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University of Southern California Dissertations and Theses
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Predictive electronic structure methods for strongly correlated systems: method development and applications to singlet fission
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Predictive electronic structure methods for strongly correlated systems: method development and applications to singlet fission
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PREDICTIVE ELECTRONIC STRUCTURE METHODS FOR STRONGLY CORRELATED SYSTEMS: METHOD DEVELOPMENT AND APPLICATIONS TO SINGLET FISSION by Xintian Feng A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (CHEMISTRY) Dec 2016 Copyright 2016 Xintian Feng Acknowledgements First of all, I would like to thank Professor Anna I. Krylov, my advisor, who has con- tributed greatly to my research progress. It was a great honor for me to be in her group, which is a friendly and cooperative environment. I appreciate her advice and support, especially the enthusiasm toward the results I obtained, which was really encouraging. Also, I would like to thank the group activities she organized: great food, hikes, trips. I also thank all the members of Anna’s group, Atanu Acharya, Anastasia Gunina, Dr. Evgeny Epifanovsky, Dr. Ilya Kaliman, Dr. Dmitry Zuev for their help throughout the projects, as well as others for stimulating discussions not only about science but also about life in the U.S. I am grateful to our collaborators on the singlet fission project, Professor Stephen E. Bradforth and Professor Mark E. Thompson for valuable discussions, Professor David Casanova for his help with the RAS-CI code in Q-Chem, Professor Anatoly Kolomeisky for enlightening discussions, as well as Professor Anatoliy V . Luzanov and Professor Spiridoula Matsika. I thank my wife Xiaoyan Wei for her understanding and support during 5 years of my Ph.D. research. Without her, I would not be able to spend enough time on my research projects. Finally, I would like to thank my family in China for supporting me and making it all possible. ii Contents Acknowledgements ii List of Tables vii List of Figures x Abbreviations xv Abstract xvii Chapter 1: Introduction and overview 1 1.1 Singlet fission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Electronic structure methods for strongly correlated systems . . . . . . 4 1.2.1 Electron correlation . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2.2 Describing electron correlation in closed-shell systems by coupled- cluster approach . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2.3 Describing Electron correlation in electronically excited and open- shell states by equation-of-motion coupled-cluster approach . . 8 1.2.4 Improving efficiency of electronic structure methods . . . . . . 10 Chapter 1 references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Chapter 2: Fission of Entangled Spins: An Electronic Structure Perspective 13 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Chapter 2 references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 Chapter 3: What We Can Learn from the Norms of One-particle Density Matrices, and What We Can’t: Some Results for Interstate Prop- erties in Model Singlet Fission Systems 37 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.3 Computational details . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.4.1 Octatetraene . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 iii 3.4.2 Butadiene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.4.3 Ethylene dimer . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 Chapter 3 references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 Chapter 4: Quantifying charge resonance and multiexciton character in cou- pled chromophores by charge and spin cumulant analysis 68 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 4.2 Electronic states of molecular dimers . . . . . . . . . . . . . . . . . . . 74 4.3 Charge cumulant indexes . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.4 Using charge cumulants to quantify charge resonance . . . . . . . . . . 81 4.5 Spin correlators for charge resonance and biexcitons . . . . . . . . . . . 85 4.6 Local excitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 4.7 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 89 4.7.1 Computational details . . . . . . . . . . . . . . . . . . . . . . . 92 4.7.2 Numeric examples . . . . . . . . . . . . . . . . . . . . . . . . 94 4.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 Chapter 4 references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 Chapter 5: A simple kinetic model for singlet fission: A role of electronic and entropic contributions to macroscopic rates 116 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 5.2 Kinetic model for SF process . . . . . . . . . . . . . . . . . . . . . . . 120 5.3 Entropy calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 5.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 Chapter 5 references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 Chapter 6: Dissecting the Effect of Morphology on the Rates of Singlet Fis- sion: Insights from Theory 144 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 6.2 Theoretical framework . . . . . . . . . . . . . . . . . . . . . . . . . . 146 6.3 Computational details . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 6.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 154 6.4.1 1,3-diphenylisobenzofuran (DPBF) . . . . . . . . . . . . . . . 154 6.4.2 1,6-diphenyl-1,3,5,-hexatriene (DPH) . . . . . . . . . . . . . . 161 6.4.3 5,12-diphenyltetracene (DPT) . . . . . . . . . . . . . . . . . . 164 6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 Chapter 6 references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 iv Chapter 7: On couplings and excimers: Lessons from studies of singlet fis- sion in covalently linked tetracene dimers 174 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 7.2 Theoretical methods and computational details . . . . . . . . . . . . . . 183 7.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 186 7.3.1 Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 7.3.2 Energetics, couplings, and rates . . . . . . . . . . . . . . . . . 187 7.3.3 The role of covalent linker in BET-B . . . . . . . . . . . . . . . 195 7.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 Chapter 7 references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 Chapter 8: Intra- vs. Inter-Molecular singlet fission in covalently linked dimer206 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 8.2 Theoretical methods and computational details . . . . . . . . . . . . . . 209 8.2.1 Raw RAS-SF energies and energy correction . . . . . . . . . . 210 8.2.2 Comparison of energies and anticipated error bars . . . . . . . . 214 8.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 216 8.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 Chapter 8 references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 Chapter 9: Cholesky representation of electron-repulsion integrals within coupled-cluster and equation-of-motion methods 232 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 9.2 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 9.2.1 Cholesky algorithm . . . . . . . . . . . . . . . . . . . . . . . . 235 9.2.2 Resolution-of-the-identity algorithm . . . . . . . . . . . . . . . 238 9.3 RI/CD CCSD and EOM-CCSD methods: Theory . . . . . . . . . . . . 239 9.3.1 Coupled-cluster equations with single and double substitutions . 239 9.3.2 EOM-EE/SF-CCSD and CD/RI EOM-EE/SF-CCSD . . . . . . 244 9.3.3 EOM-IP-CCSD and CD/RI EOM-IP-CCSD . . . . . . . . . . . 248 9.3.4 EOM-EA-CCSD and CD/RI EOM-EA-CCSD . . . . . . . . . . 249 9.4 Benchmarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 9.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 Chapter 9 references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 Chapter 10: Implementation of analytic gradients for CCSD and EOM-CCSD using Cholesky representation of electron-repulsion integrals: Theory and benchmarks 273 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 10.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 10.3 Benchmarks and Discussions . . . . . . . . . . . . . . . . . . . . . . . 277 10.3.1 Errors in optimized structures . . . . . . . . . . . . . . . . . . 287 v 10.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288 Chapter 10 references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 Chapter 11: Future work 290 Chapter 11 references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 Bibliography 294 vi List of Tables 2.1 Analysis of tetracene dimer states and multiexciton binding energy (E b , eV) at selected geometries. . . . . . . . . . . . . . . . . . . . . . . . . 24 2.2 jj jj between different dimer states and exciton binding energy (E b , eV) of tetracene (left) and pentacene (right) dimers at various geometries. The state characters (ME/EX/CR) is shown in parentheses. . . . . . . . 30 3.1 Configuration analysis of the RAS-2SF wave functions of the ME and S 1 states using localized orbitals within DMO-LCFMO framework a . . 57 4.1 Expressions for computing weights of local excitation, charge resonance, and biexcitonic contributions. . . . . . . . . . . . . . . . . . . . . . . 91 4.2 Analysis of low-lying singlet excited states in H 2 -He and (H 2 ) 2 using FCI/cc-pVDZ wave functions. E ex andf l denote excitation energy and oscillator strength, respectively. . . . . . . . . . . . . . . . . . . . . . 95 4.3 Analysis of the S 1 and ME states in ethylene dimer using the RAS(4,4)- 2SF/6-31G(d) wave functions. E ex andf l denote excitation energy and oscillator strength, respectively. . . . . . . . . . . . . . . . . . . . . . 97 4.4 Analysis of the electronic states in tetracene dimer using the RAS(4,4)- 2SF/(C:cc-pVTZ-f/H:cc-pVDZ) wave functions. E ex andf l denote ex- citation energy and oscillator strength, respectively. . . . . . . . . . . . 100 4.5 S 1 -MEjj jj for selected DPH dimers . . . . . . . . . . . . . . . . . . . 101 4.6 Analysis of the electronic states in DPH dimers a from the monoclinic form using the RAS(4,4)-2SF/(C,O:cc-pVTZ-f/H:cc-pVDZ) wave func- tions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 4.7 Analysis of the electronic states in DPH dimers a from the orthorhombic form using the RAS(4,4)-2SF/(C,O:cc-pVTZ-f/H:cc-pVDZ) wave func- tions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 4.8 Analysis of the electronic states in DPBF dimers a from form using the RAS(4,4)-2SF/(C,O:cc-pVTZ-f/H:cc-pVDZ) wave functions. . . . . . 107 4.9 Analysis of the electronic states in DPBF dimers a from form using the RAS(4,4)-2SF/(C,O:cc-pVTZ-f/H:cc-pVDZ) wave functions. . . . . . 108 5.1 Electronic energies (eV) in tetracene, pentacene, and hexacene a . . . . . 119 vii 5.2 Entropic contributions as function of the number of coordination shells. 130 5.3 Relevant thermodynamic quantities (eV) in tetracene, pentacene, and hexacene a at 298 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 5.4 Computed characteristic times for tetracene, pentacene, and hexacene. . 134 6.1 Relevant electronic energies (eV) andjj jj 2 in several DPBF dimers. . . 155 6.2 Relative rates computed for different dimers from and forms. 1:2:3:4 denotes relative rates in dimers 1-4 (see Fig. 6.2 for definition of the dimers). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 6.3 Relevant electronic energies (eV) andjj jj 2 in several DPH dimers. . . 162 6.4 Relative rates computed for different dimers from M and O forms of DPH. 1:2:3 denotes relative rates in dimers 1-3 (see Fig. 6.5 for defini- tion of different dimers). . . . . . . . . . . . . . . . . . . . . . . . . . 165 6.5 Relevant electronic energies (eV) and couplings in several DPT dimers. 167 6.6 Relative rates computed for different dimers from V and X structures at different temperatures. . . . . . . . . . . . . . . . . . . . . . . . . . . 168 7.1 Comparison of the equilibrium structures of the ground and excited states in BET-X and BET-B. . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 7.2 Relevant electronic factors and relative rates of the S 1 ! 1 ME step in dimers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 7.3 Relevant electronic factors and relative rates of the S 1 ! 1 ME and 1 ME ! 2T 1 steps in BET-B and BET-X dimers . . . . . . . . . . . . . . . . 189 7.4 Relevant electronic factors and relative rates of the S 1 ! 1 ME and 1 ME ! 2T 1 steps in BET-B and BET-X dimers at the S 1 -optimized structures. All energies are in eV . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 8.1 S 1 and T 1 vertical excitation energies (eV) for tetracene a . . . . . . . . . 215 8.2 Electronic properties of BET-B. . . . . . . . . . . . . . . . . . . . . . 218 8.3 Electronic properties of (BET-B) 2 . . . . . . . . . . . . . . . . . . . . . 221 9.1 Intermediates for CCSD calculations and estimates to store and com- pute them (closed-shell case). . . . . . . . . . . . . . . . . . . . . . . . 242 9.2 I andT intermediates for EOM-CCSD and estimated cost to store and compute them (closed-shell case). . . . . . . . . . . . . . . . . . . . . 246 9.3 Test systems used for benchmarks, converged CCSD correlation ener- gies (hartree), and number of CC iterations. . . . . . . . . . . . . . . . 254 9.4 CCSD errors and wall times (sec) using 12 cores for test1-test3 . . . . . 255 9.5 CCSD errors and wall times (sec) using 12 cores for test4-test6. . . . . 256 9.6 Wall time per CCSD iteration (sec) using 80 GB RAM. . . . . . . . . . 258 9.7 EOM-CCSD energies for the 2 lowest states in each irrep and errors in energy differences (eV), and wall times for EOM (sec) using 12 cores. . 261 viii 9.8 EOM-CCSD energies for the 2 lowest states in each irrep and errors in energy differences (eV), and wall times (sec) using 12 cores. . . . . . . 262 9.9 EOM-IP-CCSD energies (absolute errors for RI/CD) and EOM wall times (sec) for test4 (two lowest EOM roots). . . . . . . . . . . . . . . 263 9.10 Energy differences between PYPb isomers (E anitanti E antisyn ) and the corresponding errors against full CCSD. . . . . . . . . . . . . . . . 264 10.1 Norm of error (NE) between the analytic CD integral gradient, (X P ) an , and the FD CD integral gradient, (X P ) fd , computed with different CD threshold. Basis set: cc-pVDZ. . . . . . . . . . . . . . . . . . . . . . 280 10.2 Norm of error (NE) between re-constituted gradient of ERI, (j) CD , and analytic ERI derivatives, (j) CAN . Basis set: cc-pVDZ. . . . . 281 10.3 Maximum deviation (MD) between re-constituted gradient of ERI and re-constituted FD ERI gradient with fixed decomposition sequence for FD points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 10.4 Maximum deviation (MD) between re-constituted gradient of ERI and re-constituted FD ERI gradient with NOT fixed decomposition sequence for FD points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284 10.5 Norm of error (NE) between analytic CD-CCSD force and canonical analytic CCSD force at different CD threshold,. Basis set: cc-pVDZ . 286 10.6 The results of CD-CCSD/cc-pVDZ optimization using analytic gradient at ifferent CD thresholds,. The errors are computed against canonical CD-CCSD values and reported as maximum deviation (MD). MD for bondlengths and forces are denoted byjrj (pm) and byjFj (10 2 a.u.), respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288 ix List of Figures 1.1 Three-state model of singlet fission in a dimer system AB: ’State 1’ is the initially excited state S 1 /S 1 0 which consist electronic configura- tions: S 1 (A)S 0 (B), S 0 (A)S 1 (B), A + B , A B + ; ’State 2’ denotes the in- termediate singlet multi-exciton state 1 ME with dominant configuration 1 (T 1 (A)T 1 (B)); ’State 3’ corresponds to two independent triplet states. . 2 1.2 Different flavors of EOM methods. . . . . . . . . . . . . . . . . . . . 9 2.1 Relevant electronic configurations of the AB dimer in terms of molec- ular orbitals localized on individual moieties. h A and h B denote the HOMO localized on A and B, respectively, l A and l B denote the re- spective LUMOs. (a) Ground state,S 0 (A)S 0 (B). (b) Localized singly- excited configurations (EX) giving rise to singlet and triplet excitonic states,S 0 (A)S 1 (B)S 1 (A)S 0 (B) andS 0 (A)T 1 (B)T 1 (A)S 0 (B). (c) Charge-resonance (CR) configurations,A + B andA B + . (d) Configu- rations giving rise to the multiexciton (ME) manifold, 1;3;5 T 1 (A)T 1 (B). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.2 Left: Potential energy curves for parallel tetracene (top) and pentacene (bottom) dimer structures (D 2h ). Right: Weights of ME, EX, and CR configurations in the S(AB) 1 , S(AB) 1 0, and 1 ME wave functions and energy difference between 1 ME and 5 ME along the scan. . . . . . . . . 21 2.3 Top: Frontier MOs of parallel tetracene dimer (D 2h ) structure at 4 ˚ A. Bottom: MOs of the dimer at the X-ray structure. . . . . . . . . . . . . 23 2.4 jj jj and E b for tetracene (top) and pentacene (bottom) along selected scans: (a) D 2h , versus distance between two monomers; (b) C 2h , long axis, in units of the number of 6-carbon rings shifted; (c) C 2h , short axis; (d) C s rotation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 x 3.1 Stretching of octatetraene along C 4 -C 5 ( is the bond stretching param- eter,=1 corresponds to ther e ). Top: Matrix elements of the dipole op- erator (a.u.) andjj jj between the 2A g and 1B u states (left) and between 1A g and 1B u states (right). s denotes symmetrized OPDM. Bottom left: Weights of singly (C 2 1 ) and doubly (C 2 2 ) excited configurations in the 2A g wave function andjj 1Bu2Ag jj. Bottom right: Weights of singly (C 2 1 ) and doubly (C 2 2 ) excited configurations in the 1B u wave function and jj 1Bu1Ag jj. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.2 Top: Matrix elements of the dipole operator (a.u.) andjj jj between the 2A g /1B u (left) and 1A g /1B u (right) states in octatetraene along twist- ing angle . Bottom: Weights of singly (C 2 1 ) and doubly (C 2 2 ) excited configurations in the wave functions of the 1B u and 2A g states. . . . . 53 3.3 Weights of singly and doubly excited configurations in the wave func- tions of the 1B u and 2A g states of butadiene along the bond-stretching coordinate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.4 jj jj, < > and NACE for the 1A g -1B u (left), 2A g -1B u (middle), and 1A g -2A g (right) transitions along the C2-C3 stretching coordinate in butadiene. Top: All quantities are normalized to their equilibrium (=1) values. Bottom: Absolute values. . . . . . . . . . . . . . . . . . . . . 55 3.5 Ethylene dimer atZ=3.5 ˚ A andX=Y =0. . . . . . . . . . . . . . . . . 56 3.6 jj jj for the 1 ME-S 1 states as a function of the displacements along the long (X) and short (Y ) axes. . . . . . . . . . . . . . . . . . . . . . . . 57 3.7 jj jj and interstate properties for the ME-S 1 (left) and ME-S 1 0 (right) transitions computed using CASSCF. . . . . . . . . . . . . . . . . . . 58 3.8 Leading components of NAC along theX=0 scan for the ME-S 1 (left) and ME-S 1 0 (right) transitions. . . . . . . . . . . . . . . . . . . . . . . 58 3.9 Leading components of NAC along theY =0 scan for the ME-S 1 (left) and ME-S 1 0 (right) transitions. . . . . . . . . . . . . . . . . . . . . . . 59 4.1 Relevant M s =0 electronic configurations of the AB dimer expressed using molecular orbitals localized on individual moieties. h A and h B denote the HOMO localized on A and B, respectively, l A and l B de- note the respective LUMOs. (a) Ground state,S 0 (A)S 0 (B). (b) Local- ized singly excited configurations (LE) giving rise to singlet and triplet excitonic states,c 1 S 0 (A)S 1 (B) +c 2 S 1 (A)S 0 (B) andc 1 S 0 (A)T 1 (B) + c 2 T 1 (A)S 0 (B). (c) Charge-resonance (CR) configurations, A + B and A B + . (d) Configurations giving rise to the multiexciton (ME) mani- folds, 1;3;5 T 1 (A)T 1 (B) and 1 S 1 (A)S 1 (B). . . . . . . . . . . . . . . . . 72 4.2 Structures of selected dimers. (a) Perfectly stacked ethylene dimer (D 2h ). (b) Tetracene dimer from the X-ray structure. (c) DPH dimer featuring the largest coupling (dimer 1 from the orthorhombic form). (d) DPBF dimer that has the largest ME-S 1 couplings (dimer3 from the form). . 93 xi 5.1 Three-state model of singlet fission. Top left: free energies of the ini- tially excited bright (S 1 , denoted as ’State 0’), multi-exciton singlet ( 1 ME, denoted as ’State 1’), and two uncoupled triplets (T 1 +T 1 , denoted as ’State 2’). Bottom left: Electronic energy diagram. In pentacene, State 2 is lower than State 0 (E stt >0). In tetracene, it is slightly above (E stt <0). Thus, E 0 =0, E 1 =-E stt -E b , and E 2 =-E stt . Right: Cartoon illus- trating the nature of S 1 , 1 ME, and T 1 +T 1 that gives rise to entropic con- tributions. State 0 is delocalized over several chromophores (black rect- angle). The ME state can be localized on any pair of adjacent molecules (red rectangle) within the initial exciton. Our model assumes that triplet separation (second step) occurs when one of the triplet excitons hops to another chromophore (green rectangle). . . . . . . . . . . . . . . . . . 121 5.2 Crystal structure of tetracene and pentacene. Molecular arrangement in the ab plane (left and center) and in perpendicular direction (right). . . 128 6.1 Three-state model of singlet fission. ’State 0’ denotes an initially ex- cited delocalized state, ’State 1’ is a multi-exciton state, 1 T 1 (A)T 1 (B), and ’State 2’ corresponds to two independent triplets. Relevant elec- tronic energies are E stt =E[S 1 ]-2E[T 1 ], multi-exciton stabilization en- ergy, E b =E[ 5 ME]-E[ 1 ME], and Davydov’s splitting E d =E[S 1 0]-E[S 1 ]. . 146 6.2 Structure of DPBF. Unit cells of the (top left: view alongb-axis; top right: view alongc-axis) and (bottom left: view alonga-axis; bottom right: view alongc-axis) polymorphs. Dimers considered in rate calcu- lations: dimer1 (A+C), dimer2 (A+B), dimer3 (A+D), dimer4 (B+C), dimer5 (B+D). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 6.3 Active-space molecular orbitals from RAS-2SF calculations of dimer3 in the two polymorphs. Top: -DPBF; Bottom: -DPBF. From left to right: HOMO-1, HOMO, LUMO, LUMO+1; contour value: 0.02. . . . 160 6.4 Structure of DPH. Top: monoclinic form (M-DPH). Bottom: orthorhom- bic form (O-DPH). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 6.5 Four monomers from the crystal structure of DPH used in dimers’ cal- culations. Four different dimers are considered: dimer1: A+B (left), dimer2: A+C (left), dimer3: A+D (middle: M form, right: O form). . . 162 6.6 jj jj 2 for dimer1 of M-DPH as a function of displacementD along the long molecular axis.D=2.079 ˚ A in M-DPH and 4.367 ˚ A in O-DPH (see Fig. 6.4). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 6.7 The crystal structure of DPT. Left: Three dimers from DPT crystal grown by vacuum sublimation (V); Right: three dimers from DPT crys- tal grown from xylene solvent (X). . . . . . . . . . . . . . . . . . . . . 164 xii 7.1 Three-state model of singlet fission. ’State 1’ denotes an initially excited state derived from excitonic configurations, S 1 (A)S 0 (B) and S 0 (A)S 1 (B), with a possible admixture of charge resonance, A + B and A B + ; ’State 2’ is a multi-exciton state, 1 (T 1 (A)T 1 (B)), and ’State 3’ corresponds to two independent triplets. Relevant electronic energies are E sf =E[ 1 ME]- E[S 1 ], multi-exciton stabilization energy, E b =E[ 5 ME]-E[ 1 ME], and Davy- dov’s splitting E d =E[S 1 0]-E[S 1 ]. . . . . . . . . . . . . . . . . . . . . . 175 7.2 Covalently linked tetracene dimers . . . . . . . . . . . . . . . . . . . . 182 7.3 Covalently linked cofacial alkynyltetracene dimers . . . . . . . . . . . 182 7.4 The arrangement of two BET-B molecules in the crystal structure of BET-B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 7.5 Excitation energies of the S 1 , 1 ME, and 5 ME states relative to S 0 (at its equilibrium geometry) at the Franck-Condon (FC) geometry and at the optimized geometries of the S 1 and 5 ME states. Left: BET-B. Right: BET-X. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 7.6 Optimized structures of the three isomers of BET-B. BET-B(33, left) is the lowest energy one. BET-B(23, middle) and BET-B(22, right) are 3.6 and 8.6 kcal/mol higher in energy (!B97X-D/cc-pVDZ). . . . . . . . . 196 7.7 The orbital overlap versus coupling.jhH A jH B ij +jhL A jL B ij is the sum of the overlaps between the individual fragments, i.e., the HOMO of fragment A with the HOMO of fragment B and the overlap between the LUMO of fragment A with the LUMO of fragment B. . . . . . . . . . 197 7.8 The effect of the covalent linker on couplings and rates. Top: Couplings in various BET-B structures with and without the linker. Center: Rela- tiver 1 computed without couplings (energy contribution only). Bottom: Relativer 1 for different model structures. Rates are computed relative to the tetracene dimer from the crystal structure, log[r 1 =r 1 (Tc)]. . . . . 200 8.1 Two different views of the structure of the two BET-B molecules taken form crystal structure. The distance between covalently linked tetracene rings (unitsA andB) is 3.34 ˚ A; the distance betweenA andA 0 is 3.40 ˚ A. A and B rings are staggered (the angle between their long axes is 45 ), whereas A and A 0 are parallel (zero angle) but shifted along the long axis by 1.5 ˚ A (about 0.6 of benzene’s ring width). . . . . . . . . . 207 8.2 Frontier fragment orbital diagram of (BET-B) 2 . The left and right pan- els correspond to the orbitals localized on the top and bottom BET-B molecules, respectively. Orbital occupations correspond to the leading configuration of the ground-state wave function. . . . . . . . . . . . . 219 8.3 Energy levels of BET-B (left) and (BET-B) 2 (right). Bright states are marked in red; quintet ME states are marked in blue; all other states are marked in black. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 xiii 8.4 Excitation energies of (BET-B) 2 in model structures differing by the overlap between the middle tetracene moieties (A and A 0 , see Fig. 8.1). Left: The tetracene rings are perfectly stacked (shift of 0.00 ˚ A). Middle: The tetracenes are shifted along the long axis by half of benzene ring (shift of 1.23 ˚ A); this structure is similar to crystal structure. Right: The tetracenes are offset by one benzene ring along the long axis (shift of 2.46 ˚ A). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 9.1 Top: CCSD (left) and EOM-IP-CCSD (right) energies along the proton- transfer coordinate in mU-H 2 O. Bottom: Errors of RI/rimp2-aug-cc- pVTZ and CD approximations. . . . . . . . . . . . . . . . . . . . . . 265 10.1 AAD between the CD analytic energy gradient and the canonical (CAN) analytic energy gradient. . . . . . . . . . . . . . . . . . . . . . . . . . 287 xiv Abbreviations CC coupled-cluster CCSD coupled-cluster with single and double substitutions CD Cholesky decomposition CR charge resonance CS complex scaling EA electron attachment EE excitation energy ERI electron repulsion integrals EOM equation-of-motion EOM-CCSD equation-of-motion coupled-clusters single and double substitutions EX excitonic HOMO highest occupied molecular orbital LUMO lowest unoccupied molecular orbital ME multiexcitonic NAC non-adiabatic coupling xv IP ionization potential RAS-SF restricted-active-space spin-flip configuration interaction RI resolution-of-identity SF singlet fission xvi Abstract This thesis consists of two parts. The first part focuses on the electronic structure aspects of singlet fission (SF) process studied by ab initio calculations. The second part is on the improvement of the high-accuracy post Hartree-Fock methods. Our motivation for studying SF process is to understand its mechanism in order to help experimentalists to design more efficient SF materials. We compute the elec- tronic factors including energies and characters of excited states, non-adiabatic cou- plings (NAC) between these states, and analyze their effect on the rate of SF by using a kinetic model. Chapter 2 discusses the nature of correlated adiabatic wave functions of the initial excited state and dark intermediate multiexcitonic state of pentacene and tetracene. We found that the charge-resonance (CR) configurations are important for determining the NAC between these states. NACs are estimated by using the norm of one particle transition density matrix (OPDM),jj jj, based on Cauchy-Schwarz in- equality. Chapter 3 discusses the utility of the norm of OPDM and presents benchmark. We show thatjj jj contains the principal information about the changes in electronic states involved, such as varying degree of one-electron character of the transition; thus, it captures the leading trends in one-electron interstate properties. Chapter 4 analyzes xvii the computed adiabatic wave function in terms of CR and multi-exciton (ME) charac- ter by using charge and spin cumulants, which describe inter-fragment electronic cor- relation in molecular complexes. This chapter illustrates the utility of this extended excited-state structural analysis considering dimers (He-H 2 , (H 2 ) 2 and (C 2 H 4 ) 2 ), and model SF dimer systems, such as tetracene, 1,6-diphenyl-1,3,5-hexatriene (DPH) and 1,3-diphenylisobenzofuran (DPBF). In Chapter 5, we introduce a simple three-state kinetic model for SF rate and es- timate the rate of the first step of singlet fission, the transition from the lowest adi- abatic excitonic state to the singlet multiexcitonic state. We explain the experimen- tally observed 3 orders of magnitude difference in the rate of SF in tetracene and pen- tacene. Chapter 6 employs this method to explain experimentally observed differences in transition rate among polymorphs of singlet fission materials in DPBF, DPH and 5,12-diphenyltetracene (DPT). Chapter 7 and 8 focus on covalently linked dimers in which the relative orientation of chromophores can be controlled by choosing specific covalent linkers. In Chapter 7, we consider structures with coplanar and staggered arrangements of the tetracene moieties. We found that the structural relaxation in the singlet excited state leads to the increased rate of the multi-exciton formation, but impedes the dissociation of the triplet pair. We also found that the covalent linker improves the NAC between the singlet excitonic state and the multi-excitonic state, but the overall effect on SF rate is less straightforward, since the linkers may result in less favorable energetics. Chapter 8 investigates whether SF proceeds via intra or inter-molecular pathway in the aggregate of covalent linked dimers. Our simulations suggest that the intra-molecular channel dominates in the dimers with the strong though-bond coupling. xviii The second part of this thesis is dedicated to the improvement of the high-accuracy post Hartree-Fock methods. This effort is motivated by the first part. Since the SF materials are typically too large for higher levels of theory, such as coupled-cluster (CC) and equation-of-motion (EOM) level of methods, in order to study molecules relevant to SF, the computational cost of such accurate methods should be reduced. The high cost of electronic structure calculations and high storage requirements orig- inate in the two-electron repulsion integrals (ERI). In Chapter 9, we show that the CC and EOM-CC methods can be implemented in a more efficient manner by replacing ERI with resolution-of-identity (RI) or Cholesky decomposition (CD) representations of in- tegrals reducing storage requirements. This improvement allows for computationally expensive method to be applied to large molecular systems that may not be accessible by canonical methods. The benchmarks show that the introduced errors are small in both RI and CD, and in case of CD, the error can be reliably controlled by the decom- position threshold. Chapter 10 presents the implementation of the analytic CD-CCSD and CD-EOM-CCSD gradient. The error of analytic CD-CCSD energy gradient and optimized bond-lengths is slightly larger than the error of CD-CCSD energy at same decomposition threshold, because the former is unbounded. xix Chapter 1: Introduction and overview 1.1 Singlet fission Solar energy is an attractive alternative to the fossil fuel energy source because it is environmentally friendly. However, practical uses of solar energy are limited by low efficiency of current solar cells. The theoretical efficiency limit for a single junction solar cell is 33%, 1 which comes from the fact that the energy of absorbed photons above the bandgap is mainly converted into heat. One way to make the solar energy more competitive is to increase this limit up to 46% by using a process that generates two electron-hole pairs from one absorbed high-energy photon, together with utilizing low- energy photons in the usual fashion. In organic semiconductors, this can be achieved by singlet fission (SF), a spin-allowed process in which two triplet excitons are generated from one initially excited singlet exciton. 2–5 The utility of SF in the context of solar cells was illustrated by Nozik in 2006. 6 The photocurrent per photon could be doubled if the two triplets have sufficient energy for generating electron-hole pairs. In order to split the singlet excited state into two triplets, the energy matching condi- tion needs to be satisfied: E(S 1 ) 2E(T 1 ). 7 Only a few organic semiconductors meet this criterion. The examples include polyacenes and their derivatives, 8, 9 rubrenes, 10, 11 and a few other systems. The time scale of SF in these systems varies from tens of fem- toseconds to hundreds of picoseconds. Compared with conversion from singlet to triplet 1 through the intersystem crossing, the SF is faster since this process is spin-allowed. However, only a handful of molecules exhibit SF with 200% yield of triplets. The de- sign of new materials capable of efficient SF is a great challenge in this field. The SF process is illustrated in Figure 1.1, where the first step converts the initially excited states (S 1 and S 1 0) into a multi-exciton state ( 1 ME). The second step involves decoupling and spatial separation of the two triplets. The leading electronic configu- rations of S 1 /S 1 0 derived from the bright singlet state are excitonic (EX): S 1 (A)S 0 (B) and S 0 (A)S 1 (B) and charge resonance (CR) ones, A + B , A B + . The oscillator strength is shared between the two singlet excited states and their energy difference is called Davydov splitting,E D =E[S 1 0]E[S 1 ]. The initially excited state is expected to relax to the lowest excitonic excited state before the SF step. The multiexcitonic 1 ME state can be described as two singlet-coupled triplets, 1 (T 1 (A)T 1 (B)). The rate of this step is controlled by the non-adiabatic coupling between two states and energy difference, E SF =E[ 1 ME]E[S 1 ]. Figure 1.1: Three-state model of singlet fission in a dimer system AB: ’State 1’ is the initially excited state S 1 /S 1 0 which consist electronic configurations: S 1 (A)S 0 (B), S 0 (A)S 1 (B), A + B , A B + ; ’State 2’ denotes the intermediate singlet multi-exciton state 1 ME with dominant configuration 1 (T 1 (A)T 1 (B)); ’State 3’ corresponds to two independent triplet states. 2 The difficulties in developing a quantitative model for describing SF originates in the nature of the 1 ME state. Because this state is formally doubly excited relative to the ground state, it is difficult to describe by most ab initio methods. The configuration inter- action singles (CIS) and time-dependent density functional theory (TDDFT), which are the standard methods for large systems, are simply blind to 1 ME. Equation-of-motion coupled-cluster singles and doubles for excited states (EOM-EE-CCSD) can capture the 1 ME state, but it does not treat singly excited state S 1 and doubly excited state 1 ME in a balanced way. Multi-reference techniques such as CASSCF-based approaches can be used to model SF materials, but their practical applications are hindered by a system- specific active space selection, problems with treatment of dynamical correlation, and the lack of size extensivity. In Chapter 2, we present ab initio calculations of the electronic structure of two poly- acene dimers, tetracene dimer and pentacene dimer, using restricted-active-space dou- ble spin-flip configuration interaction (RAS-2SF-CI) method. This approach provides a balanced description of the states relevant to SF. We employ adiabatic framework and show that a simple diabatic representation of underlying electronic structure is not ca- pable of quantitatively accurate description of the SF process. Configurations of CR character that are present in both excitonic and multiexciton wave functions are impor- tant for the NAC between S 1 and 1 ME. Chapter 4 provides detailed analysis of the wave functions of low-lying adiabatic excited states using extended excited-state structural analysis (ESSA). We use the norms of one-particle transition density matrix (OPDM), jj jj, which provide a useful tool for studying trends in NAC, as described in Chapter 3. In Chapter 5, we set up a simple three-state kinetic model to describe the SF process and derive expression for estimating the rate of the first step: S 1 ! 1 ME transition. By 3 using these tools, we are able to explain the observed different behavior of SF in poly- morphs of 1,3- diphenylisobenzofuran (DPBF), 1,6-diphenyl-1,3,5-hexatriene (DPH), and 5,12-diphenyltetracene (DPT). These results are presented in Chapter 6. Chapter 7 and 8 discuss SF in covalently linked dimers. By using a covalent linker, the chromophores can be fixed at an orientation favorable for SF, and the coupling be- tween the states can also be enhanced by the through-bond interactions. In Chapter 7, we discuss SF in a covalently linked tetracene dimer, ortho-bis(5-ethynyltetracenyl)benzene (BET-B), and estimate the degree of the through-bond coupling. In Chapter 8, we fur- ther discuss the behavior of BET-B in bulk, focusing on the differences bewteen inter- molecular and intra-molecular SF. We show that in a model with two dimers, SF mainly occurs via an intra-molecular pathway because of the strong through-bond coupling and favorable excitation energy of intra- 1 ME states relative to the inter-molecular 1 ME man- ifold. 1.2 Electronic structure methods for strongly correlated systems 1.2.1 Electron correlation The non-relativistic electronic Schr¨ odinger equation is H el i =E i i ; (1.1) whereH el is the electronic Hamiltonian, i andE i are thei th eigenfunctions and eigen- values, called electron wavefunctions and state energies. The state that has the lowest energy is called the ground state, and the rest of the states are excited states. Eq. 1.1 4 can be solved numerically by using a linear variational approach with a complete many- electron basis set. This procedure is called full configuration interaction (full CI). The number of many-electron basis functions grows factorially with the number of electrons and orbitals. Therefore, full CI is not practical for large molecular systems. The most drastic approximation is the Hartree-Fock (HF) model, 12, 13 which assumes that every electron moves in an average Coulomb field of all other electrons and nuclei. The missing energy, which is the difference between Hartree-Fock energy and full CI, is called correlation energy. The correlation is usually divided into two non-additive parts: dynamic and static correlation. The former can be recovered partially by using wave-function based meth- ods such as Møller-Plesset perturbation theory (MP) 14 and coupled-cluster (CC) 15 ap- proaches. The static correlation can be important in cases when the ground state is a linear combination of several nearly degenerate determinants. For example, in bond- breaking problem, the highest occupied molecular orbital (HOMO) and the lowest un- occupied molecular orbital (LUMO) become degenerate when the bond is stretched. In this case, one Slater determinant is not sufficient for describing the wave function. Meth- ods for dealing with this type of correlation include multi-configurational self-consistant field (MCSCF) 16, 17 and spin-flip approachs. 18–20 1.2.2 Describing electron correlation in closed-shell systems by coupled-cluster approach In the CC formalism, 12, 21 the correlated wavefunction is given by j CC >= exp(T )j 0 > 5 where the cluster operator T is an excitation operator, and 0 > is HF reference de- terminant. The CC method can recover exact correlation energy when all substitutions are included inT . Because of high computational cost, T is usually truncated at dou- ble excitation level giving rise to the coupled-cluster method with single and double excitations (CCSD): T =T 1 +T 2 Each term can be written in the second-quantization notation as follows: T 1 = X ia t a i a + i and T 2 = 1 4 X ijab t ab ij a + b + ji wherea + andb + are creation operators,i andj are annihilation operators,t a i andt ab ij are the CC amplitudes. Here we follow the standard notation in whichijkl denote occupied molecular orbitals (MO),abcd denote virtual MOs, andpqrs denote MOs which can be either occupied or unoccupied. The separation between the two subspaces is determined by the reference determinant 0 . Because of the nature of exponential cluster operator, CCSD has contributions from higher than single and double excitations, and is size- extensive. These features make the CCSD method superior to the CI analogue with the same truncation level (CISD). 6 The CCSD wavefunction and energy are found by solving the following set of equa- tions: 0 je T He T j 0 =E CCSD a i je T (HE CCSD )e T j 0 = 0 ab ij je T (HE CCSD )e T j 0 = 0 in an iterative procedure, which involves contractions oft ab ij amplitudes with four-index ERI: t a i a i =f ia X l F (3) li t a l + X d F (1) ad t d i + X kc F (2) kc t ac ik X kc hicjjkait c k 1 2 X klc hkljjicit ac kl + 1 2 X kcd hkajjcdit cd ki and t ab ij ab ij =hijjjabi +P ab f X c t ac ij F (2) bc X k I (2a) ijkb t a k +P ij X kc I (1a) kbic t ac jk g +P ij f X c hjcjjbait c i X k t ab ik F (2) jk g + 1 2 X cd habjjcdi ~ t cd ij + 1 2 X kl t ab kl I (4) ijkl The contraction scales asO(N 6 ) and makes CCSD computationally expensive for large molecular system. The storage of double excitation amplitudes (t ab ij ) and < pqjjrs > scales asO(N 4 ), so the amount of memory increases quadratically with the system size. 7 1.2.3 Describing Electron correlation in electronically excited and open-shell states by equation-of-motion coupled-cluster ap- proach In the EOM framework, 22 the CI-type linear excitation operator R is applied to the reference-state CC wavefunction. The wavefunction of an excited electronic state is then j EOM >=Rj CC >=R exp(T )j 0 >; whereR generates determinants excited with respect to the reference determinant and is truncated usually at the same level as CC operatorT , e.g., at the double excitation level in EOM-EE-CCSD: R EOMEECCSD =r 0 + X ia r a i a + i + 1 4 X ijab r ab ij a + b + ji: As illustrated in Fig. 1.2, by using differentR operators, EOM methods can compute the energies and properties of electronically excited (EE), ionized (IP), electron-attached (EA), and spin-flipped (SF) states. Solving for the EOM-CC energies and amplitudes involves diagonization of a non- Hermitian matrix: 0 B B B B B B @ 0 H OS H OD 0 H SS E CCSD H SD 0 H DS H DD E CCSD 1 C C C C C C A 0 B B B B B B @ R 0 R 1 R 2 1 C C C C C C A =! 0 B B B B B B @ R 0 R 1 R 2 1 C C C C C C A 8 Figure 1.2: Different flavors of EOM methods. where H is the transformed Harmiltonian H = exp(T )H exp(T ), O stands for the reference,S for singly andD for doubly excited determinants. This problem is solved by using Davidson’s algorithm, 23 in which theR amplitudes are computed iteratively by using a i = (( H SS E CCSD I)R 1 ) a i + ( H SD R 2 ) a i ab ij = ( H DS R 1 ) ab ij + (( H DD E CCSD I)R 2 ) ab ij Similarly to CCSD, EOM-CCSD requires O(N 4 ) storage and scales as O(N 6 ) in the iterative procedure, which limits its application to moderate-size systems. 9 1.2.4 Improving efficiency of electronic structure methods The storage requirements of the CCSD and EOM-CCSD methods scales asO(N 4 ). If symmetry is not considered, for a calculation with 1000 basis functions, the size of ERI can be up to 1TB. Together with otherN 4 -type intermediates, the large storage require- ment makes the calculating not practical. Even if large disk is available, an extensive I/O results in deteriorated performance and poor parallel scaling. One promising way to improve the efficiency for large systems is to reduce the storage requirements. The storage of the ERI matrix scales asO(N 4 ) with the number of basis functions, (~ r): (j) = Z (~ r 1 ) (~ r 1 ) 1 j~ r 1 ~ r 2 j (~ r 2 ) (~ r 2 )d~ r 1 d~ r 2 : In order to reduce the size of ERI tensor, we employ the Cholesky decomposition (CD) and resolution-of-identity (RI): (j) M X P =1 B P B P ; (1.2) such that the N 4 -type matrix is represented by contraction of N 3 -type matrices. This is described in Chapter 9. One difference between CD and RI is that the rankM and the error of CD are controlled by user, whereas RI uses pre-established auxiliary basis functions, so the rank and error are fixed. In Chapter 10, we describe the formalism and implementation of the analytic CD gradient. 10 Chapter 1 references [1] W. Shockley and H.J. Queisser. Detailed balance limit of efficiency of p-n junction solar cells. J. Appl. Phys., 32:510–519, 1961. [2] S. Singh, W.J. Jones, W. Siebrand, B.P. Stoicheff, and W.G. Schneider. Laser generation of excitons and fluorescence in anthracene crystals. J. Chem. Phys., 42:330–342, 1965. [3] I. Paci, J.C. Johnson, X. Chen, G. Rana, D. Popovi´ c, D.E. David, A.J. Nozik, M.A. Ratner, and J. Michl. Singlet fission for dye-sensitized solar cells: Can a suitable sensitizer be found? J. Am. Chem. Soc., 128:16546–16553, 2006. [4] E. C.Greyson, B.R. Stepp, X. Chen, A.F. Schwerin, I. Paci, M. B. Smith, A. Akdag, J.C. Johnson, A.J. Nozik, J. Michl, and M.A. Ratner. Singlet exciton fission for solar cell applications: Energy aspects of interchromophore coupling. J. Phys. Chem. B, 2010. in press, asap article. [5] X. Feng, A.V . Luzanov, and A.I. Krylov. Fission of entangled spins: An electronic structure perspective. J. Phys. Chem. Lett., 4:3845–3852, 2013. [6] M. C. Hanna and A. J. Nozik. Solar conversion efficiency of photovoltaic and photoelectrolysis cells with carrier multiplication absorbers. J. App. Phys., 100:074510, 2006. [7] M. B. Smith and J. Michl. Singlet fission. Chem. Rev., 110:6891–6936, 2010. [8] S. Singh, W. J. Jones, W. Siebrand, B. P. Stoicheff, and W. G. Schneider. Laser generation of excitons and fluorescence in anthracene crystals. J. Chem. Phys., 42:330, 1965. [9] R.P. Groff, P. Avakian, and R.E. Merrifield. Coexistence of exciton fission and fusion in tetracene crystals. Phys. Rev. B, 1:815–817, 1970. [10] G.B. Piland, J.J. Burdett, D. Kurunthu, and C.J. Bardeen. Magnetic field effects on singlet fission and fluorescence decay dynamics in amorphous rubrene. J. Phys. Chem. C, 117:1224–1236, 2013. [11] W. G. Herkstroeter and P. B. Merkel. The triplet state energies of rubrene and diphenylisobenzofuran. J. Photochem., 16:331–341, 1981. [12] T. Helgaker, P. Jørgensen, and J. Olsen. Molecular electronic structure theory. Wiley & Sons, 2000. 11 [13] A. Szabo and N.S. Ostlund. Modern Quantum Chemistry: Introduction to Ad- vanced Electronic Structure Theory. McGraw-Hill, New York, 1989. [14] C. Møller and M.S. Plesset. Phys. Rev., 46:618, 1934. [15] J. Cizek. On the correlation problem in atomic and molecular systems. Calcu- lation of wavefunction components in Ursell-type expansion using quantum-field theoretical methods. J. Chem. Phys., 45:4256–4266, 1966. [16] B.O. Roos. The multiconfigurational (MC) SCF method. In Geerd H. F. Diercksen and Stephen Wilson, editors, Methods in Computational Molecular Physics, pages 161–187. D. Reidel, Dordrecht, 1983. [17] B.O. Roos. The multiconfigurational (MC) self-consistent field (SCF) theory. In B. O. Roos, editor, Lecture Notes in Quantum Chemistry: European Summer School in Quantum Chemistry, volume 58 of Lecture Notes in Chemistry, pages 177–254. Springer-Verlag, New York, 1992. [18] A. Golubeva, A.V . Nemukhin, L. Harding, S.J. Klippenstein, and A.I. Krylov. Per- formance of the spin-flip and multi-reference methods for bond-breaking in hydro- carbons: A benchmark study. J. Phys. Chem. A, 111:13264–13271, 2007. [19] P.U. Manohar and A.I. Krylov. A non-iterative perturbative triples correction for the spin-flipping and spin-conserving equation-of-motion coupled-cluster methods with single and double substitutions. J. Chem. Phys., 129:194105, 2008. [20] D. Casanova, L.V . Slipchenko, A.I. Krylov, and M. Head-Gordon. Double spin-flip approach within equation-of-motion coupled cluster and configuration interaction formalisms: Theory, implementation and examples. J. Chem. Phys., 130:044103, 2009. [21] G.D. Purvis and R.J. Bartlett. A full coupled-cluster singles and doubles model: The inclusion of disconnected triples. J. Chem. Phys., 76:1910–1918, 1982. [22] R.J. Bartlett. Coupled-cluster theory and its equation-of-motion extensions. Wiley Interdiscip. Rev.: Comput. Mol. Sci., 2(1):126–138, 2011. [23] E.R. Davidson. The iterative calculation of a few of the lowest eigenvalues and corresponding eigenvectors of large real-symmetric matrices. J. Comput. Phys., 17:87–94, 1975. 12 Chapter 2: Fission of Entangled Spins: An Electronic Structure Perspective 2.1 Introduction Singlet fission (SF), a process in which one singlet excited state splits into two triplets ultimately giving rise to four charge carriers, can be utilized in organic solar cells in- creasing their efficiency. 1 A reverse process, triplet fusion, is also of interest for solar energy; it can be used to up-convert a lower-energy part of the solar spectrum. 2 Although these phenomena have been discovered a long time ago, 3 their mechanistic understand- ing is incomplete, which hinders the design of organic photovoltaic materials for solar energy conversion. As summarized in recent reviews, 1, 4 there has been a resurgence of experimental and theoretical work investigating the mechanisms of SF in molecular solids and model compounds. Apart from the initial event (photon absorption forming initial exciton) and the final state (two uncoupled triplet-excited molecules), very little is known about the nature and evolution of the electronic states involved, mechanism of their coupling, type of nuclear motions facilitating this process, and so on. Consequently, we do not know how to op- timize molecular and material properties for optimal SF. Even molecular energy-level considerations are not obvious — although energy balance requires that E S 2E T , 13 some materials engineered to exhibit exothermic SF were found to be less efficient than, for example, tetracene, in which SF is slightly endothermic. Optimal spatial arrange- ment of individual molecules (parallel, slip-stacked, etc) is unclear, and is the impor- tance of long-range order, i.e., whether one needs to target polycrystalline or amorphous materials. 5, 6 Experimentally, a number of materials are known to exhibit SF: Acenes (tetracene, pentacene) and substituted acenes (e.g., diphenyltetracene, rubrene), perylenes, isoben- zofuran, carotenoid. 1, 4 The time scale of SF varies from tens of femtoseconds (e.g., pentacene) to tens of picoseconds (tetracene) suggesting a fast non-adiabatic transition from the initially excited bright state to the target multiexciton state (“direct” mech- anism). Recent time-resolved experiments 7 suggest that a coherent superposition of the bright singlet and dark multiexciton states is formed during the excitation, and that this state adiabatically evolves giving rise to two independent triplets. The most recent overview of the field can be found in Ref. 4 A complete theoretical description of SF process requires the following: (i) accurate electronic structure methods capable of describing electronic states involved and their couplings; (ii) coupled electronic and nuclear dynamics to describe adiabatic evolution of the initially excited state and hopping-like processes; (iii) inclusion of the environ- mental perturbation on the electronic structure as well as involvement of the bath degrees of freedom in this process. In this paper we focus on the electronic structure aspect of this problem. Although the nature of electronic states involved and their interactions is at the heart of the problem, there is still no consensus of what are essential features of the under- lying electronic structure and how to approach this problem quantitatively. Qualitative framework has been laid out by Michl and coworkers (see Ref.1, 4, 8 and references 14 therein). Critical electronic configurations are excitonic [EX, derived from the asymp- totic S 0 (A)S 1 (B) and S 1 (A)S 0 (B) states], charge-resonance [CR, often called charge transfer, A + B , A B + ], and multiexcitonic [ME, two triplets coupled into a singlet, triplet, and quintet states, 1;3;5 T 1 (A)T 1 (B)] sets. These configurations are often used as diabatic states describing the SF process. For example, Michl and co-workers frame the discussion of the SF process in terms of such diabatic states of pure EX, CR, and ME character. 1, 4, 8 As possible mechanisms of SF, they discussed either direct non-adiabatic transition from the initially excited bright singlet state (that asymptotically has pure EX character) to the state that can be described as two singlet-coupled triplets, or a tran- sition via an intermediate state of a charge-transfer (or, more precisely, CR) character. Alternatively, one can discuss “mediated” transitions between the initial and final states in which the CR configurations appear in both the initial and final states thus facilitat- ing the coupling. Nebulous and imprecise terms such as “superexchange” have been invoked to describe this possibility. 9 Such a simplified picture has been used to pre- dict qualitative trends in couplings as a function of molecular arrangements 1, 4 and has motivated the development of phenomenological few-states models based on diabatic states. 9 While such analysis of electronic structure is valuable for qualitative purposes, for a quantitative description of the SF process a more appropriate framework should be based on correlated many-electron wave functions, as advocated by Head-Gordon and co-workers. 10 These wave functions should be obtained from an appropriate elec- tronic structure method without making assumptions of what these many-electron states should be. Since ab initio calculations produce well defined adiabatic states, the elec- tronic transitions between these states are most naturally described in terms of vibronic interactions, i.e., derivative coupling that couples adiabatic states via nuclear motions. 15 In some cases, it is convenient to convert this adiabatic picture into a diabatic repre- sentation; however, as we show below, such approaches are not applicable to the SF problem. The stumbling block in developing a quantitative model that can unambiguously discriminate between different mechanisms is that this electronic structure pattern is challenging to ab initio methods. The first quantitative attempt to characterize the SF state has been undertaken by Zimmerman et al. who have shown that the state that cor- relates to two triplets is a doubly excited dark state by using CASSCF calculations on a model pentacene dimer. 11 Later, Zimmerman and coworkers have investigated this state and bright singlet states in tetracene and pentacene using QM/MM calculations with RAS-2SF method. 10, 12 Several studies interrogated the question of the degree of delocalization of the initially excited bright singlet state, possible role of excimer for- mation, as well as location of charge-transfer states. 12–16 The most important ingredient — electronic couplings between the initially excited states and multiexciton states — has been extensively discussed, 1, 4, 10, 12, 15, 17 however, no actual ab initio calculations of non-adiabatic coupling (NAC) matrix elements have been reported; instead, some studies have exploited approximate schemes of evaluating these quantities via diaba- tization, 10, 12 while others used excitonic (Davydov) splitting as a proxy for the cou- pling, or employed qualitative diabatic frameworks. 8, 15 Important predictions derived from the latter approach are that couplings promoting SF are maximized by: (i) con- tributions from CR configurations; and (ii) co-facial slip-stacked arrangements of the chromophores. 16 Here we focus on NAC and introduce an alternative scheme for estimating this cru- cial quantity from correlated many-electron wave functions. Our approach avoids dia- batization, which, as we show below, is physically inappropriate in the context of SF. In- stead, we consider the norm of reduced one-particle transition density matrix as a proxy for NAC. This approach is justified by the Cauchy-Schwarz inequality and is validated by the analysis the electronic structure at selected geometries where states character can be unambiguously assigned. In addition, we investigate whether this quantity correlates with exciton stabilization energy, which is related to the state mixing. We employ RAS-2SF method 18, 19 that is capable of describing all relevant elec- tronic states on the same footing and analyze the respective electronic wave functions in terms of monomer states, in the spirit of excimer theory 20–22 and DMO-LCFMO (dimer molecular orbital — linear combination of fragment molecular orbitals) framework. 23 Such analysis is unambiguous in two limiting situations: (i) when fragments are equiva- lent by symmetry and orbitals are completely delocalized; or (ii) when orbitals are well localized on individual fragments (in principle, orbitals can be easily localized by using various localization techniques; however, we encountered severe convergence issues in RAS-SF calculations when using localized orbitals). In more common low-symmetry situations, when orbitals are partially delocalized, assigning state characters from the wave function amplitudes becomes difficult (if not impossible); thus, an alternative way to identify a multiexciton state is needed; this can also be achieved by considering ap- propriate transition density matrices. We apply this approach to investigate the effect of relative chromophore orientations on NAC using model tetracene and pentacene dimers. Fig. 4.1 shows relevant electronic configurations of a molecular dimer in terms of frontier molecular orbitals (HOMO and LUMO) of the individual monomers. The ground-state wave function of the dimer is simply a product of the two ground-state 17 (a) A B (b) (c) (d) h A l A l B h B Tuesday, September 17, 2013 Figure 2.1: Relevant electronic configurations of the AB dimer in terms of molec- ular orbitals localized on individual moieties. h A and h B denote the HOMO lo- calized on A and B, respectively, l A and l B denote the respective LUMOs. (a) Ground state, S 0 (A)S 0 (B). (b) Localized singly-excited configurations (EX) giv- ing rise to singlet and triplet excitonic states, S 0 (A)S 1 (B) S 1 (A)S 0 (B) and S 0 (A)T 1 (B) T 1 (A)S 0 (B). (c) Charge-resonance (CR) configurations, A + B and A B + . (d) Configurations giving rise to the multiexciton (ME) manifold, 1;3;5 T 1 (A)T 1 (B). wave functions, S 0 (A)S 0 (B). Panel (b) shows localized single excitations that give rise to excitonic states, e.g., c 1 S 0 (A)S 1 (B) + c 2 S 1 (A)S 0 (B) and c 2 S 0 (A)S 1 (B) c 1 S 1 (A)S 0 (B) [we denote the adiabatic dimer states that asymptotically correlate to these excitonic states S 1 (AB) and S 1 0(AB)]. Excitonic triplet states derived from the S 0 (A)T 1 (B) andT 1 (A)S 0 (B) can be described by the same set of configurations; these are denoted as T 1 (AB) and T 1 0(AB). The values of coefficientsc 1;2 and energy splitting between the two states depend on a relative orientation of A and B and the overlap of the respective MOs. At the symmetric configuration,c 1 = c 2 = 1 p 2 and the oscillator strength of the singlet excitonic pair is carried by the ’+’ state. In real many-electron 18 adiabatic states, these excitonic configurations mix with charge-resonance (CR) config- urations,A + B andA B + shown in panel (c). Note that the presence of CR configura- tions (often referred to as charge transfer) does not imply permanent charge separation in the dimer; rather, these configurations describe the ionic character of underlying wave functions (see SI for Ref.24).These dimer states and an interplay between EX and CR contributions have been extensively discussed in the context of excimer theory. 13, 20–22 Finally, panel (d) shows electronic configurations giving rise to multiexciton states that can be described as two coupled triplet states of the monomers. Two triplets can be cou- pled to a singlet state, 1 T 1 (A)T 1 (B), triplet, or quintet. We use ME to denote states from this manifold. We note that at finite inter-fragment separation, the singlet ME configu- rations, 1 ME, can mix with all singlet configurations shown in panels (a)-(c), whereas the quintet component of the ME manifold, 5 T 1 (A)T 1 (B), can only interact with other quintet states (which lie considerably higher in energy); thus, we expect the respective adiabatic wave function to retain pure diabatic character, 5 T 1 (A)T 1 (B). 1 ME state is an example of a strongly correlated state in which two triplet states localized on separate moieties are strongly coupled (or entangled); the degree of this entanglement can be characterized by quantities derived from reduced density matrices. 25 From the methodological point of view, excitonic and CR states can, in principle, be described by standard electronic structure methods such as CIS, TD-DFT, or EOM-CCSD (see, for example, Ref.26). However, multiexciton configurations (d) are doubly-excited with respect to the ground state; thus, TD-DFT and CIS are blind to their existence (although these configurations are included in EOM-CCSD expansion, their description will not be balanced). This is the essence of the challenge faced by electronic structure in the context of SF. In principle, all these relevant states could be described within a multireference framework, however, the high cost of 19 such calculations, uncertainties with active space choice, difficulties of including dynamic correlation, and, most important, the lack of size-intensivity present significant stumbling blocks. A simple and efficient solution to this problem is offered by the spin-flip approach in which target states are described as spin-flipping excitation from a high-spin reference. 27–29 A brief inspection of the configurations in Fig. 4.1 reveals that this set can be obtained by using the quintet reference and double spin-flipping operators. 18, 19 This approach was pioneered by Zimmerman and Head-Gordon who applied RAS-2SF method to tackle this problem. 10, 12 We use the same methodology. Importantly, in our computational scheme all configurations are described on equal footing and can mix and interact giving rise to adiabatic states of different character. This allows us to address the character of the excited electronic states and the effect of structure on their mixing. To quantify the state characters in the case of delocalized orbitals, we perform simple analysis in the framework of DMO-LCFMO approach (this is only possible at symmetric configurations). 23, 26 To quantify interactions at arbitrary geometries, we introduce an approach based on transition density matrices. Let us begin with a parallel sandwich structure (D 2h ). Fig. 2.2 (left panels) shows potential energy curves of tetracene’s and pentacene’s dimer states along inter-fragment separation (see SI for Ref.24 for computational details and definitions of the scans). At large separations, the states asymptotically converge to the respective states of the monomer. The excitation energies of two singlets, S 1 (AB)/B 3g and S 1 0(AB)/B 1u , con- verge to S 1 (A/B), the two triplets (also of B 3g and B 1u symmetries) become degenerate as well, converging to T 1 (A/B). Thus, at large separation we expect to see that the wave functions of these states are dominated by excitonic configurations (row b in Fig. 4.1). The symmetry of the ME states (which are formally doubly excited with respect to the 20 Figure 2.2: Left: Potential energy curves for parallel tetracene (top) and pentacene (bottom) dimer structures (D 2h ). The states can be assigned based on their symme- try (see SI for Ref.24). Right: Weights of ME, EX, and CR configurations in the S(AB) 1 , S(AB) 1 0, and 1 ME wave functions and energy difference between 1 ME and 5 ME along the scan. This quantity can be interpreted as multiexciton binding en- ergy (E b ) and is related to the state mixing. ground state) is A g . We observe that the singlet and quintet ME states become degener- ate and both states converge to exactly 2xE(T 1 (A/B)) Note that this correct asymptotic behavior of dimer’s states can only be reproduced by size-intensive methods, such as RAS(4,4)-2SF (all states), CIS, EOM-CCSD (S 1=1 0 and T 1=1 0 only), and will be violated by, for example, CASSCF. At shorter distances, the energy of the 1 ME state is lower than that of 5 ME due to the stabilization of the singlet state by configuration interaction with other singlets. This energy difference (shown in the right panels of Fig. 2.2) can 21 be interpreted as exciton binding energy (E b ). On one hand, it is related to the degree of diabatic impurity of the 1 ME state (it should be zero when the 1 ME state is of pure multiexciton character and large when the ME state is strongly mixed with other config- urations). Thus, it can be used to evaluate the degree of the configuration interaction in the 1 ME state in conjunction with (or instead of) configuration analysis of the respective wave function. On the other hand, this quantity is relevant to the kinetics of separation of ME into the two non-interacting triplets (large values indicate an exothermic pathway for triplet-triplet separation step). Thus, while multiexciton binding energy is obviously very important, its optimal value (in terms of maximizing the SF rate) is not obvious and kinetics modeling is required. Analyzing state characters in terms of monomer configurations from Fig. 4.1 re- quires transformation of the orbitals into the localized ones. Fig. 2.3 shows the MOs of the D 2h dimer at 4 ˚ A and compares them with the MOs of the non-symmetric dimer at the X-ray structure. In the latter case, orbitals are almost completely localized on the monomers, which makes the assignment of the state characters straightforward. In order to analyze the states of the dimer at D 2h (or other symmetric configurations), we utilize linear properties of determinants and convert the leading configurations of the RAS-2SF wave functions into the localized representation by using symmetry-imposed relationships between the dimer and monomer orbitals, in the spirit of DMO-LCFMO framework. 