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 sketch for excitation problems treated by pp-RPA..... 41

3.2 Relation between pp-RPA, pp-TDA and the approaches to derive them 58

3.3 Basis set convergence test for pp-RPA excitations...... 64

3.4 Errors comparison for singlet and triplet excitations...... 70

3.5 CT and non-CT results for C2H4 ¨ C2F4 ...... 77

3.6 CT results for He2 ...... 78 4.1 Two-orbital diradical model...... 83

4.2 The pp-RPA combines DFT with wave function methods...... 85

4.3 Shape of TMM b1 orbital...... 96 5.1 Structure of acenes in one Kekule resonance form...... 101

5.2 Singlet-triplet gap for acenes...... 107

5.3 Dominant configuration contributions for acene ground state..... 111

5.4 Shape of canonical and transformed orbitals for H2 ...... 114 5.5 Shape of delocalized molecular orbitals and transformed localized or- bitals for hexacene...... 115

xii 5.6 Shape of delocalized molecular orbitals and transformed localized or- bitals for undecacene...... 116

1 5.7 Vertical excitation energy of B2u state for acenes...... 117

1 5.8 Vertical excitation energy of Ag state for acenes...... 121

1 5.9 Dominant configuration contributions for excited Ag state...... 122

5.10 Excitation energy of T2 and its relative position to S1 ...... 126

6.1 Potential energy surfaces of H3 around a conical intersection..... 132

6.2 Potential energy surfaces of NH3 around its D3h conical intersection. 133

6.3 Locations of conical intersections for NH3 with C2v symmetry.... 135

xiii List of Abbreviations and Symbols

Symbols

i, j, k, l Indices for occupied molecular orbitals.

a, b, c, d Indices for unoccupied molecular orbitals.

p, q, r, s Indices for general molecular orbitals.

σ, τ Indices for spins.

: ap Creation operator on orbital p.

ap Annihilation operator on orbital p.  Orbital energy.

|ny Dirac notation for n’th electronic state.

|pq ¨ ¨ ¨ y Slater determinant denoting a single anti-symmetrized configu- ration.

xpq|rsy Physicists’ notation for two-electron integral

xpq||rsy Physicists’ notation for anti-symmetrized two-electron integral

Abbreviations

AO atomic orbital

BS broken symmetry

BSSE basis set superposition error

CASSCF complete-active-space self-consistent field

CASPT2 complete-active-space second-order perturbation theory

CASCI complete-active-space configuration interaction

xiv CC coupled cluster

CCD coupled cluster doubles

CCSD coupled cluster singles and doubles

CCSD(T) coupled cluster singles and doubles with perturbative triples

CI configuration interaction

CIS configuration interaction singles

CISD configuration interaction singles and doubles

CISDQ configuration interaction singles, doubles and quadruples

CT charge transfer

DCC dominant configuration contribution

DFT density functional theory

DFA density functional approximation

DEA-EOM-CC double-electron-affinity equation-of-motion coupled cluster

DIP-EOM-CC double-ionization-potential equation-of-motion coupled cluster

DMRG density matrix renormalization group

EA-EOM-CC electron-affinity equation-of-motion coupled cluster

EOM equation-of-motion

EOM-CC equation-of-motion coupled cluster

EOM-CCSD equation-of-motion coupled cluster singles and doubles

FS fractional spin

FSCC Fock-space coupled cluster

FCI full configuration interaction

GGA generalized gradient approximation

HF Hartree-Fock

HMO H¨uckel molecular orbital

HOMO highest occupied molecular orbital

xv IP-EOM-CC ionization-potential equation-of-motion coupled cluster

KS-DFT Kohn-Sham density functional theory

LDA local density approximation

LR linear response

LR-CC linear response coupled cluster

LR-TDDFT linear response time-dependent density functional theory

LR-TDHF linear response time-dependent Hartree-Fock

LUMO lowest unoccupied molecular orbital

MBPT many-body perturbation theory

MO molecular orbital

MF mean field

MC-SCF multi-configurational self-consistent field

MRCI multi-reference configuration interaction

MRCC multi-reference coupled cluster

ph particle-hole

pp particle-particle

ph-RPA particle-hole random phase approximation

ph-TDA particle-hole Tamm-Dancoff approximation

pp-RPA particle-particle random phase approximation

pp-TDA particle-particle Tamm-Dancoff approximation

RDFT restricted density functional theory

RHF restricted Hartree-Fock

RPA random phase approximation

SCF self-consistent field

SF-CI spin-flip configuration interaction

SF-EOM-CC spin-flip equation-of-motion coupled cluster

xvi SF-TDDFT spin-flip time-dependent density functional theory

SOMO singly occupied molecular orbital

ST singlet-triplet

TDA Tamm-Dancoff approximation

TDDFT time-dependent density functional theory

TDDFT-P time-dependent density functional theory with pairing field

TDHF time-dependent Hartree-Fock

TEA-EOM-CC triple-electron-affinity equation-of-motion coupled cluster

TIP-EOM-CC triple-ionization-potential equation-of-motion coupled cluster

TMM trimethylenemethane

UCC unrestricted coupled cluster

UDFT unrestricted density functional theory

UHF unrestricted Hartree-Fock

VB valence bond

XC exchange-correlation

xvii Acknowledgements

I want to thank my advisor, Dr. Weitao Yang, for his support. He is a great scientist with knowledge, creativity, passion, and optimism. Although he is sometimes slow, he always keeps being rigorous. In the past five years, I learned much knowledge as well as a good scientific attitude from him. He also cares for students. I had several sever illness experiences in the last few years — shingles, iritis, and ulnar nerve surgery. Each time he asked me much about the illness and is always ready to help. He also cares about our future careers and helped me a lot in finding a post-doc position. Then I want to thank three professors, Dr. Ernest Davidson, Dr. Jianfeng Lu, and Dr. Kieron Burke. They are also great scientists and I had great collaboration with them. Among them, Dr. Davidson influenced me most. Although he is about 80 years old, he keeps being active. He is even more rigorous than Weitao. He always wants to know the answer to all puzzles and wants to give explanations to all phenomena. Therefore, we had a lot of email discussions. These discussions on some details delayed the publication of some research, but led us to a better understanding of the nature. I am also grateful to my labmates and collaborators. Dr. Helen van Aggelen and Dr. Degao Peng helped me most. They led me into the field and then we worked together with Weitao and finally got to the current stage of understanding of the pp-RPA. Furthermore, Degao is a very good friend and we had a lot of discussions

xviii on many things. Besides, Du Zhang, Neil Shenvi, Stephan Steinmann, Lin Shen, Christopher Sutton, Shubin Liu, and Adriel Dominguez are also great collaborators, and it has always been inspiring to work with them. Last but not least, I want to thank my family, especially my parents. In fact, for the past five years, they can help me very little because I cannot see them much and they know almost nothing about quantum chemistry. However, they have always been asking me about my life, my research, and my health. I sometimes feel bored and do not want to answer their questions, but I know, even though they cannot help, they do care.

xix 1

Introduction

1.1 Quantum Chemistry and Exact Hamiltonian

Quantum chemistry is a subject that uses quantum mechanics to study chemical and physical phenomena in atomic and molecular systems. It is now widely applied to many areas of chemistry and has been a powerful tool for predicting physical properties and chemical reactions, as well as explaining reaction mechanisms. As a heritage from classical mechanics, the Hamiltonian, which corresponds to the total energy of a system, is one of the most widely used keys to all quantum mechanics and quantum chemistry problems. Although it took hundreds of years to find the door to the world of quantum, it only took a few years for those brilliant scholars to figure out those keys — the matrix mechanics in 1925 by Heisenberg, Born, and Jordan [1,2], the electron spin in 1925 by Uhlenbeck and Goudsmit [3,4], the wave mechanics and Schr¨odingerequation in 1926 by Schr¨odinger[5,6,7,8], and the relativistic Dirac equation in 1928 by Dirac [9]. All of these progress provided a clearer and clearer picture of the exact quantum Hamiltonian. By 1929, the world of quantum has been so clear that Dirac claimed with great confidence that [10]

1 “The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble. It therefore becomes desirable that approximate practical methods of applying quantum mechanics should be developed, which can lead to an explanation of the main features of com- plex atomic systems without too much computation.” Ever since then, this prophetic statement has been cited thousands of times and now it is still true. Quantum chemistry, as an essential quantum many-body problem for atomic and molecular systems, has always been struggling its way towards higher and higher accuracy for larger and larger systems. Considering the exact solution based on exact Hamiltonian is still not reachable (or may never be reachable), quantum chemists live on approximations!

1.2 Two Main Approximations in Electronic Structure Theory

1.2.1 Non-relativistic Approximation

One important approximation is the non-relativistic approximation. Although the relativistic quantum chemistry is an important branch of quantum chemistry, in most molecular and biological systems that people are more interested in, the relativistic effect is not important. It is well known that relativistic effects become significant only when the speed of a particle is comparable to the speed of light, but for most elements involved in regular molecular and biological systems, their nuclear charges are not high enough to make their electrons travel so fast that is comparable to the speed of light. In the absence of any relativistic effect, the exact non-relativistic

2 Hamiltonian is (in atomic unit)

M 1 1 M Z Z 1 N 1 N 1 M N Z Hˆ “ ´ ∇2 ` A B ´ ∇2 ` ´ A , 2m A 2 |R ´ R | 2 i 2 |r ´ r | |R ´ r | A A A‰B A B i i‰j i j A i A i ÿ ÿ ÿ ÿ ÿ ÿ (1.1) in which M is the number of nuclei, N is the number of electrons, mA is the ratio of the mass of nucleus A to the mess of an electron, ZA is the nuclear charge of nucleus A, RA is the coordinate of nucleus A, and ri is the coordinate of electron i. These five terms are the kinetic term for nuclei, Coulomb repulsion term between nuclei, kinetic term for electrons, Coulomb repulsion term between electrons, and Coulomb attraction term between nuclei and electrons, respectively. It should be noted that although this is a non-relativistic Hamiltonian, those components that naturally arise in relativistic Dirac equation, especially the spin-orbit coupling effect, can be manually added into this Hamiltonian when they need to be considered.

1.2.2 Born-Oppenheimer Approximation

Another important approximation is the Born-Oppenheimer approximation. It is the basic approximation for pure electronic structure studies and in fact most relativistic calculations also adopt this approximation. Since most nuclei are much heavier than electrons and therefore move much more slowly, the electrons can be seen as moving in the field of fixed nuclei. In this sense, the degrees of freedom for electrons and nuclei can be separated. For the electrons alone, the exact non-relativistic Hamiltonian is

1 N 1 N 1 M N Z Hˆ “ ´ ∇2 ` ´ A . (1.2) ele 2 i 2 |r ´ r | |R ´ r | i i‰j i j A i A i ÿ ÿ ÿ ÿ

There are cases when the Born-Oppenheimer approximation with separated de- gree of freedom for electrons and nuclei is questionable, especially in cases when the nuclei is light, for example the proton, or when two or more electronic states

3 have similar energy. This breaking of Born-Oppenheimer approximation is often considered in non-adiabatic dynamics, which is another important field of quantum chemistry.

1.2.3 Basic Equations in Electronic Structure Theory

The quantum chemistry with both non-relativistic approximation and Born-Oppenheimer approximation is also often called electronic structure theory. It can deal with both time-dependent and time-independent problems. Now, we simply notate the Hami- tonian in Eq.(1.2) as Hˆ , then the time-dependent Schr¨odingerequation is

B i Ψptq “ Hˆ ptqΨptq, (1.3) Bt where Ψptq is the total wave function for all electrons at time t. If we are interested in the stationary states that are eigenfunctions of the Hamiltonian, we can solve the time-independent Schr¨odingerequation

ˆ HΨn “ EnΨn, (1.4)

where En is the energy of the n’th stationary state. Solving the time-independent Schr¨odingerequation and obtaining the energy for ground and electronic excited states are the main tasks of the community. As is suggested by Dirac, approximate practical methods are highly desirable, and in the past 90 years, a number of approximations have been developed.

1.3 Methods for Ground State

1.3.1 Valence Bond Theory

There are two main branches of theory to solve the time-independent Schr¨odinger equation. One is the valence bond (VB) theory, which has a deep connection with and also go beyond the famous octet rule and Lewis structure [11]. It can depict the

4 electronic nature of chemical bonds and was very popular in earlier days. Although the VB theory was later surpassed by the molecular orbital (MO) theory in computa- tion, it is still a powerful tool for qualitative analysis in organic chemistry. Moreover, recent years has seen a renaissance of the VB theory with a new face named “modern valence bond theory” [12].

1.3.2 Molecular Orbital Theory

The second branch is the popular MO theory. It basically assumes that there exist MOs, and these MOs can be generated by a linear combination of atomic orbitals (AOs). In this branch, there are famous semi-empirical H¨uckel molecular orbital (HMO) theory [13], first-principle Hartree-Fock (HF) theory [14, 15, 16, 17], and the Kohn-Sham density functional theory (KS-DFT) [18, 19].

H¨uckelMolecular Orbital Theory

The early HMO theory is used to deal with π electrons in conjugated molecules. In the basis of atomic pz electron, it simply assumes that all the diagonal element of the Hamiltonian matrix are all equal to α. For the off-diagonal elements, if the two AOs involved are spatially adjacent, then it is assumed to be β, otherwise zero. Although being a very crude semi-empirical theory, it could already unveil the qualitative picture of molecular orbitals, which can be delocalized to the whole molecule.

Hartree-Fock and Wave Function Methods

HF employs the exact Hamiltonian in Eq. (1.2) and approximates the wave function in Eq. (1.4) by a single Slater determinant, and the determinant is determined by minimizing the total energy [14, 15, 16, 17]. It takes into account the Pauli’s principle, or the antisymmetric nature of Fermions. However, it is essentially a mean field (MF) theory and assumes the electrons are independent from each other except

5 for exchange interaction. Its MOs are defined as the eigenfunctions of the MF one- particle Hamiltonian and assumed to be filled according to the Aufbau principle. As a MF theory, HF is far from being exact. It lacks electron correlation and the differences between the HF energy and the exact energy is defined as the correlation energy. In fact, the pursuit for a better description of electron correlation is center to all ground state calculations. Many methods have been derived directly on top of HF theory. They keep using the exact Hamiltonian and then improve on the approximated wave function. There- fore, these methods are often referred to as ab initio wave function methods. They include regular single-reference based approaches such as configuration interaction (CI), many-body perturbation theory (MBPT), and coupled cluster (CC) theory, as well as multi-reference methods such as multi-configurational self-consistent field (MC-SCF), multi-reference configuration interaction (MRCI), complete-active-space self-consistent field (CASSCF), complete-active-space second-order purterbation the- ory (CASPT2), and multi-reference coupled cluster (MRCC) theory. The accuracy of these ab initio wave function methods can be systematically

improved, but the cost is a main criticism. Even the cheapest HF scales as OpN 4q, with N the number of basis functions. The full configuration interaction (FCI) is in principle exact, but unfortunately, its huge computational cost with a factorial scaling limits its application to very small systems. Nowadays, the coupled cluster singles and doubles with perturbative triples (CCSD(T)) is considered as the golden standard for computational chemistry. Besides, the slow basis set convergence be- havior for ab initio wave function methods necessitates the use of a large basis set, which makes them more computationally demanding.

6 Density Functional Theory

The density functional theory (DFT) [18] was introduced in 1960s and became pop- ular in 1980s. It is based on a one-to-one mapping between the electron density and external potential. Similar to the wave function methods, DFT also contains different variations. The KS-DFT [19] is the most important variation. In KS-DFT, the density of the real system is assumed to be represented by the density of a non-interacting system, and a mapping between an imaginary non-interacting world and the real world is established. This mapping essentially frees people from using the exact Hamiltonian (Eq. (1.2)). KS-DFT mostly employs a single Slater determinant to describe the imaginary non-interaction system. It is also a MF theory but the correlations are built in the MF Hamitonian. Similar to HF, the MOs for KS-DFT are defined as the eigenfunction of the one-particle Hamitlonian. Despite the fact that neither the Hamiltonian nor the “wave function” is exact, KS-DFT is probably the most popular method in electronic structure theory thanks to its good balance between accuracy and computational cost. The essence of KS-DFT is approximating the exchange-correlation (XC) energy functional, and it is truly the most difficult part. Up till now, there are five levels of density functional approximations (DFAs) for the XC functional, which are also referred to as the Jacob’s ladder in DFT [20]. The first rung is the local density ap- proximation (LDA), which simply builds a functional mapping from the local density

ρprq to the XC energy. The second rung is the generalized gradient approximation (GGA). It adds the density gradient ∇ρprq information as a functional variable. The third rung is meta-GGA, which adds the second derivative of the density ∇2ρprq

N 2 as well as the kinetic energy density ´1{2 i |∇φiprq| . The fourth rung is hybrid functional that starts to use occupied orbitalsř in the XC functional. They mostly

7 include some percentage of HF exchange. The fifth rung includes those functionals that even adopts unoccupied orbtials. For example, some portion of the MP2 cor- relation expression can be added. However, the most famous method in the fifth level is the random phase approximation (RPA) that includes a particle-hole (ph) channel and a particle-particle (pp) channel, and the particle-particle random phase approximation (pp-RPA) will be the focus of this thesis. It is worth noting that another variation of DFT is the orbital-free DFT. It is closely related to the Thomas-Fermi model and directly approximates the kinetic energy functional from the real-space electron density. The orbital-free DFT avoids the use of orbitals and is often so cheap that can be used in dynamics calculations in large systems. However, with a poorly-behaved kinetic functional, orbital-free DFT are also less accurate than KS-DFT with even being unable to reproduce any electronic shell structure.

1.4 Methods for Excited States

Many ground state methods can also treat excited states. They include CI, MRCI, CASSCF, CASPT2, etc. Nonetheless, there are also some techniques that are specif- ically designed for excited states.

1.4.1 Delta-SCF

The ∆SCF usually combines with HF or DFT. Instead of letting the electrons occu- pying the lowest molecular orbitals and form the Aufbau configuration, ∆SCF allows non-Aufbau occupations and interprets them as excited states. However, many non- Aufbau configurations are open-shell and therefore suffers from the broken symmetry (BS) problem for spins. Fortunately, after some spin correction or spin projections, ∆SCF can often give good excitation results.

8 1.4.2 Equation-of-Motion

Equation-of-motion (EOM) is another popular technique to treat excitations. It is based on the time-independent EOM

ˆ ˆ rH, Ons|0y “ pEn ´ E0q|ny, (1.5)

ˆ where On is the excitation operator that can excite the ground state |0y to the excited state |ny. EOM often combines with wave function methods, especially the coupled cluster theory. Now equation-of-motion coupled cluster (EOM-CC) theory is widely used as an accurate excitation method although it is limited to some small systems.

1.4.3 Linear Response Theory

Linear response (LR) theory is also widely used for excited state calculations. It is based on the linear response theory and often combines with HF, DFT, and CC. The resulted theories are linear response time-dependent Hartree-Fock (LR-TDHF, or TDHF for brevity), linear response time-dependent density function theory (LR- TDDFT, or TDDFT for brevity), and linear response coupled cluster (LR-CC) the- ory. TDDFT is currently most widely used in excited states calculations owing to its good balance of accuracy and computational cost.

1.5 Current Challenges in Electronic Structure Theory

1.5.1 Challenges for Ground State

Wave function methods and DFT are widely used in ground state calculations. Wave function methods are usually more accurate and can be systematically improved, however, they suffer from huge computational cost. HF scales as OpN 4q, MBPT2 scales as OpN 5q, CCSD scales as OpN 6q, and the golden standard CCSD(T) scales as OpN 7q, with N the number of basis functions. In general, if a method scales larger

9 than OpN 4q, it is considered not suitable for large systems. Therefore, wave function methods are usually limited to small systems. The cost of different KS-DFT approximations is also varied. LDA and GGA can

scale as OpN 3q, hybrid functionals scale as OpN 4q, while the RPAs formally scale as OpN 6q. Considering the cost and accuracy, some GGAs and hybrid functionals are more widely used in applications. However, there are two great challenges for DFT, which are fractional charge error and fractional spin error. They are also more known as the (de)localization error and the static correlation error. They have been the roots of many failures for DFT.[21] These failures include bond dissociation error, transition state error, charge transfer excitation error, band gap error, etc. Great efforts have been made to solve these problems. However, no DFT method, except the pp-RPA, can elegantly address both challenges. The pursuit for a better DFA is unabated.

1.5.2 Challenges for Excited States

Many wave function based methods can be used to treat excitations. Similar to their ground state description, they are usually accurate but computationally ex- pensive. Configuration interaction singles (CIS) and TDHF are the lowest level of theory. They are inexpensive and scales as OpN 4q for a single root but they are also less accurate. Configuration interaction singles and doubles (CISD) and configura- tion interaction singles, doubles and quadruples (CISDQ) are higher level of theory within the CI branch, however, they are very expensive and suffers from size con- sistency problem. In contrast, EOM-CC is size consistent and is more widely used, but EOM-CC is also expensive. The equation-of-motion coupled cluster singles and

doubles (EOM-CCSD) scales as OpN 6q for a single root. CASSCF and CASPT2 can solve ground and excited state problem in many systems that need multi-reference descriptions. However, they heavily depend on the size of the active space, and a

10 reasonably good active space usually leads to a huge computational cost. The combination of DFT and LR gives rise to TDDFT. TDDFT is the most widely used method for excited states calculations. Same to CIS and TDHF, TDDFT

scales as OpN 4q. Nonetheless, TDDFT is often much more accurate than CIS and TDHF. However, TDDFT also faces many challenges. First, within the conventional adiabatic approximation, TDDFT can only deal with single excitations. Second, inherited from erroneous DFAs, the fractional charge error and fractional spin error are also within TDDFT. Therefore, TDDFT cannot well describe charge transfer excitation or Rydberg excitation. Third, TDDFT is based on single reference DFT, in the species with multi-reference character such as diradicals, TDDFT also fails. In summary, wave function methods are accurate but expensive, while TDDFT is inexpensive but less accurate. A method that is both accurate and inexpensive is highly desirable.

11 2

Pairing Matrix Fluctuation for Ground State Correlation

This chapter is mainly adapted from the following journal articles,

• Helen van Aggelen, Yang Yang, and Weitao Yang, Exchange-correlation en- ergy from pairing matrix fluctuation and the particle-particle random phase approximation” Phys. Rev. A 88, 030501 (2013)

• Yang Yang, Helen van Aggelen, Stephan N. Steinmann, Degao Peng, and Weitao Yang, “Benchmark tests and spin adaptation for the particle-particle random phase approximation” J. Chem. Phys. 139, 174110 (2013)

2.1 Introduction

DFT has been widely used in many applications. However, the exact density func- tional is unknown and current DFAs face great challenges. These challenges mostly include the fractional charge error, the fractional spin error, and the inability to capture long-range interactions. Furthermore, it has been known that continuous functionals of the density or the density matrix are not able to describe the piecewise-

12 linear nature of the exact energy functional [22]. Therefore, the pursuit for better functionals that can overcome these great challenges is still a hot topic. Recently, the RPA [23] has attracted increasing interests as a DFA [24, 25, 26]. The RPA has its roots in many-body theories such as Green’s function theory [27, 28] as well as the coupled cluster theory [29, 30]. In DFT, the RPA represents a sophisticated functional, obtained when coupling the adiabatic connection [31, 32] with the fluctuation dissipation theorem [33]. Therefore, the RPA forms a connection between DFT and many-body methods. It is attractive for its lower computational

cost (OpN 4q with resolution of identity [34, 35]) compared to most correlated wave function methods, and because it overcomes some failures persistent in commonly used DFAs, i.e. the long-range dispersion interaction error [36, 37] and the static correlation error [38, 39]. Previously, the term “RPA” mainly referred to the well-known ph channel of the RPA, or the ph-RPA [24, 26, 40, 41], especially the direct ph-RPA which has no exchange. However, the pp channel, or the pp-RPA, which is a natural counterpart of the ph-RPA, has less been aware of by the quantum chemistry community. Re- cently, van Aggelen et al. introduced the pp-RPA to correlation energy calculations for atomic and molecular systems [42]. By coupling the adiabatic connection with the pairing matrix fluctuation, the pp-RPA provides an approximate correlation en- ergy [42]. The difference between ph-RPA and pp-RPA can also be viewed from a diagrammatic perspective, which identifies the ph- and the pp-RPA as the sum of all “ring” diagrams and all “ladder” diagrams, respectively [28] (Figure 2.1). As the summation of all ladder diagrams, the pp-RPA is equivalent to the ladder channel of coupled cluster doubles (ladder-CCD) [29, 43, 44]. The pp-RPA has many interesting features, most notably, in contrast to ph- RPA [39], it has virtually no delocalization error for general systems, in addition to virtually no static correlation error for single-bond systems. It thus satisfies the

13 Figure 2.1: Ring diagrams (upper row) and ladder diagrams (lower row) from the second-order to the fourth-order (left to right) [28]. With exchange, the second- order ring diagram is the same as the second-order ladder diagram and they are both exact. However, the direct ph-RPA, which we abbreviate as ph-RPA, does not contain exchange and therefore is not exact even to the second order.

flat-plane condition [22, 21, 42]. This suggests that the pp-RPA can be a source of inspiration for developing new density functionals.

