, 50 e.g., in molecules with large diradical character the structures of triplet and singlet states are rather similar, in contrast to closed-shell systems. Consequently, the choice of electronic structure method is important for obtaining accurate structures. Figure 3.2 shows the optimized geometries for the two lowest spin-states of each of the di- and triradicals included in this study (see Section 3.2.1 for details). For D and H, we also show the structures computed by the standard Kohn-Sham DFT for the low-spin states (Fig. 3.12 in Appendix A). 69 1.392 1.416 1.075 1.074 126.8° 125.1° 116.6° 117.5° A 121.1° 126.9° 118.4° 110.8° 120.5° 122.3° 1.382 1.238 1.363 1.371 1.390 1.391 1.377 1.394 1.074 1.075 1.075 1.072 1.369 1.360 1.377 1.383 1.413 1.404 1.411 1.428 1.492 1.488 1.073 1.073 1.074 1.074 124.7° 125.4° 117.4° 116.9° 117.2° 117.4° 121.6° 121.6° 121.4° 121.2° 118.7° 119.1° D 1.368 1.379 1.075 1.071 121.2° 119.7° 115.2° 96.0° 123.7° 140.0° 118.1° 112.1° 1.438 1.357 1.372 1.349 1.073 1.069 1.365 1.374 1.384 1.245 1.367 1.376 1.372 1.371 1.362 1.377 1.396 1.396 1.076 1.073 1.074 1.074 1.077 1.071 121.9 ° 126.4 ° 120.3 ° 129.1 ° 116.2 ° 105.2 ° 125.9 ° 129.7 ° 116.6 ° 118.7 ° 119.2 ° 110.8 ° F C G H 133.3° 1.365 1.351 1.075 1.070 1.368 1.348 1.366 1.359 1.075 1.077 1.388 1.386 1.073 1.073 114.8° 94.9° 125.0° 139.6° 116.9° 116.0° 121.3° 113.8° B E I 1.390 1.383 1.385 1.391 1.427 1.425 1.379 1.376 1.075 1.074 1.075 1.075 1.072 1.071 127.9 ° 128.4 ° 114.3 ° 114.0 ° 121.8 ° 121.6 ° 120.9 ° 121.0 ° 122.2° 122.2° 1.406 1.408 1.388 1.385 1.428 1.433 1.357 1.354 1.075 1.075 1.074 1.074 1.073 1.073 123.0 ° 123.1 ° 117.7 ° 117.6 ° 127.2 ° 127.8 ° 117.2 ° 116.9 ° 120.9° 120.8° Figure 3.2: Structures of the four benzyne diradicals (A-D), methylene (E), and four triradicals (F-I) included in this study. Numbers in italics correspond to the optimized lowest high-spin state (M s = 1 triplets for A through E and M s = 3/2 quartets for F through I), while underlined numbers correspond to the optimized lowest energy singlet (diradicals) or doublet (triradicals) state computed by SF- TDDFT. Additional structural data for compounds D and H is presented in the appendices. Compounds will be referred to by letter designation in the remainder of this work. For species with well-separated radical centers, like p-benzyne and 1-isopropyl-4- benzyne, the SF optimized geometry of the singlet or doublet state closely resembles the optimized geometry of the respective high-spin state, indicative of weakly interact- ing unpaired electrons. For species with modest di- or triradical character, the singlet or doublet structures are markedly different from the high-spin geometries, due to the bonding interactions between the unpaired electrons. 50 70 The structures of triplet or quartet states can be accurately computed by using stan- dard Kohn-Sham DFT for high-spin states (these structures are very close to the struc- tures computed for the corresponding SF-DFT states). In cases of very weak diradi- cal/triradical character, the structures of the low-spin states can also be computed by regular Kohn-Sham DFT, however, in cases of relatively strong di-/triradical charac- ter, one needs to employ the SF-DFT approach, in order to correctly capture multi- configurational character of the underlying wave functions. The most notable example is meta-benzyne shown in Fig. 3.3, where the angle of separation between the radical carbons in the SF optimized singlet geometry is roughly 20 degrees less than in the structure optimized by regular closed-shell Kohn-Sham DFT. 1.987 Å 94.9° SF-TDDFT Singlet n u,nl = 0.26 Kohn-Sham DFT Singlet n u,nl = 0.03 1.518 Å 69.3° 2.305 Å 114.8° Kohn-Sham DFT Triplet n u,nl = 0.90 1 Figure 3.3: Optimized geometries of the lowest singlet state computed by regular Kohn-Sham DFT (left) and SF-TDDFT (center), and high-spin triplet state of meta- benzyne (B5050LYP/cc-pVTZ). Distance and angle of separation between radical sites are shown. Head-Gordon’s indices for the 3 structures are computed using SF-CCSD wave functions for the lowest singlet state. Using a method that fails to correctly describe open-shell character can lead to catas- trophically wrong structures. 4 One well studied example is meta-benzyne. 51 In this system, single-reference methods such as regular Kohn-Sham DFT and CCSD underes- timate the diradical character and yield (incorrect) bicyclic structures, whereas methods that do not include dynamic correlation (valence bond, CASSCF) exaggerate the dis- tance between the radical centers. The structure used in wave function analysis has a 71 strong effect on the bonding pattern. That is, using a wrong structure will produce an incorrect number of unpaired electrons, even if an accurate electronic structure method is used for the wave function analysis. We illustrate this point using meta-benzene. Figure 3.3 shows optimized structures for meta-benzyne. The rightmost structure is computed by B5050LYP for the high-spin triplet state in which the two electrons are completely unpaired. Because the distance between the two radical centers is large, the singlet-state wave function shows considerable diradical character as indicated by the relatively large value of n u;nl (0.9). The central structure is computed by SF- DFT/B5050LYP for the lowest singlet state. As one can see, the distance between the two radical centers is shorter by about 0.3 ˚ A, as compared to the triplet-state structure, due to a partial bond formed by the two electrons. At this geometry, the singlet-state wave function has moderate diradical character (n u;nl =0.3). The leftmost structure is computed using regular restricted Kohn-Sham DFT/B5050LYP. Because this approach is not capable of describing diradical character, the optimized structure is bicyclic, with a short distance between the two radical centers (this structure is similar to the CCSD structure reported by Crawford and co-workers 51 ). The singlet-state SF-CCSD wave function computed at this geometry shows nearly perfect closed-shell character (n u;nl =0.03). This example illustrates the importance of performing calculations at the nuclear geometry that corresponds to the correct electronic configuration of a molecule (i.e., in the case of meta-benzyne, of a singlet with moderate diradical character). 3.3.2 Head-Gordon’s Indices Along the H 2 Dissociation Curve Before proceeding to wave-function analysis in diradicals and triradicals, let us con- sider a model example, for which CCSD is exact, dissociation curve of the dihydrogen molecule. 72 0 0.5 1 1.5 2 0.5 1.5 2.5 3.5 4.5 # of unpaired electrons CCSDnu CCSDnu,nl CCSDnu - nu,nl DFTnu DFTnunl DFTnu-nunl n u (CCSD) |n u –n u,nl | (CCSD) n u,nl (B5050LYP) n u,nl (CCSD) n u (B5050LYP) |n u –n u,nl | (B5050LYP) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 4.5 4.6 4.7 4.8 4.9 5.0 5.1 5.2 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 ∆n u,nl !E (eV) R (Å) Series1 Series2 ∆E ∆E ∆n u,nl Figure 3.4: n u , n u;nl , andjn u - n u;nl j (top), and E B5050LYP - E CCSD (E) and jn B5050LYP u;nl -n CCSD u;nl j (n, bottom) computed for the lowest singlet state of H 2 along the bond-stretching coordinate using EOM-SF-CCSD and SF-TDDFT/B5050LYP with the cc-pVTZ basis set. Figure 3.4 showsn u ,n u;nl ,jn u n u;nl j, and the difference in total B5050LYP and CCSD state energies as a function of internuclear distancesR for the lowest singlet state of H 2 . Tabulated raw data is provided in the the appendices. As expected, around equilibrium the number of unpaired electrons is small. As the internuclear distanceR increases, diradical character increases, reaching 2 at the disso- ciation limit. At the equilibrium and at the dissociation limit, SF-TDDFT agrees well with SF-CCSD (exact result). However, at intermediate distances, SF-TDDFT underes- timates the number of unpaired electrons. Compare, for example, the two blue curves, which show the respectiven u;nl s. At 2 ˚ A, the exact wave function has 1 unpaired elec- tron, whereas the SF-TDDFT wave function has only 0.25. Only 0.5 ˚ A further SF-DFT 73 develops open-shell character yielding n u;nl =1. This lag is observed for n u and n u;nl . Strict SF-ADC(2) does not give exact excited states due to the zeroth-order treatment of the doubly excited determinants, but improving the description of the doubly excited configurations to first order, as in SF-ADC(2)-x and SF-ADC(3), yields exact states for H 2 , as SF-CCSD (note that only the energy differences, such as singlet-triplet gaps are exact, but not the total state energies). Regardless of the level of theory, at the dissociation limit the difference betweenn u andn u;nl is small. However, at small and intermediate distances we observe noticeable discrepancy betweenn u andn u;nl . Compared ton u;nl , around the equilibriumn u over- estimates diradical character, and at longer distances it underestimates it. As expected, maximum deviation occurs around the natural orbital occupation numbers of 0.25 and 1.75 electrons, where the quadratic nature of the expression forn u;nl suppresses and en- hances radical character, respectively. 9 The point along the dissociation curve where the n u andn u;nl curves intersect atn u =n u;nl 1 depends on the level of theory. This large discrepancy between the two quantities is only observed for the singlet state —n u and n u;nl equal exactly 2 for the high-spin reference and low-spin triplet states at every point along the dissociation curve from 0.74-5.00 ˚ A (raw data is provided in the appendices). The lower panel of Figure 3.4 shows the difference between the SF-CCSD and SF- DFT total energies and the respective n u;nl . At large internuclear separation (> 3 ˚ A), the two methods yield identical n u;nl and the respective potential energy curves are nearly parallel. However, at shorter distances (less than 3 ˚ A), where we observe large discrepancies between the number of unpaired electrons computed by SF-CCSD and SF-DFT, we also observe large non-parallelity errors in the SF-DFT potential energy curve. This example illustrates that the errors in SF-DFT energies originate from errors in underlying densities. 74 3.3.3 EOM-SF-CCSD energies and wave function character in di- radicals and triradicals Table 5.16 shows energy separations, n u;nl , andhS 2 i for the lowest states of di- and triradical benzynes and methylene. n u andjn n j for each state are provided in the the appendices. All states considered suffer very little from spin-contamination. For high-spin states, n u;nl is very close to the ideal values of 2 unpaired electrons for triplet states and 3 unpaired electrons for quartet states. In contrast, n u;nl for singlets and doublets depends very much on the nature of the di- or triradical ground state. For species with a singlet ground state, n u;nl of the ground state ranges from closed-shell values close to zero (for singlets with modest radical character) to values close to two (for open-shell singlets and strong diradical character). The same is true for triradicals with doublet ground states; namely, closed-shell doublets like those observed in F and G have n u;nl values close to 1, while open-shell doublets such as the ground state of I have n u;nl values close to 3. We observe that radical character (n u;nl ) increases and E becomes more positive as the distance between the radical carbons increases from A through D. Natural orbitals of the lowest singlet/doublet and triplet/quartet states of methylene and the di- and triradical benzynes are shown in Figure 3.5. With the exception of methylene, D, H, and I, all di- and triradical frontier orbitals are of the type, consis- tent with the molecular orbital patterns reported by Krylov and Cristian. 50 The natural orbitals in D, H, and I are of type. The values of n show spin-averaged occupancy of each spatial orbital. The trends in orbital occupancy are consistent with n u;nl : increased radical character is ascribed to ground states with natural orbitals n-values close to 1. Values in parentheses (n) 75 indicate spin imbalance in orbital occupancy arising due to spin-incompleteness of the underlying wave function. For diradicals, n-values are close to the ideal value of zero. Doublet tridehydrobenzynes also exhibit ideal n-values, with only one predominantly singly occupied natural orbital hosting the odd electron that gives rise to the doublet. Quartet states of the tridehydrobenzynes, and open-shell doublet and quartet states of the xylylene triradicals, feature natural orbitals that are predominantly singly occupied ( n close to 1), but suffer from some spin-imbalance. This imbalance appears to be minor and does not strongly impact other properties of these states, like thehS 2 i values or the sign and the magnitude of E. For all systems, we observe that the difference betweenn u;nl andn u is much smaller for the low-spin triplet states than for the singlet states. This is partially because in the limiting case of 2 unpaired electrons the difference between the two indices is ex- pected to vanish. For example, the two indices give identical results for the singlet state of dihydrogen at the dissociation limit and identical results for the triplet states at all distances. Yet, for many-electron electron examples, even for triplet statesn u assumes values that are slightly larger than 2. This can be attributed to the fact that n u;nl sup- presses contributions from the dynamic correlation. Thus, the second reason for much smaller discrepancy between the two indices for the triplet states relative to the singlets can be attributed to the smaller dynamic correlation, which is characteristic of the triplet wave functions. 19 Interestingly, the high-spin reference states show larger difference betweenn u;nl andn u than the respective low-spin components of the same multiplicity. 76 Table 3.1: EOM-SF-CCSD energy gaps (eV) and wave function properties of low- est two states (singlets and M s = 0 triplets for diradicals, M s = 1/2 doublets and quartets for triradicals). Molecule E n u;nl hS 2 i Singlet/Doublet Triplet/Quartet Singlet/Doublet Triplet/Quartet CH 2 0.94 0.25 2.00 0.00 2.00 A -2.46 0.16 2.00 0.00 2.00 B -1.90 0.26 2.00 0.01 2.01 C -0.24 1.45 2.00 0.01 2.01 D 0.24 2.00 2.01 0.01 2.04 F -2.21 1.28 3.06 0.88 3.99 G -2.00 1.16 3.02 0.82 3.83 H 0.41 3.01 3.02 0.79 3.83 I -0.14 3.01 3.01 0.78 3.80 3.3.4 Molecular Magnets: Natural Orbitals versus Molecular Or- bitals Two binuclear copper diradicals, CUAQAC02 and CITLAT (Figure S1), are included in the SF-TDDFT wave function analysis benchmark study. Their experimental exchange- coupling constants (equal to E within the Heisenberg-Dirac-van-Vleck model, which assumes weak interactions between the unpaired electrons) are -286 and 113 cm 1 , re- spectively. 40, 41 This example illustrates that a compact representation of the wave func- tion in the basis of natural orbitals provides a more facile interpretation of the underlying electronic structure than the canonical MOs. Figure 5.2 shows singly occupied natural and canonical MOs for the triplet states of the two binuclear copper diradicals alongside their spin-difference densities. The highest canonical MOs in CUAQAC02 and CITLAT (which have 202 and 278 electrons, respectively) appear to be delocalized. Furthermore, despite low spin- contamination of the Kohn-Sham triplet reference, and MOs cannot be easily 77 A D CH 2 F Doublet Quartet G Doublet Quartet H Doublet Quartet Doublet Quartet I ! " s = 0.14 (0.01) ! " t = 0.99 (0.02) ! " s = 1.85 (0.02) ! " t = 1.00 (0.01) ! " s = 0.19 (0.01) ! " t = 0.99 (0.03) ! " s = 1.80 (0.02) ! " t = 1.00 (0.02) ! " s = 0.60 (0.04) ! " t = 0.98 (0.1) ! " s = 1.38 (0.02) ! " t = 1.00 (0.03) ! " s = 0.99 (0.04) ! " t = 1.00 (0.11) ! " s = 1.00 (0.09) ! " t = 1.00 (0.04) ! " s = 1.79 (0.01) ! " t = 0.99 (0.00) ! " s = 0.19 (0.00) ! " t = 0.99 (0.02) ! " = 0.99 (0.99) ! " = 1.85 (0.02) ! $= 0.99 (0.46) ! $= 0.98 (0.38) ! " = 0.99 (0.21) ! " =1.00 (0.99) ! " = 1.80 (0.02) ! " = 0.99 (0.35) ! " = 0.99 (0.34) ! " = 0.99 (0.38) ! $= 1.00 (0.49) ! $= 1.00 (0.80) ! $= 0.99 (0.25) ! " = 1.00 (0.35) ! " = 1.00 (0.40) ! " = 1.00 (0.33) ! " = 1.00 (0.74) ! $= 1.00 (0.37) ! $= 1.00 (0.68) ! $= 1.00 (0.40) ! $= 1.00 (0.36) ! $= 1.00 (0.30) B C Figure 3.5: EOMSF-SF-CCSD/cc-pVTZ natural orbitals of lowest singlet/doublet and triplet/quartet (low-spin) states of A-I. n =jn +n j, with n =jn n j provided in parentheses. n s and n t correspond to n values obtained from the occu- pancies of the singlet and triplet natural orbitals, respectively. matched. Thus, by considering canonical MOs only, it is difficult to ascribe overall orbital character or localization to the unpaired electrons that give rise to the diradical states. In contrast, the frontier natural orbitals obtained from the triplet density matrix have expected shapes that can be described asd xy ord yz orbitals localized on the two copper centers. The spin-density differences (shown in Fig. 5.2) are consistent with the shapes of frontier natural orbitals. 78 !2 !1 !2 !1 CUAQAC02 CITLAT SOMOs SONOs SOMOs SONOs !2 !1 !2 !1 Spin Density Spin Density Figure 3.6: Spin-difference densities, unrestricted singly occupied (SO) molecular and natural frontier orbitals of high-spin triplet states of CUAQAC02 (left) and CITLAT (right) computed by B5050LYP/cc-pVTZ. 3.3.5 Comparison between EOM-SF-CCSD, SF-TDDFT, and SF- ADC(2)-s wave functions Tables 3.2, 3.5, and 3.4 show DFT and ADC(2) energy gaps and state properties for the low-lying spin-states of di- and triradical benzynes, methylene, CUAQAC02, and CIT- LAT.n u andjn n j for triradicals are provided in the appendices. Figures 3.8, 3.9, and 3.10 show natural orbitals of the relevant spin-states at each level of theory. 79 Diradicals with states affected by spin-contamination typically have largejn n j values. There is agreement in natural orbital character—and sometimes energetics— among DFT methods, and reasonable agreement with EOM-SF-CCSD for organic rad- icals. LDA and B97 fail for binuclear copper radicals with respect to energies, overesti- mating the degree of closed-shell character of the low-spin states relative to PBE50 and B5050LYP. The two non-hybrid functionals appear sufficient for describing the under- lying wave function character of di- and triradicals with respect to the shapes of natural orbital, with the notable exception of CUAQAC02 and the B97 functional, where the d yz orbitals predicted by EOM-CCSD, PBE50, B5050LYP, and LDA are not the fron- tier natural orbitals (B97 frontier natural orbitals of CUAQAC02 and CITLAT shown in Figure 3.14 in Appendix E). LDA and B97 appear to systematically underestimate diradical character of singlet states as reflected by lowern u;nl and n s values, relative to PBE50 and B5050LYP. SF-ADC(2) has been used to compute the energy splittings and Head-Gordon’s in- dices of CH 2 and A-D (Tables 3.2 and 3.4). For these molecules, the singlet-triplet energy splittings (E) computed with SF-ADC(2) agree very favorably with the ones obtained at the EOM-SF-CCSD level of theory. The largest deviation is 0.08 eV found for CH 2 . Inspecting Head-Gordon’s index n u;nl for these molecules (Table 3.2), we note a good overall agreement with the EOM-SF-CCSD values. The SF-ADC(2) values are consistently slightly larger by about 0.15. Further detailed investigations of the quality of the wave functions obtained with SF-ADC approaches, including higher orders of perturbation theory, will be the topic of future work. Natural orbitals of the binuclear copper complexes are of d xy and d yz type and well-localized on copper centers. The unpaired electrons of CITLAT (the copper di- radical with a positive E) exhibit some through-space anti-bonding interaction with 80 A D CH 2 ! " s = 0.14 (0.01) ! " t = 0.99 (0.03) ! " s = 1.85 (0.00) ! " t = 0.99 (0.03) ! " s = 0.18 (0.00) ! " t = 0.99 (0.02) ! " s = 1.81 (0.00) ! " t = 0.99 (0.00) ! " s = 0.58 (0.01) ! " t = 0.99 (0.04) ! " s = 1.40 (0.01) ! " t = 0.99 (0.05) ! " s = 0.99 (0.51) ! " t = 0.99 (0.50) ! " s = 0.99 (0.52) ! " t = 0.99 (0.53) ! " s = 1.77 (0.00) ! " t = 0.99 (0.03) ! " s = 0.21 (0.00) ! " t = 0.99 (0.04) B C Figure 3.7: ADC(2)s/cc-pVTZ natural orbitals of lowest singlet and triplet (low- spin) states of CH 2 and A-D. n = jn +n j, with n = jn n j provided in parentheses. n s and n t correspond to n values obtained from the occupancies of the singlet and triplet natural orbitals, respectively. p-orbitals of neighboring oxygen atoms in the bonding plane. This interaction is absent in CUAQAC02 (the copper diradical with a negative E), in which singly occupied nat- ural orbitals can be described as thed yz orbitals that give rise to- type interactions and bond orders greater than three between transition metal nuclei in more strongly interacting bimetallic complexes. 52 3.3.6 Adiabatic Singlet-Triplet and Doublet-Quartet Gaps All energy differences between low-lying spin states reported here are vertical energy gaps, computed at the equilibrium geometries of the ground state by optimizing the lowest energy SF-TDDFT/cc-pVTZ state. In order to assess the absolute accuracy of the methods by comparison against experimental values, one needs to consider adia- batic gaps, as was done in previous benchmark studies. 22, 24, 25, 53 As a guidance for readers, here we provide a compilation of adiabatic energy gaps for the benchmark 81 Table 3.2: SF-TDDFT and ADC(2)-s energy splittings (eV) and wave function properties of lowest singlet and M s = 0 triplet states of diradicals. Molecule Method E n u;nl hS 2 i Singlet Triplet Singlet Triplet CH 2 PBE50 1.03 0.30 2.00 0.01 2.02 B5050LYP 0.30 0.08 2.00 0.01 2.01 LDA 0.93 0.33 2.00 0.01 2.01 B97 0.23 0.04 2.00 0.01 2.00 ADC(2) 1.02 0.33 2.01 – – A PBE50 -2.38 0.09 2.00 0.03 2.02 B5050LYP -2.66 0.04 2.00 0.02 2.02 LDA -2.86 0.01 2.00 0.01 2.02 B97 -2.84 0.01 2.00 0.02 2.01 ADC(2) -2.46 0.32 2.17 – – B PBE50 -1.85 0.19 2.00 0.07 2.11 B5050LYP -2.04 0.12 2.00 0.05 2.07 LDA -2.23 0.01 2.00 0.02 2.03 B97 -2.22 0.03 2.00 0.03 2.05 ADC(2) -1.97 0.41 2.17 – – C PBE50 -0.19 1.47 2.00 0.03 2.03 B5050LYP -0.23 1.29 2.00 0.02 2.02 LDA -0.52 0.30 2.00 0.01 2.02 B97 -0.40 0.63 2.00 0.02 2.02 ADC(2) -0.24 1.58 2.18 – – D PBE50 0.24 2.02 2.02 0.33 2.10 B5050LYP 0.16 2.01 2.02 0.23 2.10 LDA 0.20 2.00 2.00 0.02 2.03 B97 0.44 2.00 2.00 0.66 1.47 ADC(2) 0.20 2.24 2.24 – – compounds considered here (with the exception of D, CUAQAC02, and CITLAT) re- ported elsewhere 22, 24, 25, 53 and are summarized in Tables 3.6 and 3.7. Note that in the reported experimental values zero-point energies are subtracted. As one can see, EOM- SF-CCSD(dT) is within 1 kcal/mol from the experiment. EOM-SF-CCSD delivers con- sistent performance and is very close to EOM-SF-CCSD(dT). This comparison justifies using EOM-SF-CCSD vertical gaps as the benchmark for SF-DFT and ADC(2). 82 Table 3.3: SF-TDDFT energy splittings (cm 1 ) and wave function properties of lowest singlet and M s = 0 triplet states of binuclear copper diradicals. Molecule Method E n u;nl hS 2 i Singlet Triplet Singlet Triplet CUAQAC02 PBE50 -123 1.97 2.00 0.01 2.01 B5050LYP -148 1.96 2.00 0.01 2.01 LDA -1420 0.47 2.00 0.01 2.01 B97 71 2.59 2.69 1.07 1.77 CITLAT PBE50 100 2.00 2.00 0.01 2.02 B5050LYP 77 2.00 2.00 0.01 2.01 LDA 411 1.96 2.00 0.01 2.01 B97 228 1.98 2.00 0.01 2.01 83 Table 3.4: n u andn u;nl at the SF-TDDFT and ADC(2)-s/cc-pVTZ levels of theory. Values are provided for the lowest singlet and M s = 0 triplet states of each diradical benchmark system. Results for four density functionals are compared. Molecule Method Triplet Singlet n u n u;nl jn u n u;nl j n u n u;nl jn u n u;nl j CH 2 PBE50 2.014 2.000 0.014 0.447 0.303 0.144 B5050LYP 2.010 2.000 0.010 0.217 0.079 0.138 LDA 2.008 2.000 0.008 0.469 0.332 0.137 B97 2.013 2.000 0.013 0.166 0.045 0.121 ADC(2) 2.114 2.009 0.105 0.578 0.326 0.252 A PBE50 2.016 2.000 0.016 0.246 0.091 0.155 B5050LYP 2.012 2.000 0.012 0.171 0.044 0.127 LDA 2.010 2.000 0.010 0.070 0.007 0.063 B97 2.014 2.000 0.014 0.095 0.013 0.082 ADC(2) 3.133 2.171 0.962 1.458 0.323 1.135 B PBE50 2.070 2.003 0.067 0.399 0.192 0.207 B5050LYP 2.045 2.001 0.044 0.302 0.116 0.186 LDA 2.020 2.000 0.020 0.092 0.011 0.081 B97 2.033 2.001 0.032 0.154 0.030 0.124 ADC(2) 3.146 2.174 0.972 1.546 0.406 1.140 C PBE50 2.019 2.000 0.019 1.272 1.470 0.198 B5050LYP 2.014 2.000 0.014 1.129 1.286 0.157 LDA 2.010 2.000 0.010 0.450 0.304 0.146 B97 2.014 2.000 0.014 0.693 0.627 0.066 ADC(2) 3.159 2.179 0.980 2.377 1.577 0.800 D PBE50 2.161 2.019 0.142 2.164 2.019 0.145 B5050LYP 2.142 2.015 0.127 2.118 2.010 0.108 LDA 2.019 2.000 0.019 2.018 2.000 0.018 B97 2.055 2.002 0.053 2.047 2.002 0.045 ADC(2) 3.796 2.244 1.552 3.785 2.237 1.548 CUAQAC02 PBE50 2.010 2.000 0.010 1.845 1.973 0.128 B5050LYP 2.009 2.000 0.009 1.814 1.961 0.147 LDA 2.004 2.000 0.004 0.567 0.466 0.101 B97 2.815 2.685 0.130 2.754 2.589 0.165 CITLAT PBE50 2.011 2.000 0.011 1.964 1.998 0.034 B5050LYP 2.010 2.000 0.010 1.951 1.996 0.045 LDA 2.006 2.000 0.006 1.804 1.959 0.155 B97 2.009 2.000 0.009 1.861 1.978 0.117 84 CH 2 A B C D PBE50 B5050LYP LDA B97 ! " s = 0.11 (0.00) ! " t = 1.00 (0.01) ! " s = 1.89 (0.00) ! " t = 1.00 (0.01) ! " s = 0.07 (0.00) ! " t = 1.00 (0.02) ! " s = 1.93 (0.00) ! " t = 1.00 (0.01) ! " s = 0.03 (0.00) ! " t = 1.00 (0.00) ! " s = 1.97 (0.00) ! " t = 1.00 (0.00) ! " s = 0.04 (0.00) ! " t = 1.00 (0.09) ! " s = 1.96 (0.00) ! " t = 1.00 (0.08) ! " s = 0.04 (0.00) ! " t = 1.00 (0.00) ! " s = 1.96 (0.00) ! " t = 1.00 (0.01) ! " s = 1.94 (0.00) ! " t = 1.00 (0.03) ! " s = 0.06 (0.00) ! " t = 1.00 (0.02) ! " s = 0.17 (0.00) ! " t = 1.00 (0.00) ! " s = 1.83 (0.00) ! " t = 1.00 (0.01) ! " s = 0.13 (0.00) ! " t = 1.00 (0.02) ! " s = 1.87 (0.00) ! " t = 1.00 (0.03) ! " s = 0.22 (0.00) ! " t = 1.00 (0.00) ! " s = 1.78 (0.00) ! " t = 1.00 (0.01) ! " s = 0.33 (0.00) ! " t = 1.00 (0.05) ! " s = 1.66 (0.00) ! " t = 1.00 (0.05) ! " s = 0.62 (0.01) ! " t = 1.00 (0.00) ! " s = 1.38 (0.01) ! " t = 1.00 (0.01) ! " s = 0.55 (0.00) ! " t = 1.00 (0.03) ! " s = 1.44 (0.00) ! " t = 1.00 (0.03) ! " s = 1.00 (0.02) ! " t = 1.00 (0.00) ! " s = 1.00 (0.01) ! " t = 1.00 (0.01) ! " s = 1.00 (0.90) ! " t = 1.00 (0.92) ! " s = 1.00 (0.91) ! " t = 1.00 (0.90) ! " s = 1.00 (0.52) ! " t = 1.00 (0.49) ! " s = 1.00 (0.49) ! " t = 1.00 (0.53) ! " s = 1.00 (0.46) ! " t = 1.00 (0.48) ! " s = 1.00 (0.47) ! " t = 1.00 (0.44) ! " s = 1.77 (0.00) ! " t = 1.00 (0.01) ! " s = 0.23 (0.00) ! " t = 1.00 (0.01) ! " s = 1.92 (0.00) ! " t = 1.00 (0.17) ! " s = 0.08 (0.00) ! " t = 1.00 (0.17) ! " s = 1.78 (0.00) ! " t = 1.00 (0.02) ! " s = 0.22 (0.00) ! " t = 1.00 (0.02) ! " s = 1.90 (0.00) ! " t = 1.00 (0.03) ! " s = 0.10 (0.00) ! " t = 1.00 (0.04) Figure 3.8: Frontier natural orbitals of lowest singlet and M s = 0 triplet states of A-E with four DFT functionals. n =jn +n j, with n =jn n j provided in parentheses. n s and n t correspond to n values obtained from the occupancies of the singlet and triplet natural orbitals, respectively. 