23 For example, the leading term of the ground-state wave function is: jH 1 H 1 HH >= 1 4 j(h A +h B )(h A +h B )(h A h B )(h A h B )>= ::: =jh A h A h B h B > (2.1) whereH 1 andH denote the HOMO-1 and HOMO of the dimer (see Fig. 2.3). Out of 16 terms, only four (permutationally equivalent) ones survive due to the Pauli principle. 22 The results of the analysis are summarized in Table 2.1 (tetracene only) and in Fig. 2.2 (tetracene and pentacene). For comparison, Table 2.1 also shows the breakdown of the wave function for the dimer taken from the X-ray structure. In addition to the configuration analysis, Table 2.1 also shows an effective number of unpaired electrons for each state,N odd , computed using the following expression: N odd = 2(Tr[ ]Tr[ + ]) = 2(NTr[ + ]) (2.2) where is the one-particle density matrix for the corresponding state. For singlet states, this is equivalent to Yamaguchi’s index. 30 Eq. (2.2) can also be used to quantify the degree of quantum entanglement. 25 It is small for ground-state closed-shell wave func- tions, but is close to 4 for the ME states. Figure 2.3: Top: Frontier MOs of parallel tetracene dimer (D 2h ) structure at 4 ˚ A. Bottom: MOs of the dimer at the X-ray structure. As one can see, the 5 ME state is indeed a pure multiexcitonic state both asymptoti- cally and at short inter-fragment separations. The 1 ME state, however, can be described 23 Table 2.1: Analysis of tetracene dimer states and multiexciton binding energy (E b , eV) at selected geometries. State E b ME EX CR N a odd D 2h , 3.7 ˚ A 0.22 1 ME/A g 0.93 - 0.06 3.6 S 1 (AB)/B 3g - 0.43 0.43 S 1 0(AB)/B 1u - 0.42 0.42 5 ME/A g 1.00 - - 4.0 D 2h , 6.0 ˚ A 0.00 1 ME/A g 1.00 - - 4.0 S 1 (AB)/B 3g - 0.84 - S 1 0(AB)/B 1u - 0.82 0.04 5 ME(A g ) 1.00 - - 4.0 X-ray 0.03 1 ME 0.60 - 0.35 3.9 S 1 (AB) - 0.39 0.37 S 1 0(AB) - 0.61 0.13 5 ME 1.00 - - 4.0 a Effective number of unpaired electrons using Yamaguchi’s index. 30 as pure multiexcitonic state only at large separations. At shorter distances, the ME char- acter is only 80%, consistently with significant stabilization of 1 ME relative to 5 ME (E b =0.55 eV for D 2h structure at 3.4 ˚ A and 0.22 eV at 3.7 ˚ A). We note that the differ- ences in the ME weight are consistent with the computed number of unpaired electrons (N odd ) — whereas the 5 ME state always has 4 unpaired electrons,N odd in the 1 ME state is reduced at small distances, when this state acquires some CR character. The singlet states, S 1 (AB) and S 1 0(AB), are almost an equal mixture of the EX and CR contributions. As expected, at large separations, the CR character is reduced, as asymptotically these ionic configurations are very high in energy. We also note that this CR character can only be deduced by configuration analysis in localized orbitals; it will not be determined by the attachment-detachment density analysis 31 employed 24 in Refs. 10, 12 that can only identify true charge-transfer states with permanent charge separation (see SI for Ref.24). The wave function analysis of the low-symmetry tetracene dimer (taken at the con- figuration in the crystal) shows 60% ME character (59% in pentacene) in the 1 ME state illustrating significant mixing with the CR configurations; however, the CR character in the S 1 and S 1 0 is reduced relative to the D 2h structure. Most importantly, Table 2.1 and Fig. 2.2 clearly illustrate that it is not possible to represent quantitatively this electronic structure by a toy few-state model based on pure diabatic states of the ME, CR, and EX characters. The contributions of these types do not add up to 100% illustrating that interactions with other states are not insignificant even in these relatively compact RAS-SF wave functions. As the correlation treatment is improved, the deviations should increase. Thus, while analyzing the states involved in SF in terms of these configurations is useful for insight, it cannot be used for developing quantitative models; rather, as emphasized by Head-Gordon and co-workers, 10 the description of SF process should be based on adiabatic many-electron wave functions. The key quantity related to the rate of populating the 1 ME state (either via non- adiabatic transitions from the bright state or by creating the population in the ME state coherently, during the excitation processes) is NAC matrix element: < i (r;R)jr Q j f (r;R)>; (2.3) where i (r;R) and f (r;R) are electronic wave function of the initial and final states [e.g., S 1=1 0(AB) and 1 ME], andQ denotes a particular nuclear displacement. This so- called derivative coupling originates in the break-down of Born-Oppenheimer approx- imation that leads to the parametric dependence of the electronic wave functions on 25 nuclear positions. To evaluate rates, this coupling (which is a vector in the space of nuclear displacements) should be contracted with a respective component of the nu- clear momentum,r Q (R). The calculations of NAC are rarely available in electronic structure codes, thus, a common strategy is to convert the problem from adiabatic into diabatic representation in which derivative coupling can be neglected and the states are coupled by the electronic Hamiltonian. Such approaches have been fruitful in various contexts (e.g., in charge-transfer processes 32, 33 ), however, the SF problem does not lend itself to this strategy, as discussed below. It is tempting to develop a simple energy or configuration-selection based diabati- zation scheme (in energy-based approaches, splittings between interacting states can be used to parameterize the transformation, whereas in configuration-based schemes, one can use weights of specific configurations to determine the transformation). However, as the analysis in Table 2.1 illustrates, the 4 lowest states do not constitute a closed manifold even at these 3 geometries, making such diabatization attempts futile. A fur- ther insight can be gained by symmetry analysis. As suggested by the energies (notable E b at 3.4 ˚ A, D 2h ) and wave function analysis (Table 2.1), one should expect interac- tion between the 1 ME state and the S 1 /S 1 0 states. Moreover, since the state composition changes significantly along the D 2h scan, one should expect to see variations in the in- teraction strength (and, consequently, in the couplings) along this coordinate. However, these three adiabatic states are of different symmetry, which means that they should not be mixed by a valid diabatization transformation. So should we consider these states as non-interacting (along D 2h scans) or not? The key to correctly resolving this issue is to realize that the symmetry of NAC is determined by the symmetries of i (r;R) and f (r;R) — in our example, the 1 ME/A g and 1 S 1 0(AB)/B 1u states can be coupled by any B 1u vibrations (and there are 22 of them in the tetracene dimer!). Thus, the states 26 can interact at D 2h configurations and the magnitude of interaction depends on the inter- fragment separation, however, no physically meaningful diabatization can be performed along a D 2h scan. Furthermore, evaluation of the probabilities of non-adiabatic tran- sitions using, for example, a Landau-Zener type model, should be done considering nuclear motions “orthogonal” to the inter-fragment motion along D 2h . In other words, while inter-fragment separation along a D 2h scan should have a significant effect on the magnitude of NAC, the coordinates promoting the transitions (in a Landau-Zener sense) are of different symmetry. Thus, diabatizing the states along the D 2h scan is meaning- less, despite the fact that this motion may be (and most likely is) important in facilitat- ing the non-adiabatic transitions by sampling the configurations where the magnitude of derivative couplings along the coordinates promoting the transition is large. Thus, we need to consider an alternative approach of estimating the trends in NAC. We base our approach on reduced one-particle transition density matrices. Since the derivative coupling is a one-electron operator, only the states whose wave functions can be connected by one-electron excitation can be coupled. This reasoning has been used by Michl 1 who pointed out that the ME configurations from row (d) in Fig. 4.1 differ by two-electron excitations from the excitonic configurations, but are related by one-electron excitation to the CR ones (the last 4 terms from Fig. 4.1 are connected by one-electron excitations, but their contributions to one-particle density matrix will cancel out due to the spin symmetry — EX configurations are singlet-coupled on A and B, whereas the ME configurations are triplet-coupled on the monomers and then singlet-coupled in the dimer). 27 Thus, a quantity related to this difference can be used as a proxy for NAC. Such quantity can be easily obtained from the one-particle transition density matrix: if pq < i jp + qj f > (2.4) pq is all what is needed to compute a coupling element described by a one-electron operator ^ A: < i j ^ Aj f >=Tr[ if A] (2.5) pq also provides a measure of a one-electron character in the transition betweenj i > andj f > which is exploited in attachment-detachment density analysis and other ap- proaches based on density matrices. 31, 34–36 Specifically, Tr[ + ] can be interpreted as the number of electrons associated with one electron excitation connecting the two states. For example, this quantity is one for purely one-electron excitation (e.g., for any Hartree-Fock-CIS states) and is zero (no one-electron character) for purely doubly excited states. We also note that: Tr[ + ] =Tr[ + ] = X pq pq pq jj jj 2 (2.6) and, using the Cauchy-Schwarz inequality: jTr[ A]jjj jjjjAjj (2.7) Thus,jj jj can be considered as the best measure of the magnitude of a one-electron interstate property when matrix representation of the operator ^ A is not available. In the context of condensed phase where one expect significant modulations in vibronic 28 matrix elements due to finite-temperature sampling one can assume random fluctuations in the matrix elements of ^ A; thus, it is reasonable to assume that statistically averaged NAC is proportional to (or, more precisely, symbatic with)jj jj. An attractive property of this quantity is that it is independent of the orbital basis (i.e., matrix-invariant) and does not require the transformation of the wave functions into a localized FMO basis. Thus, it can be uniformly applied at arbitrary fragment configurations. We note that for a Hermitian operatorA, one may consider using symmetrized . More detailed analysis of formal properties of transition density matrices (partially based on Ref.37) will be given elsewhere. Table 2.2 shows the computed values ofjj jj for selected dimer configurations (at which the wave function composition can be analyzed in terms of EX, CR, and ME con- tributions) computed for leading terms of the RAS(4,4)-2SF wave functions (see SI for Ref.24 for details). The trends injj jj are consistent with changes in state composition (and also with exciton stabilization energies). For example, S 1=1 0- 1 MEjj jj becomes zero at large inter-fragment separations when 1 ME looses its CR configurations. Like- wise, the D 2 structure, which has less CR mixing and smaller exciton stabilization, is characterized by much smaller values ofjj jj. We also note thatjj jj for S 0 -S 1=1 0 is close to 1 when the excited states have less CR character. Finally,jj jj for S 0 - 1 ME is always small, which can be used to identify the ME state when there is no symmetry and the orbitals are not well localized. 29 Table 2.2: jj jj between different dimer states and exciton binding energy (E b , eV) of tetracene (left) and pentacene (right) dimers at various geometries. The state characters (ME/EX/CR) is shown in parentheses. State f/i E b S 0 S 1 S 1 0 State f/i E b S 0 S 1 S 1 0 D 2h , 3.7 ˚ A 0.22 D 2h , 3.7 ˚ A 0.20 S 1 (0.0/0.4/0.4) 0.87 S 1 (0.0/0.4/0.4) 0.87 S 1 0 (0.0/0.4/0.4) 0.87 0.60 S 1 0 (0.0/0.5/0.3) 0.87 0.56 1 ME (0.9/0.0/0.1) 0.11 0.68 0.53 1 ME (0.9/0.0/0.1) 0.11 0.65 0.61 D 2h , 6.0 ˚ A 0.00 D 2h , 6 ˚ A 0.00 S 1 (0.0/0.8/0.0) 0.86 S 1 (0.0/0.8/0.0) 0.87 S 1 0 (0.0/0.8/0.0) 0.91 0.58 S 1 0 (0.0/0.8/0.0) 0.87 0.47 1 ME (1.0/0.0/0.0) 0.09 0.02 0.00 1 ME (1.0/0.0/0.0) 0.00 0.02 0.00 C s , 30 0.00 C s , 30 0.00 S 1 (0.0/0.8/0.1) 0.88 S 1 (0.0/0.6/0.2) 0.83 S 1 0 (0.0/0.7/0.1) 0.88 0.57 S 1 0 (0.0/0.6/0.2) 0.83 0.48 1 ME (0.9/0.0/0.1) 0.07 0.18 0.27 1 ME (0.9/0.0/0.1) 0.09 0.27 0.35 D 2 , 40 -0.01 D 2 , 40 -0.01 S 1 (0.0/0.8/0.0) 0.85 S 1 (0.0/0.0/0.8) 0.87 S 1 0 (0.0/0.8/0.0) 0.90 0.58 S 1 0 (0.0/0.1/0.7) 0.86 0.48 1 ME (1.0/0.0/0.0) 0.09 0.01 0.15 1 ME (1.0/0.0/0.0) 0.01 0.04 0.25 X-Ray 0.03 X-Ray 0.04 S 1 (0.0/0.4/0.4) 0.82 S 1 (0.0/0.2/0.6) 0.80 S 1 0 (0.0/0.6/0.1) 0.82 0.45 S 1 0 (0.0/0.7/0.1) 0.83 0.50 1 ME (0.6/0.0/0.3) 0.18 0.42 0.34 1 ME (0.6/0.0/0.3) 0.14 0.54 0.29 30 Figure 2.4:jj jj and E b for tetracene (top) and pentacene (bottom) along selected scans: (a) D 2h , versus distance between two monomers; (b) C 2h , long axis, in units of the number of 6-carbon rings shifted; (c) C 2h , short axis; (d) C s rotation. Fig. 2.4 showsjj jj and E b for tetracene and pentacene along selected coordinates (see SI for Ref.24 for scan definitions). Contrary to previous studies, 1, 4, 8 we observe largest state mixing and couplings at parallel (D 2h ) cofacial arrangement, whereas slip-stacked structures are characterized by the reduced values of jj jj. We also note sizable variations in both and E b . Particularly intriguing are out-of-phase variations in couplings between 1 ME and the two excitonic singlet states (e.g., along parallel-displaced C 2h scans). Multiexciton binding energy also depends on the relative orientation and the dependence does not always mirror the trends injj jj (see SI for Ref.24); thus, these two quantities could, in principle, be tweaked independently to maximize the overall SF rate. Importantly, at some configurations (e.g., “propeller” structure along D 2 scan at 40 ), we observe negative E b suggesting possibility of exothermic pathways for triplet-triplet separation step. This sensitivity ofjj jj and 31 E b to chromophore arrangements opens the way to a rational design of chromophore structures provided that optimal values for both parameters can be deduced from a kinetic model of the SF process. To summarize, based on the analysis of the correlated wave functions of the relevant states combined with calculations of various energy differences (Davydov splitting,E b ) and calculations ofjj jj (a proxy for NAC), we conclude that: Simple diabatization approaches are not applicable to the SF problem; conse- quently, diabatic representation of the underlying electronic structure is not ca- pable of quantitatively accurate description of the process. Instead, one should employ reliable adiabatic wave functions and NACs. Configurations of CR character that are present (with relatively small weights) in both excitonic and multiexciton wave functions are important for the couplings between the states. This effect is trivially described within simple adiabatic pic- ture and does not justify invoking “intermediate states”, “two-electron couplings”, and “super-exchange”. This can change in polar solvents where adiabatic states of CT character may become relevant. Norms of one-particle transition density matrices,jj jj, provide a useful tool for studying trends in NAC. We observe only a limited correlation between various energy splittings andjj jj. Thus, the practice of using Davydov splittings or or- bital overlaps as a proxy for NAC is of limited utility. Some of the conclusions derived from model few-states diabatic Hamiltonians should be revised — e.g., we observe large couplings at parallel configurations. 32 Modes that promote non-adiabatic transitions are, in general, different from the modes modulating the strength of NAC. Calculations of rates of non-adiabatic transitions require consideration of the former, whereas the latter are important for understanding what structural motifs can lead to more efficient SF. 33 Chapter 2 references [1] M. B. Smith and J. Michl. Singlet fission. Chem. Rev., 110:6891–6936, 2010. [2] T.F. Schulze, J. Czolk, Y .-Y . Cheng, B. F¨ uckel, R.W. MacQueen, T. Khoury, M.J. Crossley, B. Stannowski, K. Lips, U. Lemmer, A. Colsmann, and T.W. Schmidt. Efficiency enhancement of organic and thin-film silicon solar cells with photo- chemical upconversion. J. Phys. Chem. C, 116:22794–22801, 2012. [3] S. Singh, W.J. Jones, W. Siebrand, B.P. Stoicheff, and W.G. Schneider. Laser generation of excitons and fluorescence in anthracene crystals. J. Chem. Phys., 42:330–342, 1965. [4] M.B. Smith and J. Michl. Recent advances in singlet fission. Annu. Rev. Phys. Chem., 64:361–368, 2013. [5] S.T. Roberts, R.E. McAnally, J.N. Mastron, D.H. Webber, M.T. Whited, R.L. Brutchey, M.E. Thompson, and S.E. Bradforth. Efficient singlet fission found in a disordered acene film. J. Am. Chem. Soc., 134:6388–6400, 2012. [6] C. Ramanan, A.L. Smeigh, J.E. Anthony, T.J. Marks, and M.R. Wasielewski. Com- petition between singlet fission and charge separation in solution-processed blend films of 6,13-bis(triisopropylsilylethynyl)- pentacene with sterically-encumbered perylene-3,4:9,10- bis(dicarboximide)s. J. Am. Chem. Soc., 134:386–397, 2012. [7] W-L. Chan, M. Ligges, and X-Y . Zhu. The energy barrier in singlet fission can be overcome through coherent coupling and entropic gain. Nat. Chem., 4:840–845, 2012. [8] J.C Johnson, A.J. Nozik, and J. Michl. The role of chromophore coupling in singlet fission. Acc. Chem. Res., 46:1290–1299, 2013. [9] T.C. Berkelbach, M.S. Hybertsen, and D.R. Reichman. Microscopic theory of singlet exciton fission. I. General formulation. J. Chem. Phys., 138:114102, 2012. [10] P.M. Zimmerman, C.B. Musgrave, and M. Head-Gordon. A correlated electron view of singlet fission. Acc. Chem. Res., 46:1339–1347, 2012. [11] P.M. Zimmerman, Z. Zhang, and C.B. Musgrave. Singlet fission in pentacene through multi-exciton quantum states. Nature Chem., 2:648–652, 2010. 34 [12] P.M. Zimmerman, F. Bell, D. Casanova, and M. Head-Gordon. Mechanism for singlet fission in pentacene and tetracene: From single exciton to two triplets. J. Am. Chem. Soc., 133:19944–19952, 2011. [13] T.S. Kuhlman, J. Kongsted, K.V . Mikkelsen, K.B. Møller, and T.I. Sølling. Inter- pretation of the ultrafast photoinduced processes in pentacene thin films. J. Am. Chem. Soc., 132:3431–3439, 201o. [14] S. Sharifzadeh, P. Darancet, L. Kronik, and J.B. Neaton. Low-energy charge- transfer excitons in organic solids from first-principles: The case of pentacene. J. Phys. Chem. Lett., 4:2917–2201, 2013. [15] R.W.A. Havenith, H.D. de Gier, and R. Broer. Explorative computational study of the singlet fission process. Mol. Phys., 110:2445–2454, 2012. [16] D.N. Congreve, J. Lee, N.J. Thompson, E. Hontz, S.R. Yost, P.D. Reusswig, M.E. Bahlke, S. Reineke, T. Van V oorhis, and M.A. Baldo. External quantum efficiency above 100% in a singlet-exciton-fissionbased organic photovoltaic cell. Science, 340:334–337, 2013. [17] P.J. Vallett, J.L. Snyder, and N.H. Damrauer. Tunable electronic coupling and driving force in structurally well defined tetracene dimers for molecular singlet fission: A computational exploration using density functional theory. J. Phys. Chem. A, 117:10824–10838, 2013. [18] D. Casanova, L.V . Slipchenko, A.I. Krylov, and M. Head-Gordon. Double spin-flip approach within equation-of-motion coupled cluster and configuration interaction formalisms: Theory, implementation and examples. J. Chem. Phys., 130:044103, 2009. [19] F. Bell, P.M. Zimmerman, D. Casanova, M. Goldey, and M. Head-Gordon. Re- stricted active space spin-flip (RAS-SF) with arbitrary number of spin-flips. Phys. Chem. Chem. Phys., 15:358–366, 2013. [20] J.N. Murrell and J. Tanaka. Theory of electronic spectra of aromatic hydrocarbon dimers. Mol. Phys., 7:363–380, 1964. [21] R.S. Mulliken and W.B. Person. Molecular Complexes. Wiley-Interscience, 1969. [22] A.L.L. East and E.C. Lim. Naphthalene dimer: Electronic states, excimers, and triplet decay. J. Chem. Phys., 113:8981–8994, 2000. [23] P.A. Pieniazek, A.I. Krylov, and S.E. Bradforth. Electronic structure of the benzene dimer cation. J. Chem. Phys., 127:044317, 2007. 35 [24] X. Feng, A.V . Luzanov, and A.I. Krylov. Fission of entangled spins: An electronic structure perspective. J. Phys. Chem. Lett., 4:3845–3852, 2013. [25] A.V . Luzanov and O.V . Prezhdo. High-order entropy measures and spin-free quan- tum entanglement for molecular problems. Mol. Phys., 105:2879–2891, 2007. [26] K. Diri and A.I. Krylov. Electronic states of the benzene dimer: A simple case of complexity. J. Phys. Chem. A, 116:653–662, 2011. [27] A.I. Krylov. Size-consistent wave functions for bond-breaking: The equation-of- motion spin-flip model. Chem. Phys. Lett., 338:375–384, 2001. [28] A.I. Krylov. The spin-flip equation-of-motion coupled-cluster electronic structure method for a description of excited states, bond-breaking, diradicals, and triradi- cals. Acc. Chem. Res., 39:83–91, 2006. [29] A.V . Luzanov. One-particle approximation in valence scheme superposition. Theor. Exp Chem., 17:227–233, 1982. [30] K. Takatsuka, T. Fueno, and K. Yamaguchi. Distribution of odd electrons in ground-state molecules. Theor. Chim. Acta, 48:175–183, 1978. [31] M. Head-Gordon, A. M. Grana, D. Maurice, and C. A. White. Analysis of elec- tronic transitions as the difference of electron attachment and detachment densities. J. Phys. Chem., 99:14261 – 14270, 1995. [32] J.E. Subotnik, S. Yeganeh, R.J. Cave, and M.A. Ratner. Constructing diabatic states from adiabatic states: Extending generalized Mulliken-Hush to multiple charge centers with Boys localization. J. Chem. Phys., 129:244101, 2008. [33] R.J. Cave, S.T. Edwards, and J.A. Kouzelos. Reduced electronic spaces for mod- eling donor/acceptor interactions. J. Phys. Chem. B, 114:14631–14641, 2010. [34] A.V . Luzanov, A.A. Sukhorukov, and V .E. Umanskii. Application of transition density matrix for analysis of excited states. Theor. Exp. Chem., 10:354–361, 1976. [35] A.V . Luzanov and V .F. Pedash. Interpretation of excited states using charge- transfer number. Theor. Exp. Chem., 15:338–341, 1979. [36] A.V . Luzanov and O.A. Zhikol. Excited state structural analysis: TDDFT and related models. In J. Leszczynski and M.K. Shukla, editors, Practical aspects of computational chemistry I: An overview of the last two decades and current trends, pages 415–449. Springer, 2012. [37] A.V . Luzanov. Covariant models for electronic structure. In M.M. Mestechkin, editor, Many-body problem in quantum chemistry, pages 53–64. Naukova Dumka, Kiyv, 1987. 36 Chapter 3: What We Can Learn from the Norms of One-particle Density Matrices, and What We Can’t: Some Results for Interstate Properties in Model Singlet Fission Systems 3.1 Introduction Reduced density matrices (DMs) 1–4 are widely used in many-body theories, both for quantitative and qualitative purposes. They allow one to compress the information con- tained in many-electron wave functions such that only essential details are preserved. The utility of DMs stems from the basic quantum-mechanical result due to the indis- tinguishability of the electrons. One can easily show that in order to compute an ex- pectation value of a one-particle operator, ^ A = P pq A pq p + q, only a one-particle DM (OPDM), , is needed: < j ^ Aj >=Tr[A ] = X pq A pq qp (3.1) 37 where qp < jp + qj > (3.2) Here and below we employ second quantization formalism; p + and q denote creation and annihilation operators, respectively, and the sums run over all basis functions. Any orthonormal spin-orbital basis can be used; Eq. (3.1) is orbital-invariant. Likewise, two- particle DMs can be used to compute expectation values of two-electron operators, and so forth. In a similar fashion, the so-called transition DMs can be defined and used to compute interstate properties such as couplings and transition dipole moments: < i j ^ Aj f >=Tr[A if ] (3.3) if qp < i jp + qj f > (3.4) wherei andf denote the initial and final state, respectively. Since the focus of this paper is on interstate properties and transition DMs, the superscriptif will be dropped in the presentation below. Eqns. (3.1) and (3.3) are routinely used in electronic structure codes to compute one- electron properties. By virtue of these equations, the code for properties calculations is the same for various approximate wave functions. Once wave function amplitudes are obtained, one can compute the required DMs and discard the amplitudes thus avoiding storage bottlenecks in the subsequent calculations. 38 We note that for properties associated with real-valued Hermitian operators and real- valued wave functions, such as dipole moment and nonadiabatic couplings (NACs), only the symmetric part of OPDM, s , contributes to the computed matrix elements: s pq 1 2 ( pq + qp ) (3.5) OPDMs are also very useful for the interpretation and qualitative wave function analysis. OPDM defined by Eq. (3.2) gives the expansion coefficients for representing electron density in the basis of molecular (or atomic) orbitals: (r) = X pq pq p (r) q (r) (3.6) Thus, the trace of equals the number of the electrons. Consequently, OPDMs can be used to define and compute quantities such as atomic charges, bond orders, and so forth, which are exploited in Mulliken and L¨ owdin population analyses, 5 as well as in more sophisticated Atom-in-Molecule, Natural Bond Orbital, and Natural Resonance theories. 6–8 OPDMs also contain information about the correlation, which can be used for the analysis of the electronic structure. For example, Yamaguchi and Head-Gordon indexes 9, 10 and the related particle-hole occupancies 11 allow one to quantify the effective number of unpaired electrons. 39 In the context of excited states, OPDMs can be used to describe the character of the electronic transitions in terms of the orbitals involved, as done in the attachment- detachment analysis, natural transition orbitals, and other approaches. 12–15 These orbital-invariant schemes allow one to focus on the essential features of wave functions; they become critically important when the wave function includes multiple amplitudes with similar weights (see, for example, Ref.16). Furthermore, one can use OPDMs for assigning the character of the transitions as charge-transfer or local excitations. 14, 15, 17–19 Transition OPDMs can be used for visualizing excitons, computing average particle- hole distances, and more. 11, 19 The focus of this work is on a quantity that has not received much of attention yet, the norm of OPDM: jj jj s X pq ( pq ) 2 = p Tr[ + ]; (3.7) where the orthonormal basis is implied. Note that: jj jj 2 =jj s jj 2 +jj as jj 2 ; (3.8) where as pq 1 2 ( pq qp ). Thus, the norm of symmetrized is always smaller than jj jj. For pure one-electron excitations: jj s jj =jj as jj = jj jj p 2 = 1 2 (3.9) jj jj was recently used to analyze trends in NACs in the context of singlet fission 20–22 and to explain the behavior of resonance wave functions in calculations using complex absorbing potentials (CAPs). 23 In this paper we provide more detailed description of the 40 formalism and illustrate its utility by several examples, with an emphasis on interstate properties such as transition dipole moments ( tr ) and NACs. NAC matrix elements are the central quantities in nonadiabatic processes: < i (r;R)jr R f (r;R)>; (3.10) where i (r;R) and f (r;R) are electronic wave functions of the initial and final states. We note that this coupling is a vector in the space of nuclear displacements; in rates’ calculations, it should be contracted with a respective component of the nuclear mo- mentum,r R ( denotes nuclear wave function). The structure of this vector and its relation to the topology of conical intersections has been analyzed in detail by David Yarkony and co-workers. 24–26 Eq. (3.10) is never used for practical calculations, 27–32 instead one can easily show that: 33 < i (r;R)jr R f (r;R)>= < i jr R ^ H el j f > U f U i ; (3.11) where U i and U f are electronic energies of the initial and final states. This can be derived from the Schroedinger equation by either simple differentiation or by using per- turbation theory and Taylor expansion around a fixed nuclear geometry. In derivation of Eq. (3.11), we employ Hellman-Feynman theorem; thus, for approximate solutions of electronic Schroedinger equations additional terms containingr R may appear; these so-called Pulay terms can be tackled by the response theory (or by the Lagrangian tech- nique), as done in analytic nuclear gradient formalism. 34 Such theoretically complete formalism and its computer implementation for MR-CI wave functions have been first developed by David Yarkony. 27–29, 35, 36 More recently, this formalism has been applied 41 to the CIS and TDDFT methods; 30–32 analytic derivative matrix elements have also been computed for equation-of-motion coupled-cluster wave functions in order to parameter- ize vibronic Hamiltonian. 37 The only part of the electronic Hamiltonian that explicitly depends on nuclear coor- dinates is ^ V en : @ ^ H el @R = @ ^ V en @R = @ @R X A;i Z A jr i R A j ! (3.12) where the sum runs over all atoms and all electrons. Although ^ V nn also depends on nuclear coordinates, the respective contributions to NAC vanish because of the orthonormality of the electronic wave functions. Thus, in the case of the exact wave functions, NAC is a strictly one-electron operator and can be computed using unrelaxed OPDM given by Eq. (3.2): < i (r;R)jr R f (r;R)>= Tr[ if V R ] U f U i ; (3.13) whereV R is the matrix of the operator ^ V R en @ ^ Ven @R . Note that NAC is anti-Hermitian, < i jr j >= < j jr i >, owing to the energy difference denominator, whereas the derivative operator, ^ V R , is Hermitian (thus, a symmetrized if can be used for its evaluation). When approximate solutions are used, Eq. (3.13) needs to be modified. First, non- Hellman-Feynman terms should be accounted for by solving amplitude and orbital re- sponse equations. Their effect can be trivially folded into the DMs leading to the so- called relaxed DMs: < i (r;R)jr R f (r;R)>= 1 U f U i Tr[h R ~ ] +Tr[II R ~ ] ; (3.14) 42 whereh R andII R denote the derivatives of the one- and two-particle parts ofH el ; ~ and ~ are the relaxed one- and two-particle transition DMs, respectively. Note that when incomplete atom-centered one-electron basis sets are used, the full expression of NAC contains contributions from other parts ofH el (such as kinetic energy and electron repul- sion integrals) due to the implicit dependence of the matrix elements on the one-electron basis. Thus, the two-electron contributions to NAC are artifacts of using atom-centered incomplete basis sets; they vanish in the complete basis set limit or when using non-AO bases, such as plane-wave or grid representations. Interestingly, the correction restoring translational invariance of NACs exactly cancels out the non-Hellman-Feynman contri- bution to NACs in the CIS and TDDFT models. 32 The differences between the relaxed and non-relaxed DMs are usually small. While they are indispensable for calculating the nuclear gradient, their effect on the computed properties is insignificant and is often ignored. In particular, orbital response terms are usually excluded from the properties calculations because they introduce poor pole structure. 34 Thus, in the limit of complete basis set, pq is all what is needed to compute NAC, which is described by one-electron operator ^ V R en : < i j ^ V R en j f >=Tr[ if V R en ] (3.15) In this paper, we investigate NACs computed by full analytic formulation for several model systems relevant to singlet fission; 38, 39 we compare trends in NACs and other properties (i.e., dipole moments) with trends injj jj. We also investigate the origin of NACs in ethylene dimer. 40 43 The structure of the paper is as follows. The next section presents the formalism, followed by the computational details. The numeric examples are presented and dis- cussed in Section 3.4. Additional properties of OPDMs are described in Appendix A for Ref.41. Appendix B for Ref.41 outlines calculations of rates of nonadiabatic pro- cesses by using the Fermi Golden Rule. Appendix C for Ref.41 analyses multi-exciton wave functions for a model 4-electron-in-four-orbitals case. 3.2 Theory We begin by applying the Cauchy-Schwartz inequality to Eqns. (3.1) and (3.3): jTr[A ]j =j X pq A pq qp jjjAjjjj jj (3.16) Thus,jj jj provides a bound to the respective expectation values (or matrix elements describing interstate properties). An attractive property ofjj jj is that it is independent of the (orthonormal) orbital ba- sis (i.e., matrix-invariant). For example, it is not affected by the unitary transformations between localized and delocalized molecular orbitals. However, the Frobenius norm of is different in the MO and AO bases. Thus, one needs to be consistent when applying Eq. (3.16) and to compare the trends using either the MO or AO basis only. The dis- crepancy can be avoided by computing operator norms correctly in a non-orthonormal basis, i.e., by using: jj jj =Tr[ S + S]; (3.17) whereS is a covariant overlap matrix and is a contravariant density matrix in the AO basis. In the MO basis (or an orthogonal AO basis)S=1, thus, this equation reduces to 44 Eq. (3.7). In this paper, we always compute in the MO basis or in the orthonormal L¨ owdin AO basis. Eq. (3.16) is valid both for permanent properties, when is given by Eq. (3.2), as well as for interstate properties. Let us first discuss the latter. For interstate properties, is given by Eq. (3.4); it describes the redistribution of the electron density associated with the i! f transition (note that in general the transition density is not equal to the density differences). It was suggested 20 thatjj jj can be considered as a measure of the magnitude of a one-electron interstate property when matrix representation of the respective operator is not available. It was also speculated that in the context of condensed phase where one expects significant modulations in matrix elements due to finite-temperature sampling one can assume random fluctuations in the matrix elements of A; thus, it is reasonable to assume that statistically averaged NAC is proportional to (or, more precisely, symbatic with)jj jj. In the next section, we present numeric investigations of the validity of these assumptions. To better understand Eq. (3.16), we note that: jTr[ A]j =jj jjjjAjj cos() (3.18) where is an angle between the two multi-dimensional vectors, A and , defined by the above equation. This form illustrates that if the angle remains constant as a function of certain parameters (e.g., changing nuclear geometries), then one can readily estimate trends injTr[ A]j by consideringjj jj andjjAjj. If one of the norms is constant, then the trends in the other provide useful information (but not the other way around). Of course, if two vectors are orthogonal ( = 2 ) or if both quantities significantly change their magnitude, Eq. (3.16) is not useful. Another special case is when one quantity changes 45 its orientation and magnitude at random — in this case averaging can be performed giving rise to a constant factor. In the context of interstate properties, we note that pq provides a measure of a one-electron character in the transition betweenj i > andj f >. Specifically,Tr[ + ] can be interpreted as the number of electrons associated with one-electron excitation connecting the two states. For example, this quantity is one for purely one-electron exci- tation (e.g., for any Hartree-Fock/CIS states) and is zero (no one-electron character) for purely doubly excited states. Thus, whenj i > andj f > differ by two or more excited electrons, matrix elements of any one-electron property should be zero. Some in- teresting sum rules for the squared norms,jj jj 2 , are derived in the Appendix for Ref.41. Let us now consider Eq. (3.16) in the context of state properties, when represents OPDM of a given electronic state. A useful aspect of Eq. (3.16) is that it highlights the connection between the density and various observables. For example, in Ref.23 this equation was employed to map the trends in the spatial extent of the electronic wave function (as measured by < R 2 >) and its energy in the context of resonances and CAPs. In CAP-augmented electronic structure calculations, the resonances (metastable electronic states embedded in the ionization continuum) are stabilized by the CAP. It was observed that as the strengths of CAP increases, the size of the wave function shrinks; once the critical strength is achieved, the wave function size remains approximately constant (that is, the resonance becomes stabilized). This is clearly illustrated by the decrease in the< R 2 > value. Since the matrix representation ofR 2 does not depend on the CAP strength,<R 2 > is perfectly correlated withjj jj. On the other hand, the one-electron part of the total energy (and, in particular, the perturbation due to the CAP) depends on . Thus, Eq. (3.16) tells one that the energy perturbation 46 due to the CAP is proportional tojj jj giving rise to the linear dependence of the energy as a function of the CAP strength in the asymptotic region. These simple observations have led to a deperturbative correction that improved the robustness of CAP-augmented methods. 23, 42 It is important to distinguish the situations in which Eq. (3.16) may provide use- ful qualitative (or even quantitative) insights from the cases when its application (al- though formally correct) is useless. One can employ Eq. (3.16) to investigate the effect of changes in molecular geometry on matrix elements between states changing their character upon these displacements, while the matrix representation of the opera- tor representing the interstate property does not change much. For example, in Refs. 20–22 Eq. (3.16) was applied to quantify the degree of one-electron character (modulated by the mixing excitonic, multi-exciton, and charge-resonance configurations) in the transi- tions between the states involved in singlet fission. Below, we consider 3 lowest elec- tronic states of polyenes that change their character and mix with each other along var- ious displacements. We also compute interstate properties in ethylene dimer, a model system used to analyze the effect of chromophore orientation on the NAC in the context of singlet fission. 40 The key here is that the trends in the matrix elements are driven by the trends in the wave functions. As an example of a situation when application of Eq. (3.16) does not provide useful information, consider calculations of transition dipole moments between the ground and excited electronic states within CIS or TD-DFT/TDA formalisms. In this case,jj jj=1 for all transitions, yet some states are bright and some — dark, as dictated by the symme- try and the MOs involved in the transition (that is, cos is different for different excited states). Another example of this sort would be the behavior of the transition dipole 47 moment between the two lowest states of an ionized dimer along the interfragment dis- tance (e.g., He + 2 ). The two electronic states in this case preserve their character 43 and the transition between them remains predominantly one-electron (jj jj1); however, the magnitude of the dipole moment increases linearly 43 along the stretching coordinate because of the obvious trend in. 3.3 Computational details Calculations were performed using several theoretical approaches. First, for oc- tatetraene we employ semi-empirical PPP-FCI model ( the Pariser-Parr-Pople - approximation at the full configuration interaction level) 44 that enables quick ex- ploratory calculations with a complete account for -electron correlation. Calcula- tions for butadiene and ethylene were performed using the complete active space self- consistent field (CASSCF) method. 45 CASSCF interstate properties were computed us- ing full analytic formulation — that is, full expression for derivative coupling has been used, 29 including non-Hellman-Feynman terms, as given by Eq. (3.14). For comparison purposes, we also computed the Hellman-Feynman part of NAC for ethylene dimer (see SI for Ref.41); the differences between the two quantities are rather small and all trends are the same. In addition, RAS-2SF (restricted active space double spin-flip) calcula- tions 46–48 were performed for ethylene dimer. Only singlet states were considered in this work. The PPP-FCI calculations were performed using the conventional - parameterization scheme. 49 The alternation in core integrals (resonance integrals) +1 in the trans-octatetraene chain and other -parameters are the same as in Ref.50 The simplified geometry for the carbon backbone is used (in the equilibrium configuration all CC bonds are of the same length 1.4 ˚ A and all C-C-C angles equal 48 120 ). We employed the FCI matrix algorithm from Ref.51 implemented as Matematica notebooks. RAS-2SF and CASSCF calculations of ethylene dimer used the 4-electrons-in-4- orbitals active space and the 6-31G* basis set. An active space of 4-electrons-in-4- orbitals was also used in the CASSCF calculations of butadiene. The CASSCF calcu- lations were state averaged with four states included in the average for both butadiene and ethylene dimer. The four singlet states included in the average are the states of importance here: the ground state, the two components of the excitonic states derived from the excitation, and the doubly excited (ME) state. RAS-2SF calculations used a high-spin ROHF quintet reference. The structure of the ethylene molecule was optimized at the RI-MP2/cc-pVTZ level of theory. Then stacked dimer was constructed for the two identical monomers 3.5 ˚ A apart. This structure was used as a starting point to generate displacements along the long and short axes. CASSCF calculations were performed by Columbus. 52, 53 RAS-2SF calculations were performed with Q-Chem. 54, 55 All relevant geometries are given in SI for Ref.41. 3.4 Results and Discussion We begin by considering polyenes, whose electronic structure is somewhat similar to that encountered in singlet fission systems. In all-trans polyenes, the two lowest excited states are of the A g and B u symmetry. At equilibrium geometry, the former has pre- dominantly double-excited character, ( ) 2 , with respect to the ground state, () 2 , and is dark, whereas the latter is dominated by a one-electron transition, , and is bright; 2A g and 1B u are connected by a one-electron excitation. Changing molecular geometry 49 modifies the character of these states thus affecting interstate properties such as transi- tion dipole moments and NACs. We consider the interstate properties between the two exited states (2A g and 1B u ), as well as between the ground 1A g and the two excited states. 3.4.1 Octatetraene Our first example is octatetraene. We consider two types of displacements — stretching the C4-C5 bond (symmetry-preserving) and twisting around C4-C5. Changes in the states’ characters can be characterized by relative weights of the reference, singly, and doubly excited configurations. Fig. 3.1 shows the results for stretching along the C4-C5 bond obtained with the PPP-FCI model (see Section 7.2). Bond stretching parameter is defined as =r=r eq , wherer eq is the equilibrium bond length. We observe that the stretch has a very little ef- fect on the 1A g -1B u transition dipole moment, while the 2A g -1B u transition dipole first increases from 0.8 to 1.1 and then monotonically drops to zero. This different behav- ior can be explained by the changes in the underlying wave functions. The ground-state wave function is dominated by the reference determinant at all values of (e.g., at=1.5, the weight of the reference is 0.88). Likewise, 1B u retains its singly-excited character. However, in the 2A g state, the weight of doubly-excited configurations increases from 0.6 to 0.9. This explains different trends in the respective tr . We note thatjj jj catches the overall trend very well — it remains approximately constant for the 1A g -1B u transi- tion and monotonically declines for 2A g -1B u . However, finer variations in tr (e.g., the initial25% rise) are not reproduced byjj jj. These variations arise becausejjjj is not constant along these displacements; a simple analysis of the respective matrix elements reveals thatjjjj increases linearly with the bond stretch (in the AO basis, the matrix 50 Figure 3.1: Stretching of octatetraene along C 4 -C 5 ( is the bond stretching pa- rameter,=1 corresponds to ther e ). Top: Matrix elements of the dipole operator (a.u.) andjj jj between the 2A g and 1B u states (left) and between 1A g and 1B u states (right). s denotes symmetrized OPDM. Bottom left: Weights of singly (C 2 1 ) and doubly (C 2 2 ) excited configurations in the 2A g wave function andjj 1Bu2Ag jj. Bottom right: Weights of singly (C 2 1 ) and doubly (C 2 2 ) excited configurations in the 1B u wave function andjj 1Bu1Ag jj. of ^ is diagonally dominant with the matrix elements being roughly proportional to the Cartesian coordinates of the respective atoms). We note thatjj jj andjj s jj behave similarly. jj s jj is about a factor of 2 smaller than the norm of unsymmetrized for thejj s jjjj jj, by virtue of Eq. (3.8). 1B u -1A g transition, whereas for 1B u -2A g , the values of the two norms are close. The trends are well reproduced by either symmetrized or unsymmetrized , since the two quantities parallel each other. 51 The behavior along the twisting coordinate is shown in Fig. 3.2. Twisting lowers the symmetry (from C 2h to C 2 ) and the electronic states, strictly speaking, should not be labeled as A g /B u at twisted geometries (but A/B). However, since the state character changes smoothly (as evidenced by the wave function coefficients), we use these labels for clarity. As in the example above, the 1A g and B u states retains their characters along the torsional coordinate, whereas the 2A g state eventually becomes doubly excited (at >80 ). The trends injj jj follow the trends in wave function composition — for 1A g -1B u ,jj jj is nearly constant, whereas for 2A g -1B u , the value ofjj jj begins to decrease at about 60 reaching zero at 90 . As in the bond stretching example, the significant changes in tr follow the trends injj jj (i.e., nearly constant value of tr for 1A g -1B u and an eventual drop to zero for 2A g -1B u ), however, finer details, such as initial increase for the 2A g -1B u transition are due to the changes injjjj and are not captured byjj jj. Specifically, the twist leads to the increase ofjjjj because the extent of the molecule increases in the dimension perpendicular to the molecular plane (and because the symmetry is lowered), thus, for small twisting angles whenjj jj remains nearly constant, tr shows initial rise (which is eventually overcome by the decline in jj jj at large values of). 3.4.2 Butadiene Our second example is butadiene where we consider OPDMs and interstate properties computed using the CASSCF wave functions. We begin by considering the stretching along the central bond. The trends in the CASSCF wave functions of the 3 states are very similar to the octatetraene example. Fig. 3.3 shows the weights of single and double amplitudes in the the two excited states, 52 Figure 3.2: Top: Matrix elements of the dipole operator (a.u.) andjj jj between the 2A g /1B u (left) and 1A g /1B u (right) states in octatetraene along twisting angle. Bottom: Weights of singly (C 2 1 ) and doubly (C 2 2 ) excited configurations in the wave functions of the 1B u and 2A g states. 1B u and 2A g (the weights are defined relative to the reference Hartree-Fock determi- nant). The 1A g state remains predominantly single-configurational (the weight of the reference is0.9). As one can see, the 1B u state remains singly excited with respect to the reference, whereas 2A g becomes doubly excited. These changes in the wave functions result in the following trends injj jj: for the 1A g -1B u transition,jj jj remains constant, whereas for the 1A g -2A g and 2A g -1B u tran- sitions,jj jj decreases from approximately 1 to zero. The respective interstate matrix elements ( tr andjjNACjj E) follow this trend, as illustrated in Fig. 3.4. As in the octatetraene example, the correlation is not perfect and some fine variations are not reproduced byjj jj, however, the overall picture is correct — 1A g -1B u properties 53 Figure 3.3: Weights of singly and doubly excited configurations in the wave func- tions of the 1B u and 2A g states of butadiene along the bond-stretching coordinate. do not change along this displacement, whereas interstate properties for the 2A g -1B u and 1A g -2A g transitions significantly decrease. Non-monotonic behavior of the actual matrix elements is due to the variations in the respective matrix representations of the dipole moment and derivative operators. The changes in the dipole moment matrix along stretching and twisting coordinates is explained above. The reason behind variations of the derivative matrix elements, Eq. (3.12), is different; it is analyzed in details in the next section. The behavior along twisting coordinate is also similar to octatetraene; the results are shown in SI for Ref.41. 3.4.3 Ethylene dimer Motivated by singlet fission, we now proceed to the ethylene dimer, 39 a model system used to investigate the effect of relative chromophore orientation on the electronic cou- plings. 40 The dimer geometry is shown in Fig. 3.5. Fig. 3.6 showsjj jj computed by RAS- 2SF as a function of the displacements alongX andY coordinates defined in Fig. 3.5. 54 Figure 3.4: jj jj, < > and NACE for the 1A g -1B u (left), 2A g -1B u (middle), and 1A g -2A g (right) transitions along the C2-C3 stretching coordinate in butadi- ene. Top: All quantities are normalized to their equilibrium (=1) values. Bottom: Absolute values. At the displaced geometries (X6= 0 andY 6= 0), the symmetry of the system is C i . We consider the following states: the ground state, the doubly excited state that is similar to the 1 ME states in acenes (two local 3 excitations coupled into a singlet), and two states derived from local excitations that are similar to the S 1 and S 1 0 states in the context of singlet fission. 20 The ground, ME, and S 1 states are gerade; S 1 0 is ungerade. The RAS-2SF active space orbitals and their symmetries are shown in SI for Ref.41. In the CASSCF calculations, we analyze both S 1 and S 1 0, however, in RAS-2SF scheme only the former can be computed reliably (because S 1 0 lies relatively high in energy and includes significant contributions of hole configurations). As one can see, the magnitude ofjj jj at the perfectly stacked geometry (X=Y =0) is non-zero. The displacement along Y leads to the monotonic decrease, whereas the trend alongX is more complicated — we first observe an increase injj jj that reaches maximum at about 0.6 ˚ A, which is approximately half of the carbon-carbon bond length (1.33 ˚ A). 55 Figure 3.5: Ethylene dimer atZ=3.5 ˚ A andX=Y =0. These variation injj jj can be tracked down to varying weights of the charge- resonance (CR) terms in the S 1 wave function. The configuration analysis at the selected geometries is given in SI for Ref.41 and is summarized in Table 3.1 (see also Appendix C for Ref.41). We observe that the 5 ME state retains its pure multiexciton character (TT ). The 1 ME state shows small amount (2-4%) of CR at perfectly stacked (X=Y =0) and slightly displaced (alongX) geometries. The character of S 1 state varies more considerably. At strongly displaced geometries (e.g., Y =2, X=0), the S 1 state is almost pure excitonic (the weight of CR is 6%), however, at small displacements, the CR configurations be- come dominant ( 70%). A slight increase injj jj at small displacements along the long axis (Y =0) clearly correlates with a slight increase of the CR weight in S 1 . Interestingly, this behavior is different from the trends observed in Ref.22, where it was found that fa- vorable couplings in certain polymorph structures arise mostly due to the admixture of CR configurations into the 1 ME state, while S 1 remains purely excitonic. Fig. 3.7 shows the CASSCF couplings between the ME-S 1 and ME-S 1 0 states and jj jj along the two cuts (all values are normalized to their respective values atX=Y =0). 56 Figure 3.6:jj jj for the 1 ME-S 1 states as a function of the displacements along the long (X) and short (Y ) axes. Table 3.1: Configuration analysis of the RAS-2SF wave functions of the ME and S 1 states using localized orbitals within DMO-LCFMO framework a State X ( ˚ A) Y ( ˚ A) ME CR EX 1 ME 0 0 0.98 0.02 0.00 0.8 0 0.92 0.04 0.03 0 2 1.00 0.00 0.00 S 1 0 0 0.00 0.66 0.26 0.8 0 0.00 0.70 0.17 0 2 0.00 0.06 0.86 a See Ref.20. For the displacement along the short axis (X=0 cut), we observe a nearly perfect cor- relation betweenjjNACjj E andjj jj, however, the behavior along theY =0 cut is more complex.jj jj shows relatively small variations, and so does NAC. The fine de- tails, however, are not reproduced. For example, for the ME-S 1 pair (right panel of Fig. 3.6), NAC shows up to 1.6 times increase at the displaced geometries, whereasjj jj re- mains nearly constant. We note that the difference between the Hellman-Feynman part and the full NAC is very small, as illustrated for theY =0 cut in the SI for Ref.41. To better understand these trends, we consider the individual components of NAC along theX=0 (Fig. 3.8) andY =0 (Fig. 3.9) scans. The NAC vector can be described 57 Figure 3.7:jj jj and interstate properties for the ME-S 1 (left) and ME-S 1 0 (right) transitions computed using CASSCF. Figure 3.8: Leading components of NAC along theX=0 scan for the ME-S 1 (left) and ME-S 1 0 (right) transitions. 58 Figure 3.9: Leading components of NAC along theY =0 scan for the ME-S 1 (left) and ME-S 1 0 (right) transitions. as intra and intermolecular carbon-carbon displacements; the leading components are shown in Figs. 3.8 and 3.9. We observe that for the X=0 scan, intramolecular com- ponents of NAC are zero and that the trends in intermolecular components follow the trends in . However, along the Y =0 scan both intra and intermolecular components are non-zero and the former does not followjj jj. For example, for the ME-S 1 0 pair, the intermolecular NAC components (C1-C3 and C2-C4) follow the trends injj jj closely showing the initial decrease (in absolute values) up to0.7 ˚ A followed by an increase. However, intramolecular components (C1-C2 and C3-C4) are exactly zero at the per- fectly stacked configuration, but rise sharply and become dominant at large X; they remain nearly constant after the initial rise. Since the absolute value of NAC,jjNACjj, is a sum of all components, the trends due to the change in the states characters (that define intermolecular components) become obscured. The couplings for the ME-S 1 pair (alongY =0) show a similar trend — the intermolecular components are nearly constant (as arejj jj), whereas intramolecular ones exhibit a maximum and have larger values. 59 Thus, the discrepancy between the trends injj jj and the actual NAC can be attributed to a more complex behavior of the intramolecular components of NACs. The difference in behavior of intra and intermolecular components can be rational- ized by considering the explicit form of the coupling operator from Eq. (3.11) that is given by Eq. (3.12). Thus, the contribution to NAC due to displacements of atom A alongx A is: @ @x A ^ V en = X i Z A sign(R A r i ) (R A r i ) 2 (3.19) Consequently, for any symmetric vibration of two symmetry-equivalent atoms, such as, for example, C1C2 bond stretch (in which the two carbons move in opposite directions) at X=0, the respective terms cancel out. This is further illustrated in Fig. S7 (see SI for Ref.41) that shows Cartesian components of NAC along the two scans. That is why C1C2 and C3C4 NAC components are exactly zero along theX=0 scan. We note that vibrations in which both carbons are moving in the same direction (thus keeping the CC bond length constant) are expected to cause smaller perturbations in the wave functions and, consequently, smaller contributions to NACs. However, when the two carbons become inequivalent (such as atX6=0, see Fig. S7), the two derivatives do not cancel out giving rise to non-zero NAC. This explains the observed rise of intramolecular components of NACs along the Y =0 scan. These observations can be generalized as follows: Intramolecular components of NACs along symmetric vibrations are expected to be large, provided that the atoms involved have different local environment. Dimer calculations may be insufficient to quantify intramolecular components, e.g., the local environment in a periodic solid might differ from that in the dimer. 60 Intramolecular components depend strongly on relative orientation of the frag- ments; this dependence is driven by the trends inr R ^ V en and, therefore, it does not always follow the trends injj jj. For example, they can show large variations even whenjj jjconst. Intermolecular components follow the trends injj jj very well. Their magnitude can be as large as that of the intramolecular components. In the context of SF, an important question is whether intra or intermolecular compo- nents are more important for the overall rate of nonadiabatic transitions. To answer this question, one needs to evaluate the rate. As discussed in the Appendix for Ref.41, the to- tal rate depends on both the coupling and the nuclear momentum. Thus, the components of Franck-Condon displacements will have a crucial effect on the overall couplings. For example, if the initially excited state (e.g., S 1 in SF) has large displacements along in- termolecular coordinates, the rate will be driven by the intermolecular NACs. Thus, in order to fully understand the effect of morphology on the SF rates, one needs to consider Franck-Condon displacements, in addition to the NACs, and evaluate rates using mod- els that take both these factors into account (such as in Refs. 56, 57 ). We stress, however, that regardless whether inter or intramolecular couplings play a major role, one needs to have largejj jj in order to have large couplings (that is, largejj jj is necessary but not sufficient). 3.5 Conclusions In this paper, we investigated the utility of the norms of OPDMs for evaluating the trends in state and interstate properties. By comparing full calculations of transition dipole mo- ments and NACs withjj jj for selected model systems, we illustrate that while variations 61 ofjj jj do contribute systematically to the total matrix elements (e.g., whenjj jj=0, then the respective matrix element of a one-electron operator is also zero), the finer details might not be reproduced correctly because the magnitude of the operator might also vary giving rise to a more complex trend. Analysis of NACs computed for a model SF sys- tem, ethylene dimer, reveals that intermolecular components of NACs follow the trends injj jj well, as they are determined primarily by the characters of the two wave func- tions, however, intramolecular components depend on the relative orientation of the two moieties via the dependence in @ ^ Ven @R . Therefore, intramolecular NACs can exhibit large variations even when changes injj jj are small. In other words, large values ofjj jj are necessary but not sufficient for maximizing the couplings. For ethylene dimer, we observe large NACs at perfectly stacked geometry, contrary to the predictions derived based on model Hamiltonians. 39 However, larger values (by a factor of 1.6) are indeed observed at slip-stacked (along the long axis) geometries, in agreement with the predic- tions of Michl and co-workers. 39 Larger values of NACs at slip-stacked configurations are due to the breaking of symmetry of the local environments of the heavy atoms and not due to the wave function composition. The variations injj jj for ethylene dimer are due to a varying admixture of the CR configurations to the S 1 state, whereas the 1 ME state retains its pure multi-exciton character. This is different from the trends observed in Ref.22, where it was found that favorable couplings arise mostly due to the admixture of CR configurations into the 1 ME state. In sum,jj jj provide a useful tool for analyzing the trends in couplings, however, for quantitative calculations full derivative coupling is desirable. In order to fully un- derstand the effect of morphology on the rates of nonadiabatic transitions in molecular solids, one needs to employ models that take into account which nuclear motions are activated by electronic excitations. 62 Chapter 3 references [1] R. McWeeny. Methods of Molecular Quantum Mechanics. Academic Press, 2nd edition, 1992. [2] E.R. Davidson. Reduced Density Matrices in Quantum Chemistry. Academic Press, New York, 1976. [3] M.M. Mestechkin. Metod matritsy plotnosti v teorii molekul. Kyev, Naukova Dumka, 1977. [4] T. Helgaker, P. Jørgensen, and J. Olsen. Molecular electronic structure theory. Wiley & Sons, 2000. [5] A. Szabo and N.S. Ostlund. Modern Quantum Chemistry: Introduction to Ad- vanced Electronic Structure Theory. McGraw-Hill, New York, 1989. [6] R.F.W. Bader. Atoms in molecules — Quantum theory. Oxford University Press, Oxford, 1990. [7] F. Weinhold and C.R. Landis. Discovering Chemistry With Natural Bond Orbitals. New Jersey: John Wiley & Sons, 2012. [8] E.D. Glendening, J.K. Badenhoop, and F. Weinhold. Natural resonance theory: II. Natural bond order and valency. J. Comput. Chem., 19(6):610–627, 1998. [9] K. Takatsuka, T. Fueno, and K. Yamaguchi. Distribution of odd electrons in ground-state molecules. Theor. Chim. Acta, 48:175–183, 1978. [10] M. Head-Gordon. Characterizing unpaired electrons from the one-particle density matrix. Chem. Phys. Lett., 372:508–511, 2003. [11] A.V . Luzanov and O.V . Prezhdo. Analysis of multiconfigurational wave functions in terms of hole-particle distributions. J. Chem. Phys., 124:224109, 2006. [12] M. Head-Gordon, A. M. Grana, D. Maurice, and C. A. White. Analysis of elec- tronic transitions as the difference of electron attachment and detachment densities. J. Phys. Chem., 99:14261 – 14270, 1995. [13] A.V . Luzanov and O.A. Zhikol. Electron invariants and excited state structural analysis for electronic transitions within CIS, RPA, and TDDFT models. Int. J. Quant. Chem., 110:902–924, 2010. 63 [14] A.V . Luzanov and O.A. Zhikol. Excited state structural analysis: TDDFT and related models. In J. Leszczynski and M.K. Shukla, editors, Practical aspects of computational chemistry I: An overview of the last two decades and current trends, pages 415–449. Springer, 2012. [15] F. Plasser and H. Lischka. Analysis of excitonic and charge transfer interactions from quantum chemical calculations. J. Chem. Theory Comput., 8:2777–2789, 2012. [16] K.B. Bravaya, M.G. Khrenova, B.L. Grigorenko, A.V . Nemukhin, and A.I. Krylov. Effect of protein environment on electronically excited and ionized states of the green fluorescent protein chromophore. J. Phys. Chem. B, 115:8296–8303, 2011. [17] A.V . Luzanov. The structure of the electronic excitation of molecules in quantum- chemical models. Russ. Chem. Rev., 49:1033–1048, 1980. [18] A.V . Luzanov, A.A. Sukhorukov, and V .E. Umanskii. Application of transition density matrix for analysis of excited states. Theor. Exp. Chem., 10:354–361, 1976. [19] S. Tretiak and S. Mukamel. Density matrix analysis and simulation of electronic excitations in conjugated and aggregated molecules. Chem. Rev., 102:3171, 2002. [20] X. Feng, A.V . Luzanov, and A.I. Krylov. Fission of entangled spins: An electronic structure perspective. J. Phys. Chem. Lett., 4:3845–3852, 2013. [21] A.B. Kolomeisky, X. Feng, and A.I. Krylov. A simple kinetic model for singlet fission: A role of electronic and entropic contributions to macroscopic rates. J. Phys. Chem. C, 118:5188–5195, 2014. [22] X. Feng, A.B. Kolomeisky, and A.I. Krylov. Dissecting the effect of morphology on the rates of singlet fission: Insights from theory. J. Phys. Chem. C, 118:19608– 19617, 2014. [23] T.-C. Jagau, D. Zuev, K.B. Bravaya, E. Epifanovsky, and A.I. Krylov. A fresh look at resonances and complex absorbing potentials: Density matrix based approach. J. Phys. Chem. Lett., 5:310–315, 2014. [24] D.R. Yarkony. Diabolical conical intersections. Rev. Mod. Phys., 68:985–1013, 1996. [25] D.R. Yarkony. Conical intersections: Diabolical and often misunderstood. Acc. Chem. Res., 31:511–518, 1998. [26] D.R. Yarkony. Conical intersections: The new conventional wisdom. J. Phys. Chem. A, 105:6277–6293, 2001. 