2.2 Theory

2.2.1 Paring Matrix Fluctuation

The pairing matrix is defined as

N N κijptq ” xΨ0 |aHi ptqaHj ptq|Ψ0 y, (2.1)

N where |Ψ0 y is the N-electron ground state. The operator aHi ptq is the annihilation

i pHˆ ´νNˆq ´ i pHˆ ´νNˆq operators in the Heisenberg picture acting on orbital i, aHi ptq ” e ~ aie ~ and the term ´νNˆ, with ν the chemical potential, is added to the Hamiltonian to represent the grand potential. The N-electron state is the minimum under the new

14 Hamiltonian Hˆ ´ νNˆ with the particle number allowed to change. Because of the anti-commutation rule, we limit i ą j for simplicity. The dynamic pairing matrix fluctuation is defined as

¯ 1 Kijklpt ´ t q ”

iθ t t1 ΨN a t a t ΨN a a ΨN , a: t1 a: t1 ΨN a:a: ΨN ΨN , ´ p ´ qx 0 |r Hi p q Hj p q ´ x 0 | i j| 0 y Hl p q Hk p q ´ x 0 | l k| 0 y s| 0 y ` ˘ ´ ¯ (2.2)

1 N N with θpt´t q the Heaviside step function. Since the pairing matrix xΨ0 |aHi ptqaHj ptq|Ψ0 y “

N N xΨ0 |aiaj|Ψ0 y “ 0 in a regular non-superconducting system, this pairing matrix fluc- ¯ 1 ¯ R 1 tuation Kijklpt ´ t q is identical to the retarded Green function Kijklpt ´ t q

¯ R 1 i 1 N : 1 : 1 N ¯ 1 Kijklpt´t q ” ´ θpt´t qxΨ0 |raHi ptqaHj ptq, aH pt qaH pt qs|Ψ0 y “ Kijklpt´t q. (2.3) ~ l k

Fourier transform the pairing matrix to the energy domain,

8 ¯ 1 iEpt´t1q ¯ 1 KijklpEq “ dpt ´ t qe Kijklpt ´ t q ż´8 xΨN |a a |ΨN`2yxΨN`2|a:a: |ΨN y pxΨN |a:a: |ΨN´2yxΨN´2|a a |ΨN y “ 0 i j n n l k 0 ´ 0 l k n n i j 0 , E ωN`2 iη E ωN´2 iη n ´ n ` n ´ n ` ÿ ÿ (2.4)

N`2 N`2 N N´2 N N´2 where ωn “ En ´ E0 ´ 2ν, ωn “ E0 ´ En ´ 2ν, and η is an infinitesimal positive number. Therefore, the poles of the pairing matrix fluctuation K¯ pEq relate to the double electron addition and double electron removal energies.

2.2.2 Particle-Particle Random Phase Approximation

¯ 0 The pairing matrix fluctuation for a non-interacting reference KijklpEq is

¯ 0 θpi ´ F qθpj ´ F q θpF ´ iqθpF ´ jq KijklpEq “ δikδjl ´ δikδjl , (2.5) E ´ pi ` j ´ 2νq ` iη E ´ pi ` j ´ 2νq ` iη

15 where F denotes the Fermi level. The pairing matrix fluctuation for a real interacting system is unknown. We approximate it by the Dyson-like particle-particle random phase approximation,

K¯ pEq “ K¯ 0pEq ` K¯ 0pEqVK¯ pEq, (2.6) where V is defined as

Vijkl ”xij||kly ” xij|kly ´ xij|lky

˚ ˚ φ px1qφ px2qφkpx1qφlpx2q “ i j dx dx |r ´ r | 1 2 (2.7) ż 1 2 ˚ ˚ φ px1qφ px2qφlpx1qφkpx2q ´ i j dx dx . |r ´ r | 1 2 ż 1 2

Combing and reformulating Eqs. (2.4)(2.5)(2.6)(2.7) lead to the pp-RPA eigen- value equation AB X I 0 X “ ωN˘2 (2.8) B: C Y 0 ´I Y „  „  „  „  with

Aab,cd “δacδbdpa ` bq ` xab||cdy (2.9a)

Bab,kl “xab||kly (2.9b)

Cij,kl “ ´ δikδjlpi ` jq ` xij||kly (2.9c)

where a, b, c, d are particle indices and i, j, k, l are hole indices with restrictions that

a ą b, c ą d, i ą j and k ą l. Eigenvectors dominated by X components describe the

n N N`2 transition amplitudes of two-electron addition processes with Xab “ xΨ0 |aaab|Ψn y

n N N`2 and Yij “ xΨ0 |aiaj|Ψn y. Similarly, eigenvectors dominated by Y components

n describe the transition amplitudes of two-electron removal processes with Xab “

N : : N´2 n N : : N´2 xΨ0 |abaa|Ψn y and Yij “ xΨ0 |ajai |Ψn y.

16 2.2.3 Adiabatic Connection

The adiabatic connection is often used to connect a non-interacting reference system with the real interacting system. The non-interacting Hamiltonian is

ˆ ˆ H0 “ h ` uˆ0, (2.10)

ˆ where h is the core Hamiltonian, andu ˆ0 is the effective one-body potential. A pa- rameter λ denoting the interaction strength is then introduced into the Hamiltonian.

ˆ ˆ ˆ Hλ “ H0 ` λV ` uˆλ. (2.11)

The operatoru ˆλ is restricted to satisfyu ˆ1 “ 0. Therefore, when λ “ 0, the Hamilto- nian is the non-interacting Hamitonian, and when λ “ 1, it is the Hamiltonian for the real system. According to the Hellmann-Feynman theorem

1 BE E1 ´ E0 “ dλ Bλ ż0 1 BHˆ “ xΨλ| λ |Ψλydλ (2.12) Bλ ż0 1 Buˆ “ xΨλ|Vˆ ` λ |Ψλydλ Bλ ż0

ˆ Buˆλ Because V is a two-body operator while Bλ is a one-body operator, Eq. (2.12) can be rewritten in terms of the second-order density matrix Γλ and the first-order density matrix γλ for the system with interaction strength λ,

1 1 Bu E1 ´ E0 “ tr VΓλdλ ` tr λ γλdλ. (2.13) Bλ ż0 ż0 Since the energy for the non-interacting systems is

0 0 0 E “ tr hγ ` tr u0γ , (2.14) 17 the energy for the fully interacting system is

1 1 Bu E1 “ tr hγ0 ` tr VΓλdλ ` tr u γ0 ` tr λ γλdλ. (2.15) 0 Bλ ż0 ż0 Because the correlation energy is defined as the difference between the exact energy and the HF energy functional, which is

EHF “ tr hγ0 ` tr VΓ0, (2.16) the correlation energy is

Ec ”E1 ´ EHF (2.17) 1 1 Bu “tr VpΓλ ´ Γ0qdλ ` tr u γ0 ` tr λ γλdλ. 0 Bλ ż0 ż0 By assuming the potential is local and keeping the electron density constant along the adiabatic connection path, the last two terms in Eq. (2.17) become

1 Bu tr u γ0 ` tr λ γλdλ 0 Bλ ż0 1 Bu “tr u ρ ` tr λ ρdλ 0 Bλ (2.18) ż0

“tr u0ρ ` tr u1ρ ´ tr u0ρ

“0.

Therefore, the correlation energy is simply

1 Ec “ tr VpΓλ ´ Γ0qdλ. (2.19) ż0 2.2.4 Correlation Energy from Pairing Matrix Fluctuation

The second-order density matrix can be written with the transition pairing matrix

n,N´2 N´2 N elements χij “ xΨn |aiaj|Ψ0 y through the completeness of the N ´ 2 electron 18 wave function basis,

N : : N Γijkl “ xΨ0 |akal ajai|Ψ0 y

N : : N´2 N´2 N “ xΨ0 |akal |Ψn yxΨn |ajai|Ψ0 y

n,N´2 n,N´2 ˚ “ χji pχlk q , n ÿ and therefore the correlation energy can be written as

1 c n,N´2 n,N´2 ˚ n,N´2 n,N´2 ˚ E “ pχλ qjipχλ qlk ´ pχ0 qjipχ0 qlk Vijkldλ. n ijkl 0 ÿ ÿ ż ´ ¯

Using the residue theorem, it can be further related to the numerator of the pairing matrix fluctuation

´1 1 `i8 Ec “ eEηtr VrK¯ λpEq ´ K¯ 0pEqsdEdλ 2πi ż0 ż´i8 2.2.5 Correlation Energy from Particle-Particle Random Phase Approximation

The correlation energy obtained from pairing matrix fluctuation in combination with adiabatic connection is in principle exact. However, the pairing matrix fluctuation

K¯ λpEq for an interacting system with interaction strength λ is unknown. The pp- RPA Dyson-like equation is further extended to these systems by assuming

KλpEq “ K¯ 0pEq ` λK¯ 0pEqVK¯ λpEq. (2.20)

19 Therefore, the correlation energy based on pp-RPA is

´1 1 `i8 Ec “ tr rK¯ λpEqV ´ K¯ 0pEqVsdEdλ pp 2πi ż0 ż´i8 ´1 1 `i8 “ λtr rK¯ 0pEqVK¯ 0pEqVs ` λ2tr rK¯ 0pEqVK¯ 0pEqVK¯ 0pEqVs ` ... dEdλ 2πi 0 ´i8 ż ż ´ ¯ ´1 1 `i8 8 “ λn´1tr rpK¯ 0VqnsdEdλ 2πi 0 ´i8 n“2 ż ż ÿ ´1 `i8 8 1 1 “ pλqntr rpK¯ 0VqnsdE 2πi n 0 ´i8 n“2 ż ” ÿ ı 1 `i8 8 1 “ ´ tr rpK¯ 0VqnsdE 2πi n ´i8 n“2 ż ÿ 1 `i8 “ tr rlnpI ´ K¯ 0Vq ` K¯ 0VsdE. 2πi ż´i8

Note that the convergence factors eEη is no longer needed here, because no first-order poles are included. This equation can be used to obtain correlation energies through numerical integration. However, it can be reformulated [28] into a more elegant form and provide correlation energies by solving the pp-RPA working equation (2.8). The final form is

Npp c N`2 Epp “ ωn ´ tr A (2.21a) n ÿ

Nhh N´2 “ ´ ωn ´ tr C (2.21b) n ÿ N 1 pp 1 1 Nhh 1 “ ωN`2 ´ tr A ´ ωN´2 ´ tr C, (2.21c) 2 n 2 2 n 2 n n ÿ ÿ where Npp is the number of particle-particle pairs and Nhh is the number of hole-hole pairs. These equations are also employed for later correlation energy calculations.

20 2.2.6 Spin Separation and Spin Adaptation

In order to obtain the correlation energy, the eigenvalue problem (2.8) needs to be solved. The direct way is through matrix diagonalization. However, the dimension

2 1 1 of the pp-RPA matrix is OpN q, or more specifically, 2 NppNp ´ 1q ` 2 NhpNh ´ 1q. Therefore, the cost for diagonalization is OpN 6q. This is the most expensive step for pp-RPA calculation, and hence the formal cost for pp-RPA is OpN 6q. Furthermore, direct diagonalization also requires large memory to store the pp-RPA matrix, which scales as OpN 4q. Considering the huge memory and time cost, a simplification is desirable. In this section, a simplification based on spin analysis is presented. No matter for the particle-particle part or the hole-hole part, there are three different types of spin pairs — αα, ββ and αβ (βα pairs are absent because of index restrictions). In Eq. (2.9), matrix elements with different types of spin pairs are all zero, which naturally leads to the spin separated pp-RPA matrix

Aαα,αα 0 0 Bαα,αα 0 0 0 Aαβ,αβ 0 0 Bαβ,αβ 0 » fi 0 0 Aββ,ββ 0 0 Bββ,ββ : . —Bαα,αα 0 0 Cαα,αα 0 0 ffi — : ffi — 0 B 0 0 Cαβ,αβ 0 ffi — αβ,αβ ffi — 0 0 B: 0 0 C ffi — ββ,ββ ββ,ββ ffi – fl Then the eigenvalue problem can be decomposed into three independent eigenvalue problems

Aspin Bspin Xspin I 0 Xspin : “ ωspin , (2.22) B C Yspin 0 ´I Yspin „ spin spin „  „  „  with spin “ pαα, ααq or pαβ, αβq or pββ, ββq, and the matrix elements are

rAαα,ααsab,cd ” Aaαbα,cαdα “δacδbdpa ` b ´ 2νq ` xab||cdy

rAαβ,αβsab,cd ” Aaαbβ ,cαdβ “δacδbdpa ` b ´ 2νq ` xab|cdy

rAββ,ββsab,cd ” Aaβ bβ ,cβ dβ “δacδbdpa ` b ´ 2νq ` xab||cdy,

21 rBαα,ααsab,ij ” Baαbα,iαjα “xab||ijy

rBαβ,αβsab,ij ” Baαbβ ,iαjβ “xab|ijy

rBββ,ββsab,ij ” Baβ bβ ,iβ jβ “xab||ijy,

rCαα,ααsij,kl ” Ciαjα,kαlα “ ´ δikδjlpi ` j ´ 2νq ` xij||kly

rCαβ,αβsij,kl ” Ciαjβ ,kαlβ “ ´ δikδjlpi ` j ´ 2νq ` xij|kly

rCββ,ββsij,kl ” Ciβ jβ ,kβ lβ “ ´ δikδjlpi ` j ´ 2νq ` xij||kly,

The final eigenvalue set is the union of the three matrices’ eigenvalue sets

ω ” ωαα,αα Y ωαβ,αβ Y ωββ,ββ, (2.26) and the trace of the original A (or C) matrix is the sum of the traces of the three smaller Aspin (or Cspin) matrices

TrA “ TrAspin, (2.27a) spin ÿ

TrC “ TrCspin. (2.27b) spin ÿ

Therefore, the correlation energy in Eq.(2.21) can be written as

E “ ωN`2 ´ TrA “ ´ ωN´2 ´ TrC , c nspin spin nspin spin (2.28) spin n spin n ÿ ” ÿspin ı ÿ ” ÿspin ı

The spin separation can be implemented in both restricted and unrestricted cases. For restricted closed-shell singlet cases, the eigenvalue problem can be further sim- plified by using spin-adapted particle-particle and hole-hole pairs.

22 The αα pairs and ββ pairs occur only as triplet pairs, while αβ pairs can combine into either singlet or triplet pairs,

1 1 : : : : Singlet : ? ppαqβ ` qαpβq (2.29a) 2 1 ` δpq 1 a Triplet : ? pp: q: ´ q: p: q, (2.29b) 2 α β α β where p and q are generic orbitals that can be both occupied or both unoccupied. Singlet and triplet pair annihilations can be constructed in a similar manner. The spin adapted pp-RPA αβ matrix for the restricted case is obtained by dividing the αβ pairs into blocks according to their spin multiplicity:

As 0 Bs 0 0 At 0 Bt » : fi Bs 0 Cs 0 — 0 B: 0 C ffi — t t ffi – fl and the decoupled eigenvalue problem is

Amult Bmult Xmult I 0 Xmult : “ ωmult , (2.30) B C Ymult 0 ´I Ymult „ mult mult „  „  „  where the multiplicity mult is either singlet (s) or triplet (t). The elements in the triplet matrix are

rAtsab,cd “δacδbdpa ` b ´ 2νq ` xab||cdy

rBtsab,ij “xab||ijy

rCtsij,kl “ ´ δikδjlpi ` j ´ 2νq ` xij||kly, with the restriction that a ą b, c ą d, i ą j and k ą l. This triplet eigenvalue problem is the same as the αα and ββ cases and therefore gives the same eigenvalue set

ωt “ ωαα,αα “ ωββ,ββ (2.32a)

TrAt “ TrAαα,αα “ TrAββ,ββ (2.32b)

23 The elements in the singlet matrix are 1 rAssab,cd “δacδbdpa ` b ´ 2νq ` pxab|cdy ` xab|dcyq (2.33a) p1 ` δabqp1 ` δcdq 1 a rBssab,ij “ pxab|ijy ` xab|jiyq (2.33b) p1 ` δabqp1 ` δijq a 1 rCssij,kl “ ´ δikδjlpi ` j ´ 2νq ` pxij|kly ` xij|lkyq, (2.33c) p1 ` δijqp1 ` δklq a with the restriction that a ě b, c ě d, i ě j and k ě l. The linear combination to generate spin-adapted pairs is a unitary transforma- tion. Consequently, the eigenvalues and the traces do not change — the αβ eigenvalue set is the union of the singlet and triplet eigenvalue sets

ωαβ,αβ ” ωs Y ωt, (2.34)

and the trace of Aαβ,αβ (or Cαβ,αβ) matrix equals the sum of the traces of the singlet

As (or Cs) and triplet At (or Ct)

TrAαβ,αβ “ TrAs ` TrAt (2.35a)

TrCαβ,αβ “ TrCs ` TrCt. (2.35b)

Therefore, the correlation energy in Eqs. (2.21) and (2.28) can further be expressed as

N`2 N`2 Ec “3p ωt,n ´ TrAtq ` ωs,n ´ TrAs n n ÿ ÿ (2.36) N´2 N´2 “3p´ ωt,n ´ TrCtq ´ ωs,n ´ TrCs. n n ÿ ÿ Solving the spin-separated or spin-adapted equations and obtaining the correlation energy through Eqs. (2.28) and (2.36) is several times faster than using the original spin-unseparated formulation.

24 2.3 Benchmark Results

2.3.1 Computational Details

The pp-RPA with spin separation and spin adaptation is implemented in QM4D [45]. The initial self-consistent field (SCF) HF or KS-DFT calculation were also computed using the QM4D package. The PBE reference [46] was used for the G2/97 enthalpies of formation database and the DBH24 reaction barrier database. Both the PBE and HF references were tested and compared for the nonbonded interaction databases. Basis set convergence was tested along the cc-pVXZ series, X=D, T, Q, 5, for se- lected systems in G2/97, and with the aug-cc-pVXZ series, X=D, T, Q, for selected reaction barriers. The basis set convergence of pp-RPA for non-bonded interactions was assessed with aug-cc-pVXZ series, X=D, T, Q (spherical harmonic atomic or- bitals) in a locally modified version of CFOUR [47]. After balancing the accuracy and the computational cost, calculations for G2/97, DBH24 and nonbonded interac- tions adopted the cc-pVTZ, aug-cc-pVTZ and aug-cc-pVDZ basis sets, respectively. The QM4D program uses Cartesian atomic orbitals and removes basis functions with angular momentum higher than “f”. For enthalpies of formation and reaction bar- riers tests, we used cc-pVXZ-RI auxiliary basis sets (basis functions with angular momentum higher than “f” also truncated) in the post-KS pp-RPA to facilitate the atomic orbital to molecular orbital two-electron integral transformation. The ph-RPA calculations were carried out in the same way, except that no RI auxiliary basis sets were used. Geometries for the G2/97 benchmark set are taken from Ref. [48], which were optimized using the MP2(full)/6-31G* method. Geometries for the DBH24 set are taken from Ref. [49]. Geometries for the HB6/04, CT7/04, DI6/04 and WI9/04 nonbonded interaction sets are taken from Ref. [50].

25 -225 -58450 -150 -47800

ph Atomization E

ph Atomization E

pp Atomization E

-58500 pp Atomization E

ph Total E -250 -175 -47850

ph Total E

pp Total E

pp Total E

-58550

-275 -200 -47900

-58600

-300 -225 -47950

-58650

-325 -250 -48000

-58700 Total energy (kcal/mol) Total energy (kcal/mol)

-350 -275 -48050

-58750 Atomization energy (kcal/mol) Atomization energy (kcal/mol)

-375 -58800 -300 -48100

2 3 4 5 2 3 4 5

X in cc-pVXZ X in cc-pVXZ

Figure 2.2: Basis set convergence for the total energy and atomization energy of

HCN (left) and H2O(right). The total energies for HCN and H2O converge very slowly with respect to the basis set. The atomization energies, which are plotted with equal intervals between tick marks on the left axis, converge much faster and can be considered converged with the cc-pVTZ basis set.

2.3.2 Enthalpies of Formation Basis Set Convergence Test

We choose HCN and H2O to assess the basis set convergence of total and atomization energies (Figure 2.2). For both molecules and both types of RPA, the total energy converges slowly and shows as large as « 10 kcal/mol and 20 kcal/mol differences between the cc-pVQZ and the cc-pV5Z basis sets for the pp-RPA and ph-RPA, re- spectively. By contrast, due to systematic error cancellation, the atomization energy shows about or less than 5 kcal/mol differences between the cc-pVTZ and the cc- pVQZ basis sets. Considering the balance of accuracy and large computational cost, we adopted the cc-pVTZ basis set for benchmarking enthalpies of formation.

Results

Enthalpies of formation allow for a direct comparison with experimental results and are therefore often used to benchmark electronic structure methods. The closely re- lated zero-point-energy-free atomization energies are somewhat more straightforward to compare to high-level computations and it is therefore customary to report both.

26 In the present work, we investigate the performance of the pp- and ph-RPA for the atomization energies and enthalpies of formation for the G2/97 database [51, 52].

The smallest molecule of G2/97 is H2 and the largest in terms of atoms and number of electrons, are C4H10 and SiCl4, respectively. Among the whole set, the maximum error (MaxE) for the pp-RPA is -31.1 kcal/mol (C2F4), whose absolute value is half of that for the ph-RPA (63.2 kcal/mol,

SiF4). The mean signed error (MSE) for the pp-RPA is -1.9 kcal/mol which is much smaller than that for the ph-RPA (21.7 kcal/mol). The mean unsigned error (MUE) is 8.3 kcal/mol and 21.7 kcal/mol for the pp-RPA and ph-RPA. These indi- cate that the error for the pp-RPA is fluctuating around the reference values while the ph-RPA systematically overestimates enthalpies of formation. This behavior also emerges when we plot the signed error for both RPAs with respect to the number of atoms in a molecule (Figure 2.3). In contrast to the ph-RPA, the pp-RPA enthalpies of formation show no systematic drift with respect to the number of atoms. The systematic underbinding of the ph-RPA is well known, although previous conclu- sions were based on data limited to small molecules [26, 53, 54, 55]. This error has been ascribed to the ph-RPA’s insufficient description of the short-range correlation, which may be important when the number of electron pairs changes [25]. Since the number of atoms in a molecule is roughly correlated to the number of electron pairs formed, the increasing error in Figure 2.3 is in agreement with this argument. The G2/97 database is often divided into the G2-1 (small molecules) and G2-2 (large molecules) subsets. The G2-2 subset can be further divided into 5 subsets, namely non-hydrogen systems, hydrocarbons, substituted hydrocarbons, inorganic hydrides and radicals. Results are shown in Table 2.1. For the small molecules of the G2-1 subset, the ph-RPA has a relatively low error (MUE = 10.9 kcal/mol) compared to the remaining subsets. The pp-RPA is already much better than the ph-RPA for this set (MUE = 6.8 kcal/mol). Neither RPA performs well for non-

27 pp-RPA error

60

ph-RPA error

30

0 Error (kcal/mol)Error

-30

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Number of atoms Figure 2.3: Signed error with respect to number of atoms for each species in G2/97 enthalpies of formation. Red dots stand for the pp-RPA and black-white squares stand for the ph-RPA. The cc-pVTZ basis set was adopted. The pp-RPA shows smaller errors within about ˘30 kcal/mol and the error has a nearly constant trend with respect to number of atoms, while the ph-RPA has errors within about 0-60 kcal/mol and growing with respect to number of atoms.

hydrogen systems: the pp-RPA underestimates the enthalpies of formation by 14.6 kcal/mol while the ph-RPA overestimates them by 35.9 kcal/mol. For the two or- ganic subsets (hydrocarbons and substituted hydrocarbons), the pp-RPA performs significantly better than the ph-RPA (MUE = 8 and 27 kcal/mol, respectively). De- tailed inspections show that the pp-RPA is good at describing non-cyclic organic compounds. For (strained) cyclic compounds such as methylene cyclopropane, bicy- clobutane and spiropentane, the error reaches over 10 kcal/mol. For aromatic cyclic systems such as benzene, furan and pyridine, the error even exceeds 20 kcal/mol. When dividing the G2/97 database into three exclusive subsets for molecules con-

28 Table 2.1: Mean Signed Errors (MSE) and Mean Unsigned Errors (MUE) (in kcal/mol) of pp-RPA and ph-RPA for subsets in G2/97

MSE MUE subset pp-RPA ph-RPA pp-RPA ph-RPA G2-1 4.7 10.8 6.8 10.9 Non-hydrogen systems -14.6 35.9 14.6 35.9 Hydrocarbons -5.6 26.9 7.2 26.9 Substituted hydrocarbons -3.8 28.2 8.5 28.2 Inorganic hydrids 7.7 2.1 7.7 2.1 Radicals 0.9 19.4 5.1 19.4 Total -1.9 22.7 8.3 22.7 taining only single bonds, molecules with double bonds and molecules featuring a triple bond, the pp-RPA is found to have a low MSE for single bonds (2.2 kcal/mol) but still a substantial MUE (7.9 kcal/mol), while for the ph-RPA both figures of merit are about 19.6 kcal/mol. This indicates that for single bonded systems the pp-RPA over- and underestimates heats of formations to a similar extent. This is no longer true in molecules with multiple bonds: with mostly negative errors, the MSE of the pp-RPA amounts to -8.6 and -4.4 kcal/mol for double and triple bonds, respectively. This indicates that the pp-RPA may be less accurate for multiple-bond systems. However, with a mean signed and unsigned error of 0.9 kcal/mol and 5.1 kcal/mol, respectively, the pp-RPA describes radicals very well. Only NO2, a non-hydrogen system with a double bond, is problematic for the pp-RPA. Just like for closed-shell systems, the ph-RPA overestimates the enthalpies of formation for radicals.