85 PBE50 B5050LYP LDA CUAQAC02 CITLAT ! " s = 0.92 (0.00) ! " t = 1.00 (0.00) ! " s = 1.08 (0.00) ! " t = 1.00 (0.00) ! " s = 0.90 (0.00) ! " t = 1.00 (0.01) ! " s = 1.10 (0.00) ! " t = 1.00 (0.01) ! " s = 0.28 (0.00) ! " t = 1.00 (0.00) ! " s = 1.72 (0.00) ! " t = 1.00 (0.00) ! " s = 0.98 (0.00) ! " t = 1.00 (0.00) ! " s = 1.02 (0.00) ! " t = 1.00 (0.00) ! " s = 0.97 (0.00) ! " t = 1.00 (0.00) ! " s = 1.03 (0.00) ! " t = 1.00 (0.01) ! " s = 0.90 (0.00) ! " t = 1.00 (0.00) ! " s = 1.10 (0.00) ! " t = 1.00 (0.00) Figure 3.9: Natural orbitals of lowest singlet and M s = 0 triplet states of CUAQAC02 and CITLAT with the PBE50, B5050LYP-collinear, and LDA DFT functionals. n =jn +n j, with n =jn n j provided in parentheses. n s and n t correspond to n values obtained from the occupancies of the singlet and triplet natural orbitals, respectively. 86 Table 3.5: SF-TDDFT/cc-pVTZ energy splittings (eV) and wave function proper- ties of lowest M s = 1/2 doublet and quartet states of triradicals. Results for four density functionals are shown. Molecule Method E n u;nl hS 2 i Doublet Quartet Doublet Quartet F PBE50 - 2.18 1.17 3.01 0.87 3.96 B5050LYP -2.44 1.09 3.00 0.84 3.88 LDA -2.80 1.01 3.00 0.78 3.80 B97 -2.83 1.02 3.00 0.80 3.81 G PBE50 -2.64 1.09 3.00 0.79 3.80 B5050LYP -2.82 1.06 2.01 0.79 3.79 LDA -3.16 1.01 3.00 0.76 3.75 B97 -3.16 1.01 3.00 0.78 3.77 H PBE50 0.39 3.06 3.05 1.42 3.81 B5050LYP 0.23 3.04 3.04 0.99 4.10 LDA 0.28 3.00 3.00 0.79 3.80 B97 0.46 3.01 3.01 1.12 3.64 I PBE50 -0.38 3.01 3.04 1.79 3.21 B5050LYP -0.14 3.01 3.03 1.77 3.16 LDA 0.03 3.00 3.00 1.25 3.29 B97 0.19 3.00 3.00 1.74 2.91 Table 3.6: Adiabatic energy gaps (eV) between the lowest singlet and triplet states of methylene (CH 2 ) and A–C. Method CH 2 A B C EOM-SF-CCSD(dT) 0.420 a -1.619 a -0.892 a -0.172 a EOM-SF-CCSD 0.447 a -1.578 a -0.782 a -0.147 a SF-ADC(3) 0.420 b -1.525 b -1.555 b -0.123 b PBE50 0.478 a -1.591 a -0.776 a -0.116 a B5050LYP -0.249 a -1.883 a -0.984 a -0.164 a LDA 0.511 a -2.058 a -1.259 a -0.454 a B97 -0.394 a -1.968 a -1.216 a -0.343 a Experiment 0.390 b -1.628 0.013 b -0.911 0.014 b -0.165 0.016 b a From Ref. 22. b From Ref. 24. 87 F G H I Doublet Quartet Doublet Quartet Doublet Quartet Doublet Quartet PBE50 B5050LYP LDA B97 ! " =1.00 (0.33) ! " = 1.00 (0.34) ! " = 1.00 (0.35) ! " = 1.84 (0.00) ! " = 1.00 (1.00) ! " = 1.00 (1.00) ! " = 1.88 (0.00) ! " = 1.00 (0.37) ! " = 1.00 (0.32) ! " = 1.00 (0.32) ! " = 1.00 (1.00) ! " = 1.97 (0.00) ! " = 1.00 (0.34) ! " = 1.00 (0.33) ! " = 1.00 (0.34) ! " = 1.95 (0.00) ! " = 1.00 (1.00) ! " = 1.00 (0.45) ! " = 1.00 (0.22) ! " = 1.00 (0.35) ! " = 1.89 (0.01) ! " = 0.99 (0.99) ! " = 1.00 (0.37) ! " = 1.00 (0.33) ! " = 1.00 (0.30) ! " = 1.00 (0.99) ! " = 1.91 (0.01) ! " = 1.00 (0.36) ! " = 1.00 (0.34) ! " = 1.00 (0.30) ! " = 1.97 (0.00) ! " = 1.00 (1.00) ! " = 1.00 (0.24) ! " = 1.00 (0.33) ! " = 1.00 (0.44) ! " = 1.96 (0.00) ! " = 1.00 (0.99) ! " = 1.00 (0.39) ! " = 1.00 (0.37) ! " = 1.00 (0.24) ! " = 1.00 (0.75) ! " = 1.00 (0.59) ! " = 1.00 (0.29) ! " = 1.00 (0.52) ! " = 1.00 (0.37) ! " = 1.00 (0.17) ! " = 1.00 (0.43) ! " = 1.00 (0.42) ! " = 1.00 (0.19) ! " = 1.00 (0.73) ! " = 1.00 (0.44) ! " = 1.00 (0.15) ! " = 1.00 (0.36) ! " = 1.00 (0.35) ! " = 1.00 (0.30) ! " = 1.00 (0.11) ! " = 1.00 (0.07) ! " = 1.00 (0.97) ! " = 1.00 (0.81) ! " = 1.00 (0.25) ! " = 1.00 (0.02) ! " = 1.00 (0.64) ! " = 1.00 (0.60) ! " = 1.00 (0.97) ! " = 1.00 (1.00) ! " = 1.00 (0.05) ! " = 1.00 (0.04) ! " = 1.00 (0.99) ! " = 2.00 (0.00) ! " = 2.00 (0.00) ! " = 1.00 (0.08) ! " = 1.00 (0.01) ! " = 1.00 (1.00) ! " = 1.00 (0.99) ! " = 2.00 (0.00) ! " = 2.00 (0.00) ! " = 1.00 (0.75) ! " = 1.00 (0.71) ! " = 1.00 (0.46) ! " = 1.00 (0.47) ! " = 1.00 (0.30) ! " = 1.00 (0.24) ! " = 1.00 (1.00) ! " = 1.00 (0.08) ! " = 1.00 (0.05) ! " = 1.00 (0.99) ! " = 2.00 (0.00) ! " = 2.00 (0.00) Figure 3.10: Natural orbitals of lowest doublet and M s = 1/2 quartet states of F-I with four density functionals. n =jn +n j, with n =jn n j provided in parentheses. 88 Table 3.7: Adiabatic energy gaps (eV) between the lowest doublet and quartet states of F–I. Method F G H I CCSD(dT) – -1.375 b – – EOM-SF-CCSD -1.786 a -1.193 b 0.460 c -0.182 c B5050LYP 1.515 a 2.084 a – – a From Ref. 50. b From Ref. 25. c From Ref. 53. 89 3.4 Conclusion Analysis of the natural orbitals derived from the one-particle density matrices provides insight into the extent and type of radical character for a variety of di- and triradical species. This analysis affords quantitative comparisons of electronic properties beyond energy differences computed by different methods. In particular, using these tools one can compare the performance of wave function and DFT methods. In agreement with earlier benchmark studies, we observe good agreement between EOM-SF-CCSD and SF-DFT (when using recommended functionals). The comparison of natural orbitals and their occupations computed for the lowest singlet and triplet and doublet and quartet states indicates good agreement between EOM-SF-CCSD and TDDFT in most cases. We observe very good agreement among DFT functionals with regard to the character of frontier natural orbitals, however, the respective occupations (and, consequently, the effective number of unpaired electrons) vary. SF-ADC(2) results agree favorably with SF-EOM-CCSD both with respect to the relative energies of singlet and triplet states as well as with respect to the corresponding wave function character of open-shell singlets, indicating its potential as benchmark method for larger molecular systems for which SF-EOM-CCSD is no longer feasible. Our study represents the first investigation of di- and triradicals focusing on under- standing state characters, rather than energies alone. For systems like the large copper- containing diradicals considered here, the canonical Kohn-Sham or Hartree-Fock or- bitals fail to represent the correct bonding pattern even for high-spin states. In contrast, natural frontier orbitals, their occupations, and Head-Gordon’s index allow one to obtain a clear picture of the underlying electronic structure. 90 3.5 Appendix A: Equilibrium structures All relevant Cartesian geometries are provided at the end of this document. Fig. 3.11 shows structures of the two binuclear copper diradicals included in this study. Fig. 3.12 presents extended structural data for compounds D and H. CUAQAC02 CITLAT Figure 3.11: Structures 40, 41 of the two binuclear copper diradicals included the SF-TDDFT benchmark set. 1.426 1.369 1.363 1.348 1.377 1.378 1.440 1.413 1.407 1.360 1.411 1.416 1.490 1.492 1.485 1.073 1.073 1.073 1.078 1.074 1.074 113.4° 124.7° 125.0° 122.8° 117.4° 117.2° 115.2° 117.2° 117.3° 121.3° 121.6° 121.6° 122.5° 121.4° 121.4° 112.1° 118.7° 119.0° D H 1.455 1.390 1.387 1.356 1.385 1.384 1.431 1.427 1.426 1.377 1.379 1.376 1.075 1.075 1.075 1.075 1.075 1.075 1.072 1.072 1.071 113.5° 127.9 ° 128.1 ° 122.5° 114.3 ° 114.2 ° 121.0° 121.8 ° 121.8 ° 120.2° 120.9 ° 120.9 ° 122.3° 122.2° 122.2° Figure 3.12: Expanded structural data for D and H. Numbers in plain font corre- spond to the singlet (for D) or doublet (for H) state optimized by Kohn-Sham DFT. Italics denotes the optimized lowest high-spin state (triplet for D and quartet for H) optimized by Kohn-Sham DFT. Underlined numbers correspond to the optimized lowest SF-TDDFT state (triplet for D and quartet for H). 91 3.6 Appendix B: Wave function analysis in meta- benzyne Singlet <S 2 > = 0.01; 0.01; 0.01 n u,nl = 0.03; 0.26; 0.90 Triplet <S 2 > = 2.01; 2.01; 2.02 n u, nl = 2.01; 2.00; 2.00 3.92 ; 1.90 ;0.53 eV Kohn-Sham DFT Singlet SF-TDDFT Singlet ! 1 n occ = 0.4880 " 1 n occ = 0.5104 " 2 n occ = 0.5090 ! 2 n occ = 0.4839 Kohn-Sham DFT Triplet ! 1 n occ = 0.5269 " 1 n occ = 0.4705 " 2 n occ = 0.5486 ! 2 n occ = 0.4381 ! 1 n occ = 0.4874 " 1 n occ = 0.5107 " 2 n occ = 0.5107 ! 2 n occ = 0.4842 ! 1 n occ = 0.9576 " 1 n occ = 0.9829 " 2 n occ = 0.0333 ! 2 n occ = 0.0127 ! 1 n occ = 0.8907 " 1 n occ = 0.9071 " 2 n occ = 0.1019 ! 2 n occ = 0.0907 ! 1 n occ = 0.7766 " 1 n occ = 0.7947 " 2 n occ = 0.2175 ! 2 n occ = 0.2026 Figure 3.13: EOM-SF-CCSD/cc-pVTZ frontier natural orbitals of lowest singlet and triplet states ofm-benzyne at three geometries. Red corresponds to the struc- ture optimized by Kohn-Sham DFT for closed-shell singlet singlet configuration. Pink corresponds to the SF-DFT optimized structure of the lowest singlet state. Blue corresponds to the structure optimized by Kohn-Sham DFT for the triplet state. Orbital occupancy of each natural orbital is shown below orbital pictures. At each geometry, wave function properties computed for the lowest singlet state are presented in purple, while properties computed for the triplet state are pre- sented in green. 92 3.7 Appendix C: H 2 example and comparison of two Head-Gordon’s indices Table 3.8: E tot , n u and n u;nl , and their difference =jn u n u;nl j computed by EOM-SF-CCSD/cc-pVTZ. Values are provided for three states of H 2 at varying internuclear distances,R: the high-spin reference and the lowest singlet and triplet excited states. R ( ˚ A) E tot (hartree) Reference Triplet Singlet n u n u;nl n u n u;nl n u n u;nl 0.74 -1.1723 2.000 2.000 0.000 2.000 2.000 0.000 0.071 0.007 0.064 0.84 -1.1672 2.000 2.000 0.000 2.000 2.000 0.000 0.085 0.010 0.075 0.94 -1.1549 2.000 2.000 0.000 2.000 2.000 0.000 0103 0.015 0.087 1.04 -1.1392 2.000 2.000 0.000 2.000 2.000 0.000 0.126 0.024 0.102 1.14 -1.1222 2.000 2.000 0.000 2.000 2.000 0.000 0.156 0.039 0.118 1.24 -1.1054 2.000 2.000 0.000 2.000 2.000 0.000 0.195 0.061 0.134 1.34 -1.0892 2.000 2.000 0.000 2.000 2.000 0.000 0.243 0.096 0.147 1.44 -1.0744 2.000 2.000 0.000 2.000 2.000 0.000 0.303 0.148 0.156 1.54 -1.0610 2.000 2.000 0.000 2.000 2.000 0.000 0.375 0.221 0.153 1.64 -1.0493 2.000 2.000 0.000 2.000 2.000 0.000 0.459 0.322 0.137 1.74 -1.0393 2.000 2.000 0.000 2.000 2.000 0.000 0.556 0.450 0.106 1.84 -1.0309 2.000 2.000 0.000 2.000 2.000 0.000 0.661 0.602 0.059 1.99 -1.0210 2.000 2.000 0.000 2.000 2.000 0.000 0.829 0.859 0.030 2.24 -1.0104 2.000 2.000 0.000 2.000 2.000 0.000 1.111 1.285 0.174 2.49 -1.0048 2.000 2.000 0.000 2.000 2.000 0.000 1.356 1.606 0.250 2.74 -1.0020 2.000 2.000 0.000 2.000 2.000 0.000 1.547 1.800 0.253 2.99 -1.0008 2.000 2.000 0.000 2.000 2.000 0.000 1.686 1.903 0.216 3.39 -1.0000 2.000 2.000 0.000 2.000 2.000 0.000 1.830 1.971 0.141 3.79 -0.9997 2.000 2.000 0.000 2.000 2.000 0.000 1.910 1.992 0.045 4.19 -0.9997 2.000 2.000 0.000 2.000 2.000 0.000 1.953 1.998 0.045 4.59 -0.9996 2.000 2.000 0.000 1.987 2.000 0.013 1.987 2.000 0.013 4.99 -0.9996 2.000 2.000 0.000 1.993 2.000 0.007 1.993 2.000 0.007 93 Table 3.9: E tot , n u and n u;nl , and their difference = jn u n u;nl j at the B5050LYP/cc-pVTZ level of theory. Values are provided for the lowest singlet and triplet excited states of diatomic H 2 at varying internuclear distances,R. R ( ˚ A) E tot (hartree) Triplet Singlet n u n u;nl n u n u;nl 0.74 -0.9844 2.000 2.000 0.000 0.046 0.003 0.042 0.84 -0.9813 2.000 2.000 0.000 0.050 0.004 0.046 0.94 -0.9716 2.000 2.000 0.000 0.057 0.005 0.052 1.04 -0.9589 2.000 2.000 0.000 0.066 0.006 0.059 1.14 -0.9450 2.000 2.000 0.000 0.077 0.009 0.069 1.24 -0.9309 2.000 2.000 0.000 0.092 0.012 0.080 1.34 -0.9171 2.000 2.000 0.000 0.111 0.018 0.093 1.44 -0.9041 2.000 2.000 0.000 0.136 0.028 0.108 1.54 -0.8920 2.000 2.000 0.000 0.167 0.043 0.124 1.64 -0.8809 2.000 2.000 0.000 0.205 0.065 0.140 1.74 -0.8709 2.000 2.000 0.000 0.252 0.099 0.153 1.84 -0.8620 2.000 2.000 0.000 0.308 0.147 0.162 1.99 -0.8507 2.000 2.000 0.000 0.410 0.252 0.158 2.24 -0.8367 2.000 2.000 0.000 0.625 0.532 0.092 2.49 -0.8278 2.000 2.000 0.000 0.871 0.904 0.033 2.74 -0.8225 2.000 2.000 0.000 1.114 1.270 0.156 2.99 -0.8195 2.000 2.000 0.000 1.328 1.556 0.227 3.39 -0.8173 2.000 2.000 0.000 1.587 1.821 0.234 3.79 -0.8165 2.000 2.000 0.000 1.756 1.934 0.178 4.19 -0.8162 2.000 2.000 0.000 1.862 1.977 0.115 4.59 -0.8161 2.000 2.000 0.000 1.927 1.993 0.065 4.99 -0.8160 2.000 2.000 0.000 1.966 1.998 0.031 94 3.8 Appendix D: Comparison of two Head-Gordon’s in- dices for diradicals and triradicals Table 3.10: n u andn u;nl , and their difference =jn u n u;nl j computed by EOM- SF-CCSD/cc-pVTZ. Values are provided for three states of each benchmark sys- tem: the high-spin reference and the lowest high-spin and low-spin states (singlet and triplet states of diradicals, and doublet and quartet states of triradicals). Molecule Reference Triplet/Quartet Singlet/Doublet n u n u;nl n u n u;nl n u n u;nl CH 2 2.156 2.016 0.140 2.040 2.001 0.039 0.454 0.253 0.201 A 3.196 2.181 1.015 2.086 2.002 0.084 0.414 0.156 0.358 B 3.211 2.186 1.025 2.106 2.002 0.104 0.522 0.258 0.264 C 3.204 2.184 1.020 2.071 2.001 0.070 1.337 1.452 0.885 D 3.939 2.272 1.667 2.163 2.007 0.156 2.135 2.003 0.132 F 4.170 3.197 0.973 3.311 3.061 0.250 1.699 1.278 0.421 G 4.121 3.175 0.946 3.148 3.016 0.132 1.516 1.164 0.352 H 4.640 3.249 1.391 3.230 3.018 0.212 3.211 3.015 0.196 I 4.613 3.233 1.380 3.206 3.012 0.194 3.211 3.014 0.197 95 Table 3.11: n u and n u;nl and their difference =jn u n u;nl j computed by SF- TDDFT/cc-pVTZ with four different functionals. Values are provided for the low- est high-spin and low-spin states of each triradical. Molecule Method Quartet Doublet n u n u;nl n u n u;nl F PBE50 3.127 3.010 0.117 1.409 1.166 0.243 B5050LYP 3.083 3.004 0.079 1.300 1.095 0.205 LDA 3.032 3.001 0.031 1.076 1.006 0.070 B97 3.048 3.001 0.047 1.137 1.018 0.119 G PBE50 3.033 3.001 0.032 1.259 1.090 0.169 B5050LYP 3.024 3.000 0.024 1.200 1.056 0.144 LDA 3.001 3.000 0.001 1.065 1.006 0.059 B97 3.018 3.000 0.018 1.094 1.011 0.083 H PBE50 3.258 3.045 0.213 3.285 3.055 0.230 B5050LYP 3.229 3.036 0.193 3.218 3.033 0.185 LDA 3.029 3.001 0.028 3.026 3.001 0.025 B97 3.102 3.007 0.095 3.083 3.006 0.077 I PBE50 3.226 3.036 0.190 3.150 3.015 0.135 B5050LYP 3.188 3.026 0.162 3.129 3.012 0.117 LDA 3.012 3.000 0.012 3.016 3.000 0.016 B97 3.068 3.003 0.065 3.046 3.001 0.045 96 3.9 Appendix E: Results for CUAQAC02 and CITLAT with the B97 functional ! " s = 1.07 (0.17) ! " t = 1.00 (0.06) ! " s = 0.93 (0.17) ! " t = 1.00 (0.24) ! " s = 0.93 (0.00) ! " t = 1.00 (0.00) ! " s = 1.07 (0.00) ! " t = 1.00 (0.00) CITLAT CUAQAC02 Figure 3.14: SF-TDDFT/cc-pVTZ frontier orbitals of CUAQAC02 and CITLAT with the B97 functional. 97 3.10 Appendix F: Cartesian geometries FCI/TZ2P optimized triplet geometry of methylene, taken from C.D. Sherrill, M.L. Leininger, T.J. Van Huis, and H.F. Schaefer III, J. Chem. Phys. 108, 1040 (1998). 1 C -0.0000000000 0.0000000000 0.1068122905 2 H -0.9894459137 -0.0000000000 -0.3204368715 3 H 0.9894459137 0.0000000000 -0.3204368715 Nuclear Repulsion Energy: 6.1594 hartrees SF-TDDFT/B5050LYP/cc-pVTZ optimized geometry of the singlet ground state of ortho-benzyne. 1 C 1.4415113810 0.1157913314 0.0000144407 2 C 0.6193523170 1.2126922890 0.0000260342 3 C -0.6186423498 1.2130252042 -0.0000309537 4 C -1.4414474984 0.1166184055 -0.0000114902 5 H -2.5136361905 0.1185569684 0.0000019649 6 C -0.6974645040 -1.0595063406 0.0000336841 7 H -1.2160483000 -2.0013077712 0.0000682412 8 C 0.6968511812 -1.0599063264 -0.0000017881 9 H 1.2148886400 -2.0020086190 -0.0000143281 10 H 2.5137018329 0.1170995972 0.0000078440 Nuclear Repulsion Energy: 189.5511 hartrees Kohn-Sham B5050LYP/cc-pVTZ optimized geometry of triplet ortho-benzyne. 1 C 1.3948997004 0.1074061035 -0.0000018385 2 C 0.6915152399 1.2744018112 -0.0000035476 3 C -0.6907792883 1.2747996531 -0.0000007757 4 C -1.3948371084 0.1082099806 0.0000043144 5 H -2.4701749437 0.1142959908 0.0000106121 6 C -0.6890135656 -1.0897959941 0.0000179646 7 H -1.2262341351 -2.0201141176 0.0000284500 8 C 0.6883838973 -1.0901925283 0.0000149753 9 H 1.2250659790 -2.0208215112 0.0000246210 10 H 2.4702407339 0.1128653507 -0.0000011268 98 Nuclear Repulsion Energy: 187.2006 hartrees Kohn-Sham B5050LYP/cc-pVTZ optimized geometry of singlet meta-benzyne. 1 C -0.6109841449 -0.7588517785 -0.0000143575 2 C -1.7091287787 0.0003255099 -0.0000194729 3 H -2.7826124852 0.0005139222 -0.0000191124 4 C -0.6107233206 0.7591336627 -0.0000158147 5 C 0.7017225216 1.1594626575 -0.0000146432 6 H 1.0651023572 2.1661823335 -0.0000119760 7 C 1.4841872582 -0.0002220113 -0.0000219957 8 H 2.5610262913 -0.0004070756 -0.0000227568 9 C 0.7013225186 -1.1596371011 -0.0000119387 10 H 1.0643537317 -2.1664820690 -0.0000069900 Nuclear Repulsion Energy: 192.8945 hartrees SF-TDDFT/B5050LYP/cc-pVTZ optimized geometry of the singlet ground state of meta-benzyne. 1 C -0.6804935700 -0.9934314339 -0.0000136727 2 C -1.5919504832 0.0003016204 -0.0000011662 3 H -2.6617084775 0.0004866197 -0.0000356336 4 C -0.6801509685 0.9937493002 -0.0000024980 5 C 0.6682311816 1.1607973471 -0.0000062013 6 H 1.1076641699 2.1400289960 -0.0000548499 7 C 1.4254196489 -0.0002104031 0.0000283463 8 H 2.5025092016 -0.0003939165 -0.0000120058 9 C 0.6678271763 -1.1609635391 -0.0000019555 10 H 1.1069180702 -2.1403465405 -0.0000594211 Nuclear Repulsion Energy: 189.4409 hartrees Kohn-Sham B5050LYP/cc-pVTZ optimized geometry of triplet meta-benzyne. 1 C -0.7278058220 -1.1524474757 -0.0000149070 2 C -1.4648570798 0.0002860296 -0.0000237394 3 H -2.5401265303 0.0004731601 -0.0000153309 4 C -0.7274099904 1.1527629308 -0.0000183937 99 5 C 0.6370013075 1.2103692488 -0.0000166454 6 H 1.1713580675 2.1413000279 -0.0000152826 7 C 1.3168148893 -0.0001914357 -0.0000164497 8 H 2.3920873831 -0.0003791798 -0.0000148326 9 C 0.6365828477 -1.2105207396 -0.0000140342 10 H 1.1706208767 -2.1416345162 -0.0000094424 Nuclear Repulsion Energy: 187.5007 hartrees SF-TDDFT/B5050LYP/cc-pVTZ optimized geometry of the singlet ground state of para-benzyne. 1 H 2.1410131571 1.2227517483 0.0000000000 2 C 1.1990073665 0.7078869029 0.0000000000 3 C 1.1990073665 -0.7078869029 0.0000000000 4 H 2.1410131571 -1.2227517483 0.0000000000 5 C 0.0000000000 -1.3307797356 0.0000000000 6 C -1.1990073665 -0.7078869029 0.0000000000 7 H -2.1410131571 -1.2227517483 0.0000000000 8 C -1.1990073665 0.7078869029 0.0000000000 9 H -2.1410131571 1.2227517483 0.0000000000 10 C 0.0000000000 1.3307797356 0.0000000000 Nuclear Repulsion Energy: 187.6786 hartrees Kohn-Sham B5050LYP/cc-pVTZ optimized geometry of triplet para-benzyne. 1 H 2.1394304380 1.2526740674 0.0000000000 2 C 1.2202886830 0.6957870853 0.0000000000 3 C 1.2202886830 -0.6957870853 0.0000000000 4 H 2.1394304380 -1.2526740674 0.0000000000 5 C 0.0000000000 -1.3069756474 0.0000000000 6 C -1.2202886830 -0.6957870853 0.0000000000 7 H -2.1394304380 -1.2526740674 0.0000000000 8 C -1.2202886830 0.6957870853 0.0000000000 9 H -2.1394304380 1.2526740674 0.0000000000 10 C 0.0000000000 1.3069756474 0.0000000000 Nuclear Repulsion Energy: 187.5031 hartrees Kohn-Sham B5050LYP/cc-pVTZ optimized geometry of 100 singlet 1-(2-dehydroisopropyl)-4-dehydrobenzyne. 1 C 1.9511737840 -1.2839502992 0.2020372604 2 C 2.7247354995 -0.1033462963 0.4062165949 3 C 2.0503457164 1.0906071868 0.0131778002 4 C 0.7121215280 1.1583399195 -0.1388715503 5 H 0.2479857147 2.1148288581 -0.2825732019 6 C -0.1091425785 -0.0219908342 -0.0657492452 7 C 0.6108400217 -1.2627621640 0.0559899315 8 H 0.0671062659 -2.1873500607 0.0598862635 9 H 2.4374946464 -2.2415323086 0.2898888756 10 H 2.6155369461 2.0060426830 -0.0472925108 11 C -1.4660909553 0.0266206410 -0.1358145018 12 C -2.2475898979 1.2852634573 -0.2916549988 13 C -2.3451403588 -1.1752534573 -0.0772634864 14 H -2.9712311519 1.3635424419 0.5136576779 15 H -2.8219427619 1.2429547345 -1.2127871423 16 H -1.6558924597 2.1831483921 -0.3052613248 17 H -3.1265358662 -1.0172246786 0.6587963348 18 H -1.8355397200 -2.0924142576 0.1606847940 19 H -2.8477853675 -1.3039861059 -1.0319921732 Nuclear Repulsion Energy: 401.0059 hartrees SF-TDDFT/B5050LYP/cc-pVTZ optimized geometry of the lowest-energy singlet state of 1-(2-dehydroisopropyl)-4- dehydrobenzyne. 1 C 1.9529250052 -1.3086044934 0.0825931027 2 C 2.6219830999 -0.1258021539 0.0290999304 3 C 2.0485819163 1.0974516733 -0.1277810501 4 C 0.6662998093 1.1419336629 -0.1508420849 5 H 0.1864618097 2.0978437864 -0.2379761130 6 C -0.1098126455 -0.0238709999 -0.0496287615 7 C 0.5711798030 -1.2472370359 0.0582993218 8 H 0.0172814691 -2.1634554816 0.1328330844 9 H 2.4570704761 -2.2539446544 0.1607739593 10 H 2.6265730413 1.9991566270 -0.2101724491 11 C -1.5363027397 0.0327446707 -0.0651693422 12 C -2.2350659497 1.3358140138 -0.2283394341 13 C -2.3421500030 -1.2132243625 0.0311840847 14 H -3.3081454513 1.2096989724 -0.1998135077 101 15 H -1.9863703967 1.8166590381 -1.1725053827 16 H -1.9667655147 2.0424227067 0.5542187667 17 H -3.3964187579 -0.9900340657 0.1168093405 18 H -2.0617737577 -1.8160215450 0.8910575508 19 H -2.2151022088 -1.8499925074 -0.8435656186 Nuclear Repulsion Energy: 402.3556 hartrees SF-TDDFT/B5050LYP/cc-pVTZ optimized geometry of the triplet ground state of 1-(2-dehydroisopropyl)-4- dehydrobenzyne. 1 C 1.9449472009 -1.3085722770 0.0774478319 2 C 2.6224552828 -0.1300979782 -0.0221842734 3 C 2.0413532633 1.0972420353 -0.1398806041 4 C 0.6641855343 1.1438727421 -0.1582921661 5 H 0.1838125663 2.0987365313 -0.2505214847 6 C -0.1154339823 -0.0234347420 -0.0596999879 7 C 0.5682552975 -1.2479904646 0.0577466224 8 H 0.0128695459 -2.1627575770 0.1350027555 9 H 2.4496329355 -2.2525574898 0.1680262945 10 H 2.6205686161 1.9987224994 -0.2163537081 11 C -1.5301804957 0.0316718900 -0.0789325344 12 C -2.2322232797 1.3346605644 -0.2069846403 13 C -2.3352134689 -1.2120546271 0.0262567785 14 H -3.3047287049 1.2017057610 -0.2173337099 15 H -1.9569488366 1.8591755722 -1.1202398400 16 H -1.9934626431 2.0101570658 0.6129718520 17 H -3.3938680785 -0.9960153103 0.0005189690 18 H -2.1331860401 -1.7544348826 0.9484619783 19 H -2.1223857078 -1.9064914611 -0.7849347359 Nuclear Repulsion Energy: 402.8251 hartrees Kohn-Sham B5050LYP/cc-pVTZ optimized geometry of triplet 1-(2-dehydroisopropyl)-4-dehydrobenzyne. 1 C 1.9478539012 -1.3123501503 0.0748590402 2 C 2.6309113333 -0.1300802633 -0.0214833921 3 C 2.0444319723 1.1015503872 -0.1354372101 4 C 0.6678901768 1.1490096452 -0.1540786942 5 H 0.1857477247 2.1033651798 -0.2418648725 102 6 C -0.1155866797 -0.0230968198 -0.0589740072 7 C 0.5717150792 -1.2526598987 0.0556860753 8 H 0.0147372190 -2.1667721452 0.1286907191 9 H 2.4531616396 -2.2563813501 0.1619919773 10 H 2.6241852639 2.0030392520 -0.2082828521 11 C -1.5259552287 0.0315583634 -0.0783690244 12 C -2.2351349992 1.3369320130 -0.2190296611 13 C -2.3377303954 -1.2149019781 0.0365890113 14 H -3.3059566769 1.1958724964 -0.2608428301 15 H -1.9391336806 1.8655304963 -1.1211241311 16 H -2.0269508765 2.0046686679 0.6138631650 17 H -3.3951001615 -0.9910239578 0.0516794967 18 H -2.1080664153 -1.7697382422 0.9424087598 19 H -2.1605701913 -1.8929838441 -0.7952061724 Nuclear Repulsion Energy: 402.0753 hartrees SF-TDDFT/B5050LYP/cc-pVTZ optimized geometry of the doublet ground state of 1,2,4-tridehydrobenzyne. 1 C 0.4078012498 -1.1785638269 -0.0000216578 2 C -0.9425946299 -0.9097600453 -0.0000036777 3 C -1.3618866787 0.4218537038 -0.0000200954 4 C -0.2835286142 1.2727492155 -0.0000004105 5 C 0.9168342375 0.9439870472 -0.0000072426 6 C 1.4725566185 -0.3143855008 -0.0000318068 7 H 2.5074167122 -0.5888237776 0.0000531261 8 H -1.6607441280 -1.7086008970 -0.0000526725 9 H -2.3982932003 0.6996413610 -0.0000780122 Nuclear Repulsion Energy: 180.7441 hartrees Kohn-Sham B5050LYP/cc-pVTZ optimized geometry of quartet1,2,4-tridehydrobenzyne. 