64 [27] B.H. Lengsfield III, P. Saxe, and D.R. Yarkony. On the evaluation of nonadia- batic coupling matrix-elements using SA-MCSCF/CI wave functions and analytic gradient methods.1. J. Chem. Phys., 81:4549–4553, 1984. [28] P. Saxe, B.H. Lengsfield III, and D.R. Yarkony. On the evaluation of non-adiabatic coupling matrix-elements for large-scale CI wavefunctions. Chem. Phys. Lett., 113:159–164, 1985. [29] H. Lischka, M. Dallos, P.G. Szalay, D.R. Yarkony, and R. Shepard. Analytic eval- uation of nonadiabatic coupling terms at the MR-CI level. I. Formalism. J. Chem. Phys., 120:7322–7329, 2004. [30] E. Tapavicza, G.D. Bellchambers, J.C. Vincent, and F. Furche. Ab initio non- adiabatic molecular dynamics. Phys. Chem. Chem. Phys., 15:18336, 2013. [31] S. Fatehi, E. Alguire, Y . Shao, and J.E. Subotnik. Analytic derivative couplings between configuration-interaction-singles states with built-in electron-translation factors for translational invariance. J. Chem. Phys., 135:234105, 2011. [32] X. Zhang and J.M. Herbert. Analytic derivative couplings for spin-flip configura- tion interaction singles and spin-flip time-dependent density functional theory. J. Chem. Phys., 141:064104, 2014. [33] W.D. Domcke, D.R. Yarkony, and H. K¨ oppel, editors. Conical intersections. Elec- tronic structure, dynamics and spectroscopy. World Scientific Pbul Co Pte Ltd, 2004. [34] T. Helgaker, S. Coriani, P. Jørgensen, K. Kristensen, J. Olsen, and K. Ruud. Re- cent advances in wave function-based methods of molecular-property calculations. Chem. Rev., 112:543–631, 2012. [35] B.H. Lengsfield III and D.R. Yarkony. Nonadiabatic interactions between potential energy surfaces: Theory and applications. Adv. Chem. Phys., 82:1–72, 1992. [36] M. Dallos, H. Lischka, R. Shepard, D.R. Yarkony, and P.G. Szalay. Analytic eval- uation of nonadiabatic coupling terms at the MR-CI level. II. Minima on the cross- ing seam: Formaldehyde and the photodimerization of ethylene. J. Chem. Phys., 120:7330–7339, 2004. [37] T. Ichino, J. Gauss, and J.F. Stanton. Quasidiabatic states described by coupled- cluster theory. J. Chem. Phys., 130(17):174105, 2009. [38] M. B. Smith and J. Michl. Singlet fission. Chem. Rev., 110:6891–6936, 2010. [39] M.B. Smith and J. Michl. Recent advances in singlet fission. Annu. Rev. Phys. Chem., 64:361–368, 2013. 65 [40] J.C Johnson, A.J. Nozik, and J. Michl. The role of chromophore coupling in singlet fission. Acc. Chem. Res., 46:1290–1299, 2013. [41] S. Matsika, X. Feng, A.V . Luzanov, and A.I. Krylov. What we can learn from the norms of one-particle density matrices, and what we can’t: Some results for inter- state properties in model singlet fission systems. J. Phys. Chem. A, 118:11943– 11955, 2014. [42] D. Zuev, T.-C. Jagau, K.B. Bravaya, E. Epifanovsky, Y . Shao, E. Sundstrom, M. Head-Gordon, and A.I. Krylov. Complex absorbing potentials within EOM- CC family of methods: Theory, implementation, and benchmarks. J. Chem. Phys., 141(2):024102, 2014. [43] P.A. Pieniazek, S.A. Arnstein, S.E. Bradforth, A.I. Krylov, and C.D. Sherrill. Benchmark full configuration interaction and EOM-IP-CCSD results for proto- typical charge transfer systems: Noncovalent ionized dimers. J. Chem. Phys., 127:164110, 2007. [44] T. Amos and M. Woodward. Configuration interaction wavefunctions for small systems. J. Chem. Phys., 50:119–123, 1969. [45] B.O. Roos, P.R. Taylor, and P.E.M. Siegbahn. A complete active space SCF method (CASSCF) using a density matrix formulated super-CI approach. Chem. Phys., 48:157–173, 1980. [46] D. Casanova, L.V . Slipchenko, A.I. Krylov, and M. Head-Gordon. Double spin-flip approach within equation-of-motion coupled cluster and configuration interaction formalisms: Theory, implementation and examples. J. Chem. Phys., 130:044103, 2009. [47] D. Casanova and M. Head-Gordon. Restricted active space spin-flip configuration interaction approach: Theory, implementation and examples. Phys. Chem. Chem. Phys., 11:9779–9790, 2009. [48] F. Bell, P.M. Zimmerman, D. Casanova, M. Goldey, and M. Head-Gordon. Re- stricted active space spin-flip (RAS-SF) with arbitrary number of spin-flips. Phys. Chem. Chem. Phys., 15:358–366, 2013. [49] K. Ohno. Some remarks on the Pariser-Parr-Pople method. Theor. Chim. Acta, 2:219–227, 1964. [50] S. Ramasecha and Z.G. Soos. Optical excitations of even and odd polyenes with molecular PPP correlations. Synth. Metals, 1984. [51] A.V . Luzanov, A.L. Wulfov, and V .O. Krouglov. A wavefunction operator ap- proach to the full-CI problem. Chem. Phys. Lett., 197:614–619, 1992. 66 [52] I. Shavitt R. M. Pitzer M. Dallos Th. M¨ uller P. G. Szalay F. B. Brown R. Ahlrichs H. J. B¨ ohm A. Chang D. C. Comeau R. Gdanitz H. Dachsel C. Ehrhardt M. Ernzer- hof P. H¨ ochtl S. Irle G. Kedziora T. Kovar V . Parasuk M. J. M. Pepper P. Scharf H. Schiffer M. Schindler M. Sch¨ uler M. Seth E. A. Stahlberg J.-G. Zhao S. Yabushita Z. Zhang M. Barbatti S. Matsika M. Schuurmann D. R. Yarkony S. R. Brozell E. V . Beck H. Lischka, R. Shepard, B. Sellner F. Plasser J.-P. Blaudeau, M. Rucken- bauer, and J. J. Szymczak. COLUMBUS, an ab initio electronic structure program, release 7.0, 2012. [53] H. Lischka, R. Shepard, R.M. Pitzer, I. Shavitt, M. Dallos, Th. M¨ uller, P.G. Szalay, M. Seth, G.S. Kedziora, S. Yabushita, and Z. Zhang. High-level multireference methods in the quantum-chemistry program system COLUMBUS: Analytic MR- CISD and MR-AQCC gradients and MR-AQCC-LRT for excited states, GUGA spin-orbit CI and parallel CI density. Phys. Chem. Chem. Phys., 3:664–673, 2001. [54] A.I. Krylov and P.M.W. Gill. Q-Chem: An engine for innovation. WIREs Comput. Mol. Sci., 3:317–326, 2013. [55] Shao, Y .; Gan, Z.; Epifanovsky, E.; Gilbert, A.T.B.; Wormit, M.; Kussmann, J.; Lange, A.W.; Behn, A.; Deng, J.; Feng, X., et al. Advances in molecular quantum chemistry contained in the Q-Chem 4 program package. Mol. Phys., 113:184–215, 2015. [56] A.F. Izmaylov, D. Mendive-Tapia, M.J. Bearpark, M.A. Robb, J.C. Tully, and M.J. Frisch. Nonequilibrium Fermi golden rule for electronic transitions through coni- cal intersections. J. Chem. Phys., 135:234106, 2011. [57] M.H. Lee, B.D. Dunietz, and E. Geva. Calculation from first principles of in- tramolecular golden-rule rate constants for photo-induced electron transfer in molecular donor-acceptor systems. J. Phys. Chem. C, 117:23391–23401, 2013. 67 Chapter 4: Quantifying charge resonance and multiexciton character in coupled chromophores by charge and spin cumulant analysis 4.1 Introduction To derive physical insight from multi-configurational correlated wave functions, one needs to be able to condense the information contained in multi-dimensional wave func- tion amplitudes into a compact form, such that the essential features of electronic struc- ture become apparent. This can be achieved, for example, by analyzing reduced density matrices (DMs). 1–4 In the context of excited states, transition DMs or difference of the respective state DMs can be utilized (see, for example, Refs. 5, 6 for recent reviews of various DM-based approaches). The state DMs are used to compute physical properties, such as observ- ables corresponding to various operators, e.g., dipole moment, spin, spatial extend of the density. 4 The transition DMs are also related to physical observables, such as oscillator strengths, absorption cross sections, and electronic couplings between the states. 4 One 68 can also introduce operators that do not correspond to physical observables, but relate to important chemical concepts, such as atomic charges, local spin, bond orders, resonance structures, effective number of unpaired electrons, particle-hole distance (exciton delo- calization), etc. 7–12 The respective expectation values allow one to dissect the character of underlying wave functions; they provide valuable interpretation tools. Excited-state character is often assigned by identifying the leading wave function amplitude, e.g., the largest coefficient in the CIS (configuration interaction singles), TDDFT (time-dependent density functional theory), ADC (algebraic diagrammatic con- struction), or EOM-CC (equation-of-motion coupled-cluster) wave function, and ana- lyzing the respective molecular orbitals (MOs). 6, 13–15 Unless special steps are taken, these MOs are, most often, delocalized canonical Hartree-Fock orbitals. Obviously, such analysis is only limited to the case when the excited state is dominated by a tran- sition of a single electron and neglects correlation effects (such as contributions from double excitations in EOM-CCSD). The assignment becomes problematic when two or more configurations appear with comparable weights. For example, a linear combina- tion of just two excitations involving two target MOs of a mixed Rydberg-valence char- acter may give rise to either pure Rydberg, or pure valence, or a mixed Rydberg-valence state. 14 Likewise, the separation of covalent versus charge-resonance (CR) contributions is not straightforward. Furthermore, such MO-based analysis is not orbital invariant and may lead to basis-dependent artifacts. Several overlapping approaches addressing these issues have been intro- duced, 5, 13, 16–22 see Refs. 5, 6 for recent reviews. Historically, the first usage of DMs (i.e., natural orbitals) for the interpretation of excited states has been reported within the CIS framework. 23 Later, transition DMs have been exploited to analyze the localization 69 of excited states 16 and for quantifying CR. 17 Based on these ideas, excited-state struc- tural analysis (ESSA) was developed to compute fragment excitation indexes and charge transfer (CT) numbers for CIS-like wave functions. 16–18 Later, a similar approach has been reported by Plasser and co-workers. 13, 22 Recently, ESSA was extended beyond the TDA approximation to tackle the RPA and TDDFT/RPA cases. 5, 21 In addition to transition DMs, the differences of the state DMs can be utilized in a similar fashion. 16, 18 Physically, the low-lying states often have one-electron excitation character, even when they involve several interacting electronic configurations. One-electron excitation character means that no more than one electron is promoted by excitation. Formally, it means that the norm of the respective transition one-particle DM (OPDM) equals one. 24 One-electron character of the excitation does not mean a pure physical character. 14, 15 For example, an excited state which is a linear combination of n! and ! excitations is still a singly excited state. 15 Likewise, states of a mixed Rydberg-valence character are singly excited. 14 Plasmonic excitations, that are referred to as “collective excitations”, are singly excited states; their collective character refers to the fact that the respective wave functions are linear combination of many singly excited determi- nants. 25–27 Similarly, delocalized excitations in molecular aggregates (such as excitonic bands in molecular solids and polymers) are singly excited states. 28–30 Regardless of the correlation level employed, the analysis of such singly excited states is straightforward. One simply needs to invoke ESSA (or a similar analysis) of OPDMs computed using underlying wave functions. Several recent applications of this technique are given in Refs. 31–33 Technically, this approach is exploited in, for example, Q-Chem 34 where the same wave function analysis module based on Refs. 6, 22, 31, 35 tack- les OPDMs computed from the CIS, TDDFT, ADC, 36 and EOM-CC 15 wave functions. This analysis can characterize the degree of CT and CR in excited states, an effective 70 number of configurations involved in an excitation, exciton delocalization (in terms of average particle-hole distance), and so on. However, such analyses are not applicable to the states of a doubly excited character. For these states, the initial and final states differ by the states of two electrons. Conse- quently, for purely doubly excited states, the norm of transition OPDM is zero. Even when such states have some singly excited character (giving rise to a non-zero norm of OPDM), the information about doubly excited part of the wave function is not explicitly present in OPDM. Thus, it is desirable to extend ESSA to quantify the character of dou- bly excited states. Moreover, in some cases one may wish to describe the character of a particular state on its own, and not relative to another state. For example, in strongly correlated systems, such as molecular magnets, the ground state may develop a rather complicated character and it may be desirable to analyze its wave function in terms of dominant configurations of various physical types. To achieve these goals, here we present an extension of ESSA aiming to quantify electronic structure of complex wave functions based on reduced state DMs rather than transition and difference DMs. States of doubly excited character are well known in molecules, semiconductors, and nanostructures. In one-photon spectroscopy, they are optically dark, unless, of course, they are mixed with singly excited states. They can be accessed by a two-photon ex- citation or via radiationless relaxation from higher excited states. Our interest in these states was sparked by singlet fission, 37, 38 a process that may improve the efficiency of solar cells by generating more than two charge carriers per each absorbed photon. This enhancement is achieved by converting the initially excited bright state (a single sin- glet exciton) into a dark state that can be described as two singlet-coupled triplet states (hence, triplet biexciton) localized on two adjacent chromophores. A reverse process, 71 triplet-triplet annihilation, is also of interest in the context of non-linear materials for frequency up-conversion. 39 (a) A B (b) (c) (d) hA lA lB hB |AB> |A * B>, |AB * > |A + B - >, |A - B + > |A * B * > Monday, April 13, 2015 Figure 4.1: Relevant M s =0 electronic configurations of the AB dimer expressed using molecular orbitals localized on individual moieties. h A and h B denote the HOMO localized on A and B, respectively,l A andl B denote the respective LUMOs. (a) Ground state, S 0 (A)S 0 (B). (b) Localized singly excited configurations (LE) giving rise to singlet and triplet excitonic states,c 1 S 0 (A)S 1 (B) +c 2 S 1 (A)S 0 (B) and c 1 S 0 (A)T 1 (B) +c 2 T 1 (A)S 0 (B). (c) Charge-resonance (CR) configurations,A + B and A B + . (d) Configurations giving rise to the multiexciton (ME) manifolds, 1;3;5 T 1 (A)T 1 (B) and 1 S 1 (A)S 1 (B). The salient aspects of electronic structure of singlet fission 24, 40 and triplet-triplet annihilation 39 are summarized in Fig. 4.1. The initially excited state (i.e., the lowest bright singlet state) can be described as a linear combination of the local excitations (LE) of individual fragments plus some CR configurations. Following Refs., 24, 40 we de- note these states as S 1 (AB) and S 1 0(AB). Asymptotically, these states correlate to the S 1 states of the monomers. The ME states are dominated by doubly excited (with respect to the closed-shell ground state) configurations. In the ME manifold, 1 ME state of the TT character (two singlet-coupled triplet states) is the most important one, as it can couple 72 to the initially excited singlet state, S 1 (AB) or S 1 0(AB) (note that EX and ME configura- tions cannot be coupled by a one-electron operator, consequently, it is the admixture of CR configurations in the adiabatic S 1 (AB) and ME states that facilitates the coupling). As one can see, the key state in singlet fission, the 1 ME state, has dominant doubly excited character. Consequently, this state cannot be described by standard electronic structure methods, such as CIS, TDDFT, or even EOM-EE-CCSD. A practical approach to this sort of electronic structure is based on the spin-flip method. 41–43 In particular, this state (as well as other states from Fig. 4.1) can be described by double spin-flip ansatz starting from a high-spin quintet reference. 44, 45 The RASCI-2SF method has been successfully employed to model various aspects of singlet fission. 40, 46–50 In this paper, we present a new method based on charge and spin cumulants that allow one to unambiguously identify ME states in weakly coupled chromophores and to quantify their character, e.g., the mixing of pure ME and CR configurations. The latter play a crucial role in controlling the magnitude of electronic couplings between the initially excited bright state and the ME state, 24, 40 which facilitates a crucial initial step in singlet fission. These tools can also be used for molecular magnets. 51 The method is based on state DMs rather than on transition DMs. The structure of the paper is as follows. The next section describes relevant elec- tronic states of a bichromophore. Section 4.3 introduces charge cumulants character- izing electronic charge correlations. In Section 4.4 charge cumulants are employed to estimate charge transfer and charge resonance contributions in dimer wave functions. Section 4.5 presents the spin-cumulant technique and spin-component analysis for dou- bly excited states. Section 4.6 supplements the analysis by the local excitation indexes, thus presenting a complete analysis scheme for the correlated dimer states of mixed 73 character. Section 7.3 presents numeric examples. Our final remarks are given in Sec- tion 4.8. Appendices present derivations of programmable expressions, as well as an algorithm for computing charge and spin cumulants for general CI wave functions and an orbital fragment localization algorithm based on Mulliken charges. 4.2 Electronic states of molecular dimers Our approach is based on super-molecular wave functions. That is, we first calculate adiabatic wave functions of a dimer and then analyze it in terms of the wave functions (or properties) of individual chromophores. This should be contrasted to excitonic-model type of approaches in which the dimer states are constructed from the wave functions of individual fragments (see, for example, Refs. 52–54 ). Here we describe electronic wave functions of weakly coupled chromophores in terms of the electronic states of the individual moieties. In this case, the fragment’s wave functions can be made strongly orthogonal to each other. In the case of molecu- lar dimers, the separation between the two fragments is straightforward. In covalently linked chromophores, one needs to define a separation between the two units using physical considerations. In the discussion below, we employ the representation using orthonormal MOs localized on the individual chromophores. In this representation, the configurations corresponding to different fragment states are orthogonal. Orbitals can be localized after the SCF step, prior to correlated calculations. In practical calculations, delocalized canonical MOs are most commonly used, so the orbitals can be localized a posteriori and the resulting transformation can be applied to the wave function am- plitudes and DMs transforming them into a localized representation. For the methods such as CIS, TDDFT, ADC, EOM-CC that are invariant with respect to orbital rotations within either an occupied or virtual subspace (occupied and virtual orbital spaces are 74 defined by the choice of the reference determinant 0 ), such transformation does not affect energies or observables of the respective states. Fig. 4.1 shows relevant electronic configurations of a molecular dimer in terms of frontier molecular orbitals (HOMO and LUMO) of the individual monomers,A andB. The ground-state wave function of the dimer is an antisymmetrized product of the two ground-state wave functions,S 0 (A)S 0 (B)jABi. In the ground state, the number of electrons in each fragment isN A andN B , and the total number of electrons in the dimer is: N = N A +N B . Panel (b) shows localized single excitations (LE)S 0 (A)S 1 (B) jAB i andS 1 (A)S 0 (B)jA Bi; they give rise to excitonic states, S 1 (AB) and S 1 0(AB) using the notations from Refs. 40 Collectively, these configurations can be denoted as j LE i: j LE i =c A jA Bi +c B jAB i (4.1) In real many-electron adiabatic states, these excitonic configurations mix with charge- resonance (CR) configurations, A + B and A B + shown in panel (c). Note that the presence of the CR configurations (often misleadingly referred to as charge transfer) does not imply a permanent charge separation in the dimer; rather, these configurations describe the ionic character of underlying wave functions. The collective contribution of the CR configurations can be denoted asj CR i: j CR i =t AB jA + B i +t BA jA B + i (4.2) Finally, panel (d) shows electronic configurations giving rise to multiexciton (ME) states that can be described as two coupled excited states of the monomers. Here we are concerned about the state that can be described as two triplets coupled to a singlet state, 1 T 1 (A)T 1 (B). The same determinants give rise to 1 S 1 (A)S 1 (B), which, in principle, 75 can be coupled with other singlet states including the 1 TT biexciton. The respective contribution of such configurations to the total wave function can be denoted asj ME i: j ME i =d AB jA B i (4.3) As evident from Fig. 4.1, the shorthand notations employed here hide multi-determinant nature of these states which arises due to spin-adaptation. For example,j ME i com- prises all 4 determinants from row (d) in Fig. 4.1, with the weights determined by the spin-coupling rules. Using the above notations and assuming strong orthogonality between fragments’ MOs, a general adiabatic wave function of the dimer can be written as: j ij LE i +j CR i +j ME i = c A jA Bi +c B jAB i +t AB jA + B i +t BA jA B + i +d AB jA B i (4.4) Here we neglected the contributions of multiple CR terms (such asjA n+ B n i) and higher than double excitations. Within this approximation, the respective coefficients are normalized to 1: jc A j 2 +jc B j 2 +jt AB j 2 +jt BA j 2 +jd AB j 2 = 1 (4.5) such that the respective values can be used to quantify the weights of particular contri- butions in the total wave function, e.g.,jc A j 2 +jc B j 2 gives the total LE character. In the case of a non-Hermitian theory such as EOM-CC, the states are bi-orthonormal; thus the weights should be defined as products of the respective left and right amplitudes. 15, 55, 56 76 4.3 Charge cumulant indexes The cumulant of two operators, ^ U and ^ V , is defined as: UV h j ^ U ^ Vj ih j ^ Uj jih j ^ Vj i =h ^ U ^ Vih ^ Uih ^ Vi; (4.6) whereh ^ Vi denotes an expectation value of operator ^ V using a state wave function of interest, e.g., Eq. (4.4). By rewriting the above equation as follows: UV =h( ^ Uh ^ Ui)( ^ Vh ^ Vi)i (4.7) it becomes clear that the cumulant is non zero when the fluctuations of ^ U and ^ V (de- viations from the average values) are correlated. For example, the charge cumulants introduced below describe correlated (in a statistical sense) fluctuations of electrons. We begin by reviewing the definition and properties of charge cumulants. A special two-electron charge density had first appeared in the classic work of Ruedenberg 57 as the generalized exchange density x . Later, the corresponding cumulant indexes (often called the generalized bond order indexes) have become exploited in the interpretation and analysis of electronic structure calculations (see details in Refs. 58–62 ). Here we employ a localized orthonormal MO basis set, which can be obtained from the canonical orbitals using standard localization techniques (occupied and virtual or- bitals need to be localized separately, such that the separation between the occupied and virtual subspaces is preserved; 63 this restriction arises only because in our analysis 77 we use wave function amplitudes, in addition to DMs). The full basis comprises the orthogonal MOs localized on fragments A and B: f r gff A p g;f B q gg (4.8) We note that all definitions below can be formulated using atomic orbitals (AOs), such as symmetrically orthogonalized AOs that are used in the definition of L¨ owdin charges and local spin operators. 64 For DM-based calculations, such as spin and charge cumulants, using orthogonalized AOs is straightforward as the respective indexes are orbital-invariant. However, for calculations that require explicit wave functions, it is more convenient to employ localized MOs that preserve the invariance of the wave function (such as localization in each RAS subspace for RAS-CI or localization within occupied and virtual subspaces in EOM-CC) since in this case one does not need to invoke appropriate projectors for the wave function amplitudes. We now can define the local number operators: ^ N A = X p2A p + p ^ N B = X p2B p + p; (4.9) where p + and p denote the creation and annihilation operators corresponding to spin orbital p . These number operators are just the local charge operators that can be used to compute the total number of electrons on individual fragments in a given electronic state: q A =h ^ N A i and q B =h ^ N B i: (4.10) 78 We also defineN A;B , the number of electrons on the respective fragments in the refer- ence state. The charge operators satisfy the completeness relationship: ^ N A + ^ N B = X p p + p = ^ N (4.11) Consequently: N A +N B =q A +q B (4.12) Using these quantities, the total change in charge (relative to the reference state) on the individual fragments: q X =N X q X ; X =A;B (4.13) q A =q B (4.14) One can quantify the charge transfer between the fragments by: AB 1 2 (q A q B ) = N A N B 2 q A q B 2 =N A q A ; (4.15) which will be utilized in section 4.4. Using the number operators from Eq. (4.9), one can define charge cumulant AB (A6=B) as: AB h ^ N A ^ N B ih ^ N A ih ^ N B i: (4.16) As a fluctuation index, AB was introduced in Ref.58. The definition of the diagonal cumulant index, AA is slightly different: AA h( ^ N A ) 2 ih ^ N A i 2 h ^ N A i: (4.17) 79 As follows from Eq. (4.7), AB describes correlated (in a statistical sense) fluctua- tions of atomic charges. For a pure CR state, AB =-1. Since in atoms connected by a covalent bond, the decrease of density on one atom involves the increase of the density on another one, - AB can be interpreted as a bond index. 58, 62 The physical significance of such definition becomes clear when one considers an asymptotic behavior of the elec- trostatic part of the diatomic molecular Hamiltonian — at large interatomic separations, the attractive interaction between the pair of atoms assumes the following form 62 (see also Ref.57, Eq. (6.41)): ^ H AB Z A Z B R AB + 1 R AB ( ^ N A q A )( ^ N B q B ) (4.18) Using the above charge cumulant definitions, we can now define full charge- cumulant matrix: = 0 B B @ AA AB AB BB 1 C C A (4.19) AA / BB and AB are not independent but: AA + AB =q A BB + AB =q B (4.20) These sum rules can be easily verified by using Eqns. (4.10) and (4.11). For example, AA + AB =h ^ N A ( ^ N A + ^ N B )ih ^ N A i(h ^ N A i +h ^ N B i)h ^ N A i = h ^ N A ^ Nih ^ N A ih ^ Nih ^ N A i =h ^ N A i =q A (4.21) 80 where we used ^ Nj i =Nj i. More general cumulant identities can be found in Ref.61. In Section 4.4, we show that the charge cumulant matrices can be used to compute the CT numbers and the weights of CR configurations. Since ^ N A ^ N B is a two-electron operator, its calculation requires two-particle DM (TPDM): h ^ N A ^ N B i = X p2A;q2B hp + pq + qi = X p2A;q2B pqpq ; (4.22) where pqrs =hp + q + sri (4.23) In Appendix A for Ref.65, we derive the contributions to this expectation value from the separable part of TPDM and show that a non-separable part of TPDM is required to obtain excited-state CR contributions in dimers whose ground state is localized and sep- arable. Appendix B for Ref.65 presents a strategy for calculating the-matrix without an explicit evaluation of TPDM (see also Ref.61). The expressions for general CI wave functions are given in Appendix C for Ref.65. 4.4 Using charge cumulants to quantify charge reso- nance Local charge operators can be used to compute the number of electrons on individual fragments, as specified by Eq. (4.10). In a localized basis, OPDM can be represented as follows: pq = 0 B B @ AA pq AB pq BA pq BB pq 1 C C A (4.24) 81 Using the normalization condition, Tr[ ] = N, one can easily show that for an asymptotically separated dimer in a state when there is no CR between the fragments, q A = N A andq B = N B . If the net amount of AB electrons has transferred from A to B , then: q A =N A AB and q B =N B + AB (4.25) AB should be related to the coefficients on the respective configurations from Eq. (4.2). In the case of symmetric dimers,t AB =t BA ; thus, even for a pure CR configuration, j CR i, AB = 0. Thus, the state OPDM alone is insufficient for quantifying CR. In the case of perfectly symmetric dimers, this problem can be circumvented by using DMO- LCFMO (dimer molecular orbital—linear combination of fragment molecular orbitals) framework, 66 as was done in Ref.40. A more general approach 17, 18 quantifies the CR character in CIS wave functions by using the so-called CT numbers (see also Ref. 21 ). For singly excited states, the CR numbers can be computed from transition OPDMs. 5, 6 Here we define the CT numbers for a general dimer wave functions given by Eq. (4.4). Instead of transition DMs, we employ state DMs (OPDM and a sub-block of TPDM). To do so, let us examine the action of a local charge operators, Eq. (4.9), onj CR i in which the the coefficients are normalized:jt AB j 2 +jt BA j 2 = 1. Using: ^ N A jA + i = (N A 1)jA + i (4.26) 82 we obtain: ^ N A CR = (N A 1)t AB jA + B i + (N A + 1)t BA jA B + i ^ N B CR = (N B + 1)t AB jA + B i + (N B 1)t BA jA B + i: (4.27) From these, the fragments populations are: q A = (N A 1)jt AB j 2 + (N A + 1)jt BA j 2 =N A jt AB j 2 +jt BA j 2 (4.28) Likewise,q B =N B +jt AB j 2 jt BA j 2 . We can then compute AB defined by Eq. (4.15) as: AB =jt AB j 2 jt BA j 2 (4.29) We now employ charge cumulants to quantify the degree of CR, which is given by the respective weights of CR configurations. Thus, we define CR numbers as: w CR A!B jt AB j 2 ; w CR B!A jt BA j 2 ; (4.30) Using these numbers: w CR A!B w CR B!A = AB (4.31) and the total CR weight is then: w CR w CR A!B +w CR B!A (4.32) Consequently, the individual CR contributions can be computed as follows: w CR A!B = w CR AB 2 and w CR B!A = w CR + AB 2 (4.33) 83 We now introduce the expression forw CR : w CR = ( AB ) 2 AB (4.34) This important relation follows from Eqns. (4.16), (4.27) and definitions (4.30)-(4.32). Thus, Eqns. (4.33) and (4.34) provide a direct way for estimating CR weights. By using the result from Appendix A for Ref.65, we obtain: w CR = 1 4 (N A N B ) 2 + 1 4 (q A q B )(N B N A q B ) + X p2A;q2B ~ ABAB pqpq ; (4.35) where ~ denotes a non-separable part of TPDM. For the identical fragments (N A =N B ) we obtain: w CR = 1 4 (q 2 B q A q B ) + X p2A;q2B ~ ABAB pqpq (4.36) Thus, we can see that for symmetric homodimers (q A = q B ) it is only the contribution from ~ that gives rise to non-zero CR. We note that Eqns. (4.33) and (4.15) are valid for a general wave function, Eq. (4.4), since the LE configurations do not contribute to the CR or CT indexes. Finally, it is instructive to consider matrices for the two limiting cases, a pure LE state and a pure CR state in the case of a symmetric dimer: (LE) = 0 B B @ N 2 0 0 N 2 1 C C A (4.37) (CR) = 0 B B @ N 2 + 1 1 1 N 2 + 1 1 C C A (4.38) 84 Thus, unlike local charges defined by charge operators, the cumulant matrices provide a clear-cut distinction between these two different types of electronic structure. 4.5 Spin correlators for charge resonance and biexci- tons Further information about the nature of an electronic state is provided by spin cumulants called spin correlators. Spin correlators for molecules had made their first appearance in Ref.67 where the Penney-Dirac bond order was introduced. Later, they have been applied to large conjugated systems and polymers. 68 Local spin operators and spin cor- relators have been used in the context of diradicals and molecular magnets. 64, 69, 70 We will follow the technique described in Appendix C for Ref.65 (see also Refs. 71, 72 ). Spin correlators enable the calculation of the weight of the multiexciton component, j ME i (more precisely, its TT-contribution to Eq. (4.4)). Here we need to distinguish between the two spin types of the overall singlet doubly excited state, Eq. (4.3). One set of configurations may be formed from the monomer excitations producing the double excitation of an SS-type such that the two monomers are in their singlet excited states. These configurations would asymptotically correlate with the S 1 (A)S 1 (B) state; thus, their energies are expected to be much higher than the contributions of a TT type which correlates to T 1 (A)T 1 (B). In the TT configuration, the two monomers are in their triplet excited states. Thus, we refine Eq. (4.3) as follows: j ME i =j SS i +j TT i; (4.39) 85 where j SS id SS AB jA S=0 B S=0 i (4.40) j TT id TT AB jA S=1 B S=1 i (4.41) As discussed in the Introduction, our interest here is in aj TT i-type of multi-exciton state. Since there is no inter-fragment spin coupling inj LE i andj SS i configurations, we compute inter-fragment spin correlators for thej CR i +j TT i part of the wave function. Using spin-adaptation rules: jA + B i = 1 p 2 jA + " B # ijA + # B " i (4.42) and jA S=1 B S=1 i = 1 p 3 jA "" B ## i +jA ## B "" ijA S=1 "# B S=1 "# i ; (4.43) where the last term denotes the total of four Slater determinants coupled into theM s = 0 local triplet excitations. For the dimer singlet states, it is sufficient to define only a z-component of spin correlator matrixZ Spin : Z Spin = 0 B B @ Z AA Z AB Z AB Z BB 1 C C A (4.44) 86 where the individual spin correlators are: Z AA =h( ^ S A z ) 2 i Z BB =h( ^ S B z ) 2 i Z AB =h ^ S A z ^ S B z i (4.45) Here ^ S A;B z are local spin operators acting on fragmentsA andB, respectively, e.g.: ^ S A z = 1 2 X p2A p + " p " p + # p # = 1 2 ^ N A ^ N A (4.46) In the case of the dimer singlet states, the elementary sum rules are valid, 33 e.g.,Z AA + Z AB = 0. Thus,Z Spin has the following form: Z Spin =Z AA 0 B B @ 1 1 1 1 1 C C A (4.47) We can now compute individual terms from Eq. (4.45) forj CR i +j TT i. We find thatZ AA = 1 4 and 2 3 forj CR i andj TT i, respectively. Thus, for the total wave function, Eq. (4.4), we have: Z AA = w CR 4 + 2jd TT AB j 2 3 (4.48) Let us define the TT weight as: w TT jd TT AB j 2 (4.49) Then: w TT = 3 Z AA 2 w CR 8 ; (4.50) 87 wherew CR is defined by Eq. (4.32). Eq. (4.50) allows us to identify the TT state and to quantify the weight of the ME configurations in the total wave function. 4.6 Local excitations We now proceed to estimate the remaining weights. The local excitation weights are defined as: w LE A;B jc A;B j 2 (4.51) and the weight of double SS configurations is: w SS AB =jd SS AB j 2 (4.52) From the normalization condition: w LE A +w LE B +w SS AB = 1w CR w TT AB (4.53) If the weight of SS configurations is small (as expected for the states relevant for singlet fission), then the total LE weight (w A +w B ) can be computed fromw CR andw TT AB by using the normalization requirement, Eq. (4.5). In a more general case, the total weight of LE configurations can be estimated by transforming the wave function amplitudes into a localized basis and summing up the squares of all amplitudes localized on individual fragments. For example, for a wave function with single and double excitations: w LE A = X ia2A jr a i j 2 + 1 4 X ijab2A jr ab ij j 2 (4.54) 88 Again, in the case of a non-Hermitian theory, such as EOM-CCSD, products of left and right amplitudes should be computed. 15, 55, 56 Thus,w SS can be computed as: w SS AB = 1w LE A w LE B w CR A!B w CR B!A w TT AB (4.55) 4.7 Results and discussion We now have a full set of indexes needed to characterize dimers’ excited states: fw LE A ;w LE B ;w CR A!B ;w CR B!A ;w SS AB ;w TT AB g: Table 4.1 summarizes the equations for calculating these quantities and how they relate to the coefficients of the wave function, Eqns. (4.4) and (4.39). The calculations ofw LE requires the explicit wave function amplitudes expressed in a fragment-localized basis; all other indexes can be computed from the reduced state DMs. In derivations of Eqns. (4.4) and (4.5), it is assumed that the ground-state wave function,jABi, is separable and can be represented by an anti-symmetrized product of general many-electron wave functions of the fragments. Thus, the configurations in Eq. (4.4) represent local and CR excitations with respect to the separable ground- state wave function. The calculations of quantities defined by DMs do not depend on the exact structure of the ground-state wave function. However, in calculations using wave function amplitudes, Eq. (4.54), in the present paper we assume that the ground state,jABi, is dominated by a single Slater determinant. When this is not the case and the ground state wave function features significant correlation effects (such as large contributions of excited determinants), one also needs to consider the mixing of the reference determinant into other states (the respective weight can be denoted byw 0 ). In 89 principle, the calculation can be formulated in a more general way, that is, assuming that jABi represent a general separable wave function of the dimer. In the case of EOM- EE, such formulation is straightforward — one simply needs to consider only the EOM amplitudes when computing w LE , however, for RAS-CI or EOM-SF additional steps are necessary. 90 Table 4.1: Expressions for computing weights of local excitation, charge resonance, and biexcitonic contributions. Quantity Property How to compute Relationship to Eq. (4.4) w LE A Weight of LE on A P ia2A jr a i j 2 + 1 4 P ijab2A jr ab ij j 2 jc A j 2 w LE B Weight of LE on B P ia2B jr a i j 2 + 1 4 P ijab2B jr ab ij j 2 jc B j 2 AB Net CT between A and B N A q A jt AB j 2 jt BA j 2 w CR Total weight of CR terms ( AB ) 2 AB jt AB j 2 +jt BA j 2 w CR A!B Weight ofB!A CR w CR AB 2 jt AB j 2 w CR B!A Weight ofA!B CR w CR + AB 2 jt BA j 2 w TT AB Weight of 1 TT ME terms 3( Z AA 2 w CR 8 ) jd TT AB j 2 w SS AB Weight of 1 SS ME terms w SS AB = 1w LE w CR w TT AB jd SS AB j 2 91 We note that the LE indexes, w LE A;B , and CR numbers, w CR X!Y , are similar to the ESSA excitation indexes, l A;B and the CT numbers, l X!Y . While in the case of CIS, they give identical answers, they can be computed for any correlated wave functions and not only for singly excited states. The bi-excitonic indexes, w SS AB and w TT AB , are introduced in this work and do not have a counterpart in ESSA. We note that these indexes cannot be computed for CIS/TDDFT wave function; furthermore, they cannot be defined from transition OPDM. Below we illustrate the utility of the extended ESSA scheme by considering low- lying excited states of several model systems, such as H 2 -He, sandwich and T-shaped (H 2 ) 2 structures, ethylene, as well as tetracene, DPH (1,6-diphenyl-1,3,5-hexatriene), and DPBF (1,3-diphenylisobenzofuran) dimers. 4.7.1 Computational details The calculations for small dimers were performed using full configuration interaction (FCI). For other systems, the RAS-CI approach was used. All calculations were per- formed using production-level RAS-SF code in Q-Chem. 34, 73 In H 2 -He and (H 2 ) 2 calculations, r HH =0.71144 ˚ A. In H 2 -He (C 1v structure), the He atom is located at 3.33 ˚ A from the HH bond midpoint. In the parallel (H 2 ) 2 structure (D 2h symmetry), the monomers are 3.00 ˚ A apart. In the T-shaped structure (C 2v ), the distance between the bond midpoints of the two monomers is 3.49 ˚ A. Ethylene dimer calculations were performed using the basis set and the geometries from Ref.24. We considered a perfectly stacked sandwich (D 2h structure) as well as two displaced (C 2h ) structures — one along the CC bond (X-displacement) and one — in a 92 perpendicular direction (Y -displacement). We also included a structure with two non- equivalent fragments in which one of the ethylene molecules is rotated along the long axis by 90 (C 2v ). Tetracene dimer calculations were performed using the selected geometries from Ref.40. DPH and DPBF dimers calculations were performed using the geometries from Ref.47. The calculations of tetracene, DPH, and DPBF dimers employed a mixed basis set: 40 the cc-pVTZ basis with f-functions removed for C/O and the cc-pVDZ basis for H. Fig. 4.2 shows selected structures. Figure 4.2: Structures of selected dimers. (a) Perfectly stacked ethylene dimer (D 2h ). (b) Tetracene dimer from the X-ray structure. (c) DPH dimer featuring the largest coupling 47 (dimer 1 from the orthorhombic form). (d) DPBF dimer that has the largest ME-S 1 couplings 47 (dimer3 from the form). The calculations for ethylene, tetracene, DPH, and DPBF were performed with RAS(4,4)-2SF using a high-spin ROHF quintet reference. We note that due to the lack of dynamic correlation, RAS-2SF energies of the excitonic singlet dimer states, S 1 (AB) and S 1 0(AB), are overestimated relative to the ME state. The state ordering can be cor- rected by using a simple correction, as was done in Refs. 40, 46, 47 Here we report raw, uncorrected RAS-2SF energies. The state composition is not affected by the correc- tion. The RAS wave function was computed using orthogonal fragment orbitals. RAS1, RAS2 and RAS3 spaces were localized separately by maximizing the Mulliken charges 93 on each fragment (see Appendix D for Ref.65). To univocally label the fragment or- bitals within the RAS2 space as occupied or virtual, we split the ROHF frontier singly occupied orbitals defining RAS2 into the two pairs with lower and higher energies and localize them separately. Core electron excitations and transitions to the highest virtual orbitals (72, 72, and 84 in tetracene, DPH, and DPBF dimers, respectively) were not in- cluded in the RAS-2SF calculations of tetracene, DPH, and DPBF dimers. We note that for all examples,w CR computed using the charge cumulants as specified by Eq. (4.34) are within 0.5 % from the values computed from the RAS-SF wave function amplitudes. 4.7.2 Numeric examples Table 4.2 shows the results for two small model systems, H 2 -He and (H 2 ) 2 . For the latter, we consider two structures, symmetric sandwich and a T-shaped one in which the two fragments are not equivalent. As one can see, the two lowest states and the fourth state in H 2 -He are the excited states localized on H 2 . The third state corresponds to the He + !H 2 charge-transfer state. In the T-shaped H 2 dimer, we observe three LE states (the lowest one is localized on the top, and the second and third one — on the handle moiety). The contribution of CR configurations in these states is low. Because the two monomers are not equivalent, the two CR states correspond to CT between the two fragments: (A 1 )=0.986 and (B 2 )=0.983. Consequently, these two states also have large dipole moments — 5.79 a.u. and 6.38 a.u., respectively. The dipole moments in the LE states are much smaller, i.e., 0.02 a.u. for the two lowest LE states and 0.41 a.u. for the third LE state. Finally, the two highest states are the ME states. As expected, the TT state is considerably lower 94 Table 4.2: Analysis of low-lying singlet excited states in H 2 -He and (H 2 ) 2 using FCI/cc-pVDZ wave functions. E ex and f l denote excitation energy and oscillator strength, respectively. System State Char E ex , eV (f l ) w LE A w LE B w CR A!B w CR B!A w SS AB w TT AB H 2 -He a + LE 13.96 (0.53) 0.99 0.00 0.00 0.00 0.01 0.00 + LE 21.41 0.99 0.00 0.00 0.00 0.01 0.00 + CR 24.72 0.00 0.01 0.00 0.99 0.00 0.00 + LE 29.41 0.98 0.00 0.01 0.00 0.01 0.00 (H 2 ) 2 b T-shaped B 2 LE 13.91 (0.51) 0.98 0.00 0.00 0.00 0.02 0.00 A 1 LE 13.94 (0.55) 0.00 0.98 0.00 0.00 0.02 0.00 A 1 CR 16.83 (0.02) 0.01 0.00 0.98 0.00 0.00 0.00 B 2 CR 17.28 0.01 0.00 0.00 0.98 0.00 0.01 A 1 LE 21.16 0.01 0.89 0.00 0.08 0.02 0.00 A 1 TT 21.42 0.00 0.00 0.00 0.03 0.00 0.97 B 1 SS 27.57 0.00 0.00 0.00 0.00 0.98 0.01 (H 2 ) 2 parallel B 1g LE 13.47 0.45 0.45 0.04 0.04 0.01 0.00 B 2u LE 14.07 (1.00) 0.46 0.46 0.02 0.02 0.02 0.00 B 3u CR 17.02 (0.04) 0.02 0.02 0.48 0.48 0.00 0.00 B 1g CR 17.21 0.04 0.04 0.46 0.46 0.00 0.00 A g TT 21.32 0.04 0.04 0.01 0.01 0.00 0.90 A g SS 27.55 0.02 0.02 0.15 0.15 0.66 0.01 a LabelsA andB correspond to the H 2 and He fragments, respectively. b LabelsA and B mark the top and the handle monomer, respectively. in energy (about 7 eV) than the SS state; it also includes 3% of the CR configurations. The dipole moment of the 1 TT state is 0.11 a.u. In the symmetric (H 2 ) 2 , the lowest two states are dark and bright excitonic LE pair with about 4% of CR configurations. Two CR states are considerably higher in energy. 95 Due to symmetry, there is no net CT in these states; the total dipole moment in these states is zero. The upper CR state includes 8% of LE character. Finally, there are two ME states. The TT state includes 8 % of LE character and 2 % of CR. 96 Table 4.3: Analysis of the S 1 and ME states in ethylene dimer using the RAS(4,4)-2SF/6-31G(d) wave functions.E ex and f l denote excitation energy and oscillator strength, respectively. Structure State Character E ex , eV (f l ) w LE A w LE B w CR A!B w CR B!A w SS AB w TT AB D 2h X = Y =0 B 1g LE 9.30 0.34 0.34 0.14 0.14 0.03 0.00 A g ME 8.51 0.00 0.00 0.01 0.01 0.00 0.98 C 2h X=0.8 Y =0 A g LE 9.65 0.37 0.37 0.09 0.09 0.03 0.05 A g ME 8.50 0.01 0.01 0.03 0.03 0.00 0.93 C 2h X=0.0 Y =0.8 A g LE 9.50 0.37 0.37 0.12 0.12 0.03 0.00 A g ME 8.55 0.00 0.00 0.01 0.01 0.00 0.99 C 2h X=0.0 Y =2.0 A g LE 10.05 0.45 0.45 0.03 0.03 0.04 0.00 A g ME 8.65 0.00 0.00 0.00 0.00 0.00 1.00 C 2v X=0.0 Y =0.0 A 1 LE 9.94 (0.02) 0.35 0.58 0.00 0.03 0.04 0.00 A 1 ME 8.67 0.00 0.00 0.00 0.00 0.00 1.00 97 Table 4.3 shows the results for ethylene dimer, a model system used to investigate the effect of structure on the electronic couplings governing the S 1 ! 1 ME transition 24, 74 as well as excitonic interactions. 75 These numbers are in a semi-quantitative agreement (within several %) with those reported in Ref.24 where a more approximate analysis was conducted. At a perfectly stacked configuration (with the fragments 3.5 ˚ A apart), the lowest singlet state is of the ME character, with the admixture of 2 % of the CR configurations. The LE state has about 28% of CR. Such states composition gives rise to a large value ofjj jj (norm of transition OPDM) and, consequently, substantial non- adiabatic coupling between S 1 and 1 ME. 24 In the structure displaced by 0.8 ˚ A along the molecule axis the ME state acquires 6% of the CR resonance character and 2 % of LE character, whereas the weight of CR configurations in the LE states drops to 18 %. The net effect onjj jj is small, it remains nearly constant. The displacements perpendicular to molecular axis result in the ME state developing a pure TT character and in a sig- nificant reduction in the CR configurations in the LE state, which is consistent with the reduced interactions between the two fragments. This leads to the decrease injj jj and non-adiabatic coupling. The last entry in Table 4.3 corresponds to a structure with two non-equivalent fragments in which the planes of the two ethylene molecules are perpen- dicular to each other. In this structure, the fragments are essentially decoupled and the S 1 and 1 ME states have their pure asymptotic character, LE and 1 TT. Our next example is a tetracene dimer. The results for the S 1 /S 1 0, S 2 /S 2 0, and the ME states are collected in Table 8.1 (the dimer states notations are as in Refs. 24, 40, 46, 47 and are described in the Introduction). At the perfectly stacked configuration with two frag- ments 3.7 ˚ A apart, we observe considerable mixing of the LE and CR configurations in the four excitonic states. The ME state includes 7% of CR. At larger separation (6.0 ˚ A), all states acquire their pure asymptotic character — LE, CR, and ME. In the structures 98 displaced along the long molecular axis, the characters of states change considerably. In the half-ring displaced structure, the ME state is pure ME, and the lowest excitonic state is almost pure LE. However, at 1-ring displaced structure, the LE, CR, and ME configurations become mixed again. In the X-ray structure, the two fragments are not equivalent. We observe that all four singly excited states share both LE and CR char- acter. The ME state includes 3% of CR. These patterns are consistent with the analysis of the wave functions and couplings between the states reported in Ref.40; however, the quantitative measures of the weight of CR is different. For example, the weight of the CR configuration in the ME state in the X-ray structure computed using the present scheme is much smaller than estimated in Ref.40. This observation again emphasizes the importance of using the robust orbital-invariant metrics quantifying the interactions between many-determinantal adiabatic states (such asjj jj) rather than relying on simple proxies such as the diabatic state composition and weights of CR configurations. 40 As the next example, we consider selected DPH dimers from Ref.47. The six struc- tures, three representative dimers from the monoclinic (M) and orthorombic (O) forms, feature very different S 1 -ME couplings, as summarized in Table 4.5. The analysis of the electronic wave functions is given in Tables 4.6 and 4.7. As we can see, the trends injj jj are consistent with changes in the composition of the ME state — larger couplings are observed for smallerw TT AB , however, the variations in the wave function appear to be quite small in magnitude. 99 Table 4.4: Analysis of the electronic states in tetracene dimer using the RAS(4,4)- 2SF/(C:cc-pVTZ-f/H:cc-pVDZ) wave functions. E ex and f l denote excitation en- ergy and oscillator strength, respectively. Structure State Char E ex , eV (f l ) w LE A w LE B w CR A!B w CR B!A w SS AB w TT AB D 2h R=3.7 ˚ A B 3g (S 1 ) LE 3.11 0.30 0.30 0.19 0.19 0.02 0.00 B 1u (S 1 0) CR 4.00 (0.34) 0.22 0.22 0.25 0.25 0.04 0.00 B 1u (S 2 ) CR 4.13 (0.38) 0.23 0.23 0.24 0.24 0.04 0.00 B 3g (S 2 0) CR 4.64 0.17 0.17 0.31 0.31 0.02 0.00 A g ( 1 ME) TT 3.13 0.00 0.00 0.03 0.03 0.00 0.93 D 2h R=6.0 ˚ A B 3g (S 1 ) LE 3.82 0.47 0.47 0.00 0.00 0.06 0.00 B 1u (S 1 0) LE 3.93 (0.73) 0.46 0.46 0.00 0.00 0.07 0.00 B 1u (S 2 ) CR 4.87 0.00 0.00 0.50 0.50 0.00 0.00 B 3g (S 2 0) CR 4.87 0.00 0.00 0.50 0.50 0.00 0.00 A g ( 1 ME) TT 3.34 0.00 0.00 0.00 0.00 0.00 1.00 C 2h x=0.5 ring B g (S 1 ) LE 3.82 0.45 0.45 0.02 0.02 0.06 0.00 A u (S 1 0) CR 4.09 (0.29) 0.18 0.18 0.30 0.30 0.04 0.01 B g (S 2 ) CR 4.18 0.02 0.02 0.47 0.47 0.00 0.01 A u (S 2 0) LE 4.27 (0.41) 0.27 0.27 0.20 0.20 0.06 0.00 A g ( 1 ME) TT 3.41 0.00 0.00 0.00 0.00 0.00 1.00 C 2h x=1 ring B g (S 1 ) LE 3.53 0.32 0.32 0.15 0.15 0.05 0.00 A u (S 1 0) LE 4.09 (0.59) 0.39 0.39 0.07 0.07 0.08 0.00 A u (S 2 ) CR 4.36 (0.12) 0.07 0.07 0.42 0.42 0.01 0.00 B g (S 2 0) CR 4.69 0.14 0.14 0.34 0.34 0.02 0.00 A g ( 1 ME) TT 3.29 0.00 0.00 0.01 0.01 0.00 0.98 X-ray a S 1 LE 4.04 (0.39) 0.70 0.04 0.02 0.16 0.06 0.01 S 1 0 LE 4.16 (0.20) 0.14 0.65 0.00 0.13 0.07 0.01 S 2 CR 4.41 (0.16) 0.06 0.21 0.01 0.67 0.02 0.02 S 2 0 CR 5.09 (0.03) 0.03 0.02 0.94 0.00 0.01 0.01 1 ME TT 3.74 0.00 0.00 0.00 0.04 0.00 0.96 a For definition ofA andB labels, see the structure from Ref.40. 100 Table 4.5: S 1 -MEjj jj for selected DPH dimers. 47 Structure Form jj jj Dimer1 M 0.104 Dimer1 O 0.018 Dimer2 M 0.001 Dimer2 O 0.006 Dimer3 M 210 5 Dimer3 O 410 4 101 Table 4.6: Analysis of the electronic states in DPH dimers a from the monoclinic form using the RAS(4,4)-2SF/(C,O:cc- pVTZ-f/H:cc-pVDZ) wave functions. Structure State Character E ex , eV (f l ) w 0 w LE A w LE B w CR A!B w CR B!A w SS AB w TT AB Dimer1 (A+B) S 0 GS 0.00 0.85 0.07 0.07 0.00 0.00 0.01 0.00 S 1 LE 4.72 (0.02) 0.00 0.44 0.48 0.00 0.02 0.06 0.00 S 1 0 LE 5.13 (4.72) 0.00 0.49 0.42 0.00 0.02 0.07 0.00 S 2 CR 5.30 (0.06) 0.00 0.00 0.03 0.00 0.90 0.00 0.06 S 2 0 CR 5.93 0.00 0.05 0.02 0.93 0.00 0.00 0.01 1 ME TT 5.24 (0.01) 0.00 0.00 0.00 0.01 0.06 0.00 0.94 Dimer2 (A+C) S 0 GS 0.00 0.85 0.07 0.07 0.00 0.00 0.01 0.00 S 1 LE 4.92 0.00 0.46 0.46 0.00 0.00 0.07 0.00 S 1 0 LE 4.93 (4.85) 0.00 0.46 0.46 0.00 0.00 0.07 0.00 S 2 CR 5.85 (0.02) 0.00 0.00 0.00 0.50 0.50 0.00 0.00 S 2 0 CR 5.86 0.00 0.00 0.00 0.49 0.49 0.00 0.01 1 ME TT 5.23 0.00 0.00 0.01 0.01 0.00 0.00 0.99 Dimer3 (A+D) S 0 GS 0.00 0.83 0.08 0.08 0.00 0.00 0.01 0.00 S 1 LE 4.73 (5.52) 0.00 0.46 0.46 0.00 0.00 0.07 0.00 S 1 0 LE 4.84 0.00 0.46 0.46 0.00 0.00 0.07 0.00 S 2 CR 6.76 0.00 0.00 0.00 0.50 0.50 0.00 0.00 S 2 0 CR 6.78 0.00 0.00 0.00 0.50 0.50 0.00 0.00 1 ME TT 4.95 0.00 0.00 0.00 0.00 0.00 0.00 1.00 a For the definition of dimers and labels, see Ref.47. 102 Table 4.7: Analysis of the electronic states in DPH dimers a from the orthorhombic form using the RAS(4,4)-2SF/(C,O:cc- pVTZ-f/H:cc-pVDZ) wave functions. Structure State Character E ex , eV (f l ) w 0 w LE A w LE B w CR A!B w CR B!A w SS AB w TT AB Dimer1 (A+B) S 0 GS 0.00 0.85 0.07 0.07 0.00 0.00 0.01 0.00 S 1 LE 4.88 (0.02) 0.00 0.43 0.50 0.00 0.01 0.06 0.00 S 1 0 LE 5.15 (4.57) 0.00 0.50 0.42 0.00 0.01 0.07 0.00 S 2 CR 5.53 (0.04) 0.00 0.00 0.02 0.00 0.96 0.00 0.02 S 2 0 CR 6.14 0.00 0.04 0.03 0.92 0.00 0.00 0.01 1 ME TT 5.42 0.00 0.00 0.00 0.01 0.02 0.00 0.97 Dimer2 (A+C) S 0 GS 0.00 0.85 0.07 0.07 0.00 0.00 0.01 0.00 S 1 LE 4.93 (4.69) 0.00 0.47 0.47 0.00 0.00 0.06 0.00 S 1 0 LE 5.13 0.00 0.46 0.46 0.00 0.00 0.07 0.00 S 2 CR 6.52 (0.01) 0.00 0.00 0.00 0.50 0.50 0.00 0.00 S 2 0 CR 6.52 0.00 0.00 0.00 0.49 0.49 0.00 0.01 1 ME TT 5.41 0.00 0.00 0.00 0.00 0.00 0.00 1.00 Dimer3 (A+D) S 0 GS 0.00 0.85 0.07 0.07 0.00 0.00 0.01 0.00 S 1 LE 4.97 (0.75) 0.00 0.48 0.46 0.00 0.00 0.06 0.00 S 1 0 LE 5.10 (3.96) 0.00 0.45 0.48 0.00 0.00 0.07 0.00 S 2 CR 6.61 0.00 0.00 0.00 1.00 0.00 0.00 0.00 S 2 0 CR 6.70 0.00 0.00 0.00 0.00 0.99 0.00 0.01 1 ME TT 5.41 0.00 0.00 0.00 0.00 0.00 0.00 1.00 a For the definition of dimers and labels, see Ref.47. 103 Finally, we consider the DPBF dimers from Ref.47. The calculations 47 revealed that the couplings vary among various structures taken from the X-ray structure of the two DPBF polymorphs (called and forms), which explained the experimentally observed differences in singlet fission rates and yields. Here we perform the analysis of the wave functions in order to assess whether the trends in couplings can be explained by variations in state characters. We considered 3 dimers that contribute the most into singlet fission rate. Dimer3 from the form (shown in Fig. 4.2) features the largest value ofjj jj (0.014). The same dimer in the form has practically zero coupling. The next largest coupling was observed in dimer1 (jj jj equals 0.006 and 0.005 in the and forms, respectively). The results for the two forms are collected in Tables 4.8 and 4.9. As one can see, the 1 ME state in all structures retains its pure 1 TT character. Thus, the observed varia- tions injj jj should be due to changes in the S 1 state. However, as Tables 4.8 and 4.9 show, there is no significant CR contributions in the S 1 states in any of the dimers; thus, based on the present analysis, the differences in couplings cannot be attributed to the mixing of the CR configurations. Interestingly, we do observe large contributions of the SS configurations in the S 1 and S 2 state. The presence of these configurations, which asymptotically correlate with much higher excited state, is due to the correlation effects in the ground state (the analysis for S 0 is also given in the tables). As one can see, the weight of the reference determinant in the ground state is0.8. Consequently, 0 mixes into other states as well. Overall, the weights presented in Tables 4.8 and 4.9 do not explain the variations in the computedjj jj, which again highlights the complexity of the many-electron wave functions and the limitations of a simple diabatic framework of singlet fission. 104 4.8 Conclusion We presented a formalism for the excited-state analysis of electronic states in bi- chromophoric systems such as molecular dimers or covalently linked chromophores. We use super-molecular approach based on the wave function of the dimer. The main assumption of the analysis is that the coupling between the chromophores is weak and, consequently, the ground-state wave functions can be effectively factorized. The ap- proach is an extension of ESSA 16–18 and is based on state rather than transition DMs. Specifically, we generalized ESSA indexes quantifying the weights of LE and CR con- figurations to general correlated wave functions such the definition of these indexes is not restricted to singly excited states. We also introduced new indexes designed to iden- tify and characterize multi-excitonic states. The weights of the CR and TT excitations can be computed using charge and spin cumulants. Our approach requires localized orbitals. For DM-based calculations, such as spin and charge cumulants symmetrically orthogonlaized AOs can be used, as in the defintion of L¨ owdin charges and local spin operators. 64 In princicple, L¨ owdin orbitals can also be used for evaluating the weights or local excitations by using Eq. (4.54) and appropriate projectors for the wave function amplitudes. Using localized MOs that preserve the invariance of the wave funcion (such as localization within each RAS supspace) simplifies the calculations. In the case when the SS ME contributions can be neglected, the calculations only require reduced DMs, while in a more general case the information about the wave function amplitudes is also required. This scheme is general and is applicable to both symmetric and non-symmetric systems, however, in its present form it is limited to bi-chromoforic systems only. The utility of the approach is illustrated by calculations on small model dimers (H 2 -He and H 2 -H 2 ) as well as larger systems (ethylene, tetracene, DPH, and DPBF dimers). The 105 caluclaions reveal that, while in some cases variations in couplings can be explained by various degree of mixing of CR configurations into the S 1 and 1 ME states, in other cases this simple diabatic picture fails to explain the computed trends. This again emphasizes the importance of using the robust orbital-invariant metrics quantifying the interactions between many-determinantal adiabatic states (such asjj jj) rather than relying on simple proxies such as the diabatic state composition and weights of CR configurations. 40 106 Table 4.8: Analysis of the electronic states in DPBF dimers a from form using the RAS(4,4)-2SF/(C,O:cc-pVTZ-f/H:cc- pVDZ) wave functions. Structure State Character E ex , eV (f l ) w 0 w LE A w LE B w CR A!B w CR B!A w SS AB w TT AB Dimer1 (A+B) S 0 GS 0.00 0.84 0.08 0.08 0.00 0.00 0.00 0.00 S 1 LE 4.36 (0.65) 0.00 0.46 0.46 0.00 0.00 0.08 0.00 S 1 0 LE 4.50 (0.74) 0.00 0.46 0.46 0.00 0.00 0.08 0.00 S 2 CR 5.57 (0.11) 0.00 0.09 0.09 0.40 0.40 0.02 0.00 S 2 0 CR 5.57 0.00 0.00 0.00 0.50 0.50 0.00 0.00 1 ME TT 3.87 0.00 0.00 0.00 0.00 0.00 0.00 1.00 Dimer2 (A+C) S 0 GS 0.00 0.84 0.08 0.08 0.00 0.00 0.00 0.00 S 1 LE 4.38 (0.52) 0.00 0.90 0.01 0.01 0.00 0.08 0.00 S 1 0 LE 4.39 (0.73) 0.00 0.01 0.90 0.00 0.01 0.08 0.00 S 2 CR 5.19 0.00 0.01 0.00 0.61 0.37 0.00 0.01 S 2 0 CR 5.20 (0.01) 0.00 0.01 0.01 0.37 0.60 0.00 0.01 1 ME TT 3.87 0.00 0.00 0.00 0.00 0.00 0.00 1.00 Dimer3 (A+D) S 0 GS 0.00 0.84 0.08 0.08 0.00 0.00 0.11 0.00 S 1 LE 4.40 (0.69) 0.00 0.15 0.77 0.00 0.00 0.08 0.00 S 1 0 LE 4.41 (0.62) 0.00 0.77 0.15 0.00 0.00 0.08 0.00 S 2 CR 6.33 0.00 0.00 0.00 0.00 1.00 0.00 0.00 S 2 0 CR 6.37 0.00 0.00 0.00 1.00 0.00 0.00 0.00 1 ME TT 3.87 0.00 0.00 0.00 0.00 0.00 0.00 1.00 a For the definition of dimers and labels, see Ref.47. 107 Table 4.9: Analysis of the electronic states in DPBF dimers a from form using the RAS(4,4)-2SF/(C,O:cc-pVTZ-f/H:cc- pVDZ) wave functions. Structure State Character E ex , eV (f l ) w 0 w LE A w LE B w CR A!B w CR B!A w SS AB w TT AB Dimer1 (A+B) S 0 GS 0.00 0.84 0.08 0.08 0.00 0.00 0.02 0.00 S 1 LE 4.38 (0.65) 0.00 0.46 0.46 0.00 0.00 0.08 0.00 S 1 0 LE 4.53 (0.73) 0.00 0.46 0.46 0.00 0.00 0.08 0.00 S 2 CR 5.60 (0.12) 0.00 0.09 0.10 0.11 0.68 0.02 0.00 S 2 0 CR 5.61 0.00 0.02 0.01 0.76 0.21 0.00 0.00 1 ME TT 3.92 0.00 0.00 0.00 0.00 0.00 0.00 1.00 Dimer2 (A+C) S 0 GS 0.00 0.84 0.08 0.08 0.00 0.00 0.00 0.00 S 1 LE 4.41 (0.55) 0.00 0.00 0.91 0.00 0.01 0.08 0.00 S 1 0 LE 4.41 (0.70) 0.00 0.91 0.00 0.01 0.00 0.08 0.00 S 2 CR 5.22 0.00 0.01 0.00 0.69 0.29 0.00 0.01 S 2 0 CR 5.23 (0.01) 0.00 0.01 0.01 0.28 0.69 0.01 0.01 1 ME TT 3.91 0.0 0.00 0.00 0.00 0.00 0.00 1.00 Dimer3 (A+D) S 0 GS 0.00 0.84 0.08 0.08 0.00 0.00 0.00 0.00 S 1 LE 4.42 (0.61) 0.00 0.78 0.14 0.00 0.00 0.08 0.00 S 1 0 LE 4.43 (0.69) 0.00 0.14 0.78 0.00 0.00 0.08 0.00 S 2 CR 6.23 0.00 0.00 0.00 0.00 1.00 0.00 0.00 S 2 0 CR 6.26 0.00 0.00 0.00 1.00 0.00 0.00 0.00 1 ME TT 3.91 0.00 0.00 0.00 0.00 0.00 0.00 1.00 a For the definition of dimers and labels, see Ref.47. 108 Chapter 4 references [1] R. McWeeny. Methods of Molecular Quantum Mechanics. Academic Press, 2nd edition, 1992. [2] E.R. Davidson. Reduced Density Matrices in Quantum Chemistry. Academic Press, New York, 1976. [3] M.M. Mestechkin. Metod matritsy plotnosti v teorii molekul. Kyev, Naukova Dumka, 1977. [4] T. Helgaker, P. Jørgensen, and J. Olsen. Molecular electronic structure theory. Wiley & Sons, 2000. [5] A.V . Luzanov and O.A. Zhikol. Excited state structural analysis: TDDFT and related models. In J. Leszczynski and M.K. Shukla, editors, Practical aspects of computational chemistry I: An overview of the last two decades and current trends, pages 415–449. Springer, 2012. [6] F. Plasser, M. Wormit, and A. Dreuw. New tools for the systematic analysis and visualization of electronic excitations. I. Formalism. J. Chem. Phys., 141:024106, 2014. [7] R.F.W. Bader. Atoms in molecules — Quantum theory. Oxford University Press, Oxford, 1990. [8] F. Weinhold and C.R. Landis. Discovering Chemistry With Natural Bond Orbitals. New Jersey: John Wiley & Sons, 2012. [9] E.D. Glendening, J.K. Badenhoop, and F. Weinhold. Natural resonance theory: II. Natural bond order and valency. J. Comput. Chem., 19(6):610–627, 1998. [10] K. Takatsuka, T. Fueno, and K. Yamaguchi. Distribution of odd electrons in ground-state molecules. Theor. Chim. Acta, 48:175–183, 1978. [11] M. Head-Gordon. Characterizing unpaired electrons from the one-particle density matrix. Chem. Phys. Lett., 372:508–511, 2003. [12] A.V . Luzanov and O.V . Prezhdo. Analysis of multiconfigurational wave functions in terms of hole-particle distributions. J. Chem. Phys., 124:224109, 2006. [13] A. Dreuw and M. Head-Gordon. Single-reference ab initio methods for the calcu- lation of excited states of large molecules. Chem. Rev., 105:4009 – 4037, 2005. 109 [14] H. Reisler and A.I. Krylov. Interacting Rydberg and valence states in radicals and molecules: Experimental and theoretical studies. Int. Rev. Phys. Chem., 28:267– 308, 2009. [15] A.I. Krylov. Equation-of-motion coupled-cluster methods for open-shell and elec- tronically excited species: The hitchhiker’s guide to Fock space. Annu. Rev. Phys. Chem., 59:433–462, 2008. [16] A.V . Luzanov, A.A. Sukhorukov, and V .E. Umanskii. Application of transition density matrix for analysis of excited states. Theor. Exp. Chem., 10:354–361, 1976. [17] A.V . Luzanov and V .F. Pedash. Interpretation of excited states using charge- transfer number. Theor. Exp. Chem., 15:338–341, 1979. [18] A.V . Luzanov. The structure of the electronic excitation of molecules in quantum- chemical models. Russ. Chem. Rev., 49:1033–1048, 1980. [19] M. Head-Gordon, A. M. Grana, D. Maurice, and C. A. White. Analysis of elec- tronic transitions as the difference of electron attachment and detachment densities. J. Phys. Chem., 99:14261 – 14270, 1995. [20] S. Tretiak and S. Mukamel. Density matrix analysis and simulation of electronic excitations in conjugated and aggregated molecules. Chem. Rev., 102:3171, 2002. [21] A.V . Luzanov and O.A. Zhikol. Electron invariants and excited state structural analysis for electronic transitions within CIS, RPA, and TDDFT models. Int. J. Quant. Chem., 110:902–924, 2010. [22] F. Plasser and H. Lischka. Analysis of excitonic and charge transfer interactions from quantum chemical calculations. J. Chem. Theory Comput., 8:2777–2789, 2012. [23] I. Fischer-Hjalmars and J. Kowalewski. Simplified non-empirical excited state calculations. Theor Chim Acta, 27:197, 1972. [24] S. Matsika, X. Feng, A.V . Luzanov, and A.I. Krylov. What we can learn from the norms of one-particle density matrices, and what we can’t: Some results for inter- state properties in model singlet fission systems. J. Phys. Chem. A, 118:11943– 11955, 2014. [25] E.B. Guidez and C.M. Aikens. Plasmon resonance analysis with configuration interaction. Phys. Chem. Chem. Phys., 16:15501–15509, 2014. [26] E.B. Guidez and C.M. Aikens. Quantum mechanical origin of the plasmon: from molecular systems to nanoparticles. Nanoscale, 6:11512–11527, 2014. 110 [27] C.M. Krauter, J. Schirmer, C.R. Jacob, M. Pernpointner, and A. Dreuw. Plasmons in molecules: Microscopic characterization based on orbital transitions and mo- mentum conservation. J. Chem. Phys., 141:104101, 2014. [28] S. Sharifzadeh, P. Darancet, L. Kronik, and J.B. Neaton. Low-energy charge- transfer excitons in organic solids from first-principles: The case of pentacene. J. Phys. Chem. Lett., 4:2917–2201, 2013. [29] C.J. Bardeen. The structure and dynamics of molecular excitons. Annu. Rev. Phys. Chem., 65:127–148, 2014. [30] A.V . Luzanov. Analysis of the exciton states of polyconjugated systems by the transition density matrix method. J. Struct. Chem., 43:711, 2002. [31] S.A. B¨ appler, F. Plasser, M. Wormit, and A. Dreuw. Exciton analysis of many- body wave functions: Bridging the gap between the quasiparticle and molecular orbital pictures. Phys. Rev. A, 90:052521, 2014. [32] J. Wahl, R. Binder, and I. Burghardt. Quantum dynamics of ultrafast exciton re- laxation on a minimal lattice. Comp. Theo. Chem., 1040-1041:167–176, 2014. [33] L. Blancafort and A.A. V oityuk. Exciton delocalization, charge transfer, and elec- tronic coupling for singlet excitation energy transfer between stacked nucleobases in DNA: An MS-CASPT2 study. J. Chem. Phys., 140:095102, 2014. [34] Y . Shao, Z. Gan, E. Epifanovsky, A.T.B. Gilbert, M. Wormit, J. Kussmann, A.W. Lange, A. Behn, J. Deng, X. Feng, D. Ghosh, M. Goldey, P.R. Horn, L.D. Jacob- son, I. Kaliman, R.Z. Khaliullin, T. Kus, A. Landau, J. Liu, E.I. Proynov, Y .M. Rhee, R.M. Richard, M.A. Rohrdanz, R.P. Steele, E.J. Sundstrom, H.L. Wood- cock III, P.M. Zimmerman, D. Zuev, B. Albrecht, E. Alguires, B. Austin, G.J.O. Beran, Y .A. Bernard, E. Berquist, K. Brandhorst, K.B. Bravaya, S.T. Brown, D. Casanova, C.-M. Chang, Y . Chen, S.H. Chien, K.D. Closser, D.L. Crittenden, M. Diedenhofen, R.A. DiStasio Jr., H. Do, A.D. Dutoi, R.G. Edgar, S. Fatehi, L. Fusti- Molnar, A. Ghysels, A. Golubeva-Zadorozhnaya, J. Gomes, M.W.D. Hanson- Heine, P.H.P. Harbach, A.W. Hauser, E.G. Hohenstein, Z.C. Holden, T.-C. Ja- gau, H. Ji, B. Kaduk, K. Khistyaev, J. Kim, J. Kim, R.A. King, P. Klunzinger, D. Kosenkov, T. Kowalczyk, C.M. Krauter, K.U. Laog, A. Laurent, K.V . Lawler, S.V . Levchenko, C.Y . Lin, F. Liu, E. Livshits, R.C. Lochan, A. Luenser, P. Manohar, S.F. Manzer, S.-P. Mao, N. Mardirossian, A.V . Marenich, S.A. Maurer, N.J. May- hall, C.M. Oana, R. Olivares-Amaya, D.P. O’Neill, J.A. Parkhill, T.M. Perrine, R. Peverati, P.A. Pieniazek, A. Prociuk, D.R. Rehn, E. Rosta, N.J. Russ, N. Ser- gueev, S.M. Sharada, S. Sharmaa, D.W. Small, A. Sodt, T. Stein, D. Stuck, Y .-C. Su, A.J.W. Thom, T. Tsuchimochi, L. V ogt, O. Vydrov, T. Wang, M.A. Watson, J. Wenzel, A. White, C.F. Williams, V . Vanovschi, S. Yeganeh, S.R. Yost, Z.