Reaction Energies from G2/97

Chemically relevant transformations conserving the number of electron pairs might provide a view complementary to the enthalpies of formation. Therefore, we exam- ine 19 reactions involving organic compounds to investigate the performance of the pp- and ph-RPA. The reactions are divided into four groups, namely, hydrocarbon

29 reactions, substituted hydrocarbon isomerization reactions, substitution reactions and addition reactions (Table 2.2 and 2.3). Although the ph-RPA does not predict accurate enthalpies of formation, it describes the enthalpy changes in chemical re- actions rather well (MUE = 2.3 kcal/mol), on par with the pp-RPA (MUE = 2.4 kcal/mol). For the four addition reactions, where a double (or triple) bond is con- verted to two single bonds, the pp-RPA yields significantly larger errors. Already in the enthalpies of formation we have observed the qualitatively different behavior for single and double bonds for the pp-RPA. Our speculation is that the approximate pairing interactions fail to describe intra-electron pair correlation on an equal foot- ing with inter-electron pair correlation. The reasonably “constant” performance for the ph-RPA on the other hand, can be understood considering that the number of electron pairs does not change during these reactions and therefore the major source of error for enthalpies of formation does not play any role. In the hydrocarbon reactions, the enthalpy difference between allene and propyne, has been used to assess the reliability of density functionals for determining the poly- yne vs. cumulene stability, a very tricky energy difference in general[56]. Even though the sign is correct for both RPAs (in contrast to typical density functional approxi- mations such as B3LYP), the pp-RPA has a large error (2.8 kcal/mol, >200%), overly stabilizing the triple bond of propyne compared to the double bonds in allene. How- ever, the ph-RPA is not affected by such a problem, in agreement with the analysis of the G2/97 set with respect to the bond types. A similar preference for the electron localized geometry is the isomerization of 2-butyne to the more stable, conjugated butadiene. In contrast to these reactions involving a changing degree of electron de- localization, both RPAs perform excellently for the isomerization energy of butane and the isodesmic reaction energies for n-alkanes. For both types of reactions typical density functionals fail dramatically, most likely because of an inaccurate treatment of weak interactions.[57, 58, 59, 60]

30 Table 2.2: Reaction energies (in kcal/mol) among molecules taken from the G2/97 database. Basis set: cc-pVTZ. Numbers in the parenthesis indicate the error.

Type Reaction Benchmark pp-RPA ph-RPA

Hydrocarbon CH3CCHÑCH2CCH2 1.3 4.1(2.8) 1.8(0.5) Reactions CH3CCCH3ÑCH2CHCHCH2 -8.5 -3.7(4.8) -5.4(3.1) CH3CH2CH2CH3ÑCH3CH(CH3)CH3 -2.1 -2.5(-0.4) -1.7(0.4) C3H8+CH4Ñ2C2H6 2.7 3.3(0.6) 2.3(-0.4) C4H10+2CH4Ñ3C2H6 5.5 7.1(1.6) 5.0(-0.5) Substituted C2H4O(oxirane)ÑCH3CHO -27.1 -24.5(2.6) -24.9(2.2) Hydrocarbon C2H5OHÑCH3OCH3 12.2 12.6(0.4) 11.8(-0.4) Isomerization C2H5SHÑCH3SCH3 2.2 2.2(0.0) 3.4(1.2) (CH3)2CHOHÑC2H5OCH3 13.5 13.8(0.3) 12.5(-1.0) (CH3)2NHÑCH3CH2NH2 -6.9 -8.3(-1.4) -8.0(-1.1) Substitution NCCN+C2H6Ñ2CH3CN -17.2 -15.9(1.3) -13.8(3.4) Reactions H2NNH2+C2H6Ñ2CH3NH2 -13.7 -12.8(0.9) -11.2(2.5) Cl2+C2H6Ñ2CH3Cl -19.1 -19.6(-0.5) -15.2(3.9) Si2H6+C2H6Ñ2CH3SiH3 -13 -10.7(2.3) -7.8(5.2) HOOH+C2H6Ñ2CH3OH -43.4 -44.3(-0.9) -37.4(6.0) Addition HCl+C2H4ÑC2H5Cl -17.2 -25.0(-7.8) -19.1(-1.9) Reactions HCN+C2H2ÑCH2CHCN -42.5 -47.4(-4.9) -43.8(-1.3) HF+C2H2ÑCH2CHF -22.3 -26.7(-4.4) -26.6(-4.3) HCl+C2H2ÑCH2CHCl -23.2 -31.6(-8.4) -28.2(-5.0)

2.3.3 Reaction Barriers Basis Set Convergence Test

Figure 2.4 shows the basis set dependence of the barrier heights of H ` OH Ñ O ` H2 and HCN Ñ HNC along the aug-cc-pVXZ basis sets series, with X = D, T, Q. For

H ` OH Ñ O ` H2, both RPAs converge well with nearly flat behaviors except for

Table 2.3: Mean Signed Errors (MSEs) and Mean Unsigned Errors (MUEs) (in kcal/mol) of of reaction energies calculated by pp-RPA and ph-RPA using the cc- pVTZ basis set.

MSE MUE Type pp-RPA ph-RPA pp-RPA ph-RPA Hydrocarbon Reactions 1.9 0.6 2.0 1.0 Substitued Hydrocarbon Isomerizations 0.4 0.2 0.9 1.2 Substitution Reactions 0.6 4.2 1.2 4.2 Addition Reactions -6.4 -3.1 6.4 3.1 All -0.6 0.7 2.4 2.3

31 25 55

ph-RPA Foward

pp-RPA Foward

20 50

ph-RPA Backward

pp-RPA Backward

15 45

ph-RPA Foward

pp-RPA Foward

ph-RPA Backward

10 40

pp-RPA Backward

5 35 Barrier Barrier Heights (kcal/mol) Barrier Barrier Heights (kcal/mol)

0 30

2 3 4 2 3 4

X in aug-cc-pVXZ X in aug-cc-pVXZ Figure 2.4: Basis set convergence for forward and backward reactions on H ` OH Ñ O ` H2 and HCN Ñ HNC. The pp-RPA converges well for both cases, while the ph-RPA still has rather large differences between aug-cc-pVTZ and aug- cc-pVQZ.

the backward reaction calculated by ph-RPA, which yields a 3 kcal/mol difference between aug-cc-pVTZ and aug-cc-pVQZ. While the pp-RPA has a similar behavior for HCN Ñ HNC, the ph-RPA has a “bump” at the aug-cc-pVTZ basis of about 4 kcal/mol. These results emphasize that reaction barriers can be very sensitive to basis sets. Nevertheless, considering the computational cost, we choose the aug-cc- pVTZ basis set for the following reaction barrier calculations and expect the results to reflect the correct relative performance.

Results

Benchmark values in the DBH24 reaction barrier test set are best estimates either from experiments or highly accurate theoretical methods [61]. The overall perfor- mance for the DBH24 database is very similar for the pp-RPA and the ph-RPA (see Table 2.4 and 2.5): the pp-RPA has a slightly smaller mean signed error (-1.11 kcal/mol) than ph-RPA (-1.65 kcal/mol), while the ph-RPA has a slightly smaller mean unsigned error (2.48 kcal/mol vs. 3.19 kcal/mol). Among the four subsets in DBH24, the pp-RPA has the largest mean unsigned error (5.56 kcal/mol) for the

32 Table 2.4: DBH24 reaction barriers (in kcal/mol) calculated by pp-RPA and ph-RPA using the aug-cc-pVTZ basis set. Numbers in the parenthesis indicate the error.

Database Reaction Benchmark pp-RPA ph-RPA 17.13 22.34(5.21) 17.50(0.37) H ` N O Ñ OH ` N 2 2 82.47 65.65(-16.82) 75.49(-6.98) 18.00 21.29(3.29) 18.78(0.78) HATBH6 H ` ClH Ñ HCl ` H 18.00 21.29(3.29) 18.78(0.78) 6.75 3.04(-3.71) 0.16(-6.59) CH ` FCl Ñ CH F ` Cl 3 3 60.00 58.93(-1.07) 57.80(-2.2) 13.41 12.00(-1.41) 10.87(-2.54) Cl´ ¨ ¨ ¨ CH Cl Ñ ClCH ¨ ¨ ¨ Cl´ 3 3 13.41 12.00(-1.41) 10.87(-2.54) 3.44 2.11(-1.33) 1.38(-2.06) NSBH6 F´ ¨ ¨ ¨ CH Cl Ñ FCH ¨ ¨ ¨ Cl´ 3 3 29.42 26.17(-3.25) 26.46(-2.96) -2.44 -6.76(-4.32) -5.89(-3.45) OH´ ` CH F Ñ HOCH ` F´ 3 3 17.66 13.79(-3.87) 13.20(-4.46) 14.36 17.33(2.97) 14.23(-0.13) H ` N Ñ HN 2 2 10.61 10.34(-0.27) 10.54(-0.07) 1.72 3.95(2.23) 2.31(0.59) UABH6 H ` C H Ñ CH CH 2 4 3 2 41.75 43.49(1.74) 43.20(1.45) 48.07 49.24(1.17) 50.07(2.00) HCN Ñ HNC 32.82 32.55(-0.27) 35.50(2.68) 6.7 0.4(-6.3) 4.2(-2.5) OH ` CH Ñ CH ` H O 4 3 2 19.6 16.6(-3.0) 11.3(-8.3) 10.7 13.9(3.2) 7.3(-3.4) HTBH6 H ` OH Ñ O ` H 2 13.1 11.4(-1.7) 12.7(-0.4) 3.6 5.5(1.9) 2.8(-0.8) H ` H S Ñ H ` HS 2 2 17.3 14.4(-2.9) 18.6(1.3)

HATBH6 subset, which includes three heavy-atom transfer reactions. This is mainly due to the H ` N2O Ñ OH ` N2 reaction, in which the pp-RPA overestimates the forward barrier by 5.21 kcal/mol and underestimates the backward barrier by 16.82 kcal/mol. The reason is two-fold: first, as is shown before, the pp-RPA has difficulties predicting enthalpies of formation for some compounds with double bonds and triple bonds, i.e. the reactants and products are not well described with a 22 kcal/mol error for the reaction energy. Second, the pp-RPA does not describe the spin-unpolarized bond-stretching of double and triple bonds well [42], leading to a large error for the

33 Table 2.5: Mean Signed Errors (MSEs) and Mean Unsigned Errors (MUEs) (in kcal/mol) of DBH24 reaction barriers and its four subsets calculated by pp-RPA and ph-RPA

MSE MUE Database pp-RPA ph-RPA pp-RPA ph-RPA HATBH6 -1.64 -2.30 5.56 2.95 NSBH6 -2.60 -3.00 2.60 3.00 UABH6 1.26 1.09 1.44 1.15 HTBH6 -1.47 -2.37 3.14 2.79 DBH24 -1.11 -1.65 3.19 2.48 transition state. The NSBH6 subset includes three nucleophilic substitution reac-

´ ´ tions, and both RPAs perform well except for the OH ` CH3F Ñ HOCH3 ` F reaction, which might suffer from delocalization errors in the PBE reference determi- nant. For UABH6, which includes three unimolecular and association reactions, both RPAs perform well. For HTBH6, which consists of three hydrogen transfer reactions, both methods give accurate reaction barriers, except for OH ` CH4 Ñ CH3 ` H2O, where they both underestimate the energy of the transition state. In conclusion, despite some tricky cases, both the pp-RPA and ph-RPA generally provide reliable reaction barriers.

2.3.4 Nonbonded Interactions Basis Set Convergence Test

To assess the basis set convergence of the pp-RPA, we chose a strong hydrogen bonded system (HF ´ HF), a charge-transfer complex (H2O ´ ClF), a dipole-dipole and a “pure” van der Waals dimer (HCl ´ HCl and CH4 ´ Ne, respectively). Overall, the interaction energies are similar with different basis (aug-cc-pVXZ with X= D, T, Q), although in some cases the interaction energy does not change monotonically with respect to the basis (see Figure 2.5). Considering the high computational cost, we adopt the aug-cc-pVDZ basis set and correct for the basis set superposition er-

34 0

HF-HF

-1

H O-ClF

2

HCl-HCl

CH -Ne

-2 4

-3

-4

-5 Interaction energy (kcal/mol)

-6

2 3 4

X in aug-cc-pVXZ Figure 2.5: Basis set convergence for nonbonded interactions calculated by pp- RPA. Different basis sets give similar interaction energies.

ror (BSSE) according to the Boys-Bernardi counterpoise correction [62]. Table 2.6 confirms that counterpoise corrected aug-cc-pVDZ results overall agree well with the aug-cc-pVQZ results and therefore we expect them to reflect the correct relative performance of the ph- and pp-RPA.

Results

Results for nonbonded interactions are shown in Table 2.6. Both HF reference and PBE reference are investigated for the two RPAs. Overall, with the aug-cc-pVDZ basis, the HF reference gives slightly better results than the PBE reference for both RPAs. As the work of Cencek and Szalewicz nicely demonstrated [63], this behavior is a direct consequence of the exponential, instead of 1/r decay of the exchange- correlation potential of semi-local density functionals. Therefore, the weak electron density regions are very poorly described, which in turn is considerably more prob- lematic for functionals that depend on the virtual orbitals than pure density func-

35 Table 2.6: Mean Signed Errors (MSEs) and Mean Unsigned Errors (MUEs) (in kcal/mol) of HB6/04, CT7/04, DI6/04 and WI9/04 nonbonded interaction by pp- RPA and ph-RPA. If not stated otherwise, the aug-cc-pVDZ basis set has been applied and except for aug-cc-pVQZ the basis set superposition error is corrected for.

ph-RPA pp-RPA Database HF PBE HF PBE HF/augQZ1 MSE MUE MSE MUE MSE MUE MSE MUE MSE MUE HB8/04 1.14 1.14 1.82 1.82 -0.03 0.48 -0.63 0.76 -0.38 0.42 CT7/04 1.70 1.70 4.15 4.15 0.26 0.26 -0.24 1.22 -0.10 0.37 DI6/04 1.29 1.29 1.57 1.57 0.38 0.40 -0.34 0.37 -0.25 0.25 WI9/04 0.32 0.32 0.29 0.29 0.17 0.17 -0.04 0.15 -0.01 0.04 Total 1.05 1.05 1.86 1.86 0.20 0.31 -0.28 0.60 -0.15 0.25 1.Calculations were performed with spherical harmonic basis functions. BSSEs were not corrected. Results reported are based on selected systems that are within computational capability. tionals. After BSSE correction, the pp-RPA gives a slightly smaller deviation than the ph-RPA. The two RPAs perform similarly for all four types of interactions, with the pp-RPA using the Hartree-Fock determinant gives the best agreement with the benchmark data. Therefore, it can be concluded that the pp-RPA describes weak interactions equally well as the more popular ph-RPA. This may be due to the exact second-order energy expansion of the ladder diagram [42], which is commonly be- lieved to dominate the van der Waals interaction. The relationship between the two types of RPA in describing the asymptotic van der Waals awaits further investigation.

2.3.5 Conclusion

The benchmark tests on the G2/97 enthalpies of formation, DBH24 reaction barriers, and four nonbonded interaction databases demonstrate that the pp-RPA performs significantly better than the ph-RPA for enthalpies of formation: in contrast to the increasing error of the ph-RPA with the number of atoms in a molecule, the pp-RPA has a nearly constant error. For reaction enthalpies, barriers heights and non bonded interactions, the pp-RPA and ph-RPA perform essentially equally well. These bench- mark tests indicate that the pp-RPA is a promising method even for larger systems,

36 although systems with multiple bonds tend to be relatively problematic. The gen- eral success of the pp-RPA suggests that the pairing interaction in conjunction with the adiabatic connection formalism forms a promising framework for developing new density functionals.

37 3

Pairing Matrix Fluctuation for Excited States Calculation

This chapter is mainly adapted from the following journal articles,

• Yang Yang, Helen van Aggelen, and Weitao Yang, “Double, Rydberg and charge transfer excitations from pairing matrix fluctuation and particle-particle random phase approximation” J. Chem. Phys. 139, 224105 (2013)

• Yang Yang, Degao Peng, Jianfeng Lu, and Weitao Yang, “Excitation energies from particle-particle random phase approximation: Davidson algorithm and benchmark studies” J. Chem. Phys. 141, 124104 (2014)

3.1 Introduction

The accurate description of excited states is an important and challenging topic. In- formation on excitation energies and oscillator strengths is necessary for explaining and predicting excitation spectra. Theoretical studies are also particularly help- ful in determining the dynamics of electronically excited states. Many theoreti- cal approaches have been developed for studying excited states. Full and multi-

38 reference CI, methods with perturbative corrections to CIS (including CIS-MP2 [64] and CIS(D) [65]), CASSCF and CASPT2 [66], EOM-CC and LR-CC [67, 68] are generally accurate but computationally expensive. For large molecules, only a few single-determinant reference approaches are more applicable. These approaches in- clude CIS [69], TDHF [70] and TDDFT [71, 72, 73, 74]. CIS and TDHF are also known as the Tamm-Dancoff approximation (TDA) and the RPA, respectively. They have the same single-determinant reference — the Hartree-Fock ground state, which is a poor first-order approximation with no correlation and overestimated HOMO- LUMO gaps. At the same time, excitation operators of these two methods are limited to particle-hole excitations. Therefore, CIS and TDHF tend to overestimate excitation energies and are only capable of capturing single excitations. Further- more, TDHF often suffers from instabilities for triplet states [75, 76], which makes it much less used. TDDFT is based on Kohn-Sham reference states and is more accurate in predicting excitation energies than CIS and TDHF. However, within the adiabatic approximation, in which the exchange-correlation kernel is frequency- independent, TDDFT also can only capture single excitations [77, 78]. Because of their incorrect long-range behavior [75], approximate exchange-correlation kernels also have difficulties describing Rydberg excitations. Moreover, because of their delocalization/self-interaction error, TDDFT greatly underestimates charge transfer (CT) excitations and has no 1/R Coulomb interaction character, with R the separa- tion distance [74, 75, 79, 80]. Therefore, an efficient method that can accurately deal with single, double, Rydberg and CT excitations all together is particularly valuable and highly desirable. Besides above traditional methods that use N-electron ground states as the start- ing point, many non-N-electron-ground-state reference methods have also been devel- oped to investigate excitation problems. These methods are mostly in the framework of EOM-CC [81]. For example, the spin-flip (SF-) EOM-CC [82, 83] method uses an

39 N-electron high-spin triplet reference, the ionization-potential/electron-attachment

(IP/EA-) EOM-CC [84, 85] use pN ˘1q-electron ground states, the double ionization- potential/electron-attachment (DIP/DEA-) EOM-CC [86, 87] use pN ˘ 2q-electron ground states and the triple ionization-potential/electron-attachment (TIP/TEA-)

EOM-CC [88] can use pN ˘ 3q-electron ground states. Such EOM-CC methods in- volving electron number changes have roots in Fock-Space coupled cluster (FSCC) theory [89, 90, 91, 92, 93, 94, 95, 96] and similarity transformed EOM-CC [97, 98]. A similarity among these non-N-electron-ground-state reference methods is that the N-electron ground state and excited states are constructed with the same proce- dure, therefore, these methods are believed to have a balanced treatment between the ground state and excited states. Furthermore, the change of references provides much more choices in solving excitation problems. However, these well-developed variants EOM-CC are also computationally expensive. Fortunately, SF-TDDFT [99], which only uses a single-determinant high-spin triplet reference provides a com- putationally more efficient alternative to SF-EOM-CC and has found its practical use in the prediction of double excitations [100]. However, there are currently no single-determinant alternatives to EOM-CC methods based on ground states with different electron number. Here we present such a single-determinant alternative to (DIP/DEA-) EOM-CC. We use the pairing matrix fluctuation and pp-RPA to investigate excitation prob- lems. The pairing matrix fluctuation has been applied to the investigation of Auger Spectroscopy [101, 102]. However, it has never been used to investigate neutral exci- tations. We start from single-determinant pN ˘ 2q-electron references and from the pairing matrix fluctuation for these references, we obtain information on transitions both to the ground state and to the excited states of the N-electron system. With

40 Figure 3.1: The pp-RPA starts from pN ¯2q-electron references and targets neutral states by adding or removing two electrons.

this information, excitation energies can be determined (Figure 3.1).

N N N N´2 N N´2 `2e `2e En ´ E0 “pEn ´ E0 q ´ pE0 ´ E0 q “ ωn ´ ω0 for N ´ 2 reference, (3.1a)

N N N N`2 N N`2 ´2e ´2e En ´ E0 “pEn ´ E0 q ´ pE0 ´ E0 q “ ωn ´ ω0 for N ` 2 reference. (3.1b)

Although the exact pairing matrix fluctuation for interacting systems, which should give exact excitation energies, is unknown, the pp-RPA and the particle-particle Tamm-Dancoff approximation (pp-TDA) provide useful simple approximations. Un- like CIS, TDHF and TDDFT approaches, which essentially adopt the particle-hole channel of interactions to solve excitation problems, this new approach adopts the particle-particle channel. It performs well in describing double, Rydberg, CT, and single excitations excited from HOMO.