1 C 0.3932475944 -1.2268136221 -0.0000109582 2 C -0.9414297089 -0.9551357241 -0.0000321669 3 C -1.3037585872 0.3925642894 -0.0000374411 4 C -0.3256874138 1.3441983257 -0.0000178442 5 C 1.0174656750 1.0124303706 0.0000056488 6 C 1.4036614094 -0.2989946728 0.0000048673 7 H 2.4425658284 -0.5814337018 0.0000247445 103 8 H -1.6874745136 -1.7277904330 -0.0000458660 9 H -2.3410287167 0.6790724479 -0.0000534337 Nuclear Repulsion Energy: 178.3616 hartrees SF-TDDFT/B5050LYP/cc-pVTZ optimized geometry of doublet 1,3,5-tridehydrobenzyne. 1 C 1.0206237067 -1.0026086876 -0.0000146277 2 C 1.3624476314 0.3334248254 0.0001184823 3 C 0.2705267515 1.1397316235 0.0001059496 4 C -1.0761297666 1.0571492779 0.0000123738 5 C -1.1346338929 -0.2907513314 -0.0000566472 6 C -0.3090889738 -1.3682041060 0.0002451652 7 H -0.6621349712 -2.3794383037 -0.0001759089 8 H 2.3671601349 0.7046295719 -0.0000902849 9 H -1.8387460716 1.8062977070 -0.0000945273 Nuclear Repulsion Energy: 180.5429 hartrees Kohn-Sham B5050LYP/cc-pVTZ optimized geometry of quartet 1,3,5-tridehydrobenzyne. 1 C 0.9871556242 -0.9282796419 0.0000183148 2 C 1.3857590498 0.3849152353 0.0000158947 3 C 0.3686051443 1.3046005727 0.0000040068 4 C -1.0250390180 0.9519513291 -0.0000561614 5 C -1.3535102084 -0.3759088356 -0.0000146268 6 C -0.3888812968 -1.3442602118 0.0000091943 7 H -0.6254949093 -2.3924185949 -0.0000185385 8 H 2.4197927466 0.6716298574 0.0000565705 9 H -1.7683625837 1.7280008666 0.0000353205 Nuclear Repulsion Energy: 176.2277 hartrees Kohn-Sham B5050LYP/cc-pVTZ optimized geometry of doublet 2-dehydro-meta-xylylene. 1 H 0.0000002186 -2.8238938175 0.0000393496 2 C 0.0000005532 -1.7491364332 0.0000027160 3 C -1.1984417350 -1.0707818324 -0.0004899241 4 H -2.1260374202 -1.6148558795 -0.0008380255 104 5 C -1.2165584431 0.3604941935 -0.0005952274 6 C 0.0000000000 1.1582823688 -0.0001116155 7 C 1.2165584438 0.3604943675 0.0004541519 8 C 1.1984423914 -1.0707822176 0.0004520789 9 H 2.1260384040 -1.6148556360 0.0008443423 10 C -2.3716301286 1.0700168955 -0.0011281373 11 C 2.3716296951 1.0700171510 0.0009789808 12 H -2.3243042340 2.1411891044 0.0008598377 13 H -3.3385223798 0.5964192982 0.0001664174 14 H 2.3243039685 2.1411898489 -0.0007415020 15 H 3.3385225892 0.5964199378 0.0000659116 Nuclear Repulsion Energy: 311.9074 hartrees SF-TDDFT/B5050LYP/cc-pVTZ optimized geometry of doublet 2-dehydro-meta-xylylene. 1 H 0.0000017376 -2.7929729389 -0.0000478382 2 C 0.0000010758 -1.7185951045 -0.0000187006 3 C -1.2011686591 -1.0471856469 0.0000869614 4 H -2.1292908076 -1.5896613801 -0.0000238248 5 C -1.2452315088 0.3772197084 0.0000224414 6 C -0.0000000393 0.9787584786 0.0000400452 7 C 1.2452309381 0.3772221041 0.0000780007 8 C 1.2011698124 -1.0471851439 -0.0000563481 9 H 2.1292915920 -1.5896606363 0.0000480786 10 C -2.4445601291 1.0828042877 -0.0003394750 11 C 2.4445589511 1.0828037525 0.0003525154 12 H -2.4497557213 2.1538890786 -0.0005988227 13 H -3.3832715903 0.5644464048 -0.0004900605 14 H 2.4497571380 2.1538902759 0.0005238248 15 H 3.3832691357 0.5644441095 0.0003825566 Nuclear Repulsion Energy: 312.3902 hartrees SF-TDDFT/B5050LYP/cc-pVTZ optimized geometry of the quartet ground state of 2-dehydro-meta-xylylene. 1 H -0.0000013340 -2.7905494365 -0.0000587638 2 C -0.0000014788 -1.7160402408 0.0000077023 3 C -1.2024524020 -1.0461871428 -0.0004628876 4 H -2.1293635652 -1.5902063763 -0.0005806530 105 5 C -1.2472591814 0.3789751923 0.0004235094 6 C 0.0000006634 0.9862011035 0.0001003796 7 C 1.2472582548 0.3789762474 -0.0002042236 8 C 1.2024514893 -1.0461850940 0.0005204487 9 H 2.1293624990 -1.5902042207 0.0005461827 10 C -2.4407696591 1.0798632570 0.0009594450 11 C 2.4407716525 1.0798653512 -0.0008218026 12 H -2.4497254232 2.1511442268 -0.0011157962 13 H -3.3795556110 0.5617077232 -0.0007311511 14 H 2.4497286760 2.1511468028 0.0009594297 15 H 3.3795573448 0.5617099566 0.0004175348 Nuclear Repulsion Energy: 312.4758 hartrees Kohn-Sham B5050LYP/cc-pVTZ optimized geometry of quartet 2-dehydro-meta-xylylene. 1 H 0.0000004108 -2.7927646935 -0.0000034111 2 C 0.0000002962 -1.7182377492 -0.0000013105 3 C -1.2050132047 -1.0475304321 0.0001621163 4 H -2.1311336637 -1.5928273429 0.0002534650 5 C -1.2486909300 0.3788853333 0.0002703770 6 C 0.0000000259 0.9898185673 0.0000037811 7 C 1.2486910644 0.3788855796 -0.0002641385 8 C 1.2050136390 -1.0475302066 -0.0001621613 9 H 2.1311342188 -1.5928269197 -0.0002562892 10 C -2.4429312570 1.0809317533 0.0003341912 11 C 2.4429312750 1.0809321778 -0.0003284348 12 H -2.4528303716 2.1528260819 -0.0010661735 13 H -3.3826340012 0.5634140663 -0.0009024779 14 H 2.4528303121 2.1528265435 0.0010439671 15 H 3.3826341139 0.5634145906 0.0008758534 Nuclear Repulsion Energy: 312.0848 hartrees SF-TDDFT/B5050LYP/cc-pVTZ optimized geometry of the doublet ground state of 5-dehydro-meta-xylylene. 1 C 0.0000002806 -1.7405341016 -0.0001058693 2 C -1.2163739587 -1.1452610002 -0.0002901030 3 H -2.1399364311 -1.6938804824 -0.0001090586 4 C -1.2375228460 0.2873891842 0.0000705228 106 5 C 0.0000010682 0.9583298144 0.0003716647 6 H 0.0000007267 2.0332302958 0.0005865470 7 C 1.2375233428 0.2873876918 0.0004768108 8 C 1.2163739633 -1.1452605799 0.0002500521 9 H 2.1399382309 -1.6938796426 -0.0002138459 10 C 2.4379138997 0.9780643786 0.0004461613 11 C -2.4379138485 0.9780631273 0.0001107288 12 H 3.3763357577 0.4582679282 -0.0010892637 13 H 2.4579468564 2.0509775845 -0.0017533007 14 H -2.4579499178 2.0509799719 -0.0004872290 15 H -3.3763347923 0.4582685262 -0.0017230546 Nuclear Repulsion Energy: 313.6173 hartrees Kohn-Sham B5050LYP/cc-pVTZ optimized geometry of quartet 5-dehydro-meta-xylylene. 1 C 0.0000001483 -1.7462323144 -0.0001623904 2 C -1.2156857955 -1.1433606913 -0.0000099949 3 H -2.1366190147 -1.6957184224 -0.0001369422 4 C -1.2358712392 0.2842134931 0.0003343057 5 C 0.0000000961 0.9549694632 0.0003856577 6 H 0.0000001332 2.0299042106 0.0005568693 7 C 1.2358714698 0.2842135450 0.0002643438 8 C 1.2156860652 -1.1433606599 -0.0000524180 9 H 2.1366193233 -1.6957183455 -0.0002009031 10 C 2.4364026841 0.9802303458 0.0002775114 11 C -2.4364025668 0.9802302363 0.0003764038 12 H 3.3764092807 0.4633219186 -0.0012705556 13 H 2.4528528059 2.0530644244 -0.0010486022 14 H -2.4528524854 2.0530638447 -0.0013051856 15 H -3.3764085735 0.4633216478 -0.0014673371 Nuclear Repulsion Energy: 313.7070 hartrees Experimental geometry of CUAQAC02, taken from P. de Meester, S. R. Fletcher, and A. C. Skapski, J. C. S. Dalton , 2575 (1973). 1 Cu -0.0812046245 -0.0384144562 1.3047594119 2 O 1.5906936988 1.0320230998 1.3032105641 3 C 2.1545179877 1.3768207742 0.2412281333 107 4 C 3.4549185851 2.1101482190 0.3230892497 5 H 4.1695152206 1.8962487128 -0.4370203256 6 H 3.7504290807 2.3294405319 1.1154436603 7 H 3.4002572293 2.9975852882 -0.1775190247 8 O 1.6967968432 1.1424461350 -0.9182695909 9 Cu 0.0812046245 0.0384144562 -1.3047594119 10 O 1.1564111503 -1.5673521710 -1.0431977352 11 C 1.4349390038 -2.0651082749 0.0662986445 12 C 2.3249243215 -3.2792283635 0.1135053018 13 H 3.1149896705 -3.0851508138 -0.4735114288 14 H 2.5123814002 -3.3958848086 1.1109520954 15 H 1.8966840057 -4.0317754126 -0.3403682036 16 O 1.0355805193 -1.6278586656 1.1946360561 17 O -1.5906936988 -1.0320230998 -1.3032105641 18 C -2.1545179877 -1.3768207742 -0.2412281333 19 O -1.6967968432 -1.1424461350 0.9182695909 20 C -3.4549185851 -2.1101482190 -0.3230892497 21 H -4.1695152206 -1.8962487128 0.4370203256 22 H -3.7504290807 -2.3294405319 -1.1154436603 23 H -3.4002572293 -2.9975852882 0.1775190247 24 O -1.0355805193 1.6278586656 -1.1946360561 25 C -1.4349390038 2.0651082749 -0.0662986445 26 O -1.1564111503 1.5673521710 1.0431977352 27 C -2.3249243215 3.2792283635 -0.1135053018 28 H -3.1149896705 3.0851508138 0.4735114288 29 H -2.5123814002 3.3958848086 -1.1109520954 30 H -1.8966840057 4.0317754126 0.3403682036 31 O 0.4282513024 0.0762856520 -3.4319450482 32 H 1.2425955087 -0.0598196479 -3.6712020545 33 H -0.1539213596 -0.0273961234 -4.0849259550 34 O -0.4282513024 -0.0762856520 3.4319450482 35 H -1.2425955087 0.0598196479 3.6712020545 36 H 0.1539213596 0.0273961234 4.0849259550 Nuclear Repulsion Energy: 3191.9482 hartrees Experimental geometry of CITLAT, taken from S. Youngme, J. Phatchimkun, N. Wannarit, N. Chaichit, S. Meejoo, G.A. van Albada, and J. Reedijk, Polyhedron 27, 304 (2008). 1 Cu 0.1698030922 0.1287104291 1.4509940121 108 2 O 0.1158718652 1.4371095278 0.0000000000 3 H -0.4713066048 1.7823071935 0.0000000000 4 Cu 0.1698030922 0.1287104291 -1.4509940121 5 O 0.9486663417 -0.9158725158 0.0000000000 6 H 1.7307644494 -0.8971572375 0.0000000000 7 N 0.1572733229 1.4892131465 -2.9579120007 8 C 0.0991658292 1.1995230771 -4.2667369745 9 N 0.0290818050 -0.0980936259 -4.7148040939 10 H -0.2062442714 -0.1563600203 -5.3060491102 11 C 0.1233191884 -1.3201866014 -4.0797622350 12 N 0.2317332525 -1.4023126747 -2.7423009571 13 C 0.3460798999 -2.6464100630 -2.2116958737 14 H 0.4241338558 -2.6014872862 -1.0948989225 15 C 0.3360037442 -3.7952104135 -2.9528577805 16 H 0.4779394258 -4.6794092396 -2.4087777148 17 C 0.2212344410 -3.6904302577 -4.3374837918 18 H 0.0841647030 -4.5152819317 -4.9691567172 19 C 0.0995534382 -2.4462333984 -4.9034633998 20 H -0.0107449924 -2.3824721189 -5.8787653594 21 C 0.1289777545 2.1925812326 -5.2454078989 22 H 0.0608477689 1.9255954769 -6.2998798919 23 C 0.1949981554 3.5124042417 -4.8798808404 24 H 0.4158847919 4.3114670346 -5.5250288678 25 C 0.2348162590 3.8190583519 -3.5306278038 26 H 0.2246869479 4.7905039190 -3.2510077833 27 C 0.2232564635 2.7956619337 -2.6142819692 28 H 0.4029224391 2.9491344016 -1.6002369455 29 N 0.1572733229 1.4892131465 2.9579120007 30 C 0.0991658292 1.1995230771 4.2667369745 31 N 0.0290818050 -0.0980936259 4.7148040939 32 H -0.2062442714 -0.1563600203 5.3060491102 33 C 0.1233191884 -1.3201866014 4.0797622350 34 N 0.2317332525 -1.4023126747 2.7423009571 35 C 0.3460798999 -2.6464100630 2.2116958737 36 H 0.4241338558 -2.6014872862 1.0948989225 37 C 0.3360037442 -3.7952104135 2.9528577805 38 H 0.4779394258 -4.6794092396 2.4087777148 39 C 0.2212344410 -3.6904302577 4.3374837918 40 H 0.0841647030 -4.5152819317 4.9691567172 41 C 0.0995534382 -2.4462333984 4.9034633998 42 H -0.0107449924 -2.3824721189 5.8787653594 43 C 0.1289777545 2.1925812326 5.2454078989 109 44 H 0.0608477689 1.9255954769 6.2998798919 45 C 0.1949981554 3.5124042417 4.8798808404 46 H 0.4158847919 4.3114670346 5.5250288678 47 C 0.2348162590 3.8190583519 3.5306278038 48 H 0.2246869479 4.7905039190 3.2510077833 49 C 0.2232564635 2.7956619337 2.6142819692 50 H 0.4029224391 2.9491344016 1.6002369455 51 H -3.5854705415 -0.4242737426 0.0000000000 52 C -2.6578574128 -0.3343788685 0.0000000000 53 O -2.1113130473 -0.2915923669 1.1125860265 54 O -2.1113130473 -0.2915923669 -1.1125860265 Nuclear Repulsion Energy: 4911.5415 hartrees 110 Chapter 3 References [1] L. 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Lett. 426, 196 (2006). 114 Chapter 4: An ab initio exploration of the Bergman cyclization 4.1 Introduction The enediyne (Z)-hex-3-ene-1,5-diyne (R) undergoes Bergman cyclization via a tran- sition state (TS) to form p-benzyne (P), a highly reactive diradical species that readily abstracts hydrogens (Figure 4.1). 1 This fundamental reaction can be triggered either thermally, to proceed along the singlet potential energy surface, or photo-chemically and proceed along the triplet surface. 2–7 The Bergman cyclization is of interest because enediynes might be precursors to aromatic species in the interstellar medium 8 and be- cause natural enediynes are potent antitumor antibiotics. 9–11 Figure 4.1: Bergman cyclization of (Z)-hex-3-ene-1,5-diyne. Since the discovery of natural enediynes, the reaction described in Figure 4.1 has been the subject of several theoretical studies. 12–21 The structure of the singlet reactant ( 1 R) has been determined experimentally 22 and the ground state structures of 1 R, 1 TS, 115 and 1 P have been explored with numerous theoretical methods such as density functional theory (DFT), CASSCF, CASPT2, CCSD(T) and MBPT, 12–20 while the triplet surface has only been studied using CASSCF and CASPT2. 15, 19 The electronic structure and energetics of P have been investigated extensively, with an emphasis on excited states and singlet-triplet energy gaps. The adiabatic singlet-triplet gap (E ST;a = E T - E S ) was experimentally determined to be 3.8 0.4 kcal/mol. 23 Theoretical results vary widely based on the methods used. 15, 19, 24–26 Even though the Bergman cyclization is a fundamentally important reaction, the stationary points on the singlet and triplet surfaces have not yet been fully and con- sistently characterized using reliable computational methods. This is a challenging reaction to study because of the multi-configurational nature of 1 P, and the possible multi-configurational nature of 1 TS. Due to its diradical character, P features extensive electronic degeneracies. 24, 27, 28 As a result, the ground state singlet possesses a multi- configurational wave function, making it theoretically interesting and challenging to characterize. 19, 24, 27, 29–31 Methodologies capable of capturing this multi-configurational character must be applied to appropriately account for the multiple electronic config- urations. For quantitative accuracy, which is very important for reproducing energy differences between nearly degenerate states, dynamical correlation should also be in- cluded. Here we employ the spin-flip (SF) approach, 32 which accurately describes di- radical wave functions without the need to select an active space or important config- urations. Importantly, SF method provides a balanced description of dynamical and non-dynamical correlation. The SF approach developed by Krylov was originally designed for characterizing bond breaking, but its ability to describe diradicals was soon realized. 32, 33 SF mod- els can accurately describe low spin multi-configurational states by treating them as 116 spin-flipping excitations from a single-configuration high-spin reference state. 33, 34 SF methods describe the singlet target states in the following way 33 1 Ms=0 =R 3 Ms=1 Ms=+1 (4.1) where 3 Ms=+1 is the high spin reference wave function,R Ms=1 is the spin flipping excitation operator, and 1 Ms=0 is the singlet target state wave function. Because all M s = 0 determinants can be obtained from a single spin flip excitation out of the high spin reference (Figure 4.2), allM s = 0 configurations are treated in a balanced fashion using a single reference formalism. Figure 4.2: Various ways to distribute two electrons in two nearly degenerate orbitals for P. All fourM s = 0 determinants, pictured on the right, can be obtained from the singleM s = 1 Slater determinant with just one spin flipping excitation. By using a reference wave function that is accurately described by a single config- uration, SF methods do not involve a multi-reference formalism and do not depend on a user defined active space. SF methods are size extensive, variational, and have been shown to perform very well for excited states, diradicals, triradicals, and bond break- ing, with an accuracy approaching 1 kcal/mol. 27, 32, 33, 35–38 Target-state description can be systematically improved by applying increasingly accurate models for the high-spin 117 reference state. 39 In this study, we employ the SF model based on equation-of-motion coupled-cluster with single-and-double excitations (EOM-SF-CCSD). 40–42 While EOM- SF-CCSD has been used to describe the structures and electronic states of the benzyne diradicals, this is the first SF characterization of all stationary points along the lowest lying singlet and triplet surfaces of the Bergman cyclization. To gain insight into underlying electronic structure, we also use density-based wavefunction analysis tools, 43–45 which allow us to map correlated many-body wave- functions onto the simple two-electrons-in-two-orbitals picture shown in Fig. 4.2. These tools allow us to quantify the degree of radical character associated with the reactant, transition state and product along the singlet and triplet pathways. By using natural or- bitals of the correlated one-particle density, this approach allows one to visualize the true frontier molecular orbitals. This analysis is independent of the choice of canonical orbitals used in the calculation (i.e., Hartree-Fock, Kohn-Sham, etc.) and includes cor- relation effects. The occupations of natural orbitals can be used to define the number of effectively unpaired electrons. 46–48 4.2 Computational Details We carried out all calculations using the Q-Chem electronic structure package. 49, 50 We use thelibwfa module 43, 44 of Q-Chem to compute and visualize natural orbitals and the Head-Gordon index. Orbital visualization was performed using IQmol 51 and Jmol. 4.2.1 Optimizations and Frequency Calculations We performed all optimizations and frequency calculations with the cc-pVDZ and cc- pVTZ basis sets. The results reported here are computed using the structures optimized 118 with cc-pVTZ unless otherwise noted. We optimized 1 R at the CCSD level with a re- stricted Hartree-Fock (RHF) reference wave function. Since the diradical nature of 1 TS was unknown, we optimized the structure using EOM-SF-CCSD with an unrestricted Hartree-Fock (UHF) high spin 3 B 1 (C 2v point group) reference. We also employed EOM-SF-CCSD to optimize the structure of 1 P. The SF optimization of 1 P used the high spin 3 B 2u (D 2h point group) UHF reference. We optimized the structures of 3 R, 3 TS, and 3 P using UHF-CCSD. Frequency cal- culations at the same level of theory as the geometry optimization were used to confirm all geometries and to collect zero point energy (ZPE) contributions. 4.2.2 Energy Calculations In order to calculate accurate activation barriers (E a ) and reaction energies (E rxn ), the reactant energetics must be obtained with the same method as the energetics of the tran- sition state and product. However, this is challenging because 1 R is a well behaved closed-shell species while 1 TS and 1 P are possibly open-shell multi-configurational species, making it difficult to find a method that describes all three structures with a similar accuracy. This problem is not present in the triplet pathway because 3 R, 3 TS, and 3 P are all well-represented by a single high-spin determinant and can, therefore, be accurately described by the single reference CCSD method. To overcome the methodological challenges the singlet pathway presents, we fol- lowed a protocol similar to that of Cristian et al.: 52 i.e., we used the energetics of the high spin (HS) pathway and the vertical singlet-triplet gaps (E ST;v ) to calculate the E a and E rxn of the low-spin (LS) pathway. Figure 4.3 illustrates this approach. Us- ing singlet structures that were optimized in the manner described above, we performed single-point calculations using CCSD(T) level with a restricted open-shell HF (ROHF) reference and the cc-pVTZ basis set. For 1 R, the ground state singlet total energy was 119 Figure 4.3: Determination of E a and E rxn for singlet reaction pathway. All ge- ometries were optimized using the respective ground state singlet wave function. Single-point calculations of triplet states at the singlet geometries (TS HS , P HS ) were performed using CCSD(T)/ROHF/cc-pVTZ. Singlet triplet vertical gaps were cal- culated using EOM-SF-CCSD/ROHF/cc-pVTZ calculated (R LS ). For 1 TS and 1 P, the total energy for the lowest lying triplet was cal- culated (TS HS , P HS ). All three states (R LS , TS HS , and P HS ) are single configurational and are well described by the CCSD(T) method, which we used to calculate the high- spin reaction barrier (E a HS = TS HS - R LS ) and reaction energy (E rxn HS = P HS - R LS ). We carried out EOM-SF-CCSD/ROHF/cc-pVTZ calculations to obtain accurate vertical E ST for 1 TS and 1 P. To obtainE a LS andE rxn LS , we subtracted the E ST;v for 1 TS and 1 P fromE a HS andE rxn HS , respectively: E LS a =E HS a E ST (TS) (4.2) E LS run =E HS run E ST (P) (4.3) 120 Note that the high-spin states are not geometry optimized for the high spin wave func- tion; they are vertical energy gaps computed at the singlet structures. Spin-flip calculations with an unrestricted reference may be affected by spin con- tamination. 35, 37 To mitigate spin-contamination, we used ROHF references when cal- culating energetics. For geometry optimization, we used a UHF reference since analytic gradients are not yet available for ROHF EOM-SF-CCSD. A detailed comparison of the energetics computed with restricted (RHF and ROHF) and unrestricted (UHF) reference wave functions is presented below. We also carried out single-point calculations at the CCSD/UHF/cc-pVTZ optimized triplet stationary points for the triplet pathway using the CCSD(T)/ROHF/cc-pVTZ and EOM-SF-CCSD/ROHF/cc-pVTZ levels of theory. We calculated adiabatic gaps as the difference in total energy between the SF target state corresponding to theM s = 0 com- ponent of the triplet at the triplet optimized structure and the lowest lying singlet SF target state at the singlet optimized structure. We calculated vertical gaps were calcu- lated at each of the six optimized structures by taking the difference between theM s = 0 component of the triplet and the lowest lying singlet target states. To compute and visualize natural orbitals and the number of effectively unpaired electrons, we used the libwfa module 43, 44 of Q-Chem. Natural orbitals are eigenstates of the one particle density matrix and their eigenvalues can be interpreted as the occupation numbers,n i . Using the spin-average occupation numbers, n i , several ways to compute an effective number of unpaired electrons from the one-particle density matrices have been proposed. 46–48 In this work, we make use of the n u;nl index, proposed by Head- Gordon 47 as an extension of work by Yamaguchi et al.: 46 n u;nl = X i n 2 i (2 n i ) 2 (4.4) 121 Natural orbitals along the singlet pathway correspond to the lowest-lying EOM-SF- CCSD/cc-pVDZ singlet states obtained from the high-spin triplet reference using singlet geometries. Natural orbitals along the triplet pathway were obtained from the high-spin triplet at the CCSD/cc-pVDZ level of theory using triplet geometries. In all figures, we only show-orbitals, as the shapes of paired and natural orbitals are the same. We reportn u;nl (Eq. 4.4) and natural occupations of the frontier orbitals. Because the current implementations of the SF methods are not spin-adapted, the SF states show some (usually small) spin-contamination even if ROHF references are employed. Con- sequently, and frontier orbitals (and the respective occupations) are slightly dif- ferent. Below we report spin-average natural occupation, n i , as well as the difference between the and natural occupations: n =jn +n j (4.5) n =(jn n j): (4.6) The latter quantity provides an additional measure of spin-contamination. 4.3 Results and Discussion Below we discuss the geometries and electronic structure of each species along the sin- glet and triplet pathway, followed by a discussion of the energetics of the two pathways. 4.3.1 Singlet Pathway 1 R, 1 TS, and 1 P have C 2v , C 2v , and D 2h point group symmetry, respectively (Figure 4.4). The 1 R structure reported in this study (calculated at the CCSD/cc-pVTZ level) agrees well with previous studies; our value of the critical C 1 -C 6 distance is 0.057 ˚ A larger than the experimentally derived value of 4.321 ˚ A. 22 The geometry of 1 TS is very product 122 like, which is in agreement with previous theoretical results. 15, 19, 21 The geometry of 1 P is a distorted benzene structure with the bonds between the two carbons attached to a hydrogen, r 34 and r 16 , longer than the C-C bonds containing a radical center. This distortion is reported in other theoretical studies as well. 15–17, 19, 24, 29 1 R is a well behaved closed-shell species. The highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) correspond to the C 3 - C 4 (2b 2 ) and (2a 2 ) orbitals, respectively. This suggests that a HOMO to LUMO excitation would effectively break the C 3 -C 4 -bond. The large vertical singlet-triplet gap (E ST ) of 130.9 kcal/mol (Table 4.1) supports this idea. Table 4.1: Vertical singlet-triplet gaps for each structure. Adiabatic gaps for R, TS, P (ZPE corrected). All values are in kcal/mol. A positive value indicates that the singlet is lower in energy. Structure Vertical E ST Adiabatic E ST 1 R 130.9 46.83 3 R 1.11 – 1 TS 42.78 50.14 3 TS 22.32 – 1 P 5.14 3.56 3 P 3.15 – The electronic structure of 1 TS does not show significant multi-configurational char- acter. It is dominated by a single electronic configuration, with a weight of 79% and the next leading configuration being less than 5% (Table 4.2. The number of effectively unpaired electrons for this state is 0.09 (Table 4.3). Thus, both wave function anal- ysis and Head-Gordon indices indicate that the transition state is electronically more reactant-like than to the product-like. 21 The HOMO and LUMO are the anti-symmetric (8b 1 ) and symmetric combination (10a 1 ) of C 2 and C 5 radical lobes, respectively, which suggests that there is coupling through the C 3 -C 4 sigma bond. 53, 54 123 Figure 4.4: Stationary points along singlet and triplet pathways. 