Ground and Electronic Excited States from Pairing Matrix Fluctuation and Particle-Particle Random Phase Approximation
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Ground and Electronic Excited States from Pairing Matrix Fluctuation and Particle-Particle Random Phase Approximation by Yang Yang Department of Chemistry Duke University Date: Approved: Weitao Yang, Supervisor David Beratan Patrick Charbonneau Harold Baranger Dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Chemistry in the Graduate School of Duke University 2016 Abstract Ground and Electronic Excited States from Pairing Matrix Fluctuation and Particle-Particle Random Phase Approximation by Yang Yang Department of Chemistry Duke University Date: Approved: Weitao Yang, Supervisor David Beratan Patrick Charbonneau Harold Baranger An abstract of a dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Chemistry in the Graduate School of Duke University 2016 Copyright c 2016 by Yang Yang All rights reserved except the rights granted by the Creative Commons Attribution-Noncommercial Licence Abstract The accurate description of ground and electronic excited states is an important and challenging topic in quantum chemistry. The pairing matrix fluctuation, as a counter- part of the density fluctuation, is applied to this topic. From the pairing matrix fluc- tuation, the exact electron correlation energy as well as two electron addition/removal energies can be extracted. Therefore, both ground state and excited states energies can be obtained and they are in principle exact with a complete knowledge of the pairing matrix fluctuation. In practice, considering the exact pairing matrix fluctua- tion is unknown, we adopt its simple approximation | the particle-particle random phase approximation (pp-RPA) | for ground and excited states calculations. The algorithms for accelerating the pp-RPA calculation, including spin separation, spin adaptation, as well as an iterative Davidson method, are developed. For ground states correlation descriptions, the results obtained from pp-RPA are usually compa- rable to and can be more accurate than those from traditional particle-hole random phase approximation (ph-RPA). For excited states, the pp-RPA is able to describe double, Rydberg, and charge transfer excitations, which are challenging for conven- tional time-dependent density functional theory (TDDFT). Although the pp-RPA intrinsically cannot describe those excitations excited from the orbitals below the highest occupied molecular orbital (HOMO), its performances on those single exci- tations that can be captured are comparable to TDDFT. The pp-RPA for excitation calculation is further applied to challenging diradical problems and is used to unveil iv the nature of the ground and electronic excited states of higher acenes. The pp-RPA and the corresponding Tamm-Dancoff approximation (pp-TDA) are also applied to conical intersections, an important concept in nonadiabatic dynamics. Their good description of the double-cone feature of conical intersections is in sharp contrast to the failure of TDDFT. All in all, the pairing matrix fluctuation opens up new channel of thinking for quantum chemistry, and the pp-RPA is a promising method in describing ground and electronic excited states. v To all dream pursuers, To the humanity, To the world we live in, And to the Universe that created us and we stare at in awe. vi Contents Abstract iv List of Tables xi List of Figures xii List of Abbreviations and Symbols xiv Acknowledgements xviii 1 Introduction1 1.1 Quantum Chemistry and Exact Hamiltonian..............1 1.2 Two Main Approximations in Electronic Structure Theory......2 1.2.1 Non-relativistic Approximation.................2 1.2.2 Born-Oppenheimer Approximation...............3 1.2.3 Basic Equations in Electronic Structure Theory........4 1.3 Methods for Ground State........................4 1.3.1 Valence Bond Theory.......................4 1.3.2 Molecular Orbital Theory....................5 1.4 Methods for Excited States.......................8 1.4.1 Delta-SCF.............................8 1.4.2 Equation-of-Motion........................9 1.4.3 Linear Response Theory.....................9 1.5 Current Challenges in Electronic Structure Theory..........9 vii 1.5.1 Challenges for Ground State...................9 1.5.2 Challenges for Excited States.................. 10 2 Pairing Matrix Fluctuation for Ground State Correlation 12 2.1 Introduction................................ 12 2.2 Theory................................... 14 2.2.1 Paring Matrix Fluctuation.................... 14 2.2.2 Particle-Particle Random Phase Approximation........ 15 2.2.3 Adiabatic Connection....................... 17 2.2.4 Correlation Energy from Pairing Matrix Fluctuation..... 