41 3.2 Theory

3.2.1 Particle-Particle Random Phase Approximation from Equation of Motion

The basic EOM for transitions between the pN ´ 2q-electron ground state and the N-electron states can be written as

ˆ : N N´2 : rδO, rH,O ss “ pEn ´ E0 qrδO, O s. (3.2) where Hˆ is the Hamiltonian operator

1 Hˆ “ Tˆ ` Vˆ “ xα|T |βya: a ` pαβ|V |γδqa: a: a a , (3.3) α β 2 α β δ γ αβ αβγδ ÿ ÿ and O: is an excitation operator that can excite the pN ´ 2q-electron ground state to the n’th N-electron excited state, and δO is a probing de-excitation operator. The pp-RPA assumes that

: n : : n : : On “ Xabaaab ` Yij ajai , (3.4) aąb iąj ÿ ÿ and

δOn “ abaa with a ą b (3.5a)

δOn “ aiaj with i ą j, (3.5b) where a, b, c, d are unoccupied orbitals for the pN ´ 2q-electron system and i, j, k, l are occupied ones. For the first probing de-excitation operator, when we evaluate with an SCF HF

42 pN ´ 2q-electron ground state, the EOM gives

N´2 : : N´2 n N´2 : : N´2 n xΦHF | abaa, H, acad |ΦHF yXcd ` xΦHF | abaa, H, al ak |ΦHF yYkl cąd kąl ÿ ” ” ıı ÿ ” ” ıı N N´2 N´2 : : N´2 n N´2 : : N´2 n “pEn ´ E0 q xΦHF | abaa, acad |ΦHF yXcd ` xΦHF | abaa, al ak |ΦHF yYkl cąd kąl ÿ ” ı ÿ ” ı N N´2 N´2 : : N´2 n “pEn ´ E0 q xΦHF |abaaacad|ΦHF yXcd cąd ÿ N N´2 n “pEn ´ E0 q δacδbdXcd cąd ÿ N N´2 n “pEn ´ E0 qXab. (3.6) For the second operator, it gives

N´2 : : N´2 n N´2 : : N´2 n xΦHF | aiaj, H, acad |ΦHF yXcd ` xΦHF | aiaj, H, al ak |ΦHF yYkl cąd kąl ÿ ” ” ıı ÿ ” ” ıı N N´2 N´2 : : N´2 n N´2 : : N´2 n “pEn ´ E0 q xΦHF | aiaj, acad |ΦHF yXcd ` xΦHF | aiaj, al ak |ΦHF yYkl cąd kąl ÿ ” ı ÿ ” ı N N´2 N´2 : : N´2 n “ ´ pEn ´ E0 q xΦHF |al akaiaj|ΦHF yYkl kąl ÿ N N´2 n “ ´ pEn ´ E0 q δikδjlYkl kąl ÿ N N´2 n “ ´ pEn ´ E0 qYij (3.7) The above two equations can be written into a matrix equation

AB X I 0 X “ ω (3.8) B1 C Y 0 ´I Y „  „  „  „ 

43 with

N´2 : : N´2 Aab,cd “xΦHF | abaa, H, acad |ΦHF y (3.9a) ” ” ıı N´2 : : N´2 Bab,kl “xΦHF | abaa, H, al ak |ΦHF y (3.9b) ” ” ıı 1 N´2 : : N´2 Bij,cd “xΦHF | aiaj, H, acad |ΦHF y (3.9c) ” ” ıı N´2 : : N´2 Cij,kl “xΦHF | aiaj, H, al ak |ΦHF y, (3.9d) ” ” ıı N N´2 and ωn “ En ´ E0 . Evaluation of the matrix elements gives

B “ B1: (3.10) and

Aab,cd “δacδbdpa ` bq ` xab||cdy (3.11a)

Bab,kl “xab||kly (3.11b)

Cij,kl “ ´ δikδjlpi ` jq ` xij||kly (3.11c)

This is the particle-particle random phase approximation eigenvalue problem, and it is the same as Eqs. (2.8) and (2.9) derived from pairing matrix fluctuation. Setting B equal to zero, we can get the pp-TDA and the hh-TDA equations separately

AX “ωX, (3.12)

CY “ ´ ωY. (3.13)

Similarly, excitations for N-electron system can also be obtained by investigating

the EOM for transitions between pN ` 2q-electron ground state and the N-electron states: ˆ : N N`2 : rδO, rH,O ss “ pEn ´ E0 qrδO, O s, (3.14)

44 where O: is an excitation operator that can excite the pN ` 2q-electron ground state to the n’th N-electron excited state, and δO is a probing de-excitation operator. The pp-RPA assumes

: n n On “ Xabaaab ` Yij ajai, (3.15) aąb iąj ÿ ÿ and

: : δOn “ abaa with a ą b (3.16a)

: : δOn “ ai aj with i ą j, (3.16b)

where a, b, c, d are unoccupied orbitals for the pN ` 2q-electron system and i, j, k, l are occupied ones.

For the probing de-excitation operator, when we evaluate with a SCF HF pN `2q- electron ground state, the EOM gives

N`2 : : N`2 n N`2 : : N`2 n xΦHF | abaa, rH, acads |ΦHF yXcd ` xΦHF | abaa, rH, alaks |ΦHF yYkl cąd kąl ÿ ” ı ÿ ” ı N N`2 N`2 : : N`2 n N`2 : : N`2 n “pEn ´ E0 q xΦHF | abaa, acad |ΦHF yXcd ` xΦHF | abaa, alak |ΦHF yYkl cąd kąl ÿ ” ı ÿ ” ı N N`2 N`2 : : N`2 n “ ´ pEn ´ E0 q xΦHF |acadabaa|ΦHF yXcd cąd ÿ N N`2 n “ ´ pEn ´ E0 q δacδbdXcd cąd ÿ N N`2 n “ ´ pEn ´ E0 qXab. (3.17)

45 For the second operator, it gives

N`2 : : N`2 n N`2 : : N`2 n xΦHF | ai aj, rH, acads |ΦHF yXcd ` xΦHF | ai aj, rH, alaks |ΦHF yYkl cąd kąl ÿ ” ı ÿ ” ı N N`2 N`2 : : N`2 n N`2 : : N`2 n “pEn ´ E0 q xΦHF | ai aj, acad |ΦHF yXcd ` xΦHF | ai aj, alak |ΦHF yYkl cąd kąl ÿ ” ı ÿ ” ı N N`2 N`2 : : N`2 n “pEn ´ E0 q xΦHF |ai ajalak|ΦHF yYkl kąl ÿ N N`2 n “pEn ´ E0 q δikδjlYkl kąl ÿ N N`2 n “pEn ´ E0 qYij . (3.18) The above two equations can be written into a matrix equation

AB X I 0 X “ ´ω (3.19) B1 C Y 0 ´I Y „  „  „  „  with

N`2 : : N`2 Aab,cd “xΦHF | abaa, rH, acads |ΦHF y (3.20a) ” ı N`2 : : N`2 Bab,kl “xΦHF | abaa, rH, alaks |ΦHF y (3.20b) ” ı 1 N`2 : : N`2 Bij,cd “xΦHF | ai aj, rH, acads |ΦHF y (3.20c) ” ı N`2 : : N`2 Cij,kl “xΦHF | ai aj, rH, alaks |ΦHF y (3.20d) ” ı N N`2 and ωn “ En ´ E0 . Evaluation of the matrix elements also gives

B “ B1: (3.21)

46 and

Aab,cd “δacδbdpa ` bq ` xab||cdy (3.22a)

Bab,kl “xab||kly (3.22b)

Cij,kl “ ´ δikδjlpi ` jq ` xij||kly (3.22c)

Again, we obtain the same equation as Eqs. (2.8) and (2.9). Setting B equal to zero, we can also get the pp-TDA and the hh-TDA equations separately.

3.2.2 pp-RPA-HF* from Equation of Motion

We can also evaluate the EOM with an SCF HF N-electron ground state, in which two HOMO spin orbitals, denoted with m and n, are treated as unoccupied ones.

N We call this new reference HF* reference and denote it with |anamΦHF y. As or- bital energies  for the N-electron state include contributions from m and n, these contributions should be corrected for the pN ´ 2q-electron reference. Consequently evaluation of the matrix elements gives

B “ B1:, (3.23) and

Aab,cd “δacδbdpa ` bq ` xab||cdy ´ δacpxmb||mdy ` xnb||ndyq

´ δbdpxam||cmy ` xan||cnyq ´ δadpxmb||cmy ` xnb||cnyq (3.24a)

´ δbcpxam||mdy ` xan||ndyq (3.24b)

Bab,kl “xab||kly (3.24c)

Cij,kl “ ´ δikδjlpi ` jq ` xij||kly ` δikpxmj||mly ` xnj||nlyq

` δjlpxim||kmy ` xin||knyq ` δilpxmj||kmy ` xnj||knyq (3.24d)

` δjkpxim||mly ` xin||nlyq. (3.24e)

Similarly, pp-TDA equations can also be obtained by setting B to 0.

47 The above equations can be derived in detail as follows. Evaluation of the A matrix:

N : : N Aab,cd “xanamΦHF | abaa, H, acad |anamΦHF y ” ” ıı N : : N N N (3.25) “xanamΦHF |abaaHacad|anamΦHF y ´ δacδbdxanamΦHF |H|anamΦHF y

: : N : : N N N “xaaabanamΦHF |H|acadanamΦHF y ´ δacδbdxanamΦHF |H|anamΦHF y.

When a “ c and b “ d

: : N : : N N N xaaabanamΦHF |H|aaabanamΦHF y ´ δacδbdxanamΦHF |H|anamΦHF y

occ1 occ1

“haa ` hbb ` xua||uay ` xub||uby ` xab||aby u u ÿ ÿ occ occ (3.26) “haa ` hbb ` xua||uay ` xub||uby ` xab||aby u u ÿ ÿ ´ xma||may ´ xmb||mby ´ xna||nay ´ xnb||nby

“a ` b ` xab||aby ´ xma||may ´ xmb||mby ´ xna||nay ´ xnb||nby.

When a “ c but b ‰ d

: : N : : N N N xaaabanamΦHF |H|aaadanamΦHF y ´ δacδbdxanamΦHF |H|anamΦHF y

occ1`a “hbd ` xub||udy u (3.27) ÿ “xb|f|dy ` xab||ady ´ xmb||mdy ´ xnb||ndy

“xab||ady ´ xmb||mdy ´ xnb||ndy.

48 When a ‰ c but b “ d

: : N : : N N N xaaabanamΦHF |H|acabanamΦHF y ´ δacδbdxanamΦHF |H|anamΦHF y

occ1`b “hac ` xau||cuy u (3.28) ÿ “xa|f|cy ` xab||cby ´ xam||cmy ´ xan||cny

“xab||cby ´ xam||cmy ´ xan||cny.

When a “ d (c ą d “ a ą b),

: : N : : N N N xaaabanamΦHF |H|acaaanamΦHF y ´ δacδbdxanamΦHF |H|anamΦHF y

: : N : : N “ ´ xaaabanamΦHF |H|aaacanamΦHF y

occ1`a “ ´ hbc ´ xub||ucy (3.29) u ÿ “ ´ xb|f|cy ´ xab||acy ` xmb||mcy ` xnb||ncy

“xab||cay ´ xmb||cmy ´ xnb||cny.

When b “ c (a ą b “ c ą d),

: : N : : N N N xaaabanamΦHF |H|abadanamΦHF y ´ δacδbdxanamΦHF |H|anamΦHF y

: : N : : N “ ´ xaaabanamΦHF |H|adabanamΦHF y

occ1`b “ ´ had ´ xau||duy (3.30) u ÿ “ ´ xa|f|dy ´ xab||dby ` xam||dmy ` xan||dny

“xab||bdy ´ xam||mdy ´ xan||ndy.

When a ‰ c ‰ b ‰ d

: : N : : N N N xaaabanamΦHF |H|acadanamΦHF y ´ δacδbdxanamΦHF |H|anamΦHF y “ xab||cdy. (3.31)

49 As a result,

N : : N Aab,cd “xanamΦHF | abaa, H, acad |anamΦHF y ” ” ıı “δacδbdpa ` bq ` xab||cdy ´ δacpxmb||mdy ` xnb||ndyq (3.32)

´ δbdpxam||cmy ` xan||cnyq ´ δadpxmb||cmy ` xnb||cnyq

´ δbcpxam||mdy ` xan||ndyq.

Evaluation of the C matrix:

N : : N Cij,kl “xanamΦHF | aiaj, H, al ak |anamΦHF y ” ” ıı N : : N N N (3.33) “xanamΦHF |al akHaiaj|anamΦHF y ´ δikδjlxanamΦHF |H|anamΦHF y

N N N N “xakalanamΦHF |H|aiajanamΦHF y ´ δikδjlxanamΦHF |H|anamΦHF y.

When i “ k and j “ l

N N N N xaiajanamΦHF |H|aiajanamΦHF y ´ δikδjlxanamΦHF |H|anamΦHF y

occ1 occ1

“ ´ hii ´ hjj ´ xui||uiy ´ xuj||ujy ` xij||ijy (3.34) u u ÿ ÿ

“ ´ i ´ j ` xij||ijy ` xmi||miy ` xmj||mjy ` xni||niy ` xnj||njy.

When i “ k but j ‰ l

N N N N xaialanamΦHF |H|aiajanamΦHF y ´ δikδjlxanamΦHF |H|anamΦHF y

occ1´i “ ´ hjl ´ xuj||uly u (3.35) ÿ “ ´ xj|f|ly ` xij||ily ` xmj||mly ` xnj||nly

“xij||ily ` xmj||mly ` xnj||nly

50 When i ‰ k but j “ l

N N N N xakajanamΦHF |H|aiajanamΦHF y ´ δikδjlxanamΦHF |H|anamΦHF y

occ1´j

“ ´ hik ´ xiu||kuy u (3.36) ÿ “ ´ xi|f|ky ` xij||kjy ` xim||kmy ` xin||kny

“xij||kjy ` xim||kmy ` xin||kny

When i “ l (k ą l “ i ą j),

N N N N xakaianamΦHF |H|aiajanamΦHF y ´ δikδjlxanamΦHF |H|anamΦHF y

occ1´i “hjk ` xuj||uky u (3.37) ÿ “xi|f|ky ´ xij||iky ´ xmj||mky ´ xnj||nky

“xij||kiy ` xmj||kmy ` xnj||kny

When j “ k (i ą j “ k ą l),

N N N N xajalanamΦHF |H|aiajanamΦHF y ´ δikδjlxanamΦHF |H|anamΦHF y

occ1´j

“hil ` xiu||luy u (3.38) ÿ “xi|f|ly ´ xij||ljy ´ xim||lmy ´ xin||lny

“xij||jly ` xim||mly ` xin||nly

When i ‰ k ‰ j ‰ l

N N N N xakalanamΦHF |H|aiajanamΦHF y ´ δikδjlxanamΦHF |H|anamΦHF y “ xji||lky “ xij||kly (3.39)

51 As a result,

N : : N Cij,kl “xanamΦHF | aiaj, H, al ak |anamΦHF y ” ” ıı “ ´ δikδjlpi ` jq ` xij||kly ` δikpxmj||mly ` xnj||nlyq (3.40)

` δjlpxim||kmy ` xin||knyq ` δilpxmj||kmy ` xnj||knyq

` δjkpxim||mly ` xin||nlyq

Evaluation of the B and B1 matrices:

N : : N Bab,kl “xanamΦHF | abaa, H, al ak |anamΦHF y ” ” ıı N : : : : : : : : N “xanamΦHF |abaaHal ak ´ abaaal akH ´ Hal akabaa ` al akHabaa|anamΦHF y

N : : N “ ´ xanamΦHF |abaaal akH|anamΦHF y

“ ´ xba||kly “ xab||kly (3.41)

1 N : : N Bij,cd “xanamΦHF | aiaj, H, acad |anamΦHF y ” ” ıı N : : : : : : : : N “xanamΦHF |aiajHacad ´ aiajacadH ´ Hacadaiaj ` acadHaiaj|anamΦHF y

N : : : : : : : : N : “xanamΦHF |adacHajai ´ Hadacajai ´ ajai adacH ` ajai Hadac|anamΦHF y

N : : : : : : : : N : “xanamΦHF |adacHajai ´ Hajai adac ´ adacajai H ` ajai Hadac|anamΦHF y

N : : : : : : : : N : “xanamΦHF |adacHajai ´ adacajai H ´ Hajai adac ` ajai Hadac|anamΦHF y

N : : N : “xanamΦHF | adac, H, ajai |anamΦHF y ” ” ıı ˚ ˚ “Bcd,ij “ xcd||ijy “ xij||cdy (3.42)

1: Consequently, Bab,kl “ xab||kly and B “ B .

52 3.2.3 Two-Electron Systems: An Exact Case for pp-RPA-HF* and pp-TDA-HF*

For two-electron system, such as He and H2, the pN ´ 2q-electron state has no electrons, which can be denoted by |0y. Consequently, the pp-RPA equation reduces to the pp-TDA equation. With HF* reference, this pp-TDA equation is the same as Configuration Interaction Singles and Doubles (CISD):

: : Aab,cd “x0| abaa, H, acad |0y ” ” ıı (3.43) “xab|H|cdy ´ δacδbdx0|H|0y

“xab|H|cdy.

Take the left side xab| as an example: when a and b are equal to m and n, which are the HOMO spin orbitals that are treated as unoccupied, it recovers HF ground state; when only one is equal to m or n, it recovers singly excited configurations; when neither equals to m or n, it recovers doubly excited configurations. The same

applies to the right side |cdy. Consequently the elements are the same as CISD and the solutions should be exact for two-electron systems within the basis set limit. It can be further noted that the HF* reference is not the only choice. As long as there is a given orbital basis, the A matrix can be evaluated and the eigenvalue problem can be solved. If the orbital basis set is complete, then for a two-electron system, we should always obtain exact results.

3.2.4 Oscillator Strengths from pp-TDA

In the pp-TDA, eigenvector elements are defined as

n,N N´2 N χab “ xΨ0 |aaab|Ψn y. (3.44)

: : N´2 Suppose all possible abaa|Ψ0 y span the whole N-electron space, then

: : N´2 N´2 abaa|Ψ0 yxΨ0 |aaab (3.45) aąb ÿ 53 is an identity operator and

N : : N´2 N´2 N |Ψn y “ abaa|Ψ0 yxΨ0 |aaab|Ψn y aąb ÿ (3.46) n,N : : N´2 “ χab abaa|Ψ0 y. aąb ÿ This can be used to express transition dipoles

N N n,N m,N ˚ N´2 : : N´2 xΨm|ˆr|Ψn y “ χab pχcd q xΨ0 |acadˆrabaa|Ψ0 y aąb cąd ÿ ÿ (3.47) n,N m,N ˚ N´2 : : : N´2 “ χab pχcd q xµ|r|νyxΨ0 |acadaµaνabaa|Ψ0 y. aąb cąd µν ÿ ÿ ÿ After evaluating with Wick’s theorem

N´2 : : : N´2 xΨ0 |acadaµaνabaa|Ψ0 y

“θpµ ´ F qθpν ´ F q δdµδcaδbν ´ δcµδdaδbν ´ δdµδcbδaν ` δcµδdbδaν (3.48) ´ ¯ ` θpF ´ µqδµν δacδbd ´ δadδbc , ´ ¯ the transition dipole can be expressed as

N N n,N m,N ˚ xΨm|ˆr|Ψn y “ χab pχcd q xd|r|byδac ´ xc|r|byδad ´ xd|r|ayδbc ` xc|r|ayδdb aąb cąd ÿ ÿ ´ ¯ n,N m,N ˚ ` χab pχcd q xµ|r|µypδacδbd ´ δadδbcq, aąb cąd µăF ÿ ÿ ÿ (3.49) which leads to the expression for the oscillator strength

2 f “ pω ´ ω q|xΨN |ˆr|ΨN y|2. (3.50) n 3 n 0 0 n

3.2.5 Time-Dependent Density Functional Theory with Pairing Field

Besides pairing matrix fluctuation and EOM, there is another approach of deriving the pp-RPA. This approach is derived by Peng et al. [103] and is called time-

54 dependent density functional theory with pairing field (TDDFT-P). Consider a Hamiltonian within a pairing field,

Hˆ “ Tˆ ` Vˆ ` Wˆ ` D,ˆ (3.51) in which Tˆ, Vˆ and Wˆ represent kinetic, external potential and two-electron interac- tions, respectively, and Dˆ is the external pairing field,

1 Dˆ “ dxdx1rD˚ψˆpx1qψˆpxq ` h.c.s, (3.52) 2 ż where ψˆpx1qψˆpxq stands for the pair removal part, while h.c. is short for Hermitian conjugate and it stands for the pair addition part. In presence of the above pairing field, the pairing matrix

κpx, x1q “ xΨ|ψ:px1qψ:pxq|Ψy (3.53) is not zero. In fact, a perturbative pairing field δDpy, y1; τq at time τ results in a tiny change of the pairing matrix δκpx, x1; tq at time t. If we ignore the higher-order terms and only investigate the first-order change, then through the linear response theory, we obtain a linear pp-response function

Kpx, x1; y, y1; tq “ ´iθptqxΨ|rψpx1; tqψpx; tq, ψ:py; 0qψ:py1; 0qs|Ψy (3.54) and the linear response equation with the integrated form is

δκpx, x1; tq “ dτdydy1Kpx, x1; y, y1; t ´ τqδDpy, y1; τq. (3.55) ż Note here, the creation and annihilation operators in Eq. 3.54 are all in the Heisen- berg picture.

A non-interacting system |Φsy is now assumed to hold the same electron den- sity and pairing matrix as the interacting real system at every time. Instead of

55 complicated two-electron interactions, this non-interacting system only has effective one-body normal and pairing potentials, and the exchange-correlation part exists in both potentials. The total pairing field includes both the internal pairing potential

s and the external pairing field: δD “ δDint ` δD. This non-interacting system is a mapping image from the real many-body system and has a pp-response function that is much easier to calculate. By changing the coordinate basis to orbital basis and performing Fourier transform, its pp-response function K0 is simply

0 θpp ´ F qθpq ´ F q ´ θpF ´ pqθpF ´ qq Kpq,rspωq “ pδprδqs ´ δqrδpsq . (3.56) ω ´ pp ` qq ` iη

Therefore, the linear response equation for the non-interacting system becomes two separate ones, s s δDijpωq δκijpωq “ ´ , (3.57a) ω ´ pi ` jq ` iη

s s δDabpωq δκabpωq “ . (3.57b) ω ´ pa ` bq ` iη

Apart from above response equations that build up the dependence of δκ on the total pairing field δDs, the changes of pairing matrix also in turn affect the total pairing

field, or more specifically, the internal mean-field pairing potential δDint. We use a response kernel L to represent the dependence of δDint on δκ and it can be derived that

δ2E rρ, κs L pq rs 2 dx dx dx1 dx1 φ˚ x φ˚ x1 xc φ x φ x1 , pq,rs “ x || y ` 1 2 1 2 p p 1q q p 1q ˚ 1 1 rp 2q sp 2q δκ px1, x1qδκpx2, x2q ρ ż ´ ¯ (3.58) with

xpq||rsy “ xpq|rsy ´ xpq|sry, (3.59)

and ˚ ˚ 1 1 φ pxqφ px qφrpxqφspx q xpq|rsy “ dxdx1 p q . (3.60) |r ´ r1| ż 56 Plug in the response equation and eliminate the internal pairing potential, after some rearrangement, we come to the TDDFT-P equation.

AB X I 0 X δDpp ´ ω “ ´ , (3.61) B: C Y 0 ´I Y δDhh „  „  „  „  „ 

in which

Aab,cd “ δacδbdpa ` bq ` Lab,cd, (3.62a)

Bab,ij “ Lab,ij, (3.62b)

Cij,kl “ ´δikδjlpi ` jq ` Lij,kl, (3.62c)

Xab “ δκabpωq, (3.62d)

Yij “ δκijpωq, (3.62e)

pp rδD sab “ δDabpωq, (3.62f)

hh rδD sij “ δDijpωq. (3.62g)

Note here, we have further simplified the equation by restricting a ą b, c ą d, i ą j and k ą l. In real atomic or molecular systems, we take the limit that the external pairing field D goes to 0. Therefore, setting the right-hand side of Eq. (3.61) to be zero, we obtain a generalized eigenvalue equation

AB X I 0 X “ ω . (3.63) B: C Y 0 ´I Y „  „  „  „ 

Since the response kernel L is still not well known, if we simply ignore the exchange- correlation part in L in Eq. (3.58), again we arrive at the pp-RPA equation (2.8), with matrix elements being

Aab,cd “ δacδbdpa ` bq ` xab||cdy, (3.64a)

Bab,ij “ xab||ijy, (3.64b)

Cij,kl “ ´δikδjlpi ` jq ` xij||kly. (3.64c)

57 Figure 3.2: Schematic sketch of the relation between pp-RPA, pp-TDA and the approaches to derive them

Up to now, all the derivations on pp-RPA has been presented. They are summa- rized in Figure 3.2.

3.2.6 An Iterative Davidson Method for pp-RPA Motivation

The calculation of excitations does not require the whole eigenvector spectrum. Only those transitions to the states of interest are needed , and usually, we are most inter- ested in a few low-lying states. The time cost for a direct diagonalization approach to solve the pp-RPA equation (Eq. (3.63)) is OpN 6q and the memory space cost

4 is OpN q, with N being the max(Nvir, Nocc). For problems with small number of electrons and also small basis sets, this direct diagonalization approach is applicable. However, for larger systems, the computational cost significantly increases with more electron numbers and basis functions. We now use the basic idea behind the David- son method [104] to solve the lowest few pair addition eigenroots for the pp-RPA equation with an OpN 4q time cost.

58 Derivation

The dimension of the square matrices A and C in Eq. (3.63) are NvirpNvir ´1q{2 and

NoccpNocc ´ 1q{2, respectively, and the dimension of matrix B is NvirpNvir ´ 1q{2 ˆ

NoccpNocc ´ 1q{2. There are in total pNvirpNvir ´ 1q ` NoccpNocc ´ 1qq{2 eigenroots.