1 R optimized with CCSD/cc-pVTZ. 1 TS and 1 P optimized with EOM-SF-CCSD/cc-pVTZ. 3 R, 3 TS, and 3 P optimized with CCSD/cc-pVTZ. Images generated with Jmol. 124 Table 4.2: Leading electronic configurations of singlet and triplet states at the re- spective optimized structures. [Core] 30 denotes the first 15 doubly occupied lower energy molecular orbitals. State Orbital Occupancy 1 R 90% [Core] 30 , 1b 2 2 , 1a 2 2 , 9a 1 2 , 8b 1 2 , 2b 2 2 , 2a 2 0 , 12a 1 0 3 R 78% [Core] 30 , 9a 2 , 8b 2 , 9b 2 , 10a 2 , 11a 1 ,10b 1 , 14a 0 1 TS 79% [Core] 30 , 1b 2 2 , 9a 1 2 , 1a 2 2 , 2b 2 2 , 8b 1 2 , 10a 1 0 , 2a 2 0 3 TS 80% [Core] 30 , 9a 2 , 8b 2 , 10a 2 , 9b 2 , 10b 1 , 11a 1 , 12a 0 1 P 53% [Core] 3 0, 5a g 2 , 1b 1u 2 , 1b 3g 2 , 1b 2g 2 , 5b 2u 2 , 6a g 0 , + 24%[Core] 30 , 5a g 2 , 1b 1u 2 , 1b 3g 2 , 1b 2g 2 , 5b 2u 0 , 6a g 2 3 P 77% [Core] 30 , 5a g 2 , 1b 1u 2 , 1b 3g 2 , 1b 2g 2 , 5b 2u 1 , 6a g 1 The ground state of P is a singlet that exhibits a multi-configurational wave function with two leading configurations (Table 4.2) and n u;nl = 0.28. The dominant configu- ration has the asymmetric combination of radical orbitals (5b 2u ) doubly occupied and lower in energy than the symmetric combination (6a g ), a result of through bond cou- pling. 53, 54 4.3.2 Triplet Pathway Equilibrium structures of 3 R, 3 TS, and 3 P have C 2 , C 2 , and D 2h point group symmetry, respectively. The “arms” of 3 R distort out of plane 92.3 degrees and the C 3 -C 4 bond length is 1.488 ˚ A, indicating that the double bond present in 1 R is weakened in 3 R. Similarly, 3 TS is distorted out of plane by 30 degrees and the C 3 -C 4 bond is elongated to 1.503 ˚ A. The geometry of 3 P is very similar to that of 1 P, with the exception that there is more delocalization in the triplet, as indicated by the smaller differences inr 12 and r 16 . The distortion of 3 R can be explained by analyzing the molecular orbitals of 1 R and 3 R. As mentioned above, the HOMO and LUMO of 1 R are the C 3 -C 4 and orbitals. A HOMO to LUMO excitation would result in a triplet state with a single electron in 125 both the and orbitals, reducing the bond order to one. Without the bond to hold the arms of 3 R in plane, they are free to rearrange in order to minimize electronic repulsion. The resulting singly occupied molecular orbitals (SOMOs), 11a and 10b, are the in-phase and out-of-phase combinations of the two p-orbitals on C 3 and C 4 . In 1 R combine to make the and orbitals, 2b 2 and 2a 2 . In 3 R, the C 3 and C 4 are nearly perpendicular, which reduces their interaction and causes the SOMOs (10b and 11a) of 3 R to be nearly degenerate (Figure 4.7). (Figure 4.4). Visualization of the unpaired spin density and frontier natural orbitals shows that in 3 R the unpaired electrons are localized to C 3 and C 4 p-type orbitals (Figures 4.7, 4.8 and 4.9). This arrangement admits some bonding interaction, which is why the C 3 -C 4 bond length is shorter than a standard C-C single bond. This process is reversed as the triplet reaction pathway proceeds and the molecule becomes more planar. The SOMOs of 3 TS are the analogous orbitals of the 3 R SOMOs. The 3 TS SOMOs are a combination of p-orbitals on C 3 -C 4 and p-orbitals on C 1 and C 6 . 3 TS is planar enough that the in-phase and out-of-phase combinations of the C 3 - C 4 p-orbitals form and orbitals, each occupied by a single electron (Figure 4.6), 3 TS, orbitals 11a and 10b). Consequentially, there is only a single bond between C 3 - C 4 , as evidenced by the bond length. Our C 3 -C 4 bond length of 1.503 ˚ A is larger than previously reported values of 1.486 and 1.461 ˚ A. 15, 19 The C 3 -C 4 bond partly explains the high energy of 3 TS. As the cyclization proceeds the interaction between electrons on the arms of the molecule increases. Out-of-plane electrons will have more interaction as the molecule becomes more planar and in-plane electrons interact more as the two arms are brought closer together. In-phase and out-of- phase orbitals that were nearly degenerate for 3 R are now energetically separated. 3 TS MOs that bring in-phase electron densities together, such as orbitals 9a, 8b, and 10b, 126 are stabilized, while orbitals that bring out-of-phase densities together, such as orbitals 10a, 9b, and 11a, are destabilized (Figure 4.4). Altogether, there is a net destabilization, resulting in the high energy of 3 TS. This is supported by analysis of natural orbitals in Figure 4.9. In contrast to 1 P, the electronic structure of 3 P is dominated by a single configura- tion, unlike 1 P. The SOMOs of 3 P are the in-phase and out-of-phase combinations of the radical lobes on C 2 and C 5 , with the out-of-phase combination being lower in energy. 4.3.3 Wavefunction Properties and Natural Orbitals Comparing energy-ordered MOs (Figures 4.5 and 4.6) and population-ordered natural orbitals (Figures 4.8 and 4.9), frontier orbitals agree well, with respect to both character and ordering. Average occupations, n, show how many electrons are assigned to these orbitals. For the singlet product, n shows higher occupation of the natural orbitals aris- ing from the in-phase radical bonding lobes and lower occupation of the natural orbital corresponding to out-of-phase lobes (relative to 1 TS), consistent with increased diradi- cal character. For the triplet wave functions (Figure 4.9), we see that both frontier NOs are singly occupied at each stationary point on the reaction pathway. While character of MOs and natural orbitals is consistent, energy-ordering of MOs is not always indicative of the population-ordering of natural orbitals. This is particularly true of doubly-occupied orbitals, where one can reorder natural orbitals that have nearly degenerate occupations. Table 4.3 summarizes the wavefunction properties. hS 2 i and n values in Fig- ures 4.8 and 4.9 indicate very little spin contamination of the states. n values are high for frontier orbitals along the triplet pathway because these orbitals are associated with theM s = 1 triplet, and are therefore occupied by a single electron. Along the singlet 127 pathway,n u;nl indicates slight radical character even for 1 R and 1 TS, with a significant amount of radical character (n u;nl = 0.28) observed for 1 P. Table 4.3: Wavefunction properties of the reactant, transition state, and product in the singlet and triplet pathways of Bergman cyclization. State n u;nl hS 2 i 1 R 0.10 0.05 3 R 2.21 2.00 1 TS 0.09 0.03 3 TS 2.25 2.00 1 P 0.28 0.01 3 P 2.19 2.02 4.3.4 Energetics Our activation energy (H z ) and energy of reaction (H rxn ) of the cyclization reaction of 1 R are within 1 kcal/mol of Jones and Bergman’s original findings of 32 and 14 kcal/mol, respectively (Figure 4.10). 1 In addition, while our H z is slightly greater than the value of 28.1 1.6 kcal/mol reported by Wenthold and Squires, our H rxn is in excellent agreement with their value of 13 3 kcal/mol. 55 Our cyclization energetics differ from those reported by Roth et al (Table 4.4). 56 To the best of our knowledge, no experimental results exist for the cyclization of 3 R, so we can only compare with other theoretical studies. For the triplet cyclization, our H z value is significantly higher and our H rxn is considerably less exothermic than the previous results (Table 4.4). 15, 19 Our characterization of 3 TS is fundamentally different, as evidenced by significant difference in the C 3 -C 4 bond length, and produces a higher activation energy than the previous studies of Dong and Clark. 15, 19 Alternatively, if our absolute energy for 3 R is much lower than absolute energies found by Dong and 128 Table 4.4: H z and H rxn values for S 0 and T 1 cyclizations alongside available experimental and previous theoretical values. 1 H z 1 H rxn 3 H z 3 H rxn Expt 32 a , 28.2 b , 28.1 1.6 c 14 a , 8.5 b , 13 3 c – – This work 32.75 13.76 26.59 -33.29 CCSD(T)/6-31G(d/p) 28.5 8 – – CASMP2/6-31G* 24.22 -4.92 21.42 -41.8 CASPT2(12,12)/ANO-L 2.6 12.2 22.2 -37 B3LYP/6-31G** 31.2 3.3 23.7 -45.2 a Ref. 1 b Ref. 56 (Roth, 1994) c Ref. 55 (Wenthold, 1994) Clark, but the absolute energies of 3 TS and 3 P are similar, it would explain the relatively large activation barrier and less exothermic reaction enthalpy calculated in this study. Table 4.1 lists vertical and adiabatic singlet-triplet gaps for each structure. Our cal- culated singlet-triplet adiabatic gap value of 3.56 kcal/mol is in excellent agreement with Wenthold and Squire’s experimental value of 3.8 0.4 kcal/mol. 23 4.3.5 Method Comparison In this study, we explored multiple methods, basis sets, and reference wave functions. The results are compiled in Tables 4.5 and Table 4.6 in the Appendices. In Table 4.5, the energies are for the singlet reaction, which required a more complicated high-spin correction method (see Figure 4.3 for details of correction). The values in Table 4.6 are for the triplet reaction and are calculated by applying a ZPE correction to the raw energies of each triplet structure and then taking the difference (i.e., H(P) - H(R) = H rxn ). For the singlet reaction, basis set effects are significant, i.e. increasing the basis set from cc-pVDZ to cc-pVTZ results in approximately a 3 and 5 kcal/mol difference in the 129 1 H z and 1 H rxn , respectively. However, the effect is less pronounced in the triplet re- action. The differences in basis-set sensitivity between the singlet and triplet surfaces is expected; electron coupling and the multi-configurational nature of the singlet increases the importance of dynamical correlation which is better described with a larger basis set. It is also possible that the additional valence polarization functions in the triple zeta basis allow for better characterization of the long range interactions between the triple bond containing arms of R and TS as well as the through bond coupling in P. The inclu- sion of triple excitations (CCSD vs CCSD(T)) produces a difference of1 kcal/mol for the singlet surface. We do see a relatively large difference in the CCSD and CCSD(T) values of 3 H z but not in the values for H rxn . This underscores the beauty of using CCSD(T) results on the high spin pathway in conjunction with SF singlet-triplet gaps to characterize the low spin pathway. The inclusion of triple excitations (T) is impor- tant for quantitative accuracy and the balanced nature of the SF approach captures these effects in a balanced way. 4.4 Conclusions The singlet and triplet pathways of the Bergman cyclization were characterized using reliable and robust coupled-cluster methods, CCSD, CCSD(T), and EOM-SF-CCSD methods. We found that the singlet pathway has a barrier of 32.75 kcal/mol and is endothermic by 13.76 kcal/mol. We determined that the triplet pathway has a barrier of 26.59 kcal/mol and is exothermic by 33.29 kcal/mol. We found that 1 TS is dominated by a single, reactant-like, electronic configuration but is geometrically very similar to 1 P. Both 3 R and 3 TS were found to have C 2 geometries. The analysis of frontier orbitals of each stationary point allowed us to explain the large energetic costs of the triplet cyclization. 130 Additionally, this study illustrates the effectiveness of SF methods in characteriz- ing reactions involving both closed and open-shell species. Occupancy-ordered frontier natural orbitals generally agree with energy-ordered MOs. Density-based wave function analysis reveals the slight open-shell character of 1 R and 1 TS (n u;nl = 0.10 and 0.09 un- paired electrons, respectively), and the more pronounced radical character of 1 P (n u;nl = 0.28). 131 Figure 4.5: Transformation of frontier molecular orbitals along the singlet reaction coordinate. Orbitals were obtained using the HF/cc-pVTZ triplet reference state at the UCCSD/cc-pVDZ optimized structures. 132 Figure 4.6: Transformation of frontier molecular orbitals along the triplet reaction coordinate. Orbitals were obtained using the HF/cc-pVTZ triplet reference state at the UCCSD/cc-pVDZ optimized structures. 133 Figure 4.7: Unpaired spin densities for triplet optimized structures. From right to left: 3 R, 3 TS, 3 P. Isovalue of 0.075. IQmol used for visualization. 134 " = 0.10 (0.02) " = 1.89 (0.03) " = 0.09 (0.03) " = 1.94 (0.03) " = 0.21 (0.03) " = 1.81 (0.01) " = 2.00 (0.04) " = 2.00 (0.04) " = 2.00 (0.02) " = 2.00 (0.02) " = 2.00 (0.00) " = 1.93 (0.01) " = 1.97 (0.01) " = 2.00 (0.02) " = 1.99 (0.00) " = 2.00 (0.00) " = 1.95 (0.03) " = 1.99 (0.01) " = 1.99 (0.00) " = 1.99 (0.00) " = 2.00 (0.02) Figure 4.8: EOM-SF-CCSD/cc-pVDZ frontier natural orbitals of lowest singlet states of the reactant (left), transition state (middle), and product (right) in the singlet pathway of Bergman cyclization.-orbitals are shown. n =jn +n j, with n =jn n j provided in parentheses. 135 ! " = 0.98 (0.95) ! " = 0.98 (0.95) ! " = 0.98 (0.93) ! " = 1.01 (0.92) ! " = 0.99 (0.96) ! " = 1.00 (0.97) ! " = 1.92 (0.00) ! " = 1.92 (0.00) ! " = 1.93 (0.02) ! " = 1.95 (0.00) ! " = 1.94 (0.02) ! " = 1.89 (0.01) ! " = 1.92 (0.00) ! " = 1.92 (0.02) ! " = 1.93 (0.02) ! " = 1.95 (0.00) ! " = 1.91 (0.00) ! " = 1.91 (0.01) ! " = 1.94 (0.00) ! " = 1.95 (0.00) ! " = 1.95 (0.00) Figure 4.9: CCSD/cc-pVDZ frontier natural orbitals of high-spin triplet states of the reactant (left), transition state (middle), and product (right) in the triplet path- way of Bergman cyclization. -orbitals are shown. n = jn +n j, with n = jn n j provided in parentheses. 136 Figure 4.10: Energetic diagram along S 0 and T 1 pathways. Relative electronic energies are shown and ZPE corrected energies are in parenthesis. All values are in kcal/mol. The 3 R energy shown is the R adiabatic E ST . The 3 TS and 3 P energies relative to 1 R were calculated by adding the TS and P adiabatic E ST;a , to the energy of 1 TS and 1 P, respectively. 137 4.5 Appendix A: Key values: S 0 Pathway 138 Table 4.5: Comparison of key values for S 0 reaction with multiple combinations of method, basis, and spin treatment. Values are from single point calculations (ZPE corrected) on optimized structures in the manner described above. Ref- erence wavefunctions used for each calculation are indicated. When adopting a restricted approach, RHF was used for reactant and ROHF was used for TS and product. This is indicated in the table as RHF/ROHF. Note: Spin flip target state energies are all calculated at the CCSD level even though reference wave functions are optimized at CCSD(T) level. H.S denotes high spin, see Figure 4.3 for details. Total energy of the singlet reactant has units of a.u, all other values are in units of kcal/mol. Method Basis Reference H s (R) 1 H z (H.S.) 1 H rxn (H.S.) Vert. E TS (TS) E TS (TS) 1 H z 1 H rxn CCSD cc-pVDZ RHF/ROHF -230.136568 73.80 11.94 43.10 5.23 30.70 6.71 CCSD cc-pVDZ UHF -230.136568 73.74 11.88 42.98 5.21 30.77 6.67 CCSD cc-pVTZ RHF/ROHF -230.416548 76.20 16.97 42.04 4.84 34.16 12.12 CCSD cc-pVTZ UHF -230.416548 76.13 16.89 41.95 4.81 34.18 12.08 CCSD(T) cc-pVDZ RHF/ROHF -230.174304 72.24 12.84 43.10 5.23 29.14 7.60 CCSD(T) cc-pVDZ UHF -230.174304 72.45 12.86 42.98 5.21 29.48 7.65 CCSD(T) cc-pVTZ RHF/ROHF -230.471452 74.81 18.38 42.04 4.84 32.76 13.53 CCSD(T) cc-pVTZ UHF -230.471452 75.03 18.41 41.95 4.81 33.09 13.60 139 4.6 Appendix B: Key Values: T 1 Pathway 140 Table 4.6: Comparison of the impact that method, basis set, and spin restrictions have on total energies (a.u) and reaction energetics for the T 1 reaction. Values for triplet H z and H rxn are calculated from ZPE corrected total energies and are in units of kcal/mol. The spin heading refers to the treatment of the reference wavefunction where R = restricted open-shell HF and U = unrestricted HF. Method Basis Reference E T (R) E T (TS) E T (P) 3 H z 3 H rxn CCSD cc-pVDZ ROHF -230.061728 -230.008426 -230.119157 33.45 -36.04 CCSD cc-pVDZ UHF -230.063096 -230.008722 -230.119254 34.12 -35.24 CCSD cc-pVTZ ROHF -230.338612 -230.282007 -230.390813 35.52 -32.76 CCSD cc-pVTZ UHF -230.340023 -230.282253 -230.390933 36.25 -31.95 CCSD(T) cc-pVDZ ROHF -230.096907 -230.056873 -230.155435 25.12 -36.73 CCSD(T) cc-pVDZ UHF -230.095837 -230.052881 -230.155401 26.96 -37.38 CCSD(T) cc-pVTZ ROHF -230.390356 -230.348047 -230.443434 26.55 -33.31 CCSD(T) cc-pVTZ UHF -230.389099 -230.343853 -230.443377 28.39 -34.06 141 4.7 Appendix C: Zero Point Energies Table 4.7: Zero Point Energy values for all structures optimized with cc-pVDZ basis set. Calculated with same method as respective geometry optimization. ZPE kcal/mol a.u. 1 R 44.135 0.0703 3 R 41.184 0.0656 1 TS 44.016 0.0701 3 TS 41.587 0.0662 1 P 46.962 0.0748 3 P 46.921 0.0747 Table 4.8: Zero Point Energy values for all structures optimized with cc-pVTZ basis set. Calculated with same method as respective geometry optimization. ZPE kcal/mol a.u. 1 R 45.210 0.0720 3 R 42.033 0.0670 1 TS 44.850 0.0715 3 TS 42.265 0.0674 1 P 47.603 0.0759 3 P 47.586 0.0758 142 4.8 Appendix D: Cartesian Geometries 1 R optimized at the CCSD/cc-pVTZ (RHF) level of theory. C 0.6789839454 0.0000000000 1.1164673871 H 1.2072792747 0.0000000000 2.0761984111 C -0.6789839454 0.0000000000 1.1164673871 H -1.2072792747 0.0000000000 2.0761984111 C -1.4918870754 0.0000000000 -0.0739007634 C -2.2194127129 0.0000000000 -1.0586761896 H -2.8410074521 0.0000000000 -1.9380455665 C 1.4918870754 0.0000000000 -0.0739007634 C 2.2194127129 0.0000000000 -1.0586761896 H 2.8410074521 0.0000000000 -1.9380455665 3 R optimized at the CCSD/cc-pVTZ (UHF) level of theory. C 0.6161557707 -0.4171175166 0.7868417918 H 0.6824880037 -1.2822653593 1.4585851996 C 1.7060746322 -0.1199005307 -0.0361227786 C 2.6748903608 0.1445176228 -0.7609375275 H 3.5115460918 0.3773635960 -1.3972741134 C -0.6161557707 0.4171175166 0.7868417918 H -0.6824880037 1.2822653593 1.4585851996 C -1.7060746322 0.1199005307 -0.0361227786 C -2.6748903608 -0.1445176228 -0.7609375275 H -3.5115460918 -0.3773635960 -1.3972741134 1 TS optimized at the EOM-SF-CCSD/cc-pVTZ (UHF) level of theory. C -0.6962171091 0.0000000000 1.2426144619 H -1.2270361976 0.0000000000 2.1982298597 C 0.6962171091 0.0000000000 1.2426144619 H 1.2270361976 0.0000000000 2.1982298597 C 1.3805872931 0.0000000000 -0.0047728123 C 1.0058693857 0.0000000000 -1.2260459303 H 1.3000571832 0.0000000000 -2.2692425278 C -1.3805872931 0.0000000000 -0.0047728123 C -1.0058693857 0.0000000000 -1.2260459303 H -1.3000571832 0.0000000000 -2.2692425278 143 3 TS optimized at the CCSD/cc-pVTZ (UHF) level of theory. C 0.0172611334 -1.3751179922 0.0146488037 C -0.1493968878 -1.0367031089 -1.2211447755 H -0.5891271484 -1.3556275905 -2.1627052487 C 0.1908140060 -0.7349506826 1.2150859297 H 0.6299345349 -1.1847909153 2.1106173498 C -0.1908140060 0.7349506826 1.2150859297 H -0.6299345349 1.1847909153 2.1106173498 C -0.0172611334 1.3751179922 0.0146488037 C 0.1493968878 1.0367031089 -1.2211447755 H 0.5891271484 1.3556275905 -2.1627052487 1 P optimized at the EOM-SF-CCSD/cc-pVTZ (UHF) level of theory. C 1.2241688795 -0.7164516875 0.0000000000 H 2.1806312511 -1.2475283865 0.0000000000 C 0.0000000000 -1.3617239033 0.0000000000 C -1.2241688795 -0.7164516875 0.0000000000 H -2.1806312511 -1.2475283865 0.0000000000 C -1.2241688795 0.7164516875 0.0000000000 H -2.1806312511 1.2475283865 0.0000000000 C 0.0000000000 1.3617239033 0.0000000000 C 1.2241688795 0.7164516875 0.0000000000 H 2.1806312511 1.2475283865 0.0000000000 3 P optimized at the CCSD/cc-pVTZ (UHF) level of theory. 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MMs are also of interest in the context of quantum computing and quantum information storage. 3–5 Since the 1990’s much attention has been given to the design and study of organometallic complexes contain- ing d- and f-block elements—including terbium, manganese, chromium, nickel, and copper—whose electronic structure (and consequent magnetic properties) make them suitable candidates for use in magnetic materials. 6 Suitable candidates for magnetic ap- plications should have one or more unpaired, weakly interacting electrons. Here, we limit ourselves to systems containing only two unpaired electrons (diradicals). 7–11 Elec- tronic configurations that can be generated for two electrons in two orbitals are shown in Figure 5.1. 149 (i) ! + + - (ii) (iii) (iv) -! (v) Figure 5.1: Wave functions of diradicals that are eigenfunctions ofS 2 (only config- urations with positive spin projections are shown). Wave function (i) corresponds to the high-spin M s = 1 triplet state. Wave functions (ii)-(v) correspond to the low- spin states: M s = 0 singlets and triplets. Note that all M s = 0 configurations can be formally generated by a spin-flipping excitation of one electron from the high-spin M s = 1 configuration. The triplet state of a diradical can be represented by either a single Slater determi- nant (high-spin M s =1 configuration (i)) or by a linear configuration of two M s =0 deter- minants with equal weights (configuration (ii)). Configurations (iii)-(v) give rise to three singlet states. The energy gaps, relative state ordering, and relative weights of the Slater determinants (i.e., coefficients in Fig. 5.1) depend on the nature and energy separation of the respective frontier molecular orbitals (MOs). The character of the MOs also de- termines the character of the wave function, e.g., whether it is predominantly covalent (i.e., two electrons residing on different parts of a molecule) or ionic. Detailed analysis of different types of diradical electronic structure can be found in classic papers. 7, 8 In the context of MMs, the two states of interest are the lowest singlet and triplet states. In systems like binuclear copper complexes, one expects these two states to have covalent wave functions in which the unpaired electrons are localized on the two metal centers: s;t (1; 2) [ Cu1 (1) Cu2 (2) Cu2 (1) Cu1 (2)] [(1)(2)(1)(2)] (5.1) with little-to-no contributions from ionic configurations [ Cu1 (1) Cu1 (2) + Cu2 (1) Cu1 (2)] [(1)(2)(1)(2)] (5.2) 150 In these expressions, Cu1 and Cu2 denote orbitals localized on the two copper cen- ters, such as copperd-orbitals (perhaps including small contributions from the nearest ligand atoms). If the actual MOs hosting the unpaired electrons are delocalized and can be described as (nearly degenerate) bonding and antibonding combinations of Cu1 and Cu2 (as is the case in the MMs studied here), the triplet states are described by config- urations (i) and (ii) and have pure covalent character, as t in Eq. (5.1). The character of the lowest singlet state can vary, depending on the exact weights of configurations (iii) through (iv). A purely covalent singlet wave function, s of Eq. (5.1), corresponds to configuration (v) with =1. A smaller value of gives rise to the ionic configu- rations mixed into the wave function. This happens when the interaction between the two centers stabilizes the bonding MO relative to the antibonding one, either due to through-space or through-bond interactions. The ionic configurations can also appear in the singlet wave functions due to mixing with configuration (iii), which, in contrast to (ii), has pure ionic character. Since all configurations, (iii)-(v), can contribute into the singlet state, the correct description of this state requires an electronic structure method that treats (iii)-(v) on an equal footing. In order to describe relative energies of singlet and triplet states, the method should provide a balanced and unbiased description of all four M s =0 configurations from Figure 5.1. From a theoretical perspective, the search for promising MMs begins with first- principle calculations of the relevant terms in the phenomenological spin Hamilto- nian. 2, 12 Of all terms in the spin Hamiltonian, 13–15 the most energetically significant one is the exchange-coupling interaction between unpaired, spatially separated elec- trons. The sign and magnitude of electronic exchange-coupling between atomic spin centers determines whether a molecule will behave ferromagnetically or antiferromag- netically upon exposure to an external magnetic field, and thus, whether the molecule is 151 suitable for application in a magnetic material. In bimetallic diradical complexes, like those considered here, the exchange-coupling constant equals the energy difference be- tween the lowest singlet and triplet states. 16 Thus, exchange-coupling terms of the spin Hamiltonian can be computed by ab initio methods as singlet-triplet energy gaps. This approach can be generalized to systems with more unpaired electrons. 17, 18 The main challenge in applying this simple strategy is in the multiconfigurational character of the low-spin wave functions, which becomes evident by inspecting Fig. 5.1. The high-spin states, such as M s =1 triplet (i), can be well represented by a single Slater determinant and, therefore, their energies can be reliably computed by standard single- reference methods, i.e., coupled-cluster theory or DFT. In contrast, wave functions of low-spin states (ii)-(v) require at least two Slater determinants. Consequently, they are poorly described by single-reference methods. This imbalance in the description of high-spin and low-spin states results in large errors in the computed singlet-triplet gaps. Several strategies have been employed for describing the open-shell states and exchange-coupling in MMs. Historically, the most popular are the broken symme- try (BS) methods 19–21 and spin-restricted Kohn-Sham (REKS/ROKS) methods. 