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University of Pennsylvania ScholarlyCommons Publicly Accessible Penn Dissertations 2017 Tddft Derivative Couplings And Other Topics In Quantum Chemistry Qi Ou University of Pennsylvania, [email protected] Follow this and additional works at: https://repository.upenn.edu/edissertations Part of the Chemistry Commons Recommended Citation Ou, Qi, "Tddft Derivative Couplings And Other Topics In Quantum Chemistry" (2017). Publicly Accessible Penn Dissertations. 2509. https://repository.upenn.edu/edissertations/2509 This paper is posted at ScholarlyCommons. https://repository.upenn.edu/edissertations/2509 For more information, please contact [email protected]. Tddft Derivative Couplings And Other Topics In Quantum Chemistry Abstract Photochemical reactions, which involve both the ground and excited electronic states of a molecule, can promote processes otherwise inaccessible by normal reactions. In general, photochemical reactions may be classified as adiabatic or nonadiabatic depending on whether the reaction takes place on the same adiabatic potential energy surface or not. From research over the last two decades, we now understand that many processes in nature turn out to be nonadiabatic { including charge transfer, electronic excitation quenching, and spin-forbidden transitions. The efficiency of such ocessespr depends critically on the electron-nuclear interaction, which is quantified by the derivative coupling between the two involved states. The first part of the work (chapters 3-6) presented here mainly focuses on understanding the electron-nuclear interaction using the electronic structure theory. Two approaches are developed calculating the derivative couplings between the excited states within the time-dependent density functional theory. The behavior of the derivative couplings around a conical intersection is analyzed for two real molecules: benzaldehyde and protonated formaldamine. The second part of this work (chapters 7-8) focuses on understanding the electron-electron interaction in the framework of Green's function. Detailed working equations are derived for the GW approximation, which is used to calculate the electron attachment/detachment energy, and the Bethe-Salpeter equation, which is used to obtain the electron excitation energies of a system. Degree Type Dissertation Degree Name Doctor of Philosophy (PhD) Graduate Group Chemistry First Advisor Joseph E. Subotnik Keywords conical intersection, derivative coupling, electronic structure theory, nonadiabatic, tddft Subject Categories Chemistry This dissertation is available at ScholarlyCommons: https://repository.upenn.edu/edissertations/2509 TDDFT DERIVATIVE COUPLINGS AND OTHER TOPICS IN QUANTUM CHEMISTRY Qi Ou A DISSERTATION in Chemistry Presented to the Faculties of the University of Pennsylvania in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy 2017 Supervisor of Dissertation Joseph E. Subotnik, Professor of Chemistry Graduate Group Chairperson Gary A. Molander, Hirschmann-Makineni Professor of Chemistry Dissertation Committee: Andrew M. Rappe, Blanchard Professor of Chemistry Marsha I. Lester, Edmund J. Kahn Distinguished Professor Zahra Fakhraai, Assistant Professor of Chemistry ACKNOWLEDGMENT First of all, I would like to express my special appreciation and thanks to my wonderful advisor Joseph Subotnik, you have been a tremendous mentor to me. I would like to thank you for encouraging my research as well as my life. I could not sit here and write my thesis without your encouragement and guidance during the past five years. As the song says, \You raise me up, to more than I can be." I would also like to thank my thesis committee members, Andrew Rappe, Marsha Lester, and Zahra Fakhraai, for their brilliant comments and suggestions. Thanks also to Yihan Shao and Zhengting Gan at Q-Chem, Inc., we would not have accomplished anything without their help. I also have to thank all the members from Subotnik's group, for their help and friendship along the way. I want to give special thanks to Shervin Fatehi, Ethan Alguire, Xinle Liu, Wenjie Dou, who help me a bunch with my research a. Also I would like to thank Yubo Qi from Rappe's group, who has been a sincere friend to me over the years. Finally, I want to offer my sincerest thanks to my family, for the love and support of my parents and my husband, and also for the birth of my dear little son. You mean everything to me. No matter where I am, what I am doing, I love you from the bottom of my heart, forever. ii ABSTRACT TDDFT DERIVATIVE COUPLINGS AND OTHER TOPICS IN QUANTUM CHEMISTRY Qi Ou Joseph E. Subotnik Photochemical reactions, which involve both the ground and excited electronic states of a molecule, can promote processes otherwise inaccessible by normal reactions. In gen- eral, photochemical reactions may be classified as adiabatic or nonadiabatic depending on whether the reaction takes place on the same adiabatic potential energy surface or not. From research over the last two decades, we now understand that many processes in nature turn out to be nonadiabatic { including charge transfer, electronic excitation quenching, and