Among them, NvirpNvir ´1q{2 eigenvectors normalize to 1, while the rest NoccpNocc ´ 1q{2 normalize to ´1.

T T XI XI ´ YI YI “ ˘1 (3.65)

If a real eigenvector normalizes to 1, it is for a pair addition process and if normalizes

to ´1, it is for a pair removal process. When we calculate excitations with pp-RPA, we usually start with an pN ´ 2q-electron system and then add two electrons back. Therefore, the eigenvalues of our interests are those lowest ones with normalization to 1. We now simplify the notation and write the pp-RPA equation (3.63) as

Mu “ ωWu, (3.66) where M is the pp-RPA matrix rA, B; B:, Cs and W is the diagonal matrix rI, 0; 0, ´Is.

Suppose for an exact eigenpair(ωk,uk), we approximate it by(˜ωk,u˜k) such that u˜k is a linear combination of v1, v2,..., vn:

c1 c2 uk « ˜uk “ v1v2 ¨ ¨ ¨ vn » . fi “ Vc, (3.67) . — ffi “ ‰ —cnffi — ffi – fl where v’s are basis vectors and they are orthornormalized with respect to W,

T T rV WVsij “ δij sgnpvi Wviq, (3.68)

and c’s are linear combination coefficients. Therefore, the pp-RPA eigenvalue equa-

59 tion can be approximated by

MVc “ ω˜kWVc. (3.69)

Multiply VT to the left, we obtain

˜ T Mc “ ω˜kpV WVqc, (3.70)

with

M˜ “ VT MV. (3.71)

The matrix M˜ in Eq.(3.71) has dimension n, which is the number of basis vectors and it is usually much smaller than the original matrix M. By solving the eigenvalue problem (Eq.(3.70)), we are able to obtain the approximate eigenvalueω ˜k and the coefficients c, and therefore the approximate eigenvector u˜k through Eq.(3.67). To test whether u˜k is a good approximation, we calculate the residual rk,

rk “ M˜uk ´ ω˜kW˜uk. (3.72)

If the norm of rk is within a given threshold (|rk ă |), we consider the result to be converged, otherwise, we need to expand the basis vector space to help obtain a better approximation.

Suppose the difference between uk and u˜k is

t “ uk ´ u˜k (3.73)

Then the original eigenvalue problem can be written as

Mp˜uk ` tq “ ωkWp˜uk ` tq (3.74) pM ´ ωkWqt “ ´pM˜uk ´ ωkW˜ukq

Assume the approximated eigenvalueω ˜k is already a good approximation to the real one ωk, we would then need to get t by solving(recall the definition of rk in Eq. 60 (3.72))

pM ´ ω˜kWqt “ ´rk, (3.75)

To strictly solve the difference vector t, we need to calculate the inverse of M´ω˜kW, but it is expensive to do so. Fortunately, following Davidson’s suggestion, a good preconditioner P can be constructed using the diagonal part of M and the diagonal metric matrix W, since usually, the orbital energy parts in A and C are much larger than the other matrix elements,

Pij “ δijpMii ´ ω˜kWiiq. (3.76)

The inverse for P is easy to calculate and thus the i-th element for the approximated difference vector t is ´1 ti « ∆ui “ ´ rP rksi (3.77) rr s “ ´ k i . Mii ´ ω˜kWii As a good approximation to t, the newly calculated vector ∆u can be added to the existing basis vector space V to help further obtain a better approximation to the exact eigenroot pair until finally converged. It is worth noting that the way to augment such a basis vector space is not limited to the Davidson approach only. Other methods to expand the subspace, such as the Jacobi-Davidson approach [105, 106], are also applicable. However, because our goal here is simply to lower the computational cost to run benchmark tests rather than to carry out a method comparison, we only implemented the original Davidson flavor of this subspace expansion.

Detailed work flow

Now let us present our algorithm with a detailed work flow.

1. Perform an SCF calculation for the pN ´ 2q-electron system with HF or a chosen DFT functional.

61 2. Generate an initial guess for the basis vector set. Because we aim for the lowest n pair addition eigenroots, a good initial guess can be generated by sorting

the sum of any two virtual orbital energies εab “ a ` b and getting the lowest m ones (m ě n). Suppose the l’th lowest value is εcd, then the l’th initial basis vector elements are

rvlspq “ δcpδdq. (3.78)

3. Calculate matrix-vector product MV using the approach in Ref. [107]. This is one of the most expensive steps in the whole calculation: for each basis vector, the

time cost is OpN 4q and memory cost is OpN 2q. 4. Calculate vector-vector product M˜ “ VT pMVq. For this step, the time and memory costs are both OpN 2q. 5. Solve the reduced generalized eigenvalue problem (Eq. (3.70)) to obtain ap- proximated eigenvaluesω ˜ and coefficients c. Because the number of basis vectors is much smaller than N, the cost in this step is negligible. 6. Sort the approximated eigenpairs (˜ω,Vc) and pick up the lowest n pair addition ones according to the normalization constraint Eq. (3.65). 7. Calculate residual vectors using Eq. (3.72). Note here, instead of calculating

M˜uk “ MpVcq, we calculate pMVqc, in which MV is already calculated in step 3. Therefore, the cost is also negligible. 8. Calculate the norm of the residual vectors, if all converged, exit. Otherwise, for non-converged roots, calculate the approximated difference vectors ∆u using Eq. (3.77). 9. Orthogonalize ∆u with respect to V using the generalized Gram-Schmidt method. If it is not numerically zero, calculate the self-product with metric W. Normalize it and add the to basis space V. 10. Go back to step 3 and continue the loop.

62 Davidson method for pp-TDA

The pp-TDA is simply solving the equation

Au “ ωu. (3.79)

The matrix A is a diagonally dominated Hermitian matrix. Therefore, the pp-TDA equation can be solved using the canonical Davidson algorithm [104].

3.3 Results

3.3.1 Benchmark Regular Single Excitations Computational Details

We implemented the above Davidson iterative method on the spin-separated and spin-adapted pp-RPA and pp-TDA equations [108] in QM4D package [45]. Then we use it to benchmark excitation energies calculated with pp-RPA and compare with TDDFT results calculated with Gaussian 09 [109]. We use molecules with the number of atoms larger than five in Thiel’s [110, 111] and Tozer’s [112] test sets. If a molecule exists in both test sets, we use Thiel’s geometry and reference values. Limited by the convergence difficulty in the ground state calculations (especially calculations with the PBE functional) for some pN ´ 2q-electron systems, we present a relatively complete comparison and discussion with different functionals for 15 molecules with the aug-cc-pVDZ basis set. There are 24 singlet excitations and 19 triplet excitations in this set. For each excitation, we perform pp-RPA and pp-TDA calculations with HF, B3LYP and PBE references. We also compare these results with the well-known computationally efficient methods including CIS, TDHF, TD- B3LYP and TD-PBE in order to further assess the performance for our approaches.

For the rest molecules, because the PBE functional connot converge their pN ´ 2q- electron systems and methods with HF references (CIS, TDHF, pp-RPA-HF) always give unsatisfactory results, we only present converged pp-RPA-B3LYP results and

63 7.0

1Ag 1Ag 1Ag

8.0

1Bu 1Bu 7.0 1Bu

3Bu 3Bu 3Bu

7.5

3Ag 3Ag 3Ag

6.5

7.0

6.5

6.5

6.0

6.0

6.0

5.5

3.0

5.0 ExcitationEnergy (eV) ExcitationEnergy (eV) ExcitationEnergy (eV)

2.5

2.5

3.5

3.0 2.0

cc-pVDZ cc-pVTZ cc-pVQZ aug-cc-pVDZ cc-pVDZ cc-pVTZ cc-pVQZ aug-cc-pVDZ aug-cc-pVTZ cc-pVDZ cc-pVTZ cc-pVQZ aug-cc-pVDZ aug-cc-pVTZ

Basis Set Basis Set Basis Set (a) butadiene by pp-RPA-HF (b) butadiene by pp-RPA-B3LYP (c) butadiene by pp-RPA-PBE

8.5 7.0

1B2 7.0

8.0

1A1 6.5

3B2 6.5

7.5

3A1

6.0

6.0 7.0

5.5

6.5

5.5

5.0

6.0

5.0

1B2 1B2

4.5 5.5

1A1 1A1

4.5

3B2 3B2 5.0

4.0 ExcitationEnergy (eV) ExcitationEnergy (eV) ExcitationEnergy (eV)

3A1 3A1

4.0

4.5

3.5

3.5

4.0

3.0

cc-pVDZ cc-pVTZ cc-pVQZ aug-cc-pVDZ cc-pVDZ cc-pVTZ cc-pVQZ aug-cc-pVDZ aug-cc-pVTZ cc-pVDZ cc-pVTZ cc-pVQZ aug-cc-pVDZ aug-cc-pVTZ

Basis Set Basis Set Basis Set (d) furan by pp-RPA-HF (e) furan by pp-RPA-B3LYP (f) furan by pp-RPA-PBE Figure 3.3: Basis set convergence test for pp-RPA. All calculations show the lowest two singlet and lowest two triplet excitations. Calculations with B3LYP and PBE references converge fast, with aug-cc-pVDZ showing converged results. Calculations with HF references converge much slower. The importance of diffuse function is also observed in the convergence behavior, with aug-cc-pVDZ basis providing lower excitation energies than cc-pVQZ. compare with TD-B3LYP calculations. Note here, all these excitations chosen are all HOMO excitations or excitations with HOMO excitation characters. HOMO excitations are usually the most important low-lying excitations that are of interest, therefore, although we will miss those non-HOMO excitations using the canonical pp-RPA treatment, in general, we only miss a few important low-lying excitations in the systems we tested.

Basis Set Convergence Test

Basis set convergence is tested along the cc-pVXZ series, X=D,T,Q, as well as the aug-cc-pVXZ series, X=D,T. The QM4D program uses Cartesian atomic orbitals and removes basis functions with angular momentum higher than “f”. We choose butadiene and furan as test molecules. For each molecule, two lowest singlet and two

64 lowest triplet excitations are investigated. The results are shown in Figure 3.3. Because of an SCF convergence failure, aug-cc-pVTZ results for HF reference are missing. It can be seen that for the two DFT references, the excitation energies decrease from cc-pVDZ to cc-pVQZ. The energies further decrease when it comes to aug-cc-pVDZ. The difference between the results of aug-cc-pVTZ and aug-cc-pVDZ are very small(«0.02 eV). Therefore, we can consider that the excitation energy is already converged for the aug-cc-pVDZ basis set. Note here, even though cc-pVTZ and cc-pVQZ have more contracted Gaussian-type orbitals(CGTOs) than aug-cc-pVDZ, they do not reach the basis set convergence for excitation energies. The reason is that excited states are more diffuse than the ground state and in order to describe a balanced ground state and excited states well, adding diffuse functions is more crucial and more efficient than adding angular-momentum functions. For calculations with HF references, we can observe the similar convergence trend. However, it converges much slower than DFT refer- ences. With the aug-cc-pVTZ result missing, we cannot guarantee the aug-cc-pVDZ result is converged. It is very likely that more basis functions are needed consider- ing its slow convergence behaviour from the figure. Since the aug-cc-pVDZ basis is already converged for DFT references and considering the computational cost, we decide to use aug-cc-pVDZ basis in all the rest calculations, even though we cannot guarantee a convergence with the HF reference.

Functional Performance

The calculation results for the 15 molecules are shown in Table 3.1. The pp-RPA with HF references (pp-RPA-HF) accidentally has a 0 eV mean signed error, while the error for (pp-RPA-B3LYP and pp-RPA-PBE) are 0.04 eV and -0.14 eV, respectively. As to mean absolute errors, they are 0.92 eV, 0.40 eV and 0.38 eV, respectively. These numbers mean that for pp-RPA, the calculated results distribute around the

65 benchmark values with very little systematic bias. However, when we look at the mean absolute error, pp-RPA-HF has such a large one that even doubles those of pp-RPA-B3LYP or pp-RPA-PBE. Therefore, pp-RPA-HF is not as accurate as pp- RPA-DFT to treat lowest few single excitations. Moreover, when we analyse data from pp-RPA-HF, it happens in many cases that there lie many excitations with incorrect symmetry below the targeted excitations. This also happens sometimes in TDHF or CIS calculations, which might result from the problem with HF references. For pp-RPA-B3LYP and pp-RPA-PBE, the mean absolute errors are much smaller which means that they are much more accurate. The excitation spectrum is also clean without inserted wrong symmetry states and makes it easier to analyze for these DFT references.

66 Table 3.1: Vertical excitation energies (in eV) from pp-RPA, pp-TDA, CIS, TDHF and TDDFT. Numbers in the parenthesis indicate the error. TDHF instabilities are characterized by imaginary excitation energy, but are denoted with negative numbers and thus having large negative errors.

Molecule Exci Ref RPA-HF RPA-B3LYP RPA-PBE TDA-HF TDA-B3LYP TDA-PBE CIS TDHF TDB3LYP TDPBE 3 Ethene B1u 4.5 3.92(-0.58) 3.62(-0.88) 3.48(-1.02) 3.90(-0.60) 3.58(-0.92) 3.44(-1.06) 3.59(-0.91) 0.74(-3.76) 4.05(-0.45) 4.24(-0.26) 1 Ethene B1u 7.8 6.26(-1.54) 8.45(0.65) 8.85(1.05) 6.24(-1.56) 8.45(0.65) 8.86(1.06) 7.71(-0.09) 7.36(-0.44) 7.38(-0.42) 7.39(-0.41) 3 Butadiene Bu 3.2 3.22(0.02) 2.53(-0.67) 2.28(-0.92) 3.11(-0.09) 2.31(-0.89) 2.02(-1.18) 2.63(-0.57) -2.18(-5.38) 2.79(-0.41) 2.95(-0.25) 3 Butadiene Ag 5.08 5.60(0.52) 6.07(0.99) 5.85(0.77) 5.49(0.41) 5.85(0.77) 5.59(0.51) 4.33(-0.75) 2.95(-2.13) 4.85(-0.23) 4.99(-0.09) 1 Butadiene Bu 6.18 5.49(-0.69) 6.57(0.39) 6.51(0.33) 5.38(-0.80) 6.38(0.20) 6.28(0.10) 6.19(0.01) 5.90(-0.28) 5.56(-0.62) 5.44(-0.74) 1 Butadiene Ag 6.55 5.92(-0.63) 6.44(-0.11) 6.11(-0.44) 5.82(-0.73) 6.30(-0.25) 5.95(-0.60) 7.39(0.84) 7.26(0.71) 6.49(-0.06) 6.11(-0.44) 3 Hexatriene Bu 2.4 2.59(0.19) 1.88(-0.52) 1.61(-0.79) 2.49(0.09) 1.68(-0.72) 1.36(-1.04) 2.11(-0.29) -2.58(-4.98) 2.12(-0.28) 2.28(-0.12) 3 Hexatriene Ag 4.15 5.20(1.05) 4.86(0.71) 4.47(0.32) 5.10(0.95) 4.64(0.49) 4.19(0.04) 3.57(-0.58) 1.56(-2.60) 3.94(-0.21) 4.04(-0.11) 1 Hexatriene Ag 5.09 5.42(0.33) 4.99(-0.10) 4.49(-0.60) 5.34(0.25) 4.86(-0.23) 4.32(-0.77) 6.88(1.79) 6.71(1.62) 5.50(0.41) 5.02(-0.07) 1 Hexatriene Bu 5.1 5.03(-0.07) 5.29(0.19) 5.13(0.03) 4.94(-0.16) 5.09(-0.01) 4.88(-0.22) 5.33(0.23) 5.05(-0.05) 4.60(-0.50) 4.44(-0.66) 3 Octetraene Bu 2.2 2.15(-0.05) 1.49(-0.71) 1.21(-0.99) 2.07(-0.13) 1.31(-0.89) 0.99(-1.21) 1.79(-0.41) -2.74(-4.94) 1.71(-0.49) 1.87(-0.33) 3 Octetraene Ag 3.55 4.75(1.20) 4.00(0.45) 3.57(0.02) 4.66(1.11) 3.80(0.25) 3.30(-0.25) 3.01(-0.54) -1.12(-4.67) 3.26(-0.29) 3.36(-0.19) 1 Octetraene Ag 4.47 5.02(0.55) 4.08(-0.39) 3.53(-0.94) 4.95(0.48) 3.96(-0.51) 3.37(-1.10) 6.39(1.92) 6.32(1.85) 4.80(0.33) 4.17(-0.30) 1 Octetraene Bu 4.66 4.58(-0.08) 4.50(-0.16) 4.29(-0.37) 4.50(-0.16) 4.31(-0.35) 4.05(-0.61) 4.73(0.07) 4.47(-0.19) 3.96(-0.70) 3.78(-0.88) 1 Decapentaene Bu 4.27 4.69(0.42) 4.60(0.33) 4.40(0.13) 4.61(0.34) 4.40(0.13) 4.13(-0.14) 4.88(0.61) 4.63(0.36) 4.07(-0.20) 3.86(-0.41) 3 Cyclopropene B2 4.34 4.22(-0.12) 4.08(-0.26) 3.91(-0.43) 4.14(-0.20) 3.89(-0.45) 3.66(-0.68) 3.46(-0.88) 0.67(-3.67) 3.70(-0.64) 3.79(-0.55) 67 1 Cyclopropene B2 7.06 5.87(-1.19) 7.29(0.23) 7.31(0.25) 5.80(-1.26) 7.12(0.06) 8.15(1.09) 6.67(-0.39) 6.40(-0.66) 6.09(-0.97) 5.91(-1.15) 3 Cyclopentadiene B2 3.25 3.12(-0.13) 2.66(-0.59) 2.53(-0.72) 3.03(-0.22) 2.48(-0.77) 2.31(-0.94) 2.49(-0.76) -2.06(-5.31) 2.74(-0.51) 2.91(-0.34) 3 Cyclopentadiene A1 5.09 5.45(0.36) 5.33(0.24) 5.07(-0.02) 5.36(0.27) 5.15(0.06) 4.86(-0.23) 4.29(-0.80) 2.94(-2.15) 4.75(-0.34) 4.87(-0.22) 1 Cyclopentadiene B2 5.55 5.16(-0.39) 5.46(-0.09) 5.45(-0.10) 5.07(-0.48) 5.31(-0.24) 5.26(-0.29) 5.43(-0.12) 5.12(-0.43) 4.95(-0.60) 4.88(-0.67) 1 Cyclopentadiene A1 6.31 6.01(-0.30) 6.42(0.11) 6.16(-0.15) 5.94(-0.37) 6.33(0.02) 6.05(-0.26) 7.87(1.56) 7.78(1.47) 6.40(0.09) 5.78(-0.53) 3 Norbornadiene A2 3.72 3.77(0.05) 3.68(-0.04) 3.58(-0.14) 3.57(-0.15) 3.36(-0.36) 3.20(-0.52) 2.87(-0.85) -1.48(-5.20) 3.10(-0.62) 3.16(-0.56) 3 Norbornadiene B2 4.16 4.49(0.33) 4.76(0.60) 4.51(0.35) 4.28(0.12) 4.41(0.25) 4.11(-0.05) 3.27(-0.89) -1.32(-5.48) 3.63(-0.53) 3.78(-0.38) 1 Norbornadiene A2 5.34 3.95(-1.39) 5.36(0.02) 5.34(0.00) 3.75(-1.59) 5.05(-0.29) 4.99(-0.35) 5.53(0.19) 5.30(-0.04) 4.70(-0.64) 4.40(-0.94) 1 Norbornadiene B2 6.11 4.58(-1.53) 6.84(0.73) 6.99(0.88) 4.38(-1.73) 6.51(0.40) 6.64(0.53) 7.02(0.91) 6.81(0.70) 5.28(-0.83) 4.94(-1.17) 3 Furan B2 4.17 3.82(-0.35) 3.49(-0.68) 3.37(-0.80) 3.74(-0.43) 3.33(-0.84) 3.17(-1.00) 3.28(-0.89) -1.14(-5.31) 3.70(-0.47) 3.89(-0.28) 3 Furan A1 5.48 5.79(0.31) 5.37(-0.11) 5.12(-0.36) 5.71(0.23) 5.22(-0.26) 4.93(-0.55) 4.87(-0.61) 4.07(-1.41) 5.18(-0.30) 5.25(-0.23 1 Furan B2 6.32 5.08(-1.24) 6.57(0.25) 6.58(0.26) 5.00(-1.32) 6.42(0.10) 6.40(0.08) 6.28(-0.04) 5.97(-0.35) 5.93(-0.39) 5.88(-0.44) 1 Furan A1 6.57 6.61(0.04) 6.88(0.31) 6.65(0.08) 6.54(-0.03) 6.80(0.23) 6.56(-0.01) 7.89(1.32) 7.77(1.20) 6.58(0.01) 6.28(-0.29) 1 Pridazine B1 3.78 1.51(-2.27) 2.68(-1.10) 2.91(-0.87) 1.27(-2.51) 2.33(-1.45) 2.50(-1.28) 4.93(1.15) 4.72(0.94) 3.60(-0.18) 3.15(-0.63) 1 Pridazine A2 4.32 2.91(-1.41) 3.41(-0.91) 3.48(-0.84) 2.60(-1.72) 2.90(-1.42) 2.91(-1.41) 6.13(1.81) 5.95(1.63) 4.20(-0.12) 3.53(-0.79) 3 s-tetrazine B3u 1.89 3.28(1.39) 2.19(0.30) 1.86(-0.03) 2.94(1.05) 1.64(-0.25) 1.24(-0.65) 2.40(0.51) 1.89(0.00) 1.47(-0.42) 1.15(-0.74) 3 s-tetrazine Au 3.52 5.38(1.86) 3.67(0.15) 3.16(-0.36) 4.91(1.39) 2.91(-0.61) 2.30(-1.22) 4.67(1.15) 4.38(0.86) 3.15(-0.37) 2.54(-0.98) 1 s-tetrazine B3u 2.24 3.78(1.54) 2.73(0.49) 2.41(0.17) 3.53(1.29) 2.33(0.09) 1.96(-0.28) 3.54(1.30) 3.32(1.08) 2.27(0.03) 1.85(-0.39) 1 s-tetrazine Au 3.48 5.59(2.11) 3.91(0.43) 3.40(-0.08) 5.13(1.65) 3.18(-0.30) 2.58(-0.90) 5.71(2.23) 5.55(2.07) 3.54(0.06) 2.87(-0.61) 3 Formaldehyde A2 3.5 1.65(-1.85) 3.15(-0.35) 3.40(-0.10) 1.53(-1.97) 2.85(-0.65) 3.04(-0.46) 3.67(0.17) 3.34(-0.16) 3.10(-0.40) 2.97(-0.53) 1 Formaldehyde A2 3.88 2.00(-1.88) 3.68(-0.20) 3.97(0.09) 1.88(-2.00) 3.43(-0.45) 3.68(-0.20) 4.49(0.61) 4.31(0.43) 3.83(-0.05) 3.71(-0.17) 3 Acetone A2 4.05 3.10(-0.95) 4.13(0.08) 4.24(0.19) 2.92(-1.13) 3.78(-0.27) 3.83(-0.22) 4.42(0.37) 4.13(0.08) 3.68(-0.37) 3.52(-0.53) Molecule Exci Ref RPA-HF RPA-B3LYP RPA-PBE TDA-HF TDA-B3LYP TDA-PBE CIS TDHF TDB3LYP TDPBE 1 Acetone A2 4.4 3.38(-1.02) 4.56(0.16) 4.66(0.26) 3.20(-1.20) 4.25(-0.15) 4.31(-0.09) 5.15(0.75) 4.98(0.58) 4.31(-0.09) 4.14(-0.26) 3 Benzoquinone B1g 2.51 4.74(2.23) 2.93(0.42) 2.44(-0.07) 4.07(1.56) 2.05(-0.46) 1.53(-0.98) 3.31(0.80) 3.01(0.50) 1.93(-0.58) 1.42(-1.09) 1 Benzoquinone B1g 2.78 4.98(2.20) 3.14(0.36) 2.65(-0.13) 4.36(1.58) 2.36(-0.42) 1.85(-0.93) 4.00(1.22) 3.84(1.06) 2.43(-0.35) 1.87(-0.91) 1 DMABN A1 4.56 6.14(1.58) 4.96(0.40) 4.56(0.00) 6.04(1.48) 4.70(0.14) 3.90(-0.66) 5.45(0.89) 5.16(0.60) 4.52(-0.04) 4.19(-0.37) 1 DMABN B1 4.25 5.68(1.43) 4.68(0.43) 4.24(-0.01) 5.58(1.33) 4.42(0.17) 4.21(-0.04) 5.54(1.29) 5.35(1.10) 4.39(0.14) 3.97(-0.28) Total MSE 0.00 0.04 -0.14 -0.16 -0.24 -0.44 0.31 -0.95 -0.32 -0.49 Total MAE 0.92 0.40 0.38 0.89 0.43 0.60 0.79 1.82 0.37 0.49 Singlets MSE -0.23 0.10 -0.04 -0.38 -0.16 -0.30 0.84 0.62 -0.24 -0.56 Singlets MAE 1.08 0.36 0.34 1.08 0.34 0.54 0.89 0.83 0.33 0.56 Triplets MSE 0.29 -0.05 -0.27 0.12 -0.34 -0.62 -0.36 -2.93 -0.42 -0.41 Triplets MAE 0.71 0.46 0.44 0.64 0.54 0.67 0.67 3.08 0.42 0.41 68 pp-RPA v.s. pp-TDA

Unlike the results we observed for small molecules, in large systems, there is a rela- tively larger difference between pp-RPA and pp-TDA. The excitation energies calcu- lated from pp-TDA are always lower than their corresponding pp-RPA. Therefore, the mean signed error shifted from close to zero to -0.16 eV „ -0.44 eV depending on the chosen reference. Even with such a shift, the mean absolute error does not change much for HF reference or B3LYP reference. However, for pp-TDA-PBE, both the mean signed error and mean absolute error are much larger than pp-RPA-PBE. These suggest that pp-TDA gives excitation energies slightly lower than pp-RPA and its performance is similar or slightly worse than pp-RPA. This observation is very different from TDDFT, where TDA results are often better than RPA results.