21–23 Both approaches suffer from imbalance in their treatment—and sometimes, outright exclusion—of important configurations depicted in Figure 5.1. For example, in BS approach all singlet and triplet configurations are scrambled. While spin projection allows one to formally separate singlet and triplet manifolds, it does not distinguish between open- and closed-shell singlet states (iii)-(v). Despite belonging to entirely different states (which may even have different spatial symmetry), these configurations remain scrambled in spin-projected BS solutions. Although broken-symmetry solutions might represent charge density reasonably well, they result in erroneous spin-density. 12 152 Consequently, the quality of broken-symmetry description of properties that are deter- mined by charge density or by spin density can be rather different. The drawback of the REKS/ROKS scheme is that closed- and open-shell states are described by different sets of equations and that the effect of different dynamic correlation in the two manifolds is not fully accounted for. 24 A more balanced and robust alternative is the spin-flip (SF) family of methods. 25–30 Within SF approach, a single-determinant high-spin triplet state is used as a reference from which one computes the manifold of low-spin states arising from configurations (ii)–(iv) in Figure 5.1 by using spin-flipping excitations. The performance of various variants of SF method for organic polyradicals has been extensively benchmarked, fo- cusing on the energy gaps between the electronic states and their structures. 9, 28–36 Re- cently, we investigated the performance of SF methods in terms of characters of the underlying wave functions. 11 Specifically, we compared how different functionals and wave function methods describe frontier orbitals, their occupations, and the number of effectively unpaired electrons. Towards this end, we employed density-based wave function analysis tools. 37–39 By using natural orbitals and their occupations, this anal- ysis allows one to map correlated multiconfigurational wave functions into a simple two-electrons-in-two-orbitals picture from Figure 5.1. The goal of this study is to investigate the performance of SF methods for MM. In contrast to previously studied organic di- and tri-radicals, typical singlet-triplet gaps in MM are much smaller, within several hundreds of wave numbers, which calls for sub- chemical accuracy (1 kcal/mol = 350 cm 1 ). Additional challenges arise due to the pres- ence of heavy atoms, which might require using special basis sets and/or effective core potentials (ECPs). Although encouraging results of SF calculations for several MMs with transition metals have been reported before, 17, 18, 40–43 no systematic studies of the 153 effect of ECP (and/or basis sets) on multiplet splittings within the SF-TDDFT frame- work has been carried out. In the present work, we focus on a set of eight binuclear copper complexes whose magnetic properties have been extensively studied. 21, 40, 44–53 Given the above considerations and their large sizes, binuclear copper diradicals rep- resent a stringent test for the theory. We report the results of the EOM-SF-CCSD and SF-TDDFT calculations of singlet-triplet gaps (which are equal to exchange-coupling within the assumptions of Heisenberg-Dirac-van-Vleck (HDvV) model 13–15 ) and the bonding pattern (the shapes of frontier natural orbitals and the extent of interaction be- tween the formally unpaired electrons). We analyze the effects of basis sets, ECPs, TDDFT kernel, and molecular geometry. The structure of the article is as follows: the next section provides a brief theoretical overview of SF methods, the HDvV model 13–15 for exchange-coupling, and density- based analysis. The following section outlines computational details. We then illustrate the effects of the above mentioned variables on computed singlet-triplet gaps and wave function properties (frontier natural orbitals, their occupation, and the number of effec- tively unpaired electrons) of the eight binuclear copper compounds. In conclusion, we comment on the applicability of the HDvV model for this family of diradicals and the importance of density-based analysis in the description of open-shell systems. 5.2 Theoretical Methods 5.2.1 Diradicals and spin-flip approach As described above, the electronic states of diradicals, 7, 8, 10, 54 such as bicopper MMs, arise from two electrons distributed in two nearly degenerate MOs. For same-spin elec- trons, there is only one possible arrangement: M s 1. For the M s = 0 states, more configurations can be generated, as illustrated in Fig. 5.1. All configurations depicted in 154 Figure 5.1 can contribute to diradical spin-states of interest, which makes choosing an electronic structure method that treats all states on an equal footing, as SF does, of the utmost importance. Because in the SF approach there us no assumptions of the relative importance of configurations from Figure 5.1, these calculations allow one to unam- biguously determine the state character, e.g., to quantify the extent of the interaction between the unpaired electrons and the weights of ionic configurations in the respec- tive wave functions. Thus, SF methods are capable of distinguishing whether a given MM candidate falls within the regime of spatially separated, weakly-interacting spin moments, as assumed within the HDvV model. To describe diradicals within the SF framework, one begins with the high-spin M s =1 triplet state, which is well represented by a single Slater determinant. The low-spin states (singlets and the M s =0 triplets) appear as single excitations from the high-spin reference state. Thus, in the SF approach, a high-spin state is used as the reference state and the target manifold of states is generated by applying a linear spin-flipping excitation operator to the reference: s;t Ms=0 = ^ R Ms=1 t Ms=1 : (5.3) Using different approaches to describe correlation in the reference state gives rise to different SF methods. 25–30, 55 Our primary focus here is SF-TDDFT approach, 28, 29, 56–58 which is very attractive owing to its low computational cost. We compare the per- formance of several functionals chosen on the basis of the results of the previous stud- ies. 28, 29 For the benchmarking purposes, we also report the results of EOM-SF-CCSD, 27 a highly accurate wave-function method including advanced treatment of dynamical cor- relation. A density-based wave function analysis allows quantitative comparison of the character of underlying wave functions computed by different methods. 155 5.2.2 The Heisenberg-Dirac-van-Vleck Hamiltonian for localized, weakly interacting electrons The key assumption used to construct phenomenological spin Hamiltonians is is that the unpaired electrons are spatially separated and weakly interacting. The resulting magnetic Hamiltonian is: ^ H SPIN = X A ^ S A D A ^ S A + X A ^ S A A A ^ I A X AB ^ S A J ^ S B (5.4) where the ^ S are local electronic spin operators on nucleiA orB, the ^ I are nuclear spin moments, and D, A and J represent the zero-field splitting, hyperfine, and exchange- coupling interaction, respectively. 2, 12 Magnetic parameters D, A, and J are extracted from experiment, typically by fitting electron paramagnetic resonance, magnetic sus- ceptibility, or other macroscopic properties to a model equation. 15, 59–62 Solving this Hamiltonian allows one to predict magnetic behavior of the system in the presence or absence of the magnetic field. The parameters of Eg. (5.4)—such as hyperfine couplings, zero-field splittings, exchange-couplings— can be extracted from experimental measurements or computed from correlated many-electron wave functions or DFT. 2, 12 All phenomenological terms in Eq. (5.4) can be computed using existing relativistic and non-relativistic electronic structure methods. 20, 21, 40, 42, 53, 63–67 In this work, we focus on the third, non-relativistic term, which gives rise to the HDvV Hamiltonian: 13–15 ^ H HDvV = X A<B J AB ^ S A ^ S B : (5.5) The HDvV Hamiltonian describes weakly coupled localized electrons neglecting hy- perfine couplings and zero-field splittings. By solving Eq. (5.10), one can relate the energies of different spin states (singlets and triplets, in the case of diradicals) to the 156 experimental magnetic parameter, J, known as the exchange-coupling constant. 16 The sign of J determines whether a molecular magnet behaves ferromagnetically or antifer- romagnetically in an external magnetic field. Analytic solutions of Eq. (5.10) are used to construct a formula connecting macroscopic magnetic susceptibility with exchange- couplings. 68 This formula is used to extract the latter from the temperature-dependent measurements of magnetic susceptibility. Thus, all experimental values of J derived from these measurements equal to the true energy gaps only if the key assumption of the HDvV model, that the spins are weakly interacting and the contributions of the ionic configurations into the total wave function are negligible, hold true. Eq. (5.10) assumes two (or more) weakly interacting, spatially separated electronic spins localized on two (or more) atomic centers,A andB. Obviously, the configurations arising from paired (ionic) electronic configurations are not admitted. The justification for this choice of the model Hamiltonian is that for the weakly interacting unpaired electrons, the ionic configurations are much higher in energy than covalent triplet and singlet configurations of Eq. (5.1). What happens when ionic configurations are mixed into the singlet states in AF species? In this case, the energy of the siglet diradical state is stabilized via configu- ration interaction, leading to an increased singlet-triplet gap. While the macroscopic properties, such as temperature dependence of magnetic susceptibility will reflect this phenomenon, the the application of the HDvV-based model to fit the data would pro- duce J values that are too large, because larger values ofJ are necessary to describe an increased singlet-triplet gap in the absence of ionic configurations in the model. Thus, one should expect that for strongly AF species the experimentally derivedJ values are systematically overestimated. 157 J is directly related to the energies of spin states via the Land´ e interval rule. 16 With an electronic structure method that offers a balanced treatment of open and closed-shell states, one can calculate J as an energy difference between the spin states: E(S)E(S 1) =J AB (5.6) In diradicalsS andS 1 are triplet and singlet states: E(S) is the energy of the lowest- energy triplet state and E(S-1) is the energy of the lowest singlet state. A positive J gives rise to a ferromagnetic ground state, while negative J gives rise to an antiferromagnetic ground state. Note that a different form of Eq. (5.10) might be used, 17, 18 with a factor of 2 in front of the sum, giving rise to a different relationship between energy gaps and exchange-coupling. Thus, for meaningful comparisons between different studies, it is important to carefully check which form was used (and of course it is always safer to report energy gaps between physical states, e.g., E ST in the case of diradicals). 5.2.3 Quantifying radical character and bonding patterns by density-based analysis Due to the complexity of many-body wave functions and arbitrariness in orbital choices, the interpretation of realistic wave functions (and often even of Kohn-Sham states) in terms simple two-electron-in-two orbitals picture (as in Figure 5.1) is not straightfor- ward. In particular, in large open-shell systems canonical Hartree-Fock of Kohn-Sham orbitals often provide a poor representation of the frontier MOs even for relatively sim- ple high-spin states. 11 Figure 5.2 illustrates the difference between canonical Kohn- Sham orbitals and spin-density difference for copper diradicals: while the excess spin density exhibits an expected pattern (unpaired electrons are localized on d-orbitals of 158 copper), the two formally singly occupied canonical MOs are delocalized over the lig- ands. In this case, even assigning the state characters and verifying the correctness of the solution of the self-consistent field (SCF) procedure are problematic. !2 !1 !2 !1 CUAQAC02 CITLAT SOMOs SONOs SOMOs SONOs !2 !1 !2 !1 Spin Density Spin Density Figure 5.2: Spin difference densities, unrestricted singly occupied (SO) molecular and natural frontier orbitals of triplet CUAQAC02 (left) and CITLAT (right) at the B5050LYP/cc-pVTZ level of theory. CUAQAC02 has 202 electrons while CITLAT has 278, making their low-lying states and associated orbital surfaces numerous and complex. Reproduced with permission from Ref. 11. This problem is effectively addressed by using density-based analysis: 37–39 Natural orbitals and their occupations provide a clear and unambiguous picture of the essential electronic structure of open-shell systems. Natural orbitals, which are eigen-functions 159 of the one-particle density matrix, afford most compact representation of the wave func- tions. The respective eigen-values can be interpreted as occupations. Thus, by comput- ing natural orbitals for a given one-particle density matrix one can obtain frontier MOs representing, for example, the unpaired electrons in diradicals. The shapes of these or- bitals allow one to asses the extent of localization of the unpaired electrons and their through-space and through bond interactions with each other and with other moieties. We note that when two frontier orbitals are exactly singly occupied (such as in perfect diradicals), their choice is not unique and any linear combination provides a legitimate representation (for example, in H 2 with a completely broken bond, one can consider ei- ther a pair of localized atomic-like orbitals or a delocalized bonding-antibonding pair). By using natural occupations, one can define and compute the number of effectively unpaired electrons. In this work, we use two indices,n u andn u;nl , proposed by Head- Gordon: 37 n u = X i min( n i ; 2 n i ) (5.7) n u;nl = X i n 2 i (2 n i ) 2 (5.8) In both equations, the sum runs over all natural orbitals and the contributions of the doubly occupied and unoccupied orbitals are exactly zero. While both formulas give identical answers for the limiting cases (such as a two-electron triplet state),n u;nl deliv- ers more physically meaningful and consistent results for more complex wave functions. By comparing occupations of the frontier natural orbitals andn u;nl one can assess how well a two-electrons-in-two-orbitals picture represents the real wave function. We recently applied this approach to study prototypical di- and triradicals. 11 Here, we extend this approach 11 to a set of 8 binuclear copper diradicals. We quantify the degree of radical character using Head-Gordon’s indices, 37 Eqs. (5.7) and (5.8), and 160 visualize natural frontier orbitals to characterize the localization and the interactions between the unpaired electrons. Visual inspection of natural orbitals, paired with the analysis of the respective natural occupations and the computed number of effectively unpaired electrons, obtained from a variety of SF methods, provides a robust way of val- idating the applicability of the HDvV model (and the Land´ e interval rule) for describing this family of MM candidates. 5.3 Computational Details Figure 5.3 shows the experimental structures used in calculations of singlet-triplet gaps of benchmark systems. Table 5.1 provides their associated experimental exchange cou- pling constants. Counterions were removed from all structures. The ferrocene group in the experimental PATFIA complex was also removed (as in previous studies 40, 53 ), unless otherwise specified. Experimental geometries are used in all EOM-SF-CCSD and SF-TDDFT calcula- tions, unless otherwise specified. To test possible effects of uncertainty of the structure, we also considered optimized structures of the high-spin triplet states of BISDOW, PAT- FIA (without the ferrocene group), and CITLAT. These optimizations were performed using the!B97X-D 69 functional and all-electron cc-pVTZ basis. The results for opti- mized structures are presented in the appendices. In the tables below, we report singlet-triplet gaps, E ST , defined as: E ST =E S E T : (5.9) Within the HDvV model, E ST =J, by virtue of Eq. (5.6). We compared the following DFT functionals in the SF-TDDFT calculations of E ST : LDA (with Slater exchange and VWN correlation), several members of the 161 Becke-exchange/LYP correlation family: BLYP, 70, 71 B3LYP, 72 and B5050LYP (50% Hartree-Fock + 8% Slater + 42% Becke for exchange and 19% VWN + 81% LYP for correlation). 29 Of the P86 correlation (with Becke exchange) and PW91 families, we chose the BP86, B3P86, 73, 74 PW91, and B3PW91 functionals. 75–78 From the PBE fam- ily, we selected the PBE, PBE0 (75% PBE and 25% Hartree-Fock exchange, 100% PBE correlation), PBE50 (50% PBE and 50% Hartree- Fock exchange and 100% PBE cor- relation), and!PBEh (80% PBE, 20% Hartree-Fock exchange and long-range Hartree- Fock exchange, 100% PBE correlation) functionals. 79, 80 Of the Minnesota family of functionals, we chose GAM (GGA), 81 MN15-L (meta-NGA), 82 and MN15 (hybrid with 44% Hartree-Fock exchange and MN15 correlation). 83 All of these functionals (with the exception of GAM, MN15-L and MN15, which are recent additions to the Minnesota family) have been extensively benchmarked with SF-TDDFT for organic polyradicals, wherein hybrid functionals such as PBE50, B5050LYP, and PBE0 were shown to produce relative state energies approaching chemical accuracy. 28, 29 In this pa- per, we present a similar error analysis of collinear and non-collinear SF-TDDFT with the above functionals in calculating energy gaps of binuclear copper complexes. We also present EOM-SF-CCSD/cc-pVDZ energy splittings for selected complexes. The core electrons were frozen in all EOM calculations. For all SF-TDDFT and EOM- SF-CCSD calculations, we report only electronic energy separations between singlet and M s = 0 triplet states (E ST ). Because of the similarity of the electronic structure of the singlet and triplet states in the case of very weakly interacting electrons, we expect that both the structures and vibrational frequencies of the two states are very close. We performed density and wave function analysis using the libwfa module 38 con- tained in the Q-Chem electronic structure package. 84, 85 We analyzed singlet and triplet states of all eight benchmark complexes at the PBBE50/cc-pVDZ level of theory, and 162 four of the eight complexes (BISDOW, CUAQAC02, XAMBUI, and CITLAT) at the EOM-SF-CCSD/cc-pVDZ level of theory. PBE50, B97, and LDA functionals were used with the non-collinear SF-TDDFT kernel 29, 86, 87 in the analysis of CUAQAC02. Collinear SF-TDDFT with the B5050LYP functional, which was recommended in the original SF-TDDFT paper, 28 was also used in the density analysis of CUAQAC02. We also present frontier natural orbitals of the full experimental structure of the PATFIA complex, which includes a ferrocene group, at the collinear SF-TDDFT/cc-pVDZ level with the B5050LYP functional. We compare these results to those obtained from the simplified structure without the ferrocene group. The performance of Dunning’s cc-pVDZ and cc-pVTZ all-electron basis sets are compared for all eight complexes and several functionals. Table 5.12 in Appendix B provides an atom-by-atom description of the effective core potentials (ECPs) and matching basis functions used in our study. Included are LANL2DZ (a non-relativistic ECP calibrated with Hartree-Fock total energies), 88 SRSC (a relativistic ECP fitted with all electron-eigenvalues and charges), 89 and CRENBL (a relativistic ECP fitted with Hartree-Fock valence orbital energies). 90, 91 Also included is the ECP10MDF pseudopotential, 92, 93 a relativistic ECP designed to reproduce valence energy spectra. ECP10MDF has been shown to perform well in the EOM-CCSD analysis of small cop- per compounds. 94, 95 We performed all calculations with the Q-Chem electronic structure package. 84, 85 Molecular orbitals were rendered using IQmol 96 and natural orbitals were rendered us- ing Jmol. 163 CITLAT (Complex 8) YAFZOU (Complex 7) Cu 2 Cl 6 2- (Complex 5) XAMBUI (Complex 6) CAVXUS (Complex 3) PATFIA (Complex 4) BISDOW (Complex 1) CUAQAC02 (Complex 2) C O Cl Cu N H Figure 5.3: Eight binuclear copper complexes included in this study. Structures are denoted with their Cambridge Structural Database names. 97, 98 Complexes 1,3, 4, 6, and 7 have a charge of 2+; Complex 2 is neutral; Complex 5 has charge of 2-; and Complex 8 has charge of 1+. Table 5.1: Experimental exchange-coupling constants for eight binuclear copper diradicals shown in Figure 5.3. Complex J (cm 1 ) Reference BISDOW -382 45 CUAQAC02 -286 46 CA VXUS -19 47, 48 PATFIA -11 49 Cu 2 Cl 2 6 0, -40 50 XAMBUI 2 51 YAFZOU 111 52 CITLAT 113 44 164 5.4 Results and discussion 5.4.1 The comparison of singlet-triplet gaps computed by different methods We begin by comparing computed singlet-triplet gaps against experimentally derived exchange-coupling constants. Mean absolute errors (MAEs) for each DFT functional and EOM-SF-CCSD are presented in Figure 5.4. Tabulated mean errors (ME), mean absolute errors ( MAE ), and standard deviations of the error ( STD ) are provided in the appendices for the non-collinear kernel, the PBE0, PBE50, and B5050LYP functionals, and various all-electron basis sets and ECPs. Hybrid functionals—in particular, LRC-!PBEh, PBE0, PBE50, and B5050LYP— outperform LDA and GGA functionals and approach the accuracy of EOM-SF-CCSD. As will be shown in subsequent sections, errors against the experiment are non-uniform among antiferromagnetic (AF) and ferromagnetic (F) complexes and the functionals and kernels that yield the closest agreement with experiment and EOM-SF-CCSD vary de- pending on the sign of J (i.e., whether the complex exhibits a singlet or triplet ground state). Although the lowest MAE is observed for the MN15 functional, energy differ- ences between the high-spin triplet reference and the M s =0 component of the triplet state are often large (on the order of 2 eV) and spin-contamination of target states high. The non-collinear kernel outperforms the collinear SF-TDDFT kernel for all func- tionals, with the exception of hybrids PBE50 and B5050LYP, where the collinear kernel shows modest improvement. For GGA functionals, use of the collinear kernel is tanta- mount to Koopmans’ estimate of the gaps using Kohn-Sham orbital energies, as the spin- flipped configurations are not coupled in the absence of the Hartree-Fock exchange. 28 Consequently, energy errors are high for these functionals, relative to results obtained 165 with the non-collinear kernel, and spin contamination of target states significant (see appendices). The scatter plot in Figure 5.5 illustrates the singlet-triplet gaps of all eight complexes computed by several hybrid functionals and EOM-SF-CCSD. An idealized line show- ing perfect agreement between experimental values ofJ and theoretical singlet-triplet gaps is also shown. All five selected functionals capture the qualitative trend and repro- duce the sign of the exchange-coupling. The errors are relatively small for complexes with near-zero experimental exchange-coupling and for complexes with a triplet ground state. We observe better agreement with experiment when hybrid functionals with 50% HF exchange are applied to molecules with a triplet ground state. For most function- als, the discrepancy between theoretical and experimental values are quite large for the two complexes with singlet ground states, BISDOW and CUAQAC02. Table 5.16, in which EOM-SF-CCSD/cc-pVDZ results for exchange-coupling are presented, shows the same trend. Again, we note good agreement with experiment for complexes that ex- hibit near-zero or ferromagnetic coupling, while substantial discrepancy is observed for anti-ferromagnetic complexes. It will be shown in Section 5.4.5 that this discrepancy arises not from electronic structure method failure, but the limitations of the HDvV model in the case of the singlet ground states of more strongly AF complexes. 