spin-forbidden transitions. The efficiency of such processes depends critically on the electron-nuclear interaction, which is quantified by the derivative coupling between the two involved states. The first part of the work (chapters 3{6) presented here mainly focuses on understand- ing the electron-nuclear interaction using the electronic structure theory. Two approaches are developed calculating the derivative couplings between the excited states within the time-dependent density functional theory. The behavior of the derivative couplings around a conical intersection is analyzed for two real molecules: benzaldehyde and protonated formaldamine. The second part of this work (chapters 7{8) focuses on understanding the electron-electron interaction in the framework of Green's function. Detailed working equations are derived for the GW approximation, which is used to calculate the electron attachment/detachment energy, and the Bethe-Salpeter equation, which is used to obtain the electron excitation energies of a system. iii TABLE OF CONTENTS ACKNOWLEDGMENT . ii ABSTRACT . iii LIST OF TABLES . viii LIST OF ILLUSTRATIONS . xvi CHAPTER 1 : Introduction . 1 1.1 Born-Oppenheimer approximation and its breakdown . 1 1.2 Quantum chemistry excited states methods . 4 1.3 Many-body perturbation theory . 10 CHAPTER 2 : Outline . 18 CHAPTER 3 : Electronic Relaxation in Benzaldehyde evaluated via TDDFT and Localized Diabatization: Intersystem Crossings, Conical Intersec- tions, and Phosphorescence . 21 3.1 Introduction . 21 3.2 Evaluation of the optimized geometries and excitation energies . 24 3.3 S1/T2 spin-orbit coupling and intersystem-crossing rate . 27 3.4 Conical intersections and localized diabatization methods . 31 3.5 Phosphorescence lifetime of benzaldehyde. 39 3.6 Conclusion and outlook . 41 3.7 Acknowledgments . 42 CHAPTER 4 : Derivative Couplings between TDDFT Excited States Obtained by Direct Differentiation in the Tamm-Dancoff Approximation . 44 iv 4.1 Introduction . 44 4.2 Analytic derivation for TDDFT/TDA derivative couplings . 47 4.3 Comparison with finite-difference . 61 4.4 Application to benzaldehyde . 61 4.5 Conclusion . 68 4.6 Acknowledgments . 69 CHAPTER 5 : Derivative Couplings between TDDFT Excited States in the Random- Phase Approximation Based on Pseudo-Wavefunctions: Behavior around Conical Intersections . 70 5.1 Introduction . 70 5.2 Analytic derivation for TDDFT/RPA derivative couplings . 73 5.3 Comparison with finite-difference . 85 5.4 Geometric Phase and Branching Plane for RPA Conical Intersections . 86 5.5 Application to Protonated Formaldimine . 92 5.6 The Chernyak-Mukamel expression and the transition density matrix accord- ing to response theory . 95 5.7 Conclusion . 98 5.8 Acknowledgments . 99 CHAPTER 6 : First-Order Derivative Couplings between Excited States from Adi- abatic TDDFT Response Theory . 100 6.1 Introduction . 100 6.2 Theory . 103 6.3 Numerical Examples . 125 6.4 Conclusion . 127 6.5 Acknowledgments . 129 v CHAPTER 7 : A Comparison between GW and Wave-Function Based Approaches: Calculating the Ionization Potential and Electron Affinity for 1D Hubbard Chains . 130 7.1 Introduction . 130 7.2 Theory . 132 7.3 Results . 142 7.4 Discussion . 145 7.5 Conclusion . 147 7.6 Acknowledgments . 148 CHAPTER 8 : A Comparison between BSE and Configuration Interaction Approaches for Solving a Quantum Chemistry Problem: Calculating the Excita- tion Energy for Finite 1D Hubbard Chains . 156 8.1 Introduction . 156 8.2 Theory . 158 8.3 Results . 180 8.4 Discussion . 184 8.5 Conclusion . 187 CHAPTER 9 : Conclusion and future work . 190 APPENDIX . 192 BIBLIOGRAPHY . 207 vi LIST OF TABLES TABLE 3.1 : Experimental and computational results for the key geometric pa- rameters (in A˚ for bond length and degree for angles) of benzalde- hyde in different optimized geometries . 27 TABLE 3.2 : Experimental and theoretical results for the adiabatic excitation en- ergies (in eV) of S1,T1 and T2 states . 27 TABLE 3.3 : Comparison in eV of S0,S1,T1and T2 relative energies at different optimized geometries for TDDFT/!B97X (versus CASPT2 refer- ence energies ) . 28 TABLE 4.4 : Derivative couplings between the S1 and S4 states of LiH as com- puted by finite difference (FD), analytical theory (full-DC), and DC without Pulay terms (NP) . 62 TABLE 4.5 : Out-of-plane angles for the rel-DH, DH, and NP vectors at θ = 30◦. 66 TABLE 4.6 : Magnitudes of derivative coupling vectors between T1 and T2 states of benzaldehyde and their angles relative to the gradient differ- ence at θ = 30◦. Full-DC: The complete derivative couplings given by Eqn. 4.58. ETF-DC: Corrected derivative couplings given by R[Q] 1 [Q] Eqn. 4.58 when setting S = 2 S . rel-DH: All terms that con- tain D¯ IJ in Eqn. 4.58. DH: All terms that contain DIJ in Eqn. 4.53. NP: Derivative couplings without all Pulay terms. Note that only the full-DC, ETF-DC, and projected rel-DH vectors are effectively orthogonal to the energy gradient difference in properly scaled coor- dinates. 67 TABLE 4.7 : Circulations of derivative couplings vectors around the T1/T2 conical intersection point of benzaldehyde.
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