-Q. You, 111 I.Y . Zhang, X. Zhang, Y . Zhou, B.R. Brooks, G.K.L. Chan, D.M. Chipman, C.J. Cramer, W.A. Goddard III, M.S. Gordon, W.J. Hehre, A. Klamt, H.F. Schaefer III, M.W. Schmidt, C.D. Sherrill, D.G. Truhlar, A. Warshel, X. Xu, A. Aspuru- Guzik, R. Baer, A.T. Bell, N.A. Besley, J.-D. Chai, A. Dreuw, B.D. Dunietz, T.R. Furlani, S.R. Gwaltney, C.-P. Hsu, Y . Jung, J. Kong, D.S. Lambrecht, W.Z. Liang, C. Ochsenfeld, V .A. Rassolov, L.V . Slipchenko, J.E. Subotnik, T. Van V oorhis, J.M. Herbert, A.I. Krylov, P.M.W. Gill, and M. Head-Gordon. Advances in molec- ular quantum chemistry contained in the Q-Chem 4 program package. Mol. Phys., 113:184–215, 2015. [35] F. Plasser, S.A. B¨ appler, M. Wormit, and A. Dreuw. New tools for the systematic analysis and visualization of electronic excitations. I. Applications. J. Chem. Phys., 141:024107, 2014. [36] A. Dreuw and M. Wormit. The algebraic diagrammatic construction scheme for the polarization propagator for the calculation of excited states. WIREs Comput. Mol. Sci., 5:82–95, 2015. [37] M. B. Smith and J. Michl. Singlet fission. Chem. Rev., 110:6891–6936, 2010. [38] M.B. Smith and J. Michl. Recent advances in singlet fission. Annu. Rev. Phys. Chem., 64:361–368, 2013. [39] T.F. Schulze, J. Czolk, Y .-Y . Cheng, B. F¨ uckel, R.W. MacQueen, T. Khoury, M.J. Crossley, B. Stannowski, K. Lips, U. Lemmer, A. Colsmann, and T.W. Schmidt. Efficiency enhancement of organic and thin-film silicon solar cells with photo- chemical upconversion. J. Phys. Chem. C, 116:22794–22801, 2012. [40] X. Feng, A.V . Luzanov, and A.I. Krylov. Fission of entangled spins: An electronic structure perspective. J. Phys. Chem. Lett., 4:3845–3852, 2013. [41] A.V . Luzanov. One-particle approximation in valence scheme superposition. Theor. Exp Chem., 17:227–233, 1982. [42] A.I. Krylov. Size-consistent wave functions for bond-breaking: The equation-of- motion spin-flip model. Chem. Phys. Lett., 338:375–384, 2001. [43] A.I. Krylov. The spin-flip equation-of-motion coupled-cluster electronic structure method for a description of excited states, bond-breaking, diradicals, and triradi- cals. Acc. Chem. Res., 39:83–91, 2006. [44] D. Casanova, L.V . Slipchenko, A.I. Krylov, and M. Head-Gordon. Double spin-flip approach within equation-of-motion coupled cluster and configuration interaction formalisms: Theory, implementation and examples. J. Chem. Phys., 130:044103, 2009. 112 [45] F. Bell, P.M. Zimmerman, D. Casanova, M. Goldey, and M. Head-Gordon. Re- stricted active space spin-flip (RAS-SF) with arbitrary number of spin-flips. Phys. Chem. Chem. Phys., 15:358–366, 2013. [46] A.B. Kolomeisky, X. Feng, and A.I. Krylov. A simple kinetic model for singlet fission: A role of electronic and entropic contributions to macroscopic rates. J. Phys. Chem. C, 118:5188–5195, 2014. [47] X. Feng, A.B. Kolomeisky, and A.I. Krylov. Dissecting the effect of morphology on the rates of singlet fission: Insights from theory. J. Phys. Chem. C, 118:19608– 19617, 2014. [48] P.M. Zimmerman, F. Bell, D. Casanova, and M. Head-Gordon. Mechanism for singlet fission in pentacene and tetracene: From single exciton to two triplets. J. Am. Chem. Soc., 133:19944–19952, 2011. [49] P.M. Zimmerman, C.B. Musgrave, and M. Head-Gordon. A correlated electron view of singlet fission. Acc. Chem. Res., 46:1339–1347, 2012. [50] D. Casanova. Electronic structure study of singlet-fission in tetracene derivatives. J. Chem. Theory Comput., 10:324–334, 2014. [51] E.R. Davidson and A.E. Clark. Model molecular magnets. J. Phys. Chem. A, 106:7456–7461, 2002. [52] R.D. Harcourt, G.D. Scholes, and K.P. Ghiggino. Rate expressions for excitation transfer. II. Electronic considerations of direct and through-configuration exciton resonance interactions. J. Chem. Phys., 101:10521–10525, 1994. [53] S.M. Parker and T. Shiozaki. Quasi-diabatic states from active space decomposi- tion. J. Chem. Theory Comput., 10:3738–3744, 2014. [54] A. Sisto, D.R. Glowacki, and T.J. Martinez. Ab initio nonadiabatic dynamics of multichromophore complexes: A scalable graphical-processing-unit-accelerated exciton framework. Acc. Chem. Res., 47:2857–2866, 2014. [55] R.J. Bartlett. Coupled-cluster theory and its equation-of-motion extensions. WIREs Comput. Mol. Sci., 2(1):126–138, 2012. [56] A.V . Luzanov. Excited state structural analysis for correlated many-electron sys- tems. Funct. Mat., 21:125–129, 2014. [57] K. Ruedenberg. The physical nature of the chemical bond. Rev. Mod. Phys., 34:326, 1962. 113 [58] M.S. de Giambiagi, M. Giambiagi, and F.E. Jorge. Bond index - relation to 2nd- order density-matrix and charge fluctuations. Theor. Chim. Acta, 68:337–341, 1985. [59] A.B. Sannigrahi. Ab-initio molecular-orbital calculations of bond index and va- lency. Adv. Quantum Chem., pages 301–351, 1992. [60] A. Torre, L. Lain, R. Bochicchio, and R. Ponec. Topological population analysis from higher order densities II. The correlated case. J. Math. Chem., 32:241–248, 2002. [61] A.V . Luzanov and O.V . Prezhdo. Irreducible charge density matrices for analysis of many-electron wave functions. Int. J. Quant. Chem., 102:582–601, 2005. [62] I. Mayer. Bond order and valence indices: A personal account. J. Comput. Chem., 28:204–221, 2007. [63] S.F. Boys. Construction of some molecular orbitals to be approximately invariant for changes from one molecule to another. Rev. Mod. Phys., 32:296, 1960. [64] A.E. Clark and E.R. Davidson. Local spin. J. Chem. Phys., 115:7382–7340, 2001. [65] A.V . Luzanov, D. Casanova, X. Feng, and A.I. Krylov. Quantifying charge res- onance and multiexciton character in coupled chromophores by charge and spin cumulant analysis. J. Chem. Phys., 142:224104, 2015. [66] P.A. Pieniazek, A.I. Krylov, and S.E. Bradforth. Electronic structure of the benzene dimer cation. J. Chem. Phys., 127:044317, 2007. [67] W.G. Penney. The electronic structure of some polyenes and aromatic molecules. III. Bonds of fractional order by the pair method. Proc. Roy. Soc. A., 158:306–324, 1937. [68] A.A. Ovchinnikov, I.I. Ukrainskii, , and G.V . Kventsel. Theory of one-dimensional Mott semiconductors and the electronic structure of long molecules with conju- gated bonds. Sov. Phys. Usp., 15:575, 1973. [69] E.R. Davidson and A.E. Clark. Local spin II. Mol. Phys., 100:373–383, 2002. [70] E.R. Davidson and A.E. Clark. Local spin III: Wave function analysis along a reaction coordinate, h atom abstraction, and addition processes of benzyne. J. Phys. Chem. A, 106:6890–6896, 2002. [71] A.V . Luzanov and O.V . Prezhdo. High-order entropy measures and spin-free quan- tum entanglement for molecular problems. Mol. Phys., 105:2879–2891, 2007. 114 [72] A.V . Luzanov. Some spin and spin-free aspects of coulomb correlation in molecules. Int. J. Quant. Chem., 112:2915–2923, 2012. [73] A.I. Krylov and P.M.W. Gill. Q-Chem: An engine for innovation. WIREs Comput. Mol. Sci., 3:317–326, 2013. [74] J.C Johnson, A.J. Nozik, and J. Michl. The role of chromophore coupling in singlet fission. Acc. Chem. Res., 46:1290–1299, 2013. [75] G.D. Scholes. Energy transfer and spectroscopic characterization of multichro- mophre assemblies. J. Phys. Chem., 100:18731–18739, 1996. 115 Chapter 5: A simple kinetic model for singlet fission: A role of electronic and entropic contributions to macroscopic rates 5.1 Introduction Singlet fission (SF) is a non-adiabatic process in which one singlet excited state splits into two triplets that ultimately give rise to four charge carriers. 1 If this process is har- nessed in solar cells, their efficiency can be increased beyond Shockley-Queisser limit by efficiently utilizing higher-energy photons. Although this phenomenon was discov- ered a long time ago, 2 its mechanistic understanding is incomplete, which hinders the design of organic photovoltaic materials for solar energy conversion. Recent reviews 1, 3 summarize experimental and theoretical work investigating the mechanisms of SF in molecular solids and model compounds. The electronic structure aspects of SF have received considerable attention. 1, 3–12 The important quantities are energy levels (lowest bright singlet states of individual chromophores should be about twice higher than T 1 ) and electronic couplings between 116 an initial excitonic state and a dark multi-exciton state that eventually splits into two independent triplets. These factors depend strongly on the relative orientations of the individual molecules, 4, 13 which is likely to be responsible for observed effects of mor- phology on the rates and yields of SF. One can connect the rates and electronic quantities by modeling complicated non- adiabatic dynamics encompassing several interacting electronic states. Simple esti- mates can be obtained using Landau-Zener type of approaches. 7 Two studies 14, 15 pre- sented calculations aiming at complete macroscopic description of SF process from first- principles employing approximate electronic structure models. In stark contrast to the large number of electronic structure studies, only recently a possible role of entropy in promoting SF has been noted. 16 Phenomenological kinetic models of SF and a reverse process, triplet-triplet annihi- lation, have been developed, with an emphasis on explaining magnetic field effects. 17, 18 Kinetic models of varying complexity are often employed in analyzing the experimental data. 16, 19–22 Our motivation is to develop a simple kinetic model that will allow us to connect electronic structure calculations with the experimental observables. We aim at develop- ing a theoretical framework that will explain the observed trends in SF yield and, more importantly, enable screening of various structures with respect to their efficiency in performing SF. In our previous work, 13 we discussed essential features of the underlying electronic structure and calculations of quantities relevant to SF process. We frame the discussion in terms of correlated many-electron adiabatic states that are coupled by non-adiabatic 117 (derivative) couplings. The electronic states involved are an initially excited bright sin- glet state (of a mixed excitonic and charge-resonance character), a dark singlet multi- exciton (or bi-exciton) state (ME or 1 (TT), two singlet-coupled triplets with a variable admixture of charge-resonance configurations) coupled to the singlet state by a non- adiabatic coupling, and two independent triplets (T 1 +T 1 ) that lost their coherence. The most basic quantity relevant to SF is electronic energy difference between the initially excited singlet state and twice the energy of separated triplets: E stt =E[S 1 ] 2E[T 1 ] (5.1) For efficient solar energy conversion, these two states should be approximately isoen- ergetic, small electronic energy mismatch being compensated by vibrational motion. In pentacene, E stt >0, and the overall SF process is exoergic, however, in tetracene 2E T >E S . The rate of electronic transition from the bright singlet state to 1 ME is pro- portional to non-adiabatic coupling, which is related to the norm of one-electron tran- sition density matrix,jj jj. The third important quantity is multi-exciton stabilization (or binding) energy, E b , that we define as energy difference between the 1 ME and 5 ME states (the former is stabilized by configuration interaction with other singlet configu- rations, whereas the latter maintains pure diabatic character of two entangled triplets). Since fission results in a loss of decoherence yielding two independent triplets, E b can be interpreted as a minimal energy required for the separation of two triplets to oc- cur. Although E stt is dominated by the S 1 and T 1 excitation energies of the isolated monomers, it is also affected by Davydov’s splitting and, therefore, can be modulated by local environment. jj jj and E b are also sensitive to the local environment. Thus, all three quantities can be tuned by structural modifications of the chromophores af- fecting their packing. Table 5.1 summarizes relevant electronic energies for tetracene, 118 pentacene, and hexacene. We note that calculations ofE stt involve larger error bars than calculations ofE b , due to error cancellation (the two ME states have the same character, thus, the dynamic correlation effects are quite similar and cancel out). Table 5.1: Electronic energies (eV) in tetracene, pentacene, and hexacene a . System E stt (monomer b ) E stt (bulk c ) E stt , exp E d b Tetracene -0.342 -0.303 -0.18 e 0.028 Pentacene 0.281 0.245 0.21 e 0.038 Hexacene 0.803 0.625 0.74 f /0.32 g 0.049 a See Ref.13 for details of computational protocol. b SOS-CIS(D) values. c Computed for the AB dimer asE[S 1 (AB)]E[ 5 ME(AB)] by using corrected RAS- 2SF values. d RAS-2SF values for AB dimer computed asE[ 5 ME]E[ 1 ME]. e From Ref.1. f Solution (Ref.23). g Film (Ref.24). While most electronic structure studies have been focusing on characterizing rele- vant electronic energies and couplings, one needs to operate with Gibbs free energies in order to understand driving forces in SF process and to compute rates. There is a thermodynamic driving force if the overall change in the free energy is negative, even if electronic energy differences are unfavorable (as, for example, in tetracene). Structure of molecular solids such as different packing can affect the free energies of each state via entropic contributions. A possible role of entropy has been recently emphasized by Zhu and co-workers. 16 In this paper, we discuss calculations of the entropic contribu- tions using a minimal kinetic model for SF that captures essential physics of the process. The model explains the observed difference in the SF rate between tetracene, pentacene, and 5,12-diphenyltetracene (DPT) and makes a prediction about hexacene. Consistently with experimental observations, the model predicts weak temperature dependence of the 119 ME formation rate in tetracene as well as a reduced rate of this step in solutions and in isolated dimers. The structure of the paper is as follows. Section 5.2 presents the kinetic model, Section 5.3 describes calculations of entropic contributions; the discussion is presented in Section 5.4. 5.2 Kinetic model for SF process The singlet fission is a complex process encompassing many states. Our goal is not to take into account all microscopic details, but rather to develop a simple conceptual picture of the process that would provide a semi-quantitative description of dynamics; thus, we choose the minimalist kinetic scheme that has only 3 states. It means that many states are combined into these states. Fig. 6.1 shows energy diagram for such minimal three-state model (see Table 5.1 for electronic energies). We assume that the process begins with populating the bright state (denoted by State 0). The second step is population of the multi-exciton state (State 1); the rates for the forward and backward reactions are denoted asr 1 andr 1 , respectively. The third step is production of independent triplets (State 2); the forward and backward rates arer 2 andr 2 . We choose to frame our discussion in terms of rates rather than rate constants (k 1 =k 1 andk 2 =k 2 ) in order to account for possible concentration effects and to simplify the calculation of characteristic times for SF. In a single-molecule framework employed here the rates are identical to the respective rate constants (i.e., one can rewrite all the equations usingk’s instead ofr’s), however, the situation in solutions is different and the rates should include both rate constants and the respective concentrations. Thus, our formulation is appropriate both for molecular solids and for solutions. 120 State 0: S1/S1’ State 1: 1 ME State 2: T1+T1 r1/r-1 r2/r-2 Sunday, November 10, 2013 State 0: S1/S1’ State 1: 1 ME State 2: T1+T1 Eb ESTT Thursday, November 7, 2013 S 1 1 ME T 1 + T 1 Thursday, January 23, 2014 Figure 5.1: Three-state model of singlet fission. Top left: free energies of the ini- tially excited bright (S 1 , denoted as ’State 0’), multi-exciton singlet ( 1 ME, denoted as ’State 1’), and two uncoupled triplets (T 1 +T 1 , denoted as ’State 2’). Bottom left: Electronic energy diagram. In pentacene, State 2 is lower than State 0 (E stt >0). In tetracene, it is slightly above (E stt <0). Thus, E 0 =0, E 1 =-E stt -E b , and E 2 =-E stt . Right: Cartoon illustrating the nature of S 1 , 1 ME, and T 1 +T 1 that gives rise to entropic contributions. State 0 is delocalized over several chromophores (black rectangle). The ME state can be localized on any pair of adjacent molecules (red rectangle) within the initial exciton. Our model assumes that triplet separation (second step) occurs when one of the triplet excitons hops to another chromophore (green rectangle). Using a linear free energy arguments, 25 the rates are related to the free energies of the three states (see Fig. 6.1): G 0 = 0 (5.2) G 1 =E stt E b TS 1 = stt b (5.3) G 2 =E stt TS 2 = stt (5.4) 121 where electronic energies are computed relative to State 0 and the calculations of respec- tive entropic contributions are described in Section 5.3. stt and b are now free energy differences that incorporate both electronic and entropic contributions: stt =E stt +TS 2 (5.5) b =E b +TS 1 TS 2 (5.6) stt characterizes overall driving force for SF, whereas b is multi-exciton stabilization (or binding) free energy. Relevant free energy differences are: GG 2 G 0 = stt (5.7) G 1 G 1 G 0 = stt b (5.8) G 2 G 2 G 1 = b (5.9) For the SF process to be thermodynamically possible, the total Gibbs free energy change, G, should be negative, which means that stt must be positive. As one can see from Eq. (5.5), negative E stt (endoergic SF) can be surpassed by sufficiently large entropic gain in State 2. G 1 or G 2 may, in principle, be positive, provided that G < 0. However, large positive value of either G 1 or G 2 would mean that the respective step is too slow for the overall process to happen on a realistic time scale, even though the overall reaction is thermodynamically allowed (just like diamonds-to-graphite transition). For example, positive ME stabilization free energy, as defined in Eq. (5.6), will slow down the second step. b depends on the difference of entropic terms for State 1 (bound multi-exciton state) and State 2 (two independent triplets), negative values mean that the 1 ME state is unbound and the second step is fast. 122 The quantity of interest is the total time to reach State 2. It can be computed as first-passage time, a powerful theoretical tool that was widely used in many chemical, physical and biological processes: 26 = 1 r 1 + 1 r 2 + r 1 r 1 r 2 = 1 r 1 + 1 r 2 + 1 K eq 1 r 2 = 1 r 1 + 1 r 2 1 + 1 K eq 1 (5.10) whereK eq 1 is an equilibrium constant for step 1. As one can see, whenK eq 1 >>1, the total time is just the sum of the inverses of individual rates. However, ifK eq 1 is small, the relative contribution of step 2 into the total time becomes more pronounced. Efficient SF will be identified by small . Thus, our aim is to investigate the dependence of on E stt , , E b , as well as entropic factors for the three states. We note that this model aims to describe the efficiency of the initial SF steps and will not be able to describe long-time (nanosecond) kinetics that involves diffusion, trapping, and recombination of independent triplet excitons. 19 The model is not suitable for describing magnetic field effects on SF. 17 However, this theoretical framework can be extended to take these processes into account. The ratios of forward and backward rates are determined by free energy differences satisfying the detailed balance condition: r 1 r 1 =exp[(G 0 G 1 )] =exp[( stt + b )] (5.11) r 2 r 2 =exp[(G 1 G 2 )] =exp[( b )] (5.12) 123 where = 1 k B T . Absolute rates depend on respective activation energies and prefactors. It is convenient to write down the expression for absolute rates as following: r 1 =r 1 (0)exp[ b ] (5.13) r 1 =r 1 (0)exp[( 1) b ] (5.14) r 2 =r 2 (0)exp[ b ] (5.15) r 2 =r 2 (0)exp[(1) b ] (5.16) In these equations ratesr i (0) (i = 1;1; 2;2) correspond to a hypothetical case when the ME stabilization energy is zero ( b =0). This procedure chooses a zero energy, and it is done for convenience only. Our calculations obviously do not depend on the choice of the reference state. In addition, parameter 01 describes the relative position of the transition state along the reaction coordinate (e.g., for = 1 2 ) the transition state is half-way between the reactants and products). For simplicity, we assumed that is the same for both steps. This approach allows us to analyze the effect of local structure by focusing on b . Thus, all information about activation energies and prefactors is now contained in r i (0). We note that detailed balance condition requires that: r 1 (0) r 1 (0) =exp( stt ) (5.17) Using Fermi Golden Rule and the analysis in Ref.13, one could argue that ratesr 1 (0) and r 1 (0) are proportional tojj jj 2 (or, more precisely, to jj jj 2 (Estt+E b ) 2 ) that describes the coupling between S 1 and 1 ME. Strong coupling would correspond to fast rates in both directions and relatively weak dependence on the temperature, while in the case of weak coupling these rates are small and strongly temperature-dependent. Transition 124 ratesr 2 (0) andr 2 (0) describe Dexter energy transfer between perfectly iso-energetic chromophores ( b = 0); they fall off quickly with distance (it is generally assumed that Dexter transfer is operational within 10-20 ˚ A). One can anticipate that r 2 (0) and r 2 (0) would be very similar in homologous compounds that share similar structure (e.g., crystalline tetracene and pentacene). The explicit expressions for transition rates allow us to estimate the average times for SF process: = exp[ b ] r 1 (0) + exp[ b ] r 2 (0) + exp[( 1) b ]r 1 (0) r 1 (0)r 2 (0) (5.18) Analysis of Eq. (5.18) suggests that there is an optimal value for b at which the singlet fission is the fastest. A more positive ME stabilization energy lowers the first and third terms, while it also increases the second term. Physically it means that for large and positive b (bound ME state), the first transition (from State 0 to State 1) is faster, while the second transition (from State 1 to State 2) is getting slower, suggesting that there is some optimal value of the multi-exciton stabilization energy. Similar arguments can be made for negative b (unbound ME) — a more negative value would impede the first step, but speed up the second. The optimal condition for b can be easily obtained by taking a derivative of with respect to the total binding energy. To illustrate this, let us assume that = 1 2 . Then by minimizing the time for SF with respect to the total binding energy we obtain: x 2 = r 2 (0) r 1 (0) + r 1 (0) r 1 (0) ; (5.19) where we defined an auxiliary function: x =exp b 2 (5.20) 125 Solving Eq. (5.19) yields the prediction for the optimal binding energy in terms of transition ratesr i (0): b =k B Tln r 2 (0) r 1 (0) + r 1 (0) r 1 (0) = stt +k B Tln r 2 (0) r 1 (0) (5.21) We note that, unlike the overall rate, the value of optimal b depends strongly on the free energy landscape, i.e., the value of parameter. By choosing=1 instead of 0.5, the minimization of Eq. (5.18) yields: x 4 = r 2 (0) r 1 (0) (5.22) instead of Eq. (5.19) giving rise to the optimal value of b that does not depend on stt : b = k B T 2 ln r 2 (0) r 1 (0) (5.23) Let us now analyze Eqns. (5.21) and (5.23). The second term in Eq. (5.21) is the same (up to a factor of two) as Eq. (5.23). Whenr 2 (0) > r 1 (0), this term is positive. Thus, in the case of =1 (transition state on the product side), optimal b is positive; larger r 2 (0) mean more positive values. This can be easily rationalized by analyzing Eq. (5.18): large r 2 (0) will reduce the magnitude of the second term, while positive b will make the first term smaller (the third term is zero for =1). Consequently, if r 2 (0)<r 1 (0), negative b (unbound ME) is desired. Ifr 1 (0) = r 2 (0), then the optimal value for b is zero — this means that the overall rate for the first step, r 1 , should be equal tor 2 to achieve shortest. 126 In the case of=0.5 (transition state between the reactants and products), optimal b depends on stt . For example, ifr 2 (0)r 1 (0), then: b stt (5.24) First, we note that in this case b is negative (unbound multi-exciton). Thus, if overall thermodynamic driving force for SF is large, then fastest rates are achieved at relatively large and negative multi-exciton stabilization energies. However, for small stt , b should be less negative, to make the first step sufficiently fast. The effect of the second term is exactly as in the case of =1 — if r 2 (0) >> r 1 (0), more positive b is required to achieve optimal rate by speeding up the first step. For another limiting case of =0, one can show that the optimal ME stabilization energy must be very large and positive. For all other cases the optimal ME stabiliza- tion energy can be understood by interpolating between the three limiting behaviors explained above. 5.3 Entropy calculations Below we provide an illustrative calculation of entropic contributions. As will become evident, the calculation invokes several assumptions; thus, it is of a semi-quantitative value. To estimate relative entropic contributions, we rely on the following simple rea- soning (similar to that of Ref.16). As illustrated in Fig. 6.1, State 0 is delocalized 7, 8 (or effectively delocalized owing to the ultra-fast exciton hopping 14 ), however, State 1 is localized on two neighboring chromophores. Thus, to estimate entropy of State 1, one needs to count by how many ways one can choose a pair of neighboring chromophores from the initially prepared delocalized exciton. In State 2, two triplets are separated; 127 thus, the entropic contribution can be estimated by counting a number of ways in which a multi-exciton state can separate within the Dexter radius. In polycrystalline materials, the initial exciton is delocalized, as evidenced by the noticeable spectral shift (relative to solution) and also supported by electronic structure calculations. 7, 8 The most thorough study of exciton delocalization is presented in Ref.8 where it was found that low-lying singlet excited states exhibit an average electron-hole distance greater than 6 ˚ A. From the figures visualizing the exciton, the exciton appears to be delocalized over about 13 molecules (in pentacene). Also, in agreement with molecular orbital considerations, the delocalization is most prominent in the ab plane. 1 1 ME: This state is localized on two adjacent monomers, where one of them must be in the shell of delocalized region where the radius is a (points inside the black hexagon, can be either black or red points). The total number of possible pairs P is total number of small edges in this picture: The total number of monomers inside the delocalized region is N, and the total number of shell is n. N=7 when n=1. The number of monomer of the i th shell is N s =6i. N is the total number of monomers: ܰ ൌ ͳ σ ܰ ௦ ୀ ൌ ͵ ݊ ଶ ͵݊ ͳ For one monomer, the total number of possible pair is 6, which is the number of edges that connect it. Therefore, 6N is the total number of pairs in which the edges inside the black hexagon is doubly counted. The number of edges that are outside the black hexagon is ܲ ௦ ା ൌ ͵ ൈ ʹ ൈ ܰ ௦ െ ൌ ͳʹ݊ , where we have 6 points that have 3 outer edges, and the rest (Ns-6) have 2 outer edges. ܲ ௦ ା is counted only once, so the total number of edges ܲ ൌ ሺ ܰ ܲ ௦ ା ሻ Ȁ ʹ ൌ ͻ ݊ ଶ ͳͷ ݊ ൌ ͵ ሺ ݊ ͳ ሻ ሺ ͵ ݊ ʹ ሻ a 1 10Å 6Å 15Å 1 ME: This state is localized on two adjacent monomers, where one of them must be L Q W K H G HO R FDO L ]HG UHJ L R Q )L UV W O HW ¶ V FR Q V L G HU W K H ' FDVH Z K HUHW K H H[ FL W HG monomer is the black point in the center of the following picture: Shell Total # of monomers in delocalized region Radius (Å) # of 1 ME pairs Entropy contribution* (eV) 1 7 6 30 0.088 2 19 10 72 0.111 3 37 15 132 0.127 n 3n 2 +3n+1 ~5n 9n 2 +15n+6 0.026ln(9n 2 +15n+6) * T=298K Details of how to calculate the total numbers of pairs and monomers are shown in last page. 2 In case of 3D, we consider the number of layer (l) to be 1, 3, 5, ..., . If the radius of delocalized/hopping region is about 15~20Å, I think l=3 is reasonable as shown in the following picture: Since we only count those pairs that are adjacent, the number of possible 1 ME pairs (P 12 ) between layer 1 and layer 2 is the total number of monomers in delocalized region within one layer as shown in previous page. 15Å layer 2 layer 1 layer Shell Total # of monomers Radius (Å) P in one layer P 12 Total P Entropy contribution* (eV) 3 3 111 15 132 37 470 0.160 n 9n 2 +9n+3 ~5n 9n 2 +15n+6 3n 2 +3n+1 33n 2 +51n+20 0.026ln(33n 2 +51n+20) 5 3 185 15 132 37 808 0.174 n 15n 2 +15n+5 ~5n 9n 2 +15n+6 3n 2 +3n+1 57n 2 +87n+34 0.026ln(57n 2 +87n+34) * T=298K Figure 5.2: Crystal structure of tetracene and pentacene. Molecular arrangement in the ab plane (left and center) and in perpendicular direction (right). Fig. 5.2 shows crystal structure of solid pentacene (tetracene is very similar, except for inter-layer distance). In the ab plane, each molecule has 6 nearest neighbors. The average distance between the chromophores in the ab plane is similar for tetracene and pentacene and is about 6 ˚ A. The distance between the planes is 15 ˚ A in pentacene (and is shorter in tetracene). 128 We assume that both steps of SF occur in the ab plane. We begin by calculating the number of chromophore molecules, N, contained in n coordination shells. Since ith coordination shell adds 6i molecules, then N = 3n 2 + 3n + 1 (5.25) Based on the distance between the chromophores centers, n can easily be connected with a spatial extent. To estimate entropic contribution to State 1, we compute the number of unique adja- cent chromophore pairs withinn coordination shells. As one can see from Fig. 5.2, the total number of possible pairs for one molecule is 6 (the number of edges that connect it to the nearest neighbors). Therefore, 6N is the total number of pairs in which the edges inside the black hexagon are doubly counted. The number of edges that are outside the black hexagon isP + s = 3 6 + 2 (N s 6) = 12n + 6, where we have 6 points that have 3 outer edges, and the rest (N s 6) have 2 outer edges. P + s is counted only once, so the total number of edgesP = (6N +P + s )=2. Thus, the total number of pairs is: 1 = 9n 2 + 15n + 6 (5.26) To estimate entropy for State 2, we consider a number of possibilities for a triplet exciton to hop within the area determined by the Dexter radius. Let N t denote the number of molecules within the Dexter radius (10-20 ˚ A) that comprisesn shells. Then 2 = N t (N t 1) 2 2 = (3n 2 + 3n + 1)(3n 2 + 3n) (5.27) 129 Here a factor of 2 accounts for the two triplets formed from one ME state. We note that 2 depends quadratically on the number of molecules within the Dexter radius, whereas 1 depends linearly on the number of molecules covered by the singlet exciton delocalization. This is important because for the second step to be thermodynamically favorable, 2 should be greater than 1 (E b is positive, see Table 5.1). We note that if the 1 ME state is mobile and can hop, this will contribute towards increasing the effective radius and, therefore, the increase in 2 . Table 5.2 collects the results of entropic contributions for States 1 and 2 as a function of the number of coordination shells (we note that, in general,n should be different for State 1 and State 2 because of differences in the underlying physical processes). Table 5.2: Entropic contributions as function of the number of coordination shells. n R, ˚ A N a 1 b 2 TS 1 , eV c TS 2 , eV c 1 6 7 15 42 0.070 0.096 2 10 19 72 342 0.110 0.150 3 15 37 132 1,332 0.125 0.185 4 20 61 210 3,660 0.137 0.211 5 25 91 306 8,190 0.147 0.231 6 30 127 420 16,002 0.155 0.249 a Using Eq. (5.26). b Using Eq. (5.27). c k B =8.617310 5 eV/K; k B T=0.026 eV at 298 K. 5.4 Discussion We begin by analyzing the thermodynamic driving force for the overall SF process. Clearly, the entropic contribution is always beneficial for SF (TS 2 >0). We note that such entropic contributions depend critically on the concentration of chromophore 130 molecules; thus, they will not be operational in dilute solutions, which is partially re- sponsible for low SF yields in isolated covalently linked dimers. 1, 27 We note that in a recent study efficient SF has been reported in pentacene solutions; 28 the kinetics re- vealed that the process is diffusion-limited and the yield shows a strong concentration dependence. Whereas it is not surprising that the overall process occurs on a nanosec- ond timescale (time required for an excited chromophore to find another one to form an excimer, a solution analog of State 0), the kinetics of the ME formation (Step 1 in our model) is 3 orders of magnitude slower than in solid pentacene (Ref.28 reports 400-530 ps). Based on the results collected in Table 5.2, and taking 10-25 ˚ A as a radius within which triplet separation may occur, we expect values of 0.2 eV , which is sufficient to make SF in tetracene possible. We note that the minimal number of shells required to make G <0 (in tetracene) is 3 (corresponding to TS 2 =0.185 eV). In pentacene, entropic contributions increase the driving force for SF. We note that possible mobility of the 1 ME state and/or triplet separation between the planes will further increase this value. We note that despite large uncertainties involved in calculations of entropic contri- butions, only a limited variations inTS 1 andTS 2 are possible. TS 2 has a lower bound determined by endothermicity of SF in tetracene. G 2 or G 1 may, in principle be positive; however, too large positive values would make individual steps too slow for the process to occur on a realistic time scale. Thus, we first consider the range ofTS 1 that would result in G 2 <0 and G 1 <0. Eq. (5.9) means that the entropic contributions should make the multi-exciton un- bound, to promote triplet separation. Using TS 2 =0.185 eV , G 2 <0, but G 1 >0. Using TS 2 =0.211 eV (n=4) or 0.231 eV (n=5), both conditions can be satisfied with 131 TS 1 =0.155 eV (n=6). The resulting thermodynamic quantities for these two combina- tions ofTS 1 andTS 2 are summarized in Table 5.3. To consider the case when G 1 >0 (corresponding to a small yield of 1 ME), we also performed calculations forTS 2 =0.211 eV (n=4) andTS 1 =0.147 eV (n=5), as well asTS 2 =0.185 eV (n=3) andTS 1 =0.147 eV (n=5). The results of these two calculations, which correspond to smaller efficient delocalization of S 1 , are also given in Table 5.3. Table 5.3: Relevant thermodynamic quantities (eV) in tetracene, pentacene, and hexacene a at 298 K. stt b b (=0.5) b (=1) TS 1 = 0.155 & TS 2 =0.211 Tetracene 0.031 -0.028 0.147 0.089 Pentacene 0.421 -0.018 -0.423 0.001 Hexacene 0.531 -0.007 -0.571 -0.020 TS 1 = 0.155 & TS 2 =0.231 Tetracene 0.052 -0.048 0.126 0.089 Pentacene 0.441 -0.038 -0.443 0.001 Hexacene 0.551 -0.027 -0.592 -0.020 TS 1 = 0.147 & TS 2 =0.211 Tetracene 0.031 -0.036 0.147 0.089 Pentacene 0.421 -0.026 -0.423 0.001 Hexacene 0.531 -0.015 -0.571 -0.020 TS 1 = 0.147 & TS 2 =0.185 Tetracene 0.005 -0.010 0.173 0.089 Pentacene 0.395 0.0002 -0.397 0.001 Hexacene 0.505 0.011 -0.545 -0.020 a Computed using experimentalE stt , see Table 5.1. b is computed using Eqns. (5.21) and (5.23) and the same parameters as in the rate calculations. Let us now analyze Eq. (5.18) in more detail. As illustrated by Table 5.3, the calcu- lated b are very similar in all three compounds, consequently, the values ofexp(0:5 b ) 132 are also very close. Using b computed with TS 1 =0.147 eV and TS 2 =0.211 eV , we arrive at the following: t = 2:05 r t 1 (0) + 1:11 r t 2 (0) (5.28) p = 1:65 r p 1 (0) + 0:61 r p 2 (0) (5.29) h = 1:33 r h 1 (0) + 0:75 r h 2 (0) (5.30) A reasonable assumption is thatr 2 (0) is very similar in all acenes (and is probably fast compared tor 1 (0), at least in tetracene). Thus, three orders of magnitude difference in the rates of SF in tetracene and pentacene should be due to the respectiver 1 (0). Using a linear free energy approach, 25 which argues that the activation energy for a process is proportional to the free energy difference of the reaction, one can write: r 1 (0)exp( stt ); (5.31) where is a coefficient determining the relationship betweenE a and G. We note that within this approachr 2 (0) should be the same for the 3 compounds, as it is defined as a hypothetic rate when b =0. To account for variations in coupling, we multiplyr 1 (0) by the respective jj jj 2 (Estt+E b ) 2 (jj jj 2 equals 0.115, 0.166, and 0.193 for tetracene, pentacene, and hexacene, respectively). Using = 0:5 and the computedE stt ,E b , andjj jj 2 , we obtain: r t 1 (0) :r p 1 (0) :r h 1 (0) = 1 : 1077 : 4814 (5.32) Note that since we use the same entropic factors for all three compounds, these ratios are determined byE stt ,E b , andjj jj 2 alone and, therefore, do not depend on the specific choices ofTS 1 andTS 2 (but they do depend on T). 133 By takingr t 1 (0) 10 10 s 1 andr 2 (0) 10 13 s 1 , one can explain the experimentally observed difference 1, 27 between tetracene and pentacene, and to make a prediction about hexacene: t 201ps (5.33) p 214fs (5.34) h 103fs (5.35) The computed times for different choices of TS 1 and TS 2 are summarized in Ta- ble 5.4. As these results demonstrate, the computed times are quite robust within the reasonable range of variations ofTS 1 andTS 2 . We also note that variations of have negligible effect on computed. Table 5.4: Computed characteristic times for tetracene, pentacene, and hexacene. TS 1 , eV TS 2 , eV t p h 0.155 0.211 171 ps 202 fs 112 fs 0.155 0.231 256 ps 243 fs 94 fs 0.147 0.211 201 ps 214 fs 103 fs 0.147 0.185 121 ps 192 fs 141 fs These simple calculations suggest that: The first step is rate-determining in tetracene; it becomes much faster in pentacene and hexacene due to increased thermodynamic driving force ( stt ) for the SF pro- cess. One can expect that already in pentacene the rates of Step 1 and Step 2 might become comparable. 134 Within our simple model, the rate of the first step in hexacene becomes very fast; thus, we expect that the second state might become rate-limiting. We expect that the rate of SF in hexacene will be comparable (yet slightly faster) to that in pentacene. Computedjj jj 2 are very similar in all three compounds, thus, the relative strength of couplings is unlikely to be responsible for very different rates of SF in tetracene and in pentacene. Of course, the presented calculations should be regarded as of a qualitative value, owing to the simplicity of the model and numerous assumptions. Yet, they present a framework for analyzing the growing data on the kinetics and yields of SF in various sys- tems. Moreover, they highlight relative significance of different factors (singlet-triplet energy gaps, entropic factors, and couplings) for the overall rate of SF. Using the above values of r 1 (0) and r 2 (0) we compare the computed b with the optimal values, as predicted by Eq. (5.21). The results for = 0:5 and = 1 are summarized in Table 5.3. We note that in tetracene optimal b is positive, consistently with a very slow first step owing to small thermodynamic drive due to small stt . One can easily see that while the calculations of rates are relatively insensitive to the value of parameter that characterizes the free energy landscape, the optimal value of b is strongly-dependent. When using=0.5, b are 1-2 orders of magnitude larger than the actual b . When using=1 (transition state at the product side), the computed optimal multi-exciton stabilization energies [using Eq. (5.23)] are 0.089 eV , 0.001 eV , and - 0.020 eV for tetracene, pentacene, and hexacene, respectively; these values are much closer to the actual b . Without more detailed knowledge of the free energy surface, it is unclear whether these systems are close to the optimal state. 135 To further test our model, we estimated the rate of SF in DPT; this material shows very efficient SF, with the fast and slow rate components of 1.3 ps and 105 ps, respec- tively. 19 We employTS 1 =0.147 eV andTS 2 =0.211 eV . Admittedly, this is a very crude estimate, as one may expect that the entropic contributions in an amorphous solid could be rather different. While the physical nature of excitons in amorphous solids is dif- ferent from crystalline materials, simple estimates suggest that their effective radius of delocalization (for the purpose of counting microstates in entropy calculations) may be similar. The spectra of solid DPT and DPT in solutions are identical, 19 indicating the localized nature of the initial exciton. However, as shown by the ab initio molecular dynamics simulations, 14 the initially excited state in DPT hops frequently around (about once per 100 fs) giving rise to effective delocalization (for the purpose of calculating entropic contributions), i.e., in the course of 1 ps the exciton will cover a sphere of the radius of about 20 ˚ A, which is indeed very close to the exciton size in tetracene and pentacene. Using experimentalE stt =0.08 eV and computed (using the same protocol)E b =0.022 eV andjj jj 2 =0.140, we arrive at DPT =0.6 ps, which is in a semi-quantitative agree- ment with the experimental trend. Based on our model, efficient SF in DPT can be attributed to an improved energy gap (E stt ) which increases thermodynamic drive and also increases the coupling. Using our estimates of the entropic terms, we evaluate anticipated differences in the rate of the first step,r 1 , for covalently linked dimers or in solutions (assuming that an excimer-like dimer is already formed). Assuming that all electronic factors (E stt , E b , 136 andjj jj 2 ) are the same in the dimers as in the respective solids (which is of course rather a crude approximation), the ratio is (using linear free energy approach and=0.5): r dimer 1 :r solid 1 =exp[0:5(TS 1 )] 0:03 0:05 (5.36) Thus, due to the entropic factors alone, the dimers (either covalently linked or excimers formed in solutions) are expected to exhibit S 1 !ME rates that are about 20-30 times slower than in respective pristine solids. Of course, to evaluate the actual rates, one should also take into account changes in the electronic factors. When applied to co- valently linked dimers from Ref.27, our model reproduces the observed three orders of magnitude drop in SF rates; the analysis of different components reveals that both electronic (couplings, variations in energies) and entropic factors are important. As a final remark, our model predicts a very weak temperature dependence of the first step, S 1 !ME, in tetracene. Using the same parameters as in the rates calculations above (and 1 =306 and 2 =3,660, see Table 5.2), the model predicts that the charac- teristic time of the first step in tetracene will increase only by a factor of 2 when the temperature is lowered by 100 K. The second step, however, is more sensitive and the respective time will increase by a factor of 15. These results are in a qualitative agree- ment with recent experimental observations. 29, 30 The week dependence can be easily rationalized by writing the rate as: rexp( G k B T ) =exp( S k B )exp( E k B T ) (5.37) Thus, when electronically a reaction is nearly isoergic (E 0), the relative importance of the temperature-dependent second term is small. In our calculation, we assumed that 137 exciton delocalization is not affected by temperature. However, at low temperatures, su- perradiant emission from tetracene has been reported by Bardeen and coworkers 31 and attributed to an increase in the coherence length. Thus, one may expect an increase of exciton delocalization that will enhance the relative importance of the entropic contri- bution. This would result in reducing the temperature dependence even further. 5.5 Conclusions We employed a simple three-state model for the SF process to interrogate the relative significance of different factors. By considering series of three acenes (tetracene, pen- tacene, and hexacene), we (i) explained the experimentally observed three orders of magnitude difference in the rate of SF in tetracene and pentacene; (ii) predicted that the rate of SF in hexacene will be slightly faster than in pentacene. This trend is driven by the increased thermodynamic drive for SF (Gibbs free energy difference of the initial excitonic state and two separated triplets). The important role of entropy was discussed; the anticipated magnitude of entropic contributions was illustrated by simple calcula- tions. The entropy is crucially important; it allows to overcome an unfavorable elec- tronic energy difference in tetracene. Entropy also facilitates the separation of the two bound triplets (multi-exciton state, 1 ME) into two independent triplets. The entropy in- creases both in the first (S 1 ! 1 ME) and in the second ( 1 ME!2T 1 ) steps. In the former, the entropy increase is due to a delocalized nature of the initial exciton and a localized nature of a multi-exciton state (there are multiple ways in which one can chose a dimer from the delocalized exciton). We note that although completely different in underlying physics, either delocalization of an excited state in an ordered polycrystalline solid or a high mobility of a localized exciton in disordered solids that enables sampling a cer- tain volume during exciton’s lifetime lead to the same final result in terms of counting 138 microstates in entropy calculations. In the second step, the entropy increases due to multiple possible ways in which the ME state can separate into two triplets. We note that the leading factor in computing these entropic contributions is due to the structure of the respective solids. We anticipate that in dilute solutions the entropic drive for the first step will not be operational resulting in a slower kinetics (at least 20-30 times slower) and reduced yields. Thus, in compounds in which SF is endoergic, such as tetracene, SF is expected to be significantly impeded. Pentacene, in which SF is exoergic and does not require additional entropic drive, does exhibit SF in solutions, 28 however, the rate of the multi- exciton formation from an excimer-like dimer is three orders of magnitude slower than in solid pentacene. We predict that SF is likely to be slow and inefficient in isolated covalently bound dimers of tetracene derivatives, even if the coupling elements are opti- mized, unless of course the substituents or solvent change the electronic energies of the S 1 and ME states such that the ME state drops below S 1 . Low yields (3-5%) and three orders of magnitude slower SF rates have been reported for covalently linked tetracene dimers; 27 our model reproduces this trend in the rate. Another important conclusion is that for an efficient SF, the two thermodynamic quantities, b and stt , may need to be balanced; furthermore, r 1 (0) : r 2 (0) ratio also affects optimal b . We illustrate that under certain conditions there is an optimal rela- tionship between b and stt . For example, for large b the first transition (from State 0 to State 1) is faster, while the second transition (from State 1 to State 2) is getting slower, suggesting that there is some optimal value of the multi-exciton stabilization energy for a given stt . Estimates of optimal ME stabilization energies in the three acenes suggest that their b might be too small; thus, the rates of SF in these compounds can be further optimized 139 by designing materials with larger multi-exciton stabilization energies. We note that b values can be tuned by either electronic or entropic contributions. Our estimate of SF rate in DPT is in semi-quantitative agreement with the experi- mental trend; the calculations suggest that the increased efficiency of DPT relative to tetracene is due to the reduced energy gap (E stt ). The model also explains weak temperature dependence 29, 30 of the S 1 !ME step in tetracene; this is a consequence of a significant entropic drive of this step. 140 Chapter 5 references [1] M. B. Smith and J. Michl. Singlet fission. Chem. Rev., 110:6891–6936, 2010. [2] S. Singh, W.J. Jones, W. Siebrand, B.P. Stoicheff, and W.G. Schneider. Laser generation of excitons and fluorescence in anthracene crystals. J. Chem. Phys., 42:330–342, 1965. [3] M.B. Smith and J. Michl. Recent advances in singlet fission. Annu. Rev. Phys. Chem., 64:361–368, 2013. [4] J.C Johnson, A.J. Nozik, and J. Michl. The role of chromophore coupling in singlet fission. Acc. Chem. Res., 46:1290–1299, 2013. [5] P.M. Zimmerman, C.B. Musgrave, and M. Head-Gordon. A correlated electron view of singlet fission. Acc. Chem. Res., 46:1339–1347, 2012. [6] T.S. Kuhlman, J. Kongsted, K.V . Mikkelsen, K.B. Møller, and T.I. Sølling. Inter- pretation of the ultrafast photoinduced processes in pentacene thin films. J. Am. Chem. Soc., 132:3431–3439, 201o. [7] P.M. Zimmerman, F. Bell, D. Casanova, and M. Head-Gordon. Mechanism for singlet fission in pentacene and tetracene: From single exciton to two triplets. J. Am. Chem. Soc., 133:19944–19952, 2011. [8] S. Sharifzadeh, P. Darancet, L. Kronik, and J.B. Neaton. Low-energy charge- transfer excitons in organic solids from first-principles: The case of pentacene. J. Phys. Chem. Lett., 4:2917–2201, 2013. [9] R.W.A. Havenith, H.D. de Gier, and R. Broer. Explorative computational study of the singlet fission process. Mol. Phys., 110:2445–2454, 2012. [10] D.N. Congreve, J. Lee, N.J. Thompson, E. Hontz, S.R. Yost, P.D. Reusswig, M.E. Bahlke, S. Reineke, T. Van V oorhis, and M.A. Baldo. External quantum efficiency above 100% in a singlet-exciton-fissionbased organic photovoltaic cell. Science, 340:334–337, 2013. [11] P.J. Vallett, J.L. Snyder, and N.H. Damrauer. Tunable electronic coupling and driving force in structurally well defined tetracene dimers for molecular singlet fission: A computational exploration using density functional theory. J. Phys. Chem. A, 117:10824–10838, 2013. 141 [12] D. Casanova. Electronic structure study of singlet-fission in tetracene derivatives. J. Chem. Theory Comput., 10:324–334, 2014. [13] X. Feng, A.V . Luzanov, and A.I. Krylov. Fission of entangled spins: An electronic structure perspective. J. Phys. Chem. Lett., 4:3845–3852, 2013. [14] W. Mou, S. Hattori, P. Rajak, F. Shimojo, and A. Nakano. Nanoscopic mechanisms of singlet fission in amorphous molecular solid. Appl. Phys. Lett., 102:173301, 2013. [15] T.C. Berkelbach, M.S. Hybertsen, and D.R. Reichman. Microscopic theory of singlet exciton fission. I. General formulation. J. Chem. Phys., 138:114102, 2012. [16] W-L. Chan, M. Ligges, and X-Y . Zhu. The energy barrier in singlet fission can be overcome through coherent coupling and entropic gain. Nat. Chem., 4:840–845, 2012. [17] A. Suna. Kinematics of exciton-exciton annihilation in molecular crystals. Phys. Rev. B, 1:1716–1739, 1970. [18] R.C. Johnson and R.E. Merrifield. Effects of magnetic fields on the mutual anni- hilation of triplet excitons in anthracene crystals. Phys. Rev. B, 1:896–902, 1970. [19] S.T. Roberts, R.E. McAnally, J.N. Mastron, D.H. Webber, M.T. Whited, R.L. Brutchey, M.E. Thompson, and S.E. Bradforth. Efficient singlet fission found in a disordered acene film. J. Am. Chem. Soc., 134:6388–6400, 2012. [20] G.B. Piland, J.J. Burdett, D. Kurunthu, and C.J. Bardeen. Magnetic field effects on singlet fission and fluorescence decay dynamics in amorphous rubrene. J. Phys. Chem. C, 117:1224–1236, 2013. [21] W.B. Whitten and S. Arnold. Pressure modulation of exciton fission in tetracene. Phys. Stat. Sol., 74:401–407, 1976. [22] J. Lee, P. Jadhav, P.D. Reusswig, S.R. Yost, N.J. Thompson, D.N. Congreve, E. Hontz, T. Van V oorhis, and M.A. Baldo. Singlet exciton fission photovoltaics. Acc. Chem. Res., 46:1300–1311, 2013. [23] H. Angliker, E. Rommel, and J. Wirz. Electronic spectra of hexacene in solution. Chem. Phys. Lett., 87:208–212, 1982. [24] M. Watanabe, Y . Chang., S. Liu, T. Chao, K. Goto, M.M. Islam, C. Yuan, Y . Tao, T. Shinmyozu, and T.J. Chow. The synthesis, crystal structure and chargetransport properties of hexacene. Nat. Chem., 4:574–578, 2012. 142 [25] E.V . Anslyn and D.A. Dougherty. Modern Physical Organic Chemistry. University Science Books, 2006. Chapter 8. [26] N.G. van Kampen. Stochastic Processes in Physics and Chemistry. Elsevier, 2 edition, 2001. [27] J.J. Burdett and C.J. Bardeen. The dynamics of singlet fission in crystalline tetracene and covalent analogs. Acc. Chem. Res., 46:1312–1320, 2013. [28] B.J. Walker, A.J. Musser, D. Beljonne, and R.H. Friend. Singlet exciton fission in solution. Nature Chem., 5:1019–1024, 2013. [29] J.J. Burdett, D. Gosztola, and C.J. Bardeen. The dependence of singlet exciton relaxation on excitation density and temperature in polycrystalline tetracene thin films: Kinetic evidence for a dark intermediate state and implications for singlet fission. J. Chem. Phys., 135:214508, 2011. [30] M.W.B. Wilson, A. Rao, K. Johnson, S. G´ elinas, R. di Pietro, J. Clark, and R.H. Friend. Temperature-independent singlet exciton fission in tetracene. J. Am. Chem. Soc., 135(44):16680–16688, 2013. [31] J.J. Burdett, A.M. M¨ uller, D. Gosztola, and C.J. Bardeen. Excited state dynamics in solid and monomeric tetracene: The roles of superradiance and exciton fission. J. Chem. Phys., 133:144506, 2010. 143 Chapter 6: Dissecting the Effect of Morphology on the Rates of Singlet Fission: Insights from Theory 6.1 Introduction Singlet fission (SF), a process in which one singlet excited state is converted into two coupled triplet states, is of interest in the context of organic solar cell technology. 1, 2 Al- though the phenomenon is known for quite some time, 3 the mechanistic details are still unclear. Thus, despite vigorous research, 1, 2 the design principles for materials capable of efficient SF remain elusive. Many theoretical studies of SF interrogated electronic structure aspects of the prob- lem 1, 2, 4–14 focusing on energies of the relevant states and estimating electronic cou- plings using diabatization schemes or one-electron approximations. Several studies at- tempted to estimate the rates of SF by using simple Landau-Zener type of approaches 7 or more sophisticated dynamics simulations. 15–17 Recently, we introduced a theoreti- cal framework 18, 19 that combines high-level ab initio calculations of electronic factors with a simple 3-state kinetic model of SF process. Our approach is based on corre- lated adiabatic wave functions and the respective transition density matrices that give 144 rise to non-adiabatic couplings. The model was used to explain the observed trends in SF rates in acenes and to interrogate the relative importance of electronic and entropic contributions. 19 A kinetic model for the the first step of singlet fission, a generation of the multi-exciton state, has been developed by Van V oorhis and co-workers; 20 their approach is based on a Marcus-like rate expression and employs constrained DFT 21 for calculations of the diabatic and adiabatc states and the couplings. Several factors are contributing to the efficiency of SF. First, the individual chro- mophores should have properly aligned energy levels, such that E(S 1 ) 2E(T 1 ). Sec- ond, the interaction between the chromophores in a molecular solid is also critically important. The arrangement of the molecules in a solid affects the energy levels, exci- ton delocalization, and the electronic couplings between the relevant states. While the significance of each of these factors for the SF process is obvious, it is not clear what exactly is the best arrangement. Several recent experimental studies have illustrated the significance of the morphology on the rates and yields of SF. 4, 22–28 For example, covalently linked dimers of tetracene show significantly decreased rates and yields. 22 Likewise, SF in solution is much slower than in the bulk. 29 While these observations can be explained by unfavorable couplings and entropic contributions, 19 the reported differences in the yields of SF in different forms of the same molecular solid 25–28 are more puzzling. Several examples are listed below. Efficient SF has been reported in amorphous 5,12-diphenyltetracene (DPT), however, the crystalline form shows no fis- sion. 23 Bardeen and coworkers reported 1.5 difference in SF rates in the two poly- morphs of 1,6-diphenyl-1,3,5-hexatriene (DPH). Finally, recent studies by Michl and coworkers 25–27 presented strikingly different yields (and rates) in the two forms of 1,3- diphenylisobenzofuran (DPBF): while the so-called form shows 125% yield of triplet excitons (at 300 K), the second form,, shows less than 10%. The difference becomes 145 even more pronounced at low temperature. The structural, spectroscopic, and physi- cal properties of the two forms are similar; the calculations reported in Ref.26 further confirmed that the electronic structure of the two polymorphs is indeed very similar. In this work, we investigate three systems, DPBF, DPH, and DPT. We explain the observed differences in the SF efficiency in DPBF and DPH by using our recently de- veloped theoretical framework. 18, 19 We also make predictions about SF rates in the two crystalline forms of DPT. 23, 30 6.2 Theoretical framework Eb ESTT Ed 1 ME S1 S1’ Wednesday, April 23, 2014 Figure 6.1: Three-state model of singlet fission. ’State 0’ denotes an initially excited delocalized state, ’State 1’ is a multi-exciton state, 1 T 1 (A)T 1 (B), and ’State 2’ cor- responds to two independent triplets. Relevant electronic energies are E stt =E[S 1 ]- 2E[T 1 ], multi-exciton stabilization energy, E b =E[ 5 ME]-E[ 1 ME], and Davydov’s splitting E d =E[S 1 0]-E[S 1 ]. Fig. 6.1 summarizes relevant electronic states and salient features of our model (see Refs. 18, 19 for details). SF is initiated by the absorption of a photon producing initially excited delocalized singlet state. In a dimer, its wave function can be described as a 146 linear combination of the excitonic (S 1 (A)S 0 (B) and S 0 (A)S 1 (B)) and charge-resonance (A B + andA + B ) configurations: c 1 S 1 (A)S 0 (B) +c 2 S 0 (A)S 1 (B) +c 3 A B + +c 4 A + B (6.1) The oscillator strength is spread over several such excitonic states (two in the dimer, denoted by S 1 (AB) and S 1 0(AB), following the notations from Ref.18); their energies differ by the so-called Davydov splitting, E d . We assume that the initially excited state relaxes to the lowest excitonic singlet state prior to SF; thus, in our calculations we use energy and electronic coupling of the lowest excitonic state of a dimer, whether it happens to be bright or dark. In the case when the Davydov splitting is small giving rise to substantial thermal populations of the higher states, we include the respective rates into the total rate calculation using Boltzmann’s weights: BFP (S 1 ) :P (S 1 0) =e E d (6.2) State 1 is the multi-exciton state ( 1 ME); it can be described as two triplet states localized on the adjacent individual chromophores coupled into a singlet state, 1 T(A)T(B). State 2 is two uncoupled triplets. Recently, we introduced a simple three-state model for the rate of SF process. 19 Electronic energy diagram is shown in Fig. 6.1. The relevant energies are: E stt (the difference between the initial singlet state and two independent triplets, E stt = E[S 1 ] 2E[T 1 ]) and E b (multi-exciton stabilization energy, 5 ME- 1 ME). The rates of the first (State 0! State 1) and the second (State 1! State 2) steps are denoted by 147 r 1 =r 1 and r 2 =r 2 (note that these notations differ from the Merrifield triplet recom- bination model 31 ). This minimalist model of SF can be further extended by including decay channel (characterized by rater d ), as discussed below. Using a linear free energy approach, 32, 33 which argues that the activation energy for a process is proportional to the free energy difference of the reaction, one can connect the rates and the free energies of the three states: G 0 = 0 (6.3) G 1 =E stt E b TS 1 (6.4) G 2 =E stt TS 2 (6.5) whereTS 1 andTS 2 denote entropic contributions (relative to State 0) to the Gibbs free energies of States 1 and 2. This model is, admittedly, very simple and relies on numerous approximations; its validity will be judged by a careful comparison against the experi- mental trends. We note that linear free energy approach has been successfully utilized to rationalize and predict trends in a large variety of processes in organic chemistry giving rise to such important relationships as Hammett, Taft, Grunwald-Winstein, Swain-Scott, and Bronsted scales. 32, 33 Recently, the assumptions behind linear free energy relation- ships have been analyzed and justified by first-principle calculations for series of S N2 reactions. 34 Linear free energy approach works the best when applied to series of suf- ficiently similar compounds/processes. Since the polymorphs of molecular solids have very similar structural and physical properties, this is a good case for applying a linear free energy relationship. 148 The entropy is crucially important for singlet fission; 19, 35 for example, it allows to overcome an unfavorable electronic energy difference in tetracene. Entropy also facili- tates the separation of the two bound triplets (multi-exciton state, 1 ME) into two inde- pendent triplets. In solids, the entropy increases both in the first (S 1 ! 1 ME) and in the second ( 1 ME!2T 1 ) steps. 19 In our previous work, 19 we estimated the entropic contribu- tions for crystalline acenes. The calculation depends on the degree of the delocalization of State 0; thus, the magnitude of entropic contribution depends on the morphology. However, when comparing rates in homologically similar compounds (such as acenes) or polymorphs of the same compound, it is reasonable to assume that the entropic con- tributions are similar because: (i) the number of the nearest neighbors is the same for the polymorphs considered in the present paper; (ii) the lattice parameters and, conse- quently, the average separation between the chromophores are very close; and (iii) the spectroscopic properties of different forms are very similar suggesting a similar degree of delocalization of the initial exciton. Under this assumption, the relative rates are determined by the electronic energy differences alone (and, of course, the couplings). In the present study we focus on estimating the rate of the first step, S 1 (AB)! 1 ME and S 1 0(AB)! 1 ME: r 1 [S 1 (AB)! 1 ME] jj jj E stt +E b 2 e (Estt+E b ) (6.6) r 1 [S 1 0(AB)! 1 ME] jj 0 jj E stt +E b +E d 2 e (Estt+E b +E d ) (6.7) where is a proportionality coefficient (we use 0.5, as in the previous study 19 ) and = 1 kT .jj jj andjj 0 jj are the norms of symmetrized one-particle transition density ma- trices between 1 ME-S 1 and 1 ME-S 1 0, respectively. jj jj E is a proxy for a non-adiabatic 149 matrix coupling element (see Refs. 18, 36 for details and benchmarks); the rate is pro- portional to its square, as follows from the Fermi Golden Rule expression for rates of non-adiabatic transitions. The overall rate is computed as Boltzmann-averagedr 1 and r 1 0: r = BFr 1 [S 1 ! 1 ME] +r 1 [S 1 0! 1 ME] BF + 1 (6.8) whereBF is defined by Eq. (6.2). We note thatjj jj is not equivalent to the coupling; thus, it cannot be used to com- pute absolute rates of non-adiabatic transitions. However, it contains information about state interactions (e.g., mixing of CR configurations into the S 1 and ME states) and the changes injj jj along different displacements correlate well with the magnitude of the full non-adiabatic coupling, as illustrated by a recent benchmark study. 36 Finally, we note that the rates are computed for fixed geometries of dimers taken from the crystal structures; thus, the effect of nuclear motions is not included. While it may be important for absolute rate calculations, we expect that the frequencies of the promoting modes are relatively similar in the polymorphs leading to the cancellation of the Golden Rule like prefactors in the rates expressions. This limitation can be lifted by including classical or quantum estimates of Franck-Condon factors. 37–39 In Ref.19, this 3-state model was used to to compute the first-passage time, 40 : = 1 r 1 + 1 r 2 + r 1 r 1 r 2 = 1 r 1 + 1 r 2 1 + 1 K eq 1 (6.9) Note that this minimalist model focuses on the rates alone and does not describe yields. Efficient SF can be identified by small that reduces the losses due to the competing channels (radiationless or radiative relaxation, exciton trapping, etc). In order to make 150 more quantitative connection between the rates and the experimental yields, we extend the model by introducing decay channels (from State 0 and State 1) characterized by rater d . For simplicity, we assume the same yields (Y 2 =Y 1 ) for Step 1 and Step 2: Y =Y 1 Y 2 = (Y 1 ) 2 = r 1 r 1 +r d 2 = 1 1 +x 2 (6.10) where x = r d r 1 (6.11) describes the competition between the decay and SF rates: whenr d r 1 ,Y 1 1. The effective rate of the first step is also affected by the decay, since the residence time in State 0 is now equal to: 1 = 1 r 1 +r d (6.12) Thus, the rate of the first step that takes into account the decay channel, r d 1 , can be written as: r d 1 =r 1 (1 +x) (6.13) Note that while lowering the yield, the decay accelerates the effective rate out of State 0. In Section 6.4.1, we use experimental data on the yields of SF in DPBF to evaluate x. We then use x to evaluate r d 1 . The assumptions of equal yields is, of course, very crude and is used here only to illustrate the overall effect of the decay channel. It can be relaxed and refined when more detailed experimental data becomes available. 151 6.3 Computational details We use RAS-2SF method 41, 42 that is capable of describing both the excitonic and mul- tiexciton states in the same framework. Following the protocol developed in Ref.19, we employ the cc-pVTZ basis 43 from which f-functions were removed, cc-pVTZ(-f). RAS- 2SF calculations are performed using ROHF quintet reference and use 4-electrons-in-4- orbitals active space. The structures of different forms of crystalline DPBF, DPH, and DPT are taken from Refs. 23, 26, 28, 30 From these structures, we selected monomers and dimers for electronic structure calculations, as described in the respective sections below. We also performed geometry optimizations for the monomers at the RI-MP2/cc-pVTZ level of theory; at these structures RAS-SF vertical excitation energies were computed and compared with the experimental data in solutions. All relevant Cartesian geometries are given in SI for Ref.44. All calculations are performed using Q-Chem electronic structure package. 