5.4.2 SF-TDDFT: The effect of ECP and basis sets Tables 5.3 – 5.6 provide singlet-triplet gaps calculated with four hybrid functionals, and seven ECP/basis set combinations. For the PBE0 and LRC-!PBEh functionals, results obtained with the collinear kernel are tabulated in the appendices. The use of smaller basis sets and ECPs affords computational savings, which is attractive for large sys- tems, however, careful benchmarking is required to determine the effect of additional approximations on the computed quantities. We find that regardless of functional, there 166 57 51 58 65 68 78 86 99 431 494 500 500 581 38 60 60 1738 2078 MAE ST Gap (cm -1 ) Noncollinear Kernel Collinear Kernel Figure 5.4: Mean absolute error (MAE) in the singlet-triplet gap for the eight cop- per benchmark systems. Error relative to experimental values of the exchange- coupling constant, J, are presented for fifteen density functionals and EOM-SF- CCSD. The all-electron cc-pVDZ basis set was used for all atoms. Table 5.2: EOM-SF-CCSD/cc-pVDZ singlet-triplet gaps (computed using the X- ray structures) and experimental exchange-coupling constants, in cm 1 . Complex EOM-CCSD E ST hS 2 i a Exp.J BISDOW -208 b 2.010 -382 CUAQAC02 -180 c 2.006 -286 Cu 2 Cl 2 6 -16 c 2.009 0, -40 YAFZOU 86 c 2.022 111 a hS 2 i of the reference high-spin triplet. b Cholesky decomposition with a tolerance of 1e-2. Virtual orbitals frozen above 4.5 eV . c Cholesky decomposition with a tolerance of 1e-3. is little difference in quality of results when a triple-zeta, rather than double-zeta, ba- sis set is used, justifying the choice of a smaller DZ-quality basis for calculations of exchange-coupling in binuclear copper diradicals. Use of an ECP (regardless of ECP 167 Expt. (cm -1 ) Theory (cm -1 ) PBE50 LRC-!PBEh PBE0 B3LYP B5050LYP (col.) Figure 5.5: Theoretical singlet-triplet gaps computed with the PBE0, LRC-!- PBEh, B3LYP, and B5050LYP functionals (y-axis) versus experimentally derived values of the exchange-coupling parameter (x-axis). The inset in the bottom right zooms into the region spanning the range of -50 to 10 cm 1 . Dashed diagonal line marks perfect agreement between theoretical and experimental values. The all- electron cc-pVDZ basis was applied to all atoms. The collinear SF-TDDFT kernel was used with the B5050LYP functional, and non-collinear SF-TDDFT kernel with all other functionals. choice) appears to yield slightly better agreement with experiment for AF complexes, while little-to-no effect is observed for F complexes. Table 5.3: Singlet-triplet gaps (cm 1 ) for Complexes 1-8 calculated using SF- TDDFT with the LRC-!PBEh method and the non-collinear TDDFT kernel. BASIS COMPLEX 1 2 3 4 5 6 7 8 cc-pVDZ -444 -319 -24 -35 39 0 198 231 ECP10MDF/ cc-pVDZ -416 -296 -24 -44 40 0 184 213 LANL2DZ -510 -532 -20 38 -2 -1 244 263 EXP.J -382 -286 -19 -11 (0, -40) -2 111 113 168 5.4.3 SF-TDDFT: Collinear versus non-collinear kernel Figure 5.4 and Tables 5.3 – 5.6 present error analysis and values of the singlet-triplet gap computed with the collinear SF-TDDFT kernel and using the geometries from crys- tal structures. Figure 5.6 compares the singlet-triplet gap calculated for the BISDOW (AF), PATFIA (weakly AF), and CITLAT (F) complexes using the collinear and non- collinear SF-TDDFT kernels. The ECP10MDF/cc-pVDZ basis is used, where the ECP is applied only to copper atoms. Errors are non-uniform with respect to both functional and kernel choice, a result that is consistent with findings for other complexes not in- cluded in Figure 5.6. For BISDOW, the most anti-ferromagnetic complex in our study, the modest Hartree-Fock exchange of the PBE0 and LRC-!PBEh functionals paired with the non- collinear kernel appears to yield results in closest agreement with the experimental val- ues ofJ. However, even with the PBE0 and LRC-!PBEh functionals, the discrepancy is always larger than for ferromagnetic complexes. Exchange-coupling between cop- per centers in CITLAT, the most ferromagnetic complex in our study, is best captured by functionals with a higher percentage of Hartree-Fock exchange, such as that found in PBE50 and B5050LYP. The sign of the weakly AF complex, PATFIA, is captured by both the PBE0 and LRC-!PBEh functionals, regardless of kernel choice, while the magnitude is better represented by the PBE50 or B5050LYP functionals and the non- collinear kernel. 169 (D) (C) -382 -158 -213 Complex1 -11 15 -60 Complex4 113 112 75 Complex8 Experiment Non-Collinear Collinear -600 -400 -200 0 200 J (cm -1 ) -382 -142 -205 Complex1 -11 9 -57 Complex4 113 98 74 Complex8 Experiment Non-Collinear Collinear -600 -400 -200 0 200 J (cm -1 ) (A) (B) Figure 5.6: Singlet-triplet gaps for Complexes 1-8 calculated using the PBE0 (A), LRC-!PBEh (B), PBE50 (C), and B5050LYP (D) density functionals. The perfor- mances of the non-collinear and collinear TDDFT kernels are compared in (C) and (D). The ECP10MDF pseudopotential and matching cc-pVDZ-PP basis set were applied to Cu atoms. The cc-pVDZ basis was used for all non-Cu atoms. 5.4.4 The effect of geometry on the computed gaps To investigate the possible effect of geometry on the computed singlet-triplet gaps, we carried out geometry optimizations for three of the eight benchmark complexes: BIS- DOW, PATFIA, and CITLAT. The resulting structures are shown in Fig. 5.10 in Ap- pendix D. The most important structural parameter is the distance between the radical sites (Cu atoms). We found that the internuclear Cu distance of the optimized triplet geometries of BISDOW and CITLAT changed very little, with an increase of 0.03 ˚ A and a decrease of 0.02 ˚ A, respectively. We observed a larger change in the optimized 170 PATFIA complex (0.19 ˚ A), possibly due to the absence of the ferrocene group in the simplified structure, a topic that is examined more thoroughly in the next section. Table 5.7 summarizes exchange-coupling constants computed for the two types of geometries, four hybrid functionals and three ECP/basis combinations. Overall, we find that singlet-triplet gaps computed at a given level of theory changes very little with subtle changes in nuclear geometry, indicating that possible uncertainties in the experi- mentally derived geometries cannot account for the discrepancies between the computed and experimental exchange-couplings. Only for PATFIA, we observe a slightly better agreement between theory and experiment when using optimized structure. Overall, these results justify using crystal structures for exchange-couplings calculations. 5.4.5 Wavefunction analysis Energies alone are not sufficient to assess the performance of electronic structure meth- ods. 11 To gain further insight into performance of SF-TDDFT, we now investigate the characters of the underlying electronic states and compare relevant quantities (such as the number of effectively unpaired electrons) computed by SF-TDDFT and EOM-SF- CCSD. Table 5.8 presents density-based analysis of the lowest singlet and triplet SF- TDDFT states of all eight complexes at the PBE50/cc-pVDZ level of theory. Figure 5.7 shows the associated frontier natural orbitals. As expected, for all triplet states the number of effectively unpaired electrons as given byn u;nl is exactly 2 (n u gives slightly larger values, similarly to our previous study or organic diradicals 11 ). We observe that for the BISDOW and PATFIA complexes— for which the recommended SF-TDDFT methods and EOM-SF-CCSD consistently dis- agree with experiment—the number of unpaired electrons (i.e., values ofn u andn u;nl ) in the ground-state singlets are less than 2, indicative of weaker diradical character and mixing in of the ionic configurations. These effects are outside of the domain of the 171 HDvV Hamiltonian. The weakly AF complexes Cu 2 Cl 2 6 and PATFIA, for which top- performing SF-TDDFT methods disagree with experiment in both magnitude and sign, also exhibit weaker diradical character. CA VXUS (a weakly AF complex), XAMBUI (a complex that exhibits no exchange-coupling between radical centers), and the fer- romagnetic YAFZOU and CITLAT complexes all have singlet states with the n u;nl values at the ideal diradical value of 2. These are also the four complexes for which top-performing DFT functionals and EOM-SF-CCSD agree well with the experimental exchange-coupling values. Spin-contamination appears to be minor: for most complexes the deviation from the exacthS 2 i values are 0.01-0.03. In CA VCUS, the deviation is slightly larger (0.07 for the singlet state). The largest spin-contamination is observed in PATFIA where the hS 2 i of the singlet state is 0.36. Spin-balance is also evidenced by the natural orbitals occupations: jn n j values of exactly zero for BISDOW, CUAQAC02, Cu 2 Cl 2 6 , XAMBUI, and CITLAT. The frontier natural orbitals depicted in Figure 5.7 are all of the d xy or d yz type, exhibiting weak interaction with p-orbitals of nearest ligand atoms. Mixing of open- and closed-shell configurations is symmetry-allowed for CA VXUS, PATFIA, and YAF- ZOU. When spin-traced occupation numbers are exactly equal, both localized and delo- calzied frontier orbitals provide a valid representation of the density. Thus, the apparent localization of the frontier orbitals in CA VXUS, PATFIA, and the M s = 0 triplet state of CITLAT does not result from the spatial or spin symmetry breaking. We note that populatons associated with frontier natural orbitals are consistent with the computed number of the unpaired electrons, meaning that electronic structure of these compounds maps well into two-electrons-in-two-orbitals problem. 172 The d yz bonding/anti-bonding pair of frontier orbitals observed in CUAQAC02 is unique among the other copper diradicals considered in our study. The natural orbital orientation suggests a level of electron-electron interaction that might preclude the use of the HDvV Hamiltonian for modeling exchange-coupling, and lead to the observed imperfect diradical character of the singlet ground state. CUAQAC02 CAVXUS PATFIA BISDOW ! " s = 1.00 (0.33) ! " t = 1.00 (0.32) ! " s = 1.00 (0.75) ! " t = 1.00 (0.75) ! " s = 0.90 (0.00) ! " t = 1.00 (0.00) XAMBUI YAFZOU CITLAT Cu 2 Cl 6 2- ! " s = 1.00 (0.32) ! " t = 1.00 (0.33) ! " s = 1.00 (0.76) ! " t = 1.00 (0.75) ! " s = 1.10 (0.00) ! " t = 1.00 (0.00) ! " s = 0.99 (0.00) ! " t = 1.00 (0.00) ! " t = 1.00 (0.02) ! " s = 0.91 (0.00) ! " t = 1.00 (0.00) ! " s = 1.01 (0.00) ! " t = 1.00 (0.00) ! " t = 1.00 (0.02) ! " s = 1.09 (0.00) ! " t = 1.00 (0.01) ! " s = 0.92 (0.00) ! " t = 1.00 (0.00) ! " s = 1.08 (0.00) ! " t = 1.00 (0.00) ! " s = 0.98 (0.00) ! " t = 1.00 (0.00) ! " s = 1.02 (0.00) ! " t = 1.00 (0.00) ! " s = 1.03 (0.00) ! " s = 0.97 (0.00) Figure 5.7: Frontier natural orbitals of lowest singlet and M s = 0 triplet states of copper diradicals at the PBE50/cc-pVDZ level of theory. With the exception of YAFZOU, for other systems only orbitals of the triplet states are shown, since spatial extent of relevant paired and natural orbitals—and the appearance of frontier natural orbitals associated with the lowest singlet and triplet states— does not differ. n =jn +n j, withjn n j provided in parentheses. n s and n t correspond to n values obtained from the occupancies of the singlet and triplet natural orbitals, respectively. 173 Using the interesting case of CUAQAC02, Table 5.9 and Figure 5.8 illustrate the agreement between the SF-TTDFT/PBE50, SF-TDDFT/B5050LYP, and EOM-SF- CCSD, not just for the relative energetics of the states of interest, but their associated densities. While LDA and B97 fail to accurately describe state energetics and the de- gree of diradical character associated with the singlet (and, in the case of B97, the triplet) states, the shapes of the frontier natural orbitals predicted by LDA are consistent with EOM-SF-CCSD, PBE50 and B5050LYP. The main difference between LDA and other methods is in natural occupations (and, consequently, the number of effectively un- paired electrons): LDA’s singlet state has much stronger closed-shell character. Shown in Appendix G, B97 singly occupied natural orbitals exhibit a-type interaction with equatorial nearest-neighbor oxygen atoms, meaning that the lowest energy triplet state returned by this functional is altogether inconsistent with the other four methods. Us- ing standard troubleshooting tools, we were not able to find an SCF solution with this functional that corresponds to the same bonding pattern as predicted by other function- als. Using a wrong reference state leads to large errors in the singlet triplet gaps and large spin-contamination (Table 5.9). This case illustrates the utility of using natural orbitals for detecting problematic situations with SCF convergence (as one can see from Fig. 5.2, the inspection of canonical orbitals is not very instructive for these highly non-Koopmans systems). Finally, we consider the case of PATFIA when the full experimental structure that includes the ferrocene group is used in the SF-TDDFT calculation. Results at the B5050LYP/cc-pVDZ level of theory (with the collinear TDDFT kernel) are summa- rized for simplified PATFIA and PATFIA + Fe(C 5 H 5 ) 2 in Table 5.10 and Figure 5.9. The observed localization of unpaired electron density on the copper atoms in PATFIA + (C 5 H 5 ) 2 , and the consistency in the computed values of the singlet-triplet gap,n u and 174 EOM-SF-CCSD PBE50 B5050LYP LDA ! " s = 0.92 (0.00) ! " t = 1.00 (0.00) ! " s = 1.08 (0.00) ! " t = 1.00 (0.00) ! " s = 0.90 (0.00) ! " t = 1.00 (0.01) ! " s = 1.10 (0.00) ! " t = 1.00 (0.01) ! " s = 0.28 (0.00) ! " t = 1.00 (0.00) ! " s = 1.72 (0.00) ! " t = 1.00 (0.00) ! " s = 0.85 (0.07) ! " t = 0.95 (0.06) ! " s = 1.06 (0.05) ! " t = 0.96 (0.07) Figure 5.8: Frontier natural orbitals of lowest singlet and M s = 0 triplet states of CUAQAC02 at the SF-TDDFT/cc-pVTZ and EOM-SF-CCSD/cc-pVDZ levels of theory. Natural orbital surfaces obtained with three different density functionals are compared. The collinear TDDFT kernel was used with the B5050LYP func- tional. n =jn +n j, withjn n j provided in parentheses. n s and n t corre- spond to n values obtained from the occupancies of the singlet and triplet natural orbitals, respectively. n u;nl , suggest that the ferrocene group indeed does not interact with unpaired electrons on the copper centers, and that the ubiquitous but thus-far unsubstantiated use of the simplified structure in theoretical calculations of exchange-coupling is valid. 5.5 Conclusions We report a benchmark study of eight binuclear copper diradicals, focusing on the singlet-triplet energy gaps (which are related to exchange-coupling constants within the HDvV model) and the associated wave functions. We carefully compare SF-TDDFT methods with selected functionals against a high-level wave function based method, 175 PATFIA ! " t = 1.00 (0.05) ! " t = 1.00 (0.05) PATFIA + Fe(C₅H₅)₂ ! " s = 1.00 (0.28) ! " t = 1.00 (0.26) ! " s = 1.00 (0.28) ! " t = 1.00 (0.26) ! " s = 1.00 (0.12) ! " s = 1.00 (0.12) Figure 5.9: Frontier natural orbitals of lowest singlet and M s = 0 triplet states of simplified PATFIA and the full experimental structure, PATFIA + Fe(C 5 H 5 ) 2 . Natural orbitals were obtained at the B5050LYP/cc-pVDZ level of theory with the collinear TDDFT kernel applied. n =jn +n j, withjn n j provided in paren- theses. n s and n t correspond to n values obtained from the occupancies of the singlet and triplet natural orbitals, respectively. EOM-SF-CCSD. In agreement with previous calibration studies of organic di- and tri- radicals, 11, 28, 29 we find that hybrid functionals outperform LDA and GGA-type func- tionals. The non-collinear kernel vastly improves the quality of SF-TDDFT results for GGA and LDA functionals, with modest improvement for most hybrids. Performance of DZ and TZ-quality Dunning’s basis sets are comparable, while the use of various ECPs has little effect on the computed value of E ST , suggesting that computational savings can be achieved without sacrificing quality of results when extending the approaches outlined in this work to larger binuclear (or trinuclear) copper complexes. 176 Small variations in nuclear geometries—specifically, a slight increase or decrease in the internuclear separation between the Cu atoms that host the unpaired electrons—does not impact the computed value of E ST , validating the use of experimentally deter- mined geometries (or slightly modified symmetrized geometries) in studies of exchange- coupling. Furthermore, the presence of the ferrocene group that is often excluded in ab initio studies of exchange-coupling in the PATFIA complex did not alter the value of E ST or the underlying density, justifying the use of the simplified structure. We employ density-based analysis to gain insight into the interaction between the unpaired electrons in copper diradicals and, in so doing, discover that only five of the eight binuclear copper complexes exhibit perfect or near-perfect diradical character in their lowest singlet and triplet states. This finding suggests that while the HDvV Hamil- tonian might be an appropriate model for exchange-coupling in CA VXUS, PATFIA, Cu 2 Cl 2 6 , XAMBUI, and CITLAT, quality of results should be expected to suffer for complexes like BISDOW and CUAQAC02, wherein ionic configurations are mixed into the singlet ground states. For these species, the experimentally derived values ofJ are expected to be systematically overestimated. Thus, the observed discrepancies between theoretical and experimental values for AF molecules arise not due to the limitations of theoretical methods, but due to the limitations of the HDvV model. Spin-contamination appears to be minor in this class of compounds. We have illustrated that the PBE50 or B5050LYP functionals (with a modest choice of basis set and ECP) can be paired with the SF-TDDFT approach to paint a robust picture of the energetics of MM candidates, with a level of accuracy comparable to EOM-SF-CCSD. When combined with the density-based analysis of the relevant spin states, one can determine whether or not the unpaired electrons in a given molecular magnet obey the underlying physics of the HDvV Hamiltonian. Screening of molecular 177 magnet candidates based solely on degree of radical character observed in the low-lying spin states is made possible. Recent work by Head-Gordon and others extends the for- malism of SF to computation of exchange-coupling constants in complexes with greater than two radical sites and/or unpaired electrons. 17, 18 178 Table 5.4: Singlet-triplet gaps (cm 1 ) for Complexes 1-8 calculated using SF-TDDFT with the PBE0 method and the non-collinear TDDFT kernel. BASIS COMPLEX 1 2 3 4 5 6 7 8 cc-pVDZ -523 -354 -27 -35 -23 2 201 232 cc-pVTZ -506 -362 -26 -33 35 1 197 227 ECP10MDF/ cc-pVDZ -486 -326 -27 -40 23 -1 186 213 ECP10MDF/ cc-pVTZ -456 -326 -25 -40 40 1 84 204 LANL2DZ -615 -603 -21 39 0 0 250 267 SRSC -471 -449 -19 21 -10 1 191 208 CRENBL -474 -459 -19 31 -75 0 202 223 EXP.J -382 -286 -19 -11 (0, -40) -2 111 113 179 Table 5.5: Singlet-triplet gaps (cm 1 ) for Complexes 1-8 calculated using SF-TDDFT with the PBE50 method. BASIS COMPLEX 1 2 3 4 5 6 7 8 NC a C b NC a C b NC a C b NC a C b NC a C b NC a C b NC a C b NC a C b cc-pVDZ -173 -219 -121 -154 -9 -11 0 60 55 -3 0 0 93 77 120 78 cc-pVTZ -158 -211 -119 -153 -7 -11 1 -57 63 7 1 -1 90 60 118 77 ECP10MDF/ cc-pVDZ -158 -213 -115 -148 8 11 15 -60 -52 -3 -1 1 88 57 112 75 ECP10MDF/ cc-pVTZ -148 -199 -111 -143 -6 -11 -8 -56 61 15 0 0 84 56 110 74 LANL2DZ -203 -262 -208 -269 -9 -10 15 -60 -18 -85 0 0 141 80 114 86 SRSC -160 -208 -156 -198 -7 -9 -15 -46 7 – 0 0 79 60 99 73 CRENBL -161 -211 -158 -201 -7 -15 -15 -50 -35 -90 0 0 92 60 99 74 EXP.J -382 -286 -19 -11 0, -40 -2 111 113 a Non-collinear TDDFT kernel. b Collinear TDDFT kernel. 180 Table 5.6: Singlet-triplet gaps (cm 1 ) for Complexes 1-8 calculated using SF-TDDFT with the B5050LYP method. BASIS COMPLEX 1 2 3 4 5 6 7 8 NC a C b NC a C b NC a C b NC a C b NC a C b NC a C b NC a C b NC a C b cc-pVDZ -155 -212 -114 -149 -8 -11 2 -57 45 -7 -1 0 84 61 109 77 cc-pVTZ -150 -205 -115 -149 -6 -10 2 -56 53 2 -1 0 82 60 108 77 ECP10MDF/ cc-pVDZ -142 -205 -103 -140 7 11 9 -57 44 0 1 1 75 56 98 74 ECP10MDF/ cc-pVTZ -133 -191 -99 -136 -2 -10 -8 -56 52 12 1 0 73 53 92 73 LANL2DZ -204 -255 -211 -261 -7 -10 7 -31 -27 -83 0 -1 97 79 110 85 SRSC -162 -199 -158 -192 -7 -9 22 -38 -7 – 0 0 72 58 90 72 CRENBL -165 -205 -162 -199 -9 -9 6 -42 -48 -95 -2 0 79 60 94 73 EXPJ -382 -286 -19 -11 0, -40 -2 111 113 a Non-collinear TDDFT kernel. b Collinear TDDFT kernel. 181 Table 5.7: Singlet-triplet gaps (cm 1 ) for optimized geometries of Complexes 1, 4 and 8, compared with gaps obtained using the experimentally determined molecular geometries. Method Basis Complex 1 Complex 4 Complex 8 X-Ray Optimized X-Ray Optimized X-Ray Optimized PBE0 cc-pVDZ -523 -524 -35 -57 109 DRS a ECP10MDF/ ccpvdz -486 -459 -40 -61 213 233 LANL2DZ -615 -571 39 -12 267 313 LRC-!PBEh cc-pVDZ -444 -418 -35 -59 231 259 ECP10MDF/ ccpvdz -416 -394 -44 -64 213 233 LANL2DZ -510 -473 38 -15 263 306 PBE50 cc-pVDZ -173 -161 0 -24 120 113 ECP10MDF/ ccpvdz -158 -148 15 -31 112 104 LANL2DZ -203 -180 15 -16 114 134 B5050LYP cc-pVDZ -155 -152 2 -25 109 102 ECP10MDF/ ccpvdz -142 -133 9 -32 98 90 LANL2DZ -204 -165 7 -19 110 115 EXP J -382 -11 113 a SCF calculation converged to a wrong reference state. 182 Table 5.8: NC-PBE50/cc-pVDZ energy splittings (cm 1 ) and wavefunction prop- erties of lowest singlet and M s = 0 triplet states of all eight benchmark complexes. Molecule E ST n u n u;nl hS 2 i Singlet Triplet Singlet Triplet Singlet Triplet BISDOW -173 1.81 2.01 1.96 2.00 0.01 2.02 CUAQAC02 -121 1.84 2.01 1.97 2.00 0.01 2.01 CA VXUS -9 1.96 2.01 2.00 2.00 0.07 1.97 PATFIA 0 1.86 1.98 1.98 2.00 0.36 1.68 Cu 2 Cl 2 6 55 1.85 2.02 1.97 2.00 0.02 2.03 XAMBUI 0 2.00 2.01 2.00 2.00 0.01 2.02 YAFZOU 93 1.97 2.01 2.00 2.00 0.01 2.02 CITLAT 120 1.96 2.01 2.00 2.00 0.01 2.02 Table 5.9: SF-TDDFT/cc-pVTZ and EOM-SF-CCSD/cc-pVDZ energy splittings (cm 1 ) and wave function properties of lowest singlet and M s = 0 triplet spin states of CUAQAC02. Method E ST n u n u;nl hS 2 i Singlet Triplet Singlet Triplet Singlet Triplet NC-PBE50 -123 1.84 2.01 1.97 2.00 0.01 2.01 COL-B5050LYP -148 1.81 2.01 1.96 2.00 0.01 2.01 NC-LDA -1420 0.57 2.00 0.47 2.00 0.01 2.01 NC-B97 22180 2.82 71 2.59 2.69 1.07 1.77 EOM-SF-CCSD -171 1.79 1.91 1.96 2.00 0.00 2.00 Table 5.10: B5050LYP/cc-pVDZ SF-TDDFT energy splittings (cm 1 ) and wave function properties of lowest singlet and M s = 0 triplet spin states of simplified PATFIA and the full experimental structure, PATFIA + Fe(C 5 H 5 ) 2 . The collinear TDDFT kernel was applied. Molecule E ST n u n u;nl hS 2 i Singlet Triplet Singlet Triplet Singlet Triplet PATFIA -57 1.80 2.01 1.95 2.00 0.02 2.02 PATFIA + Fe(C 5 H 5 ) 2 -60 1.80 2.01 1.96 2.00 0.05 1.98 183 5.6 Appendix A: Derivation of the Heisenberg-Dirac- van-Vleck Hamiltonian and the Land´ e Interval Rule The goal is to show that the HDvV Hamiltonian, ^ H HDvV =2 X J kl ^ S k ^ S l ; (5.10) and the associated spectroscopic interval rule, E(S tot )E(S tot 1) =2J kl S tot (5.11) can be derived in a pedagogical way from first principles. We begin by considering a system of two non-interacting electrons in a potential,V . At first, we neglect spin. The corresponding Schr¨ odinger equation is ~ 2 2m e r 2 1 + ~ 2 2m e r 2 2 + [E 0 V ( ! r 1 )V ( ! r 2 )] = 0 (5.12) where ! r 1 and ! r 2 define the coordinates of electrons one and two, respectively. Since the electrons are non-interacting, we can express as linear combinations of I and II , where I = k ( ! r 1 ) m ( ! r 2 ) II = k ( ! r 2 ) m ( ! r 1 ); (5.13) satisfying E 0 =E k +E m : (5.14) k and m are real and orthonormal. Electron one is in a statek and electron two is in a statem. The only difference between I and II is that the electrons have swapped places, due to the fact that identical particles are interchangeable. 184 This degeneracy is lifted when we introduce a potential energy of interaction be- tween the two electrons, ^ V 12 . The Hamiltonian becomes ^ H =E 0 + ^ V 12 = 2 6 4 E 0 0 0 E 0 3 7 5 + 2 6 4 C 11 J 12 J 21 C 22 3 7 5 ; (5.15) where ^ H = 2 6 4 E 0 +C 11 J 12 J 21 E 0 +C 22 3 7 5 : (5.16) C 11 =C 12 andJ 21 =J 12 , therefore, the secular equation for this 2x2 problem is E 0 +C 12 E J 12 J 12 E 0 +C 12 E = 0: (5.17) Solving Eq. 5.17, we obtain an expression forE: E =E 0 +C 12 J 12 : (5.18) C 12 has the form of C 12 = Z I V 12 I d ! r 1 d ! r 2 (5.19) = Z II V 12 II d ! r 1 d ! r 2 ; (5.20) andJ 12 has the form of C 12 = Z I V 12 II d ! r 1 d ! r 2 : (5.21) 185 The eigenvectors of the Hamiltonian that includes the electron-electron interaction po- tential are then sym = 1 p 2 ( I + II ) =)E =E 0 +C 12 +J 12 (5.22) antsym = 1 p 2 ( I II ) =)E =E 0 +C 12 J 12 ; (5.23) which satisfy the probability density requirement for indistinguishable particles: j I j 2 =j II j 2 =j k ( ! r 1 ) m ( ! r 2 )j 2 (5.24) =j k ( ! r 2 ) m ( ! r 1 )j 2 : (5.25) Next, we introduce spin =1/2 and derive the appropriate wave functions. In the absence of coupling between spin and orbital angular momenta, wave functions are products of spin and spatial parts. I = k ( ! r 1 )s(1) m ( ! r 2 )s(2) (5.26) II = k ( ! r 2 )s(2) m ( ! r 1 )s(1); (5.27) wheres(1) ands(2) are spin functions of electrons one and two, respectively. s(1) can take the form of(1) or(1) (“spin-up” or “spin-down”) ands(2) can take the form of (2) or(2). Spin (i.e., Pauli) symmetry requires that symmetry of spatial components be matched with antisymmetry in spin. In other words, electrons must adopt one and one spin if spatial components are symmetric. This permits the following Slater determinants for the overall wavefunction ex- pressed as a product of spin and spatial parts: 186 k m = 1 p 2 k (1) m (1) k (2) m (2) (5.28) = 1 p 2 [ k m m k ](1)(2) (5.29) k m = 1 p 2 k (1) m (1) k (2) m (2) (5.30) = 1 p 2 [ k m m k ](1)(2) (5.31) k m = 1 p 2 k (1) m (1) k (2) m (2) (5.32) = 1 p 2 [ k ( ! r 1 ) m ( ! r 2 )(1)(2) m ( ! r 1 ) k ( ! r 2 )(1)(2)] (5.33) k m = 1 p 2 k (1) m (1) k (2) m (2) (5.34) = 1 p 2 [ k ( ! r 1 ) m ( ! r 2 )(1)(2) m ( ! r 1 ) k ( ! r 2 )(1)(2)] (5.35) Eq. 5.33 and 5.35 are not eigenfunctions of both ^ S z and ^ S 2 , but their linear combinations are: k m+ k m = 1 p 2 [ k ( ! r 1 ) m ( ! r 2 ) m ( ! r 1 ) k ( ! r 2 )] 1 p 2 ((1)(2)+(1)(2)) (5.36) k m k m = 1 p 2 [ k ( ! r 1 ) m ( ! r 2 )+ m ( ! r 1 ) k ( ! r 2 )] 1 p 2 ((1)(2)(1)(2)) (5.37) 187 Equations 5.29, 5.31, 5.36, and 5.37 preserve the overall antisymmetry requirement. Symmetries and spin expectation values are summarized in Table 5.11. Table 5.11: Spatial Sym. Spin Sym. jS z j hS 2 i Multiplicity k m antisym. symm. 1 2 3 (triplet) k m antisym. symm. 1 2 3 (triplet) k m+ k m antisym. sym. 0 2 3 (triplet) k m k m sym. antisym. 