Comparison with Conventional Computationally Efficient Methods

We presented CIS, TDHF, TD-B3LYP, TD-PBE results in the last four columns of the Table 3.1. It can be seen that CIS overestimates excitation energies. Among all the tested methods, TDHF is the only one that has stability issues: instead of real ones, the excitation energies are imaginary (shown with negative numbers in the table). When we calculate the error for TDHF, we use the negative numbers as a punishment for instability issue. Therefore, for TDHF there is a large negative signed error and large mean absolute error. For TD-B3LYP and TD-PBE, they mostly underestimate excitation energies with negative mean signed errors. Compare CIS and pp-RPA-HF, TD-B3LYP and pp-RPA-B3LYP, TD-PBE and pp-RPA-PBE, we can see the mean absolute errors are very similar in each comparison group, which suggests that our pp-RPA methods are comparable to the convention CIS and TDDFT methods for large molecules.

69 1.5

1.5

pp-RPA-B3LYP

pp-RPA-B3LYP

pp-TDA-B3LYP

pp-TDA-B3LYP

1.0 TD-B3LYP 1.0 TD-B3LYP

0.5 0.5

0.0 0.0 Error(eV) Error (eV)Error

-0.5 -0.5

-1.0 -1.0

-1.5 -1.5

2 3 4 5 6 7 8 2 3 4 5 6

Benchmark value (eV) Benchmark value (eV) (a) Singlet excitations (b) Triplet excitations Figure 3.4: Error distributions for pp-RPA-B3LYP, pp-TDA-B3LYP and TD- B3LYP for singlet and triplet excitations. It can be observed that (1) pp-RPA has nearly even distribution around the zero line with no systematic error, while TD-B3LYP tends to underestimate excitations; (2) pp-TDA excitations are lower than pp-RPA with slightly larger error; (3) For the pp- methods, most singlet excita- tions’ errors are within -0.5eV to 0.5eV, they are much better described than triplet excitations with error distributed between -1eV to 1eV.

Singlets v.s. Triplets

We investigated the error for the singlet excitation group and the triplet excita- tion group. For pp-RPA-HF and pp-TDA-HF, the singlet excitations have negative errors while triplet excitations have positive errors, and triplets excitations have slightly smaller mean absolute error. For the two DFT references, the results are op- posite: triplet excitations have more negative errors and larger mean absolute errors. Therefore, singlet excitations are better described by pp-RPA with DFT references. A B3LYP example is shown in Figure 3.4.

Non-HOMO Excitations

It had been our concern for the pp-RPA and pp-TDA methods because within our regular treatment, non-HOMO excitations cannot be captured. When the molecule gets larger, the non-HOMO excitations might play a more important role. However, according to the large systems we investigated, the lowest few excitations very com-

70 monly has some HOMO excitation characters. To our surprise, even if the HOMO excitation contribution is as small as half in a TDDFT calculation, our approach still can capture that state well with a reasonably good excitation energy. These excita- tions with significant non-HOMO excitation characters are marked with underlines in the Table 3.1. This phenomenon is very similar to double excitations for TDDFT: although TDDFT cannot capture pure double excitations, it can accurately predict the excitation energies for many excitations with both double and single excitation characters. [113] Therefore, even though our methods with regular treatments can- not capture those excitations with almost pure non-HOMO excitation characters, they are reliable in predicting most of the low-lying excitations with a significant HOMO-excitations characters. Note there is another way to capture non-HOMO ex- citations with pp-RPA: we can use the non-ground state of pN ´ 2q-electron systems as the reference, somewhat like the use of HF*.

Table 3.2: Vertical excitation energies (in eV) from pp- RPA-B3LYP and TD-B3LYP. Numbers in the parenthe- sis indicate the error. Only B3LYP results are reported for the following molecules because PBE cannot converge their corresponding pN ´ 2q-electron systems.

Molecule Exci Ref RPA-B3LYP TD-B3LYP 3 Pyrrole B2 4.48 3.79(-0.69) 4.07(-0.41) 3 Pyrrole A1 5.51 5.28(-0.23) 5.22(-0.29) 1 Pyrrole A1 6.37 6.69(0.32) 6.33(-0.04) 1 Pyrrole B2 6.57 6.80(0.23) 5.98(-0.59) Imidazole 3A1 4.69 3.94(-0.75) 4.24(-0.45) Imidazole 3A1 5.79 5.14(-0.65) 5.35(-0.44) Imidazole 1A1 6.19 6.54(0.35) 6.16(-0.04) Imidazole 1A1 6.93 7.27(0.34) 6.35(-0.58) 3 Pyridine A1 4.06 4.01(-0.05) 3.91(-0.15) 3 Pyridine B2 4.64 3.95(-0.69) 4.46(-0.18) 1 Pyridine B2 4.85 6.00(1.15) 5.45(0.60) 1 Pyridine A1 6.26 6.97(0.71) 6.19(-0.07) 1 Pyrazine B3u 3.95 4.41(0.46) 3.93(-0.02) 1 Pyrazine Au 4.81 5.15(0.34) 4.69(-0.12) 1 Pyrimidine B1 4.55 4.32(-0.23) 4.25(-0.30)

71 Molecule Exci Ref RPA-B3LYP TD-B3LYP 1 Pyrimidine A2 4.91 4.84(-0.07) 4.60(-0.31) Formamide 3A1 5.74 4.39(-1.35) 5.10(-0.64) Formamide 1A1 7.39 8.21(0.82) 7.53(0.14) Acetamide 3A1 5.88 5.49(-0.39) 5.25(-0.63) Acetamide 1A1 7.27 6.68(-0.59) 7.10(-0.17) Propanamide 3A1 5.9 6.47(0.57) 5.27(-0.63) Propanamide 1A1 7.2 7.33(0.13) 7.00(-0.20) Cytosine 1A1 4.66 4.74(0.08) 4.59(-0.07) Thymine 1A1 5.2 5.57(0.37) 4.89(-0.31) Uracil 1A1 5.35 5.90(0.55) 5.08(-0.27) Adenine 1A1 5.25 5.16(-0.09) 4.91(-0.34) 1 N-phenylpyrrole B2 4.85 4.83(-0.02) 4.51(-0.34) 1 N-phenylpyrrole A1 5.13 5.23(0.10) 4.57(-0.56) 1 N-phenylpyrrole A1 5.94 6.19(0.25) 4.87(-1.07) MSE 0.03 -0.29 MAE 0.43 0.34

Molecules with N-2 Convergence Difficulties

The starting points for pp-RPA calculations are self-converged pN ´ 2q-electron sys- tems. Most systems of interest are closed-shell systems, therefore, the correspond-

ing pN ´ 2q-electron systems sometimes show a diradical nature if the HOMO and HOMO-1 orbitals are very close. This diradical nature makes it difficult to converge pN ´ 2q-electron systems. One extreme case is molecules with high symmetry. If the HOMO and HOMO-1 happen to be degenerate, then the pN ´ 2q SCF calculation will be difficult. Even if in some cases convergence can be reached, the orbital degenerate symmetry is not conserved. This category includes molecules such as benzene and triazine. For them, we still cannot solve the problem gracefully with the current approach. However, fortunately, these highly degenerate cases are rare, especially in large molecules.

There are also some molecules without high symmetry but with pN ´ 2q con- vergence difficulties. This often happens with the PBE functional. However, when some Hartree-Fock exchange is present, which expands the HOMO-LUMO gap, SCF

72 convergence is easier to reach. In Table 3.2, we show the molecules that cannot con- verge with PBE functional but can converge with B3LYP functional. It can be seen that in this set, pp-RPA-B3LYP still has small mean signed error and TD-B3LYP has a negative error. As to the absolute error, pp-RPA-B3LYP is 0.03 eV larger than the set in Table 3.1 while TD-B3LYP is 0.03 eV smaller which means pp-RPA does not perform as well as TD-B3LYP in this set. This relatively large error for pp-RPA might be related the pN ´ 2q-electron system: PBE cannot converge, therefore, we anticipate the converged B3LYP results might also be not good enough. However, the 0.43 eV mean absolute error for pp-RPA-B3LYP is acceptable although it is larger than 0.34 eV for TD-B3LYP.

Conclusion

We carried out benchmark tests on pp-RPA and pp-TDA on HF, B3LYP and PBE references. Despite some convergence problems with pN ´2q-electron systems, we are able to accurately capture lowest few excitations for systems with converged SCF calculations. Among them, DFT references have a better performance with both clean ordered spectrum and accurate excitation energies. Excitations calculated from pp-TDA are lower than pp-RPA. Compared to TDDFT, there is no systematic bias for pp-RPA with the mean signed error close to zero. The mean absolute error with B3LYP or PBE references is similar to that of TDDFT, which suggests the pp- RPA is a comparable method to TDDFT for the test molecules. Moreover, despite some concerns for non-HOMO excitations, in many cases, excitations with relatively large non-HOMO excitation contributions are also well described in terms of the excitation energy, as long as there is also a considerably amount of HOMO excitation contribution. Therefore, the pp-RPA is also a reliable method to solve the lowest few excitations problems in large systems.

73 3.3.2 Double Excitations

Although the pp-RPA can also well describe those HOMO-related single excitations, the main strength of the pp-RPA is that it can capture double excitations accurately (Table 3.3). For beryllium, with HF and HF* references, the errors for double excita- tions are within 0.1 eV. With the B3LYP reference, the errors are slightly larger, but also within 1 eV. In this few-electron atomic system, the pp-RPA and the pp-TDA hardly show any differences (<0.01 eV). For BH, the pp-RPA also captures double ex- citations. Compared to EOM-CCSD(T) results, HF and HF* references show errors of about 0.2 eV and B3LYP references show errors of about 0.1 eV. The excitations with double excitation character in all-trans- 1,3-butadiene and 1,3,5-hexatriene are also captured. Compared to experimental data or accurate ab initio methods, HF and B3LYP references give relatively accurate results, while the HF* reference over- estimates the excitation energies by about 1.5-2 eV. The pp-RPA and the pp-TDA show some differences («0.1 eV) in these larger systems, but the differences are still too small to conclude which approximation is better.

3.3.3 Rydberg Excitations

The pp-RPA can describe Rydberg excitations well (Table 3.4). With HF or HF* references, the pp-RPA describes Rydberg excitation energies within 0.03 eV for beryllium. For open-shell lithium, in spite of some spin contamination, results are also in good agreement with experimental data, with errors smaller than 0.1 eV.

For molecules such as N2, errors are about 1.2 eV, which is better than TDLDA results (« 2 eV) [118]. Calculations with the B3LYP reference overestimate Rydberg excitations and do not perform as well as with the HF reference. A more careful investigation on Rydberg excitations has been carried out with the atomic quantum defect series. [120]

74 Table 3.3: Lowest double excitations or excitations with double excitation character (in eV)

Term Standard RPA-HF TDA-HF RPA-B3LYP RPA-HF* Be 1D 7.05 7.06 7.06 7.97 7.06 3P 7.40 7.45 7.45 7.84 7.45 BH 3Σ 5.04 5.51 5.48 5.12 5.53 1∆ 6.06 6.15 6.12 5.98 6.18 1Σ 7.20 7.10 7.11 7.05 7.22 butadiene 1 Ag 6.55 5.93 5.83 6.47 7.93 hexatriene 1 Ag 5.21 5.43 5.34 5.01 7.46

RPA is short for pp-RPA and TDA is short for pp-TDA. Standard values are experimental data for Be [114] and hexatriene [115], EOM-CCSD(T)/cc-pVQZ results for BH [116], MR-CISD(Q) results for butadiene [117].

Table 3.4: Rydberg excitations (in eV)

Transition Term Standard RPA-HF TDA-HF RPA-HF* Be 2sÑ6s 3S 8.82 8.79 8.79 8.79 2sÑ6s 1S 8.84 8.81 8.81 8.81 2sÑ6p 3P 8.89 8.87 8.87 8.87 2sÑ6p 1P 8.90 8.87 8.87 8.87 2sÑ6d 3D 8.93 8.91 8.91 8.91 2sÑ6d 1D 8.96 8.95 8.95 8.95 Li 2sÑ6s 2S 4.96 4.97 4.97 - 2sÑ6p 2P 5.01 5.05 5.05 - 2sÑ6d 2D 5.01 5.03 5.03 -

N2 3 + σg Ñ 3sσg Σg 12.0 10.97 10.39 - 1 + σg Ñ 3sσg Σg 12.2 11.07 10.69 - 1 σg Ñ 3pπu Πu 12.90 11.62 11.26 - 1 + σg Ñ 3pσu Σu 12.98 11.63 11.29 -

Standard values are all experimental data [114, 119]. Li has no hole-hole pairs and consequently RPA and TDA calculations are the same. Since our current implementation on the pp-RPA-HF* is only for closed-shell systems with non-degenerate HOMO orbitals, so no data are available for Li nor for N2 because its HF HOMO orbitals are degenerate π, which is incorrect.

75 3.3.4 CT Excitations

The pp-RPA is capable of describing CT excitations (Figure 3.5 for C2H4 ¨ C2F4 and

Figure 3.6 for He2). The computed CT excitations show exact 1/R dependence, with R the separation distance. Other non-CT excitations remain constant with respect

to R. In these two systems, because HF and DFT calculations on the pN ´2q-electron references give nearly degenerate and delocalized HOMO and LUMO orbitals and are hard to converge, we cannot perform further pp-RPA calculations. Fortunately, with the HF* reference, the HOMO orbital for the N-electron system is non-degenerate and localized and can be treated as unoccupied. Through pairing matrix fluctuation, two electrons can be added either both to the same molecule, thus describing non-CT excitations, or to a different molecule each, thus describing CT excitations.

3.3.5 Oscillator Strengths

Oscillator strengths can be calculated with the pp-TDA. As the eigenvectors X de-

scribe transition amplitudes between the pN ´ 2q-electron ground state and all the N-electron states, we can calculate all the N-electron wavefunctions and transition

dipoles between any two N-electron states. A test on H2 agrees well with TDHF results (Table 3.5).

3.3.6 Excitations from (N+2)-Electron References

We also performed preliminary tests on the two electron-removal part with pN ` 2q- electron references (Table 3.6). On the whole, the results has relatively large errors. However, from these two limited cases, it seems that the HF reference is better for HOMO excitations, while the B3LYP reference is better for lower-orbital excitations.

76 19

18

17

16

15

14

13

Excitationenergy (eV) 12

11

10

5 6 7 8 9 10

Distance (Ang)

Figure 3.5: CT and non-CT excitations calculated with pp-RPA-HF* for C2H4 ¨ C2F4 (The lowest non-CT excitation is not shown). Non-CT excitations are denoted with black solid lines and they show a constant behavior. CT excitations are denoted with red dashed lines and they increase when distance increases. This increasing behavior is “parallel” to the dotted blue line, which is a shifted 1/R reference.

Table 3.5: Oscillator strengths (in A.U.) for H2 calculated with the pp-TDA-HF* and TDHF

Transition pp-TDA-HF* TDHF HOMOÑ HOMO+1 0.28 0.29 HOMOÑ HOMO+2 0 0 HOMOÑ HOMO+3 0.35 0.35 HOMOÑ HOMO+4 0 0 HOMOÑ HOMO+5 0.03 0.04 HOMOÑ HOMO+6 0.82 0.83 HOMOÑ HOMO+7 0.83 0.83 HOMOÑ HOMO+8 0 0

Both calculations are performed using the 6-31++G** basis set. The pp-TDA shows good agree- ment with TDHF.

77 1.2

1.0

0.8

CT1

0.6

CT2

CT3

1/R

0.4 Excitationenergy (eV)

0.2

0.0

10 15 20 25 30 35 40

Distance (Ang)

Figure 3.6: CT excitations calculated with pp-RPA-HF* for He2. Excitation en- ergies for the three CT states at 9 Aare˚ all set to zero. The 1/R reference is also shifted to zero at 9 A.˚ All these three CT excitations show an exact 1/R behavior.

Table 3.6: Excitations for S and O atoms (in eV)

Configuration Term Expt RPA-HF TDA-HF RPA-B3LYP TDA-B3LYP S 3s23p4 1D 1.15 1.07 1.13 1.04 1.11 3s23p4 1S 2.75 1.93 2.26 1.71 2.08 3s3p5 3P˝ 8.93 13.15 13.08 10.17 10.08 O 2s22p4 1D 1.97 1.49 1.59 1.43 1.56 2s22p4 1S 4.19 2.70 3.11 2.39 2.83 2s2p5 3P˝ 15.66 19.28 19.16 14.87 14.71

Experimental values are from Ref. [114]. All calculations start with pN `2q-electron references and use the cc-pVQZ basis set.

78 3.3.7 Conclusion

The pairing matrix fluctuation and its simple approximation – pp-RPA has been ap- plied to excitations problems. The pp-RPA can be seen as either a single-reference counterpart to the DIP/DEA-EOM-CC, or a particle-particle channel counterpart of CIS, TDHF, and TDDFT. The pp-RPA adopts a non-ground-state single-determinant starting point, and therefore has a similar philosophy to SF-TDDFT. The canonical pp-RPA can describe HOMO-dominated single, double, Rydberg, and CT excita- tions accurately. The excitations dominated from non-HOMOs can may be reached by adopting more flexible references. The pp-RPA has a computational cost simi- lar to TDDFT. Therefore, the pp-RPA is promising for general practical excitation calculations.

79 4

Application to Diradicals

This chapter is mainly adapted from the following journal article,

• Yang Yang, Degao Peng, Ernest Davidson, and Weitao Yang, “Singlet-triplet energy gaps for diradicals from particle-particle random phase approximation” J. Phys. Chem. A 119, 4923 (2015)

4.1 Introduction

Diradicals are molecules with two electrons in degenerate or nearly degenerate molec- ular orbitals [121, 122]. They can be short-lived reactive species that play an im- portant role in chemical reactions, such as carbenes [123]. They can also be rela- tively stable species that can possibly be used in material sciences, such as graphene fragments [124]. In a diradical, the various possibilities of alignments for the two non-bonding electrons give rise to its versatile chemistry behaviors. Therefore, it is of particular interest to study diradicals both experimentally and theoretically. In a simplest two-orbital diradical model, there are six possible occupation con- figurations corresponding to six determinants (Figure 4.1). Through linear combi-

80 nation, six spin-adapted configurations can be generated. Of these six spin-adapted ones, there are three triplet states with the same energy and three singlet states usually with different energies. In a simple HF analysis, the alignment of these four energy levels depends on the orbital energies, the coulomb repulsion energy within the same orbital and between two orbitals, as well as the exchange energy between the two orbitals. In general, if we ignore the symmetry and consider all the possible configuration interactions within the two-orbital two-electron model, there is a general form for the three singlet states,

¯ cab ¯ |Ψy “caa|aa¯y ` cbb|bby ` ? |aby ´ |ab¯ y 2 ` ˘ c ` c c ´ c c “ aa bb |aa¯y ` |b¯by ` aa bb |aa¯y ´ |b¯by ` ?ab |a¯by ´ |ab¯ y (4.1) 2 2 2 ` ˘ ` ˘ ` ˘ “C1Φ1 ` C2Φ2 ` C3Φ3,

with |aa¯y ` |b¯by Φ1 “ ? , (4.2a) 2

|aa¯y ´ |b¯by Φ2 “ ? , (4.2b) 2

|a¯by ´ |ab¯ y Φ3 “ ? . (4.2c) 2

Under a transform, for example, the following one,

|ay ` |by |Ay “ ? , (4.3a) 2

|ay ´ |by |By “ ? , (4.3b) 2

81 the three component terms become

|AA¯y ` |BB¯y Φ1 “ ? , (4.4a) 2

|AB¯y ´ |AB¯ y Φ2 “ ? , (4.4b) 2

|AA¯y ´ |BB¯y Φ3 “ ? . (4.4c) 2

The form of the first term is unchanged and so are the triplet terms not shown here. This holds true for general unitary transforms. These transformed orbitals are useful for understanding and analysing results in specific systems, although these orbitals do not necessarily diagonalize any one-electron Hamiltonian and cannot be assigned orbital energies from eigenvalues. In most practical cases, the two lowest configurations are involved with the three triplet states and a singlet state. This singlet state can be either a closed-shell singlet (for example, methylene [125]), or possibly be an open-shell singlet (for example, cyclobutadiene). The energy difference between the triplets and the low-lying singlet differs from species to species, and can be used to predict the electronic properties and chemical reactivities of a given diradical. Despite the long history of diradical problems [122, 126, 127, 128], it still remains a hot topic to pursue a theoretical method that can efficiently and accurately de- scribe a diradical system, especially its open-shell singlet state. Because of the energy near-degeneracy of the orbitals that diagonalize the one-body effective Hamiltonian, the open-shell singlet state has strong static correlation. In other words, the state has strong multi-configuration nature, and in order for it to be theoretically well de- scribed, multi-reference methods are required. The commonly used multi-reference wave function methods include CASSCF, CASPT2 [66], MR-CI [129], and MR-CC [130, 131]. These methods are generally accurate. However, they are computation-

82 Figure 4.1: Two-orbital diradical model. Of the six possible configurations, there are two singlets, two triplets and two broken-symmetry configurations. The broken- symmetry configurations can linear combine and form a singlet and a triplet. The true singlet wavefunctions can be obtained through configuration interaction of the singlet configurations. ally demanding and therefore limited to relatively small systems. Another category of methods that are commonly used is usually called the broken-symmetry (BS) approach [132, 133, 134, 135]. In this approach, instead of directly describing the challenging open-shell singlet state, one uses unrestricted density functional theory (UDFT) or unrestricted coupled cluster theory (UCC) to describe one of the broken- symmetry states (Figure 4.1), which is roughly half open-shell singlet contaminated by half triplet state and energetically lies approximately half way in between the triplets and the open-shell singlet state. The amount of spin-contamination can be further corrected by spin-projection [136, 137]. Despite its low cost and great popularity, the broken-symmetry approach is an indirect method of obtaining the open-shell singlet, whose SCF calculation only targets on the spin-contaminated BS state. Moreover, the spin projected DFT is also reported to lead to degraded poten- tial energy surfaces [138].