45, 46 The computed adiabatic wave functions are used to calculate one-particle density matrices and, consequently, to estimate non-adiabatic coupling elements, as described in Ref.18. While RAS-2SF produces qualitatively correct wave functions, the absolute excitation energies are not sufficiently accurate, due to the lack of dynamic correlation. Consequently, energies of the excitonic states are overestimated; thus, S 1 - 1 ME gaps are of a poor quality. However, energy differences such as Davydov splittings and multi- exciton stabilization energy are reproduced much better due to error cancellation, as confirmed by comparisons with higher-level methods, such as SOS-CIS(D) (see SI of 152 Ref.18). To obtain better estimates of the S 1 - 1 ME gaps, we use an empirical correction based on the experimental excitation energies of the individual chromophores: E corr stt =E comp stt + (E[S 1 ] 2 E[T 1 ]) (6.14) E[S 1 ] =E exp [S 1 ]E comp [S 1 ] (6.15) E[T 1 ] =E exp [T 1 ]E comp [T 1 ] (6.16) Note that while these energy gaps are crucially important for absolute rates or when comparing the rates in different compounds, in the calculations of the relative rates of the polymorphs of the same compound, the correction (almost) entirely cancels out, as one can see from Eq. (8.1). When calculating couplings fromjj jj and E, it is more appropriate to use an unshifted value of E stt , as it is consistent with the Hamiltonian that defines non-adiabatic coupling. All calculations presented in the main manuscript use unshifted E stt in the couplings calculations. In this case, the energy corrections exactly cancels out from the relative rate calculations, that is, relative rates do not depend on the absolute values of E stt , but only on their differences. We expect that the differences in E stt between different dimer structures of the same compound are reproduced much better than the respective absolute values. In the SI for Ref.44, we also present values computed with couplings with the shifted gap (this leads to a small change in the relative rates). 153 Figure 6.2: Structure of DPBF. Unit cells of the (top left: view alongb-axis; top right: view alongc-axis) and (bottom left: view alonga-axis; bottom right: view alongc-axis) polymorphs. Dimers considered in rate calculations: dimer1 (A+C), dimer2 (A+B), dimer3 (A+D), dimer4 (B+C), dimer5 (B+D). 6.4 Results and Discussion 6.4.1 1,3-diphenylisobenzofuran (DPBF) Fig. 6.2 shows the structure of DPBF. The main structural differences between the and forms are the length of the c-axis (20.271 ˚ A and 19.423 ˚ A in the and forms, 154 respectively) and angle (106.215 and 93.534 ) of the unit cell, as shown in Fig. 6.2. In our calculations, we consider five different dimers: dimer1 is A+C, dimer2 is A+B, dimer3 is A+D, dimer4 is B+C, dimer5 is B+D. There is the same number of dimers of type 1, 2, and 3; and twice as many dimers of type 4 and 5 (dimers are counted by considering different types of nearest neighbors for each molecule in the unit cell). Table 6.1 summarizes relevant electronic factors, E stt (corrected), E b , E d ,jj jj and jj 0 jj, the respective raw energies are given in SI (Table S1) for Ref.44. Table 6.1: Relevant electronic energies (eV) andjj jj 2 in several DPBF dimers. Structure E stt E b E d jj jj 2 jj 0 jj 2 Dimer 1 -0.494 0.006 0.002 0.008 0.006 -0.518 0.005 0.002 0.008 0.005 Dimer 2 -0.506 0.003 0.141 0.003 8.310 4 -0.533 0.003 0.144 0.003 7.210 4 Dimer 3 -0.462 0.000 0.008 0.018 0.014 -0.485 0.000 0.011 8.510 6 8.510 6 Dimer 4 -0.467 0.000 0.016 4.710 5 4.710 5 -0.492 0.000 0.017 4.010 5 4.010 5 Dimer 5 -0.474 0.000 0.032 1.710 6 7.810 7 -0.493 0.000 0.020 1.110 6 1.210 6 We observe that in all dimers, SF is more endothermic (by about 0.02 eV) in the form. In agreement with calculations from Ref.26, we observe the largest Davydov splitting in dimer2. Both in dimer1 and dimer2, the couplings are larger for the S 1 state than for S 1 0. The magnitude of the couplings in the two forms of dimer1 and dimer2 is similar. Interestingly, dimer3 (whose structure is neither stacked nor herring-bone like) shows the largest difference between the two polymorphs as well as the largest coupling (for the form). Dimer4 and dimer5 feature smaller couplings (dimer5 is excluded from the rate calculations below). Note that the trends in Davydov splitting and E b do 155 not follow the trends injj jj, e.g., E b is zero for dimer3 that has the largest coupling. This observation further illustrates the importance of computing the coupling, rather than using energy differences as a proxy. Based on Davydov’s splittings, for dimer1 and dimer3 both singlet states should be included in the rate calculations at room temperature (Boltzmann factors are 1-1.5), whereas for dimer2 only the lowest excitonic state should be considered. At lower tem- peratures, however, the contributions from the higher excitonic states become insignifi- cant. Table 6.2 summarizes the results for rate calculations and the respective Boltzmann factors. First, for each form, we compute the rates relative to the dimer2 rate. Then for each dimer, we compute the relative rates for the and forms. As one can see, dimer3 shows the fastest rate in the form for both S 1 and S 1 0. In, dimer1 and dimer2 contribute the most. At lower temperature, the differences are significantly enhanced due to the difference in E stt between the two forms. 156 Table 6.2: Relative rates computed for different dimers from and forms. 1:2:3:4 denotes relative rates in dimers 1-4 (see Fig. 6.2 for definition of the dimers). Form Dimer S 1 ! 1 ME rate S 1 0! 1 ME rate Boltzmann factor Total rate T=298 K 1:2:3:4 3.37:1:11.5:0.028 0.98:1:3.81:0.014 1.1/247/1.4/1.9 2.99:1:10.8:0.03 1:2:3:4 3.47:1:0.06:0.024 0.95:1:0.003:0.013 1.1/277/1.5/1.9 2.86:1:0.006.8:0.03 : 1 1.48:1 1.77:1 1.59:1 : 2 1.52:1 1.71:1 1.52:1 : 3 3048:1 2274:1 2702:1 : 4 1.76:1 1.73:1 1.76:1 : all 5.34:1 5.03:1 5.77:1 T=77 K 1:2:3:4 7.94:1:119:0.217 1:1201:24:0.102 1.4/2.410 9 /3.4/11.6 7:49:1:129:0.256 1:2:3:4 9.15:1:0.074:0.210 1:1320:0.025:0.110 1.4/3.810 9 /5.4/13.5 8.07:1:0.089:0.246 : 1 6.15:1 7.39:1 6.58:1 : 2 7.09:1 6.73:1 7.09:1 : 3 11300:1 7118:1 10400:1 : 4 7.34:1 6.82:1 7.38:1 : all 81.3:1 6.71:1 96.8:1 157 To compute relative rates, we add the rates for each type of dimers (multiplied by the number of the dimers of each type) for the and forms and then scale the result by the: ratio for dimer2 (the rates for each form are computed relative to dimer2). At 298 K, this yields: r 1 () :r 1 () = (2:99+1+10:8+0:032)1:52 : (2:86+1+0:006+0:0262) = 5:8 : 1 (6.17) which is in a semiquantitative agreement with the experimental ratio, 2.7 in bulk crystals and 3.4 in films. 25, 26 We note that using Boltzmann averaging leads to a small but noticeable difference, e.g., the ratios computed only for the S 1 and S 1 0 states are 5.3:1 and 5.0:1, respectively. At 77 K, the computed ratio is much larger: r 1 () : r 1 ()=97; this is because SF in the form is more endothermic than in. The experimental rates at lower temperature have not been reported, however, our result is in agreement with the observed increased difference in the SF efficiency, e.g., at 300 K, the yield of triplets is 125% and 10% for the and forms, respectively, whereas at 77 K, the yield for the form increases up to 200%. The yield of triplets for the form at 15 K is less than 5%. Using the extension of the model described in Section 6.2, Eqns. (6.10)-(6.13), the experimental total yields of triplets, Y , result in the following yields for step one: Y 1 ()=0.84 andY 1 ()=0.22 at 298 K, andY 1 ()=1 andY 1 ()=0.15 at 77 K. Note that the difference in yields Y 1 for the two forms (as well as differences at two different temperatures) are less pronounced than in the total triplet yields. Using these yields, we estimate the values of parameter x for the two forms by Eq. (6.11). At 298 K, x()=0.27 andx()=3.5, and at 77 K —x()=0 andx()=5.7. Now we can use these values ofx derived from the experimental yields to correct the rates of the first step (or, 158 more precisely, the rates out of State 0) to take into account the decay channel using Eq. (6.13). At 298 K, we obtainr d 1 ():r d 1 ()=2.1:1 which is in a better agreement with the experiment than the r 1 values presented above. At 77 K, our calculations predict the following ratio:r d 1 ():r d 1 ()=14.5:1. We emphasize that explicit values for the rates ratios depend on the yield in each step. However, for all conditions we expect that the decay processes will modify the rate for the form much less than that of the form because of the higher yield of the former. In summary, our calculations explain the observed difference in rates and efficiency of SF in the two polymorphs of DPBF. The calculations also explain larger difference at lower temperature. The analysis of different components attributes the difference to a particularly favorable chromophore arrangement in dimer3 of the form, in contrast to the form. The difference in rates of the two forms of dimer3 is dominated by the couplings and can be explained by more extensive orbital delocalization in the form (see Fig. 6.3); however, the energetic contribution is also important (SF is about 0.02 eV less endothermic in the form). The next significant contribution to the rate comes from dimer1; in this case, the form is also more efficient than the form, however, the difference is much smaller (1.5:1). In this case, the couplings are similar and the difference in rates is mostly due to different E stt . This finding is in agreement with the conclusions from Ref.20 that emphasized the importance of the overall thermodynamic drive (i.e., E stt ) for the fast singlet fission. 159 HOMO-1 HOMO LUMO LUMO+1 -DPBF -DPBF Figure 6.3: Active-space molecular orbitals from RAS-2SF calculations of dimer3 in the two polymorphs. Top:-DPBF; Bottom:-DPBF. From left to right: HOMO-1, HOMO, LUMO, LUMO+1; contour value: 0.02. 160 To better understand the nature of coupling, we analyzed the RAS-2SF wave func- tions of dimer1 and dimer3 following the same strategy as in Ref.18. Leading electronic configurations of the 1 ME, 5 ME, and S 1 states are given in SI (Tables S5-S8) for Ref.44. As one can see, in both forms 5 ME state is of purely multi-exciton character and does not include CR contributions. However, in dimer3 the 1 ME state has30% of CR char- acter in the form, and no CR character in the form. The S 1 state of dimer3 shows small admixture (2 %) of CR configurations in the form. Interestingly, dimer1’s states show no CR character in either form, which is consistent with smaller magnitude of the couplings. Thus, the form of dimer3 has largest couplings because of efficient mixture of CR configurations into the 1 ME state. 6.4.2 1,6-diphenyl-1,3,5,-hexatriene (DPH) Figure 6.4: Structure of DPH. Top: monoclinic form (M-DPH). Bottom: or- thorhombic form (O-DPH). 161 Figure 6.5: Four monomers from the crystal structure of DPH used in dimers’ calculations. Four different dimers are considered: dimer1: A+B (left), dimer2: A+C (left), dimer3: A+D (middle: M form, right: O form). Fig. 6.4 shows the structure of the two polymorphs of DPH, monoclinic form and orthorhombic form; Fig. 6.5 shows the individual chromophores used to construct dif- ferent dimers. We considered three types of dimers: dimer1 is A+B (herring-bone like structure, as in acenes), dimer2 is A+C (stacked structure), and dimer3 is A+D (in-plane non-stacked structure). There are four dimer1, two dimer2, and two dimer3, which will be taken into account in the overall rate calculation. As explained above, the dimers are counted by considering different types of nearest neighbors for each molecule in the unit cell. Table 6.3: Relevant electronic energies (eV) andjj jj 2 in several DPH dimers. Structure Form E stt E b E d jj jj 2 Dimer 1 M -0.108 0.015 0.443 0.104 O -0.106 0.008 0.192 0.018 Dimer 2 M 0.044 0.006 0.145 0.001 O -0.082 0.009 0.197 0.006 Dimer 3 M 0.102 0.000 0.117 2.310 5 O -0.040 0.000 0.123 4.010 4 Table 6.3 summarizes relevant energies (the raw values are given in SI, Table S9, for Ref.44). In this case, Davydov’s splitting is sufficiently large for all three dimers, such that only the S 1 ! 1 ME rate needs to be considered. The results of rate calculations 162 are given in Table 6.4. We note that dimer1 (whose structure is similar to the herring- bone structure of dimers from acenes) provides dominant contributions in both M and O at 298 K. Dimer2 (stacked structure) follows dimer1 closely (at 298 K). Dimer3 shows small contributions at room temperature. In dimer1, E stt is nearly the same in both forms, thus, the difference in rates is dominated by the coupling. The situation is different in dimer2 and dimer3 where SF is exothermic in M and endothermic in O. Differences in couplings in dimer1 from the M and O crystals can be rationalized based on the results of Ref.18, where the effects of orientation were investigated for model acenes structures. Fig. 6.6 shows the S 1 - 1 ME coupling along the inter-fragment slipping coordinate (D, the displacement along the long molecular axis). As observed in Ref.18, varies strongly along this coordinate. As clearly seen from Fig. 6.6, drops by a factor of 2 asD varies from 2.1 ˚ A (its value in M-DPH) to 4.4 ˚ A (the value ofD in O-DPH). Thus, it is the difference in the slipping coordinate of the two forms, rather than the distance between the planes of the chromophores that controls the coupling. The MOs of the two forms are shown in SI (Fig. S1) for Ref.44; M-DPH shows larger delocalization. The wave function analysis of the ME and S 1 states of dimer1 (see SI, Tables S11-S12, for Ref.44) reveals that the weights of CR configurations are almost twice higher in the M form, i.e., the 1 ME and S 1 states have 21% and 9% of CR character in M, whereas in the O form the CR weights in these states are only 12% and 4%. Thus, as in the case of dimer3 of DPBF, larger couplings in dimer1 of M-DPH are due to a noticeable mixing of the CR configurations into the 1 ME state. Using the same algorithm and assuming that all dimers are equal, we obtain r 1 (M):r 1 (O)=6.1:1 (at 298 K), which is in a semiquantitative agreement with the ex- perimental ratio of 1.5:1. As seen from Table 6.4, all dimers exhibit a similar trend, but for different reasons: in dimer1, couplings are responsible for faster rate in the M 163 Figure 6.6:jj jj 2 for dimer1 of M-DPH as a function of displacementD along the long molecular axis.D=2.079 ˚ A in M-DPH and 4.367 ˚ A in O-DPH (see Fig. 6.4). form, whereas in dimer2 and dimer3, E stt drive the difference. At lower temperatures, we observe the change in the relative contributions of dimer1, dimer2, and dimer3. Our calculations predict an increased difference in rates between the two forms at low tem- perature. 6.4.3 5,12-diphenyltetracene (DPT) Figure 6.7: The crystal structure of DPT. Left: Three dimers from DPT crystal grown by vacuum sublimation (V); Right: three dimers from DPT crystal grown from xylene solvent (X). 164 Table 6.4: Relative rates computed for different dimers from M and O forms of DPH. 1:2:3 denotes relative rates in dimers 1-3 (see Fig. 6.5 for definition of differ- ent dimers). Form Dimer S 1 ! 1 ME rate 298 K M 1:2:3 1.98:1:0.111 O 1:2:3 1.53:1:0.153 M:O 1 6.56:1 M:O 2 5.07:1 M:O 3 3.67:1 M:O all 6.10:1 200 K M 1:2:3 0.50:1:0.18 O 1:2:3 1.20:1:0.21 M:O 1 6.88:1 M:O 2 16.5:1 M:O 3 14.3:1 M:O all 9.96:1 100 K M 1:2:3 0.008:1:0.83 O 1:2:3 0.58:1:0.55 M:O 1 7.96:1 M:O 2 590:1 M:O 3 892:1 M:O all 401:1 DPT is a promising SF system. Interestingly, an amorphous DPT film shows a high triplet quantum yield of 122% (Ref.23), whereas a vapor-grown DPT crystal shows almost no SF (Bradforth, private communication). Two different crystal structures of DPT have been reported: a crystal grown by vac- uum sublimation 23 and from xylene solvent. 30 Fig. 6.7 shows representative dimer structures taken from the two X-ray structures (V and X, respectively) and the three 165 dimers used in the rate calculations. The relevant electronic properties are collected in Table 6.5; raw energies are given in the SI (Table S14) for Ref.44. Dimer1 and dimer2 are slip-stacked, dimer3 is not. There is an equal number of the dimers of type 1 and 2, and 4 times as many of dimer3. Note that in dimer1 and dimer2 in V, the DPT moieties are offset by 0.5 and 1.5 rings, whereas in X the chromophores are offset by 0.25 and 2 rings. As observed for all acenes in Ref.18, half-integer ring offset structures feature smalljj jj and largejj 0 jj, whereas structures offset by the integer number of rings have largejj jj and smalljj 0 jj. The results for dimer1 and dimer2 in Table 6.5 follow this trend. The small magnitude of couplings (between the 1 ME-S 1 pair) in all V dimers explains the lack of efficient SF in vapor-grown crystals of DPT. Thus, based on the couplings alone, one would expect that S 1 ! 1 ME rates will be larger in X. For dimer1, E stt also favors the X form, however, in dimer2 SF is more exothermic in the V form (by about 0.06 eV). However, dimer3 has several orders of magnitude larger couplings (for both S 1 and S 0 1 ) in the V form. Thus, dimer3 will only contribute towards the total rate in V; however, its contribution is expected to be less that those of dimer1 and dimer2. The relative rates are summarized in Table 6.6. As expected from the observed trends in couplings, the X form shows 50 times fasterr 1 for the S 1 ! 1 ME transition than the V form. Interestingly, the S 1 0! 1 ME rate is also faster in X. However, the Boltzmann- averaged rates become nearly equal in the both forms, slightly favoring V. This reversal of the trend is due to large differences in E d in dimer2. Whereas E d in dimer2 from the V form is very small (as expected for half-integer ring offset structures), it is significant for the X form. Thus, it makes the contribution of the S 1 0! 1 ME transition significant in V, but not in X. This, combined with contributions from dimer3, reverses the ratio. Thus, although electronic couplings are clearly more favorable in X, the overall rates appear to be close in both forms. We note that this result depends strongly on the computed 166 Davydov’s splitting. Unlike DPBF, our model predicts that the temperature dependence of the V:X rates ratio is weak. To summarize, our calculations predict small difference in SF rates in the two forms of DPT at all temperatures. However, given the uncertainties in the computed values and the strong effect ofE d on the ratio, the X form may show faster SF rate than the V form, owing to more favorable couplings in the former. Note that given inefficient SF in V, it is unlikely that the SF rate in X would become as fast as in the amorphous DPT. 23 It would be extremely interesting to measure SF rates in the two forms experimentally; such measurements will aid further development of the theoretical framework. Table 6.5: Relevant electronic energies (eV) and couplings in several DPT dimers. Structure Form E stt E b E d jj jj jj 0 jj Dimer 1 V 0.590 0.004 0.202 0.001 0.093 X 0.615 0.014 0.374 0.118 0.029 Dimer 2 V 0.711 0.025 0.004 0.001 0.139 X 0.639 0.022 0.285 0.140 0.098 Dimer 3 V 0.680 0.000 0.048 0.018 0.014 X 0.816 0.000 0.047 2.910 5 2.410 7 6.5 Conclusions We investigate SF process in three different compounds, DPBF, DPH, and DPT. Each compound forms two different types of molecular crystals. Although the polymorphs exhibit very similar structural, physical, and spectroscopic properties, the efficiency of SF is different, as illustrated by the experimental studies 25, 26, 28 of DPBF and DPH. We conducted electronic structure calculations of relevant electronic factors and estimated relative rates of the first step of SF using a simple kinetic model. 19 The computed ratios are in a good agreement with the experimental results for DPBF and DPH, which 167 Table 6.6: Relative rates computed for different dimers from V and X structures at different temperatures. Form Dimer S 1 ! 1 ME rate S 1 0! 1 ME rate Total rate T=298 K V 1:2:3 1:14.3:1.77 1:0.58:0.006 1:459:1.86 X 1:2:3 1:2:0.006 1:1.27:7.2810 7 1:2:0.005 V:X 1 1:368 1:11.1 1:133 V:X 2 1:51.3 1:24.2 1:0.58 V:X 3 1:1.16 1:0.001 1:0.35 V:X all 1:49.5 1:15.7 1:0.86 T=77 K V 1:2:3 1:4.1110 4 :220 7858:197:1 1:3.4810 6 :255 X 1:2:3 1:12:199 1:0.052:2.8410 10 1:12:199 V:X 1 1:2617 1:1.2110 6 1:2617 V:X 2 1:0.77 1:2.5110 6 1:0.009 V:X 3 1:2366 1:2.71 1:2042 V:X all 1:50.5 1:1.2510 6 1:0.61 further validates our theoretical framework. 18, 19 The analysis of the calculations reveals that: (i) there is more than one pair of adjacent chromophores that contribute to SF; (ii) not only slip-stacked configurations show efficient fission; (iii) larger couplings in can be explained by orbital delocalization and efficient mixing of the CR configurations into the 1 ME state (and, to a smaller extent, to S 1 ); (iv) both electronic couplings and energies are responsible for different rates. Notably, the recent study of Van V oorhis and coworkers 20 has arrived to the same conclusion, that the thermodynamic drive (overall exothermicity of the singlet fission) plays a crucial role in determining the rates of the process. The calculations also explain the increased difference of the rates in the DPBF poly- morphs at low temperature; this is due to the difference in electronic energy gaps be- tween the singlet and triplet states in the two crystal forms. In DPH, both couplings and 168 E stt contribute to the difference; the model predicts similar temperature dependence as in DPBF. For DPT, our model predicts similar rates for the two forms, although the X form features much more favorable electronic couplings. We note that relatively small changes inE d for DPT may change this conclusion in favor of faster SF in the X form. The combination of calculations and experimental measurements of rates and yields in various polymorphs at different temperatures will help to further understand the relative importance of different factors contributing to the efficiency of SF. 169 Chapter 6 references [1] M. B. Smith and J. Michl. Singlet fission. Chem. Rev., 110:6891–6936, 2010. [2] M.B. Smith and J. Michl. Recent advances in singlet fission. Annu. Rev. Phys. Chem., 64:361–368, 2013. [3] S. Singh, W.J. Jones, W. Siebrand, B.P. Stoicheff, and W.G. Schneider. Laser generation of excitons and fluorescence in anthracene crystals. J. Chem. Phys., 42:330–342, 1965. [4] J.C Johnson, A.J. Nozik, and J. Michl. The role of chromophore coupling in singlet fission. Acc. Chem. Res., 46:1290–1299, 2013. [5] P.M. Zimmerman, C.B. Musgrave, and M. Head-Gordon. A correlated electron view of singlet fission. Acc. Chem. Res., 46:1339–1347, 2012. [6] T.S. Kuhlman, J. Kongsted, K.V . Mikkelsen, K.B. Møller, and T.I. Sølling. Inter- pretation of the ultrafast photoinduced processes in pentacene thin films. J. Am. Chem. Soc., 132:3431–3439, 201o. [7] P.M. Zimmerman, F. Bell, D. Casanova, and M. Head-Gordon. Mechanism for singlet fission in pentacene and tetracene: From single exciton to two triplets. J. Am. Chem. Soc., 133:19944–19952, 2011. [8] S. Sharifzadeh, P. Darancet, L. Kronik, and J.B. Neaton. Low-energy charge- transfer excitons in organic solids from first-principles: The case of pentacene. J. Phys. Chem. Lett., 4:2917–2201, 2013. [9] R.W.A. Havenith, H.D. de Gier, and R. Broer. Explorative computational study of the singlet fission process. Mol. Phys., 110:2445–2454, 2012. [10] D.N. Congreve, J. Lee, N.J. Thompson, E. Hontz, S.R. Yost, P.D. Reusswig, M.E. Bahlke, S. Reineke, T. Van V oorhis, and M.A. Baldo. External quantum efficiency above 100% in a singlet-exciton-fissionbased organic photovoltaic cell. Science, 340:334–337, 2013. [11] P.J. Vallett, J.L. Snyder, and N.H. Damrauer. Tunable electronic coupling and driving force in structurally well defined tetracene dimers for molecular singlet fission: A computational exploration using density functional theory. J. Phys. Chem. A, 117:10824–10838, 2013. 170 [12] D. Casanova. Electronic structure study of singlet-fission in tetracene derivatives. J. Chem. Theory Comput., 10:324–334, 2014. [13] D. Beljonne, H. Yamagata, J.L. Br´ edas, F.C. Spano, and Y . Olivier. Charge-transfer excitations steer the Davydov splitting and mediate singlet exciton fission in pen- tacene. Phys. Rev. Lett., 110:226402, 2013. [14] T. Zeng, R. Hoffmann, and N. Ananth. The low-lying electronic states of pen- tacene and their roles in singlet fission. J. Am. Chem. Soc., 2014. in press, dx.doi.org/10.1021/ja500887a. [15] W. Mou, S. Hattori, P. Rajak, F. Shimojo, and A. Nakano. Nanoscopic mechanisms of singlet fission in amorphous molecular solid. Appl. Phys. Lett., 102:173301, 2013. [16] T.C. Berkelbach, M.S. Hybertsen, and D.R. Reichman. Microscopic theory of singlet exciton fission. I. General formulation. J. Chem. Phys., 138:114102, 2012. [17] A.V . Akimov and O.V . Prezhdo. Nonadiabatic dynamics of charge transfer and singlet fission at the pentacene/C 60 interface. J. Am. Chem. Soc., pages 1599– 1608, 2014. [18] X. Feng, A.V . Luzanov, and A.I. Krylov. Fission of entangled spins: An electronic structure perspective. J. Phys. Chem. Lett., 4:3845–3852, 2013. [19] A.B. Kolomeisky, X. Feng, and A.I. Krylov. A simple kinetic model for singlet fission: A role of electronic and entropic contributions to macroscopic rates. J. Phys. Chem. C, 118:5188–5195, 2014. [20] S.R. Yost, J. Lee, M.W.B. Wilson, T. Wu, D.P. McMahon, R.R. Parkhurst, N.J. Thompson, D.N. Congreve, A. Rao, K. Johnson, M.Y . Sfeir, M.G. Bawendi, T.M. Swager, R.H. Friend, M.A. Baldo, and T. Van V oorhis. A transferable model for singlet-fission kinetics. Nature Chem., 6:492, 2014. [21] Q. Wu, C. Cheng, and T. Van V oorhis. Configuration interaction based on con- strained density functional theory: a multireference method. J. Chem. Phys., 127:164119, 2007. [22] A. M. M¨ uller, Y . S. Avlasevich, W. W. Schoeller, Klaus¨ ullen, and C. J. Bardeen. Exciton fission and fusion in bis(tetracene) molecules with different covalent linker structures. J. Am. Chem. Soc., 129:14240–14250, 2007. [23] S.T. Roberts, R.E. McAnally, J.N. Mastron, D.H. Webber, M.T. Whited, R.L. Brutchey, M.E. Thompson, and S.E. Bradforth. Efficient singlet fission found in a disordered acene film. J. Am. Chem. Soc., 134:6388–6400, 2012. 171 [24] C. Ramanan, A.L. Smeigh, J.E. Anthony, T.J. Marks, and M.R. Wasielewski. Com- petition between singlet fission and charge separation in solution-processed blend films of 6,13-bis(triisopropylsilylethynyl)- pentacene with sterically-encumbered perylene-3,4:9,10- bis(dicarboximide)s. J. Am. Chem. Soc., 134:386–397, 2012. [25] J.C. Johnson, A.J. Nozik, and J. Michl. High triplet yield from singlet fission in a thin film of 1,3-diphenylisobenzofuran. J. Am. Chem. Soc., 132:16302–16303, 2010. [26] J. Ryerson, J.N. Schrauben, A.J. Ferguson, S.C. Sahoo, P. Naumov, Z. Havlas, J. Michl, A.J. Nozik, and J.C. Johnson. Two thin film polymorphs of the singlet fis- sion compound 1,3-diphenylisobenzofuran. J. Phys. Chem. C, 118:12121–12132, 2014. [27] J.N. Schrauben, J.Ryerson, J. Michl, and J.C. Johnson. Mechanism of singlet fission in thin films of 1,3-diphenylisobenzofuran. J. Am. Chem. Soc., 136:7363– 7373, 2014. [28] R.J. Dillon, G.B. Piland, and C.J. Bardeen. Different rates of singlet fission in mon- oclinic versus orthorhombic crystal forms of diphenylhexatriene. J. Am. Chem. Soc., 135:17278–17281, 2013. [29] B.J. Walker, A.J. Musser, D. Beljonne, and R.H. Friend. Singlet exciton fission in solution. Nature Chem., 5:1019–1024, 2013. [30] C. Kitamura, C. Matsumoto, N. Kawatsuki, A. Yoneda, T. Kobayashi, and H. Naito. Crystal structure of 5,12-diphenyltetracene. Anal. Sci. Soc. Japan, 22:2, 2006. [31] R.C. Johnson and R.E. Merrifield. Effects of magnetic fields on the mutual anni- hilation of triplet excitons in anthracene crystals. Phys. Rev. B, 1:896–902, 1970. [32] E.V . Anslyn and D.A. Dougherty. Modern Physical Organic Chemistry. University Science Books, 2006. Chapter 8. [33] N. Streidl, B. Denegri, O. Kronja, and H. Mayr. A practical guide for estimating rates of heterolysis reactions. Acc. Chem. Res., 43:1537–1549, 2010. [34] E. Rosta and A. Warshel. Origin of linear free energy relationships: Exploring the nature of the off-diagonal coupling elements in S N2 reactions. J. Chem. Theory Comput., 8:3574–3585, 2012. [35] W-L. Chan, M. Ligges, and X-Y . Zhu. The energy barrier in singlet fission can be overcome through coherent coupling and entropic gain. Nat. Chem., 4:840–845, 2012. 172 [36] S. Matsika, X. Feng, A.V . Luzanov, and A.I. Krylov. What we can learn from the norms of one-particle density matrices, and what we can’t: Some results for inter- state properties in model singlet fission systems. J. Phys. Chem. A, 118:11943– 11955, 2014. [37] O.V . Prezhdo and P.J. Rossky. Evaluation of quantum transition rates from quantum-classical molecular dynamics simulations. J. Chem. Phys., 107:5863– 5878, 1997. [38] A.F. Izmaylov, D. Mendive-Tapia, M.J. Bearpark, M.A. Robb, J.C. Tully, and M.J. Frisch. Nonequilibrium Fermi golden rule for electronic transitions through coni- cal intersections. J. Chem. Phys., 135:234106, 2011. [39] M.H. Lee, B.D. Dunietz, and E. Geva. Calculation from first principles of in- tramolecular golden-rule rate constants for photo-induced electron transfer in molecular donor-acceptor systems. J. Phys. Chem. C, 117:23391–23401, 2013. [40] N.G. van Kampen. Stochastic Processes in Physics and Chemistry. Elsevier, 2 edition, 2001. [41] D. Casanova, L.V . Slipchenko, A.I. Krylov, and M. Head-Gordon. Double spin-flip approach within equation-of-motion coupled cluster and configuration interaction formalisms: Theory, implementation and examples. J. Chem. Phys., 130:044103, 2009. [42] F. Bell, P.M. Zimmerman, D. Casanova, M. Goldey, and M. Head-Gordon. Re- stricted active space spin-flip (RAS-SF) with arbitrary number of spin-flips. Phys. Chem. Chem. Phys., 15:358–366, 2013. [43] T.H. Dunning. Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogen. J. Chem. Phys., 90:1007–1023, 1989. [44] X. Feng, A.B. Kolomeisky, and A.I. Krylov. Dissecting the effect of morphology on the rates of singlet fission: Insights from theory. J. Phys. Chem. C, 118:19608– 19617, 2014. [45] Shao, Y .; Fusti-Molnar, L.; Jung, Y .; Kussmann, J; Ochsenfeld, C.; Brown, S.; Gilbert, A.T.B.; Slipchenko, L.V .; Levchenko, S.V .; O’Neill, D.P.; et al. Ad- vances in methods and algorithms in a modern quantum chemistry program pack- age. Phys. Chem. Chem. Phys., 8:3172–3191, 2006. [46] A.I. Krylov and P.M.W. Gill. Q-Chem: An engine for innovation. WIREs Comput. Mol. Sci., 3:317–326, 2013. 173 Chapter 7: On couplings and excimers: Lessons from studies of singlet fission in covalently linked tetracene dimers 7.1 Introduction Singlet fission (SF), a process in which one singlet excited state is converted into two triplet states, is of interest in the context of organic photovoltaic technology. 1, 2 By converting one absorbed high-energy photon into two lower-energy triplet excitons, the efficiency of a solar cell can be improved by reducing losses in the blue part of the solar spectrum. The reverse process, triplet-triplet annihilation producing an emitting singlet state, is exploited in frequency-upconversion materials that can harvest the lower-energy part of the solar spectrum. 3, 4 Owing to its technological significance, the mechanism of SF has been vigorously investigated. 1, 2 Yet, the design principles for materials capable of efficient SF remain elusive. As illustrated in Fig. 7.1, SF involves at least two steps, conversion of the initially excited singlet state (S 1 and S 1 0) into a multi-exciton state ( 1 ME) followed by the decou- pling and spatial separation of the two triplets. Initial singlet state is derived from the 174 S1(A)S0(B) 1 [T1(A)T1(B)] T1(A) + T1(B) + r1 r2 State 2: 1 ME State 3: T1+T1 Eb ESF State 1: S1/S1’ Ed Figure 7.1: Three-state model of singlet fission. ’State 1’ denotes an initially excited state derived from excitonic configurations, S 1 (A)S 0 (B) and S 0 (A)S 1 (B), with a pos- sible admixture of charge resonance, A + B and A B + ; ’State 2’ is a multi-exciton state, 1 (T 1 (A)T 1 (B)), and ’State 3’ corresponds to two independent triplets. Rele- vant electronic energies are E sf =E[ 1 ME]-E[S 1 ], multi-exciton stabilization energy, E b =E[ 5 ME]-E[ 1 ME], and Davydov’s splitting E d =E[S 1 0]-E[S 1 ]. bright singlet state of individual chromophores, S 1 (A)S 0 (B) and S 0 (A)S 1 (B); depend- ing on the structure, it can be either localized on one chromophore or delocalized over several neighboring molecules. The electronic configuration of the 1 ME state can be de- scribed as two triplet excitations residing on the nearby chromophores and coupled into a singlet state. From a quantum mechanical point of view, these two steps entail coupled electron-nuclear dynamics in a system comprising several (at least two) chromophores. Several other processes can compete with these two steps: radiationless relaxation from the initially excited or the ME state restoring the ground-state chromophore, annihila- tion of the triplets forming a singlet excitonic state, formation of charge-transfer states. In addition to these electronic processes, adiabatic nuclear dynamics can be important, e.g., the formation of excimers on the S 1 surface or structural relaxation of the ME state can lead to significant changes in energies and couplings thus affecting the rates. 175 SF rates are controlled by energetics and couplings. 5, 6 To make SF thermodynam- ically feasible, the energy of the S 1 state of the individual chromophore should be ap- proximately double the energy of the T 1 state. This criterion is satisfied in large conju- gated molecules such as acenes (starting from tetracene), carotenoids, etc. 1, 2 Michl and coworkers noted the connection between the energetic drive for SF and diradical charac- ter, i.e., that triplet states can be lowered by increasing diradical character. 7 This design principle 8, 9 has been used to engineer new systems capable of SF; 10–14 other suggested molecules await experimental validation. 8, 9 Electronically endoergic SF (E sf >0) is possible due to the entropic factor. 5, 15 Note that the energies of the states can be affected by interactions between the chromophores in a molecular solid (e.g., Davydov’s splitting lowers the energy of the S 1 state) as well as excited-state structural relaxation (e.g., excimer formation). Electronic couplings, which facilitate non-adiabatic transitions between the states, depend on the underlying wave functions and are strongly affected by morphology. 1, 2, 16–18 The necessary condi- tion to have a non-zero coupling is some degree of the overlap between the frontier orbitals. However, the non-zero overlap alone is not sufficient. Owing to the one- electron nature of the derivative operator, 18 the coupling between pure excitonic and the ME configurations is zero, however, mixing the charge-resonance(CR) configura- tions into the respective adiabatic wave functions leads to non-zero couplings. 1, 2, 17 Not all structures with large overlap have large mixing of CR configurations. A simple 4- electron-in-4-orbitals model 1 predicts zero couplings at perfectly stacked configuration and suggests that parallel-displaced structures are optimal for SF. 1, 2 However, calcula- tions of non-adiabatic couplings (NACs) in a model system showed that although the largest couplings are observed at the slip-stacked arrangement, the couplings are also 176 quite large at perfectly stacked geometries. 17, 18 The role of other configurations (be- yond the HOMO-LUMO active space) may also be important. As a recent example illustrating the complexity of the electronic structure, consider dimers from the crystals of the 1,3-diphenylisobenzofuran (DPBF) polymorphs. 10–12 The calculations revealed, unexpectedly, that the largest couplings are exhibited by non-stacked (i.e., coplanar) dimers. 19 In the case of the covalently linked dimers, the contribution of the linker to the couplings needs to be considered. Even if the arrangement of the chromophores that maximizes the couplings can be found, it does not necessarily translate into fast SF rates, because such a configuration may have poor energetics. Furthermore, the compet- ing channels may be adversely affected by structural variations. Additional complexity arises due to the multi-step nature of the process. For example, lowering the energy of the ME state, which can be achieved by increasingE b , speeds up the first step, but slows down the second (see Fig. 7.1). In order to connect the electronic factors with macroscopic rates, we introduced a simple 3-state kinetic model 5 based on Fermi’s Golden Rule and linear free energy approach, 20, 21 which argues that the activation energy for a process is proportional to the free energy difference of the reaction. The essential details of the model are outlined in Fig. 7.1 (here and in Ref.22 we redefined some of the quantities relative to Ref.5, to follow the standard convention of the signs of chemical reactions and to improve the clarity; however, the model itself is unchanged). The relevant electronic quantities are: E sf , E d , E b , and parametersjj jj andjj 0 jj, which are proportional to the NACs between the S 1 /S 1 0 and the 1 ME state. The protocol for computing these quantities is described in Sec. 7.2. E sf , the energy difference between the lowest S 1 state and 1 ME, determines the energetic drive for the formation of the multi-exciton state. Energy splitting between the excitonic pair, S 1 and S 1 0 (Davydov’s splitting, E d ), determines 177 Boltzmann’s populations of the states from the initially excited singlet manifold (if the population of S 1 0 is significant, then the contribution of this state to SF should be also included). Energy difference between the ME state and two uncoupled triplets (multi- exciton stabilization energy, E b ) arises due to the stabilization of the 1 ME state relative to the two independent triplets through the configuration interaction with other singlet states. E b is a minimal energy that needs to be overcome for the ME state to acquire a pure (T 1 T 1 ) character, before the two triplets can separate. While large positive E b is beneficial for the 1 ME state formation (step 1) because they lower E sf , it constitutes a penalty for the triplet separation step (step 2). The rates for the first step, S 1 (AB)! 1 ME and S 1 0(AB)! 1 ME, are: r 1 r[S 1 (AB)! 1 ME] jj jj E 2 e E sf ; (7.1) r 1 0r[S 1 0(AB)! 1 ME] jj 0 jj E +E d 2 e (E sf E d ) ; (7.2) where = 1 k b T and is a parameter from the free energy relationship and E is the energy difference between the raw (uncorrected) RAS-SF energies of the S 1 and 1 ME states. 19 The rate for the second step, 1 ME!2T 1 , is: r 2 r[ 1 ME!T 1 +T 1 ]e E b : (7.3) As in the previous studies, 5, 19 we use=0.5. The quantity of interest is the total time to reach State 3, which can be computed as first-passage time: 23 = 1 r 1 + 1 r 2 + r 1 r 1 r 2 = 1 r 1 + 1 r 2 + 1 K eq 1 r 2 ; (7.4) 178 whereK eq 1 is an equilibrium constant for step 1. WhenK eq 1 >>1, the total time is just the sum of the inverses of individual rates: 1 r 1 + 1 r 2 : (7.5) Efficient SF is identified by small. Thus, we investigate variations ofr 1 andr 2 in dif- ferent tetracene dimers. Our model does not attempt to compute absolute rates. Instead, we compute rates relative to a reference system, so that the exact expressions for the prefactors in Eqns. (7.2)-(7.3) are not required. This strategy also takes advantage of error cancellation, i.e., one may expect that the errors of an electronic structure method applied to homologically similar compounds are systematic and, therefore, cancel out in the relative rate calculations. We note that the essential features of our model, the expo- nential dependence on energy differences and the quadratic dependence on the coupling, are also present in the kinetic model developed by Van V oorhis and co-workers. 6 One difference between our model and that of Van V oorhis (or Marcus rate expression 24 ) is that we do not account for the kinetics in the inverted region that may occur for strongly exoergic processes. Thus, our model, in its current formulation, is only valid for endo- ergic and weakly exoergic SF. The physical foundations behind such phenomenological rate theories, the limits of their applicability, and their utility in understanding general trends and interpreting the experimental observations are reviewed in Ref.25. Our strategy is as follows. We first compute relevant energies (E sf , E b , E d ) and couplings (jj jj andjj 0 jj) for the dimers. The details of the electronic structure protocol are given in Sec. 7.2. We then compute the rates relative to the reference system (in this paper, two tetracenes in the orientation taken from the crystal structure). Since faster rates reduce the competition with parasitic processes, we equate faster rates with larger yields. 179 In the above variant, the model neglects triplet-triplet annihilation, radiationless relaxation to the ground state, geometric relaxation, and contributions to SF beyond dimers. As discussed below, some of these effects can be accounted for. For example, the effect of structural relaxation can be evaluated by computing relevant electronic pa- rameters at the relaxed structures. The variations in the rates of radiationless relaxation leading to S 0 can be investigated by analyzing the respective couplings. The utility of this approach has been illustrated by applications to several SF sys- tems. 5, 19 The model has explained the relative rates in tetracene and pentacene and attributed three orders of magnitude difference to more favorable energetics (the cou- plings are very similar). 5 The model predicted that the rate of SF in hexacene will be comparable to that in pentacene. A later experimental study 26 reported 5-6 times slower rate of the S 1 population decay relative to pentacene. Although our model did not predict slower kinetics, the predicted rate (91-141 fs) was in qualitative agreement with the ex- perimentally determined one (530 fs). The model has also explained 19 the differences in the SF rates in the polymorphs of DPBF 11 and DPH (1,6-diphenyl-1,3,5-hexatriene). 27 The focus of this work is on covalently linked tetracene dimers. Tetracene is a par- ticularly interesting system because of slight endoergicity of SF. In the context of solar energy harvesting, small E sf is desirable because it minimizes efficiency losses due to the energy-levels mismatch. Since the rates are very sensitive to E sf , small variations in states’ energies and the couplings due to the interactions between the chromophores may lead to a significant effect. Thus, understanding the relative rates in various tetracene- based systems presents a stringent test for a theoretical approach. Early experimental studies of tetracene dimers were conducted by Bardeen and co- workers 28 who reported low yields (3-5%) and three orders of magnitude slower SF 180 rates (relative to tetracene) for the structures shown in Fig. 7.2. This finding can be ra- tionalized by a small overlap between the individual chromophores, which are coplanar. This group also investigated a face-to-face tetracene dimer in which the two moieties are attached to a xanthene bridge 29 and observed no evidence of SF, despite the strong electronic coupling between the fragments. The lack of SF was attributed to the forma- tion of an excimer state lying too low in energy thus impeding the endoergicity of the generation of free triplets. 29 Formation of excimers on a picosecond timescale has also been observed in face-to-face and slip-stacked dimers of perylene. 30–32 Interestingly, increasing the offset between the chromophores impeded the excimer formation, but did not result in SF. 31 From the theoretical perspective, a possible role of excimers in SF and their energetics have been discussed by Zimmerman et al. 33 Contrary to the findings in covalently linked dimers, excimer formation in dilute so- lutions of pentacene and tetracene does not block the SF channel but rather serves as an intermediate step. 34, 35 The formation of excimer-like structure, described as frustrated photodimerization, has been invoked to explain the apparent lack of temperature depen- dence of the first SF step in tetracene and strong temperature dependence of the second SF step. 36 Recently, efficient SF has been observed in covalently linked pentacene dimers; 37 this study suggested that the through-bond coupling between the linked chromophores might be important. Dimers in which the tetracene moieties are not coplanar were recently synthesized at USC. 22 Fig. 7.3 shows the structures of the two cofacial alkynyltetracene dimers called BET-B and BET-X. As one can see, the tetracene moieties are staggered, which suggests larger overlap than in the structures from Fig. 7.2. In BET-X, which is very similar to the face-to-face dimer from Ref.29, the rings are almost parallel, whereas in 181 Figure 7.2: Covalently linked tetracene dimers from Ref.28. Figure 7.3: Covalently linked cofacial alkynyltetracene dimers from Ref.22. BET-B they are not. The experiments have shown that in neat film BET-B undergoes efficient SF within 1 ps with 154%10% efficiency. Interestingly, similar rates of the 1 ME state appearance were observed in neat films, solution, PMMA, and in BET-B doped DPT (diphenyltetracene) film. 22 For the second step, the separation of triplets, possible triplet acceptors are needed. Thus, in PMMA where the dimers are isolated and immobilized, the 1 ME state simply decays to the ground state. However, in neat films and in the doped DPT films the 1 ME state can break into two independent triplets. The rates of the second step were 0.230.05 ps 1 and 1501 ps 1 for BET-B in neat film and in doped DPT film, respectively. BET-X was found to be photochemically unstable in neat films and DPT; 22 in solution, the initially excited singlet state in 182 BET-X is shorter lived than the experiment time resolution (200 fs). The earliest-time transient-absorption spectra in BET-X were found to be similar to those observed in BET-B and assigned to the 1 ME state, suggesting similar initial dynamics, but with faster formation of the 1 ME state. 22 The calculations below analyze the electronic factors in these dimers and provide explanations to the observed trends. The structure of the paper is as follows. Section 7.2 outlines the computational protocols. The results of calculations are presented and discussed in Section 7.3. Specifically, we discuss relevant structures in the ground and excited states and analyze energetics and the couplings on the SF rates. The effect of the covalent linker is also investigated. Our concluding remarks are given in Sec. 7.4. 7.2 Theoretical methods and computational details We employ the same protocol as in Refs. 5, 17, 19 For the sake of clarity, we rewrote the rate expressions, Eqns. (8.1)-(7.3), using slightly different quantities; however, the es- sential features of the model are unchanged. As shown in Fig. 7.1, we focus on the following states: initially excited excitonic pair, S 1 and S 1 0, the 1 ME state of a pre- dominant 1 (T 1 T 1 ) character, and separated triplets, T 1 +T 1 . The gap between S 1 and S 1 0 (Davydov’s splitting) disappears when the two fragments are not interacting (e.g., at infinite separations between the fragments); in such a case, these states are just the S 1 states of the monomers. We use an adiabatic framework 17 and compute all rele- vant states using the RAS-2SF (restricted-active-space double spin-flip) configuration interaction method. 38, 39 In this approach, different types of configurations (excitonic, charge-resonance, multi-excitonic) can mix and interact leading to complex adiabatic wave functions. To characterize the couplings between the resulting adiabatic states, 183 we use density-matrix based formalism 17, 18 in which the NAC between the two states is proportional to the norm of the respective transition-density matrix,jj jj: pq =h i jp + qj f i; (7.6) jj jj = s X pq 2 pq : (7.7) The RASCI-2SF method has been successfully employed to model various aspects of SF. 5, 17, 19, 40–43 Following the protocol developed in Ref.5, we employ the cc-pVTZ basis 44 from which f-functions are removed, cc-pVTZ(-f). The RAS-2SF calculations were per- formed using the ROHF quintet references and the 4-electrons-in-4-orbitals active space. The key electronic energies, E sf , E b , and E d , were computed as follows: E sf =E( 1 ME)E(S 1 ); (7.8) E d =E(S 1 0)E(S 1 ); (7.9) E b =E( 5 ME)E( 1 ME): (7.10) To account for dynamical correlation, the raw RAS-2SF E sf energies were corrected by a simple energy-difference correction 17, 19 (note that in relative rate calculations of tetracene dimers, the correction cancels out exactly). The rates were computed using Eqns. (8.1)-(7.3) relative to the tetracene A-C dimer structure. 17 In the tables below, logarithmic relative rates are reported. In this paper, we neglect entropic contributions and focus exclusively on the electronic factors. The dimer structures in Figs. 7.2 and 7.3 were optimized by !B97X-D with the cc-pVDZ basis set. The crystalline BET-B/X structures (denoted as ’cry’) are based 184 on the X-ray structures from Ref. 22 for which the CH bonds were optimized using the same level of theory. The excimer structures for BET-B and BET-X were optimized by TDDFT with!B97X-D/cc-pVDZ following the S 1 and T 1 states. The structure of the 5 ME state was optimized by!B97X-D/cc-pVDZ. The degree of overlap between the frontier orbitals of tetracenes in various struc- tures was computed as follows. First, fragment’s molecular orbitals were computed by the explicit polarization (XPOL) method, 45, 46 where each tetracene moiety in a dimer (Tc) or a linked dimer (BET-B/X) is treated as one fragment (the linkers are removed in these calculations), with electrostatic inter-fragment interaction described by the L¨ owdin charges. Then the orbital overlap between the relevant MOs of the two tetracene units is calculated ashH A jH B i andhL A jL B i, where H A;B and L A;B are the HOMO and the LUMO of fragments A and B, respectively. To analyze the effect of the covalent linker, we prepared additional model structures as follows. Starting from the above BET-X and BET-B structures, we replaced the linkers (benzene or xanthene) by the hydrogen atoms (the CH bonds were optimized with !B97X-D/cc-pVDZ). These structures are denoted ‘TcCC-BET-B/X’. Similarly, we prepared the structures in which the benzene/xanthene moiety and the two ethynyl groups were replaced by hydrogens; these structures are denoted ‘Tc-BET-B/X’. These model structures are shown in Fig. S2 in SI of Ref.47. All calculations were performed with the Q-Chem electronic structure program. 48, 49 185 7.3 Results and discussion 7.3.1 Structures The most important difference between the structures from Figs. 7.2 and 7.3 is that in the former the tetracene rings are coplanar, whereas in the latter they are staggered. We considered several model structures of BET-B and BET-X (see section 7.2 for details). The first set was taken from the crystal structures of BET-B and BET-X (which has two types of crystals, BET-X 1 and BET-X 2 ). In the discussion below, these struc- tures are denoted by ‘cry’. The second set, denoted by ‘i’, consists of the fully optimized structures (for BET-B, we considered 3 conformers). The main difference between the two sets of structures is the distance and the angle between the tetracene rings. Interest- ingly, in BET-B fully optimized dimers the tetracene moieties are closer to each other and more parallel, relative to the crystal structure. This is accompanied by a slight bend- ing of the CC linkers. The reason for this becomes apparent when analyzing the crystal structure of BET-B shown in Fig. 7.4 — as one can see, the molecules are packed such that the tetracene rings from the neighboring dimers form a nearly perfect -stacked motif. Thus, in the crystal, the inter-dimer dispersion interaction, which favors more open dimer structures, compensates intra-dimer dispersion favoring structures with bent linkers and shorter intra-dimer tetracene distances. The dimer structures derived from the crystal structure are representative of the neat film, whereas the fully optimized structures provide realistic models for dimers in solu- tions, PMMA, and DPT. In addition to the ground-state structures, we also considered excited-state optimized geometries for the S 1 , T 1 , and 5 ME states, which are represen- tative of excimers. 186 Figure 7.4: The arrangement of two BET-B molecules in the crystal structure of BET-B. 22 Table 7.1 summarizes the key features of the ground-state and excited-state struc- tures. Specifically, we consider the distance between the tetracene moieties (d), the angle between their long axes (), the bent angle of the ethynyl linkers (), and the de- formation of the xanthene bridge ( ); see SI of Ref.47 for the exact definitions of these parameters. We note that twist angle has a large effect on the electronic couplings. 17 The differences between the structures are subtle. The distance between the tetracene planes decreases slightly in the isolated ground state and excimer structures of BET-B relative to the crystal structure. We also note a large change in between the crystal structure of BET-B and the S 1 excimer — the tetracene units become better aligned. In the 5 ME excimer, the angle increases again. 7.3.2 Energetics, couplings, and rates Table 7.2 presents the results for the structures shown in Fig. 7.2. As one can see, all structures show the decrease in r 1 for up to three orders of magnitude. The analysis of the individual components shows that energetics in these linked dimers is more fa- vorable than in tetracene (E sf is less endoergic), however, the couplings are more than 30 times smaller. We also note that Davydov’s splittings are rather large, which, once 187 Table 7.1: Comparison of the equilibrium structures of the ground and excited states in BET-X and BET-B. See SI of Ref.47 for the definition of the structural parameters. System d, ˚ A Tc 2.73 3 N/A NA BET-B(cry) 3.34 45 5 NA BET-B(i) 3.29 22 14 NA BET-B(S 1 ) 3.30 22 14 NA BET-B( 5 ME) 3.28 33 8 NA BET-B(T 1 ) 3.23 22 15 NA BET-X(cry1) 3.14 9 8 180 BET-X(cry2) 3.66 1 8 179 BET-X(i) 3.21 32 8 177 BET-X(S 1 ) 3.17 36 10 175 BET-X( 5 ME) 3.42 28 8 158 BET-X(T 1 ) 3.28 35 9 162 again, illustrates that strong electronic interactions between the chromophores do not imply favorable S 1 / 1 ME couplings. Experimentally, three orders of magnitude slower rates for dimers I-III relative to tetracene were observed. 28 The calculations based on electronic factors alone reproduce this trend, but show a smaller effect. We note that in- cluding the entropic effect will result in additional slowdown (the entropy contribution is proportional to the exciton delocalization and the number of possible triplet acceptors 5 ). Table 7.3 presents the results for BET-B and BET-X structures from Ref. 22 We consider the structures of the dimer from the X-ray (denoted by ‘cry’) and of fully optimized isolated dimers (denoted by ‘i’). The former are representative of the neat film structures, whereas the latter provide realistic models for dimers in solutions, PMMA, or DPT. The calculations reveal that the couplings are favorable in all dimer structures. Cou- plings are larger in fully optimized dimer structures relative to the ones from the crystal 188 Table 7.2: Relevant electronic factors and relative rates of the S 1 ! 1 ME step in dimers from Ref.28. All energies are in eV . System E sf E b E d jj jj 2 log h r 1 r 1 (Tc) i Tc 0.396 0.021 0.140 0.084 0 cis-I 0.136 0.001 0.098 5.210 4 -0.72 trans-I 0.134 0.001 0.093 3.810 4 -0.84 cis-II 0.141 0.005 0.121 1.710 3 -0.23 trans-II 0.159 0.001 0.122 7.310 4 -0.72 cis-III 0.102 0.000 0.029 5.810 7 -3.44 trans-III 0.098 0.000 0.027 1.410 3 -0.04 Ground-state energy differences (!B97X-D/cc-pVDZ): E(trans-I) - E(cis-I) = 0.36 kcal/mol; E(cis-II) - E(trans-II) = 0.20 kcal/mol; E(cis-III) - E(trans-III) = 0.37 kcal/mol. Table 7.3: Relevant electronic factors and relative rates of the S 1 ! 1 ME and 1 ME!2T 1 steps in BET-B and BET-X dimers from Ref.22. All energies are in eV . System E sf E b E d jj jj 2 log h r 1 r 1 (Tc) i log h r 2 r 2 (Tc) i Tc 0.396 0.021 0.140 0.084 0 0 BET-B(cry) 0.035 0.152 0.313 0.044 1.90 -1.11 BET-B(i) 0.315 0.098 0.323 0.087 0.41 -0.65 BET-X(cry1) 0.447 0.069 0.553 0.172 0.13 -0.38 BET-X(cry2) 0.364 0.043 0.356 0.121 0.31 -0.77 BET-X(i) 0.020 0.312 0.226 0.195 0.96 -2.47 structures because of better overlap between the rings. The analysis of the couplings shows that the presence of the linker is important (more on this below). E sf is more favorable in BET-X(i) > BET-B(cry) > BET-B(i) > BET-X(cry2) > Tc > BET-X(cry(1). Better energetics in BET-B relative to Tc are mostly due to the lowering triplet energy and, consequently, the 1 ME state (this can be seen from the raw energies shown in SI of Ref.47). For example, relative to Tc, in BET-B(cry) the S 1 state is lower by 0.14 eV , whereas 1 ME is lowered by 0.5 eV . As one can see, a more parallel arrangement of the tetracene moieties in BET-B(i) leads to a larger Davydov splitting, 189 and, consequently, preferential stabilization of S 1 and less favorable E sf . Likewise, the BET-X structures show large E d and less favorable E sf . The trends in couplings and energetics often oppose each other, since the couplings often increase when the overlap between the tetracenes increases. For example, BET-B(i) and BET-X(cry) exhibit larger couplings and less favorable energetics than BET-B(cry). Likewise, the couplings in all three BET-X structures are larger than in the reference tetracene system. The rate of the first step of intra-dimer SF in BET-B(cry) is faster by almost two or- ders of magnitude than in tetracene suggesting that SF occurs on pico to sub-picosecond timescale, in agreement with the experimental observations. 22 We note that for such fast process one may need to consider the contribution to fission from the initially ex- cited state, S 1 0. Due to more favorable energetics, these rates are even faster (see SI of Ref.47). We note that the intra-dimer rate of SF is somewhat higher (up to 10 times) in BET-B(cry) than in BET-B(i), suggesting that one may expect faster rates in neat films. However, to make unambiguous comparison between the neat film and isolated dimer, the contributions from inter-dimer fission should be considered. We will address this question in future work. On the basis of the data from Table 7.3, both crystal structures of BET-X are ex- pected to haver 1 that are more than an order of magnitude slower than in BET-B(cry), but slightly faster than in neat tetracene. Thus, one would expect that BET-X may also undergo SF. Experimentally, BET-X crystals were found to be photochemically unsta- ble, 22 suggesting that additional photochemical channels are operational. Thus, in the discussion below we focus on the structures of isolated dimers and experimental results for PMMA and solution. As discussed above, transient absorption of BET-B reveals the formation of the 1 ME state with2 ps time constant in PMMA and in DPT. 22 In the latter, independent triplets 190 were also detected. In BET-X, it was found that the initially excited singlet state is shorter lived than the experiment time resolution (200 fs). The earliest-time transient- absorption spectra in BET-X were found to be similar to those observed in BET-B and assigned to the 1 ME state, suggesting faster formation of the 1 ME state. Below we analyze whether triplets could, in principle, be harvested from BET-X and conclude that this system is not favorable for SF due to possible trapping in an excimer-type structure, which slows down the second step (r 2 ) and facilitates radiationless relaxation to the ground state. We first analyze the rates of the second step in BET-B(i) and BET-X(i) at the Franck- Condon geometries. Owing to relatively large multi-exciton stabilization energy, the second step is impeded in both BET-B and BET-X (note that in this calculation, we only consider energy factor; thus, the direct comparison of the rate relative to tetracene is not possible). In isolated dimers, r 2 in BET-X(i) is 10 times slower than in BET- B(i). A slower second step suggests reduced yields due to the competing processes, TT annihilation and radiationless relaxation to the ground state. In order to assess relative rates of the radiationless relaxation to the ground state, we computedjj jj 2 for the 1 ME!S 0 transition for BET-B(i) and BET-X(i). The respective values are 0.076 and 0.160; thus, one may expect twice higher rates of the 1 ME!S 0 transition in BET-X(i). Thus, a combination of slowerr 2 and higher rates for the radia- tionless relaxation to the ground state might be responsible for a fast excited-state decay in BET-X. Finally, we consider the effect of excimer formation on SF rates. We computed opti- mized geometries on the S 1 surface for BET-B and BET-X (see Table 7.1). As expected, both structures are characterized by shorter distances between the tetracene moieties relative to the respective crystal structures. BET-X shows larger relaxation relative to 191 BET-B (the structures in Table 7.1 suggest that a larger geometric relaxation in BET-X is mostly due to a more flexible bridging moiety). Table 7.4 collects relevant electronic factors and rates. As expected, excimer formation leads to the increase in E d and lower- ing of the S 1 state. Unexpectedly, we found that E sf actually becomes more favorable, because of the considerable relaxation of the 1 ME state along this displacement. In order to validate the RAS-2SF results, we also computed relaxation of the T 1 state at the excimer structure by TDDFT. We found that the computed energy drop of the ME state is indeed roughly equal to twice the value for the triplet. We found that at the S 1 -excimer structure, the rates of the first step, the ME formation, are comparable to or faster than the rates computed at the Franck-Condon geometries. However, the rates of the second step, triplet separation, are considerably impeded, due to the ME stabiliza- tion and the increase in E b . Thus, the excimer serves a trap of the transient 1 ME state. Again, we observe that the decrease in r 2 is two orders of magnitude larger in BET- X(S 1 ) than in BET-B(S 1 ). The analysis of the couplings (see Table 7.4) suggests that the excimer formation leads to the faster rate of radiationless relaxation and the effect is more pronounced in BET-X(S 1 ). Thus, the calculations suggest that triplet harvesting from BET-X would be hampered by faster rates of radiationless relaxation and slower rates of the triplet separation step relative to BET-B. Both effects are enhanced by the excimer formation. 192 Table 7.4: Relevant electronic factors and relative rates of the S 1 ! 1 ME and 1 ME! 2T 1 steps in BET-B and BET-X dimers at the S 1 -optimized structures. All energies are in eV . System E sf E b E d jj jj 2 jj jj 2 jj jj 2 log h r 1 r 1 (Tc) i log h r 2 r 2 (Tc) i S 1 - 1 ME 1 ME-S 0 S 1 -S 0 BET-B(S 1 ) 0.439 0.229 0.771 0.195 0.014 0.224 0.21 -1.77 BET-X(S 1 ) -0.070 0.475 0.757 0.191 0.172 0.222 3.28 -3.85 193 An interesting question is what is the impact of the geometric relaxation of the 1 ME state. As one can see from Table 7.4, the 1 ME state is considerably stabilized at the S 1 - excimer geometry. We expect that it may relax even further. This geometric relaxation of the 1 ME state will increase energetic penalty for the separation of triplets. We were not able to optimize the structure of the 1 ME state because the analytic gradients are not available for the RAS-2SF method. Instead, we optimized the geometry of the 5 ME state by using!B97X-D/cc-pVDZ. This structure corresponds to a pure 1 T 1 T 1 state (we note that it is very similar to the C-DFT optimized structure of T 1 T 1 ), but it does not capture the effect of other electronic configurations, which are present in the adiabatic 1 ME state. The energy diagram in Fig. 7.5 shows energies of the S 1 , 1 ME, and 5 ME states relative to S 0 (at its equilibrium geometry) at the Frank-Condon geometry and the optimized geometries of the S 1 and 5 ME states. As one can see, the relaxation of the ME state can be quite substantial, up to 0.85 eV (BET-B), which likely will impede the separation of the triplets. Thus, if the lifetime of the ME state is sufficiently long and if the structure is conductive of excimers’ formation, the ME state can be trapped in an excimer-like structure. Another observation is that the structures of the S 1 and ME excimers are apparently rather different, leading to different energy ordering of the relevant states. For example, in the S 1 -excimer structure, the ME state is stabilized more in BET-X relative to BET-B, whereas the ME excimer is lower in BET-B. The comparison of the optimized geometries of the 5 ME state (see Table 7.1) reveals that the distance between the tetracene moieties is slightly larger in BET-X and BET-B and that the xanthene bridge is strongly bent. We note that our results also lend an indirect support to the mechanistic hypothe- ses put forward by Schmidt and coworkers. 36 They posited that in solid tetracene a 194 Figure 7.5: Excitation energies of the S 1 , 1 ME, and 5 ME states relative to S 0 (at its equilibrium geometry) at the Franck-Condon (FC) geometry and at the optimized geometries of the S 1 and 5 ME states. Left: BET-B. Right: BET-X. multi-exciton state relaxes, forming an excimer-like structure and acquiring small os- cillator strength via intensity borrowing from S 1 . The structural relaxation, which they described as frustrated photodimerization, lowers the energy of the ME state below ver- tical S 1 energy making the first SF step temperature independent and resulting in strong temperature dependence of the second step. As discussed below, at the excimer-like structures, both BET-X and BET-B show non-zero oscillator strength, 0.001, for the S 0 $ 1 ME transition. As one can see from Fig. 7.5, unconstrained optimization on the 5 ME surface (using DFT) results in the 1 ME state dropping by 0.53 eV below the verti- cal energy S 1 state in BET-B. In solid tetracene, the relaxation of the ME state is likely to be incomplete. Based on the computed energetics for a model Tc dimer (see Table 7.3), relaxation of 0.4 eV would be sufficient to support Schmidt’s hypothesis. 