0 0 1 (singlet) Consider the expression forhS 2 i for each of these total wave functions: hS 2 i =h(s 1 +s 2 ) 2 i =hs 1 2 +s 2 2 + 2s 1 s 2 i (5.38) =S tot (S tot + 1); (5.39) wheres 1 ands 2 are the spin moments associated with each electron, ands 1 2 = s 2 2 = 1 2 ( 1 2 + 1) = 3 4 . S tot will take the value of either 1 or 0. IfS tot = 0 (a symmetric spatial function), then 2s 1 s 2 = 3 4 . If S tot = 1 (an antisymmetric spatial function), then 2s 1 s 2 = + 1 2 . Recall that the Hamiltonian describing the energy of two interacting electrons (Eq. 5.15) yielded an eigenvalue of the form in Eq. 5.18. For the case where the spatial part of the wavefunction is symmetric, E =E 0 +C 12 +J 12 : (5.40) Similarly, for the case where the spatial part of the wave function is antisymmetric, E =E 0 +C 12 J 12 : (5.41) 188 Ignoring theE 0 term, which does not depend on symmetry or describe interaction, we can write E =C 12 1 2 J 12 2J 12 (^ s 1 ^ s 2 ); (5.42) where we recover Eq. 5.40 in the case ofS tot = 0 and Eq. 5.42 in the case ofS tot = 1. In magnetism, we are concerned only with the term in E that depends on the orien- tation of the spins,2J 12 (^ s 1 ^ s 2 ), and we can use it to write the Hamiltonian ^ H =2J kl (^ s k ^ s l ); (5.43) which is a potential that couples spins of two electrons in statesk andl. For an extended system of more than two electrons, ^ H =2 X J kl (^ s k ^ s l ): (5.44) The above expression is the HDvV Hamiltonian (Eq. 5.10) that we set out to obtain. We know that for two spin moments ^ s 1 and ^ s 2 , ^ s 2 tot = (^ s 1 + ^ s 2 ) (^ s 1 + ^ s 2 ) (5.45) =s 1 2 +s 2 2 + 2s 1 s 2 (5.46) =S tot (S tot + 1): (5.47) Therefore, ^ s 1 ^ s 2 = 1 2 [S tot (S tot + 1)s 1 2 s 2 2 ]; (5.48) wheres 1 2 =s 2 2 =s(s + 1). The right hand side of Eq. 5.48 becomes 1 2 [S tot (S tot + 1) 2s(s + 1)]: (5.49) 189 Therefore, we can expand Eq. 5.10 as ^ H HDvV =2 X J kl (^ s k ^ s l ) =2 X J kl [ 1 2 S tot (S tot + 1) 2s(s + 1)] (5.50) =J kl [S tot (S tot + 1)s(s + 1)]; (5.51) where Eq. 5.51 is the expression for eigenenergies,E(S tot ), that depend only on scalars S tot ,J kl ands. Now, we can express energies of adjacent spin states,E(S tot ) andE(S tot 1), as: E(S tot ) =J kl [S tot (S tot + 1)s(s + 1)] (5.52) =J kl [S tot 2 +S tot 2s 2 2s] (5.53) E(S tot 1) =J kl [(S tot 1)(S tot + 1 1)s(s + 1)] (5.54) =J kl [S tot 2 S tot 2s 2 2s]: (5.55) SubtractingE(S tot 1) fromE(S tot ) gives the energy difference between adjacent spin states, E s;s1 , and their relation toJ kl . E s;s1 =E(S tot )E(S tot 1) (5.56) =J kl [S tot (S tot + 1)s(s + 1)] [J kl [(S tot 1)(S tot + 1 1)s(s + 1)]] (5.57) =2J kl S tot : (5.58) Therefore, J kl = E(S tot )E(S tot 1) 2S tot ; (5.59) which is the Land´ e interval rule (Eq. 5.11) that we set out to recover. 190 5.7 Appendix B: ECP Details 191 Table 5.12: ECPs with matching basis sets and core electron approximations for Cu, Cl, C, O, N and H. Atom LANL2DZ SRSC CRENBL ECP10MDF Core Valence Core Valence Core Valence Core Valence Cu [Ne] (3s, 3p, 2d) [Ne] (6s, 5p, 3d) [Ne] (7s, 6p, 6d) [Ne] cc-pVDZ-PP, cc-pVTZ-PP Cl [Ne] (2s, 2p) None 6-311G* [Ne] (4s, 4p) None cc-pVDZ, cc-pVTZ O, N, C None 6-31G None 6-311G* [He] (4s, 4p) None cc-pVDZ, cc-pVTZ H None 6-31G None 6-311G* None 6-311G* None cc-pVDZ, cc-pVTZ 192 5.8 Appendix C: Error Analysis Table 5.13: Mean error ( ME ), mean absolute error ( MAE ), and standard devia- tion of the error ( STD ) in the calculated values of the exchange coupling constants of AF complexes 1-5 at different levels of theory a . Errors are reported relative to experiment in cm 1 . BASIS PBE0 PBE50 B5050LYP ME MAE STD ME MAE STD ME MAE STD cc-pVDZ -45 52 62 +98 98 90 +102 102 96 cc-pVTZ -31 61 75 +104 104 94 +101 104 97 ECP10MDF/ cc-pVDZ -24 49 60 +87 92 104 +111 111 98 ECP10MDF/ cc-pVTZ -14 46 58 +105 105 101 +110 110 107 LANL2DZ -92 128 170 +63 63 70 +59 59 71 SRSC -38 63 85 +81 83 94 +85 85 88 CRENBL -52 68 84 +72 74 99 +72 75 96 a Calculations performed with non-collinear TDDFT kernel using experimental structures. 193 Table 5.14: Mean error ( ME ), mean absolute error ( MAE ), and standard de- viation of the error ( STD ) in the calculated values of the singlet-triplet gap of F complexes 6-8 at different levels of theory. Errors are reported relative to experi- ment in cm 1 . BASIS PBE0 PBE50 B5050LYP ME MAE STD ME MAE STD ME MAE STD cc-pVDZ +70 70 62 -10 10 8 -11 11 14 cc-pVTZ +66 67 60 -6 9 14 -12 12 14 ECP10MDF/ cc-pVDZ +57 59 54 -9 9 12 -17 17 18 ECP10MDF/ cc-pVTZ +21 40 62 -11 11 14 -20 20 19 LANL2DZ +97 98 86 +10 11 18 -6 6 7 SRSC +58 59 52 -16 16 15 -21 21 19 CRENBL +66 68 60 -12 12 9 -18 18 14 a Calculations performed with non-collinear TDDFT kernel using experimental structures. Table 5.15: Mean error ( ME ), mean absolute error ( MAE ), and standard devi- ation of the error ( STD ) in the calculated values of the singlet-triplet gap of all complexes at different levels of theory. Errors are reported relative to experiment in cm 1 . BASIS PBE0 PBE50 B5050LYP ME MAE STD ME MAE STD ME MAE STD cc-pVDZ -2 58 83 +57 65 88 +59 68 94 cc-pVTZ +6 63 82 +63 68 91 +61 70 95 ECP10MDF/ cc-pVDZ +7 53 68 +51 61 93 +63 76 100 ECP10MDF/ cc-pVTZ -1 44 58 +62 70 97 +61 76 105 LANL2DZ -21 117 168 +43 44 60 +35 39 64 SRSC -2 61 86 +45 58 88 +45 61 87 CRENBL -7 68 94 +41 51 86 +38 54 87 a Calculations performed with non-collinear TDDFT kernel using experimental structures. 194 5.9 Appendix D: Optimized structures 5.17 (5.14) Å 3.43 (3.24) Å Figure 5.10: Optimized triplet geometries of BISDOW (top left), PATFIA (middle), and CITLAT (top right). Interatomic Cu-Cu distances in parentheses correspond to experimental geometries. All three complexes were optimized at the !B97X- D/cc-pVTZ level of theory. 195 5.10 Appendix E: EOM-SF-CCSD calculations at opti- mized structures Table 5.16: EOM-SF-CCSD/cc-pVDZ singlet-triplet gaps and experimental ex- change coupling constants for three binuclear copper diradicals at optimized ge- ometries. Cholesky decomposition with a threshold of 1e-3 was used for two- electron integral calculations. The frozen core approximation was applied. Ge- ometries were optimized at the!B97X-D/cc-pVTZ level of theory. Complex E ST (cm 1 ) hS 2 i a Exp.J (cm 1 ) BISDOW -215 2.017 -382 PATFIA -77 2.008 -11 CITLAT 123 2.011 113 a hS 2 i of the reference high-spin triplet. 196 5.11 Appendix F: Singlet-Triplet Gaps with the Collinear TDDFT Kernel Table 5.17: Singlet-triplet gaps (cm 1 ) for Complexes 1-8 calculated using SF- TDDFT with the LRC-!PBEh method and the collinear TDDFT kernel. Exper- imentally determined molecular geometries were used. BASIS COMPLEX 1 2 3 4 5 6 7 8 cc-pVDZ -733 -519 -44 -280 -188 -1 90 60 ECP10MDF/ cc-pVDZ -725 -498 -44 -274 -173 -1 109 66 LANL2DZ -846 -866 -38 -217 -265 -2 129 75 EXP.J -382 -286 -19 -11 (0, -40) -2 111 113 Table 5.18: Singlet-triplet gaps (cm 1 ) for Complexes 1-8 calculated using SF- TDDFT with the PBE0 method and the collinear TDDFT kernel. Experimentally determined molecular geometries were used. BASIS COMPLEX 1 2 3 4 5 6 7 8 cc-pVDZ -882 -582 -56 -290 -229 0 104 79 cc-pVTZ -853 -600 -53 -282 -219 -2 105 74 ECP10MDF/ cc-pVDZ -866 -558 -55 -280 -211 1 123 83 ECP10MDF/ cc-pVTZ -809 -564 -50 -273 -195 -1 56 73 LANL2DZ -1048 -269 -48 -248 -315 -2 146 108 SRSC -826 -198 -42 -217 -285 -2 117 67 CRENBL -823 -201 -68 -218 -365 -2 111 67 EXP.J -382 -286 -19 -11 (0, -40) -2 111 113 197 5.12 Appendix G: Wavefunction Analysis: B97 Frontier natural orbitals of the CUAQAC02 Complex ! " s = 1.07 (0.17) ! " t = 1.00 (0.06) ! " s = 0.93 (0.17) ! " t = 1.00 (0.24) ! " s = 0.93 (0.00) ! " t = 1.00 (0.00) ! " s = 1.07 (0.00) ! " t = 1.00 (0.00) CITLAT CUAQAC02 Figure 5.11: SF-TDDFT/cc-pVTZ frontier orbitals of CUAQAC02 with the B97 functional (non-collinear TDDFT kernel). 198 5.13 Appendix H: Cartesian geometries Experimental geometry of BISDOW (Complex 1), taken from O. Castillo; I. Muga; A. Luque; J. M. Gutierrez-Zorrilla; J. Sertucha; P. Vitoria; P. Roman, Polyhedron 18, 1235 (1999). Cu -2.35254640 0.57677107 0.86384176 O -1.07833077 -0.90965212 1.01996557 O -0.96187584 1.42020443 -0.27484590 C -0.03492785 -0.66732598 0.37029459 O -1.46679874 1.88201362 2.60682569 N -3.73778003 1.95253258 0.60794323 C -3.57908589 3.14559856 0.03840417 C -4.61404624 4.04749925 -0.08057997 C -5.85665045 3.66888992 0.35738664 C -6.03598320 2.44856997 0.95095211 C -4.95514302 1.60222814 1.06881648 C -5.00093520 0.29479636 1.73917028 N -3.83897000 -0.37705060 1.76213853 C -3.77611505 -1.55137619 2.39582430 C -4.86428748 -2.09810499 3.03720592 C -6.04580551 -1.41965076 3.01871321 C -6.14361504 -0.21036119 2.34917081 H 6.96035423 -0.28296881 -2.38042154 H 6.82249448 1.67143857 -3.36395826 H 4.73282136 2.97028678 -3.48570631 H 2.94548681 1.87621280 -2.31503499 H -2.70370579 3.40223394 -0.20212432 H -4.49867539 4.86708750 -0.43102206 H -6.51174152 4.27324389 0.36162699 H -6.85732915 2.20590734 1.29095458 H -2.94548681 -1.87621280 2.31503499 H -4.73282136 -2.97028678 3.48570631 H -6.82249448 -1.67143857 3.36395826 H -6.96035423 0.28296881 2.38042154 H -0.81875344 1.60541487 2.91912108 H -1.95320317 1.58829473 3.04098458 O 0.96187584 -1.42020443 0.27484590 Cu 2.35254640 -0.57677107 -0.86384176 O 1.07833077 0.90965212 -1.01996557 199 C 0.03492785 0.66732598 -0.37029459 O 1.46679874 -1.88201362 -2.60682569 H 0.81875344 -1.60541487 -2.91912108 H 1.95320317 -1.58829473 -3.04098458 N 3.73778003 -1.95253258 -0.60794323 C 3.57908589 -3.14559856 -0.03840417 C 4.61404624 -4.04749925 0.08057997 C 5.85665045 -3.66888992 -0.35738664 C 6.03598320 -2.44856997 -0.95095211 C 4.95514302 -1.60222814 -1.06881648 C 5.00093520 -0.29479636 -1.73917028 N 3.83897000 0.37705060 -1.76213853 H 6.85732915 -2.20590734 -1.29095458 H 6.51174152 -4.27324389 -0.36162699 H 4.49867539 -4.86708750 0.43102206 H 2.70370579 -3.40223394 0.20212432 C 3.77611505 1.55137619 -2.39582430 C 4.86428748 2.09810499 -3.03720592 C 6.04580551 1.41965076 -3.01871321 C 6.14361504 0.21036119 -2.34917081 Optimized $\omega$B97X-D/cc-pVTZ Kohn-Sham triplet geometry of BISDOW (Complex 1). Cu 2.5833491351 -0.0009904119 0.1291850887 O 1.1170753802 1.3282000412 0.0327516014 C 0.0001048572 0.7718040969 0.0067932958 O -1.1164196952 1.3291297510 -0.0094867315 C -0.0001048570 -0.7718040965 -0.0067932958 O 1.1164196952 -1.3291297510 0.0094867315 O -1.1170753800 -1.3282000408 -0.0327516014 Cu -2.5833491348 0.0009904123 -0.1291850888 N -4.0682174830 1.2998301754 0.0702348396 C -3.9369618909 2.6242574996 0.1044843851 H -2.9259219211 3.0047210936 0.0681946756 C -5.2869911318 0.7377507924 0.1316192026 C -5.0320586571 3.4619974192 0.1983743645 H -4.8953106009 4.5320752067 0.2295681032 C -6.4251174566 1.5180750115 0.2168712828 H -7.4049524306 1.0689921489 0.2557787966 C -6.2941061480 2.8976613004 0.2476047560 H -7.1722495965 3.5236083586 0.3199871198 200 C -5.2854287130 -0.7431853280 0.1312140901 N -4.0661698135 -1.3025307697 0.0559523575 C -3.9317743705 -2.6269546064 0.0843456738 H -2.9201774419 -3.0041171469 0.0345072850 C -5.0246994071 -3.4661520878 0.1905159932 H -4.8862571322 -4.5363149421 0.2138596566 C -6.4211580088 -1.5251490333 0.2329657269 H -7.4014618567 -1.0789117305 0.2883650366 C -6.2868171354 -2.9044466256 0.2604672494 H -7.1626398125 -3.5319713878 0.3467616510 N 4.0682174832 -1.2998301750 -0.0702348396 C 3.9369618911 -2.6242574991 -0.1044843852 H 2.9259219213 -3.0047210932 -0.0681946756 C 5.2869911320 -0.7377507920 -0.1316192027 C 5.0320586573 -3.4619974188 -0.1983743645 H 4.8953106011 -4.5320752062 -0.2295681033 C 6.4251174568 -1.5180750110 -0.2168712828 H 7.4049524308 -1.0689921485 -0.2557787967 C 6.2941061482 -2.8976613000 -0.2476047560 H 7.1722495967 -3.5236083582 -0.3199871199 C 5.2854287132 0.7431853284 -0.1312140902 N 4.0661698137 1.3025307701 -0.0559523576 C 3.9317743707 2.6269546069 -0.0843456738 H 2.9201774421 3.0041171473 -0.0345072851 C 5.0246994073 3.4661520882 -0.1905159933 H 4.8862571324 4.5363149425 -0.2138596566 C 6.4211580090 1.5251490337 -0.2329657270 H 7.4014618569 1.0789117309 -0.2883650366 C 6.2868171356 2.9044466260 -0.2604672495 H 7.1626398127 3.5319713882 -0.3467616510 O -2.5412578917 0.0194927722 -2.4748985207 H -2.5544074104 0.7915396360 -3.0420283833 H -2.5437428892 -0.7442447095 -3.0533586924 O 2.5412578919 -0.0194927718 2.4748985207 H 2.5544074106 -0.7915396356 3.0420283832 H 2.5437428894 0.7442447099 3.0533586924 Experimental geometry of CUAQAC02 (Complex 2), taken from P. de Meester; S. R. Fletcher; A. C. Skapski, J. Am. Chem. Soc. Dalton Trans. 23, 2575 (1973). Cu 5.64300000 0.71900000 0.55500000 201 C 5.31800000 0.18300000 -2.22700000 C 4.66600000 0.32500000 -3.56500000 C 7.78300000 2.12100000 -0.62600000 C 8.47800000 3.39400000 -1.03200000 H 5.31600000 0.37700000 -4.40700000 H 3.90600000 0.75400000 -3.60500000 H 4.45700000 -0.59100000 -3.96300000 H 8.72300000 3.29700000 -2.00000000 H 7.79600000 4.11900000 -0.80200000 H 9.34900000 3.46000000 -0.59300000 H 3.90800000 1.43900000 2.40700000 H 3.50700000 2.33800000 1.33300000 O 4.80000000 0.75400000 -1.24200000 O 6.79300000 0.51600000 2.17200000 O 6.68500000 2.26700000 0.00400000 O 4.86400000 -1.02900000 0.93000000 O 4.11000000 1.77500000 1.64200000 Cu 7.52500000 -0.71900000 -0.55500000 O 6.37500000 -0.51600000 -2.17200000 C 7.85000000 -0.18300000 2.22700000 O 8.30400000 1.02900000 -0.93000000 C 5.38500000 -2.12100000 0.62600000 O 8.36800000 -0.75400000 1.24200000 O 6.48300000 -2.26700000 -0.00400000 O 9.05800000 -1.77500000 -1.64200000 C 8.50200000 -0.32500000 3.56500000 C 4.69000000 -3.39400000 1.03200000 H 9.26000000 -1.43900000 -2.40700000 H 9.66100000 -2.33800000 -1.33300000 H 7.85200000 -0.37700000 4.40700000 H 9.26200000 -0.75400000 3.60500000 H 8.71100000 0.59100000 3.96300000 H 4.44500000 -3.29700000 2.00000000 H 5.37200000 -4.11900000 0.80200000 H 3.81900000 -3.46000000 0.59300000 Experimental geometry of CAVXUS (Complex 3), taken from J. Sletten, Acta. Chem. Scand., Ser. A 37, 569 (1983). Cu 0.55900000 1.49200000 7.08000000 O -0.56900000 0.46400000 8.32700000 O -0.99600000 0.72200000 5.68000000 202 N 2.07100000 0.01800000 7.02400000 N 1.67800000 2.52100000 5.75300000 N -0.24300000 3.34400000 7.67900000 C -1.61200000 -0.08200000 7.84800000 C 2.93900000 0.12500000 8.25800000 C 2.28100000 -0.35700000 9.53500000 C 1.54000000 -1.36600000 6.84400000 C 2.61000000 -2.46300000 6.74900000 C 2.89500000 0.40600000 5.83800000 C 3.03700000 1.90800000 5.77300000 C 1.05500000 2.43000000 4.39300000 C 1.69700000 3.27900000 3.31900000 C 1.69700000 3.93400000 6.25600000 C 0.27900000 4.30600000 6.64200000 C 0.23900000 3.79200000 9.02600000 C -0.27800000 3.01100000 10.20000000 C -1.75200000 3.29700000 7.61100000 C -2.44900000 4.64300000 7.82400000 H 3.20100000 1.09200000 8.37800000 H 3.77400000 -0.33400000 8.06600000 H 2.93000000 -0.15600000 10.26400000 H 2.22600000 -1.20400000 9.51200000 H 1.44200000 0.24500000 9.69600000 H 1.01600000 -1.29300000 5.92600000 H 0.91900000 -1.42700000 7.54200000 H 3.25300000 -2.22900000 5.95400000 H 3.05200000 -2.63000000 7.55600000 H 2.08100000 -3.29900000 6.52100000 H 3.72100000 -0.02200000 5.91100000 H 2.48300000 0.04500000 5.03300000 H 3.47700000 2.07300000 5.08900000 H 3.57800000 2.29600000 6.60600000 H 1.07500000 1.56000000 4.08300000 H 0.16300000 2.60800000 4.40900000 H 2.55500000 3.18700000 3.24600000 H 1.29100000 3.12100000 2.42400000 H 1.65600000 4.19100000 3.57200000 H 2.04700000 4.50300000 5.55700000 H 2.32900000 3.96800000 7.10200000 H -0.27300000 4.25700000 5.81200000 H 0.27600000 5.14900000 7.14500000 H 0.14800000 4.85900000 9.32800000 203 H 1.07700000 3.58900000 8.98800000 H -0.33100000 1.82800000 10.13600000 H 0.05400000 3.38800000 10.93000000 H -1.30600000 3.07600000 10.33400000 H -1.99200000 2.65300000 8.35000000 H -1.94100000 3.03100000 6.70500000 H -2.10700000 5.17100000 6.94600000 H -3.14700000 4.52500000 7.68300000 H -2.29000000 4.92600000 8.74700000 C -1.83900000 0.08200000 6.32800000 O -2.45500000 -0.72200000 8.49600000 O -2.88100000 -0.46400000 5.84900000 Cu -4.01000000 -1.49200000 7.09600000 N -5.52200000 -0.01800000 7.15200000 N -5.12800000 -2.52100000 8.42300000 N -3.20700000 -3.34400000 6.49700000 C -6.38900000 -0.12500000 5.91900000 C -4.99000000 1.36600000 7.33200000 C -6.34500000 -0.40600000 8.33800000 C -6.48700000 -1.90800000 8.40400000 C -4.50500000 -2.43000000 9.78300000 C -5.14700000 -3.93400000 7.92000000 C -3.73000000 -4.30600000 7.53500000 C -3.69000000 -3.79200000 5.15000000 C -1.69900000 -3.29700000 6.56500000 C -5.73200000 0.35700000 4.64100000 H -6.65100000 -1.09200000 5.79800000 H -7.22400000 0.33400000 6.11000000 C -6.06000000 2.46300000 7.42700000 H -4.46700000 1.29300000 8.25000000 H -4.36900000 1.42700000 6.63400000 H -7.17100000 0.02200000 8.26500000 H -5.93400000 -0.04500000 9.14400000 H -6.92800000 -2.07300000 9.08700000 H -7.02900000 -2.29600000 7.57000000 C -5.14700000 -3.27900000 10.85700000 H -4.52600000 -1.56000000 10.09300000 H -3.61400000 -2.60800000 9.76700000 H -5.49700000 -4.50300000 8.61900000 H -5.77900000 -3.96800000 7.07400000 H -3.17800000 -4.25700000 8.36400000 H -3.72700000 -5.14900000 7.03100000 204 C -3.17300000 -3.01100000 3.97600000 H -3.59900000 -4.85900000 4.84800000 H -4.52800000 -3.58900000 5.18800000 C -1.00200000 -4.64300000 6.35200000 H -1.45800000 -2.65300000 5.82600000 H -1.50900000 -3.03100000 7.47100000 H -6.38000000 0.15600000 3.91300000 H -5.67700000 1.20400000 4.66400000 H -4.89300000 -0.24500000 4.48000000 H -6.70400000 2.22900000 8.22200000 H -6.50200000 2.63000000 6.62000000 H -5.53100000 3.29900000 7.65500000 H -6.00600000 -3.18700000 10.93000000 H -4.74200000 -3.12100000 11.75200000 H -5.10700000 -4.19100000 10.60400000 H -3.12000000 -1.82800000 4.04000000 H -3.50400000 -3.38800000 3.24600000 H -2.14400000 -3.07600000 3.84200000 H -1.34300000 -5.17100000 7.23000000 H -0.30400000 -4.52500000 6.49300000 H -1.16000000 -4.92600000 5.42900000 Simplified experimental structure of PATFIA (Complex 4) without the ferrocene group, taken from M. Julve; M. Verdaguer; A. Gleizes; M. Pohiloche-Levisalles; O. Kahn, Inorg. Chem. 23, 3808 (1984). Cu 0.000000 0.000000 0.000000 Cu 0.925000 1.720000 -2.581000 H -1.006000 -0.187000 -3.472000 H 2.069000 0.732000 2.139000 O -0.181000 0.318000 -1.874000 O 2.524000 0.972000 -1.839000 O 1.892000 -0.349000 -0.158000 H 1.253000 0.938000 3.462000 C 1.203000 0.934000 2.503000 H 0.915000 1.792000 2.191000 H 0.122000 -2.063000 2.097000 N 0.218000 -0.118000 2.058000 N -1.942000 0.181000 0.393000 H -2.227000 0.999000 0.145000 H -2.405000 -0.431000 -0.076000 205 N 2.064000 3.355000 -3.079000 N -0.499000 2.585000 -3.664000 H -0.800000 2.003000 -4.281000 H -1.190000 2.803000 -3.128000 H 3.773315 -0.253545 -0.829399 H 1.627000 4.936000 -4.249000 C 1.138000 4.370000 -3.637000 H -1.504000 -0.959000 -2.201000 H 2.790000 3.400000 -1.215000 C 3.051000 2.978000 -4.104000 H -0.712000 4.452000 -4.377000 H 0.812000 4.925000 -2.911000 C 2.731000 0.106000 -0.917000 H -1.948000 0.525000 -2.439000 C -0.001000 3.794000 -4.333000 C -1.230000 -0.106000 -2.540000 H 2.292000 4.800000 -1.712000 H 3.655000 4.224000 -2.228000 C 2.761000 4.003000 -1.962000 H 3.308000 2.060000 -3.979000 H 2.663000 3.083000 -4.977000 H 3.825000 3.539000 -4.028000 H 0.259000 3.573000 -5.242000 C -1.030000 0.395000 2.631000 H -1.136000 0.061000 3.533000 H -0.989000 1.366000 2.672000 C -2.185000 -0.009000 1.814000 H -2.960000 0.509000 2.080000 H -2.380000 -0.944000 1.984000 C 0.639000 -1.379000 2.522000 H 0.514000 -1.423000 3.474000 H 1.568000 -1.499000 2.316000 Optimized $\omega$B97X-D/cc-pVTZ Kohn-Sham triplet geometry of the simplified PATFIA (Complex 4) structure. Cu 1.7087227025 -0.1792834903 0.1608894488 O 1.1510877085 1.5186111353 0.8977506371 C 0.0259987040 2.0401407554 1.0578341471 H 0.0146802968 3.0227985586 1.5416296736 O -1.0863580177 1.5872962809 0.7045453660 Cu -1.7153965803 -0.1621452984 0.1726197439 206 O -0.0071181378 -1.0311286136 0.5412325888 C 0.0030057456 -2.2756009208 1.2155946223 H 0.9333350080 -2.4176420955 1.7706126488 H -0.7895151328 -2.3283409390 1.9661011991 H -0.1154330288 -3.1118706257 0.5180707573 N -2.7336423434 -1.9056433478 -0.1370361317 H -2.5876357917 -2.5933835980 0.5932900883 H -2.3832137993 -2.3341014358 -0.9887331535 C -4.1779826589 -1.6063992118 -0.2812385809 H -4.5995491335 -1.5088882249 0.7167569764 H -4.7040744188 -2.4177316843 -0.7819519177 C -4.3021627019 -0.3226480645 -1.0701279691 H -3.9509652816 -0.4827683024 -2.0904271954 H -5.3399705359 0.0104030531 -1.1255563887 N -3.4558063585 0.7337142466 -0.4641119051 C -3.1877799322 1.7885317544 -1.4640335437 H -2.5934196416 2.5748372585 -1.0094502121 H -2.6523323100 1.3669152040 -2.3133245196 H -4.1278307542 2.2120268177 -1.8243815287 C -4.1337542238 1.3399052215 0.7052696848 H -3.4822453447 2.0864572595 1.1483079253 H -4.3596974913 0.5819780017 1.4521592151 H -5.0663404102 1.8100631123 0.3869947656 N 2.6201735674 -1.8968553219 -0.4678369142 H 2.4357916814 -2.7001126866 0.1223649203 H 2.2456395244 -2.1388704123 -1.3807648385 C 4.0798535321 -1.6552639611 -0.5721153213 H 4.5006453777 -1.7281982560 0.4282594133 H 4.5713720983 -2.4004889581 -1.1950634155 C 4.2637661191 -0.2692572306 -1.1472915868 H 3.8892250618 -0.2476605094 -2.1715119771 H 5.3175409713 0.0140447135 -1.1748259443 N 3.4875316595 0.7156442408 -0.3589983072 C 3.2835211482 1.9414419703 -1.1598403187 H 2.7641842796 2.6793528630 -0.5581710474 H 2.6879045539 1.7117465834 -2.0420692295 H 4.2450530380 2.3463113052 -1.4814884068 C 4.1992063212 1.0681887976 0.8912164078 H 3.5810602997 1.7434307266 1.4745666489 H 4.3989122355 0.1783492095 1.4853492250 H 5.1497006857 1.5522941282 0.6575385510 207 Experimental structure of PATFIA including the ferrocene group, taken from M. Julve; M. Verdaguer; A. Gleizes; M. Pohiloche-Levisalles; O. Kahn, Inorg. Chem. 23, 3808 (1984). Cu -2.12986 -1.03325 0.57368 Cu -1.20483 0.68621 -2.00729 Fe 3.39024 -0.45780 0.94495 O -2.31122 -0.71618 -1.30040 O 0.39423 -0.06124 -1.26598 O -0.23800 -1.38309 0.41582 N -1.91186 -1.15181 2.63117 N -4.07216 -0.85241 0.96659 H -4.35689 -0.03505 0.71825 H -4.53547 -1.46468 0.49696 N -0.06583 2.32123 -2.50520 N -2.62923 1.55103 -3.09038 H -2.93019 0.96944 -3.70753 H -3.32049 1.76884 -2.55437 C 2.04553 -1.38459 -0.24804 C 2.50561 -2.30974 0.76005 H 2.00021 -2.75959 1.39687 C 3.90669 -2.37635 0.55843 H 4.49809 -2.87785 1.07231 C 4.26863 -1.57172 -0.53326 H 5.12639 -1.46917 -0.87995 C 3.09046 -0.94148 -1.01764 H 3.03668 -0.34043 -1.72577 C 2.50756 1.08995 1.96977 H 1.61975 1.36091 1.92305 C 2.99897 0.17528 2.85000 H 2.50791 -0.29253 3.48683 C 4.43570 0.05777 2.61888 H 5.01867 -0.50211 3.07867 C 4.77700 0.91705 1.60095 H 5.63753 1.05402 1.27393 C 3.57011 1.57947 1.11903 H 3.49871 2.18501 0.41582 C -3.15977 -0.63834 3.20407 H -3.26612 -0.97292 4.10644 H -3.11923 0.33247 3.24587 C -4.31512 -1.04253 2.38776 208 H -5.08996 -0.52456 2.65330 H -4.50978 -1.97815 2.55741 C -1.49146 -2.41228 3.09588 H -1.61635 -2.45719 4.04743 H -0.56175 -2.53279 2.88935 H -2.00825 -3.09641 2.67051 C -0.92720 -0.09942 3.07621 H -0.87703 -0.09567 4.03513 H -1.21538 0.75836 2.76395 H -0.06112 -0.30151 2.71231 C 0.60126 -0.92726 -0.34394 C -0.99239 3.33620 -3.06334 H -1.31806 3.89159 -2.33800 H -0.50343 3.90206 -3.67557 C -2.13113 2.75985 -3.75917 H -1.87172 2.53979 -4.66891 H -2.84219 3.41853 -3.80343 C 0.92071 1.94399 -3.53050 H 1.69482 2.50536 -3.45428 H 0.53267 2.04953 -4.40337 H 1.17773 1.02633 -3.40510 C 0.63093 2.96943 -1.38891 H 1.52523 3.19024 -1.65446 H 0.66000 2.36614 -0.64145 H 0.16221 3.76584 -1.13812 C -3.35978 -1.13983 -1.96672 H -3.63441 -1.99237 -1.62741 H -3.13654 -1.22067 -2.89860 Experimental structure of Cu$_2$Cl$_6ˆ{2-}$ (Complex 5), taken from O. Castell; J. Miralles; R. Caballol, Chem. Phys. 179, 377 (1994). Cu -0.44198447 1.04236051 -0.03573258 Cl 1.34164711 -0.39956189 0.13607601 Cu 2.94244268 1.24888730 0.03573258 Cl 1.15881110 2.69080969 -0.13607601 Cl -1.89798505 -0.68257133 0.07504862 Cl -2.09088015 2.57782327 -0.21207639 Cl 4.59133836 -0.28657547 0.21207639 Cl 4.39844326 2.97381913 -0.07504862 209 Experimental structure of XAMBUI (Complex 6), taken from C. Lopez; R. Costa; F. Illas; C. de Graaf; M. M. Turnbull; C. P. Landee; E. Espinosa; I. Mata; E. Molins, Dalton Trans. 