83 In the past few years, some new methods have been developed and shown their strength in describing diradical systems. One of them is the fractional-spin method [139] or the variational fractional-spin method [140]. They are able to directly obtain the open-shell singlet state without spin contamination using a fractionally occupied non-Aufbau state. This category of methods is the fractional-spin version of ∆SCF

approach [32, 132] and it can obtain good singlet-triplet (ST) gaps (ES ´ ET ) for many types of diradicals [139, 140]. However, due to the limitations of current DFAs, the (variational) fraction-spin method suffers from DFA’s intrinsic static correlation error [21, 141] and therefore cannot well describe diradicals with disjoint features [140]. Another category of new methods uses the concept of spin flip [82, 83]. These methods include SF-EOM-CC [142, 143, 144], spin-flip configuration interaction (SF- CI) [145, 146], restricted active space spin-flip configuration interaction (RAS-SF- CI)[147, 148] and SF-TDDFT with either collinear [99] or non-collinear kernel[149, 150, 151]. These spin-flip methods start with an open-shell high-spin triplet and obtain singlet states through spin-flip excitation operators. With an unrestricted initial reference, these spin-flip methods can obtain improved results over the BS approach for diradicals [99, 146, 147, 152, 153, 154, 155] with tolerably small amount of spin contamination. Among these methods, SF-CIS and SF-TDDFT are able

to achieve low OpN 4q scaling, which makes them applicable to large systems. The introduction of non-collinear kernel [149, 150, 151] improved the original collinear SF-TDDFT, although at the cost of being more functional sensitive [156]. With a restricted open-shell calculation and by applying spin-adapted operators and the tensor reference, one can formulate spin-pure and spin-complete SF-TDDFT [157, 158, 159].

84 Figure 4.2: pp-RPA combines DFT with wave function methods. The pN ´ 2q- electron system is described by single reference DFT. The two electrons added in can form any two-electron configuration within the virtual space, resembling a subspace configuration interaction singles and doubles (CISD).

4.2 Methods

In a diradical system, when we remove the two non-bonding electrons, the remaining

(N ´ 2)-electron system is usually a well-behaved closed-shell singlet species with- out dramatic static correlation. Therefore, the (N ´ 2)-electron reference can be well described by some well-known density functionals, such as B3LYP [160, 161] or PBE [46]. In contrast, the two non-bonding electrons, which dominate the chemical behavior of a diradical, show strong static correlation and need more accurate multi- reference descriptions, such as CASSCF, CASPT2 or MR-CI. Through a seamless

combination of DFT for the (N ´2)-electron reference and a subspace CI for the two non-bonding electrons (Figure 4.2), the pp-RPA is expected to describe diradicals well with a low cost.

+ We report our results on four diatomic diradicals (NH, OH , NF, O2), four + + carbene-like diradicals (CH2, NH2 , SiH2, PH2 ), three disjoint diradicals

(. CH2CH2CH2 ¨ , . CH2CH2CH2CH2CH2CH2 ¨ , . CH2CH2CH2CH2CH2C(CH3)H ¨ ), two

85 four-electron diradicals (cyclobutadiene (C4H4) and TMM (C(CH2)3)), and three benzyenes (o-, m-, p-). ST gaps for these diradicals are calculated through pp-RPA with HF, B3LYP and PBE functionals, which include different percentage of HF exchange. The aug-cc-pVDZ basis set, which has been shown in our previous paper [162] to give converged excitation energies is adopted in all pp-RPA calculations.

Vertical ST gaps Egv are directly calculated with Eq. (3.1), while adiabatic gaps

Ega are calculated from Egv, plus a correction on the potential energy curves ob- tained from SCF calculations for the N-electron system. Two flavors of correction can be employed, either on the triplet energy curve (4.5a) or on the singlet energy curve (4.5b), depending on which N-electron spin state can be better described by the functional used.

Ega “ ES,Sgeo ´ ET,T geo “Egv,Sgeo ` pET,Sgeo ´ ET,T geoq, (4.5a)

Ega “ ES,Sgeo ´ ET,T geo “Egv,T geo ` pES,Sgeo ´ ES,T geoq. (4.5b)

In most cases, the N-electron triplet state is better described by single reference methods. Therefore, the correction based on equation (4.5a) is adopted unless specif- ically specified.

4.3 Results and Discussions

4.3.1 Diatomic Diradicals

The test set of diatomic diradicals are adopted from a previous fractional-spin study [139]. This set of diradicals was also well investigated and benchmarked earlier by spin-flip methods [152, 150]. There are four diradicals in this set: NH, OH+, NF and O2, which are two pairs of isoelectronic species. Although their geometries are simple and characterized by a single bond length, their electronic structures are non-trivial. For these diradicals, the configurations Φ2 and Φ3 in Eq. (4.2) and (4.4) span a degenerate irreducible representation, generating a two-fold degenerate singlet

86 1∆ state. The triplet 3Σ state is below the 1∆ state, while the singlet 1Σ state is above it. The calculated singlet-triplet (1∆ ´3 Σ) gaps are summarized in Table 4.1. Overall, there is a good agreement between pp-RPA calculations and experimental results. With the aug-cc-pVDZ basis set, the pp-RPA with HF reference (pp-RPA-

HF, or pp-HF for short), which has no dynamic correlations for the (N ´2)-reference, also performs well for these small molecules with a small underestimation of the ST gaps. The pp-RPA with DFT references performs better and the mean absolute error for pp-RPA-B3LYP is as small as 2.8 kcal/mol. These pp-RPA results are better than SF-CIS (MAE 6.3 kcal/mol) and comparable with (V)FS-PBE (MAE 3.4 kcal/mol). However, SF-CIS(D)/cc-pVDZ and SF-TDLDA/TZ2P with non-collinear kernel (NC-SF-TDLDA) have mean absolute error even below 2 kcal/mol, which shows their good performances in these very small diradicals.

4.3.2 Carbene-like Diradicals

Carbene and carbene-like diradicals are important species that are widely used in organic chemistry [123]. Here we choose a well-studied set [139, 140, 152, 156] for

+ + testing purpose. This set includes CH2, NH2 , SiH2 and PH2 , which are also two pairs of isoelectrionic molecules. Unlike the previous diatomic diradicals, carbene- like diradicals have non-degenerate frontier orbitals — one σ orbital and one π orbital — with different symmetry, which means the open-shell singlet (σ1π1) and closed- shell singlet (σ2 or π2) cannot further mix through configuration interaction. Take

3 methylene as an example. The lowest state is triplet B1 with one electron occupying

1 σ and another one occupying π. Then the closed-shell singlet A1 comes next with mostly the σ orbital doubly occupied and a small contribution from π orbital doubly

1 occupied. The open-shell singlet B1 with singly occupied σ and π is the third state

1 and finally there is another closed-shell A1 which is mostly the π orbital doubly occupied. In spite of the relatively simple orbital picture, the energy alignment might

87 change greatly depending on the center atom and the surrounding environments. For

+ 3 1 example, in PH2 and SiH2, the order of B1 and the first A1 is switched compared + to CH2 and NH2 , resulting in a negative ST gap; Chen et. al. [125] also have

1 theoretically attempted to lower the energy of the π-doubly-occupied A1 state to make it the ground state by a proper choice of ligand groups.

88 Table 4.1: Adiabatic singlet-triplet gaps (in kcal/mol) for diatomic diradicalsa

Exptb pp-HF pp-B3LYP pp-PBE (V)FS-PBEc SF-CISd SF-CIS(D)d SF-LDAe NH 35.9 30.9 38.5 40.5 41.1 40.7 37.1 35.6 OH+ 50.5 45.5 52.3 54.2 54.8 53.6 50.2 48.8 NF 34.3 28.6 28.7 28.3 34.0 40.8 35.3 30.3 O2 22.6 23.1 23.6 23.5 26.2 33.4 24.6 23.7 MAE 4.1 2.8 3.8 3.4 6.3 1.1 1.8

a The geometries for pp-RPA calculations are adopted from Ref. [139]. Aug-cc-pVDZ basis set is used for all pp-RPA calculations. The adiabatic gap correction is based on Eq.(4.5a); b Experimental values from Ref. [163]; c (Variational) fractional-spin results from Ref. [140] with 6-311++G(2d,2p) basis set; d Spin-flip CI results from Ref. [152] with cc-pVQZ basis set; e Spin-flip TDLDA with non-collinear kernel results from Ref. [150] with TZ2P basis set.

Table 4.2: Adiabatic singlet-triplet gaps (in kcal/mol) for carbene-like diradicalsa

b c d f f g g g

89 Ref.1 Ref.2 pp-HF pp-B3LYP pp-PBE (V)FS-PBE SF-CIS SF-CIS(D) SF-LDA SF-PBE SF-B3LYP 1 A1 9.0 9.7 -2.4 3.7 5.9 15.7 20.4 14.1 11.8 12.3 0.4 1 CH2 B1 31.7 32.5 26.3 32.0 33.5 36.2 43.2 38.0 30.7 32.3 23.2 1 e 2 A1 - 58.3 45.6 52.9 54.8 - 83.0 68.1 61.7 64.9 46.1 1 A1 29.0 28.9 12.8 23.1 25.7 35.5 38.6 30.9 29.6 31.3 17.3 + 1 NH2 B1 43.6 43.0 36.5 42.0 43.7 47.8 49.6 45.2 40.8 43.3 31.8 1 e 2 A1 - 76.5 62.3 70.2 72.5 - 100.9 83.8 81.2 86.1 62.9 1 A1 -21.0 -20.6 -24.9 -28.7 -28.3 -16.1 -11.6 -17.9 -21.5 -22.2 -30.1 1 SiH2 B1 23.5 24.1 24.0 26.3 26.4 24.0 39.2 31.0 19.0 20.3 16.3 1 e 2 A1 - 57.0 51.7 54.0 52.7 - 79.4 70.9 52.4 56.2 46.6 1 A1 -17.0 -18.3 -25.2 -24.6 -23.5 -12.8 -8.9 -15.7 -19.3 -19.0 -29.2 + 1 PH2 B1 27.0 27.6 28.9 30.1 30.2 28.8 41.0 33.5 22.4 24.5 18.4 1 e 2 A1 - 65.6 60.6 61.1 59.9 - 95.8 76.9 58.8 64.6 51.5 MAE1h 6.8 4.3 3.5 4.2 10.7 4.2 2.4 1.9 9.7 MAE2i 7.6 4.3 3.7 - 15.5 6.2 3.2 2.7 10.8

a The geometries for pp-RPA calculations are adopted from Ref. [152]. Aug-cc-pVDZ basis set is used for all pp-RPA calculations. The adiabatic gap correction is b + based on Eq.(4.5a); Experimental values for CH2 from Ref. [164]. MR-CI values for NH2 from Ref. [165]. Experimental values for SiH2 from Refs. [166, 167]. + c d Experimental values for PH2 from Ref. [168]; EOM-SF-CCSD(dT)/aug-cc-pVQZ results from Ref. [156]; (Variational) fractional-spin results from Ref. [140] with e 1 6-311++G(2d,2p) basis set; (Variational) fractional-spin methods are not able to capture the 2 A1 state; f. Spin-flip CI results from Ref. [152] with TZ2P basis set; g 1 Spin-flip TDDFT with non-collinear kernel results from Ref. [156] with cc-pVTZ basis set. h. Due to limited data in Reference 1, 2 A1 data is not counted in when i 1 MAE is calculated; All excitation values are taking into account when comparing with Reference 2. Error for (V)FS-PBE is not calculated because of the lack of 2 A1 data. Table 4.3: Vertical singlet-triplet gaps (in kcal/mol) for disjoint diradicalsa

Ref.b pp-HF pp-B3LYP pp-PBE (V)FS-PBEc SF-LDAd SF-PBEd SF-B3LYPd SF-ωPBEhd . CH2CH2CH2 ¨ 1.8 2.4 4.4 5.4 20.4 - - - - 1 A1 0.0 0.0 0.1 0.1 - 0.0 -0.1 0.0 0.1 1 . CH2(CH2)4CH2 ¨ B1 145.5 79.9 147.4 159.2 - 33.6 36.8 66.3 126.8 1 2 A1 150.1 80.6 149.0 161.9 - 36.1 38.9 68.1 129.2 1 A1 -0.2 -0.3 0.1 0.4 - -1.5 -1.2 -1.1 -0.6 1 . CH2(CH2)4C(CH3)H ¨ 2 A1 131.1 66.1 129.0 139.8 - 26.9 23.2 54.8 113.3 1 3 A1 144.3 78.3 142.1 152.3 - 46.4 44.6 74.5 127.6

a For pp-RPA calculations, the geometry of . CH2CH2CH2 ¨ is adopted from Ref. [140] and the geometries of . CH2(CH2)4CH2 ¨ and . CH2(CH2)4C(CH3)H ¨ are b adopted from Ref. [156]. Aug-cc-pVDZ basis set is used for all pp-RPA calculations; CASPT2(2,4)/6-311++G(2d,2p) result for . CH2CH2CH2 ¨ from Ref. [140]. c EOM-SF-CCSD(dT)/6-311G(d) results for . CH2(CH2)4CH2 ¨ and . CH2(CH2)4C(CH3)H ¨ from Ref. [156]; (Variational) fractional-spin result from Ref. [140] with 6-311++G(2d,2p) basis set; d SF-TDDFT with non-collinear kernel results from Ref. [156] with cc-pVTZ basis set.

Table 4.4: Singlet-triplet gapsa (in kcal/mol) for four-π-electron diradicalsb 90

Ref1c Ref2d pp-HF pp-B3LYP pp-PBE FS-PBEe SF-CISf SF-CIS(D)f SF-LDAg SF-PBEg SF-B3LYPg SF-ωPBEhg 1 B1 16.7 16.0 17.1 16.3 15.6 21.5 23.5 23.6 15.6 17.2 17.0 16.4 1 TMM A1 19.1 19.5 26.4 33.9 35.5 - 20.4 20.6 13.0 13.3 15.9 18.8 1 2 A1 - 79.5 76.6 107.3 114.4 - 151.7 82.3 30.5 33.9 46.6 68.7 Cyclobutadiene -8.1 7.2 6.7 6.5 16.5 ------

a Adiabatic gaps for TMM (correction based on Eq.(4.5a)) and vertical gaps for square cyclobutadiene; b For pp-RPA calculations, the geometry of TMM and Cyclobu- tadiene are adopted from Refs. [152] and [135], respectively. Aug-cc-pVDZ basis set is used for all pp-RPA calculations; c MCQDPT2(10,10)/cc-pVTZ results for TMM from Ref. [152]. CASSCF/MkCCSD result for Cyclobutadiene from Ref. [135]; d EOM-SF-CCSD(dT)/6-311G(d) results from Ref. [156]; e (Variational) fractional-spin results from Ref. [140] with 6-311++G(2d,2p) basis set; f Spin-flip CI results from Ref. [152] with cc-pVQZ basis set; g SF-TDDFT with non-collinear kernel results from Ref. [156] with cc-pVTZ basis set. Table 4.5: Adiabatic Singlet-triplet gaps (in kcal/mol) for benzynes a

Exptb pp-HF pp-B3LYP pp-PBE SF-CISc SF-CI(D)c SF-LDAd SF-PBEd SF-B3LYPd SF-ωPBEhd o- -37.5 -45.6 -37.4 -e -23.2 -35.7 -47.5 -44.3 -46.9 -43.6 m- -21.0 -35.5 -22.1 -e -3.82 -19.4 -29.0 -27.7 -26.1 -20.8 p- -3.8 -4.0 -0.6 -8.9 -0.32 -2.1 -10.5 -9.6 -6.9 -4.1

a The o- and m- benzyne geometries for pp-RPA calculations are adopted from Ref. [152]. However, the p-benzyne geometry is adopted form Ref. [156] optimized by SF-CCSD method, which gives better optimized structure in terms of DFT total energy. Aug-cc-pVDZ basis set is used for all pp-RPA calculations. The adiabatic gap correction is based on Eq.(4.5b); b Experimental values from Ref. [169]; c Spin-flip CI results from Ref. [152] with cc-pVQZ basis set; d SF-TDDFT with non-collinear kernel results from Ref. [156] with cc-pVTZ basis set; e No results for PBE reference because of an SCF convergence failure for the (N ´ 2)-electron system. 91 The pp-RPA results for these carbene-like diradicals are summarized in Table 4.2. For methylene, the first adiabatic ST gap had been a puzzle for a long time in the history [170, 171], which was finally agreed by experiments and calculations to be as small as only 9 kcal/mol [172, 173]. The pp-RPA with B3LYP and PBE references reasonably predicts the adiabatic gap to be 3.7 and 5.9 kcal/mol, respectively. For the HF reference, the gap is predicted to be negative (-2.4 kcal/mol), which shows that here the HF reference is not as good as DFT references. The (variational) fractional- spin method overestimates the gap (15.7 kcal/mol) and so do SF-CI (20.4 kcal/mol) and SF-CISD (14.1 kcal/mol). The SF-TDDFT with non-collinear kernels greatly depend on functionals, with hybrid B3LYP functional greatly underestimating the gap while local LDA and PBE functional well predicting the gap. For the next

3 1 gap between B1 and B1 states, the results of pp-RPA with DFT references (32.0 kcal/mol for B3LYP and 33.5 kcal/mol for PBE), together with SF-TDLDA (30.7 kcal/mol) and SF-TDPBE (32.3 kcal/mol), is even closer to the reference value (31.7 kcal/mol) than that of SF-CIS(D) (38.0 kcal/mol). SF-CIS and SF-B3LYP still have

3 1 large systematic errors. The gap between B1 and the second 1A is similar, with pp-PBE and SF-TDLDA performing the best. It is also worth noting that many works [173, 174, 175] have shown that at the optimized singlet geometry for methylene, the singlet energy is very close to or even lower than the triplet energy. Therefore, a very small or negative vertical gap is expected. With pp-RPA, we also observed this phenomenon, with -7.3 and -4.5 kcal/mol vertical ST gap obtained with B3LYP and PBE references, respectively. For other carbene-like molecules, the results are similar to methylene. Interest- ingly, the pp-RPA results with B3LYP reference — more HF exchange than PBE and less HF exchange than HF — mostly lie between the results with PBE and those with HF references. This phenomenon implies the amount of HF exchange and its related virtual orbital energy alignments may play an important role in achieving

92 good accuracy. Overall, for this test set, pp-RPA with DFT references — with no or a small amount of HF exchange — perform well with 4.3 kcal/mol MAE for B3LYP reference and 3.5 or 3.7 kal/mol for PBE reference. Although they are still not as good as SF-TDLDA (2.4 or 3.2 kcal/mol) and SF-TDPBE (1.9 or 2.7 kcal/mol), they are much better than SF-CIS (10.7 or 15.5 kcal/mol) and SF-TDB3LYP (9.7 or 10.8 kcal/mol), and similar to or slightly better than (V)FS-PBE (4.2 kcal/mol) and SF-CIS(D) (4.2 kcal/mol or 6.2 kcal/mol).

4.3.3 Disjoint Diradicals

Disjoint diradicals are problematic for the (variational) fractional-spin method be- cause of the static correlation error intrinsic to commonly used DFAs [21, 140, 141].

For . CH2CH2CH2 ¨ , the reference vertical ST gap is 1.8 kcal/mol. Fractional-spin LDA (12.8 kcal/mol) and fractional-spin PBE (20.4 kcal/mol) greatly overestimate the gaps [140]. In contrast, pp-RPA with DFT references give reasonably small gap, with 4.4 and 5.4 kcal/mol for B3LYP and PBE, respectively (Table 4.3). The pp- RPA-HF also reasonably predicts a vertical gap of 2.4 kcal/mol. The advantage of pp-RPA is even more clear for two longer disjoint diradicals, i.e. . CH2(CH2)4CH2 ¨

and . CH2(CH2)4C(CH3)H ¨ . The pp-RPA with DFT references can not only well describe the lowest singlet-triplet gap, but also well predict the higher excitations with charge transfer character.

These disjoint diradicals can be viewed as two electrons connected by a (CH2)n bridge. If we ignore the bridge, the systems resemble the stretched H2 molecule. It is well known that the stretched H2 has strong static correlation and most single refer- ence methods will fail to describe the stretching process [141]. Meanwhile, it is also a challenge for DFAs to describe the charge transfer excitation in stretched H2 because of common DFAs’ large delocalization error. Similarly, with an additional (CH2)n bridge, these long disjoint diradicals become even more challenging. Fortunately, the

93 subspace CISD treatment frees pp-RPA from static correlation and enables it to well

3 1 describe the lowest A1 and A1 states. The full coulomb kernel enables pp-RPA to correctly capture the asymptotic behavior of charge-transfer excitations[176]. For SF-TDDFT, the employment of high spin references and spin-flip excitation opera-

1 3 tors also frees it from static correlation error, thus the A1- A1 gap is well described by all functionals. However, when it comes to excitations, the erroneous DFA kernels in SF-TDDFT account for the greatly underestimated charge transfer excitations. Only when the long-range corrected functional (for example, ωPBEh) is employed can the problem be relieved.

4.3.4 Four-Electron Diradicals

Diradicals with four π electrons also have a long history. They were discussed in detail in Borden and Davidson’s work [177]. The non-Kekule molecule trimethylen- emethane (TMM) is one of the examples [178, 179, 180, 181, 182, 183, 184]. In a single reference picture, TMM has four electrons occupying four π atomic orbitals. If

with a D3h geometry, these four π orbitals form one molecular orbital lying at the bot- tom, one at the top and two degenerate ones in the middle. Therefore, two electrons form a pair and occupy the lowest one and two remaining electrons occupy the two degenerate orbitals, forming a diradical. A further mix of open-shell and closed-shell

singlet determinants is allowed with D3h geometry because of the proper symmetry of the degenerate orbitals. Therefore, the picture is much like the diatomic cases. According to Hund’s rule, which is obeyed in this case, the triplets show lower energy

than the singlets. When the D3h symmetry is not preserved, the state degeneracy is broken. One state finds its global minimum at a twisted non-planar geometry and

1 this state is assigned the symbol B1. The other state has a local minimum with the

1 restricted planar geometry and it is assigned as the A1 state [179]. The calculated adiabatic ST gaps for TMM are summarized in Table 4.3. It can be seen that the

94 1 3 1 lowest adiabatic ST gap ( B1- A2) is well predicted by pp-RPA with errors even less

1 3 1 than 1 kcal/mol. However, for the second ST gap ( A1- A2), the pp-RPA greatly overestimates the results. This overestimation is very likely to be related to the two bonding electrons treated with DFT. Even though in a single-reference picture,

these two electrons form a pair and occupy a b1 orbital that seems to play a less

important role in determining the chemical properties of the diradical, this b1 orbital

1 is very sensitive to the chemical environment at the optimized A1 geometry. With a restricted B3LYP SCF calculation for the singlet neutral system, in which the four

2 2 electrons line up as 1b11a2, the b1 orbital mostly localizes as a π bond between the central atom and the closest carbon atom (Figure 4.3(a)). However, a calculation for the (N ´ 2)-electron reference shows it to be much more diffuse onto the other two farther carbon atoms (Figure 4.3(b)). An unrestricted triplet calculation for the neutral system gives pictures between these two cases (Figures 4.3(c) and 4.3(d)). Overall, from the most localized one to the most diffuse one, the order is neutral singlet orbital, neutral triplet beta orbital, neutral triplet alpha orbital and finally the (N ´ 2)-electron singlet orbital. The frozen b1 orbital for the (N ´ 2)-electron reference might be too diffuse compared to the real picture, making the (N ´ 2)- electron core not well described. This inaccurate description of the core leads to the large error. Interestingly, in contrast, this b1 orbital is not quite sensitive to chemical

1 environment at the optimized B1 geometry (Figure 4.3(e) to 4.3(h)). Therefore, the

1 3 1 frozen core should be close to the real picture and the vertical gap ( B1- A2) at the

1 optimized B1 is well described, finally making the adiabatic gap also well predicted by pp-RPA. Spin-flip TDDFT with LDA, PBE and B3LYP functionals can also well predict

1 3 1 1 the lowest B1- A2 gap. However, they greatly underestimate the two higher A1-

3 1 A2 gaps. Unlike pp-RPA, SF-TDDFT can correlate the lower non-HOMO excitation configurations, and therefore the error for SF-TDDFT mostly lies in the functional

95 (a) (b) (c) (d)

(e) (f) (g) (h)

Figure 4.3: Orbital picture of the b1 orbital occupied by the two bonding π elec- trons calculated with B3LYP functional. (a)Neutral singlet calculation at optimized 1 1 A1 geometry (b)(N ´ 2)-electron singlet calculation at optimized A1 geometry 1 (c)Neutral triplet calculation (alpha orbital) at optimized A1 geometry (d)Neutral 1 triplet calculation (beta orbital) at optimized A1 geometry (e)Neutral singlet calcu- 1 lation at optimized B1 geometry (f)(N ´2)-electron singlet calculation at optimized 1 1 B1 geometry (g)Neutral triplet calculation (alpha orbital) at optimized B1 geom- 1 etry (h)Neutral triplet calculation (beta orbital) at optimized B1 geometry

that is employed. When the long-range corrected functional ωPBEh with a larger portion of Hartree-Fock exchange is employed, the error gets smaller. Another typical four-π-electron system is cyclobutadiene[142, 144, 185, 186, 187, 188, 189, 190, 191]. It has similar single-reference orbital picture to TMM. However, this molecule violates the Hund’s rule — the open-shell singlet is lower in energy than the triplets, even though the single-reference frontier orbitals are degenerate. This is even true for the vertical gap of the square cyclobutadiene [192]. This phenomenon

is related to the D4h symmetry and its Φ2 and Φ3 are not degenerate and are of different symmetry. A pictorial explanation is given by Borden and Davidson [177] by

considering the molecule’s D4h symmetry and the possibility to localize degenerate MOs. The reference vertical ST gap for square cyclobutadiene is negative (-8.1 kcal/mol). Unfortunately, although much better than the fractional-spin method (16.5 kcal/mol), pp-RPA also predicts positive ST gaps (6.5-7.5 kcal/mol) whatever reference is used. This may also be related to the inaccurate frozen description of the

96 two bonding π electrons, which should be described with more explicit correlation together with the two non-bonding electrons.