7.3.3 The role of covalent linker in BET-B In this section, we further analyze the effect of the structure on the energetics and cou- plings. In particular, we compare the trends in couplings with those in orbital overlaps and also consider the role of the covalent linkers. To extend the set of representative structures, we consider three optimized structures shown in Fig. 7.6, in addition to the crystal structure of BET-B. The structures differ by the degree of the overlap between 195 the tetracene moieties. We denote them as 33, 22, and 23. The lowest-energy one is BET-B(33); it is the same structure as BET-B(i) in the discussion above. It is most sim- ilar to the BET-B(cry). In addition, we included the S 1 -optimized excimer structures in the analysis (denoted as ‘ex’). Orbital overlaps for these structures are given in Table S2 in SI of Ref.47. As ex- pected, the excimer structure features the largest overlap between the frontier orbitals. Conversely, the X-ray structure in which the rings are less parallel, has smaller overlap than 33. However, the differences between the three optimized structures, 33, 23, and 22, are less obvious. For example, BET-B(33) has smaller overlap than 22 and 23, due to the interplay between different types of displacements and nodal structures of the MOs. 17 Fig. 7.7 shows the correlation of the computed overlaps with the couplings. As one can see, the correlation is not perfect. The differences in overlaps between the two limiting cases, such as the X-ray and excimer structures, correlate with the differences in the couplings. However, for other structures the trends in couplings differ from the computed overlaps. Thus, the orbital overlaps alone should not be taken as a reliable proxy for the electronic couplings. Figure 7.6: Optimized structures of the three isomers of BET-B. BET-B(33, left) is the lowest energy one. BET-B(23, middle) and BET-B(22, right) are 3.6 and 8.6 kcal/mol higher in energy (!B97X-D/cc-pVDZ). Fig. 7.8 shows the couplings and rates computed for model structures in which the covalent linker is removed (see section 7.2). The top panel shows the couplings, whereas 196 Figure 7.7: The orbital overlap versus coupling.jhH A jH B ij +jhL A jL B ij is the sum of the overlaps between the individual fragments, i.e., the HOMO of fragment A with the HOMO of fragment B and the overlap between the LUMO of fragment A with the LUMO of fragment B. the middle and bottom panels show rates relative to the tetracene dimer from the crystal structure (log[r 1 =r 1 (Tc)]). As one can see, the couplings are larger in the presence of the linker. The effect is most pronounced in the structure with the smallest overlap, BET- B(cry). For this structure, we also observe that both the benzene ring and the -CC- part of the linker are significant. Our results on the role of the covalent linker are in line with the experimental findings for pentacene dimers, 37 which showed that the through-bond interaction is important. Such through-bond interactions are turned off when the linkers are removed and the couplings are driven by the overlaps. The effect of the linker on the rate is more complex, because of its effect on energetics. The energy contribution from the linker disfavors SF, as one can see from the middle panel of Fig. 7.8, which shows the energy contribution to the relative rates. The overall rate (Fig. 7.8, bottom panel) shows that the linker slows down SF relative to the unlinked tetracenes at the identical geometric arrangement. 197 Finally, we would like to discuss the nature of the 1 ME state in BET-B and BET-X in crystal and excimer-like structures. In contrast to the 5 ME state, only asymptoti- cally 1 ME can be described as pure 1 (T 1 T 1 ). At typical chromophore orientations, the adiabatic wave function of 1 ME contains small contributions (4-6%) from other singlet configurations, such as CR and excitonic configurations. 17, 43 Indirectly, the wave func- tion composition can be inferred from variations inE b (larger values are indicative of an increased mixing of other singlet configurations into the 1 ME state),jj jj (admix- ture of CR configurations leads to increase injj jj), and transition dipole moments (the oscillator strength of the S 0 $ 1 ME transition is zero for the pure 1 (T 1 T 1 ) state). We observe relatively largeE b in BET-X and BET-B, especially at the excimer-like struc- tures. We also found that at the x-ray structure, the oscillator strength is 0.0004 and 0.0007 for BET-B and BET-X, respectively. At the S 1 excimer geometries, the oscillator strength increases to0.001 for both species. This small oscillator strength acquired by the 1 ME state is likely responsible for weak steady-state emission observed in BET-X in rigid media. 22 The wave function analysis 43, 50 shows that the weight of the 1 (T 1 T 1 ) configuration in the multi-exciton state is roughly 80% (at the crystal structure). Thus, the weight of other singlet configurations is indeed larger than observed in unlinked pairs of chromophores (4-6%). 17, 43 Such large mixture of other configurations explains why in BET-B, the transient absorption of the 1 ME state is noticeably different from the absorption of a pure triplet state. 22 7.4 Conclusion We presented electronic structure calculations of electronic factors controlling SF rates in several covalently linked tetracene dimers. In the dimers with coplanar moieties, the calculations predict slower rates relative to neat tetracene, in qualitative agreement 198 with experimental findings. 28 In contrast, the BET-X and BET-B dimers in which the tetracene rings are staggered, 22 the electronic factors are favorable for SF. The calcula- tions predict faster (relative to neat tetracene) intra-fragment SF (i.e., formation of the 1 ME state) in both BET-X and BET-B. Relative to the coplanar dimers, 28 the computed rate of SF in BET-B is faster by more than 3 orders of magnitude. Fast SF in BET-B is confirmed by the recent experimental study. 22 Early excited-state dynamics in BET- X is similar to that in BET-B. The calculations reveal that the second step of SF, the separation of the 1 ME state into independent triplets, is significantly impeded in BET- X. Furthermore, almost perfect sandwich-like arrangement of the tetracene moieties in BET-X leads to the increased couplings with the ground state. In addition, the excimer formation, which is more favorable in BET-X, enhances both effects. Thus, the calcula- tions suggest that the BET-X type of structures, which are more conductive to excimers formation, lead to a slower rate of the second step and more efficient radiationless re- laxation (to S 0 ). We also analyzed the effect of the fragments orbital overlaps and covalent linkers on the couplings and rates of SF. We found a limited correlation between the overlaps and couplings. In all considered structures, the presence of the linker leads to larger cou- plings, however, the effect on the overall rate is less obvious, since the linkers generally result in less favorable energetics. This complex behavior once again illustrates the im- portance of integrative approaches 5, 6 that evaluate the overall rate, rather than focusing on specific electronic factors, such as energies or couplings. Using such kinetic mod- els is important because couplings and energies are affected by morphology in different ways. 199 Figure 7.8: The effect of the covalent linker on couplings and rates. Top: Cou- plings in various BET-B structures with and without the linker. Center: Relative r 1 computed without couplings (energy contribution only). Bottom: Relative r 1 for different model structures. Rates are computed relative to the tetracene dimer from the crystal structure, log[r 1 =r 1 (Tc)]. 200 Chapter 7 references [1] M. B. Smith and J. Michl. Singlet fission. Chem. Rev., 110:6891–6936, 2010. [2] M.B. Smith and J. Michl. Recent advances in singlet fission. Annu. Rev. Phys. Chem., 64:361–368, 2013. [3] T.F. Schulze, J. Czolk, Y .-Y . Cheng, B. F¨ uckel, R.W. MacQueen, T. Khoury, M.J. Crossley, B. Stannowski, K. Lips, U. Lemmer, A. Colsmann, and T.W. Schmidt. Efficiency enhancement of organic and thin-film silicon solar cells with photo- chemical upconversion. J. Phys. Chem. C, 116:22794–22801, 2012. [4] V . Gray, D. Dzebo, M. Abrahamsson, B. Albinsson, and K. Moth-Poulsen. Triplet- triplet annihilation photon-upconversion: towards solar energy applications. Phys. Chem. Chem. Phys., 16:10345–10352, 2014. [5] A.B. Kolomeisky, X. Feng, and A.I. Krylov. A simple kinetic model for singlet fission: A role of electronic and entropic contributions to macroscopic rates. J. Phys. Chem. C, 118:5188–5195, 2014. [6] S.R. Yost, J. Lee, M.W.B. Wilson, T. Wu, D.P. McMahon, R.R. Parkhurst, N.J. Thompson, D.N. Congreve, A. Rao, K. Johnson, M.Y . Sfeir, M.G. Bawendi, T.M. Swager, R.H. Friend, M.A. Baldo, and T. Van V oorhis. A transferable model for singlet-fission kinetics. Nature Chem., 6:492, 2014. [7] A. Akdag, Z. Havlas, and J. Michl. Search for a small chromophore with efficient singlet fission: Biradicaloid heterocycles. J. Am. Chem. Soc., 132:16302–16303, 2010. [8] T. Minami and M. Nakano. Diradical character view of singlet fission. J. Phys. Chem. Lett., 3:145–150, 2012. [9] T. Minami, S. Ito, and M. Nakano. Fundamental of diradical-character-based molecular design for singlet fission. J. Phys. Chem. Lett., 4:2133–2137, 2013. [10] J.C. Johnson, A.J. Nozik, and J. Michl. High triplet yield from singlet fission in a thin film of 1,3-diphenylisobenzofuran. J. Am. Chem. Soc., 132:16302–16303, 2010. [11] J. Ryerson, J.N. Schrauben, A.J. Ferguson, S.C. Sahoo, P. Naumov, Z. Havlas, J. Michl, A.J. Nozik, and J.C. Johnson. Two thin film polymorphs of the singlet fis- sion compound 1,3-diphenylisobenzofuran. J. Phys. Chem. C, 118:12121–12132, 2014. 201 [12] J.N. Schrauben, J.Ryerson, J. Michl, and J.C. Johnson. Mechanism of singlet fission in thin films of 1,3-diphenylisobenzofuran. J. Am. Chem. Soc., 136:7363– 7373, 2014. [13] T. Zeng, N. Ananth, and R. Hoffman. Seeking small molecules for singlet fission: A heteroatom substitution strategy. J. Am. Chem. Soc., 136:12638–12647, 2014. [14] J. Wen, Z. Havlas, and J. Michl. Captodatively stabilized biradicaloids as chro- mophores for singlet fission. J. Am. Chem. Soc., 137:165–172, 2015. [15] W-L. Chan, M. Ligges, and X-Y . Zhu. The energy barrier in singlet fission can be overcome through coherent coupling and entropic gain. Nat. Chem., 4:840–845, 2012. [16] J.C Johnson, A.J. Nozik, and J. Michl. The role of chromophore coupling in singlet fission. Acc. Chem. Res., 46:1290–1299, 2013. [17] X. Feng, A.V . Luzanov, and A.I. Krylov. Fission of entangled spins: An electronic structure perspective. J. Phys. Chem. Lett., 4:3845–3852, 2013. [18] S. Matsika, X. Feng, A.V . Luzanov, and A.I. Krylov. What we can learn from the norms of one-particle density matrices, and what we can’t: Some results for inter- state properties in model singlet fission systems. J. Phys. Chem. A, 118:11943– 11955, 2014. [19] X. Feng, A.B. Kolomeisky, and A.I. Krylov. Dissecting the effect of morphology on the rates of singlet fission: Insights from theory. J. Phys. Chem. C, 118:19608– 19617, 2014. [20] E.V . Anslyn and D.A. Dougherty. Modern Physical Organic Chemistry. University Science Books, 2006. Chapter 8. [21] N. Streidl, B. Denegri, O. Kronja, and H. Mayr. A practical guide for estimating rates of heterolysis reactions. Acc. Chem. Res., 43:1537–1549, 2010. [22] N.V . Korovina, S. Das, Z. Nett, X. Feng, J. Joy, A.I. Krylov, S.E. Bradforth, and M.E. Thompson. Singlet fission in a covalently linked cofacial alkynyltetracene dimer. J. Am. Chem. Soc., 138:617–627, 2016. [23] N.G. van Kampen. Stochastic Processes in Physics and Chemistry. Elsevier, 2 edition, 2001. [24] R.A. Marcus. Chemical and electrochemical electron-transfer theory. 15:155, 1964. 202 [25] B. Peters. Common features of extraordinary rate theories. J. Phys. Chem. C, 119:6349–6356, 2015. [26] E. Busby, T.C. Berkelbach, B. Kumar, A. Chernikov, Y . Zhong, H. Hlaing, X.-Y . Zhu, T.F. Heinz, M.S. Hybertsen, M.Y . Sfeir, D.R. Reichman, C. Nuckolls, and O. Yaffe. Multiphonon relaxation slows singlet fission in crystalline hexacene. J. Am. Chem. Soc., 136:10654–10660, 2014. [27] R.J. Dillon, G.B. Piland, and C.J. Bardeen. Different rates of singlet fission in mon- oclinic versus orthorhombic crystal forms of diphenylhexatriene. J. Am. Chem. Soc., 135:17278–17281, 2013. [28] J.J. Burdett and C.J. Bardeen. The dynamics of singlet fission in crystalline tetracene and covalent analogs. Acc. Chem. Res., 46:1312–1320, 2013. [29] H. Liu, V .M. Nichols, L. Shen, S. Jahansouz, Y . Chen, K.M. Hanson, C.J. Bardeen, and X. Li. Synthesis and photophysical properties of a ”face-to-face” stacked tetracene dimer. Phys. Chem. Chem. Phys., 17:6523–6531, 2015. [30] J.M. Giaimo, J.V . Lockard, L.E. Sinks, A.M. Scott, T.M. Wilson, and M.R. Wasielewski. Excited singlet states of covalently bound, cofacial dimers and trimers of perylene-3,4:9,10-bis(dicarboximide)s. J. Phys. Chem. A, 112:2322– 2330, 2008. [31] R.J. Lindquist, K.M. Lefler, K.E. Brown, S.M. Dyar, E.A. Margulies, R.M. Young, and M.R. Wasielewski. Energy flow dynamics within cofacial and slip-stacked perylene-3, 4-dicarboximide dimer models of -aggregates. J. Am. Chem. Soc., 136:14192–14923, 2014. [32] E.A. Margulies, L.E. Shoer, S.W. Eaton, and M.R. Wasielewski. Excimer forma- tion in cofacial and slip-stacked perylene-3,4:9,10-bis(dicarboximide) dimers on a redox-inactive triptycene scaffold. Phys. Chem. Chem. Phys., 16, 2014. 23735- 23742. [33] P.M. Zimmerman, Z. Zhang, and C.B. Musgrave. Singlet fission in pentacene through multi-exciton quantum states. Nature Chem., 2:648–652, 2010. [34] B.J. Walker, A.J. Musser, D. Beljonne, and R.H. Friend. Singlet exciton fission in solution. Nature Chem., 5:1019–1024, 2013. [35] H.L. Stern, A.J. Musser, S. Gelinas, P. Parkinson, L.M. Herz, M.J. Bruzek, J. An- thony, R.H. Friend, and B.J. Walker. Identification of a triplet pair intermediate in singlet exciton fission in solution. Proc. Nat. Acad. Sci., 112:7656–7661, 2015. [36] M.J.Y . Tayebjee, R.G.C.R. Cladyy, and T.W. Schmidt. The exciton dynamics in tetracene thin films. Phys. Chem. Chem. Phys., 14:14797–14805, 2013. 203 [37] J. Zirzlmeier, D. Lehnherr, P.B. Coto, E.T. Chernick, R. Casillas, B.S. Basel, M. Thoss, R.R. Tykwinski, and D.M. Guldi. Singlet fission in pentacene dimers. Proc. Nat. Acad. Sci., 112:5325–5330, 2015. [38] D. Casanova, L.V . Slipchenko, A.I. Krylov, and M. Head-Gordon. Double spin-flip approach within equation-of-motion coupled cluster and configuration interaction formalisms: Theory, implementation and examples. J. Chem. Phys., 130:044103, 2009. [39] F. Bell, P.M. Zimmerman, D. Casanova, M. Goldey, and M. Head-Gordon. Re- stricted active space spin-flip (RAS-SF) with arbitrary number of spin-flips. Phys. Chem. Chem. Phys., 15:358–366, 2013. [40] P.M. Zimmerman, F. Bell, D. Casanova, and M. Head-Gordon. Mechanism for singlet fission in pentacene and tetracene: From single exciton to two triplets. J. Am. Chem. Soc., 133:19944–19952, 2011. [41] P.M. Zimmerman, C.B. Musgrave, and M. Head-Gordon. A correlated electron view of singlet fission. Acc. Chem. Res., 46:1339–1347, 2012. [42] D. Casanova. Electronic structure study of singlet-fission in tetracene derivatives. J. Chem. Theory Comput., 10:324–334, 2014. [43] A.V . Luzanov, D. Casanova, X. Feng, and A.I. Krylov. Quantifying charge res- onance and multiexciton character in coupled chromophores by charge and spin cumulant analysis. J. Chem. Phys., 142:224104, 2015. [44] T.H. Dunning. Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogen. J. Chem. Phys., 90:1007–1023, 1989. [45] J.M. Herbert, L. D. Jacobson, K.U. Lao, and M.A. Rohrdanz. Rapid computation of intermolecular interactions in molecular and ionic clusters: self-consistent po- larization plus symmetry-adapted perturbation theory. Phys. Chem. Chem. Phys., 14:7679–7699, 2012. [46] L. D. Jacobson and J. M. Herbert. An efficient, fragment-based electronic structure method for molecular systems: Self-consistent polarization with perturbative two- body exchange and dispersion. J. Chem. Phys., 134:094118, 2011. [47] X. Feng and A.I. Krylov. On couplings and excimers: Lessons from studies of singlet fission in covalently linked tetracene dimers. Phys. Chem. Chem. Phys., 18:7751–7761, 2016. 204 [48] Y . Shao, Z. Gan, E. Epifanovsky, A.T.B. Gilbert, M. Wormit, J. Kussmann, A.W. Lange, A. Behn, J. Deng, X. Feng, D. Ghosh, M. Goldey, P.R. Horn, L.D. Jacob- son, I. Kaliman, R.Z. Khaliullin, T. Kus, A. Landau, J. Liu, E.I. Proynov, Y .M. Rhee, R.M. Richard, M.A. Rohrdanz, R.P. Steele, E.J. Sundstrom, H.L. Wood- cock III, P.M. Zimmerman, D. Zuev, B. Albrecht, E. Alguires, B. Austin, G.J.O. Beran, Y .A. Bernard, E. Berquist, K. Brandhorst, K.B. Bravaya, S.T. Brown, D. Casanova, C.-M. Chang, Y . Chen, S.H. Chien, K.D. Closser, D.L. Crittenden, M. Diedenhofen, R.A. DiStasio Jr., H. Do, A.D. Dutoi, R.G. Edgar, S. Fatehi, L. Fusti- Molnar, A. Ghysels, A. Golubeva-Zadorozhnaya, J. Gomes, M.W.D. Hanson- Heine, P.H.P. Harbach, A.W. Hauser, E.G. Hohenstein, Z.C. Holden, T.-C. Ja- gau, H. Ji, B. Kaduk, K. Khistyaev, J. Kim, J. Kim, R.A. King, P. Klunzinger, D. Kosenkov, T. Kowalczyk, C.M. Krauter, K.U. Laog, A. Laurent, K.V . Lawler, S.V . Levchenko, C.Y . Lin, F. Liu, E. Livshits, R.C. Lochan, A. Luenser, P. Manohar, S.F. Manzer, S.-P. Mao, N. Mardirossian, A.V . Marenich, S.A. Maurer, N.J. May- hall, C.M. Oana, R. Olivares-Amaya, D.P. O’Neill, J.A. Parkhill, T.M. Perrine, R. Peverati, P.A. Pieniazek, A. Prociuk, D.R. Rehn, E. Rosta, N.J. Russ, N. Ser- gueev, S.M. Sharada, S. Sharmaa, D.W. Small, A. Sodt, T. Stein, D. Stuck, Y .-C. Su, A.J.W. Thom, T. Tsuchimochi, L. V ogt, O. Vydrov, T. Wang, M.A. Watson, J. Wenzel, A. White, C.F. Williams, V . Vanovschi, S. Yeganeh, S.R. Yost, Z.-Q. You, I.Y . Zhang, X. Zhang, Y . Zhou, B.R. Brooks, G.K.L. Chan, D.M. Chipman, C.J. Cramer, W.A. Goddard III, M.S. Gordon, W.J. Hehre, A. Klamt, H.F. Schaefer III, M.W. Schmidt, C.D. Sherrill, D.G. Truhlar, A. Warshel, X. Xu, A. Aspuru- Guzik, R. Baer, A.T. Bell, N.A. Besley, J.-D. Chai, A. Dreuw, B.D. Dunietz, T.R. Furlani, S.R. Gwaltney, C.-P. Hsu, Y . Jung, J. Kong, D.S. Lambrecht, W.Z. Liang, C. Ochsenfeld, V .A. Rassolov, L.V . Slipchenko, J.E. Subotnik, T. Van V oorhis, J.M. Herbert, A.I. Krylov, P.M.W. Gill, and M. Head-Gordon. Advances in molec- ular quantum chemistry contained in the Q-Chem 4 program package. Mol. Phys., 113:184–215, 2015. [49] A.I. Krylov and P.M.W. Gill. Q-Chem: An engine for innovation. WIREs Comput. Mol. Sci., 3:317–326, 2013. [50] D. Casanova and A.I. Krylov. Quantifying local excitation, charge resonance, and multiexciton character in correlated wave functions of multichromophoric systems. J. Chem. Phys., 144:014102, 2016. 205 Chapter 8: Intra- vs. Inter-Molecular singlet fission in covalently linked dimer 8.1 Introduction Singlet fission (SF), a process in which one singlet excited state is converted into two coupled triplet states, is of interest in the context of organic photovoltaic technology. 1, 2 Although design principles for materials capable of efficient SF remain elusive, it is clear that morphology of a molecular solid is an important factor affecting the two steps in the SF photophysical mechanism, that is, the formation of an intermediate state of multiexcitonic character and its splitting into two non-interacting triplets. Due to the inherent difficulties with controlling morphology in molecular solids, the idea of using covalently linked dimers, in which the relative orientation of the two chromophore moieties can be tuned up at will by the rational design of the linker, shows much promise in the search for efficient SF materials. 3–13 Among different chro- mophores capable of SF, tetracene is an interesting system because of slight endoergicity of SF: close level matching is more attractive from the efficiency point of view. 206 Recently, dimers with staggered tetracene moieties have shown promising SF abil- ity. In particular, ortho-bis(5-ethynyltetracenyl)benzene (BET-B) was found to exhibit efficient SF on the time scale of 1-10 ps in neat films, in solution, and when doped into DPT; 8 these rates are an order of magnitude faster than the SF rate measured in neat tetracene. Fig. 8.1 shows the structure of BET-B. On the basis of calculations, 9 fast SF rates were attributed to the improved energetics and couplings relative to the neat tetracene. The analysis of electronic states illuminated the role of the covalent linker: it facilitates through-bond interactions between the tetracene moieties leading to the no- ticeable changes in the wave-function composition of the relevant states relative to the unlinked tetracenes at the same configuration. This system demonstrated that by us- ing the linked dimers scaffold, one can control both through-space and through-bond interactions between the chromophores. Thus, one may be able to tune the electronic properties of the individual chromophores dimers to maximize SF rates. Figure 8.1: Two different views of the structure of the two BET-B molecules taken form crystal structure. The distance between the covalently linked tetracene rings (unitsA andB) is 3.34 ˚ A; the distance betweenA andA 0 is 3.40 ˚ A.A andB rings are staggered (the angle between their long axes is 45 ), whereas A and A 0 are parallel (zero angle) but shifted along the long axis by 1.5 ˚ A (about 0.6 of benzene’s ring width). 207 One interesting aspect of BET-B revealed by calculations 9 is that the distance and the overlap between the rings depends strongly on the competition between inter and intra-molecular dispersion interactions between the -systems on the tetracene moieties. For example, geometry optimization of isolated BET-B results in the structure in which the two ethynyl units in the linker are slightly bent and the two rings are more parallel than in the X-ray structure (Fig. 8.1). The driving force for this distortion is intra-molecular dispersion interaction. Interestingly, in the X-ray structure the distance between the tetracene moieties from the adjacent BET-B molecules (A and A 0 ) is nearly the same as between the rings from the same BET-B molecule (A and B). Also, the adjacent tetracene moieties from the two different BET-B molecules are more parallel than the intra-molecular pair. These structural features of (BET-B) 2 lead to an important mechanistic question: whether SF in the neat film of BET-B involves intra- or inter-molecular fission, or both. While intra-molecular SF has been characterized in several systems, 10–14 the competition between intra- and inter-molecular pathways in dimers’ aggregates has not been investigated. In this paper, we analyze the electronic structure of the two BET-B molecules and assess the relative importance of intra- versus inter-molecular SF. As in our previous work, 9, 15–17 we employ adiabatic framework for calculating relevant states and their couplings. In the context of SF, relevant electronic states of a system comprising two chromophores (A and B) are derived from excitonic (EX, or local excitations, LE), charge-resonances (CR), and multiexciton (ME) configurations. 15 We analyze the char- acter of many-electron adiabatic states in terms of EX, CR, and ME configurations using wave-function analysis scheme based on localized orbitals and spin-correlators. 18, 19 We address the following questions: 208 What is the degree of delocalization of the relevant electronic states and what impact it has on the SF properties? How electronic states of the single BET-B molecule are affected by inter- molecular interactions? We consider energies, couplings, and the characters of the states. How do the SF mechanism and rates in the dimer compare to those in isolated BET-B? 8.2 Theoretical methods and computational details The structures of BET-B and (BET-B) 2 were taken from the crystal structure and posi- tions of all hydrogens were optimized by!B97X-D. BET-B has C 2 symmetry, whereas the dimer has C i symmetry. The Cartesian geometries of all structures are given in supplemental information (SI) of Ref.20. We employ adiabatic framework for calculating all relevant states by using the restricted-active-space spin-flip configuration interaction (RASCI-SF or RAS-SF) method 21–23 with single, double, and quadrupole spin-flip. In the systems with two chromophores (such as tetracene dimer and BET-B molecule), we use RAS-2SF and a quintet reference state. In the system with four teracenes, (BET-B) 2 , we use RAS-4SF and a high-spin reference state with eight unpaired electrons. We quantify the electronic couplings between the states by using respective one-particle transition density matri- ces. 15, 24 We then analyze the characters of the adiabatic states in terms of EX, CR, and ME configurations using wave-function analysis tool based on localized orbitals and spin correlators. 18, 19 The weights of different configurations in RAS-SF calculations 209 are used to compute state-specific energy corrections, 18, 25 as described below (section 8.2.1). As in our previous studies of SF, 9, 17 we estimate the effects of changes in electronic structure on the SF rates by using a simple kinetic model. 16 The rate of 1 ME state forma- tion is given by the following expression derived by using linear free energy approach and Fermi Golden Rule: r 1 r[S 1 (AB)! 1 ME] jj jj E 2 e E sf ; (8.1) where = 1 k b T and is a parameter from the free energy relationship and E is the energy difference between the raw (uncorrected) RAS-SF energies of the S 1 and 1 ME states. 17 The rates are computed relative to the reference system, a single BET-B molecule. When several target 1 ME states are available (as it happens in (BET-B) 2 where the two lowest 1 ME states are degenerate), the respective rates are summed. When the gap between the lowest excitonic pair (EX 1 and EX 2 ) is comparable with kT, the rates from the both states are computed and averaged using Boltzmann’s populations. The cc-pVDZ basis was used in all calculations reported here. We note that this is a smaller basis than in Refs. 8, 9, where we employed a mixed basis (cc-pVTZ-f on car- bons and cc-pVDZ on hydrogens). Core electrons were frozen in RAS-SF calculations. All calculations were performed using the Q-Chem electronic structure program. 26, 27 8.2.1 Raw RAS-SF energies and energy correction The RAS-SF configuration expansion contains all important determinants needed to de- scribe EX/LE, CR, and ME configurations; thus, it provides a balanced description of 210 electronic states relevant to SF. Although RAS-SF wave-functions are of good qual- ity, the dynamic correlation is largely missing. Consequently, the absolute excitation energies are not sufficiently accurate. In our previous work, 9, 15–17 we used simple em- pirical correction to account for missing dynamical correlation. Here we employ a more rigorous energy correction scheme based on state-specific wave-function composition. The wave-function analysis scheme for multi-chromophore assemblies 18 allows one to decompose the total adiabatic wave-function in terms of EX/LE, CR, and ME configu- rations. The energy correction scheme in terms of wave-function composition was described in detail in Refs. 18, 25. The expression for the corrected energies is: E[ ] =E 0 [ ] + X X ! X E C [ X ]; (8.2) whereX corresponds to a specific type of excitation contribution andE C [ X ] are cor- rection energies associated with each X contribution with ! X weight. The different types of excitations considered here are: EX/LE, CR, and the ME configurations. Corrected energies for BET-B For a bi-chromophore system, the weights of different configurations are given by 18 : LE indices ! LE A and ! LE B , CR numbers ! CR AB and ! CR BA , and ME indices ! TT (triplet- triplet) and! SS (singlet-singlet). We note that in BET-B the two tetracene moieties are equivalent by symmetry; thus, the weights of LE and CR configurations are symmetric and it is sufficient to consider their sums,! LE and! CR . For low-lying excited singlet states of a bichromophoric system, Eq. (8.2) assumes the following form: E[ ] =E 0 [ ] +! LE E C [LE] +! TT E C [TT ] +! SS E C [SS] +! CR E C [CR] (8.3) 211 Based on our previous studies, 18, 19 the SS contributions in the decomposition of low- lying states are usually rather small. Here we approximate the correction to these contri- butions (E C [SS]) as twice the correction of singlet local excitons (E C [SS] = 2E C [LE]). Higher contributions like QQ (quintet-quintet) are not considered. Hence: E[ ] =E 0 [ ] + (! LE + 2! SS )E C [LE] +! TT E C [TT ] +! CR E C [CR] (8.4) To obtain the correction energy to the CR contributions, we compare the energies of RAS-2SF CR diabatic states (computed as described in our previous work 25 ) to the ex- citation energy of the CT state computed with constrained DFT (C-DFT) with!B97X- D. The C-DFT state corresponds to the A + B configuration, while the RAS-2SF CR diabatic states for symmetric dimers correspond to “+” and “” linear combinations of the A + B and A B + configurations. Therefore, in order to compare C-DFT with the RAS-2SF energies, we take the average of the two lowest RAS-2SF CR diabatic states: E C [CR] =E !B97XD CT 1 2 E RAS2SF CR1 +E RAS2SF CR2 =1:693 eV (8.5) The correction energies to the TT contributions are obtained from the energies of the lowest quintet state (Q 1 ) of BET-B as: E C [TT ] =E !B97XD Q 1 E RAS2SF Q 1 =0:462 eV (8.6) 212 Finally, the energy correction to LE contributions is obtained by enforcing the RAS-2SF corrected energy for the lowest EX singlet state of BET-B to be equal to the excitation energy of the lowest excited singlet computed with!B97X-D: E C [LE] = E !B97XD S 1 (E RAS2SF S 2 +! TT S 2 E C [TT ] +! CR S 2 E C [CR]) ! LE S 2 + 2! SS S 2 =0:939 eV (8.7) where E !B97XD S 1 is the excitation energy of the lowest excited singlet of the BET-B molecule computed with!B97X-D, and the S 2 label corresponds to the lowest EX sin- glet of BET-B obtained with RAS-2SF (see Table S1). Corrected energies for (BET-B) 2 The expression of corrected energies for (BET-B) 2 is: E[ ] =E 0 [ ] + (! LE + 2! SS )E C [LE] +! TT E C [TT ] +! CR E C [CR] +! CR 0 E C [CR 0 ] (8.8) where CR 0 corresponds to charge resonances between the two BET-B molecules and E C [CR 0 ] is obtained as: E C [CR 0 ] = E !B97XD S 1 (E RAS4SF S 7 + (! LE S 7 + 2! SS S 7 )E C [LE] +! TT S 7 E C [TT ] +! CR S 7 E C [CR]) ! CR 0 S 7 =1:522 eV (8.9) where here E !B97XD S 1 is the excitation energy of the lowest excited singlet of (BET- B) 2 computed with!B97X-D, and theS 7 label corresponds to the lowest EX singlet of (BET-B) 2 obtained with RAS-4SF (see Table S1). Higher configurations with the TTS form have been included in the! TT weights. 213 8.2.2 Comparison of energies and anticipated error bars Here we discuss anticipated error bars of the corrected energies. Let us revisit ter- tacene, to illustrate the origin of the problem. Table 8.1 shows excitation energies for tetracene. As one can see, RAS-SF considerably overestimates experimental energies and the errors are different for the singlet and triplet states: they are 0.88 eV and 0.28 eV , respectively, in cc-pVDZ basis (1.41 eV and 0.53 eV , in a triple-zeta quality basis). Thus, the errors in relative energies of the singlet and multiexciton states in dimers (and larger aggregates) will be significant, as illustrated by E[S 1 ]-2E[T 1 ] given in the last row of Table 8.1. In our previous work, 9, 15–17 we used simple empirical correction following the idea from Ref. 28: raw RAS-SF energies of S 1 and S 1 0 states were shifted down by 1 eV , energies of the triplets were shifted down by 0.2 eV , and energies of the ME states were shifted down by 0.4 eV (twice the value for the triplet). Since we always compute rates relative to the reference tetracene system, the correction was canceling out exactly. That is, the relative rates computed using the old fixed-value correction are identical to the rates computed using raw RAS-SF energies. In this approach, the effects of possible contributions of CR configurations and mixing of LE and TT configurations were not ac- counted for. The new energy correction scheme is based on wave-function compositions of the adiabatic states and, therefore, accounts for these effects. However, the correction relies on the reference values. We choose to use!B97X-D/cc-pVDZ as our reference method because we could not find a more reliable approach, which is still affordable. Thus, our corrected RAS-SF energies are going to be as good as!B97X-D/cc-pVDZ is. 214 Table 8.1: S 1 and T 1 vertical excitation energies (eV) for tetracene a . State !B97X-D/DZ b !B97X-D/TZ c RAS-SF/DZ b RAS-SF/TZ c exp S 1 3.198 3.144 4.078 3.989 2.583 29 T 1 1.629 1.691 1.894 1.878 1.35 30 E[S 1 ]-2E[T 1 ] -0.06 -0.24 0.29 0.23 -0.12 a : Structure optimized with!B97X-D/cc-pVDZ. b : DZ: cc-pVDZ. c : TZ: cc-pVTZ-f on heavy atoms and cc-pVDZ on hydrogens. 215 As one can see from Table 8.1, the errors of!B97X-D/cc-pVDZ against the exper- iment are 0.615 eV and 0.279 eV , respectively. These values give an estimate of the error bars in the absolute excitation energies computed using our scheme: 0.62 eV for S 1 /S 1 0, 0.28 eV for T 1 /T 1 0, and 0.56 eV for ME. The errors in the S 1 1 ME gap are much smaller, about 0.06 eV . To evaluate the accuracy of absolute excitation energies, we can compare the computed corrected value of S 1 of 2.61 eV with the experimental excitation energies of BET-B in solution (2.42 eV) and in neat film (2.32 eV). Thus, the computed S 1 state is blue shifted by about 0.2 eV . The agreement could be improved by using higher-quality reference data for computing energy correction. 8.3 Results and discussion In the context of SF, relevant electronic states of a system comprising two chromophores (A andB) are EX, CR, and ME states. 15 The former can be described as linear com- bination of locally excited states (S 1 ) of the individual fragments (c 1 S 1 (A)S 0 (B) + c 2 S 0 (A)S 1 (B)); these states carry oscillator strength of the transition giving rise to the initially excited bright state (the distribution of the oscillator strength and the energy splitting between these states depends on the relative orientation of the chromophores). Because EX states are derived from the excited states of the individual chromophores, they are often referred to as LE states. At small inter-chromophore distances, adia- batic wave-functions of the excitonic states also include CR configurations (A + B and A B + ). The ME states are derived from the simultaneously excited triplet states of the individual chromophores (T 1 (A)T 1 (B)), which can be coupled into the overall singlet, triplet, or quintet state. The quintet state usually has pure multi-excitonic character, whereas the 1 ME state can mix with other singlet configurations (EX and CR), which lowers its energy and affects the electronic couplings with other singlet states. 15, 18, 19 At 216 large inter-chromophore separation, the energy of the EX states equals the energy of the S 1 state of the individual chromophore and the energy of the ME states equals twice the energy ofT 1 . The first step of SF entails a non-adiabatic transition from the EX to the 1 ME manifold. Once 1 ME is formed, two triplets can, in principle, separate by Dexter energy transfer, 16, 31 provided that possible triplet acceptors are available. 8 The energy splitting between 1 ME and 5 ME arises due to stabilization of the 1 ME state by config- uration interaction; it represents the energy penalty that needs to be overcome for two triplets to separate. This quantity is called multiexciton binding energy. 15 This relatively simple picture becomes increasingly complex for larger chromophore aggregates. The number of the EX states increases proportionally to the number of the chromophores and the excitonic band develops. 32, 33 Importantly, these states are quite delocalized; 28, 34 the calculations in tetracene derivatives 35 and pentacene 34 show delo- calization over more than 10 molecules and an average electron-hole distance greater than 6 ˚ A. The ME states, on the other hand, are more localized. 18 Thus, many computa- tional studies of SF focus on model dimers whose structures are taken from the available X-ray structures. Here we compute the manifold of low-lying excited states in BET-B and (BET-B) 2 and describe their characters in terms of the LE, ME and CR configurations. Figure 8.1 shows labels marking the individual chromophores used throughout this study. In BET- B, A and B refer to the two tetracene rings. In (BET-B) 2 , states localized on one of the BET-B molecules are denoted by either A/B or A 0 /B 0 , whereas the inter-chromophore states are marked by mixed labels, i.e., A/B 0 , A 0 /B, etc. In the case of the BET-B molecule, we consider two lowest excitonic states (S 1 (AB) and S 10 (AB), using the notations from our previous work 9, 15–17 ) as well as the singlet and quintet ME states. Table 8.2 shows energies and properties of these states. The energies 217 were computed from the raw RAS-SF energies by using the energy correction computed by Eq. (8.4), 18 which accounts for dynamical correlation. As one can see, both excitonic states include some CR contributions. Noticeably, the 1 ME state has quite substantial contribution from the CR configurations (20%), which is responsible for non-vanishing oscillator strength and for significant coupling with the excitonic states, evaluated in terms of the squared norm of the symmetrized one-particle transition-density matrix (jj jj 2 ). 8, 9 The multiexciton binding energy for BET-B is 0.41 eV . Table 8.2: Electronic properties of BET-B. State E ex , eV f l ! LE ! CR ! TT ! SS jj jj 2 S 1 (AB) 2.69 0.307 0.80 0.14 0.00 0.06 0.056 S 1 0(AB) 3.09 0.932 0.88 0.02 0.01 0.09 0.092 1 ME 2.49 <0.001 0.00 0.20 0.79 0.00 5 ME 2.90 - 0.00 0.00 1.00 0.00 - jj jj denotes coupling with the 1 ME state. The (BET-B) 2 dimer has C i symmetry. Consequently, the canonical MOs are delo- calized over both BET-B molecules and even qualitative assignment of state characters of the computed adiabatic states cannot be easily performed by analyzing the wave- function amplitudes. However, the state characters can be unambiguously determined and quantified using a recently introduced approach based on localized MOs and spin- correlators. 18 Frontier fragment orbitals for the (BET-B) 2 system within the RAS2 space in RAS-4SF calculations mainly correspond to the HOMOs and LUMOs of the four tetracene moieties with small delocalization towards the covalent bridge (Figure 8.2). Qualitatively, the manifold of the low-lying singlet adiabatic excited states in the sys- tem with four chromophore units is derived from the 16 diabatic configurations: four EX states, six ME states, and six CR states. We computed the lowest 16 singlet adia- batic states of (BET-B) 2 and assigned their character using our wave-function analysis 218 scheme. 18 We also computed the six lowest high-spin 5 ME states. The results for all 1 ME and 5 ME states, and for the lowest seven EX/CR states are collected in Table 8.3. These states constitute the lowest-energy manifold (the next singlet state appears at 3.22 eV). Figure 8.2: Frontier fragment orbital diagram of (BET-B) 2 . The left and right panels correspond to the orbitals localized on the top and bottom BET-B molecules, respectively. Orbital occupations correspond to the leading configuration of the ground-state wave function. All computed 1 ME/ 5 ME states can be described as singlet- and quintet-coupled pairs of triplet states residing on two tetracene fragments. As one can see, the two lowest 1 ME states, which are degenerate, are of intra-molecular type. They are energetically sepa- rated from the next four 1 ME states, which are of the inter-molecular type, by a gap of0.3 eV . The intra-molecular 1 ME states have considerably larger mixing of the CR 219 configurations, which is responsible for their energy lowering and also larger couplings with the EX states. This behavior can be explained by through-bond interactions within each BET-B molecule, which lowers the energy and increases the intra-molecular cou- plings. The two lowest EX/CR states are located at 2.61 eV and 2.64 eV; EX 1 is dark and EX 2 is bright. We note that both states are delocalized and feature noticeable inter- and intra-molecular CR. The two singlet states at 2.73 eV and 2.82 eV (i.e., CR 1 and EX 3 in Table 8.3) are derived from a mixture of LE and CR on the two outer tetracene units (B and B 0 ), whereas the two highest computed states are predominantly of CR character. There are six quintet ME states, which are nearly degenerate (2.79-2.84 eV). Simi- larly to BET-B, their wave-functions are dominated by the TT configurations. The ab- sence of other configurations in these states explains small splittings of the energies, in contrast to the 1 ME manifold. Thus, through-bond interaction has much less of an effect on the 5 ME states relative to the 1 ME ones. We observe that five out of six 5 ME states have mixed intra- and inter-molecular character. The highest 5 ME state is almost pure TT 0 (inter-molecular) state. The multi-exciton binding energy of the lowest 1 ME state is 0.34-0.39 eV , which is very close to the value obtained for BET-B. In contrast, the four highest inter-molecular 1 ME states are practically degenerate with the 5 ME band, leading to the vanishing multiexciton binding energies. 220 Table 8.3: Electronic properties of (BET-B) 2 . State E ex (f l ), eV ! LE A ! LE A 0 ! LE B ! LE B 0 ! CR ! CR 0 ! TT ! TT 0 jj jj 2 jj 0 jj 2 ME 1 ( 1 TT) 2.44 0 0 0 0 0.17 0 0.81 0 0.024 0.030 ME 2 ( 1 TT) 2.44 0 0 0 0 0.17 0 0.81 0 0.024 0.031 ME 3 ( 1 TT 0 ) 2.77 0 0 0 0 0 0 0 0.98 0.009 0.019 ME 4 ( 1 TT 0 ) 2.78 0 0 0 0 0.02 0 0 0.97 0.001 0.003 ME 5 ( 1 TT 0 ) 2.78 0 0 0 0 0.01 0 0 0.98 0.021 0.042 ME 6 ( 1 TT 0 ) 2.79 0 0 0 0 0.02 0 0 0.97 0.014 0.027 5 ME 1 2.79 0 0 0 0 0 0 0.20 0.78 - - 5 ME 2 2.80 0 0 0 0 0 0 0.42 0.55 - - 5 ME 3 2.81 0 0 0 0 0 0 0.38 0.61 - - 5 ME 4 2.82 0 0 0 0 0 0 0.20 0.79 - - 5 ME 5 2.82 0 0 0 0 0 0 0.40 0.59 - - 5 ME 6 2.84 0 0 0 0 0 0 0.05 0.93 - - EX 1 2.61 0.25 0.25 0.05 0.05 0.09 0.14 0 0.02 EX 2 2.64 (0.486) 0.12 0.12 0.13 0.13 0.13 0.21 0 0.01 EX 3 2.82 0.01 0.01 0.22 0.21 0.10 0.29 0 0.02 CR 1 2.73 (0.473) 0 0 0.11 0.12 0.07 0.60 0 0.01 CR 2 2.90 0.10 0.10 0.06 0.06 0.01 0.56 0 0.01 CR 3 2.90 (0.170) 0 0 0.07 0.07 0.72 0.01 0.07 0 CR 4 2.89 0 0 0.05 0.05 0.75 0.01 0.08 0 jj jj andjj 0 jj denote couplings with the two lowest EX states (EX 1 and EX 2 ), respectively. 221 Figure 8.3 shows the energy diagram for BET-B and (BET-B) 2 . As one can see, in (BET-B) 2 , the transition to the lowest two 1 ME states from the lowest EX pair is exoergic by 0.17 eV , whereas the four inter-molecular 1 ME states are higher in energy (by 0.16 eV). Thus, most likely, SF in BET-B aggregates proceeds via intra-molecular 1 ME states, as in the isolated covalent dimer. Although we anticipate that our com- puted absolute excitation energies might have substantial errors, we estimate the errors in the S 1 1 ME gap to be about 0.06 eV (see section 8.2.2 for discussion). Thus, we believe that the relative ordering of the lowest EX states relative to the 1 ME manifold is described correctly by the present calculations. Figure 8.3: Energy levels of BET-B (left) and (BET-B) 2 (right). Bright states are marked in red; quintet ME states are marked in blue; all other states are marked in black. The interaction between the two BET-B molecules lowers the excitation energies of the lowest 1 ME and EX states in (BET-B) 2 with respect to the transition energies in BET-B by 0.05 eV and 0.08 eV , respectively. The larger effect for the excitonic state is consistent with its delocalized character. Thus, the overall driving force for SF is slightly 222 reduced in the dimer with respect to the BET-B monomer (by 0.03 eV). The couplings are also slightly reduced, probably because of the delocalized character of the excitonic states. Nevertheless, these changes are rather small to significantly affect the rates. The computed rates for the transition from the EX 1 /EX 2 pair to the two lowest ME states in (BET-B) 2 are almost the same as in BET-B: log(r 1 (BET-B) 2 /r 1 (BET-B))=-0.4. We also estimated the rate of transitions to the inter-molecular 1 ME states; as expected, the rate is considerably lower (by about 3 orders of magnitude) relative to the rate to the lowest (intra-molecular) 1 ME states. An intriguing thought is that if these states are populated, either via transitions from the lowest EX state or from higher excited states, one might expect more efficient separation of the triplets due to vanishing multiexciton binding energies. Thus, our calculations suggest that SF in neat films of BET-B is predominantly intra-molecular and is similar to that in the isolated dimer. These results explain and rationalize why the SF rates in BET-B are virtually the same in solution, neat film and when doped in DPT. 8 Although electronic structure of the dimers (and larger aggregates) of BET-B is rather complex and features many more states, the energies and character of the relevant states, the lowest excitonic and ME states, are similar to those of the isolated dimer. The manifold of the inter-molecular ME states lies higher in energy; also, because inter-molecular 1 ME states have less of CR character, they feature smaller electronic couplings. We note that relatively weak couplings for the inter-molecular 1 ME states in (BET- B) 2 might be not only because there is no through-bond interaction, which is stronger than the through-space one, 9 but also because the A and A 0 tetracene moieties are shifted 223 by about 0.6 benzene rings (Fig. 8.1), which is unfavorable for couplings. 15 In or- der to test how the interplay between through-space and through-bond inter- and intra- molecular interactions can be affected by different packing of dimers in a molecular solid or aggregate, we analyzed electronic states of (BET-B) 2 in three additional model structures varying by the offset between the A and A 0 chromophores. In the first struc- ture, the two middle tetracenes are perfectly stacked; this structure is 0.22 eV higher than the dimer form the crystal structure. In the second structure, the two BET-B molecules are shifted such that the two tetracenes are offset by exactly half of the benzene ring (this structure is similar to the crystal structure; its energy is only 0.02 eV higher). In the third structure, the tetracenes are offset by one benzene ring (this structure is 0.09 eV above the original X-ray structure). Fig. 8.4 shows the energies of the ME and EX/CR manifolds in the three structures; the detailed analysis of the state characters and energies is given in the SI of Ref.20. The first observation is that the electronic properties of the half-ring offset structure are rather similar to the model X-ray dimer. In the structures with zero and full-ring off- set tetracenes, the state ordering changes significantly. The most striking observation is broadening of the ME band and strong stabilization of the lowest excitonic state (in the structure with stacked tetracenes, this state develops significant (46%) CR character). These EX/CR states drop significantly below the ME band, which is conducive to exci- ton trapping in the excimer-like states. Based on the recent study of two different linked tetracene dimers, 8, 9 we expect that such type of molecular packing will be detrimental to SF. Interestingly, strong perturbation to the state ordering and the EX/CR band does not affect the wave-function composition of the lowest 1 ME states — they retain their pure intra-molecular character. We conclude that small variations in molecular pack- ing of BET-B-like dimers, which feature strong through-bond coupling between the two 224 linked tetracenes, will not change the balance between intra- versus inter-molecular cou- plings, that is, we expect that the lowest ME states are likely to be localized on individ- ual dimers, despite relatively good overlap between the-systems of the tetracenes from different molecules. This is a characteristic feature of BET-B deriving from the strong through-bond coupling between the chromophores. Since the strength of the through- bond interactions can be turned on and off by structural variations, as was illustrated in a recent study of pentacene dimers, 10 other isomers of BET-B with weak coupling might feature a different pattern. The calculations on different model structures suggest that the design of the SF materials based on covalently linked chromophores can indeed focus on the electronic properties of the individual molecules and electronic couplings facilitated by the linker, rather than on molecular solid morphology. Excited-state nuclear relaxation might play an important role in the SF of crystalline BET-B, by affecting state energies and electronic couplings and, ultimately, SF rates. 9 Although these effects have not been considered in the present study, the results obtained for different intermolecular displacements suggest that excited-state structural relaxation could drive the excited system into a trap state, hindering the SF process. On the other hand, as discussed above, these trap states correspond to inter-molecular excimers, while the lowest ME states are of intra-molecular nature. Hence, relaxation within the ME manifold following the initial photo-excitation might favor SF over the formation of low-energy excimers. We expect that the competition of different deactivation pathways will be strongly dependent on the activation of specific vibrational modes. 8.4 Conclusion In this paper we investigated electronic properties of the two BET-B molecules, each consisting of the two covalently linked tetracene chromophores. The analysis of the 225 Figure 8.4: Excitation energies of (BET-B) 2 in model structures differing by the overlap between the middle tetracene moieties (A and A 0 , see Fig. 8.1). Left: The tetracene rings are perfectly stacked (shift of 0.00 ˚ A). Middle: The tetracenes are shifted along the long axis by half of benzene ring (shift of 1.23 ˚ A); this structure is similar to crystal structure. Right: The tetracenes are offset by one benzene ring along the long axis (shift of 2.46 ˚ A). electronic states reveals that the lowest 1 ME states, which are well separated in energy from the higher ME states, are of local, intra-molecular character, due to strong through- bond interactions. In contrast, the excitonic states are more delocalized. The 1 ME states in which two triplets excitons reside on different molecules are not readily accessible from the lowest excitonic state (although, they might be populated from higher-excited excitonic states). The important practical implications are: (i) the electronic structure of the individual covalently linked dimer is the most essential factor for the design of efficient SF materials; (ii) the theoretical modeling of SF using small dimer systems is reasonably justified. Our calculations and wave-function analysis also provide valuable reference data for the development of excitonic models for multi-chromophore assem- blies. 36–43 226 Chapter 8 references [1] M. B. Smith and J. Michl. Singlet fission. Chem. Rev., 110:6891–6936, 2010. [2] M.B. Smith and J. Michl. Recent advances in singlet fission. Annu. Rev. Phys. Chem., 64:361–368, 2013. [3] J.J. Burdett and C.J. Bardeen. The dynamics of singlet fission in crystalline tetracene and covalent analogs. Acc. Chem. Res., 46:1312–1320, 2013. [4] H. Liu, V .M. Nichols, L. Shen, S. Jahansouz, Y . Chen, K.M. Hanson, C.J. Bardeen, and X. Li. Synthesis and photophysical properties of a ”face-to-face” stacked tetracene dimer. Phys. Chem. Chem. Phys., 17:6523–6531, 2015. [5] J.M. Giaimo, J.V . Lockard, L.E. Sinks, A.M. Scott, T.M. Wilson, and M.R. Wasielewski. Excited singlet states of covalently bound, cofacial dimers and trimers of perylene-3,4:9,10-bis(dicarboximide)s. J. Phys. Chem. A, 112:2322– 2330, 2008. [6] R.J. Lindquist, K.M. Lefler, K.E. Brown, S.M. Dyar, E.A. Margulies, R.M. Young, and M.R. Wasielewski. Energy flow dynamics within cofacial and slip-stacked perylene-3, 4-dicarboximide dimer models of -aggregates. J. Am. Chem. Soc., 136:14192–14923, 2014. [7] E.A. Margulies, L.E. Shoer, S.W. Eaton, and M.R. Wasielewski. Excimer forma- tion in cofacial and slip-stacked perylene-3,4:9,10-bis(dicarboximide) dimers on a redox-inactive triptycene scaffold. Phys. Chem. Chem. Phys., 16, 2014. 23735- 23742. [8] N.V . Korovina, S. Das, Z. Nett, X. Feng, J. Joy, A.I. Krylov, S.E. Bradforth, and M.E. Thompson. Singlet fission in a covalently linked cofacial alkynyltetracene dimer. J. Am. Chem. Soc., 138:617–627, 2016. [9] X. Feng and A.I. Krylov. On couplings and excimers: Lessons from studies of singlet fission in covalently linked tetracene dimers. Phys. Chem. Chem. Phys., 18:7751–7761, 2016. [10] J. Zirzlmeier, D. Lehnherr, P.B. Coto, E.T. Chernick, R. Casillas, B.S. Basel, M. Thoss, R.R. Tykwinski, and D.M. Guldi. Singlet fission in pentacene dimers. Proc. Nat. Acad. Sci., 112:5325–5330, 2015. 227 [11] T. Zeng and P. Goel. Design of small intramolecular singlet fission chromophores: An azaborine candidate and general small size effects. J. Phys. Chem. Lett., 7:1351–1358, 2016. [12] E.G. Fuemmeler, S.N. Sanders, A.B. Pun, E. Kumarasamy, T. Zeng, K. Miyata, M.L. Steigerwald, X.-Y . Zhu, M.Y . Sfeir, L.M. Campos, and N. Ananth. A direct mechanism of ultrafast intramolecular singlet fission in pentacene dimers. ACS Cent. Sci., 2:316–324, 2016. [13] T. Sakuma H. Sakai, Y . Araki, T. Mori, T. Wada, N.V . Tkachenko, and T. Hasobe. Long-lived triplet excited states of bent-shaped pentacene dimers by intramolecu- lar singlet fission. J. Phys. Chem. A, 120:1867–1875, 2016. [14] A.D. Chien, A.R. Molina, N. Abeyasinghe, O.P. Varnavski, T. Goodson III, and P.M. Zimmerman. Structure and dynamics of the 1 (TT) state in a quinoidal bithio- phene: Characterizing a promising intramolecular singlet fission candidate. J. Phys. Chem. C, 119:28258–28268, 2015. [15] X. Feng, A.V . Luzanov, and A.I. Krylov. Fission of entangled spins: An electronic structure perspective. J. Phys. Chem. Lett., 4:3845–3852, 2013. [16] A.B. Kolomeisky, X. Feng, and A.I. Krylov. A simple kinetic model for singlet fission: A role of electronic and entropic contributions to macroscopic rates. J. Phys. Chem. C, 118:5188–5195, 2014. [17] X. Feng, A.B. Kolomeisky, and A.I. Krylov. Dissecting the effect of morphology on the rates of singlet fission: Insights from theory. J. Phys. Chem. C, 118:19608– 19617, 2014. [18] D. Casanova and A.I. Krylov. Quantifying local excitation, charge resonance, and multiexciton character in correlated wave functions of multichromophoric systems. J. Chem. Phys., 144:014102, 2016. [19] A.V . Luzanov, D. Casanova, X. Feng, and A.I. Krylov. Quantifying charge res- onance and multiexciton character in coupled chromophores by charge and spin cumulant analysis. J. Chem. Phys., 142:224104, 2015. [20] X. Feng, D. Casanova, and A.I. Krylov. Intra- and inter-molecular singlet fission in covalently linked dimers. J. Phys. Chem. C, 2016. [21] D. Casanova and M. Head-Gordon. Restricted active space spin-flip configuration interaction approach: Theory, implementation and examples. Phys. Chem. Chem. Phys., 11:9779–9790, 2009. 228 [22] D. Casanova, L.V . Slipchenko, A.I. Krylov, and M. Head-Gordon. Double spin-flip approach within equation-of-motion coupled cluster and configuration interaction formalisms: Theory, implementation and examples. J. Chem. Phys., 130:044103, 2009. [23] F. Bell, P.M. Zimmerman, D. Casanova, M. Goldey, and M. Head-Gordon. Re- stricted active space spin-flip (RAS-SF) with arbitrary number of spin-flips. Phys. Chem. Chem. Phys., 15:358–366, 2013. [24] S. Matsika, X. Feng, A.V . Luzanov, and A.I. Krylov. What we can learn from the norms of one-particle density matrices, and what we can’t: Some results for inter- state properties in model singlet fission systems. J. Phys. Chem. A, 118:11943– 11955, 2014. [25] D. Casanova. Bright fission: Singlet fission into a pair of emitting states. J. Chem. Theory Comput., 11:2642–2650, 2015. [26] Y . Shao, Z. Gan, E. Epifanovsky, A.T.B. Gilbert, M. Wormit, J. Kussmann, A.W. Lange, A. Behn, J. Deng, X. Feng, D. Ghosh, M. Goldey, P.R. Horn, L.D. Jacob- son, I. Kaliman, R.Z. Khaliullin, T. Kus, A. Landau, J. Liu, E.I. Proynov, Y .M. Rhee, R.M. Richard, M.A. Rohrdanz, R.P. Steele, E.J. Sundstrom, H.L. Wood- cock III, P.M. Zimmerman, D. Zuev, B. Albrecht, E. Alguires, B. Austin, G.J.O. Beran, Y .A. Bernard, E. Berquist, K. Brandhorst, K.B. Bravaya, S.T. Brown, D. Casanova, C.-M. Chang, Y . Chen, S.H. Chien, K.D. Closser, D.L. Crittenden, M. Diedenhofen, R.A. DiStasio Jr., H. Do, A.D. Dutoi, R.G. Edgar, S. Fatehi, L. Fusti- Molnar, A. Ghysels, A. Golubeva-Zadorozhnaya, J. Gomes, M.W.D. Hanson- Heine, P.H.P. Harbach, A.W. Hauser, E.G. Hohenstein, Z.C. Holden, T.-C. Ja- gau, H. Ji, B. Kaduk, K. Khistyaev, J. Kim, J. Kim, R.A. King, P. Klunzinger, D. Kosenkov, T. Kowalczyk, C.M. Krauter, K.U. Laog, A. Laurent, K.V . Lawler, S.V . Levchenko, C.Y . Lin, F. Liu, E. Livshits, R.C. Lochan, A. Luenser, P. Manohar, S.F. Manzer, S.-P. Mao, N. Mardirossian, A.V . Marenich, S.A. Maurer, N.J. May- hall, C.M. Oana, R. Olivares-Amaya, D.P. O’Neill, J.A. Parkhill, T.M. Perrine, R. Peverati, P.A. Pieniazek, A. Prociuk, D.R. Rehn, E. Rosta, N.J. Russ, N. Ser- gueev, S.M. Sharada, S. Sharmaa, D.W. Small, A. Sodt, T. Stein, D. Stuck, Y .-C. Su, A.J.W. Thom, T. Tsuchimochi, L. V ogt, O. Vydrov, T. Wang, M.A. Watson, J. Wenzel, A. White, C.F. Williams, V . Vanovschi, S. Yeganeh, S.R. Yost, Z.-Q. You, I.Y . Zhang, X. Zhang, Y . Zhou, B.R. Brooks, G.K.L. Chan, D.M. Chipman, C.J. Cramer, W.A. Goddard III, M.S. Gordon, W.J. Hehre, A. Klamt, H.F. Schaefer III, M.W. Schmidt, C.D. Sherrill, D.G. Truhlar, A. Warshel, X. Xu, A. Aspuru- Guzik, R. Baer, A.T. Bell, N.A. Besley, J.-D. Chai, A. Dreuw, B.D. Dunietz, T.R. Furlani, S.R. Gwaltney, C.-P. Hsu, Y . Jung, J. Kong, D.S. Lambrecht, W.Z. Liang, C. Ochsenfeld, V .A. Rassolov, L.V . Slipchenko, J.E. Subotnik, T. Van V oorhis, 229 J.M. Herbert, A.I. Krylov, P.M.W. Gill, and M. Head-Gordon. Advances in molec- ular quantum chemistry contained in the Q-Chem 4 program package. Mol. Phys., 113:184–215, 2015. [27] A.I. Krylov and P.M.W. Gill. Q-Chem: An engine for innovation. WIREs Comput. Mol. Sci., 3:317–326, 2013. [28] P.M. Zimmerman, F. Bell, D. Casanova, and M. Head-Gordon. Mechanism for singlet fission in pentacene and tetracene: From single exciton to two triplets. J. Am. Chem. Soc., 133:19944–19952, 2011. [29] J.J. Burdett, A.M. M¨ uller, D. Gosztola, and C.J. Bardeen. Excited state dynamics in solid and monomeric tetracene: The roles of superradiance and exciton fission. J. Chem. Phys., 133:144506, 2010. [30] Y . Tomkiewicz, R.P. Groff, and P. Avakian. Spectroscopic approach to energetics of exciton fission and fusion in tetracene crystals. J. Chem. Phys., 54:4504–4507, 1971. [31] G.D. Scoles. Correlated pair states formed by singlet fission and exciton-exciton annihilation. J. Phys. Chem. A, 119:12699–12705, 2015. [32] C.J. Bardeen. The structure and dynamics of molecular excitons. Annu. Rev. Phys. Chem., 65:127–148, 2014. [33] F.C. Spano and C. Silva. H- and J-aggregate behavior in polymeric semiconduc- tors. Annu. Rev. Phys. Chem., 65:477–500, 2014. [34] S. Sharifzadeh, P. Darancet, L. Kronik, and J.B. Neaton. Low-energy charge- transfer excitons in organic solids from first-principles: The case of pentacene. J. Phys. Chem. Lett., 4:2917–2201, 2013. [35] D. Casanova. Electronic structure study of singlet-fission in tetracene derivatives. J. Chem. Theory Comput., 10:324–334, 2014. [36] T. Wesolowski, R.P. Muller, and A. Warshel. Ab initio frozen density functional calculations of proton transfer reactions in solution. J. Phys. Chem., 100:15444– 15449, 1996. [37] Q. Wu and T. Van V oorhis. Constrained density functional theory and its applica- tion in long-range electron transfer. J. Chem. Theory Comput., 2:765–774, 2006. [38] Q. Wu, C. Cheng, and T. Van V oorhis. Configuration interaction based on con- strained density functional theory: a multireference method. J. Chem. Phys., 127:164119, 2007. 230 [39] H. Zhang, J. Malrieu, H. Ma, and J. Ma. Implementation of renormalized excitonic method at ab initio level. J. Comput. Chem., 33:34, 2011. [40] S.M. Parker, T. Seideman, M. Ratner, and T. Shiozaki. Communication: Active- space decomposition for molecular dimers,. J. Chem. Phys., 139:021108, 2013. [41] S.M. Parker and T. Shiozaki. Quasi-diabatic states from active space decomposi- tion. J. Chem. Theory Comput., 10:3738–3744, 2014. [42] A. Sisto, D.R. Glowacki, and T.J. Martinez. Ab initio nonadiabatic dynamics of multichromophore complexes: A scalable graphical-processing-unit-accelerated exciton framework. Acc. Chem. Res., 47:2857–2866, 2014. [43] A. Morrison, Z. You, and J. Herbert. Ab initio implementation of the Frenkel- Davydov exciton model: A naturally parallelizable approach to computing collec- tive excitations in crystals and aggregates. J. Chem. Theory Comput., 10:5366, 2014. 231 Chapter 9: Cholesky representation of electron-repulsion integrals within coupled-cluster and equation-of-motion methods 9.1 Introduction Theoretical model chemistries 1 based on wave function methods provide the most re- liable approach to electron correlation. Among different ab initio-based techniques, 2 coupled-cluster (CC) theory holds a pre-eminent position. 3 The single-reference CC hierarchy of approximations allows one to compute highly accurate molecular struc- tures, reaction energies, and other properties for ground-state species. 2 The equation- of-motion (EOM), or linear response, approach 4–6 extends the CC formalism to a vari- ety of multi-configurational wave functions encountered in electronically excited states and various open-shell species. Unfortunately, similarly to other wave function based methods, the computational cost and hardware requirements (disk and memory) of CC and EOM-CC scale quite steeply with the number of electrons and the size of the one- electron basis set, i.e., the number of occupied (O) and unoccupied, or virtual (V ), 232 orbitals. For example, the scaling of a CCSD (coupled-cluster with single and double substitutions) calculation is O 2 V 4 , and for CCSDT (CCSD plus explicit triple excita- tions) it is O 3 V 5 . The disk usage in CC and EOM-CC calculations depends on the implementation specifics and can reachO(V 4 ); integral-direct algorithms could be em- ployed to reduce storage requirements. The high cost of electronic structure calculations originates in the two-electron part of the molecular Hamiltonian that describes electron-electron repulsion. The represen- tation of the electron-repulsion integrals (ERIs) in an atomic orbital (AO) basis gives rise to a four-index tensor: (j) = Z (~ r 1 ) (~ r 1 ) 1 j~ r 1 ~ r 2 j (~ r 2 ) (~ r 2 )d~ r 1 d~ r 2 The size of this object scales as N 4 where N is the number of basis functions i (~ r). For accurate results the size of the AO basis needs to be sufficiently large, for example a popular cc-pVTZ basis defines 30 contracted Gaussian functions per second-row atom. All electronic structure methods include contractions of ERIs with various tensors, such as reduced density matrices, wave functions amplitudes, etc. Thus, the large size of ERIs propagates through the electron structure calculations from self-consistent field up to correlated methods. Fortunately, the structure of the ERI matrix is sparse, which can be exploited in effi- cient computer implementations. It was recognized a long time ago that representing the “densities” by a linear expansion over the products of particular one-electron functions, such as (~ r 1 ) (~ r 1 ), includes linear dependencies and could be rewritten in a more compact form using a new set of basis functions. 233 There are two alternative approaches to achieve this goal, the density fitting, or resolution-of-the-identity (RI), 7–13 approximation and the Cholesky decomposition (CD). 14–20 In both approaches, the decomposed ERI matrix is represented as: (j) M X P =1 B P B P ; (9.1) whereM is the rank of the decomposition, which depends on the target accuracy. The algorithm for determiningB is different in RI and CD approaches: RI uses a predeter- mined auxiliary basis set that corresponds to the primary one-electron basis, whereas Cholesky vectors are obtained by performing the Cholesky decomposition of the actual ERI matrix. CD is thus a more general approach that can work with any primary basis and is free from externally optimized auxiliary basis sets. The Cholesky approach can be viewed as system-specific density fitting. 17–19 Decomposition shown in Eq. (10.1) produces a more compact representation of ERIs compared with the full ERI matrix, thus enabling memory and disk savings. In addi- tion, it allows one to achieve improved parallel performance of calculations involving ERI through reduced disk input-output (I/O) penalties and better CPU utilization. For example, the AO-MO integral transformation has a computational cost ofO(N 5 ) when using the canonical procedure, now only involves the transformation of the RI/Cholesky vectors and therefore requires onlyO(N 3 M) steps. The transformedB-matrices can be used to assemblehpqjjrsi integrals as needed in integral-direct implementations. How- ever, to realize the maximum potential of the method, programmable equations that involve contractions of ERIs with the amplitudes and density matrices need to be rewrit- ten. The RI/Cholesky representation by itself does not lead to a scaling reduction in CCSD and EOM equations unless special care is taken about exchange-like terms. A 234 number of strategies have been pursued to this end, 19, 21, 22 including using Cholesky decomposed wave function amplitudes 23, 24 and local correlation schemes (see, for ex- ample, Ref. 25, 26 and references therein). However, even without these more ad- vanced algorithms, computational savings due to a straightforward implementation of RI/Cholesky representations are very useful, especially in view of improved parallel scaling. We present our implementation of RI/Cholesky within the CCSD and EOM-CCSD suite of methods in the Q-Chem electronic structure package. 