13, 2322 (2005). Cu 0.883000 1.658000 1.047000 N 1.772000 0.049000 1.855000 H 1.819000 0.193000 2.743000 H 1.215000 -0.648000 1.735000 N 2.654000 2.688000 0.968000 H 2.830000 2.938000 1.825000 N -0.270000 3.270000 0.633000 H -0.868000 2.991000 0.020000 H -0.739000 3.433000 1.384000 C 3.100000 -0.388000 1.425000 H 3.051000 -0.711000 0.510000 H 3.394000 -1.122000 1.985000 C 4.100000 0.725000 1.502000 H 4.981000 0.370000 1.305000 H 4.116000 1.070000 2.409000 C 3.805000 1.862000 0.548000 H 4.590000 2.429000 0.479000 H 3.625000 1.497000 -0.332000 C 2.633000 3.936000 0.191000 H 2.484000 3.722000 -0.745000 H 3.500000 4.365000 0.260000 C 1.571000 4.904000 0.647000 H 1.566000 4.926000 1.617000 H 1.805000 5.792000 0.334000 C 0.190000 4.573000 0.165000 H -0.425000 5.255000 0.475000 H 0.181000 4.583000 -0.805000 H -1.566000 -4.926000 -1.617000 O -0.859000 1.012000 1.755000 O 0.999000 1.144000 -1.161000 C -1.353000 -0.156000 1.847000 H -2.162251 -0.270057 2.598516 H -0.181000 -4.583000 0.805000 O 0.859000 -1.012000 -1.755000 C 1.353000 0.156000 -1.847000 H -4.590000 -2.429000 -0.479000 N 0.270000 -3.270000 -0.633000 210 N -2.654000 -2.688000 -0.968000 H 0.868000 -2.991000 -0.020000 H 0.739000 -3.433000 -1.384000 H -3.500000 -4.365000 -0.260000 C -0.190000 -4.573000 -0.165000 H -2.484000 -3.722000 0.745000 H -1.805000 -5.792000 -0.334000 H -2.830000 -2.938000 -1.825000 C -2.633000 -3.936000 -0.191000 H 0.425000 -5.255000 -0.475000 C -1.571000 -4.904000 -0.647000 H 2.162251 0.270057 -2.598516 H -3.625000 -1.497000 0.332000 O -0.999000 -1.144000 1.161000 Cu -0.883000 -1.658000 -1.047000 N -1.772000 -0.049000 -1.855000 H -1.819000 -0.193000 -2.743000 H -1.215000 0.648000 -1.735000 C -3.100000 0.388000 -1.425000 H -3.051000 0.711000 -0.510000 H -3.394000 1.122000 -1.985000 C -4.100000 -0.725000 -1.502000 H -4.981000 -0.370000 -1.305000 H -4.116000 -1.070000 -2.409000 C -3.805000 -1.862000 -0.548000 Experimental structure of YAFZOU (Complex 7), taken from T. Tokii; N. Hamamura; M. Nakashima; Y. Muto, Bull. Chem. Soc. Jpn. 65, 1214 (1992). Cu 0.70500000 3.58600000 12.98500000 Cu 0.60200000 0.61700000 13.51000000 O -0.44600000 2.21900000 13.69500000 O 0.72000000 2.87600000 11.18800000 O 0.71700000 0.67000000 11.57900000 H 1.68900000 1.75100000 9.02600000 N 0.52400000 4.76500000 14.61500000 N 1.75000000 5.17700000 12.31800000 N 1.58600000 -1.16000000 13.47400000 N 0.19300000 0.00400000 15.39800000 C 0.75300000 1.67700000 10.81600000 C 0.85900000 1.41900000 9.34600000 211 C -0.10900000 4.53800000 15.74600000 C -0.18700000 5.47300000 16.77500000 C 0.38200000 6.70900000 16.60400000 C 1.05600000 6.99300000 15.41300000 C 1.67400000 8.25800000 15.11200000 C 2.28000000 8.48300000 13.92800000 C 2.35600000 7.44800000 12.92500000 C 2.96500000 7.60300000 11.68000000 C 2.95000000 6.55000000 10.78900000 C 2.34200000 5.35700000 11.13100000 C 1.75600000 6.21200000 13.20600000 C 1.10400000 5.98900000 14.44500000 C 2.26300000 -1.71100000 12.47800000 C 2.77400000 -3.00800000 12.56600000 C 2.57300000 -3.75300000 13.69100000 C 1.85200000 -3.20100000 14.76600000 C 1.56200000 -3.87400000 16.01400000 C 0.88600000 -3.26900000 16.99000000 C 0.42400000 -1.91500000 16.86900000 C -0.28400000 -1.21700000 17.84100000 C -0.78300000 0.01900000 17.62200000 C -0.52400000 0.60500000 16.32300000 C 0.66400000 -1.24900000 15.64800000 C 1.39600000 -1.88000000 14.60600000 H -0.83600000 5.26700000 17.64300000 H 0.51300000 7.20300000 17.42200000 H 1.56200000 8.88900000 15.79700000 H 2.81200000 9.26400000 13.76100000 H 3.34200000 8.43700000 11.39400000 H 3.59300000 6.49000000 10.00200000 H 2.13400000 4.51800000 10.43000000 H 2.33900000 -1.18900000 11.62100000 H 3.24600000 -3.36000000 11.79400000 H 3.15100000 -4.73900000 13.72400000 H 1.77400000 -4.96900000 15.96600000 H 0.63500000 -3.82600000 17.75800000 H -0.67000000 -1.32700000 18.75500000 H -1.26600000 0.71600000 18.26800000 H -0.93100000 1.72200000 16.15400000 H -0.53400000 3.66800000 15.91300000 H 0.80900000 0.48400000 9.18400000 H -1.47700000 2.22800000 13.23300000 212 H 0.14700000 1.85700000 8.89600000 Experimental structure of CITLAT (Complex 8), taken from S. Youngme; J. Phatchimkun; N. Wannarit; N. Chaichit; S. Meejoo; G. A. Van Albada; J.Reedijk, Polyhedron 18, 1235 (1999). H -1.219148 8.422300 12.395086 H -4.824884 8.422300 11.143813 H -0.668144 2.122420 12.384391 H -1.580903 7.327401 7.935422 H -1.937515 2.543535 8.266956 H 0.287625 5.171292 15.090135 H -1.504874 3.116251 10.459356 H -2.108325 6.013522 5.924830 H -2.440758 3.453143 6.192196 H -0.054372 6.822063 13.272046 H 0.337745 2.897271 14.576792 Cu -1.064279 6.971306 10.628332 O -0.751436 8.422300 11.899924 N -1.262633 3.707496 10.449731 N -0.697154 5.464388 11.938425 N -1.431483 5.679999 9.140707 C -1.668384 6.210604 7.914032 O -0.607388 8.422300 9.408073 O -3.372154 7.309714 10.860405 C -1.512714 4.342538 9.249793 C -1.849356 3.518837 8.174982 C -0.833693 4.155563 11.676406 C 0.026621 4.891672 14.154354 C -0.269691 5.808018 13.174725 C -2.079241 4.084816 6.946168 C -1.998220 5.469442 6.813554 C -0.528307 3.176892 12.621812 C -0.097081 3.542419 13.870946 C -3.908969 8.422300 10.971630 H 0.148940 8.422300 9.208084 H -0.668144 14.722180 12.384391 H -1.580903 9.517199 7.935422 H -1.937515 14.301065 8.266956 H 0.287625 11.673308 15.090135 H -1.504874 13.728349 10.459356 213 H -2.108325 10.831078 5.924830 H -2.440758 13.391457 6.192196 H -0.054372 10.022537 13.272046 H 0.337745 13.947329 14.576792 Cu -1.064279 9.873294 10.628332 N -1.262633 13.137104 10.449731 N -0.697154 11.380212 11.938425 N -1.431483 11.164601 9.140707 C -1.668384 10.633996 7.914032 O -3.372154 9.534886 10.860405 C -1.512714 12.502062 9.249793 C -1.849356 13.325763 8.174982 C -0.833693 12.689037 11.676406 C 0.026621 11.952928 14.154354 C -0.269691 11.036582 13.174725 C -2.079241 12.759784 6.946168 C -1.998220 11.375158 6.813554 C -0.528307 13.667708 12.621812 C -0.097081 13.302181 13.870946 Optimized $\omega$B97X-D/cc-pVTZ Kohn-Sham triplet geometry of CITLAT (Complex 8). Cu -0.0730822308 0.0887543490 1.4463209949 O 0.1963213404 1.3935693921 0.0000000000 H 1.1388404125 1.5796172815 0.0000000000 Cu -0.0730822308 0.0887543490 -1.4463209949 O -1.0697306915 -0.8929854865 0.0000000000 H -2.0145095732 -0.7424336971 0.0000000000 N -0.3194045620 -1.3725424431 -2.8182377381 C -0.2077106385 -1.2411017902 -4.1361913994 C -0.3775229840 -2.6148509399 -2.3016680362 H -0.4613016849 -2.6430449547 -1.2266080247 C -0.3297549917 -3.7513730297 -3.0649424262 H -0.3730432446 -4.7237351293 -2.6011612967 N -0.1316561306 0.0012665462 -4.7303882939 H -0.0771810535 -0.0414506000 -5.7322233869 C -0.1398518047 1.2896986011 -4.2407970530 N -0.1425212461 1.5356338365 -2.9327641994 C -0.1626320700 2.8218674030 -2.5291504919 H -0.1779485926 2.9491277063 -1.4572148956 C -0.1617656483 3.8846864627 -3.3938698797 214 H -0.1660002714 4.8925871558 -3.0100534873 C -0.1451313324 3.6225478278 -4.7604324222 H -0.1442531405 4.4276563301 -5.4818225863 C -0.1404703499 2.3216420590 -5.1898911247 H -0.1386936301 2.0811777972 -6.2436861728 C -0.2196685908 -3.6115492312 -4.4450954622 H -0.1721535683 -4.4781434081 -5.0893558134 C -0.1614646143 -2.3535873868 -4.9858415415 H -0.0771908322 -2.2161966441 -6.0547551149 N -0.3194045620 -1.3725424431 2.8182377381 C -0.2077106385 -1.2411017902 4.1361913994 C -0.3775229840 -2.6148509399 2.3016680362 H -0.4613016849 -2.6430449547 1.2266080247 C -0.3297549917 -3.7513730297 3.0649424262 H -0.3730432446 -4.7237351293 2.6011612967 N -0.1316561306 0.0012665462 4.7303882939 H -0.0771810535 -0.0414506000 5.7322233869 C -0.1398518047 1.2896986011 4.2407970530 N -0.1425212461 1.5356338365 2.9327641994 C -0.1626320700 2.8218674030 2.5291504919 H -0.1779485926 2.9491277063 1.4572148956 C -0.1617656483 3.8846864627 3.3938698797 H -0.1660002714 4.8925871558 3.0100534873 C -0.1451313324 3.6225478278 4.7604324222 H -0.1442531405 4.4276563301 5.4818225863 C -0.1404703499 2.3216420590 5.1898911247 H -0.1386936301 2.0811777972 6.2436861728 C -0.2196685908 -3.6115492312 4.4450954622 H -0.1721535683 -4.4781434081 5.0893558134 C -0.1614646143 -2.3535873868 4.9858415415 H -0.0771908322 -2.2161966441 6.0547551149 O 1.9122864545 -0.6622436998 -1.1242579786 C 2.4185555355 -0.8815298070 0.0000000000 H 3.4311512747 -1.3172547860 0.0000000000 O 1.9122864545 -0.6622436998 1.1242579786 215 Chapter 5 References [1] P.M. 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Both the systems and their properties of interest vary in complexity, the smallest and most conceptually straightforward of our examples being photodetachment from the di- and triatomic anionic copper oxides presented in Chapter 2. The work presented in Chapter 2 is limited to EOM-CCSD analysis of photodetached states of CuO , Cu 2 O , and CuO 2 . As clusters become larger and the number of correlated electrons more numerous, the energy separation between the occupied and virtual orbital subspace shrinks, posing a significant challenge when choosing a model chemistry and/or ref- erence state(s) for theoretical analysis. We encountered this challenge even for the relatively small CuO 2 . Density-based analysis paired with a spin-flip approach with multiple spin-flips, 1, 2 would go a long way toward resolving the true radical character (i.e., two vs. four and one vs. three unpaired electrons in the ground anionic and electron detached states, respectively) of CuO 2 and similar small metal oxide molecules. Yet another challenge lies in the stability of the larger Cu x O y complexes; photodisso- ciation is observed at relatively low photon energies. Both the CuO 3 and CuO 4 clusters (structures depicted in Figure 6.1) dissociate to form O 2 at 3.49 eV . A comprehensive theoretical investigation of their photodetachment spectra would first require an accu- rate computation of the photodissociation energy barrier, followed by EOM-IP-CCSD 221 ( )(O 2 ) Figure 6.1: CuO 3 (left) and two structural isomers of CuO 4 : the Cu(O 2 ) 2 complex (middle) and the O 2 -solvated OCuO structure (right). Copper atoms are shown in gold, oxygen atoms in red. The left and middle structures undergo photodisoc- ciation to yield O 2 at 355 nm. The structure on the right gives rise to a photode- tachment spectrum similar to CuO 2 , with the solvating O 2 slightly perturbing the transitions. 3 calculations at key points along the potential energy surface of the dissociation chan- nels. Stable orientations of the O 2 molecule in the O 2 -solvated structure of CuO 4 would be critical for accurate determination of the way in which the solvating O 2 molecule impacts electron detachment energies and states of OCuO . Subsequent NO and NTO analysis would reveal how the electron density associated with O 2 plays a role in the low-lying electron detached states of interest. Our prescription for density-based radical state analysis—described in Chapter 3 and applied in Chapters 4 and 5—should play a role in any future theoretical investigations of molecular magnetism. Chapter 5 is restricted to the analysis of eight copper diradi- cals, which are themselves only an example of the array of binuclear copper diradicals available for study (other examples depicted in Figure 6.2). Analysis of singlet-triplet gaps and wavefunction properties of similar copper systems will be straightforward, and aimed toward discovering the optimal ligand environment for maximizing the en- ergy difference between the triplet ground state and singlet excited state in the ideal binuclear copper magnet. Analysis of exchange-coupling in molecular magnets is not restricted to two-center radicals. For systems with three or more radical sites, visual inspection of NOs may be 222 CITLEX CITLIB CITLOH EBEFIB JEJCIK RUXDIX01 YAHYUC YAHZEN Figure 6.2: Binuclear copper diradicals not considered in the present work. Exper- imental structures 4–7 are labeled with their Cambridge Crystal Structure Database reference codes. Counter ions are shown, but would be removed in the final analy- sis. sufficient to conclude the region of localization of the radical site(s) in low-lying spin states, thus allowing direct application of the Land´ e interval rule for each unique pair of neighboring radical sites and their associatedJ’s. Where a more rigorous localization scheme is needed, a technique implemented by Mayhall and Head-Gordon 8, 9 in which spin-flipped excited states are projected onto the space of the Heisenberg Hamiltonian facilitates computing the exchange-coupling constant for polyradicals with greater than two radical sites. Interesting examples for study include the copper triradicals depicted in Figure 6.3 and the hexa-manganese cage complex depicted in Figure 6.4. The latter complex is interesting not only due to its high total spin (S = 12) and magnetization reorientation barrier (89 K), but due to the segregation of the triangular planes of Mn 223 NETSAH NETSIP Figure 6.3: Two examples of trinuclear copper triradicals. Experimental struc- tures 11 are labeled with their Cambridge Crystal Structure Database reference codes. spin sites. In particular, computational studies of exchange-coupling in this complex would commence with the goal of understanding whether the six Mn centers behave as individual spin centers, or whether the two groups of three planar manganese atoms act as two effective spin centers, with one coupling parameter,J eff . The latter is assumed in experimental models of the complex’s exchange-coupling parameter. 10 In the analysis of radical wave functions—and for magnetic applications in particular—a numerical procedure for determining whether a singlet (or doublet) wave- function arises from closed or open-shell configurations is currently unavailable in the electronic structure toolkit. For magnetic applications, it is important that radical elec- trons be spatially separated (i.e., purely of the open-shell covalent radical type). In cir- cumstances where spin and spatial symmetry mixing allows mixing of open and closed- shell singlets, it can be difficult to determine whether and electrons are spatially separated. A density-matrix-based method for computing charge and spin cumulants that is aimed at determining the degree of multi-exciton and charge-resonance charac- ter of excited states has been implemented for the restricted active space (RAS) family 224 Figure 6.4: Experimental structure of [Mn 6 O 2 (sao) 6 (O 2 CPh) 2 (EtOH) 4 ], where “sao” is short for salicylaldoxime. Mn atoms depicted in red, oxygen in green, and nitrogen in blue. J exp = 1.6 cm 1 . 10 The shaded rhombus shows the plane of symmetry that divides the two triangular groups of three Mn(III) spin centers. of spin-flip methods. 12 Once implemented within the EOM-CCSD and TDDFT frame- works, this approach will provide another, more direct means of determining whether a potential single molecule magnet obeys Heisenberg physics. Together with the ap- plication of robust open-shell electronic structure methods and NO spin state analysis described in the previous chapters, charge and spin cumulant analysis would provide another useful test of molecular magnetic potential on the road to finding alternatives to conventional magnetic materials. 225 Magnets Inc. Molecular Under New Management FabriCating devices since 4000 BCE lightweight, tunable ^ TODAY "It's been coming for a while. I was just too dense to see it." ©2017 Figure 6.5: There will likely always be a place in industry for traditional magnetic materials, which have been a part of human culture for thousands of years. Nev- ertheless, studies like those presented here bring us ever closer to the day when molecular magnetic devices can compete with their conventional predecessors. 226 Chapter 6 References [1] D. Casanova, L. V . Slipchenko, A. I. Krylov, and M. Head-Gordon, J. Chem. Phys. 130, 044103 (2009). [2] F. Bell, P. M. Zimmerman, D. Casanova, M. Goldey, and M. Head-Gordon, Phys. Chem. Chem. Phys. 15, 358 (2013). [3] H. Wu, S. R. Desai, and L. S. Wang, J. Phys. Chem. A 101, 2103 (1997). [4] S. Youngme, J. Phatchimkun, N. Wannarit, N. Chaichit, S. Meejoo, G. A. van Albada, and J. Reedijk, Polyhedron 27, 304 (2008). [5] S. Youngme, C. Chailuecha, G. A. van Albada, C. Pakawatchai, N. Chaichit, and J. Reedijk, Inorg. Chim. Acta 357, 2532 (2004). [6] G. Christou, S. P. Perlepes, K. Folting, J. C. Huffman, R. J. Webb, and D. N. Hendrickson, J. Chem. Soc., Chem. Commun. , 746 (1990). [7] S. Youngme, C. Chailuecha, G. A. Van Albada, C. Pakawatchai, N. Chaichit, and J. Reedijk, Inorg. Chim. Acta 358, 1068 (2005). [8] N.J. Mayhall and M. Head-Gordon, J. Chem. Phys. 141, 134111 (2014). [9] N.J. Mayhall and M. Head-Gordon, J. Phys. Chem. Lett. 6, 1982 (2015). [10] C. J. Milios, A. Vinslava, W. Wernsdorfer, S. Moggach, S. Parsons, S. P. Perlepes, G. Christou, and E. K. Brechin, J. Am. Chem. Soc. 129, 2754 (2007), PMID: 17309264. [11] A. Roth, J. Becher, C. Herrmann, H. G¨ orls, G. Vaughan, M. Reiher, D. Klemm, and W. Plass, Inorg. Chem. 45, 10066 (2006). [12] A. V . Luzanov, D. Casanova, X. Feng, and A. I. Krylov, J. Chem. Phys. 142, 224104 (2015). 227
Abstract (if available)
Abstract
Modeling the electronic structure of open-shell species is both essential to computational studies of excited-state processes and difficult due to the multi-configurational nature of the states involved. Choosing a reliable methodology becomes even more difficult when the systems one wishes to study have a large number of electrons, or when inclusion of relativistic effects is necessary for accurate description. The work presented here provides several diverse examples of challenging open-shell systems. For each system, we offer a detailed account of how the challenges inherent in their study were overcome for accurate description of the relevant chemical and physical properties, ranging from photodetachment energies to Heisenberg exchange-coupling interactions. ❧ The experimental photoelectron spectra of di- and triatomic copper oxide anions have been reported previously. In Chapter 2, we present an analysis of the experimental spectra of the CuO−, Cu₂O−, and CuO₂- anions using equation-of-motion coupled-cluster (EOM-CC) methods. The open-shell electronic structure of each cluster demands a unique combination of EOM-CC methods to achieve an accurate and balanced representation of the multi-configurational anionic and neutral state manifolds. Analysis of the Dyson orbitals associated with photodetachment from CuO− reveals the strong non-Koopmans character of the CuO states. A perturbative triples correction to the coupled-cluster singles and doubles ansatz is required for accurate description of electron detachment from Cu₂O−. Use of a relativistic pseudopotential and matching basis set improves quality of results in most cases. EOM-DIP-CCSD analysis of the low-lying states of CuO₂- reveals multiple singlet and triplet anionic states near the triplet ground state, adding a extra layer of complexity to the interpretation of the experimental CuO₂- photoelectron spectrum. ❧ In Chapter 3, we present an analysis of the wavefunction properties of classic open-shell systems. Density-based wave function analysis enables unambiguous comparisons of electronic structure computed by different methods and removes ambiguity of orbital choices. We use this tool to investigate the performance of different spin-flip methods for several prototypical diradicals and triradicals. In contrast to previous calibration studies that focused on energy gaps between high and low spin-states, we focus on the properties of the underlying wave functions, such as the number of effectively unpaired electrons. Comparison of different density functional and wave function theory results provides insight into the performance of the different methods when applied to strongly correlated systems such as polyradicals. We show that canonical molecular orbitals for species like large copper-containing diradicals fail to correctly represent the underlying electronic structure due to highly non-Koopmans character, while density-based analysis of the same wave function delivers a clear picture of bonding pattern. ❧ Chapter 4 extends the density-based wavefunction analysis presented in Chapter 3 to stationary points in the Bergman cyclization, an important reaction in which an enediyne cyclizes to produce a highly reactive diradical species, p-benzyne. Enediyne motifs are found in natural antitumor antibiotic compounds, such as calicheammicin and dynemicin. Understanding of the energetics of the cyclization is required in order to better control the initiation of the cyclization, which induces cell death. The singlet and triplet potential energy surfaces for the thermochemically induced Bergman cyclization of (Z)- hex-3-ene-1,5-diyne have been computed by the CCSD and EOM-SF-CCSD methods. The triplet enediyne and transition state were found to have C₂ symmetry, which contrasts with the singlet reactant and transition state that possess C₂v symmetry. The frontier orbitals of both cyclization pathways were analyzed to explain the large energetic barrier of the triplet cyclization. Reaction energies were calculated using CCSD(T)/cc- pVTZ single point calculations on structures optimized with CCSD/cc-pVDZ. The singlet reaction was found to be slightly endothermic (∆Hrxn = 13.76 kcal/mol) and the triplet reaction was found to be highly exothermic (∆Hrxn = -33.29 kcal/mol). The adiabatic singlet-triplet gap of p-benzyne, computed with EOM-SF-CCSD/cc-pVTZ, was found to be 3.56 kcal/mol, indicating a ground state singlet. ❧ Molecular magnets, defined here as organic polyradicals, can be used as building blocks in the fabrication of novel and structurally diverse magnetic light-weight materials. In Chapter 5, we present a theoretical investigation of the lowest spin states of several binuclear copper diradicals. In contrast to previous studies, we consider not only the energetics of the low-lying states (which are related to the exchange-coupling parameter within the Heisenberg-Dirac-van-Vleck model), but also the character of the diradical states themselves. We use natural orbitals, their occupations, and the number of effectively unpaired electrons to quantify bonding patterns in these systems. We compare the performance of spin-flip time-dependent density functional theory (SF-TDDFT) using various functionals and effective core potentials against the wave function based approach, equation-of-motion spin-flip coupled-cluster method with single and double substitutions (EOM-SF-CCSD). We find that SF-TDDFT paired with the PBE50 and B5050LYP functionals performs comparably to EOM-SF-CCSD, with respect to both singlet-triplet gaps and states’ characters. Visualization of frontier natural orbitals shows that the unpaired electrons are localized on copper centers, in some cases exhibiting slight through-bond interaction via copper d-orbitals and p-orbitals of neighboring ligand atoms. The analysis reveals considerable interactions between the formally unpaired electrons in the antiferromagnetic diradicaloids, meaning that they are poorly described by the Heisenberg-Dirac-van-Vleck model. Thus, for these systems the experimentally derived exchange-coupling parameters are not directly comparable with the singlet-triplet gaps. This explains systematic discrepancies between the computed singlet-triplet energy gaps and the exchange-coupling parameters extracted from experiment.
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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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New electronic structure methods for electronically excited and open-shell species within the equation-of-motion coupled-cluster framework
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Development of robust ab initio methods for description of excited states and autoionizing resonances
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Orms, Natalie Faith
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Core Title
Electronic structure analysis of challenging open-shell systems: from excited states of copper oxide anions to exchange-coupling in binuclear copper complexes
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College of Letters, Arts and Sciences
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Doctor of Philosophy
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Chemistry (Chemical Physics)
Publication Date
11/03/2017
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10/20/2017
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Bergman cyclization,copper oxide,coupled-cluster,density functional theory,diradicals,electronic structure theory,equation-of-motion,exchange-coupling,molecular magnets,OAI-PMH Harvest,open-shell,quantum chemistry,spin-flip,triradicals,wavefunction analysis
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Krylov, Anna I. (
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natalie.orms@gmail.com,orms@usc.edu
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Bergman cyclization
copper oxide
coupled-cluster
density functional theory
diradicals
electronic structure theory
equation-of-motion
exchange-coupling
molecular magnets
open-shell
quantum chemistry
spin-flip
triradicals
wavefunction analysis