18 2.2.5 Correlation Energy from Particle-Particle Random Phase Ap- proximation............................ 19 2.2.6 Spin Separation and Spin Adaptation.............. 21 2.3 Benchmark Results............................ 25 2.3.1 Computational Details...................... 25 2.3.2 Enthalpies of Formation..................... 26 2.3.3 Reaction Barriers......................... 31 2.3.4 Nonbonded Interactions..................... 34 2.3.5 Conclusion............................. 36 3 Pairing Matrix Fluctuation for Excited States Calculation 38 3.1 Introduction................................ 38 3.2 Theory................................... 42 3.2.1 Particle-Particle Random Phase Approximation from Equation of Motion............................. 42 3.2.2 pp-RPA-HF* from Equation of Motion............. 47 3.2.3 Two-Electron Systems: An Exact Case for pp-RPA-HF* and pp-TDA-HF*........................... 53 3.2.4 Oscillator Strengths from pp-TDA............... 53 viii 3.2.5 Time-Dependent Density Functional Theory with Pairing Field 54 3.2.6 An Iterative Davidson Method for pp-RPA........... 58 3.3 Results................................... 63 3.3.1 Benchmark Regular Single Excitations............. 63 3.3.2 Double Excitations........................ 74 3.3.3 Rydberg Excitations....................... 74 3.3.4 CT Excitations.......................... 76 3.3.5 Oscillator Strengths........................ 76 3.3.6 Excitations from (N+2)-Electron References.......... 76 3.3.7 Conclusion............................. 79 4 Application to Diradicals 80 4.1 Introduction................................ 80 4.2 Methods.................................. 85 4.3 Results and Discussions......................... 86 4.3.1 Diatomic Diradicals........................ 86 4.3.2 Carbene-like Diradicals...................... 87 4.3.3 Disjoint Diradicals........................ 93 4.3.4 Four-Electron Diradicals..................... 94 4.3.5 Benzynes............................. 97 4.4 Conclusion................................. 97 5 Nature of Ground and Excited States of Higher Acenes 99 5.1 Introduction................................ 99 5.2 Results................................... 103 5.2.1 Singlet-Triplet Energy Gap.................... 103 5.2.2 Singlet Ground State in the Diradical Continuum....... 106 ix 5.2.3 Lowest Bright Singlet Excitation................ 114 1 5.2.4 Doubly Excited Ag State.................... 118 5.2.5 Singlet Fission for Higher Acenes................ 125 5.3 Conclusion................................. 126 6 Application to Conical Intersections 128 6.1 Introduction................................ 128 6.2 Method.................................. 131 6.3 Results................................... 131 6.3.1 D3h H3 ............................... 131 6.3.2 D3h NH3 .............................. 133 6.3.3 C2v NH3 .............................. 134 6.4 Conclusion................................. 135 7 Outlook and Future Directions 136 7.1 Future Developments within Pairing Matrix Fluctuation....... 136 7.2 Future of Electronic Structure Theory.................. 138 Bibliography 140 Biography 160 x List of Tables 2.1 RPA errors for subsets in G2/97 database............... 29 2.2 RPA reaction energies among the molecules from G2/97 database.. 31 2.3 RPA errors for reaction energies..................... 31 2.4 RPA reaction barriers........................... 33 2.5 RPA errors for four subsets in DBH24 reaction barries........ 34 2.6 RPA errors for nonbonded interaction sets............... 36 3.1 Vertical excitation energies from pp-RPA, CIS, TDHF and TDDFT. 67 3.2 Vertical excitation energies from pp-RPA-B3LYP and TD-B3LYP.. 71 3.3 Double excitation results......................... 75 3.4 Rydberg excitation results........................ 75 3.5 Oscillator strength results........................ 77 3.6 Excitations from (N+2)-electron reference............... 78 4.1 Adiabatic singlet-triplet gaps for diatomic molecules.......... 89 4.2 Adiabatic singlet-triplet gaps for carbene-like molecules........ 89 4.3 Vertical singlet-triplet gaps for disjoint diradicals........... 90 4.4 Singlet-triplet gaps for four-π-electron diradicals............ 90 4.5 Adiabatic singlet-triplet gaps for benzynes............... 91 3 5.1 Excitation energy of B2u state for acenes............... 104 1 5.2 Excitation energy of B2u state for acenes............... 104 1 5.3 Excitation energy of Ag state for acenes................ 105 xi List of Figures 2.1 Ring and ladder diagrams........................ 14 2.2 Basis set convergence for RPA total energies and atomization energies 26 2.3 Errors for G2/97 enthalpies of formation test set............ 28 2.4 Basis set convergence for RPA reaction barriers............ 32 2.5 Basis set convergence for RPA nonbonded interaction energies.... 35 3.1 Schematic