27, 28 The implementation eliminates the storage of the most expensive four-index integrals and intermediates. As described below, in the EOM-CCSD implementation we choose to keep two smallest four-index intermediates, OOOO and OOOV . While CCSD implementations have been reported before, 29 the EOM-CCSD meth- ods have not been reimplemented using RI/CD. Below we briefly describe the algo- rithms used to produce Cholesky and RI vectors (Sections 9.2.1 and 9.2.2) and explain its implementation within CCSD and EOM-CCSD (Section 9.3). The following EOM methods have been implemented: EE/SF, IP, and EA. We discuss the performance of the new implementation in Section 9.4. 9.2 Algorithms 9.2.1 Cholesky algorithm The idea of Cholesky decomposition (CD) of ERI 14, 15, 17, 30 was proposed over 30 years ago by Beebe and Linderberg. 14 The ERI matrix in the AO basis, which is a positive- semidefinite 14 matrix, can be represented in the Cholesky-decomposed form as given by Eq. (10.1). The rank of the decomposition, M, is typically 3–10 times the number of 235 basis functionsN. 17 It depends on the decomposition threshold and is considerably smaller than the full rank of the matrix,N(N + 1)=2. 14, 17, 31 CD removes linear depen- dencies in product densities 17 (j, allowing one to approximate the original matrix to arbitrary accuracy. Decomposition threshold defined by the user is the only parameter that controls the accuracy and the rank of the decomposition. The algorithm 15, 17, 30 proceeds as follows: (1) Compute all diagonal elements of ERI:D 0 ; = (j). (2) Choose the largest diagonal element ( 0 j 0 ). () 0 here is a fixed index corresponding to the largest diagonal element. (3) Compute densities (j 0 ). (4) Compute first Cholesky vectorB 1 = (j 0 )= p ( 0 j 0 ). From this point the algorithm proceeds in an iterative manner, checking the accuracy and generating a new Cholesky vector to refine the previous-step approximation at every iteration.k is an iteration count that starts from 2 and increments after every iteration. (5) Update the residual of the diagonal by subtracting the Cholesky vector obtained in the previous iterationD (k1) ; =D (k2) ; B (k1) B (k1) . (6) Choose the largest element of the diagonal residual D k1 k1 ; k1 . If D k1 k1 ; k1 <, then terminate and return the Cholesky vectors,fB i g k1 i=1 . (7) Compute densities (j k1 ) and the corresponding residual, D (k1) ; k1 = (j k1 ) P k1 i=1 B i B i k1 . (8) Compute new Cholesky vectorB k = D k1 ; k1 = q D k1 k1 ; k1 . Repeat from step (5). Since the ERI matrix is positive-semidefinite, 8, 14, 32 it follows that: jD k1 ; j q D k1 ; D k1 ; 236 Thus, the accuracy of the decomposition is given by the largest element of diagonal residual matrix D ; at every iteration (step 6), which ensures that the error in any ERI matrix element does not exceed. Note that the algorithm does not require the calculation and storage of the full ERI matrix [O(N 4 )], which would be prohibitive for large systems. At the initialization of the algorithm only the calculation of the diagonal elements is necessary [step 1,O(N 2 )], which are updated at each iteration by subtracting newly produced Cholesky vectors to form a residual diagonal matrix (step 5). The calculation of the densities (j k1 ) [step 7,O(N 2 )] are performed at each step with subsequent calculation of the residual and the corresponding Cholesky vector (step 8). At each iteration only the calculation of newO(N 2 ) elements of the ERI matrix is required and the number of Cholesky vectors grows by one resulting in theO(MN 2 ) memory storage of all Cholesky vectors for the final decomposition. Thus, only a small fraction of about 1–5% of the full ERI matrix needs to be calculated in the decomposition procedure. 17 The most expensive step is the calculation of the residual matrix 17 (step 7), which requires (M 1) subtractions of previously obtained Cholesky vectors at each iteration [O((M 1)N 2 ) operations at each iteration], giving rise to the full complexity of the algorithm ofO(M 2 N 2 ). For correlated calculations, the Cholesky vectors obtained in the AO basis are usually transformed to the molecular orbital (MO) basis. This algorithm is implemented using our tensor algebra library 33 such that Cholesky vectors are stored as a list of two-dimensional block tensors, i.e., a list of block matrices. The library is based on virtual memory management such that block tensors are stored in RAM if sufficient memory is available or saved on disk and reloaded as necessary. Note that the generation of a new Cholesky vector [steps 5–8] does not require vectors from previous iterations (k 1 at stepk) to be in RAM; for calculation of the residual 237 matrix (step 5) they can be uploaded from disk sequentially, or even block-by-block. After all Cholesky vectors B are generated, the list of block matrices is merged to form a three-dimensional block tensorB M containing all the Cholesky vectors. 9.2.2 Resolution-of-the-identity algorithm Similar to the Cholesky decomposition, the RI approach 8–13 allows one to expand prod- uct densities (j in an auxiliary basis set: (j) X PQ C P (PjQ)C Q X PQ (jP )(PjQ) 1 (Qj) IndicesP andQ denote auxiliary basis functions and (PjQ) defines a Coulomb metric matrix: 13, 17, 20 (PjQ) = Z P (~ r 1 )Q(~ r 2 ) j~ r 1 ~ r 2 j d~ r 1 d~ r 2 The auxiliary basis expansion coefficients (C L ) are found by minimizing the differ- ence between the actual and fitted product densities, 13, 17, 18 leading to the following set of linear equations: X L (KjL)C L = (Kj) By defining new auxiliary basis coefficients B K = X L C L (LjK) 1=2 X L (Kj)(KjL) 1=2 we can rewrite approximate ERIs in a form identical to the Cholesky representation 13 as given by Eq. (10.1). 238 The accuracy and performance of RI depend on the quality of the chosen auxiliary basis set. Ideally an auxiliary basis set should be balanced between accuracy and com- pactness. Errors should be at least an order of magnitude smaller than the error due to one-electron basis set incompleteness. RankM should be no more that 2–4 times larger than the number of AO basis functions N. 7, 17, 34–39 To achieve these goals, auxiliary basis sets are usually optimized for each atom, AO basis set and level of theory (e.g., Hartree–Fock, MP2). 7, 17, 34–39 In this work we employ auxiliary basis sets developed for MP2. 9.3 RI/CD CCSD and EOM-CCSD methods: Theory 9.3.1 Coupled-cluster equations with single and double substitu- tions The exact solution of the Schr¨ odinger equation can be written as the exponential of a cluster operator ^ T operating on a reference function: 40 exact = CC =e ^ T 0 where 0 is a single Slater determinant. In CCSD, the expansion of ^ T is truncated at a two-electron excitation level: ^ T ^ T 1 + ^ T 2 239 For ^ T 1 and ^ T 2 , the expansions are 1 : ^ T 1 = X ia t a i a y i ^ T 2 = 1 4 X ijab t ab ij a y ib y j Thus, CCSD =e ^ T 1 + ^ T 2 0 The equations to determine CCSD correlation energyE CCSD and cluster amplitudes t a i ,t ab ij are derived algebraically by a projection approach such that the Schr¨ odinger equa- tion is satisfied in the subspace spanned by the reference, singly, and doubly excited determinants: E CCSD = h 0 j ^ Hj CCSD i =h 0 j ^ Hj(1 + ^ T 1 + 1 2 ^ T 2 1 + ^ T 2 ) 0 i (9.2) 0 = h a i j ^ HE CCSD j CCSD i = h a i j ^ HE CCSD j(1 + ^ T 1 + 1 2 ^ T 2 1 + ^ T 2 + ^ T 1 ^ T 2 + 1 3! ^ T 3 1 ) 0 i (9.3) 0 = h ab ij j ^ HE CCSD j CCSD i = h ab ij j ^ HE CCSD j (1 + ^ T 1 + 1 2 ^ T 2 1 + ^ T 2 + ^ T 1 ^ T 2 + 1 3! ^ T 3 1 + 1 2 ^ T 2 2 + 1 2 ^ T 2 1 ^ T 2 + 1 4! ^ T 4 1 ) 0 i (9.4) Evaluating Eq. (9.2) in terms of amplitudest a i andt ab ij yields the following expression: E CCSD =h 0 j ^ Hj 0 i + X ia f ia t a i + 1 2 X ijab hijjjabit a i t b j + 1 4 X ijab hijjjabit ab ij (9.5) 1 Throughout the paper, we adhere to the convention thatijkl denote occupied orbitals,abcd denote virtual orbitals, andpqrs denote orbitals that can be either occupied or virtual. 240 where f ia = h a i j ^ Hj 0 i =h ia + X j hijjjaji hijjjabi = hijjabihijjbai = (iajjb) (ibjja) (iajjb) = Z i (1) j (2) 1 r 12 a (1) b (2)d1d2 Once Eq. (9.5) is substituted into Eqns. (9.3) and (9.4), t a i and t ab ij amplitudes can be solved iteratively by: t a i a i =f ia X l F (3) li t a l + X d F (1) ad t d i + X kc F (2) kc t ac ik X kc hicjjkait c k 1 2 X klc hkljjicit ac kl + 1 2 X kcd hkajjcdit cd ki and t ab ij ab ij =hijjjabi +P ab ( X c t ac ij F (2) bc X k I (2a) ijkb t a k +P ij X kc I (1a) kbic t ac jk ) +P ij ( X c hjcjjbait c i X k t ab ik F (2) jk ) + 1 2 X cd habjjcdi ~ t cd ij + 1 2 X kl t ab kl I (4) ijkl where a i = f ii f aa and ab ij = a i + b j . The expressions for the intermediates are given in Table 9.1. 241 Table 9.1: Intermediates for CCSD calculations and estimates to store and com- pute them (closed-shell case). Equation Memory Flops F (1) bc =f bc + P kd hkbjjdcit d k 1 2 P kld hkljjcdit bd kl F (2) ij =f ij + P a f ja t a i + P ka hjkjjiait a k + P kab hjkjjabit a i t b k + 1 2 P kab hjkjjabit ab ik F (2) ia =f ia + P jb hijjjabit b j F (2) bc =F (1) bc P k f kc t b k P kld hkljjcdit b k t d l F (3) ki =f ki + P c F (2) kc t c i + 1 2 P jab hkjjjabit ab ij + P lc hkljjicit c l ~ t ab ij =t ab ij +P ab t a i t b j 3 4 O 2 V 2 I (1a) iajb =hiajjjbi P c hiajjbcit c j P k hikjjjbit a k 2O 2 V 2 3O 3 V 3 1 2 P kc hikjjcbi t ca jk + 2t c j t a k I (2a) ijkb =hijjjkbi 1 2 P l I (4) ijkl t b l + 1 2 P cd hkbjjcdi ~ t cd ij +P ij P c hkbjjicit c j 3 2 O 3 V 5 4 O 3 V 3 I (4) ijkl =hijjjkli + 1 2 P ab hkljjabi ~ t ab ij +P ij P a hkljjiait a j 3 4 O 4 5 8 O 4 V 2 P kc I (1a) kbic t ac jk 3O 3 V 3 P cd habjjcdi ~ t cd ij 5 8 O 2 V 4 P kl t ab kl I (4) ijkl 5 8 O 4 V 2 Memory requirements for theT amplitude update procedure in the closed-shell case are 2 : 9 8 O 4 + 3O 3 V + 6O 2 V 2 + 3 2 OV 3 + 3 8 V 4 (9.6) This estimate includes all the blocks of ERIs, necessary four-index intermediates, and two sets of T 2 amplitudes. Excluded are additional copies of T amplitudes required by the DIIS 3 procedure and lower-dimensional quantities. TheO(N 6 ) part of the total computational cost of updatingT amplitudes is 7 4 O 4 V 2 + 29 4 O 3 V 3 + 5 8 O 2 V 4 . 2 In the closed-shell case, two spin cases of integrals and amplitudes are stored: hjji and hjji; in the open-shell case there is additionally a third spin case: hjji. With applica- ble permutational symmetry taken into consideration and an assumption of no point group symmetry in the molecule, the disk requirement for ERI objects in the closed-shell case are: hijjjkli : 3 8 O 4 , hijjjkai : 3 2 O 3 V ,hijjjabi : 3 4 O 2 V 2 ,hiajjjbi :O 2 V 2 ,hiajjbci : 3 2 OV 3 ,habjjcdi : 3 8 V 4 . 3 Usually, severalT 2 vectors are stored for the DIIS algorithm. 242 Because RI and Cholesky representations of ERI use identical expressions, we begin with the following expression for anti-symmetrized integrals: hjji X P B P B P X P B P B P =P X P B P B P (9.7) Upon substituting Eq. (9.7) into Eq. (9.5), (9.3), (9.4) and corresponding intermidi- ates (Table 9.1) the equations for RI/CD CCSD can be obtained. The details on deriva- tion of equations as well as new intermediates formed for RI/CD CCSD can be found in Ref. 41. To illustrate the difference in storage requirements, consider a calculation of closed- shell naphthalene using the cc-pVTZ/rimp2-cc-pVTZ basis set. There are 68 electrons, 412 basis functions (O = 34,V = 378), 1050 auxiliary basis functions (M = 1050). Whereas conventional CCSD calculation requires 10917 Mwords (85 GB), CD/RI- CCSD requires 846 Mwords (6.6 GB). Thus, for this calculation the data set is almost 13 times smaller in the case of RI-CCSD. The number of floating point operations scales asO(N 6 ) for both CCSD and CD/RI- CCSD. The most significant contraction in CCSD, P cd habjjcdi ~ t cd ij , and its CD/RI- CCSD counterpart, take the same number of flops. In the latter case the intermedi- ateP ab P P M P da M P cb is formed on the fly thus reducing overall memory requirements. The CD/RI-CCSD equations involve fewerO 3 V 3 -type contractions, leading to a smaller prefactor (4O 3 V 3 vs. 29 4 O 3 V 3 in conventional CCSD). While this improvement is offset by the increased number and cost ofO(N 5 ) steps, in practical applications CD/RI- CCSD are superior in terms of floating point operations, memory and I/O, as illustrated by benchmark calculations in Section 9.4. 243 9.3.2 EOM-EE/SF-CCSD and CD/RI EOM-EE/SF-CCSD In the EOM-CCSD framework, the target excited-state wave functions are written as: 42, 43 j R i =Re ^ T j 0 i (9.8) h L j =h 0 je ^ T L y (9.9) The operatorsR andL are linear excitation operators: R =R 0 +R 1 +R 2 + (9.10) R n = 1 n! 2 X r abc ijk a y ib y jc y k (9.11) In EOM-EE operatorsR n are spin-conserving (M s = 0 operators), whereas in EOM-SF they involve changing the spin of an electron (M s =1). The spin-orbital form of the EOM-CCSD equations is the same in EOM-EE and EOM-SF. 43, 44 By introducing a similarity-transformed Hamiltonian H: He ^ T He ^ T (9.12) the energy and CCSD amplitude equations become: E CC = h 0 j Hj 0 i (9.13) 0 = h a i j HE CC j 0 i (9.14) 0 = h ab ij j HE CC j 0 i (9.15) ::: 244 whereE CC is the coupled-cluster energy for the reference state. Usually bothT andR are truncated at the same level, which is the single (S) and double (D) excitations in this work. Thus, in the basis of the reference (O), S, and D we have: 0 B B B B B B @ 0 H OS H OD 0 H SS E CC H SD 0 H DS H DD E CC 1 C C C C C C A 0 B B B B B B @ R 0 R 1 R 2 1 C C C C C C A =! 0 B B B B B B @ R 0 R 1 R 2 1 C C C C C C A (9.16) where on the left-hand sideE CC only appears for the diagonal elements in the diagonal blocks and! = EE CC . Because the right eigenvectors do not form an orthonormal set,R 0 =r 0 ^ 1 can be present in the target excited states: r 0 = 1 ! H OS R 1 + H OD R 2 = 1 ! ( X ia F (2) ia r a i + 1 4 X ijab hijjjabir ab ij ) (9.17) Eq. (9.16) is solved by using the generalized Davidson iterative diagonalization pro- cedure, 43 which involves the calculation of the following-vectors: a i = H SS E CC R 1 a i + H SD R 2 a i = X b F (2) ab r b i X j F (2) ij r a j X jb I (1) ibja r b j + X jb F (2) jb r ab ij 1 2 X jkb I (6) jkib r ab jk 1 2 X jbc I (7) jabc r bc ij (9.18) 245 Table 9.2: I andT intermediates for EOM-CCSD and estimated cost to store and compute them (closed-shell case). Equation Memory Flops I (1) ickb =hicjjkbi P d hkbjjcdit d i P ld hkljjcdit bd il 2O 2 V 2 3O 3 V 3 + P l P d hkljjcdit d i hkljjici t b l I (2) ijkb =hijjjkbi P l I (4) ijkl t b l + 1 2 P cd hkbjjcdi ~ t cd ij + P d [ P lc hkljjcdit c l ]t bd ij P c t bc ij f kc P ij P c hkbjjjci P ld hkljjcdit bd jl t c i + P lc hkljjjcit bc il 3 2 O 3 V 21 4 O 3 V 3 I (3) jcab =hjcjjabi P d I (5) abcd t d j + 1 2 P kl hkljjjcit ab kl P l P kd hlkjjcdit d k t ab jl P l t ab jl f lc +P ab [ P k hkajjjci 1 2 P l hkljjjcit a l P ld hkljjcdit ad jl t b k P ld hlbjjcdit ad jl ] 3 2 OV 3 33 4 O 3 V 3 I (4) ijkl =hijjjkli + 1 2 P ab hkljjabi ~ t ab ij +P ij P a hkljjiait a j 3 4 O 4 5 8 O 4 V 2 I (5) abcd =habjjcdi + 1 2 P kl hkljjcdi ~ t ab kl P ab P k hkbjjcdit a k 3 4 V 4 5 8 O 2 V 4 I (6) klic =hkljjici P d t d i hkljjcdi 3 2 O 3 V I (7) kacd =hkajjcdi P l t a l hkljjcdi 3 2 OV 3 T (1) ij = P kc r c k I (6) jkic T (2) ab = P kc r c k I (7) kabc T (3) ij = 1 2 P kab hjkjjabir ab ik T (4) ab = 1 2 P ijc hijjjbcir ac ij ab ij = H DS R 1 ab ij H DD E CC R 2 ab ij =P ab X k I (2) ijkb r a k P ij X c I (3) jcab r c i +P ij X l T (1) il t ab jl +P ab X d T (2) ad t bd ij +P ij X k r ab jk F (2) ik +P ab X c r ac ij F (2) bc +P ij P ab X kc I (1) ickb r ac jk + 1 2 X kl r ab kl I (4) ijkl + 1 2 X cd r cd ij I (5) abcd +P ij X l T (3) il t ab jl +P ab X d T (4) ad t bd ij (9.19) 246 TheI,F , andT intermediates used in Eq. (9.18) and (9.19) are collected in Tables 9.1 and 9.2. Total storage requirements for computing a-vector, including a set ofT ,R, amplitudes and all the integrals and intermediates, are: 9 8 O 4 + 9 2 O 3 V + 6O 2 V 2 + 9 2 OV 3 + 9 8 V 4 (9.20) Note that multiple sets ofR and amplitudes are required in the Davidson procedure for finding the excitation energies. Following the same procedure as in the derivation of the CD/RI-CCSD equations one arrives at the equations with RI/CD integrals, details can be found in Ref. 41. For the computation of a-vector in the Davidson iterative procedure, the storage requirement for CD/RI-EOM-EE implementation becomes: 5 2 O 2 M + 5OVM + 5 2 V 2 M + 3 2 O 4 + 3 2 O 3 V + 21 4 O 2 V 2 (9.21) For the naphthalene example the RI version of EOM-EE reduces the amount of required memory by a factor of 24 relative to the canonical implementation, that is, the conventional EOM-EE needs 30795 Mwords (241 GB), whereas CD/RI-EOM-EE uses 1275 Mwords (10 GB). The number of floating point operations in the-vector update procedure for both conventional and RI/CD implementations scales asO(N 6 ). The cost of EOM-EE is 5 8 O 2 V 4 + 3 4 O 3 V 3 + 5 8 O 4 V 2 , whereas RI/CD-EOM-EE takes 5 8 O 2 V 4 + 9O 3 V 3 + 5 4 O 4 V 2 operations. There is a larger number ofO 3 V 3 contractions in the latter case, leading to a bigger prefactor. This is the result of the on-the-fly reassembly of some fourth-order intermediates that are stored in memory in the case of conventional EOM-EE. 247 9.3.3 EOM-IP-CCSD and CD/RI EOM-IP-CCSD In EOM-IP-CCSD (EOM-CCSD for ionization potentials), the operator R is not particle-conserving: R n (N 1) = 1 n! 2 X r bc ijk ib + jc + k (9.22) In EOM-IP-CCSD, R is truncated at the two-hole-one-particle excitation level. The equations for-vectors are as follows: i = X j F (2) ij r j + X jb F (2) jb r b ij + 1 2 X jkb I (6) kjib r b jk a ij = X k r k I (2) ijka +P ij X k r a jk F (2) ik + X b r b ij F (2) ab P ij X kb I (1) jbka r b ik + 1 2 X kl I (4) ijkl r a kl + X b t ab ij T (4) b T (4) b = 1 2 X klb hkljjabir b kl whereF andI intermediates are collected in Tables 9.1 and 9.2. Memory requirements for the update procedure are: 3 4 O 4 + 3O 3 V + 11 4 O 2 V 2 (9.23) This estimate excludes any three-dimensional quantities, e.g. EOM-IP amplitudes. The CD/RI equations are derived following the same procedure as in the EOM-EE case. For the naphthalene example, memory savings achieved by using RI are limited to about 20%, that is, conventional EOM-IP requires 477 Mwords (3.7 GB), whereas CD/RI-EOM-IP, needs 382 Mwords (3.0 GB). The difference in memory requirements 248 is not as large as in the case of EOM-EE because EOM-IP does not use the OVVV and VVVV blocks of the ERIs. The number of floating point operations in the-vector update procedure for both implementations isO(N 5 ). The CD/RI scheme requires sixO(N 5 )-type contractions, the dominant contraction beingO 2 V 2 M. The canonical EOM-IP requires twoO(N 5 )- type contractions, the dominant contraction beingO 3 V 2 . Therefore, the-vector update procedure in CD/RI-EOM-IP is expected to be about three times slower than in canon- ical EOM-IP; however, some of this cost increase is offset by more favorable parallel scaling. Moreover, for fair comparison, the calculation of the intermediates should also be considered. 9.3.4 EOM-EA-CCSD and CD/RI EOM-EA-CCSD In EOM-EA (EOM for electron attachment), the operatorR is: R n (N + 1) = 1 n! 2 X r abc jk a + b + jc + k (9.24) In EOM-EA-CCSD,R is truncated at the one-hole-two-particles level and the equations for the-vectors are: a = X c F (2) ac r c + X kc F (2) kc r ac k + 1 2 X kcd I (7) kacd r cd k ab i = P ab X c F (2) ac r cb i X k F (2) ki r ab k X c I (3) icab r c + 1 2 X cd I (5) abcd r cd i +P ab X kc I (1) kbic r ac k X k T (3) k t ab ik T (3) k = 1 2 X kcd r cd k hkijjcdi 249 whereF (2) andI intermediates are given in Tables 9.1 and 9.2. The disk requirements for the EOM-EA update procedure are estimated at: 7 2 O 2 V 2 + 3OV 3 + 3 4 V 4 (9.25) Following the same procedire as for EOM-EE and EOM-IP of substituting of RI/CD decomposed integrals one arrives at RI/CD EOM-EA, details can be found in Ref. 41. The storage requirement for CD/RI-EOM-EA-vector update is: 3 2 O 2 M + 2OVM + 3 2 V 2 M + 3 4 O 2 V 2 (9.26) To again illustrate memory savings using the naphthalene example, conven- tional EOM-EA uses 20408 Mwords (159 GB), whereas CD/RI-EOM-EA needs only 360 Mwords (2.8 GB). Because the most expensive in terms of storage intermediates have been eliminated in the CD/RI implementation, the procedure requires about 57 times less memory. Similar to EOM-IP-CCSD, the number of floating point operations for both im- plementations scales asO(N 5 ). The dominantO(N 5 )-type contraction, out of two in canonical EOM-EA, is the OV 4 -type, whereas in CD/RI, which has tenO(N 5 )-type contractions, the dominant one isV 4 M-type. 9.4 Benchmarks The errors introduced by the RI and CD approximations have been extensively bench- marked for quantities like total energies, molecular structures, dipole moments, and excitation energies; 7, 15, 18, 20, 35, 37–39, 45–49 for a recent review see Ref. 19. Total energies 250 have been analyzed for density functional theory, 18–20, 35, 45 Hartree-Fock 15, 18–20, 45, 46 and MP2 methods. 7, 15, 18–20, 37, 45, 46 Typical errors in absolute energies are in a millihartree (mE h ) range [or 0.01 kcal/(mol-electron)] for common auxiliary basis sets 20, 35, 37 and for CD with a threshold of 10 4 . 18, 19, 45 The accuracy in energy differences, such as activation energies, 16 is better by a factor of 2–3 in comparison to total energies due to error cancellation. The errors in dipole moments computed with RI 37, 38 and CD 19 are below 0.01 D and are usually an order of magnitude smaller that the errors due to the incompleteness of basis sets. The RI/CD bond lengths are within 0.01 pm from the respective full calculations. 20, 38, 48 Aquilante et al. 19 have also reported vertical excitation energies (computed with CASSCF and CASPT2) that show average errors less than 0.01 eV and 0.001 eV for thresholds of 10 3 and 10 4 , respectively. The effect of the RI approximation on excitation energies within an approximate second-order coupled-cluster model, CC2, has been thoroughly investigated by K¨ ohn and H¨ attig who reported errors of 0.01 eV or less. 48 In the present paper, we focus mostly on the effect of using RI/CD representations on energy differences between different electronic states, such as electronic excitation en- ergies and ionization/electron attachment energies. We also consider energy differences along potential energy surfaces. We compare the timings for RI/CD versus canonical implementations and investi- gate the parallel performance of the code 4 . All calculations were performed on desig- nated benchmark nodes. The hardware configuration is Xeon X5675 (2 processors, 6 cores each, 3.0 GHz, 12 Mb cache), 126 GB RAM, RAID 0 4600 GB=2.2 TB. This configuration was referred to as Xeon-USC in our previous benchmark study. 33 4 During the final revision stage, a small algorithmic improvement has been implemented that resulted in5% speed-up in CD/RI CC/EOM calculations. Thus, the reported timings are roughly 5% slower than the current code. 251 We use the following test cases: 1. Phenolate form of the anionic chromophore of the photoactive yellow protein (PYPb). 50, 51 We perform CCSD calculations as well as EOM-EE/IP/EA-CCSD. We consider the energy difference between the cis- and trans-isomers and elec- tronic energy differences between various states (electronically excited, electron- attached, and ionized states). The calculations were performed with three basis sets — 6-31+G(d,p) (test1), aug-cc-pVDZ (test2), and cc-pVTZ (test3). 2. Cluster of two methylated uracils and a water molecule (test4). 33 Energy differ- ences between different electronic states are considered. 3. Tetramer of 4 nucleobases, AATT, from Ref. 52 (test5). 4. Oligoporphyrin dimer used in previous benchmarks 33 (test6). 5. Cluster of methylated uracil, mU, and water from Ref. 53. We focus on the po- tential energy profiles along the proton-transfer reaction coordinate. The following thresholds were used in CCSD and EOM-CCSD calculations 5 : T - amplitudes convergence ofjT n T n1 j 10 4 , energy convergencejE n E n1 j 10 6 , Davidson’s procedure convergencejR n j 10 5 (hereR n is the Davidson resid- ual), threshold for subspace expansion in Davidson’s procedurejR n j> 10 5 . Table 9.3 lists parameters for different benchmark examples. All electrons were active in test1; test2 was executed with and without frozen core; core electrons were frozen in test3–6. In some cases, we also employed Frozen Natural Orbitals (FNO) approximation. 54 All Cartesian geometries and relevant energies are given in supplementary materials for Ref. 5 Q-Chem’s keywords controlling the CC and EOM convergence: CC T CONV = 4, CC E CONV = 6, EOM DA VIDSON CONV = 5, EOM DA VIDSON THRESH = 5. 252 41. All calculations were performed using regular (i.e., employing non-decomposed ERI) Hartree–Fock procedure. Correlation energies in Table 9.3 are for canonical cal- culations (using full ERI) except for test5. Tables 9.4 and 9.5 presents the comparison of the canonical CCSD calculation and RI/CD approximations. We note that the errors in total CCSD energies for RI and CD/10 3 approximations are comparable (and are in a millihartree range). However, the rank of CD/10 3 is often less than that of RI giving rise to a more significant speed- up (the situation is reversed in for test3 which uses the cc-pVTZ basis). We also note that CD/10 4 leads to the rank comparable to the size of the auxiliary basis in RI (1065 versus 1099), but yields two orders of magnitude more accurate total energies (error 8:27 10 6 versus 8:10 10 4 hartree). Overall, RI/CD CCSD calculations are 10–60% faster than the canonical implemen- tation. We observe a more significant speed-up for larger calculations, e.g., compare test4, test3, and test2 versus test1, likely because the I/O penalties are more pronounced for larger jobs. For the same molecule, we observe more significant speed-up in larger bases (compare test3 versus test2 and test1), because larger bases have more linear de- pendencies. Using test4 as an example, we observe that combination of CD with FNO approximation leads to a very impressive speed-up, i.e., CD/10 3 /FNO calculation takes only 15% of the time of the full CCSD calculation. 253 Table 9.3: Test systems used for benchmarks, converged CCSD correlation energies (hartree), and number of CC itera- tions. Name Molecule Formula Symm N el Basis a # b.f. E corr N iter Test1 PYPb b;c C 9 O 3 H 7 C s 86 6-31+G(d,p) 263 -1.838 498 14 Test2 PYPb b;c C 9 O 3 H 7 C s 86 aug-cc-pVDZ 363 -1.955 977 14 Test2-fc PYPb b;d C 9 O 3 H 7 C s 86 aug-cc-pVDZ 363 -1.875 348 14 Test3 PYPb b;d C 9 O 3 H 7 C s 86 cc-pVTZ 489 -2.147 251 14 Test4 (mU) 2 H 2 O c C 12 O 5 N 4 H 18 C 1 158 6-31+G(d,p) 489 -3.334 026 12 Test5 AATT d C 20 O 4 N 14 H 22 C 1 386 6-311+G(d,p) 968 -5.975 572 e 12 Test6 Porphyrin d C 46 N 12 H 26 D 2h 272 cc-pVDZ 942 -7.995 344 11 a Ref. 55, 56 for 6-31+G(d,p), Ref. 56, 57 for 6-311+G(d,p), Ref. 58 for cc-pVDZ, cc-pVTZ and aug-cc-pVDZ. b PYPb: anti-syn conformation. c Core electrons active. d Core electrons frozen. e Calculation using conventional CCSD with RI integrals (rimp2-cc-pVDZ 37 auxiliary basis set). 254 Table 9.4: CCSD errors and wall times (sec) using 12 cores for test1-test3 Name Method Rank CCSD error CCSD wall Ratio a CD wall time b Test1 Full 1894 RI/rimp2-aug-cc-pVDZ 1099 8:10 10 4 1771 0.94 CD/10 1 135 1:49 10 1 1500 0.79 85 CD/10 2 505 1:71 10 3 1591 0.84 530 CD/10 3 715 2:74 10 4 1642 0.87 978 CD/10 4 1065 8:27 10 6 1773 0.94 2035 Test2 Full 9490 RI/rimp2-aug-cc-pVDZ 1099 7:8 10 4 5175 0.55 CD/10 3 804 2:1 10 3 4750 0.50 2345 Test2-fc Full 2870 RI/rimp2-aug-cc-pVDZ 1099 1:1 10 3 2847 0.99 CD/10 3 804 4:2 10 4 2626 0.91 2348 Test3 Full 21257 RI/rimp2-cc-pVTZ 1256 1:4 10 3 9209 0.43 CD/10 3 1629 1:9 10 4 10367 0.49 15491 a Ratio=Time(RI/CD)/Time(full) for CCSD iterations. b Wall time for Cholesky decomposition (CD) procedure. Memory settings: test1 — 20 GB, test2 — 50 GB, test3 — 80 GB 255 Table 9.5: CCSD errors and wall times (sec) using 12 cores for test4-test6. Name Method Rank CCSD error CCSD wall Ratio a CD wall time b Test4 Full 110379 RI/rimp2-aug-cc-pVDZ 2067 1:6 10 3 85425 0.77 CD/10 3 1335 6:4 10 4 88198 0.80 10301 CD/10 3 /FNO c 1335 25464 0.15 10392 Test5 Full/rimp2-cc-pVTZ/FNO d;e;f 726.4 h RI/rimp2-cc-pVTZ/FNO d 3738 699.2 h 0.96 CD/10 2 /FNO d 1688 443.6 h 0.61 15.4 h Test6 Full 4:7 10 3 211.0 h RI/rimp2-cc-pVDZ 3612 4:7 10 3 194.9 h 0.92 RI/rimp2-cc-pVDZ/FNO g 3612 63.3 h 0.30 a Ratio=Time(RI/CD)/Time(full) for CCSD iterations. b Wall time for Cholesky decomposition (CD) procedure. c Using FNO (see Ref. 54), 99.50% occupation threshold and frozen core (350 active orbitals). CCSD converges in 11 iterations. d Using FNO, 99.50% occupation threshold and frozen core (649 active orbitals). CCSD converges in 12 iterations. e From Ref. 33. f Canonical CCSD calculations using RI integrals. g Using FNO, 99.50% occupation threshold and frozen core (754 active orbitals). CCSD converges in 16 iterations. Memory settings: test4 — 48 GB, test5 — 100 GB, test6 — 100 GB. 256 Finally, let us consider two large examples, i.e., a nucleobase tetramer (AATT) 33, 52 (test5, C 1 symmetry, 966 basis functions, 38 core orbitals frozen) and the oligopor- phyrin dimer (test6, D 2h symmetry, 942 basis functions, 58 core orbitals frozen) 33 and compare them to the canonical calculations. 33 In test5 we also employ FNO approxi- mation (279 out of 830 virtuals frozen, total 649 active orbitals). First, we note signifi- cant reduction in disk requirements for both examples (e.g., 382 GB versus 2.8 TB for AATT). For test6, the first RI/CD CCSD iteration is more than twice faster than in the canonical implementation (6.33 hours for RI-CCSD versus 13.2 hours for the canonical CCSD 33 ). However, we observe a slowdown of the subsequent iterations due to the in- creasing number ofT -amplitudes that need to be handled by the DIIS procedure. The average time per iteration for oligoporphyrin is 12 hours (194.9 hours total time, 16 it- erations), although the first iteration is two times faster (6.33 hrs). We also computed oligoporphyrin in combination with FNO (130 out of 749 virtual orbitals frozen, 754 active orbitals) with time for first iteration 2.35 hours and the average time per iteration 3.95 hours. For AATT we observe a similar speed-up of RI-CCSD iterations, the first RI-CCSD iteration takes 39 hours (to be compared to 60 hours in the canonical imple- mentation 33 ). AATT computed with CD/10 2 yields a rank of 1688 and the first CCSD iteration takes 28.25 hours, to be compared with 60 hours in the canonical implementa- tion and 39 hours with RI/rimp2-cc-pVTZ. A more detailed breakdown of timings for CCSD calculations is given in supplemen- tary materials for Ref. 41 (Table S1). As expected, the evaluation of intermediate I (2i) ijab takes a significant fraction of time, especially in larger bases (it scales asO 2 V 4 ). Evalu- ation of equation for t 2 amlitudes, which contains oneO 3 V 3 (third term) and oneO 4 V 2 (the last term) contractions, is also significant and becomes dominant in an electron-rich case, test6. 257 The time used for the decomposition and integral transformation steps is also shown in Tables 9.4 and 9.5. Since the present implementation of the decomposition algorithm does not use point group symmetry, the timings for test1–test3 are relatively large. We note that for test4 (C 1 ), the time of decomposition with a threshold of 10 3 is about 12% of the total time for CCSD iterations. We investigate parallel performance using test4 (Table 9.6). The parallel scaling is improved, e.g., the canonical implementation shows a factor of 6 speed-up on 12 cores, whereas the RI/CD code is accelerated by a factor of 9. Thus, the speed-up relative to the canonical CCSD code becomes more pronounced on 12 cores, e.g., on a single CPU, RI/CD calculation is about 20% faster, whereas on 12 cores, it is almost a factor of two faster than the canonical code. This improvement in the parallel performance is due to the significant reduction of the amount of data to be handled in the CCSD calculations. Table 9.6: Wall time per CCSD iteration (sec) using 80 GB RAM. Job 1 core 4 cores 8 cores 12 cores Full 46405 14278 (3.25) 9506 (4.88) 7973 (5.82) RI/rimp2-aug-cc-pVDZ 39347 10283 (3.83) 5539 (7.10) 4342 (9.06) CD/10 3 37330 9889 (3.78) 4973 (7.51) 4185 (8.92) Table 9.7 shows EOM energies and timings for test1 and test2; the results for test2- fc and test3 are presented in Table 9.8. We note that RI and CD/10 3 give comparable errors in excitation, attachment, and ionization energies, i.e., less than 0.01-0.001 eV . These errors are consistent with those reported for the CASSCF and CASPT2 meth- ods. 19 The errors in the energies are systematically reduced with the Cholesky threshold decrease from 10 2 to 10 4 for all methods. We observe that a threshold of 10 2 yields errors of0.03 eV , which is acceptable in many situations and is less than error bars of EOM-CCSD. 258 For test1, the timings for RI/CD EOM methods are slower than of the canonical im- plementation due to the increased number of contractions, as explained in Section 9.3. However, in a larger basis (aug-cc-pVDZ versus 6-31+G*), the gap shrinks for EOM-IP (total RI/CD EOM time is almost the same as of the canonical calculation), and RI/CD EOM-EE shows 60-70% speed-up. Further increase of the basis (to cc-pVTZ) leads to an additional speed-up, i.e., RI/CD EOM-EE calculations for test3 take 25% of the full EOM-EE time. This is because the increased number of time-determining contrac- tions in RI/CD EOM-EE (7 for RI/CD EOM-EE versus 3 N 6 operations in canonical EOM) is offset by the significantly reduced disk and memory usage by RI/CD EOM that reduces I/O penalties and improves parallel scaling. For example, for test5 (AATT) EOM-EE-CCSD calculations (with frozen core and FNO) the estimated disk usage is 7.2 TB, whereas for the corresponding RI/rimp2-cc-pVTZ calculation it is only 590 GB. For test3, we observe that canonical EOM shows rather poor parallel scaling (CPU 102655 s, wall 93432 s, ratio=1.09), whereas for RI EOM we see more than a 10 fold CPU/wall ratio (CPU 228202 s, wall 21553 s, ratio=10.58), leading to an overall 5-fold speedup of Davidson iterations. Thus, RI/CD implementation of EOM not only extends the applicability of the method to larger systems that may not be accessible by canonical EOM-CCSD due to disk/memory bottlenecks, but also improves timings of the calcula- tions by removing the overheads due to large size of the data. The calculation time of the intermediates for EOM calculations is significantly re- duced for all RI/CD methods, as illustrated by test2 timings revealing that the interme- diates calculations (dominated by the VVVV block of the transformed integrals) take almost as much time as Davidson iterations. Thus, the overall CD/RI EOM-EE timings (Davidson iterations plus intermediates) are considerably faster (3-5 times) than those of the canonical code when only of few EOM roots are computed for large systems. 259 The detailed timings for RI EOM-EE-CCSD calculations are given in supplementary information for Ref. 41 (Table S2). We observe that 2 -vector update procedure takes most of total EOM time (96% for test 3). Within it, the calculation of I 1i ijab , is dominant (70% of the total EOM time for test3), as expected based onO 2 V 4 operations required to evaluate this term. Calculations of ionization energies for test4 (Table 9.9) show the errors of the same order of the magnitude, 0.001 eV and 0.0001 eV for RI and CD/10 3 , respectively, as in the PYPb examples (test1 and test2). Calculations of EOM-IP-vectors with RI/CD are slightly slower than in the canonical calculations, however the time required for calculation of intermediates is significantly smaller for RI/CD, resulting in more than 2-fold overall speedup. The speed-up for RI/CD EOM-IP is less than for RI/CD EOM- EE due to smaller size of the data used by EOM-IP, which does not involve VVVV and OVVV intermediates; thus, the canonical code shows much better parallel performance than EOM-EE (for full EOM-IP, CPU 587 s, wall 63 s, ratio=9.45; for RI EOM-IP, CPU 2067 s, wall 214 s, ratio=9.70). Using FNO (threshold 99.5%, 118 virtual orbitals frozen out of 410 total) signif- icantly improves the total EOM timings making it more than 6 times faster than the full canonical calculation. The errors introduced by the FNO approximation are larger than those due to CD (0.01 eV), but they are still acceptable for most applications. Thus, RI/CD in conjunction with FNO leads to significant reduction of both memory and computational cost requirements, with only minor losses in accuracy. 260 Table 9.7: EOM-CCSD energies for the 2 lowest states in each irrep and errors in energy differences (eV), and wall times for EOM (sec) using 12 cores. Method EOM time EOM 1A’ 2A’ 1A” 2A” Int a Iter b Total c Ratio d calls e Test1 EOM-EE-CCSD 342 3344 3686 46 3.158 eV 4.233 eV 3.860 eV 4.171 eV RI f 65 6641 6706 1.8 (2.0) 46 1:010 4 1:010 4 1:210 3 1:210 3 CD/10 2 56 6093 6149 1.7 (1.8) 46 2:810 2 3:510 2 2:510 2 2:010 2 CD/10 3 59 6276 6335 1.7 (1.9) 46 4:210 3 6:710 3 6:510 3 3:910 3 CD/10 4 64 6588 6652 1.8 (2.0) 46 2:010 4 6:010 4 1:010 3 9:010 4 EOM-EA-CCSD 351 486 837 31 3.931 eV 4.329 eV 3.700 eV 5.268 eV RI f 65 984 1049 1.3 (2.0) 31 1:010 4 < 10 4 1:310 3 1:010 4 CD/10 2 56 710 766 0.9 (1.5) 31 1:910 2 9:810 3 1:510 2 2:510 2 CD/10 3 59 788 847 1.0 (1.6) 31 7:510 3 5:610 3 2:510 3 6:610 3 CD/10 4 65 975 1040 1.2 (2.0) 31 4:010 4 1:410 3 3:010 4 1:110 3 EOM-IP-CCSD 93 46 139 26 4.328 eV 6.758 eV 2.735 eV 5.353 eV RI f 65 132 197 1.4 (2.9) 26 6:010 4 8:010 4 1:310 3 1:110 3 CD/10 2 57 108 165 1.2 (2.4) 26 1:310 2 6:910 3 2:310 3 2:810 3 CD/10 3 60 117 177 1.3 (2.5) 26 2:010 4 1:010 4 2:010 4 7:010 4 CD/10 4 65 132 197 1.4 (2.9) 26 < 10 4 < 10 4 1:010 4 < 10 4 Test2 EOM-EE-CCSD 20021 39991 60012 49 3.167 eV 4.148 eV 3.349 eV 5.666 eV RI f 139 18522 18661 0.3 (0.5) 49 1:010 4 1:010 4 1:010 3 1:110 3 CD/10 3 125 18572 18697 0.3 (0.5) 49 5:010 4 1:210 3 2:010 4 < 10 4 EOM-IP-CCSD 245 70 315 26 4.467 eV 6.772 eV 2.904 eV 5.462 eV RI f 165 211 376 1.2 (3.0) 26 9:010 4 8:010 4 1:310 3 8:010 4 CD/10 3 159 196 355 1.1 (2.8) 26 9:010 4 7:010 4 9:010 4 8:010 4 a Time for calculations of the EOM-CCSD intermediates for the Davidson procedure. b Time for EOM iterations. c Total EOM time (intermediates + Davidson iterations). d Ratio= Time(RI/CD)/Time(full). The first value is the ratio of total EOM times; the ratio for Davidson iterations is given in parentheses. e -vector update calls. f rimp2-aug-cc-pVDZ auxiliary basis. 261 Table 9.8: EOM-CCSD energies for the 2 lowest states in each irrep and errors in energy differences (eV), and wall times (sec) using 12 cores. Method EOM time EOM 1A’ 2A’ 1A” 2A” Int a Iter b Total c Ratio d calls e Test2-fc EOM-EE-CCSD 4565 6917 11482 48 3.166 eV 4.148 eV 3.344 eV 5.662 eV RI f 68 9619 9687 0.8 (1.4) 48 1:010 4 1:010 4 1:110 3 1:110 3 CD/10 3 62 8987 9049 1.3 (2.3) 48 5:010 4 1:210 3 2:010 4 < 10 4 Test3 EOM-EE-CCSD 47946 93432 141378 36 3.307 eV 4.333 eV 5.513 eV 5.690 eV RI g 170 21552 21722 0.2 (0.2) 36 3:010 4 1:010 4 4:010 4 3:010 4 CD/10 3 194 26839 27033 0.2 (0.3) 36 8:010 4 9:010 4 2:610 3 2:410 3 EOM-IP-CCSD 426 62 488 26 4.450 eV 6.795 eV 2.872 eV 5.447 eV RI g 166 214 380 0.8 (3.5) 26 1:310 3 1:310 3 1:010 3 8:010 4 CD/10 3 189 245 434 0.9 (4.0) 26 1:010 4 1:010 4 2:010 4 3:010 4 a Time for calculations of the EOM-CCSD intermediates for the Davidson procedure. b Time for EOM iterations. c Total EOM time (intermediates + Davidson iterations). d Ratio= Time(RI/CD)/Time(full). The first value is the ratio of total EOM times; the ratio for Davidson iterations is given in parentheses. e -vector update calls. f rimp2-aug-cc-pVDZ auxiliary basis. g rimp2-cc-pVTZ auxiliary basis. 262 Table 9.9: EOM-IP-CCSD energies (absolute errors for RI/CD) and EOM wall times (sec) for test4 (two lowest EOM roots). Method EOM time EOM State 1 State 2 Int a Iter b Total c Ratio d calls e EOM-IP-CCSD 5088 503 5591 10 8.421 eV 8.858 eV RI f 1866 576 2442 0.4 10 1:0 10 3 1:0 10 3 CD/10 3 2083 514 2597 0.5 10 5:0 10 4 2:0 10 4 CD/10 3 /FNO g 541 270 811 0.2 10 1:6 10 2 1:5 10 2 a Time for calculations of the EOM-CCSD intermediates for the Davidson procedure. b Time for EOM iterations. c Total EOM time (intermediates + Davidson iterations). d Ratio of total times: Time(RI/CD)/Time(Full). e Number of calls of-update procedure. f rimp2-aug-cc-pVDZ auxiliary basis. g Frozen core and FNO (threshold 99.50%) was used. To quantify the errors in energy differences along potential energy surfaces, we con- sider two examples. We begin by considering the energy differences between two PYP isomers 51 shown in Table 9.10. The energy difference between two PYPb isomers (anti- syn and anti-anti) is 4.15 kcal/mol at the CCSD/6-31+G(d,p) level of theory. The er- rors introduced by RI and CD are: 2:40 10 3 (rimp2-aug-cc-pVDZ), 6:42 10 2 (CD/10 2 ), 2:12 10 2 (CD/10 3 ), and 9:00 10 4 (CD/10 4 ) kcal/mol; the errors are considerably smaller than the errors in the total CCSD correlation energy due to error cancellation. Note that even for the crudest CD threshold (10 2 ) the error in the energy differences is quite satisfactory (0.1 kcal/mol). The error cancellation effect is more pronounced for RI where the error in energy differences is more than 2 orders of magnitude less than the error in the total energy, whereas for CD the difference is more modest (about 1 order of magnitude). Thus, in terms of the energy differences, RI is more accurate than CD/10 3 , but is still slightly less accurate than CD/10 4 . As a more challenging case, we consider scans along proton-transfer coordinate in ionized mU-H 2 O cluster from Ref. 53. Fig. 9.1 shows CCSD and EOM-IP-CCSD energies along the proton-transfer reaction coordinate computed in the 6-311+G(d,p) 263 Table 9.10: Energy differences between PYPb isomers (E anitanti E antisyn , kcal/mol) and the corresponding errors against full CCSD. Method Energy difference Error, kcal/mol Error, hartree Full 4.1531 RI/rimp2-aug-cc-pVDZ 4.1507 2:40 10 3 3:82 10 6 CD/10 2 4.0889 6:42 10 2 1:02 10 4 CD/10 3 4.1319 2:12 10 2 3:37 10 5 CD/10 4 4.1540 9:00 10 4 1:43 10 6 basis set. We note that RI features the smallest errors, both in terms of absolute values (around 10 4 –10 5 eV) and in terms of non-parallelity errors (NPEs) (4 10 5 and 5 10 5 eV for CCSD and EOM-IP-CCSD energies, respectively). This is because the auxiliary basis in RI is atom-centered and does not depend on geometry. 59 CD shows larger errors along the scan; however, the respective NPEs are small and do not exceed 0.001 eV for CD/10 3 and 0.0003 eV for CD/10 4 . We note that the range of changes in total energy along this scan is about 2 eV . Smooth behavior of the CD scans is consistent with small variations of the rank along this scan, e.g., for CD/10 3 and CD/10 4 the rank is 8342 and 11883, respectively. 9.5 Conclusions We present a new implementation of RI and Cholesky decompositions within the CCSD/EOM-CCSD suite of methods in the Q-Chem electronic structure package. 27, 28 This implementation eliminates the storage of the most expensive four-index electron repulsion integrals and intermediates, such as VVVV , OVVV and OVOV blocks of ERI, leading to a significant reduction in storage requirements and I/O overheads. The num- ber of floating-point operations is reduced for CCSD; however, it is increased by approx- imately a factor of 3 in EOM calculations (-vectors update) because the transformed 264 Figure 9.1: Top: CCSD (left) and EOM-IP-CCSD (right) energies along the proton-transfer coordinate in mU-H 2 O. Bottom: Errors of RI/rimp2-aug-cc-pVTZ and CD approximations. integrals and related intermediates, which are computed only once in canonical EOM, need to be reassembled at each Davidson iteration in the RI/CD implementation. How- ever, this undesirable increase in computations is offset by significantly reduced I/O overheads. In a shared-memory parallel setting the reduction of I/O also leads to better CPU utilization and improved parallel scalability. When the calculation of the interme- diates is included, the ratio between RI/CD and canonical EOM-EE timings is about 0.3–0.5 for moderate-size basis sets. The gains are more significant in large bases, e.g., a RI-EOM-EE-CCSD/cc-pVTZ calculation takes only 15% of the time required for the full calculation. Additional computational savings can be achieved by combining RI/CD and FNO approaches. 54 265 The accuracy of RI/CD implementations is benchmarked with an emphasis on en- ergy differences, such as excitation energies. In agreement with previous benchmarks based on the CASSCF, CASPT2, and CC2 methods, 19, 48 we observe that the errors in energy differences are smaller than the errors in total energies due to error cancellation. Typical errors in the CCSD correlation energy are less than a millihartree for the RI approximation with RI-MP2 auxiliary bases, however, the respective EOM errors are less than 0.001 eV . The accuracy of CD can be controlled by a single threshold. For a threshold of 10 4 , which results in a rank similar to RI, the errors in total energies are two orders of magnitude less than for RI; however, the errors in energy differences are roughly the same. This threshold is therefore recommended when high accuracy is re- quired. We note that errors in excitation energies are quite small when using thresholds of 10 2 and 10 3 (less than 0.04 and 0.008 eV , respectively); therefore these thresholds can be used in most calculations. This paper presents our first step towards developing reduced-scaling CC/EOM-CC codes. While the present implementation does not reduce scaling of the calculations, it affords significant computational savings thus extending the applicability of these methods to larger systems. In order to achieve further gains, additional steps should be taken. Among promising strategies 19 are a tensor hyper-contraction approach, 21, 22 local correlation schemes and pair natural orbitals, 25, 26, 60, 61 as well as reduced-rank representations of the CC/EOM amplitudes. 23, 24, 62, 63 266 Chapter 9 references [1] J.A. Pople. Theoretical models for chemistry. In D.W. Smith and W.B. McRae, ed- itors, Energy, Structure and Reactivity: Proceedings of the 1972 Boulder Summer Research Conference on Theoretical Chemistry, pages 51–61. Wiley, New York, 1973. [2] T. Helgaker, P. Jørgensen, and J. Olsen. Molecular electronic structure theory. Wiley & Sons, 2000. [3] R.J. Bartlett. The coupled-cluster revolution. Mol. Phys., 108:2905–2920, 2010. [4] A.I. Krylov. Equation-of-motion coupled-cluster methods for open-shell and elec- tronically excited species: The hitchhiker’s guide to Fock space. Annu. Rev. Phys. Chem., 59:433–462, 2008. [5] K. Sneskov and O. Christiansen. Excited state coupled cluster methods. WIREs Comput. Mol. Sci., 2:566, 2012. [6] R.J. Bartlett. Coupled-cluster theory and its equation-of-motion extensions. WIREs Comput. Mol. Sci., 2(1):126–138, 2012. [7] F. Weigend, M. H¨ aser, H. Patzelt, and R. Ahlrichs. RI-MP2: optimized auxiliary basis sets and demonstration of efficiency. Chem. Phys. Lett., 294:143–152, 1998. [8] J.L. Whitten. Coulombic potential energy integrals and approximations. J. Chem. Phys., 58:4496, 1973. [9] M.W. Feyereisen, G. Fitzgerald, and A. Komornicki. Use of approximate integrals in abinitio theory - an application in MP2 energy calculations. Chem. Phys. Lett., 208:359–363, 1993. [10] O. Vahtras, J. Alml¨ of, and M.W. Feyereisen. Integral approximations for LCAO- SCF calculations. Chem. Phys. Lett., 213:514–518, 1993. [11] A. Komornicki and G. Fitzgerald. Molecular gradients and hessians implemented in density functional theory. J. Chem. Phys., 98:1398–1421, 1993. [12] D.E. Bernholdt and R.J. Harrison. Large-scale correlated electronic structure calculations: The RI-MP2 method on parallel computers. Chem. Phys. Lett., 250:477–484, 1996. 267 [13] Y . Jung, A. Sodt, P. M. W. Gill, and M. Head-Gordon. Auxiliary basis expansions for large-scale electronic structure calculations. Proc. Nat. Acad. Sci., 102:6692– 6697, 2005. [14] N.H.F. Beebe and J. Linderberg. Simplifications in the generation and transfor- mation of two-electron integrals in molecular calculations. Int. J. Quant. Chem., 12:683–705, 1977. [15] H. Koch, A.S. de Mer´ as, and T.B. Pedersen. Reduced scaling in electronic structure calculations using Cholesky decompositions. J. Chem. Phys., 118:9481, 2003. [16] F. Aquilante, R. Lindh, and T. B. Pedersen. Unbiased auxiliary basis sets for accurate two-electron integral approximations. J. Chem. Phys., 127:114107, 2007. [17] F. Aquilante, T. B. Pedersen, and R. Lindh. Density fitting with auxiliary basis sets from Cholesky decompositions. Theor. Chem. Acc., 124:1–10, 2009. [18] F. Aquilante, L. Gagliardi, T. B. Pedersen, and R. Lindh. Atomic Cholesky decom- positions: A route to unbiased auxiliary basis sets for density fitting approximation with tunable accuracy and efficiency. J. Chem. Phys., 130:154107, 2009. [19] F. Aquilante, L. Boman, J. Bostr¨ om, H. Koch, R. Lindh, A.S. de Mer´ as, and T.B. Pedersen. Cholesky decomposition techniques in electronic structure theory. In R. Zale´ sny, M.G. Papadopoulos, P.G. Mezey, and J. Leszczynski, editors, Linear- Scaling Techniques in Computational Chemistry and Physics, Challenges and ad- vances in computational chemistry and physics, pages 301–343. Springer, 2011. [20] F. Weigend, M. Kattannek, and R. Ahlrichs. Approximated electron repulsion integrals: Cholesky decomposition versus resolution of the identity methods. J. Chem. Phys., 130:164106, 2009. [21] R.M. Parrish, E.G. Hohenstein, T.J. Mart´ ınez, and C.D. Sherrill. Tensor hyper- contraction. II. Least-squares renormalization. J. Chem. Phys., 137(22):224106, 2012. [22] E.G. Hohenstein, R.M. Parrish, C.D. Sherrill, and T.J. Mart´ ınez. Communication: Tensor hypercontraction. III. Least-squares tensor hypercontraction for the deter- mination of correlated wavefunctions. J. Chem. Phys., 137(22):221101, 2012. [23] F. Aquilante, T.K. Todorova, L. Gagliardi, T.B. Pedersen, and B.O. Roos. System- atic truncation of the virtual space in multiconfigurational perturbation theory. J. Chem. Phys., 131:034113, 2009. [24] F. Aquilante and T.B. Pedersen. Quartic scaling evaluation of canonical scaled opposite spin second-order MøllerPlesset correlation energy using Cholesky de- compositions. Chem. Phys. Lett., 449:354–357, 2007. 268 [25] C. Riplinger and F. Neese. An efficient and near linear scaling pair natural orbital based local coupled cluster method. J. Chem. Phys., 138:034106, 2013. [26] D. Kats, T. Korona, and M. Sch¨ utz. Local CC2 electronic excitation energies for large molecules with density fitting. J. Chem. Phys., 125:104106–104121, 2006. [27] Y . Shao, L. Fusti-Molnar, Y . Jung, J. Kussmann, C. Ochsenfeld, S. Brown, A.T.B. Gilbert, L.V . Slipchenko, S.V . Levchenko, D.P. O’Neill, R.A. Distasio Jr, R.C. Lochan, T. Wang, G.J.O. Beran, N.A. Besley, J.M. Herbert, C.Y . Lin, T. Van V oorhis, S.H. Chien, A. Sodt, R.P. Steele, V .A. Rassolov, P. Maslen, P.P. Koram- bath, R.D. Adamson, B. Austin, J. Baker, E.F.C. Byrd, H. Daschel, R.J. Doerksen, A. Dreuw, B.D. Dunietz, A.D. Dutoi, T.R. Furlani, S.R. Gwaltney, A. Heyden, S. Hirata, C.-P. Hsu, G.S. Kedziora, R.Z. Khalliulin, P. Klunziger, A.M. Lee, W.Z. Liang, I. Lotan, N. Nair, B. Peters, E.I. Proynov, P.A. Pieniazek, Y .M. Rhee, J. Ritchie, E. Rosta, C.D. Sherrill, A.C. Simmonett, J.E. Subotnik, H.L. Woodcock III, W. Zhang, A.T. Bell, A.K. Chakraborty, D.M. Chipman, F.J. Keil, A. Warshel, W.J. Hehre, H.F. Schaefer III, J. Kong, A.I. Krylov, P.M.W. Gill, M. Head-Gordon. Advances in methods and algorithms in a modern quantum chemistry program package. Phys. Chem. Chem. Phys., 8:3172–3191, 2006. [28] A.I. Krylov and P.M.W. Gill. Q-Chem: An engine for innovation. WIREs Comput. Mol. Sci., 3:317–326, 2013. [29] A.P. Rendell and T.J. Lee. Coupled-cluster theory employing approximate inte- grals - an approach to avoid the input/output and storage bottlenecks. J. Chem. Phys., 101:400–408, 1994. [30] F. Aquilante, L. de Vico, N. Ferr´ e, G. Ghigo, P- ˚ A Malmqvist, P. Neogr´ ady, T. B. Pedersen, M. Pitonˇ ak, M. Reiher, B. Roos, L. Serrano-Andr´ es, M. Urban, V . Verya- zov, and R. Lindh. Molcas 7: The next generation. J. Comput. Chem., 31:224–247, 2010. [31] S. Wilson. Universal basis sets and Cholesky decomposition of the two-electron integral matrix. Comp. Phys. Comm., 58:71–81, 1990. [32] G.H. Golub and C.F. Van Loan. Matrix computations. Johns Hopkins University Press, 1996. [33] E. Epifanovsky, M. Wormit, T. Ku´ s, A. Landau, D. Zuev, K. Khistyaev, P. Manohar, I. Kaliman, A. Dreuw, and A.I. Krylov. New implementation of high-level corre- lated methods using a general block-tensor library for high-performance electronic structure calculations. J. Comput. Chem., 34:2293–2309, 2013. [34] K. Eichkorn, O. Treutler, H. ¨ Ohm, M. H¨ aser, and R. Ahlrichs. Auxiliary basis sets to approximate Coulomb potentials. Chem. Phys. Lett., 242:652–660, 1995. 269 [35] K. Eichkorn, F. Weigend, O. Treutler, and R. Ahlrichs. Auxiliary basis sets for main row atoms and transition metals and their use to approximate Coulomb po- tentials. Theor. Chem. Acc., 97:119–124, 1997. [36] F. Weigend. Hartree-Fock exchange fitting basis sets for H to Rn. J. Comput. Chem., 29:167–175, 2008. [37] F. Weigend, A. K¨ ohn, and C. H¨ attig. Efficient use of the correlation consistent basis sets in resolution of the identity MP2 calculations. J. Chem. Phys., 116:3175, 2002. [38] F. Weigend. Accurate coulomb-fitting basis sets for H to Rn. Phys. Chem. Chem. Phys., 8:1057–1065, 2006. [39] A. Hellweg, C. H¨ attig, S. H¨ ofener, and W. Klopper. Optimized accurate auxiliary basis sets for RI-MP2 and RI-CC2 calculations for the atoms Rb to Rn. Theor. Chem. Acc., 117:587–597, 2007. [40] G.D. Purvis and R.J. Bartlett. A full coupled-cluster singles and doubles model: The inclusion of disconnected triples. J. Chem. Phys., 76:1910–1918, 1982. [41] E. Epifanovsky, D. Zuev, X. Feng, K. Khistyaev, Y . Shao, and A.I. Krylov. General implementation of resolition-of-identity and Cholesky representations of electron- repulsion integrals within coupled-cluster and equation-of-motion methods: The- ory and benchmarks. J. Chem. Phys., 139:134105, 2013. [42] J.F. Stanton and R.J. Bartlett. The equation of motion coupled-cluster method. A systematic biorthogonal approach to molecular excitation energies, transition probabilities, and excited state properties. J. Chem. Phys., 98:7029–7039, 1993. [43] S.V . Levchenko and A.I. Krylov. Equation-of-motion spin-flip coupled-cluster model with single and double substitutions: Theory and application to cyclobu- tadiene. J. Chem. Phys., 120(1):175–185, 2004. [44] A.I. Krylov. The spin-flip equation-of-motion coupled-cluster electronic structure method for a description of excited states, bond-breaking, diradicals, and triradi- cals. Acc. Chem. Res., 39:83–91, 2006. [45] J. Bostr¨ om, F. Aquilante, T. B. Pedersen, and R. Lindh. Ab Initio density fitting: Accuracy assessment of auxiliary basis sets from Cholesky decompositions. J. Chem. Theory Comput., 5:1545–1553, 2009. [46] V . P. Vysotskiy and L. S. Cederbaum. On the Cholesky decomposition for electron propagator methods: General aspects and application on C 60 . J. Chem. Phys., 132:044110, 2010. 270 [47] C. H¨ attig and F. Weigend. CC2 excitation energy calculations on large molecules using the resolution of the identity approximation. J. Chem. Phys., 113(13):5154– 5161, 2000. [48] A. K¨ ohn and C. H¨ attig. Analytic gradients for excited states in the coupled-cluster model CC2 employing the resolution-of-the-identity approximation. J. Chem. Phys., 119(10):5021–5036, 2003. [49] C. H¨ attig and A. K¨ ohn. Transition moments and excited-state first-order properties in the coupled-cluster model CC2 using the resolution-of-the-identity approxima- tion. J. Chem. Phys., 117(15):6939–6951, 2002. [50] T. Rocha-Rinza, O. Christiansen, J. Rajput, A. Gopalan, D.B. Rahbek, L.H. An- dersen, A.V . Bochenkova, A.A.Granovsky, K.B. Bravaya, A.V . Nemukhin, K.L. Christiansen, and M.B. Nielsen. Gas phase absorption studies of photoactive yel- low protein chromophore derivatives. J. Phys. Chem. A, 113:9442–9449, 2009. [51] D. Zuev, K.B. Bravaya, T.D. Crawford, R. Lindh, and A.I. Krylov. Electronic structure of the two isomers of the anionic form of p-coumaric acid chromophore. J. Chem. Phys., 134:034310, 2011. [52] K.B. Bravaya, E. Epifanovsky, and Anna I. Krylov. Four bases score a run: Ab initio calculations quantify a cooperative effect of h-bonding and pi-stacking on ionization energy of adenine in the AATT tetramer. J. Phys. Chem. Lett., 3:2726– 2732, 2012. [53] K. Khistyaev, A. Golan, K.B. Bravaya, N. Orms, A.I. Krylov, and M. Ahmed. Proton transfer in nucleobases is mediated by water. J. Phys. Chem. A, 117:6789– 6797, 2013. [54] A. Landau, K. Khistyaev, S. Dolgikh, and A.I. Krylov. Frozen natural orbitals for ionized states within equation-of-motion coupled-cluster formalism. J. Chem. Phys., 132:014109, 2010. [55] W.J. Hehre, R. Ditchfield, and J.A. Pople. Self-consistent molecular orbital meth- ods. XII. Further extensions of gaussian-type basis sets for use in molecular orbital studies of organic molecules. J. Chem. Phys., 56:2257, 1972. [56] T. Clark, J. Chandrasekhar, and P.V .R. Schleyer. Efficient diffuse function- augmented basis sets for anion calculations. III. The 3-21+g basis set for first-row elements, li-f. J. Comput. Chem., 4:294–301, 1983. [57] R. Krishnan, J.S. Binkley, R. Seeger, and J.A. Pople. Self-consistent molecular orbital methods. XX. A basis set for correlated wave functions. J. Chem. Phys., 72:650, 1980. 271 [58] T.H. Dunning. Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogen. J. Chem. Phys., 90:1007–1023, 1989. [59] F. Weigend and M. H¨ aser. RI-MP2: first derivatives and global consistency. Theor. Chim. Acta, 97:331–340, 1997. [60] J. Yang, Y . Kurashige, F.R. Manby, and G.K.L. Chan. Tensor factorizations of local second-order Moller-Plesset theory. J. Chem. Phys., 134:044123, 2011. [61] J.E. Subotnik and M. Head-Gordon. A local correlation model that yields intrinsi- cally smooth potential energy surfaces. J. Chem. Phys., 123:064108, 2005. [62] U. Benedikt, A.A. Auer, M. Espig, and W. Hackbusch. Tensor decomposition in post-Hartree-Fock methods. J. Chem. Phys., 134:054118, 2011. [63] F. Bell, D. Lambrecht, and M. Head-Gordon. Higher order singular value decom- position (HOSVD) in quantum chemistry. Mol. Phys., 108:2759–2773, 2010. 272 Chapter 10: Implementation of analytic gradients for CCSD and EOM-CCSD using Cholesky representation of electron-repulsion integrals: Theory and benchmarks 10.1 Introduction One of the problems with correlated electronic structure methods is the fourth-order- scaling of the storage requirements, which limits the scope of applications. The main culprit is the matrix of two-electron repulsion integrals (ERI). One way to improve the efficiency of correlated electronic structure methods is to decompose ERI using Cholesky decomposition (CD) technique into the following form: (j) M X P =1 B P B P ; (10.1) 273 where M is the rank of decomposition, which depends on the desired accuracy. The latter is controlled by served decomposition threshold. With this technique, theN 4 - type matrix is reduced toN 3 -type ones, so the calculations of larger molecular system become affordable. This technique was implemented for calculating energies and amplitudes in coupled- cluster (CC) and equation-of-motion (EOM) CC mothods in order to reduce the storage requirements. As a consequence, more efficient parallelization due to reduced I/O over- heads was achieved. The algorithm 1–4 proceeds in an iterative manner, giving rise to the following recursive expression for Cholesky vectors,B P : B J = ^ (h J jh J ) 1 2 " (jh J ) J1 X K=1 B K B K J # ; (10.2) ^ (h J jh J ) = (h J jh J ) J1 X K=1 B K J B K J ; (10.3) whereh J denotes a particular column in the (j), i.e., h J = () J . The order of h J is such that the first Cholesky vector, (jh 1 )= p (h 1 jh 1 ), corresponds to the largest diagonal element of the ERI matrix, (() 1 j() 1 ). The implementation of analytical gradients is crucially important for explorations of potential energy surfaces (PESs), optimization of the equilibrium and transition state geometries, and finding minimum energy crossing points. Therefore, implementing the analytical gradient within CD framework is important for applications. An implemen- tation of CD gradient at the density functional theory (DFT) level was presented. 5 Here we present the analytical derivative of CD-CC and CD-EOM-CC energies. Because the CC and EOM methods are much more computationally expensive than DFT, the savings due to CD are more promising. 274 10.2 Theory The CD algorithm can be extended for energy gradient calculations by differentiating Eq. (10.1): @(j) @ = X P @B P @ B P +B P @B P @ ! X P X P B P +B P X P ; (10.4) whereX P denotes the derivative of theP th Cholesky vector and is obtained by differ- entiating Eq. (10.2): X J = ^ (h J jh J ) 1 2 @(jh J ) @ J1 X K=1 X K B K J J1 X K=1 B K X K J ! 1 2 ^ (h J jh J ) 1 @(h J jh J ) @ 2 J1 X K=1 X K J B K J ! B J : (10.5) This is the exact derivative of Cholesky vector B P assuming that the number of Cholesky vectors in the expansion does not change upon small nuclear displacements. We note that Eq. (10.5) can be evaluated recursively, similar to the decomposition procedure. In particular, one can simply modify CD algorithm such that the derivative vectors, X J , are computed simultaneously with the respective Cholesky vectors, B J (alternatively, one can compute derivatives in a separate routine, but this will require more bookkeeping). AtJ th iteration of the CD procedure, the following quantities are needed for evaluatingX J : 1. Cholesky vectors and their derivatives from previous steps,fB K ;X J g J1 K=1 . 275 2. A batch of derivative integrals as needed for the following: @(jh J ) @ = ( j() J ) + ( j() J ) + (j( ) J ) + (j( ) J ) (10.6) and @(h J jh J ) @ = 2 (( ) J j() J ) + ( ) J j() J ) : (10.7) Thus, derivative integrals can be computed in batches, i.e., for a givenh J = () J , one needs a batch of ( jh J ), where ; run over all AO indexes, and (j( ) J ) and (j( ) J ), where only symmetry-unique elements are needed. Other derivative integrals needed for this step can be obtained from this batch by using permutational symmetry: ( jh J ) = ( jh J ). () J j( ) J ) = ( ) J j() J ) are included in the ( jh J ) batch. By replacing the second term of the expression for the CCSD/EOM-CCSD energy gradient: dE d = X pq h pq pq + 1 4 X pqrs <pqjjrs> pqrs + X pq ! pq S pq ; (10.8) with Cholesky vectors: X pqrs <pqjjrs> pqrs = X pqrs X P B P pr X P qs +X P pr B P qs B P ps X P qr X P ps B P qr pqrs = 4 X P X pr P pr X P pr ; (10.9) where P pr = X qs B P qs pqrs (10.10) we are able to compute the gradient of CD-CC and CD-EOM-CC analytically. 276 10.3 Benchmarks and Discussions The validation of Cholesky decomposed (CD) gradient is complicated by the fact that CD energies are computed with finite precision (determined by the CD threshold), which might lead to the errors that are not constant along the PES. Thus, the standard valida- tion against finite-difference (FD) calculations is of a limited value. To validate the implementation of the analytic CD-CCSD gradient we consider several quantities. First, we compare the analytic derivative of the CD vectors,X P , computed analyti- cally: (X P ) an = @B P @ (10.11) against the derivatives ofB P computed using the two-point FD procedure: (X P ) fd = B P ( + )B P ( ) 2x (10.12) where denote displaced geometries: = 0 x (10.13) andx is the step size for the FD procedure (x=10 5 ˚ A). In these calculations, the CD of the integrals at the FD points is performed in the same decomposition sequence as at 0 . Second, we compare the re-constituted gradient of the two-electron repulsion integral (ERI), (j) CD : (j) CD = P X i X i B i +B i X i (10.14) 277 against the canonical (CAN) analytic ERI gradient, (j) CAN . Third, we compare analytic CD-CCSD energy gradient, F CD , against the canonical analytic CCSD force, F CAN . Forth, we compare analytic F CD against the CD force computed by FD of the CD-CCSD energies. Finally, we compare the optimized bondlengths of several molecules computed with the analytic CD-CCSD gradient against the reference geome- tries computed with the canonical analytic CCSD gradient. To quantify the differences, we compute the maximum deviation (MD), norm of error (NE), and the average absolute difference (AAD) between the two vectors. NE and AAD are: NE = s X i (V i V i ref ) 2 ; (10.15) AAD = 1 3N X i jV i V i ref j; (10.16) and MD is the value of the largest element of the difference vector,V i V i ref . Hereref stands for the reference value and the sums run of the 3N Cartesian degrees of freedom. The benchmark set consists of H 2 , N 2 , C 2 H 2 , C 2 H 6 , CH 3 OH, and HCOCl. We used the following basis sets: 6-31G, 6-311G(d,p), cc-pVDZ, aug-cc-pVDZ, cc-pVTZ. The following parameters were used in the calculations: The HF and CCSD energy convergence thresholds: 10 11 a.u.; Step size (x) for FD procedure: x=10 5 ˚ A 1:89 10 5 a.u.; Convergence threshold for the maximum gradient component: 3 10 6 a.u.; Convergence threshold for the maximum atomic displacement: 1:2 10 5 a.u. 6 10 6 ˚ A; 278 Convergence threshold for the energy change in successive optimization cycles: 1 10 8 a.u. The errors introduced by the FD procedure executed with the above parameters can be evaluated by computing the difference between canonical analytical and canonical FD gradients. With the above parameters, the norm of the error is 10 8 . The benchmark results for the analytic CD integral gradient against the FD CD in- tegral gradient are given in Table 10.1. At loose decomposition thresholds, the analytic and FD CD integral gradients match each other within the accuracy of the FD procedure ( 10 8 ). However, NE becomes larger at very tight thresholds. There are two reasons for this behavior. First, at tight thresholds, we have many more Cholesky vectors and the errors are accumulated through the decomposition algorithm. Second, when very tight threshold (10 14 ) is used, we start to encounter very small diagonal matrix elements and the resulting CD integral gradient vectors have large errors caused by the division by this small diagonal element. This type of error does not show up when computing the CD vectors, because in this case the square root of the maximum diagonal element is used as the denominator. 279 Table 10.1: Norm of error (NE) between the analytic CD integral gradient, (X P ) an , and the FD CD integral gradient, (X P ) fd , computed with different CD threshold. Basis set: cc-pVDZ. molecule 10 2 10 3 10 4 10 5 10 8 10 11 10 14 H 2 4:6 10 10 9:9 10 10 2:4 10 9 8:9 10 10 4:1 10 8 4:1 10 8 4:1 10 8 N 2 2:7 10 9 5:1 10 9 2:4 10 8 1:4 10 7 2:6 10 6 4:8 10 5 5:9 10 4 C 2 H 2 1:2 10 8 2:7 10 8 5:4 10 8 1:3 10 7 4:1 10 6 5:1 10 5 2:0 10 3 CH 3 OH 3:4 10 8 5:5 10 8 1:2 10 7 3:0 10 7 6:1 10 6 8:0 10 5 7:5 10 4 280 Table 10.2: Norm of error (NE) between re-constituted gradient of ERI, (j) CD , and analytic ERI derivatives, (j) CAN . Basis set: cc-pVDZ. molecule 10 2 10 3 10 4 10 5 10 8 10 11 10 14 H 2 4:4 10 2 5:1 10 3 9:8 10 4 4:0 10 5 4:0 10 15 4:0 10 15 4:0 10 15 N 2 5:2 10 1 3:0 10 2 2:9 10 3 5:0 10 4 2:9 10 7 1:3 10 10 7:0 10 14 C 2 H 2 3:8 10 1 3:8 10 2 3:8 10 3 4:9 10 4 4:5 10 7 6:3 10 10 3:9 10 13 CH 3 OH 4:4 10 1 4:5 10 2 5:3 10 3 6:6 10 4 8:3 10 7 5:9 10 10 3:1 10 13 281 The results for the re-constituted gradient of ERI are shown in Table 10.2. We see that the error is about one order of magnitude larger than the decomposition threshold. The latter provides the bound for the maximum difference between the re-constituted and original ERIs. The difference between the re-constituted ERI gradient and the ERI gradient is not bounded by . Because of accumulation of the errors, the error in the ERI gradients is expected to be larger than the error in ERIs. Compared to Table 10.1, Table 10.2 shows a good match at tight thresholds. This is because all X vectors are contracted withB vectors to form the ERI gradient (see Eq. (10.14)) and the errors in individualX vector are summed up and cancel out. Table 10.3 and 10.4 show differences between re-constituted ERI gradients from analytic CD integral gradient and those from finite difference of ERI re-constituted by CD integrals at FD points. In Table 10.3, the CD integrals at FD points are decomposed in same sequence as it at analytic point = 0. We can see from this table that, all differences are very small, which means good agreement between CD analytic gradient and CD FD gradient. 282 Table 10.3: Maximum deviation (MD) between re-constituted gradient of ERI, (j) ;anal CD , and re-constituted FD ERI gradient with fixed decomposition sequence for FD points, (j) ;FD CD . Basis set: cc-pVDZ. Symmetry is disabled. molecule 10 2 10 3 10 4 10 5 10 8 10 11 10 14 H 2 8:0 10 11 8:0 10 11 7:8 10 11 7:2 10 11 7:2 10 11 7:2 10 11 7:2 10 11 N 2 1:4 10 10 1:4 10 10 1:3 10 10 1:9 10 10 1:5 10 10 1:5 10 10 1:5 10 10 C 2 H 2 4:9 10 10 4:9 10 10 4:9 10 10 4:9 10 10 4:9 10 10 4:9 10 10 4:9 10 10 CH 3 OH 7:1 10 10 7:1 10 10 7:1 10 10 7:1 10 10 7:1 10 10 7:1 10 10 7:1 10 10 283 Table 10.4: Maximum deviation (MD) between re-constituted gradient of ERI, (j) ;anal CD , and re-constituted FD ERI gradient with NOT fixed decomposition sequence for FD points, (j) ;FD CD . Basis set: cc-pVDZ. Symmetry is disabled. molecule 10 2 10 3 10 4 10 5 10 8 10 11 10 14 H 2 1:2 10 2 2:8 10 4 7:9 10 11 2:9 10 1 7:1 10 11 7:1 10 11 7:1 10 11 N 2 4:8 10 2 3:4 10 1 4:7 10 0 2:1 10 1 4:9 10 4 3:4 10 7 1:5 10 10 C 2 H 2 4:2 10 2 4:3 10 1 4:5 10 0 3:6 10 1 4:3 10 4 3:1 10 7 5:0 10 10 CH 3 OH 7:1 10 10 7:1 10 10 5:0 10 0 7:1 10 10 3:9 10 4 7:1 10 10 7:1 10 10 284 In Table 10.4, all CD integrals are decomposed following the standard algorithm does (i.e., the decomposition sequence at different points may be different). Since the decomposition sequence is not fixed, at different FD points, the maximum difference between two re-constituted ERIs from two CD algorithms (with and without fixing de- composition sequence) is threshold. Therefore, the maximum difference between the two re-constituted ERI integrals: (j) CD (fixed) (j) CD (unfixed) = (j) CD (fixed)(+) (j) CD (fixed)() 2 (j) CD (unfixed)(+) (j) CD (unfixed)() 2 = f CD (+) f CD () 2 where f CD = (j) CD (fixed) (j) CD (unfixed) is the difference caused by different decomposition sequence and is smaller than, so we have (j) CD (fixed) (j) CD (unfixed) 2 In cases when decomposition sequence is different at loose threshold, e.g., H 2 at = 10 5 , we can see that the maximum deviation is of the order of 10 1 10 5 =10 5 ( = 10 5 ˚ A). If the decomposition sequence does not change at FD points, e.g., CH 3 OH at = 10 2 or 10 3 , the maximum deviation is small and almost identical to the corre- sponding value in Table 10.3. 285 Table 10.5: Norm of error (NE) between analytic CD-CCSD force and canonical analytic CCSD force at different CD threshold,. Basis set: cc-pVDZ molecule 10 2 10 3 10 4 10 5 10 8 10 11 10 14 H 2 8:6 10 3 1:5 10 4 2:0 10 4 1:3 10 4 2:8 10 14 2:8 10 14 2:8 10 14 N 2 1:2 10 0 4:5 10 2 6:7 10 3 4:5 10 3 4:8 10 6 1:8 10 8 1:1 10 11 C 2 H 2 5:4 10 2 3:8 10 2 7:1 10 3 8:5 10 4 1:6 10 6 1:3 10 9 1:1 10 10 CH 3 OH 5:6 10 1 2:4 10 2 1:4 10 2 5:6 10 3 1:3 10 6 5:3 10 9 1:2 10 9 HCOCl 1:4 10 0 5:7 10 2 5:5 10 2 5:1 10 3 1:2 10 5 9:4 10 9 1:2 10 7 286 Let us now compare analytic CD versus CAN energy gradient. Fig. 10.1 shows the computed AAD and Table 10.5 shows NEs for our test set as a function of the CD threshold. As one can see, CD-CCSD energy gradient converges to the CAN value as the threshold gets tighter. For = 10 3 and = 10 4 , the AAD is around 10 5 a.u.. With = 10 5 , the AAD drops below 10 6 a.u.. Figure 10.1: AAD between the CD analytic energy gradient and the canonical (CAN) analytic energy gradient. 10.3.1 Errors in optimized structures Table 10.6 compares analytic CD-CCSD and canonical CCSD calculations of the opti- mized structures. We report MDs for optimized bondlengths and for forces computed at the initial structures. The error of optimized bondlengths and in the energy gradient are consistent with each other. When = 10 2 , the errors in optimized bondlengths of most molecules are around 10 3 a.u.. As the decomposition threshold becomes tighter, = 10 3 , 10 4 , 10 5 , the error decrease accordingly, 10 4 a.u., 10 5 a.u., 10 6 a.u.. These results are similar to those reported by Lindh and co-workers. 5 287 Table 10.6: The results of CD-CCSD/cc-pVDZ optimization using analytic gradient at ifferent CD thresholds,. The errors are computed against canonical CD-CCSD values and reported as maximum deviation (MD). MD for bondlengths and forces are denoted byjrj (pm) and byjFj (10 2 a.u.), respectively. jrj, pm MD, 10 2 a.u. Molecule 10 2 10 3 10 4 10 5 10 2 10 3 10 4 10 5 H 2 0.0095 0.0002 0.0003 0.0001 0.0061 0.0001 0.0001 0.0001 N 2 0.2080 0.0121 0.0005 0.0003 0.8601 0.0318 0.0048 0.0032 C 2 H 2 0.0170 0.0112 0.0031 0.0008 0.0399 0.0298 0.0050 0.0006 C 2 H 6 0.4980 0.0083 0.0013 0. 0.1822 0.0143 0.0009 0.0005 CH 3 OH 0.3176 0.0117 0.0103 0.0036 0.2919 0.0134 0.0079 0.0031 HCOCl 1.5495 0.0434 0.0138 0.0010 0.9654 0.0455 0.0391 0.0036 1 pm = 10 2 ˚ A. 10.4 Conclusions The differences between analytic and FD gradient are consistent with the variations in the errors in energies introduced by the CD decomposition, which determines the error in the FD gradient. The uniform convergence towards CAN value with tight thresholds and the agreement between the analytic and FD gradient with tight CD threshold provide support in favor of the correctness of the implementation. 288 Chapter 10 references [1] F. Aquilante, T. B. Pedersen, and R. Lindh. Density fitting with auxiliary basis sets from Cholesky decompositions. Theor. Chem. Acc., 124:1–10, 2009. [2] F. Aquilante, L. de Vico, N. Ferr´ e, G. Ghigo, P- ˚ A Malmqvist, P. Neogr´ ady, T. B. Pedersen, M. Pitonˇ ak, M. Reiher, B. Roos, L. Serrano-Andr´ es, M. Urban, V . Verya- zov, and R. Lindh. Molcas 7: The next generation. J. Comput. Chem., 31:224–247, 2010. [3] H. Koch, A.S. de Mer´ as, and T.B. Pedersen. Reduced scaling in electronic structure calculations using Cholesky decompositions. J. Chem. Phys., 118:9481, 2003. [4] E. Epifanovsky, D. Zuev, X. Feng, K. Khistyaev, Y . Shao, and A.I. Krylov. General implementation of resolition-of-identity and Cholesky representations of electron- repulsion integrals within coupled-cluster and equation-of-motion methods: Theory and benchmarks. J. Chem. Phys., 139:134105, 2013. [5] F. Aquilante, R. Lindh, and T.B. Pedersen. Analytic derivatives for the Cholesky representation of the two-electron integrals. J. Chem. Phys., 129:034106, 2008. 289 Chapter 11: Future work Chapter 7 reported that in covalently linked tetracene dimer where ortho-diethynyl- benzene is used as the linker, a faster rate of singlet fission (SF) can be obtained, relative to the crystalline tetracene. We were able to show that this type of linker increases the non-adiabatic coupling (NAC) between the lowest singlet excitonic excited state (S 1 ), and the singlet multiexcitonic state ( 1 ME). Since ortho-BET-B shows fast SF, one might expect that the BET-B with para-linker has similar behavior, whereas another BET-B isomer with meta-linker could have smaller NAC and, consequently, a slower SF rate. By using same techniques as in Chapter 7 we can estimate the NAC between S 1 and 1 ME in ortho-, para- and meta-BET-B by computingjj jj. What we find is that the NAC of ortho- and para-BET-B are of same order of magnitude, while in meta-BET- B the NAC is more than two orders of magnitude smaller. This is consistent with the resonance structures of para- and ortho-benzene have more resonance structures than meta-benzene, predicts faster SF in para-BET-B than meta-BET-B. The methodology employed in this thesis is based on the restricted-active-space spin-flip configuration interaction (RAS-SF-CI), in which the dynamic correlation is poorly described. To improve the excitation energy, one should use higher level of theory, such as the equation-of-motion double spin-flip coupled-cluster (EOM-DSF- CCSD) method or EOM-DSF-CCSD with perturbed triples (EOM-DSF-CCSD(T)). Be- cause of high computational cost, several techniques should be employed in order to 290 make the methods applicable to large molecular systems, such as those relevant to SF materials with the basis set we employed (cc-pVTZ-f for C atoms and cc-pVDZ for H atoms). For example, we can use Cholesky decomposition (CD) or resolution-of- identity (RI) to reduce the storage scaling of two-electron repulsion integrals (ERI) from O(N 4 ) toO(N 3 ). Another important development is the implementation of NAC. Within the EOM framework, the NAC equals to Tr[ H R ], which is simply the trace of product of two matrices: one particle transition density matrix (OPDM) and derivative of Hamiltonian with respect to nuclear coordinate. The amplitude and orbital response can be included in the relaxed OPDM, as done in the calculation of other one-electron properties, such as transition dipole moment. Chapter 9 and Chapter 10 show the implementation of the CD technique within CC and EOM-CC framework reducing the storage requirements. However, the overall stor- age scaling is stillN 4 , since the CCSD T 2 amplitudes and EOM-CCSD R 2 amplitudes scale asN 4 . One simple way to achieve further reduction is to use frozen natural orbital (FNO) technique to reduce the size of virtual molecular orbital space. 1, 2 The MP2-based FNO truncation scheme has been implemented for the CC and EOM-CC ionization po- tentials. Utilizing 99–99.5% of virtual space population leads to the 30–70% truncation of virtual space, introducing errors less than 1 kcal/mol. However FNO gradient has not yet been implemented. It is needed for geometry optimizations. Another way to improve efficiency is to exploit the sparsity ofN 4 -type tensors. For example, when the distance between two atomic orbitals is larger than a given thresh- old, the corresponding ERI element can be set directly to zero. If elements of one block of the ERI matrix are set to zero, it needs not be stored thus reducing the disk require- ments. The number of numerically significant integrals in the ERI matrix can be reduced 291 asymptotically to O(N 2 ). 3 There are existing algorithms for the AO-MP2 method, 4, 5 which allow for evaluation of AO-MP2 energy with computational cost that scales lin- early. Esmond Ng and co-workers have presented a new algorithm for sparsifying the amplitude correction for the inexact Newton iteration of CCD amplitude. 6 In this work, about 90% non-zero elements are ignored without a major effect on the convergence, so we might expect the sparsity in T 2 amplitudes can also be efficiently utilized. Once the new screening algorithm for T 2 and R 2 amplitude is established and implemented, we should gain more computational savings, thus improving the performance of CC and EOM-CC methods. 292 Chapter 11 references [1] A. Landau, K. Khistyaev, S. Dolgikh, and A.I. Krylov. Frozen natural orbitals for ionized states within equation-of-motion coupled-cluster formalism. J. Chem. Phys., 132:014109, 2010. [2] A. G. Taube and R. J. Bartlett. Frozen natural orbital coupled-cluster theory: Forces and application to decomposition of nitroethane. J. Chem. Phys., 128:164101, 2008. [3] M. H¨ aser and R. Ahlrichs. Improvements on the direct SCF method. J. Comput. Chem., 10, 1988. [4] S. A. Maurer, D. S. Lambrecht, D. Flaig, and C. Ochsenfeld. Distance-dependent schwarz-based integral estimates for two-electron integrals: Reliable tightness vs. rigorous upper bounds. J. Chem. Phys., 136, 2012. [5] S. A. Maurer, D. S. Lambrecht, J. Kussmann, and C. Ochsenfeld. Efficient distance- including integral screening in linear-scaling moller-plesset perturbation theory. J. Chem. Phys., 138, 2013. [6] J. Brabec, C. Yang, E. Epifanovsky, A.I. Krylov, and E. Ng. Reduced-cost sparcity- exploiting algorithm for solving coupled-cluster equations. J. Comput. Chem., 37:1059–1067, 2016. 293 Bibliography W. Shockley and H.J. Queisser. Detailed balance limit of efficiency of p-n junction solar cells. J. Appl. Phys., 32:510–519, 1961. S. Singh, W.J. Jones, W. Siebrand, B.P. Stoicheff, and W.G. Schneider. Laser genera- tion of excitons and fluorescence in anthracene crystals. J. Chem. Phys., 42:330–342, 1965. I. Paci, J.C. Johnson, X. Chen, G. Rana, D. Popovi´ c, D.E. David, A.J. Nozik, M.A. Ratner, and J. Michl. Singlet fission for dye-sensitized solar cells: Can a suitable sensitizer be found? J. Am. Chem. Soc., 128:16546–16553, 2006. E. C.Greyson, B.R. Stepp, X. Chen, A.F. Schwerin, I. Paci, M. B. Smith, A. Akdag, J.C. Johnson, A.J. Nozik, J. Michl, and M.A. Ratner. Singlet exciton fission for solar cell applications: Energy aspects of interchromophore coupling. J. Phys. Chem. B, 2010. in press, asap article. X. Feng, A.V . Luzanov, and A.I. Krylov. Fission of entangled spins: An electronic structure perspective. J. Phys. Chem. Lett., 4:3845–3852, 2013. M. C. Hanna and A. J. Nozik. Solar conversion efficiency of photovoltaic and pho- toelectrolysis cells with carrier multiplication absorbers. J. App. Phys., 100:074510, 2006. M. B. Smith and J. Michl. Singlet fission. Chem. Rev., 110:6891–6936, 2010. S. Singh, W. J. Jones, W. Siebrand, B. P. Stoicheff, and W. G. Schneider. Laser gen- eration of excitons and fluorescence in anthracene crystals. J. Chem. Phys., 42:330, 1965. R.P. Groff, P. Avakian, and R.E. Merrifield. Coexistence of exciton fission and fusion in tetracene crystals. Phys. Rev. B, 1:815–817, 1970. G.B. Piland, J.J. Burdett, D. Kurunthu, and C.J. Bardeen. Magnetic field effects 294 on singlet fission and fluorescence decay dynamics in amorphous rubrene. J. Phys. Chem. C, 117:1224–1236, 2013. W. G. Herkstroeter and P. B. Merkel. The triplet state energies of rubrene and diphenylisobenzofuran. J. Photochem., 16:331–341, 1981. T. Helgaker, P. Jørgensen, and J. Olsen. Molecular electronic structure theory. Wiley & Sons, 2000. A. Szabo and N.S. Ostlund. Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory. McGraw-Hill, New York, 1989. C. Møller and M.S. Plesset. Phys. Rev., 46:618, 1934. J. Cizek. On the correlation problem in atomic and molecular systems. Calculation of wavefunction components in Ursell-type expansion using quantum-field theoretical methods. J. Chem. Phys., 45:4256–4266, 1966. B.O. Roos. The multiconfigurational (MC) SCF method. In Geerd H. F. Diercksen and Stephen Wilson, editors, Methods in Computational Molecular Physics, pages 161–187. D. Reidel, Dordrecht, 1983. B.O. Roos. The multiconfigurational (MC) self-consistent field (SCF) theory. In B. O. Roos, editor, Lecture Notes in Quantum Chemistry: European Summer School in Quantum Chemistry, volume 58 of Lecture Notes in Chemistry, pages 177–254. Springer-Verlag, New York, 1992. A. Golubeva, A.V . Nemukhin, L. Harding, S.J. Klippenstein, and A.I. Krylov. Per- formance of the spin-flip and multi-reference methods for bond-breaking in hydro- carbons: A benchmark study. J. Phys. Chem. A, 111:13264–13271, 2007. P.U. Manohar and A.I. Krylov. A non-iterative perturbative triples correction for the spin-flipping and spin-conserving equation-of-motion coupled-cluster methods with single and double substitutions. J. Chem. Phys., 129:194105, 2008. D. Casanova, L.V . Slipchenko, A.I. Krylov, and M. Head-Gordon. Double spin-flip approach within equation-of-motion coupled cluster and configuration interaction formalisms: Theory, implementation and examples. J. Chem. Phys., 130:044103, 2009. G.D. Purvis and R.J. Bartlett. A full coupled-cluster singles and doubles model: The inclusion of disconnected triples. J. Chem. Phys., 76:1910–1918, 1982. R.J. Bartlett. Coupled-cluster theory and its equation-of-motion extensions. Wiley Interdiscip. Rev.: Comput. Mol. Sci., 2(1):126–138, 2011. 295 E.R. Davidson. The iterative calculation of a few of the lowest eigenvalues and corre- sponding eigenvectors of large real-symmetric matrices. J. Comput. Phys., 17:87–94, 1975. T.F. Schulze, J. Czolk, Y .-Y . Cheng, B. F¨ uckel, R.W. MacQueen, T. Khoury, M.J. Crossley, B. Stannowski, K. Lips, U. Lemmer, A. Colsmann, and T.W. Schmidt. Ef- ficiency enhancement of organic and thin-film silicon solar cells with photochemical upconversion. J. Phys. Chem. C, 116:22794–22801, 2012. M.B. Smith and J. Michl. Recent advances in singlet fission. Annu. Rev. Phys. Chem., 64:361–368, 2013. S.T. Roberts, R.E. McAnally, J.N. Mastron, D.H. Webber, M.T. Whited, R.L. Brutchey, M.E. Thompson, and S.E. Bradforth. Efficient singlet fission found in a disordered acene film. J. Am. Chem. Soc., 134:6388–6400, 2012. C. Ramanan, A.L. Smeigh, J.E. Anthony, T.J. Marks, and M.R. Wasielewski. Com- petition between singlet fission and charge separation in solution-processed blend films of 6,13-bis(triisopropylsilylethynyl)- pentacene with sterically-encumbered perylene-3,4:9,10- bis(dicarboximide)s. J. Am. Chem. Soc., 134:386–397, 2012. W-L. Chan, M. Ligges, and X-Y . Zhu. The energy barrier in singlet fission can be overcome through coherent coupling and entropic gain. Nat. Chem., 4:840–845, 2012. J.C Johnson, A.J. Nozik, and J. Michl. The role of chromophore coupling in singlet fission. Acc. Chem. Res., 46:1290–1299, 2013. T.C. Berkelbach, M.S. Hybertsen, and D.R. Reichman. Microscopic theory of singlet exciton fission. I. General formulation. J. Chem. Phys., 138:114102, 2012. P.M. Zimmerman, C.B. Musgrave, and M. Head-Gordon. A correlated electron view of singlet fission. Acc. Chem. Res., 46:1339–1347, 2012. P.M. Zimmerman, Z. Zhang, and C.B. Musgrave. Singlet fission in pentacene through multi-exciton quantum states. Nature Chem., 2:648–652, 2010. P.M. Zimmerman, F. Bell, D. Casanova, and M. Head-Gordon. Mechanism for sin- glet fission in pentacene and tetracene: From single exciton to two triplets. J. Am. Chem. Soc., 133:19944–19952, 2011. T.S. Kuhlman, J. Kongsted, K.V . Mikkelsen, K.B. Møller, and T.I. Sølling. Interpre- tation of the ultrafast photoinduced processes in pentacene thin films. J. Am. Chem. Soc., 132:3431–3439, 201o. 296 S. Sharifzadeh, P. Darancet, L. Kronik, and J.B. Neaton. Low-energy charge-transfer excitons in organic solids from first-principles: The case of pentacene. J. Phys. Chem. Lett., 4:2917–2201, 2013. R.W.A. Havenith, H.D. de Gier, and R. Broer. Explorative computational study of the singlet fission process. Mol. Phys., 110:2445–2454, 2012. D.N. Congreve, J. Lee, N.J. Thompson, E. Hontz, S.R. Yost, P.D. Reusswig, M.E. Bahlke, S. Reineke, T. Van V oorhis, and M.A. Baldo. External quantum efficiency above 100% in a singlet-exciton-fissionbased organic photovoltaic cell. Science, 340:334–337, 2013. P.J. Vallett, J.L. Snyder, and N.H. Damrauer. Tunable electronic coupling and driv- ing force in structurally well defined tetracene dimers for molecular singlet fission: A computational exploration using density functional theory. J. Phys. Chem. A, 117:10824–10838, 2013. F. Bell, P.M. Zimmerman, D. Casanova, M. Goldey, and M. Head-Gordon. Restricted active space spin-flip (RAS-SF) with arbitrary number of spin-flips. Phys. Chem. Chem. Phys., 15:358–366, 2013. J.N. Murrell and J. Tanaka. Theory of electronic spectra of aromatic hydrocarbon dimers. Mol. Phys., 7:363–380, 1964. R.S. Mulliken and W.B. Person. Molecular Complexes. Wiley-Interscience, 1969. A.L.L. East and E.C. Lim. Naphthalene dimer: Electronic states, excimers, and triplet decay. J. Chem. Phys., 113:8981–8994, 2000. P.A. Pieniazek, A.I. Krylov, and S.E. Bradforth. Electronic structure of the benzene dimer cation. J. Chem. Phys., 127:044317, 2007. A.V . Luzanov and O.V . Prezhdo. High-order entropy measures and spin-free quan- tum entanglement for molecular problems. Mol. Phys., 105:2879–2891, 2007. K. Diri and A.I. Krylov. Electronic states of the benzene dimer: A simple case of complexity. J. Phys. Chem. A, 116:653–662, 2011. A.I. Krylov. Size-consistent wave functions for bond-breaking: The equation-of- motion spin-flip model. Chem. Phys. Lett., 338:375–384, 2001. A.I. Krylov. The spin-flip equation-of-motion coupled-cluster electronic structure method for a description of excited states, bond-breaking, diradicals, and triradicals. Acc. Chem. Res., 39:83–91, 2006. A.V . Luzanov. One-particle approximation in valence scheme superposition. Theor. 297 Exp Chem., 17:227–233, 1982. K. Takatsuka, T. Fueno, and K. Yamaguchi. Distribution of odd electrons in ground- state molecules. Theor. Chim. Acta, 48:175–183, 1978. M. Head-Gordon, A. M. Grana, D. Maurice, and C. A. White. Analysis of electronic transitions as the difference of electron attachment and detachment densities. J. Phys. Chem., 99:14261 – 14270, 1995. J.E. Subotnik, S. Yeganeh, R.J. Cave, and M.A. Ratner. Constructing diabatic states from adiabatic states: Extending generalized Mulliken-Hush to multiple charge cen- ters with Boys localization. J. Chem. Phys., 129:244101, 2008. R.J. Cave, S.T. Edwards, and J.A. Kouzelos. Reduced electronic spaces for modeling donor/acceptor interactions. J. Phys. Chem. B, 114:14631–14641, 2010. A.V . Luzanov, A.A. Sukhorukov, and V .E. Umanskii. Application of transition den- sity matrix for analysis of excited states. Theor. Exp. Chem., 10:354–361, 1976. A.V . Luzanov and V .F. Pedash. Interpretation of excited states using charge-transfer number. Theor. Exp. Chem., 15:338–341, 1979. A.V . Luzanov and O.A. Zhikol. Excited state structural analysis: TDDFT and related models. In J. Leszczynski and M.K. Shukla, editors, Practical aspects of compu- tational chemistry I: An overview of the last two decades and current trends, pages 415–449. Springer, 2012. A.V . Luzanov. Covariant models for electronic structure. In M.M. Mestechkin, edi- tor, Many-body problem in quantum chemistry, pages 53–64. Naukova Dumka, Kiyv, 1987. R. McWeeny. Methods of Molecular Quantum Mechanics. Academic Press, 2nd edition, 1992. E.R. Davidson. Reduced Density Matrices in Quantum Chemistry. Academic Press, New York, 1976. M.M. Mestechkin. Metod matritsy plotnosti v teorii molekul. Kyev, Naukova Dumka, 1977. R.F.W. Bader. Atoms in molecules — Quantum theory. Oxford University Press, Oxford, 1990. F. Weinhold and C.R. Landis. Discovering Chemistry With Natural Bond Orbitals. New Jersey: John Wiley & Sons, 2012. 298 E.D. Glendening, J.K. Badenhoop, and F. Weinhold. Natural resonance theory: II. Natural bond order and valency. J. Comput. Chem., 19(6):610–627, 1998. M. Head-Gordon. Characterizing unpaired electrons from the one-particle density matrix. Chem. Phys. Lett., 372:508–511, 2003. A.V . Luzanov and O.V . Prezhdo. Analysis of multiconfigurational wave functions in terms of hole-particle distributions. J. Chem. Phys., 124:224109, 2006. A.V . Luzanov and O.A. Zhikol. Electron invariants and excited state structural anal- ysis for electronic transitions within CIS, RPA, and TDDFT models. Int. J. Quant. Chem., 110:902–924, 2010. F. Plasser and H. Lischka. Analysis of excitonic and charge transfer interactions from quantum chemical calculations. J. Chem. Theory Comput., 8:2777–2789, 2012. K.B. Bravaya, M.G. Khrenova, B.L. Grigorenko, A.V . Nemukhin, and A.I. Krylov. Effect of protein environment on electronically excited and ionized states of the green fluorescent protein chromophore. J. Phys. Chem. B, 115:8296–8303, 2011. A.V . Luzanov. The structure of the electronic excitation of molecules in quantum- chemical models. Russ. Chem. Rev., 49:1033–1048, 1980. S. Tretiak and S. Mukamel. Density matrix analysis and simulation of electronic ex- citations in conjugated and aggregated molecules. Chem. Rev., 102:3171, 2002. A.B. Kolomeisky, X. Feng, and A.I. Krylov. A simple kinetic model for singlet fis- sion: A role of electronic and entropic contributions to macroscopic rates. J. Phys. Chem. C, 118:5188–5195, 2014. X. Feng, A.B. Kolomeisky, and A.I. Krylov. Dissecting the effect of morphology on the rates of singlet fission: Insights from theory. J. Phys. Chem. C, 118:19608– 19617, 2014. T.-C. Jagau, D. Zuev, K.B. Bravaya, E. Epifanovsky, and A.I. Krylov. A fresh look at resonances and complex absorbing potentials: Density matrix based approach. J. Phys. Chem. Lett., 5:310–315, 2014. D.R. Yarkony. Diabolical conical intersections. Rev. Mod. Phys., 68:985–1013, 1996. D.R. Yarkony. Conical intersections: Diabolical and often misunderstood. Acc. Chem. Res., 31:511–518, 1998. D.R. Yarkony. Conical intersections: The new conventional wisdom. J. Phys. Chem. A, 105:6277–6293, 2001. 299 B.H. Lengsfield III, P. Saxe, and D.R. Yarkony. On the evaluation of nonadiabatic coupling matrix-elements using SA-MCSCF/CI wave functions and analytic gradient methods.1. J. Chem. Phys., 81:4549–4553, 1984. P. Saxe, B.H. Lengsfield III, and D.R. Yarkony. On the evaluation of non-adiabatic coupling matrix-elements for large-scale CI wavefunctions. Chem. Phys. Lett., 113:159–164, 1985. H. Lischka, M. Dallos, P.G. Szalay, D.R. Yarkony, and R. Shepard. Analytic eval- uation of nonadiabatic coupling terms at the MR-CI level. I. Formalism. J. Chem. Phys., 120:7322–7329, 2004. E. Tapavicza, G.D. Bellchambers, J.C. Vincent, and F. Furche. Ab initio non- adiabatic molecular dynamics. Phys. Chem. Chem. Phys., 15:18336, 2013. S. Fatehi, E. Alguire, Y . Shao, and J.E. Subotnik. Analytic derivative couplings be- tween configuration-interaction-singles states with built-in electron-translation fac- tors for translational invariance. J. Chem. Phys., 135:234105, 2011. X. Zhang and J.M. Herbert. Analytic derivative couplings for spin-flip configuration interaction singles and spin-flip time-dependent density functional theory. J. Chem. Phys., 141:064104, 2014. W.D. Domcke, D.R. Yarkony, and H. K¨ oppel, editors. Conical intersections. Elec- tronic structure, dynamics and spectroscopy. World Scientific Pbul Co Pte Ltd, 2004. T. Helgaker, S. Coriani, P. Jørgensen, K. Kristensen, J. Olsen, and K. Ruud. Recent advances in wave function-based methods of molecular-property calculations. Chem. Rev., 112:543–631, 2012. B.H. Lengsfield III and D.R. Yarkony. Nonadiabatic interactions between potential energy surfaces: Theory and applications. Adv. Chem. Phys., 82:1–72, 1992. M. Dallos, H. Lischka, R. Shepard, D.R. Yarkony, and P.G. Szalay. Analytic eval- uation of nonadiabatic coupling terms at the MR-CI level. II. Minima on the cross- ing seam: Formaldehyde and the photodimerization of ethylene. J. Chem. Phys., 120:7330–7339, 2004. T. Ichino, J. Gauss, and J.F. Stanton. Quasidiabatic states described by coupled- cluster theory. J. Chem. Phys., 130(17):174105, 2009. S. Matsika, X. Feng, A.V . Luzanov, and A.I. Krylov. What we can learn from the norms of one-particle density matrices, and what we can’t: Some results for interstate properties in model singlet fission systems. J. Phys. Chem. A, 118:11943–11955, 300 2014. D. Zuev, T.-C. Jagau, K.B. Bravaya, E. Epifanovsky, Y . Shao, E. Sundstrom, M. Head-Gordon, and A.I. Krylov. Complex absorbing potentials within EOM- CC family of methods: Theory, implementation, and benchmarks. J. Chem. Phys., 141(2):024102, 2014. P.A. Pieniazek, S.A. Arnstein, S.E. Bradforth, A.I. Krylov, and C.D. Sherrill. Benchmark full configuration interaction and EOM-IP-CCSD results for prototypical charge transfer systems: Noncovalent ionized dimers. J. Chem. Phys., 127:164110, 2007. T. Amos and M. Woodward. Configuration interaction wavefunctions for small systems. J. Chem. Phys., 50:119–123, 1969. B.O. Roos, P.R. Taylor, and P.E.M. Siegbahn. A complete active space SCF method (CASSCF) using a density matrix formulated super-CI approach. Chem. Phys., 48:157–173, 1980. D. Casanova and M. Head-Gordon. Restricted active space spin-flip configuration interaction approach: Theory, implementation and examples. Phys. Chem. Chem. Phys., 11:9779–9790, 2009. K. Ohno. Some remarks on the Pariser-Parr-Pople method. Theor. Chim. Acta, 2:219– 227, 1964. S. Ramasecha and Z.G. Soos. Optical excitations of even and odd polyenes with molecular PPP correlations. Synth. Metals, 1984. A.V . Luzanov, A.L. Wulfov, and V .O. Krouglov. A wavefunction operator approach to the full-CI problem. Chem. Phys. Lett., 197:614–619, 1992. I. Shavitt R. M. Pitzer M. Dallos Th. M¨ uller P. G. Szalay F. B. Brown R. Ahlrichs H. J. B¨ ohm A. Chang D. C. Comeau R. Gdanitz H. Dachsel C. Ehrhardt M. Ernzer- hof P. H¨ ochtl S. Irle G. Kedziora T. Kovar V . Parasuk M. J. M. Pepper P. Scharf H. Schiffer M. Schindler M. Sch¨ uler M. Seth E. A. Stahlberg J.-G. Zhao S. Yabushita Z. Zhang M. Barbatti S. Matsika M. Schuurmann D. R. Yarkony S. R. Brozell E. V . Beck H. Lischka, R. Shepard, B. Sellner F. Plasser J.-P. Blaudeau, M. Rucken- bauer, and J. J. Szymczak. COLUMBUS, an ab initio electronic structure program, release 7.0, 2012. H. Lischka, R. Shepard, R.M. Pitzer, I. Shavitt, M. Dallos, Th. M¨ uller, P.G. Szalay, M. Seth, G.S. Kedziora, S. Yabushita, and Z. Zhang. High-level multireference meth- ods in the quantum-chemistry program system COLUMBUS: Analytic MR-CISD and MR-AQCC gradients and MR-AQCC-LRT for excited states, GUGA spin-orbit 301 CI and parallel CI density. Phys. Chem. Chem. Phys., 3:664–673, 2001. A.I. Krylov and P.M.W. Gill. Q-Chem: An engine for innovation. WIREs Comput. Mol. Sci., 3:317–326, 2013. Shao, Y .; Gan, Z.; Epifanovsky, E.; Gilbert, A.T.B.; Wormit, M.; Kussmann, J.; Lange, A.W.; Behn, A.; Deng, J.; Feng, X., et al. Advances in molecular quantum chemistry contained in the Q-Chem 4 program package. Mol. Phys., 113:184–215, 2015. A.F. Izmaylov, D. Mendive-Tapia, M.J. Bearpark, M.A. Robb, J.C. Tully, and M.J. Frisch. Nonequilibrium Fermi golden rule for electronic transitions through conical intersections. J. Chem. Phys., 135:234106, 2011. M.H. Lee, B.D. Dunietz, and E. Geva. Calculation from first principles of in- tramolecular golden-rule rate constants for photo-induced electron transfer in molec- ular donor-acceptor systems. J. Phys. Chem. C, 117:23391–23401, 2013. F. Plasser, M. Wormit, and A. Dreuw. New tools for the systematic analysis and visu- alization of electronic excitations. I. Formalism. J. Chem. Phys., 141:024106, 2014. A. Dreuw and M. Head-Gordon. Single-reference ab initio methods for the calcula- tion of excited states of large molecules. Chem. Rev., 105:4009 – 4037, 2005. H. Reisler and A.I. Krylov. Interacting Rydberg and valence states in radicals and molecules: Experimental and theoretical studies. Int. Rev. Phys. Chem., 28:267–308, 2009. A.I. Krylov. Equation-of-motion coupled-cluster methods for open-shell and elec- tronically excited species: The hitchhiker’s guide to Fock space. Annu. Rev. Phys. Chem., 59:433–462, 2008. I. Fischer-Hjalmars and J. Kowalewski. Simplified non-empirical excited state calcu- lations. Theor Chim Acta, 27:197, 1972. E.B. Guidez and C.M. Aikens. Plasmon resonance analysis with configuration inter- action. Phys. Chem. Chem. Phys., 16:15501–15509, 2014. E.B. Guidez and C.M. Aikens. Quantum mechanical origin of the plasmon: from molecular systems to nanoparticles. Nanoscale, 6:11512–11527, 2014. C.M. Krauter, J. Schirmer, C.R. Jacob, M. Pernpointner, and A. Dreuw. Plasmons in molecules: Microscopic characterization based on orbital transitions and momentum conservation. J. Chem. Phys., 141:104101, 2014. C.J. Bardeen. The structure and dynamics of molecular excitons. Annu. Rev. Phys. 302 Chem., 65:127–148, 2014. A.V . Luzanov. Analysis of the exciton states of polyconjugated systems by the tran- sition density matrix method. J. Struct. Chem., 43:711, 2002. S.A. B¨ appler, F. Plasser, M. Wormit, and A. Dreuw. Exciton analysis of many-body wave functions: Bridging the gap between the quasiparticle and molecular orbital pictures. Phys. Rev. A, 90:052521, 2014. J. Wahl, R. Binder, and I. Burghardt. Quantum dynamics of ultrafast exciton relax- ation on a minimal lattice. Comp. Theo. Chem., 1040-1041:167–176, 2014. L. Blancafort and A.A. V oityuk. Exciton delocalization, charge transfer, and elec- tronic coupling for singlet excitation energy transfer between stacked nucleobases in DNA: An MS-CASPT2 study. J. Chem. Phys., 140:095102, 2014. Y . Shao, Z. Gan, E. Epifanovsky, A.T.B. Gilbert, M. Wormit, J. Kussmann, A.W. Lange, A. Behn, J. Deng, X. Feng, D. Ghosh, M. Goldey, P.R. Horn, L.D. Jacobson, I. Kaliman, R.Z. Khaliullin, T. Kus, A. Landau, J. Liu, E.I. Proynov, Y .M. Rhee, R.M. Richard, M.A. Rohrdanz, R.P. Steele, E.J. Sundstrom, H.L. Woodcock III, P.M. Zim- merman, D. Zuev, B. Albrecht, E. Alguires, B. Austin, G.J.O. Beran, Y .A. Bernard, E. Berquist, K. Brandhorst, K.B. Bravaya, S.T. Brown, D. Casanova, C.-M. Chang, Y . Chen, S.H. Chien, K.D. Closser, D.L. Crittenden, M. Diedenhofen, R.A. DiSta- sio Jr., H. Do, A.D. Dutoi, R.G. Edgar, S. Fatehi, L. Fusti-Molnar, A. Ghysels, A. Golubeva-Zadorozhnaya, J. Gomes, M.W.D. Hanson-Heine, P.H.P. Harbach, A.W. Hauser, E.G. Hohenstein, Z.C. Holden, T.-C. Jagau, H. Ji, B. Kaduk, K. Khistyaev, J. Kim, J. Kim, R.A. King, P. Klunzinger, D. Kosenkov, T. Kowalczyk, C.M. Krauter, K.U. Laog, A. Laurent, K.V . Lawler, S.V . Levchenko, C.Y . Lin, F. Liu, E. Livshits, R.C. Lochan, A. Luenser, P. Manohar, S.F. Manzer, S.-P. Mao, N. Mardirossian, A.V . Marenich, S.A. Maurer, N.J. Mayhall, C.M. Oana, R. Olivares-Amaya, D.P. O’Neill, J.A. Parkhill, T.M. Perrine, R. Peverati, P.A. Pieniazek, A. Prociuk, D.R. Rehn, E. Rosta, N.J. Russ, N. Sergueev, S.M. Sharada, S. Sharmaa, D.W. Small, A. Sodt, T. Stein, D. Stuck, Y .-C. Su, A.J.W. Thom, T. Tsuchimochi, L. V ogt, O. Vydrov, T. Wang, M.A. Watson, J. Wenzel, A. White, C.F. Williams, V . Vanovschi, S. Yeganeh, S.R. Yost, Z.-Q. You, I.Y . Zhang, X. Zhang, Y . Zhou, B.R. Brooks, G.K.L. Chan, D.M. Chipman, C.J. Cramer, W.A. Goddard III, M.S. Gordon, W.J. Hehre, A. Klamt, H.F. Schaefer III, M.W. Schmidt, C.D. Sherrill, D.G. Truhlar, A. Warshel, X. Xu, A. Aspuru-Guzik, R. Baer, A.T. Bell, N.A. Besley, J.-D. Chai, A. Dreuw, B.D. Duni- etz, T.R. Furlani, S.R. Gwaltney, C.-P. Hsu, Y . Jung, J. Kong, D.S. Lambrecht, W.Z. Liang, C. Ochsenfeld, V .A. Rassolov, L.V . Slipchenko, J.E. Subotnik, T. Van V oorhis, J.M. Herbert, A.I. Krylov, P.M.W. Gill, and M. Head-Gordon. Advances in molec- ular quantum chemistry contained in the Q-Chem 4 program package. Mol. Phys., 113:184–215, 2015. 303 F. Plasser, S.A. B¨ appler, M. Wormit, and A. Dreuw. New tools for the systematic analysis and visualization of electronic excitations. I. Applications. J. Chem. Phys., 141:024107, 2014. A. Dreuw and M. Wormit. The algebraic diagrammatic construction scheme for the polarization propagator for the calculation of excited states. WIREs Comput. Mol. Sci., 5:82–95, 2015. D. Casanova. Electronic structure study of singlet-fission in tetracene derivatives. J. Chem. Theory Comput., 10:324–334, 2014. E.R. Davidson and A.E. Clark. Model molecular magnets. J. Phys. Chem. A, 106:7456–7461, 2002. R.D. Harcourt, G.D. Scholes, and K.P. Ghiggino. Rate expressions for excitation transfer. II. Electronic considerations of direct and through-configuration exciton res- onance interactions. J. Chem. Phys., 101:10521–10525, 1994. S.M. Parker and T. Shiozaki. Quasi-diabatic states from active space decomposition. J. Chem. Theory Comput., 10:3738–3744, 2014. A. Sisto, D.R. Glowacki, and T.J. Martinez. Ab initio nonadiabatic dynamics of mul- tichromophore complexes: A scalable graphical-processing-unit-accelerated exciton framework. Acc. Chem. Res., 47:2857–2866, 2014. R.J. Bartlett. Coupled-cluster theory and its equation-of-motion extensions. WIREs Comput. Mol. Sci., 2(1):126–138, 2012. A.V . Luzanov. Excited state structural analysis for correlated many-electron systems. Funct. Mat., 21:125–129, 2014. K. Ruedenberg. The physical nature of the chemical bond. Rev. Mod. Phys., 34:326, 1962. M.S. de Giambiagi, M. Giambiagi, and F.E. Jorge. Bond index - relation to 2nd-order density-matrix and charge fluctuations. Theor. Chim. Acta, 68:337–341, 1985. A.B. Sannigrahi. Ab-initio molecular-orbital calculations of bond index and valency. Adv. Quantum Chem., pages 301–351, 1992. A. Torre, L. Lain, R. Bochicchio, and R. Ponec. Topological population analysis from higher order densities II. The correlated case. J. Math. Chem., 32:241–248, 2002. A.V . Luzanov and O.V . Prezhdo. Irreducible charge density matrices for analysis of many-electron wave functions. Int. J. Quant. Chem., 102:582–601, 2005. 304 I. Mayer. Bond order and valence indices: A personal account. J. Comput. Chem., 28:204–221, 2007. S.F. Boys. Construction of some molecular orbitals to be approximately invariant for changes from one molecule to another. Rev. Mod. Phys., 32:296, 1960. A.E. Clark and E.R. Davidson. Local spin. J. Chem. Phys., 115:7382–7340, 2001. A.V . Luzanov, D. Casanova, X. Feng, and A.I. Krylov. Quantifying charge resonance and multiexciton character in coupled chromophores by charge and spin cumulant analysis. J. Chem. Phys., 142:224104, 2015. W.G. Penney. The electronic structure of some polyenes and aromatic molecules. III. Bonds of fractional order by the pair method. Proc. Roy. Soc. A., 158:306–324, 1937. A.A. Ovchinnikov, I.I. Ukrainskii, , and G.V . Kventsel. Theory of one-dimensional Mott semiconductors and the electronic structure of long molecules with conjugated bonds. Sov. Phys. Usp., 15:575, 1973. E.R. Davidson and A.E. Clark. Local spin II. Mol. Phys., 100:373–383, 2002. E.R. Davidson and A.E. Clark. Local spin III: Wave function analysis along a re- action coordinate, h atom abstraction, and addition processes of benzyne. J. Phys. Chem. A, 106:6890–6896, 2002. A.V . Luzanov. Some spin and spin-free aspects of coulomb correlation in molecules. Int. J. Quant. Chem., 112:2915–2923, 2012. G.D. Scholes. Energy transfer and spectroscopic characterization of multichro- mophre assemblies. J. Phys. Chem., 100:18731–18739, 1996. W. Mou, S. Hattori, P. Rajak, F. Shimojo, and A. Nakano. Nanoscopic mechanisms of singlet fission in amorphous molecular solid. Appl. Phys. Lett., 102:173301, 2013. A. Suna. Kinematics of exciton-exciton annihilation in molecular crystals. Phys. Rev. B, 1:1716–1739, 1970. R.C. Johnson and R.E. Merrifield. Effects of magnetic fields on the mutual annihila- tion of triplet excitons in anthracene crystals. Phys. Rev. B, 1:896–902, 1970. W.B. Whitten and S. Arnold. Pressure modulation of exciton fission in tetracene. Phys. Stat. Sol., 74:401–407, 1976. J. Lee, P. Jadhav, P.D. Reusswig, S.R. Yost, N.J. Thompson, D.N. Congreve, 305 E. Hontz, T. Van V oorhis, and M.A. Baldo. Singlet exciton fission photovoltaics. Acc. Chem. Res., 46:1300–1311, 2013. H. Angliker, E. Rommel, and J. Wirz. Electronic spectra of hexacene in solution. Chem. Phys. Lett., 87:208–212, 1982. M. Watanabe, Y . Chang., S. Liu, T. Chao, K. Goto, M.M. Islam, C. Yuan, Y . Tao, T. Shinmyozu, and T.J. Chow. The synthesis, crystal structure and chargetransport properties of hexacene. Nat. Chem., 4:574–578, 2012. E.V . Anslyn and D.A. Dougherty. Modern Physical Organic Chemistry. University Science Books, 2006. Chapter 8. N.G. van Kampen. Stochastic Processes in Physics and Chemistry. Elsevier, 2 edi- tion, 2001. J.J. Burdett and C.J. Bardeen. The dynamics of singlet fission in crystalline tetracene and covalent analogs. Acc. Chem. Res., 46:1312–1320, 2013. B.J. Walker, A.J. Musser, D. Beljonne, and R.H. Friend. Singlet exciton fission in solution. Nature Chem., 5:1019–1024, 2013. J.J. Burdett, D. Gosztola, and C.J. Bardeen. The dependence of singlet exciton re- laxation on excitation density and temperature in polycrystalline tetracene thin films: Kinetic evidence for a dark intermediate state and implications for singlet fission. J. Chem. Phys., 135:214508, 2011. M.W.B. Wilson, A. Rao, K. Johnson, S. G´ elinas, R. di Pietro, J. Clark, and R.H. Friend. Temperature-independent singlet exciton fission in tetracene. J. Am. Chem. Soc., 135(44):16680–16688, 2013. J.J. Burdett, A.M. M¨ uller, D. Gosztola, and C.J. Bardeen. Excited state dynamics in solid and monomeric tetracene: The roles of superradiance and exciton fission. J. Chem. Phys., 133:144506, 2010. D. Beljonne, H. Yamagata, J.L. Br´ edas, F.C. Spano, and Y . Olivier. Charge-transfer excitations steer the Davydov splitting and mediate singlet exciton fission in pen- tacene. Phys. Rev. Lett., 110:226402, 2013. T. Zeng, R. Hoffmann, and N. Ananth. The low-lying electronic states of pen- tacene and their roles in singlet fission. J. Am. Chem. Soc., 2014. in press, dx.doi.org/10.1021/ja500887a. A.V . Akimov and O.V . Prezhdo. Nonadiabatic dynamics of charge transfer and sin- glet fission at the pentacene/C 60 interface. J. Am. Chem. Soc., pages 1599–1608, 306 2014. S.R. Yost, J. Lee, M.W.B. Wilson, T. Wu, D.P. McMahon, R.R. Parkhurst, N.J. Thompson, D.N. Congreve, A. Rao, K. Johnson, M.Y . Sfeir, M.G. Bawendi, T.M. Swager, R.H. Friend, M.A. Baldo, and T. Van V oorhis. A transferable model for singlet-fission kinetics. Nature Chem., 6:492, 2014. Q. Wu, C. Cheng, and T. Van V oorhis. Configuration interaction based on constrained density functional theory: a multireference method. J. Chem. Phys., 127:164119, 2007. A. M. M¨ uller, Y . S. Avlasevich, W. W. Schoeller, Klaus¨ ullen, and C. J. Bardeen. Exciton fission and fusion in bis(tetracene) molecules with different covalent linker structures. J. Am. Chem. Soc., 129:14240–14250, 2007. J.C. Johnson, A.J. Nozik, and J. Michl. High triplet yield from singlet fission in a thin film of 1,3-diphenylisobenzofuran. J. Am. Chem. Soc., 132:16302–16303, 2010. J. Ryerson, J.N. Schrauben, A.J. Ferguson, S.C. Sahoo, P. Naumov, Z. Havlas, J. Michl, A.J. Nozik, and J.C. Johnson. Two thin film polymorphs of the singlet fission compound 1,3-diphenylisobenzofuran. J. Phys. Chem. C, 118:12121–12132, 2014. J.N. Schrauben, J.Ryerson, J. Michl, and J.C. Johnson. Mechanism of singlet fis- sion in thin films of 1,3-diphenylisobenzofuran. J. Am. Chem. Soc., 136:7363–7373, 2014. R.J. Dillon, G.B. Piland, and C.J. Bardeen. Different rates of singlet fission in mon- oclinic versus orthorhombic crystal forms of diphenylhexatriene. J. Am. Chem. Soc., 135:17278–17281, 2013. C. Kitamura, C. Matsumoto, N. Kawatsuki, A. Yoneda, T. Kobayashi, and H. Naito. Crystal structure of 5,12-diphenyltetracene. Anal. Sci. Soc. Japan, 22:2, 2006. N. Streidl, B. Denegri, O. Kronja, and H. Mayr. A practical guide for estimating rates of heterolysis reactions. Acc. Chem. Res., 43:1537–1549, 2010. E. Rosta and A. Warshel. Origin of linear free energy relationships: Exploring the nature of the off-diagonal coupling elements in S N2 reactions. J. Chem. Theory Com- put., 8:3574–3585, 2012. O.V . Prezhdo and P.J. Rossky. Evaluation of quantum transition rates from quantum- classical molecular dynamics simulations. J. Chem. Phys., 107:5863–5878, 1997. 307 T.H. Dunning. Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogen. J. Chem. Phys., 90:1007–1023, 1989. Shao, Y .; Fusti-Molnar, L.; Jung, Y .; Kussmann, J; Ochsenfeld, C.; Brown, S.; Gilbert, A.T.B.; Slipchenko, L.V .; Levchenko, S.V .; O’Neill, D.P.; et al. Advances in methods and algorithms in a modern quantum chemistry program package. Phys. Chem. Chem. Phys., 8:3172–3191, 2006. V . Gray, D. Dzebo, M. Abrahamsson, B. Albinsson, and K. Moth-Poulsen. Triplet- triplet annihilation photon-upconversion: towards solar energy applications. Phys. Chem. Chem. Phys., 16:10345–10352, 2014. A. Akdag, Z. Havlas, and J. Michl. Search for a small chromophore with efficient sin- glet fission: Biradicaloid heterocycles. J. Am. Chem. Soc., 132:16302–16303, 2010. T. Minami and M. Nakano. Diradical character view of singlet fission. J. Phys. Chem. Lett., 3:145–150, 2012. T. Minami, S. Ito, and M. Nakano. Fundamental of diradical-character-based molec- ular design for singlet fission. J. Phys. Chem. Lett., 4:2133–2137, 2013. T. Zeng, N. Ananth, and R. Hoffman. Seeking small molecules for singlet fission: A heteroatom substitution strategy. J. Am. Chem. Soc., 136:12638–12647, 2014. J. Wen, Z. Havlas, and J. Michl. Captodatively stabilized biradicaloids as chro- mophores for singlet fission. J. Am. Chem. Soc., 137:165–172, 2015. N.V . Korovina, S. Das, Z. Nett, X. Feng, J. Joy, A.I. Krylov, S.E. Bradforth, and M.E. Thompson. Singlet fission in a covalently linked cofacial alkynyltetracene dimer. J. Am. Chem. Soc., 138:617–627, 2016. R.A. Marcus. Chemical and electrochemical electron-transfer theory. 15:155, 1964. B. Peters. Common features of extraordinary rate theories. J. Phys. Chem. C, 119:6349–6356, 2015. E. Busby, T.C. Berkelbach, B. Kumar, A. Chernikov, Y . Zhong, H. Hlaing, X.-Y . Zhu, T.F. Heinz, M.S. Hybertsen, M.Y . Sfeir, D.R. Reichman, C. Nuckolls, and O. Yaffe. Multiphonon relaxation slows singlet fission in crystalline hexacene. J. Am. Chem. Soc., 136:10654–10660, 2014. H. Liu, V .M. Nichols, L. Shen, S. Jahansouz, Y . Chen, K.M. Hanson, C.J. Bardeen, and X. Li. Synthesis and photophysical properties of a ”face-to-face” stacked tetracene dimer. Phys. Chem. Chem. Phys., 17:6523–6531, 2015. J.M. Giaimo, J.V . Lockard, L.E. Sinks, A.M. Scott, T.M. Wilson, and M.R. 308 Wasielewski. Excited singlet states of covalently bound, cofacial dimers and trimers of perylene-3,4:9,10-bis(dicarboximide)s. J. Phys. Chem. A, 112:2322–2330, 2008. R.J. Lindquist, K.M. Lefler, K.E. Brown, S.M. Dyar, E.A. Margulies, R.M. Young, and M.R. Wasielewski. Energy flow dynamics within cofacial and slip-stacked perylene-3, 4-dicarboximide dimer models of -aggregates. J. Am. Chem. Soc., 136:14192–14923, 2014. E.A. Margulies, L.E. Shoer, S.W. Eaton, and M.R. Wasielewski. Excimer formation in cofacial and slip-stacked perylene-3,4:9,10-bis(dicarboximide) dimers on a redox- inactive triptycene scaffold. Phys. Chem. Chem. Phys., 16, 2014. 23735-23742. H.L. Stern, A.J. Musser, S. Gelinas, P. Parkinson, L.M. Herz, M.J. Bruzek, J. An- thony, R.H. Friend, and B.J. Walker. Identification of a triplet pair intermediate in singlet exciton fission in solution. Proc. Nat. Acad. Sci., 112:7656–7661, 2015. M.J.Y . Tayebjee, R.G.C.R. Cladyy, and T.W. Schmidt. The exciton dynamics in tetracene thin films. Phys. Chem. Chem. Phys., 14:14797–14805, 2013. J. Zirzlmeier, D. Lehnherr, P.B. Coto, E.T. Chernick, R. Casillas, B.S. Basel, M. Thoss, R.R. Tykwinski, and D.M. Guldi. Singlet fission in pentacene dimers. Proc. Nat. Acad. Sci., 112:5325–5330, 2015. J.M. Herbert, L. D. Jacobson, K.U. Lao, and M.A. Rohrdanz. Rapid computation of intermolecular interactions in molecular and ionic clusters: self-consistent polariza- tion plus symmetry-adapted perturbation theory. Phys. Chem. Chem. Phys., 14:7679– 7699, 2012. L. D. Jacobson and J. M. Herbert. An efficient, fragment-based electronic structure method for molecular systems: Self-consistent polarization with perturbative two- body exchange and dispersion. J. Chem. Phys., 134:094118, 2011. X. Feng and A.I. Krylov. On couplings and excimers: Lessons from studies of singlet fission in covalently linked tetracene dimers. Phys. Chem. Chem. Phys., 18:7751– 7761, 2016. D. Casanova and A.I. Krylov. Quantifying local excitation, charge resonance, and multiexciton character in correlated wave functions of multichromophoric systems. J. Chem. Phys., 144:014102, 2016. T. Zeng and P. Goel. Design of small intramolecular singlet fission chromophores: An azaborine candidate and general small size effects. J. Phys. Chem. Lett., 7:1351– 1358, 2016. E.G. Fuemmeler, S.N. Sanders, A.B. Pun, E. Kumarasamy, T. Zeng, K. Miyata, M.L. 309 Steigerwald, X.-Y . Zhu, M.Y . Sfeir, L.M. Campos, and N. Ananth. A direct mecha- nism of ultrafast intramolecular singlet fission in pentacene dimers. ACS Cent. Sci., 2:316–324, 2016. T. Sakuma H. Sakai, Y . Araki, T. Mori, T. Wada, N.V . Tkachenko, and T. Hasobe. Long-lived triplet excited states of bent-shaped pentacene dimers by intramolecular singlet fission. J. Phys. Chem. A, 120:1867–1875, 2016. A.D. Chien, A.R. Molina, N. Abeyasinghe, O.P. Varnavski, T. Goodson III, and P.M. Zimmerman. Structure and dynamics of the 1 (TT) state in a quinoidal bithiophene: Characterizing a promising intramolecular singlet fission candidate. J. Phys. Chem. C, 119:28258–28268, 2015. X. Feng, D. Casanova, and A.I. Krylov. Intra- and inter-molecular singlet fission in covalently linked dimers. J. Phys. Chem. C, 2016. D. Casanova. Bright fission: Singlet fission into a pair of emitting states. J. Chem. Theory Comput., 11:2642–2650, 2015. Y . Tomkiewicz, R.P. Groff, and P. Avakian. Spectroscopic approach to energetics of exciton fission and fusion in tetracene crystals. J. Chem. Phys., 54:4504–4507, 1971. G.D. Scoles. Correlated pair states formed by singlet fission and exciton-exciton an- nihilation. J. Phys. Chem. A, 119:12699–12705, 2015. F.C. Spano and C. Silva. H- and J-aggregate behavior in polymeric semiconductors. Annu. Rev. Phys. Chem., 65:477–500, 2014. T. Wesolowski, R.P. Muller, and A. Warshel. Ab initio frozen density functional cal- culations of proton transfer reactions in solution. J. Phys. Chem., 100:15444–15449, 1996. Q. Wu and T. Van V oorhis. Constrained density functional theory and its application in long-range electron transfer. J. Chem. Theory Comput., 2:765–774, 2006. H. Zhang, J. Malrieu, H. Ma, and J. Ma. Implementation of renormalized excitonic method at ab initio level. J. Comput. Chem., 33:34, 2011. S.M. Parker, T. Seideman, M. Ratner, and T. Shiozaki. Communication: Active-space decomposition for molecular dimers,. J. Chem. Phys., 139:021108, 2013. A. Morrison, Z. You, and J. Herbert. Ab initio implementation of the Frenkel- Davydov exciton model: A naturally parallelizable approach to computing collective excitations in crystals and aggregates. J. Chem. Theory Comput., 10:5366, 2014. 310 J.A. Pople. Theoretical models for chemistry. In D.W. Smith and W.B. McRae, editors, Energy, Structure and Reactivity: Proceedings of the 1972 Boulder Sum- mer Research Conference on Theoretical Chemistry, pages 51–61. Wiley, New York, 1973. R.J. Bartlett. The coupled-cluster revolution. Mol. Phys., 108:2905–2920, 2010. K. Sneskov and O. Christiansen. Excited state coupled cluster methods. WIREs Com- put. Mol. Sci., 2:566, 2012. F. Weigend, M. H¨ aser, H. Patzelt, and R. Ahlrichs. RI-MP2: optimized auxiliary basis sets and demonstration of efficiency. Chem. Phys. Lett., 294:143–152, 1998. J.L. Whitten. Coulombic potential energy integrals and approximations. J. Chem. Phys., 58:4496, 1973. M.W. Feyereisen, G. Fitzgerald, and A. Komornicki. Use of approximate integrals in abinitio theory - an application in MP2 energy calculations. Chem. Phys. Lett., 208:359–363, 1993. O. Vahtras, J. Alml¨ of, and M.W. Feyereisen. Integral approximations for LCAO-SCF calculations. Chem. Phys. Lett., 213:514–518, 1993. A. Komornicki and G. Fitzgerald. Molecular gradients and hessians implemented in density functional theory. J. Chem. Phys., 98:1398–1421, 1993. D.E. Bernholdt and R.J. Harrison. Large-scale correlated electronic structure calcula- tions: The RI-MP2 method on parallel computers. Chem. Phys. Lett., 250:477–484, 1996. Y . Jung, A. Sodt, P. M. W. Gill, and M. Head-Gordon. Auxiliary basis expansions for large-scale electronic structure calculations. Proc. Nat. Acad. Sci., 102:6692–6697, 2005. N.H.F. Beebe and J. Linderberg. Simplifications in the generation and transformation of two-electron integrals in molecular calculations. Int. J. Quant. Chem., 12:683–705, 1977. H. Koch, A.S. de Mer´ as, and T.B. Pedersen. Reduced scaling in electronic structure calculations using Cholesky decompositions. J. Chem. Phys., 118:9481, 2003. F. Aquilante, R. Lindh, and T. B. Pedersen. Unbiased auxiliary basis sets for accurate two-electron integral approximations. J. Chem. Phys., 127:114107, 2007. F. Aquilante, T. B. Pedersen, and R. Lindh. Density fitting with auxiliary basis sets from Cholesky decompositions. Theor. Chem. Acc., 124:1–10, 2009. 311 F. Aquilante, L. Gagliardi, T. B. Pedersen, and R. Lindh. Atomic Cholesky decom- positions: A route to unbiased auxiliary basis sets for density fitting approximation with tunable accuracy and efficiency. J. Chem. Phys., 130:154107, 2009. F. Aquilante, L. Boman, J. Bostr¨ om, H. Koch, R. Lindh, A.S. de Mer´ as, and T.B. Pedersen. Cholesky decomposition techniques in electronic structure theory. In R. Zale´ sny, M.G. Papadopoulos, P.G. Mezey, and J. Leszczynski, editors, Linear- Scaling Techniques in Computational Chemistry and Physics, Challenges and ad- vances in computational chemistry and physics, pages 301–343. Springer, 2011. F. Weigend, M. Kattannek, and R. Ahlrichs. Approximated electron repulsion inte- grals: Cholesky decomposition versus resolution of the identity methods. J. Chem. Phys., 130:164106, 2009. R.M. Parrish, E.G. Hohenstein, T.J. Mart´ ınez, and C.D. Sherrill. Tensor hypercon- traction. II. Least-squares renormalization. J. Chem. Phys., 137(22):224106, 2012. E.G. Hohenstein, R.M. Parrish, C.D. Sherrill, and T.J. Mart´ ınez. Communication: Tensor hypercontraction. III. Least-squares tensor hypercontraction for the determi- nation of correlated wavefunctions. J. Chem. Phys., 137(22):221101, 2012. F. Aquilante, T.K. Todorova, L. Gagliardi, T.B. Pedersen, and B.O. Roos. Systematic truncation of the virtual space in multiconfigurational perturbation theory. J. Chem. Phys., 131:034113, 2009. F. Aquilante and T.B. Pedersen. Quartic scaling evaluation of canonical scaled oppo- site spin second-order MøllerPlesset correlation energy using Cholesky decomposi- tions. Chem. Phys. Lett., 449:354–357, 2007. C. Riplinger and F. Neese. An efficient and near linear scaling pair natural orbital based local coupled cluster method. J. Chem. Phys., 138:034106, 2013. D. Kats, T. Korona, and M. Sch¨ utz. Local CC2 electronic excitation energies for large molecules with density fitting. J. Chem. Phys., 125:104106–104121, 2006. Y . Shao, L. Fusti-Molnar, Y . Jung, J. Kussmann, C. Ochsenfeld, S. Brown, A.T.B. Gilbert, L.V . Slipchenko, S.V . Levchenko, D.P. O’Neill, R.A. Distasio Jr, R.C. Lochan, T. Wang, G.J.O. Beran, N.A. Besley, J.M. Herbert, C.Y . Lin, T. Van V oorhis, S.H. Chien, A. Sodt, R.P. Steele, V .A. Rassolov, P. Maslen, P.P. Korambath, R.D. Adamson, B. Austin, J. Baker, E.F.C. Byrd, H. Daschel, R.J. Doerksen, A. Dreuw, B.D. Dunietz, A.D. Dutoi, T.R. Furlani, S.R. Gwaltney, A. Heyden, S. Hirata, C.-P. Hsu, G.S. Kedziora, R.Z. Khalliulin, P. Klunziger, A.M. Lee, W.Z. Liang, I. Lotan, N. Nair, B. Peters, E.I. Proynov, P.A. Pieniazek, Y .M. Rhee, J. Ritchie, E. Rosta, C.D. Sherrill, A.C. Simmonett, J.E. Subotnik, H.L. Woodcock III, W. Zhang, A.T. Bell, 312 A.K. Chakraborty, D.M. Chipman, F.J. Keil, A. Warshel, W.J. Hehre, H.F. Schaefer III, J. Kong, A.I. Krylov, P.M.W. Gill, M. Head-Gordon. Advances in methods and algorithms in a modern quantum chemistry program package. Phys. Chem. Chem. Phys., 8:3172–3191, 2006. A.P. Rendell and T.J. Lee. Coupled-cluster theory employing approximate integrals - an approach to avoid the input/output and storage bottlenecks. J. Chem. Phys., 101:400–408, 1994. F. Aquilante, L. de Vico, N. Ferr´ e, G. Ghigo, P- ˚ A Malmqvist, P. Neogr´ ady, T. B. Ped- ersen, M. Pitonˇ ak, M. Reiher, B. Roos, L. Serrano-Andr´ es, M. Urban, V . Veryazov, and R. Lindh. Molcas 7: The next generation. J. Comput. Chem., 31:224–247, 2010. S. Wilson. Universal basis sets and Cholesky decomposition of the two-electron in- tegral matrix. Comp. Phys. Comm., 58:71–81, 1990. G.H. Golub and C.F. Van Loan. Matrix computations. Johns Hopkins University Press, 1996. E. Epifanovsky, M. Wormit, T. Ku´ s, A. Landau, D. Zuev, K. Khistyaev, P. Manohar, I. Kaliman, A. Dreuw, and A.I. Krylov. New implementation of high-level correlated methods using a general block-tensor library for high-performance electronic struc- ture calculations. J. Comput. Chem., 34:2293–2309, 2013. K. Eichkorn, O. Treutler, H. ¨ Ohm, M. H¨ aser, and R. Ahlrichs. Auxiliary basis sets to approximate Coulomb potentials. Chem. Phys. Lett., 242:652–660, 1995. K. Eichkorn, F. Weigend, O. Treutler, and R. Ahlrichs. Auxiliary basis sets for main row atoms and transition metals and their use to approximate Coulomb potentials. Theor. Chem. Acc., 97:119–124, 1997. F. Weigend. Hartree-Fock exchange fitting basis sets for H to Rn. J. Comput. Chem., 29:167–175, 2008. F. Weigend, A. K¨ ohn, and C. H¨ attig. Efficient use of the correlation consistent basis sets in resolution of the identity MP2 calculations. J. Chem. Phys., 116:3175, 2002. F. Weigend. Accurate coulomb-fitting basis sets for H to Rn. Phys. Chem. Chem. Phys., 8:1057–1065, 2006. A. Hellweg, C. H¨ attig, S. H¨ ofener, and W. Klopper. Optimized accurate auxiliary basis sets for RI-MP2 and RI-CC2 calculations for the atoms Rb to Rn. Theor. Chem. Acc., 117:587–597, 2007. 313 E. Epifanovsky, D. Zuev, X. Feng, K. Khistyaev, Y . Shao, and A.I. Krylov. General implementation of resolition-of-identity and Cholesky representations of electron- repulsion integrals within coupled-cluster and equation-of-motion methods: Theory and benchmarks. J. Chem. Phys., 139:134105, 2013. J.F. Stanton and R.J. Bartlett. The equation of motion coupled-cluster method. A systematic biorthogonal approach to molecular excitation energies, transition proba- bilities, and excited state properties. J. Chem. Phys., 98:7029–7039, 1993. S.V . Levchenko and A.I. Krylov. Equation-of-motion spin-flip coupled-cluster model with single and double substitutions: Theory and application to cyclobutadiene. J. Chem. Phys., 120(1):175–185, 2004. J. Bostr¨ om, F. Aquilante, T. B. Pedersen, and R. Lindh. Ab Initio density fitting: Ac- curacy assessment of auxiliary basis sets from Cholesky decompositions. J. Chem. Theory Comput., 5:1545–1553, 2009. V . P. Vysotskiy and L. S. Cederbaum. On the Cholesky decomposition for elec- tron propagator methods: General aspects and application on C 60 . J. Chem. Phys., 132:044110, 2010. C. H¨ attig and F. Weigend. CC2 excitation energy calculations on large molecules us- ing the resolution of the identity approximation. J. Chem. Phys., 113(13):5154–5161, 2000. A. K¨ ohn and C. H¨ attig. Analytic gradients for excited states in the coupled-cluster model CC2 employing the resolution-of-the-identity approximation. J. Chem. Phys., 119(10):5021–5036, 2003. C. H¨ attig and A. K¨ ohn. Transition moments and excited-state first-order properties in the coupled-cluster model CC2 using the resolution-of-the-identity approximation. J. Chem. Phys., 117(15):6939–6951, 2002. T. Rocha-Rinza, O. Christiansen, J. Rajput, A. Gopalan, D.B. Rahbek, L.H. Ander- sen, A.V . Bochenkova, A.A.Granovsky, K.B. Bravaya, A.V . Nemukhin, K.L. Chris- tiansen, and M.B. Nielsen. Gas phase absorption studies of photoactive yellow pro- tein chromophore derivatives. J. Phys. Chem. A, 113:9442–9449, 2009. D. Zuev, K.B. Bravaya, T.D. Crawford, R. Lindh, and A.I. Krylov. Electronic struc- ture of the two isomers of the anionic form of p-coumaric acid chromophore. J. Chem. Phys., 134:034310, 2011. K.B. Bravaya, E. Epifanovsky, and Anna I. Krylov. Four bases score a run: Ab initio calculations quantify a cooperative effect of h-bonding and pi-stacking on ionization energy of adenine in the AATT tetramer. J. Phys. Chem. Lett., 3:2726–2732, 2012. 314 K. Khistyaev, A. Golan, K.B. Bravaya, N. Orms, A.I. Krylov, and M. Ahmed. Pro- ton transfer in nucleobases is mediated by water. J. Phys. Chem. A, 117:6789–6797, 2013. A. Landau, K. Khistyaev, S. Dolgikh, and A.I. Krylov. Frozen natural orbitals for ionized states within equation-of-motion coupled-cluster formalism. J. Chem. Phys., 132:014109, 2010. W.J. Hehre, R. Ditchfield, and J.A. Pople. Self-consistent molecular orbital methods. XII. Further extensions of gaussian-type basis sets for use in molecular orbital studies of organic molecules. J. Chem. Phys., 56:2257, 1972. T. Clark, J. Chandrasekhar, and P.V .R. Schleyer. Efficient diffuse function-augmented basis sets for anion calculations. III. The 3-21+g basis set for first-row elements, li-f. J. Comput. Chem., 4:294–301, 1983. R. Krishnan, J.S. Binkley, R. Seeger, and J.A. Pople. Self-consistent molecular or- bital methods. XX. A basis set for correlated wave functions. J. Chem. Phys., 72:650, 1980. F. Weigend and M. H¨ aser. RI-MP2: first derivatives and global consistency. Theor. Chim. Acta, 97:331–340, 1997. J. Yang, Y . Kurashige, F.R. Manby, and G.K.L. Chan. Tensor factorizations of local second-order Moller-Plesset theory. J. Chem. Phys., 134:044123, 2011. J.E. Subotnik and M. Head-Gordon. A local correlation model that yields intrinsi- cally smooth potential energy surfaces. J. Chem. Phys., 123:064108, 2005. U. Benedikt, A.A. Auer, M. Espig, and W. Hackbusch. Tensor decomposition in post-Hartree-Fock methods. J. Chem. Phys., 134:054118, 2011. F. Bell, D. Lambrecht, and M. Head-Gordon. Higher order singular value decompo- sition (HOSVD) in quantum chemistry. Mol. Phys., 108:2759–2773, 2010. F. Aquilante, R. Lindh, and T.B. Pedersen. Analytic derivatives for the Cholesky representation of the two-electron integrals. J. Chem. Phys., 129:034106, 2008. A. G. Taube and R. J. Bartlett. Frozen natural orbital coupled-cluster theory: Forces and application to decomposition of nitroethane. J. Chem. Phys., 128:164101, 2008. M. H¨ aser and R. Ahlrichs. Improvements on the direct SCF method. J. Comput. Chem., 10, 1988. S. A. Maurer, D. S. Lambrecht, D. Flaig, and C. Ochsenfeld. Distance-dependent schwarz-based integral estimates for two-electron integrals: Reliable tightness vs. 315 rigorous upper bounds. J. Chem. Phys., 136, 2012. S. A. Maurer, D. S. Lambrecht, J. Kussmann, and C. Ochsenfeld. Efficient distance- including integral screening in linear-scaling moller-plesset perturbation theory. J. Chem. Phys., 138, 2013. J. Brabec, C. Yang, E. Epifanovsky, A.I. Krylov, and E. Ng. Reduced-cost sparcity- exploiting algorithm for solving coupled-cluster equations. J. Comput. Chem., 37:1059–1067, 2016. 316
Abstract (if available)
Abstract
This thesis consists of two parts.The first part focuses on the electronic structure aspects of singlet fission (SF) process studied by ab initio calculations. The second part is on the improvement of the high-accuracy post Hartree-Fock methods. ❧ Our motivation for studying SF process is to understand its mechanism in order to help experimentalists to design more efficient SF materials. We compute the electronic factors including energies and characters of excited states, non-adiabatic couplings (NAC) between these states, and analyze their effect on the rate of SF by using a kinetic model. Chapter 2 discusses the nature of correlated adiabatic wave functions of the initial excited state and dark intermediate multiexcitonic state of pentacene and tetracene. We found that the charge-resonance (CR) configurations are important for determining the NAC between these states. NACs are estimated by using the norm of one particle transition density matrix (OPDM) ║γ║, based on Cauchy-Schwarz inequality. Chapter 3 discusses the utility of the norm of OPDM and presents benchmark. We show that ║γ║ contains the principal information about the changes in electronic states involved, such as varying degree of one-electron character of the transition
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University of Southern California Dissertations and Theses
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Creator
Feng, Xintian
(author)
Core Title
Predictive electronic structure methods for strongly correlated systems: method development and applications to singlet fission
School
College of Letters, Arts and Sciences
Degree
Doctor of Philosophy
Degree Program
Chemistry
Publication Date
11/11/2016
Defense Date
10/18/2016
Publisher
University of Southern California
(original),
University of Southern California. Libraries
(digital)
Tag
Cholesky decomposition,coupled cluster,OAI-PMH Harvest,singlet fission
Language
English
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Electronically uploaded by the author
(provenance)
Advisor
Krylov, Anna I. (
committee chair
), Nakano, Aiichiro (
committee member
), Thompson, Mark E. (
committee member
)
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xintianf@usc.edu
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Feng, Xintian
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University of Southern California Dissertations and Theses
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The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...
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Tags
Cholesky decomposition
coupled cluster
singlet fission