4.3.5 Benzynes

Benzyne has three isomers (o-, m-, p-). These isomers all have closed-shell singlets as the ground state, but they still hold some diradical characters with interesting chemistry behaviors that have been theoretically studied [152, 156, 193, 194, 195]. Following the ortho, meta, para sequence, the diradical character increases, resulting in a decrease of ST gaps. The results are summarized in Table 4.5. There are some

problems with the convergence of (N ´ 2)-electron systems with the PBE functional. Therefore, some PBE data is missing. Apart from that, the pp-RPA-DFT results are in good agreement with experimental results, especially with the B3LYP reference. The pp-RPA-B3LYP results are better than the SF-CIS and SF-TDDFT with LDA, PBE and B3LYP functional. They are similar to SF-CIS(D) and SF-TD-ωPBEh. Here pp-RPA-HF give results with larger errors, probably because of the lack of

dynamic correlation for the relatively large-size pN ´ 2q-electron reference. The good performance of pp-RPA-B3LYP on these relatively larger molecules shows that it might be promising to use pp-RPA in even larger and more complex diradical systems.

4.4 Conclusion

The pp-RPA describes diradical systems by starting from an SCF pN ´ 2q-electron reference, and then adding two electrons with explicit correlation. The relatively sim-

pler pN ´ 2q system can be well described by a single-determinant DFT reference, while the two electrons are described in a subspace CI fashion. The combination of DFT with correlated wavefunction methods, as well as the treatment of all neu- tral states on the same footing [176], enables pp-RPA to describe diradicals both

97 accurately and efficiently. The bare coulomb kernel of pp-RPA ensures a correct de- scription of charge separated and charge transferred state. In this work, we showed the pp-RPA results with HF, B3LYP and PBE functionals for a variety of categories of diradicals, including diatomic diradicals, carbene-like diradicals, disjoint diradi- cals, four-π-electron diradicals and larger benzyne diradicals. The difference between different functional references shows that the amount of HF exchange affects the final results by affecting the virtual orbital energies. As to the results, except for some states for the four-π-electron diradicals with important correlation contributions from more than the two non-bonding electrons, the pp-RPA with DFT references gener- ally gives rise to good results, which shows that pp-RPA is much more accurate than SF-CIS. It is comparable or better than (variational) fractional-spin method. In some small diradical systems, such as diatomic diradicals and carbene-like dirad- icals, the error for pp-RPA is slightly larger than that of SF-TDDFT with LDA and PBE functionals. However, when it comes to more difficult disjoint diradicals and benzynes, the pp-RPA with DFT reference performs much better and it becomes comparable to SF-CIS(D) and SF-TDωPBEh. Moreover, the pp-RPA is an inexpen- sive first order theory with OpN 4q scaling that is newly applied to molecular systems. Although further exploration of functionals and kernels may further improve the re- sults and a better converging method may relieve some SCF convergence problems for the pN ´ 2q system, now the pp-RPA is already a reliable theoretical method for describing diradical systems and could be promising in solving much larger and more challenging diradical-related problems.

98 5

Nature of Ground and Excited States of Higher Acenes

This chapter is mainly adapted from the following journal article,

• Yang Yang, Ernest Davidson, and Weitao Yang, “Nature of ground and elec- tronic excited states of higher acenes”, Submitted

5.1 Introduction

Organic semiconductors can potentially replace their conventional inorganic coun- terparts with less manufacturing cost and more flexibility over a large but light substrate. They have been widely used in organic light-emitting diodes (OLEDs), organic field-effect transistors (OFETs) and organic photovoltaic cells (OPVs). Some of the organic semiconductors can undergo singlet fission, a multi exciton generation process which has the potential to dramatically increase solar cell efficiency. Acenes and their derivatives are a typical organic semiconductor material that is being in- tensively studied [196, 197, 198, 199]. With the increase of the oligomer length, the acenes display a rapid evolution of electronic structure and molecular proper-

99 ties, which greatly attracts the scientific community. The smallest naphthalene (2) and anthracene (3) are stable species that can be isolated from coal or petroleum resources. They are widely used as basic synthetic blocks [200, 201]. The larger tetracene (4) and pentacene (5) are promising organic semiconductors and have been applied as OFETs [202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212] thanks to a low vibrational reorganization energy and therefore a high hole mobility [213]. Many of the derivatives of 4 and 5 are good candidates for OLEDs [214, 215, 216, 217]. Recently, with a growing interest in solar energy and solar cells, 4 and 5 have also been made into OPV devices [218, 219, 220, 221, 222]. Furthermore, 3-5 have been shown to be able to undergo singlet fission [223, 224, 225, 226, 227, 228](1M* + 1M

Ñ 2 3M*), which in principle can raise the efficiency limit of a solar cell to more than 40% [229, 230] by converting one high energy singlet exciton to two spatially sepa- rated lower energy triplet excitons. Higher acenes with more than five linearly-fused benzene rings are believed to possess more tempting electronic structure and molec- ular properties [198, 199, 231]. However, the instability of the higher acenes remains a main obstacle to experimental research. Even though the synthesis of hexacene (6) was reported more than 70 years ago [232, 233], a scheme that is easy to reproduce was not achieved until 2007 through a photochemical process in a polymer matrix, which helps retain the highly reactive product 6 [234]. Similar schemes were also successfully applied to the synthesis of three even less stable species — heptacene (7) [235], octacene (8), and nonacene (9)[236]. Although the synthesis of decacene (10) and undecacene (11) with ten or more fused-benzene rings will be more challenging and has not yet been achieved, the exploration of higher acenes and their derivatives continues unabated. In contrast to these fruitful experimental studies, there are relatively fewer sys- tematic theoretical studies on higher acenes. The nature of the ground state of higher acenes still remains controversial. Is the

100 Figure 5.1: Structure of acenes in one Kekule resonance form. ground state a triplet, a closed-shell singlet or an open-shell singlet? How significant is the diradical character? What is the singlet-triplet (ST) gap limit for polyacene? The answer to these questions concerns the molecular properties of higher acenes, and it is an important challenge, particularly in the absence of accurate experimental characterization. Unfortunately, the large system size of higher acenes hinders the use of many accurate but expensive theoretical tools. Empirical or semi-empirical stud- ies [237, 238, 239, 240, 241, 242] are inexpensive but also less accurate, and therefore seem to be less widely accepted. One of the most acknowledged theoretical predic- tion, up to now, is based on the UDFT calculation performed more than ten years ago [243]. However, it is widely known that the utilization of broken-symmetry calcula- tion is a limitation and the breaking of spin symmetry is unphysical. Therefore, the broken-symmetry determinant of UDFT contains limited information of the entire many-electron wave function, and it may not describe the nature of the ground state. In the past ten years, some accurate but relatively inexpensive theories have been developed and applied to higher acenes [139, 140, 147, 244, 245, 246, 247, 248, 249]. Among them, the most widely accepted one seems to be the density matrix renor- malization group (DMRG) method in combination with the complete-active-space

101 configuration interaction (CAS-CI) theory [244]. It provided valuable insights into the ground state of higher acenes with a natural orbital analysis. However, the computation is still demanding, and acenes longer than 6 were computed with the minimal basis set, which leads to uncertainty to the overall quantitative accuracy. The ground state is already difficult to describe, let alone the more challeng- ing excited states [250, 251, 252, 253, 254, 255]. As a result of a“charge separa-

1 tion in disguise” [253, 256], TDDFT is known to greatly underestimate the B2u HOMOÑLUMO excitation [257, 258], which is the lowest bright excitation for acenes longer than 2. Moreover, because of a double excitation nature, the description of the

1 dark Ag state is completely unattainable by the commonly used adiabatic TDDFT [74, 77, 78, 259, 260]. Unfortunately, this dark state is particularly important to researchers as it has been proposed as a crucial intermediate in the process of singlet fission [228]. Only some expensive multi-reference methods [227, 228, 261, 262, 263]

1 have hitherto been applied to describe this Ag state for small acenes, but these methods are not feasible for higher acenes because of the computational cost. Very recently, the density functional theory based multi reference configuration interaction method (DFT/MRCI) has been applied to the excited states of higher acenes up to 9 and achieved good accuracy [255]. However, only lowest few bright excited states were investigated and the bonding nature of those states was not fully elaborated. Here we provide both qualitative and quantitative descriptions on the nature of the ground and electronic excited states of higher acenes. The quantitative calcula- tion is based on the pp-RPA, while our qualitative analysis is based on the homosym- metric diradical discussion by Salem and Rowland [264]. For a simple analogy, we use the simplest stretched H2 diradical model, as in existing literature [265].

102 5.2 Results

5.2.1 Singlet-Triplet Energy Gap

We here refer to the singlet-triplet energy gap as the transition energy from the

1 3 lowest Ag to the lowest B2u state (ET ´ ES). We use optimized geometries for the neutral N-electron molecule obtained with B3LYP/6-31G*. Three set of geome- tries are used, which are optimized by restricted singlet (R), unrestricted singlet (U), and unrestricted triplet (T), respectively. Table 5.1 shows the results from the pp-RPA-B3LYP/cc-pVDZ calculation, we predict the ground state of higher acenes to be singlet rather than triplet, and it is so even at the optimized geome- try for the neutral triplet state (T geo). This agrees with many recent predictions [147, 241, 243, 244, 245, 246, 247, 266]. With the increase of the acene size, the ST gap decreases. Compared with experimental data, our theoretical predictions consis- tently overestimate ST gaps by around 0.15 eV at geometries optimized by restricted singlet (R geo) while underestimate them at T geo. This is probably because the experimental data are possibly closer to adiabatic gaps while theoretical predictions are for vertical gaps at a fixed geometry. Therefore, these two prediction series at R and T geometries are expected to provide the upper and lower bounds for the adiabatic ST gaps. We fit the two series with the simple exponential decay formula

(E “ ae´n{b `c) and plot the result in Figure 5.2. The predicted gap limit is between 0.0eV-0.1eV, which is still positive but vanishingly small. This small ST gap also indicates a probable diradical or polyradical nature for higher acenes, which we will discuss more in later ground state discussion section.

103 3 Table 5.1: Excitation energy of B2u state for acenes

Acene 2 3 4 5 6 7 8 9 10 11 12 pp-RPA@R 2.87 1.98 1.39 0.98 0.70 0.51 0.37 0.28 0.22 0.18 0.16 pp-RPA@U 2.87 1.98 1.39 0.98 0.66 0.39 0.23 0.15 0.11 0.11 0.13 pp-RPA@T 1.99 1.35 0.90 0.59 0.39 0.25 0.17 0.12 0.09 0.09 0.11 Expt. 2.65a 1.87b 1.27c 0.86d

a Ref. [267]; b Ref. [268]; cRef. [269]; d Ref. [270];

1 Table 5.2: Excitation energy of B2u state for acenes

Acene 2 3 4 5 6 7 8 9 10 11 12 104 pp-RPA@R 4.97 3.65 2.82 2.26 1.86 1.58 1.36 1.20 1.08 0.99 0.93 pp-RPA@U 4.97 3.65 2.82 2.26 1.83 1.49 1.26 1.12 1.02 0.96 0.94 pp-RPA@T 4.30 3.18 2.46 1.98 1.64 1.40 1.23 1.11 1.02 0.96 0.93 CC2a 4.88 3.69 2.90 2.35 1.95 1.43 DFT-MRCI 4.66b 3.51b 2.74b 2.22b,2.16c 1.85b,1.80c 1.57c 1.44b,1.43c 1.34c 2.37d ,2.3e Expt. 4.66d 3.60d 2.88d 1.89f ,1.85h 1.50i ,1.70f 1.54j 1.43j 2.21f ,2.22g

a CC2/cc-pVTZ results from Ref. [257]; b Ref. [262]; cRef. [255]; d Refs. [257, 271]; e Ref. [272]; f Ref. [273]; g Ref. [274]; h Ref. [234]; i Ref. [235]; j Ref. [236]; 1 Table 5.3: Excitation energy of Ag state for acenes

Acene 2 3 4 5 6 7 8 9 10 11 12 pp-RPA@R 6.43 4.87 3.65 2.74 2.05 1.54 1.14 0.84 0.61 0.43 0.31 pp-RPA@U 6.43 4.87 3.65 2.74 1.99 1.36 0.95 0.67 0.48 0.36 0.30 pp-RPA@T 5.31 3.94 2.92 2.17 1.61 1.20 0.89 0.67 0.51 0.40 0.33 DFT-MRCIa 5.73 4.60 3.37 2.52 1.86 Other 1.95b,2.63c,2.88d

a Ref. [262]; b MRMP(12,12)/TZ result from Ref. [228]; cCASSCF(12π12e) result from Ref. [263]; d CASPT2/SA-CASSCF(14,14).ABI-L-VTZP result from Re.. [261]; 105 The ST gaps at geometries optimized by unrestricted singlet (U geo) are also plotted. It can be seen that it first coincides with R results and later resembles T results. Therefore, its ST gap decreases much faster. Since there is an essential discontinuity between the shorter acenes where U converges to R and the longer ones where it gives different results, we perform the exponential extrapolation for acenes starting from 6, whose U result begins to differ from R result. The gap limit is also vanishingly small between 0.0eV-0.1eV. It should be noted that if we adopt the whole U series for extrapolation, the gap limit is -0.02eV. We expect it unphisically too low due to the discontinuity. This phenomenon is similar to the dissociation of

H2, in which there is a break in the slope at a critical bond length with unrestricted Hartree-Fock (UHF). It is also worth noting that after removing the two outermost electrons, many diradical systems become closed-shell dications that can be well described by re- stricted DFT. The stable dication species can be perceived as a frozen core that makes a limited contribution to the dynamic correlations on both ground states and excited states. Acenes have been known to be such kind of molecules with stable dications [275]. Since pp-RPA starts with a DFT calculation on this N-2 system, its overall accuracy benefits from the stability of acene dications.

5.2.2 Singlet Ground State in the Diradical Continuum

For a diradical system, besides the ST gap, the energy gap between the open-shell singlet and closed-shell singlet is also important. In the acene case, this question is the same as whether the ground state is open-shell or closed-shell since the ground state has already mostly been predicted to be singlet. Chemists normally define open-shell and closed-shell with the help of MOs, be- cause many chemical species can be approximately described by a single Slater de- terminant based on MOs. If two outermost electrons with opposite spins occupy the

106 Figure 5.2: Vertical triplet excitation energy from pp-RPA for acenes with re- stricted singlet geometry (R), unrestricted singlet geometry (U) and triplet geometry (T). The R and T series approximately provide upper and lower bounds for the adi- abatic singlet-triplet gaps. The U series first coincides with R and then differs from R starting from 6. Because the U-R difference is an essential discontinuity for U series, it is fitted starting from 6. The constant term in the exponential fit provides the limit for polyacenes. The predicted ST gap in the limit is vanishingly small and is about 0.0 eV - 0.1 eV. same MO, it is a closed-shell singlet; if they form a singlet but occupy different MOs, it is an open-shell singlet. Our study on acenes starts with this definition. Nonethe- less, it should also be noted that this intuitive single determinant picture collapses, when a system has strong static correlation and therefore needs multi-reference for an accurate description. Many diradicals are such kind of systems. The multi-reference character, which may be reflected by the contribution from the dominant configu- ration in a CI manner, can work as an indicator of the diradical character. Using pp-RPA, which seamlessly combines DFT with the wave function-based CI [276], we are able to analyze the diradical character of higher acenes.

107 The dominant configuration contribution (DCC) of a state, which we define as the square of the largest number in its corresponding pp-RPA matrix eigenvector, works as the indicator of the multi-reference character. The larger the DCC, the less the multi-reference character, and the more likely that this state can be well described with a single reference. The DCC for acenes at different geometries are plotted in Figure 5.3. For the first acene 2, the single reference picture is completely valid because the dominant configuration with two electrons occupying HOMO contributes more than 99% to the ground state. The remaining contributions mainly come from the one with two electrons occupying LUMO rather than HOMO. With this 99% DCC, we assign the ground state of 2 to be an almost pure closed-shell singlet. As the acene fuses more benzene rings, the DCC gradually decreases. With the R geometries, it drops from 99% for 2, to 95% for 8, to 90% for 10, and eventually to 75% for dodecane (12) at an accelerating speed. Nonetheless, all of them keep the Ag symmetry. For the same acene, with a different geometry, the DCC for R is larger than that for T, and the result for U mostly lies in between, although it may finally cross over T for 11 and 12. Among all the cases, the smallest DCC happens to the U geometry for 12, where it is only 42% and almost the same as the weight of the second dominant configuration, with the remaining 16% contribution coming from other configurations. With a gradually decreasing DCC, the assignment of closed-shell and open-shell becomes tricky, so let us first consider in the two-orbital diradical model [264]. The extreme case is that the DCC is 50% with the two electrons occupying HOMO having equal contribution to the configuration with the two electrons occupying LUMO. Note that the assignment of HOMO and LUMO is not so important in this case because they could in principle be degenerate. We simply call these two configurations HOMO pair and LUMO pair, respectively. If we write the two MOs

108 as A and B, the singlet ground state can be noted as

1 ? p|AA¯y ´ |BB¯yq, (5.1) 2

with opposite signs in front of the two configurations. Viewing from the valance

bond perspective and performing the following unitary orbital transformation a “ ? ? 1{ 2pA ` Bq, b “ 1{ 2pA ´ Bq, we can rewrite Eq. (5.1) as

1 ? p|a¯by ´ |ab¯ yq, (5.2) 2

The stretched H2 at the stretched limit is the simplest case in this model and the shape of its transformed orbitals a and b during the stretch process are plotted in Figure 5.4 together with the original MOs A and B. It can been seen that a mostly localizes on one atom of the molecule while b localizes on the other, and the longer the bond stretch, the more localized these transformed orbitals are and the less overlap they have. Note that the shape of the transformed orbitals look like the singly occupied molecular orbitals (SOMOs) obtained from UDFT, which gives correct energy curve for H2 stretched limit. For this extreme case, usually we no longer define open-shell or closed-shell in terms of the completely delocalized MOs A and B. Instead, we focus on the spatially localized atomic-like orbitals a and b. Because the two electrons separately occupy a and b and get spatially disconnected, it is open-shell. Therefore, most molecules (|AA¯y) with nearly 100% DCC are closed- ? shell and have regular covalent nature, while extreme diradicals (1{ 2p|AA¯y´|BB¯yq) with 50% DCC are open-shell and have diradical covalent nature. (In valence bond theory, regular covalent has half ionic contribution while diradical covalent is purely covalent.) Those cases in-between lie in the transitional region which in this paper we call the diradical continuum. In the diradical continuum, it becomes meaningless to strictly label a species with open-shell or closed-shell, but here we can provide

109 a qualitative distinction: if we set 75% DCC as the middle of the continuum, then those cases with more than 75% DCC are on the closed-shell side while those with less are on the open-shell side. The case for acenes is highly similar, although slightly more complicated because more orbitals rather than only two may be involved. The HOMO and LUMO orbitals and their transformed orbitals are plotted in Figure 5.5 for 6 and Figure 5.6 for 11. There are also plots for HOMO-1 and LUMO+1 that will be discussed in later excited states discussion sections. For current ground state discussion, we can simply focus on HOMO and LUMO because, as is shown above, the HOMO pair and LUMO pair configurations have largest weight up to 12. The transformed orbitals for HOMO and LUMO localize on either edge of the acene molecules, and the more fused benzene rings, the more separated they are. Note how similar they are to the SOMOs in the existing literature [243]. We can roughly think of the ring fusing process of

acenes having the same effects as the stretching process of H2, although the fusing is on the long axis while the orbital localization is with respect to the perpendicular short axis. If the ground state of an acene is completely dominated by the HOMO pair configuration, for example, 2 at R geometry, it is without doubt a closed-shell species; if nearly half contribution comes from the HOMO pair and the other half from the LUMO pair, for example, 12 at U geometry, then it is open-shell and almost a full diradical. The remaining cases in between are in the diradical continuum. The ground state of 10 at its U and T geometries and the ground state of 12 at R geometry have a DCC around 75%, and they roughly lie in the middle of the continuum. Since we already know the ground state is singlet rather than triplet, the R and U geometries should be closer to the true ground state geometry than T geo. Between the R and U, because we find the ground states for acenes up to 10 have more closed-shell character, and the U calculation was more and more contaminated by the triplet state, we believe the R geo is more reliable for these acenes. For 11

110 and 12, because more and more diradical and even polyradical character are added, we expect the true geometry is somewhere between U and R results, with a ground state nature more and more to the open-shell.

Figure 5.3: First (two outermost electrons added to HOMO and HOMO) and second (two outermost electrons added to LUMO and LUMO) dominant configura- tions of the ground state for acenes with restricted singlet geometry (R), unrestricted singlet geometry (U) and triplet geometry (T). The contribution from the first dom- inant configuration (DCC) decreases with the increasing size of acenes, but up to 10, it is always larger than 0.75, which suggests that these acenes lie more on the closed shell side. The DCC is higher with an R geometry than with a T geometry. The results from U geometry mostly lie in and shift from close to R to close to T, indicating a spin symmetry broken solution. The U series finally crosses the T series and displays an almost full diradical character with similar contributions from the first and second dominant configurations. Meanwhile, the total contribution from the first two dominant configurations gradually decreases, suggesting the beginning of a polyradical nature.

Note that we use DCC based on a molecular orbital picture for the N-2 system to describe the diradical character. This is new and also unique for pp-RPA. Because of the potential strong static correlation in neutral diradical systems, previously

111 the natural orbital occupation analysis in combination with complete active space theory [244] or projected HF [248] have been used. These are wave function-based methods and provide useful information of the ground state wave function. However, the natural orbital analysis is less intuitive than molecular orbital analysis, and moreover it is considered to assign too much diradical character to the closed-shell 2 [244, 245]. In contrast, the molecular orbital based on KS-DFT is more intuitive and has been argued to be very suitable for qualitative chemical applications [277, 278, 279]. Although, strictly speaking, the transitional amplitude of pp-RPA only describes the pairing density response to a pairing field [103], it can also be perceived that pp-RPA combines seamlessly a DFT description of the N-2 core electrons and a CI wave function description of the outermost two electrons. It is thus possible to use molecular orbitals to reveal the nature of an excitation. This is similar in spirit to the well-known TDDFT, whose transitional amplitude only describes a density response to an external potential but is also used to explain the nature of an excitation based on a Kohn-Sham molecular orbital [72]. Furthermore, unlike the natural orbital analysis, the nearly closed-shell description of 2 by pp-RPA agrees well with the generally accepted consensus. Therefore, the DCC descriptor based on the molecular orbital picture of the two-electron deficient system should be reliable in describing the diradical character. What will the ground state be for even higher acenes beyond 12? First, although

in this work the optimized geometries for acenes up to 12 all collapse to the D2h point group, it does not necessarily mean that it will always keep such a high symmetry without any Jahn-Teller distortion or Peierls distortion as the number of benzene rings becomes even larger. In fact, some existing literatures have already pointed out the possibility [280]. Secondly, even if we assume the absence of the distortion

and trust a geometry with D2h symmetry, from Figure 5.3, we can see that the DCC later decreases very rapidly even at the R geometry, and meanwhile, the total weight

112 of the first two dominant configurations (HOMO pair and LUMO pair) decreases while the weight of the LUMO+1 pair increases. Usually the emergence of the LUMO+1 pair correlates with the breaking of the HOMO-1 pair, which is therefore an indirect sign for a polyradical character. Under this circumstance, it may also be challenging for current pp-RPA that assumes the electron pair occupying HOMO-1 will not break and therefore only explicitly correlates the two outermost electrons. Although for 11 and 12, we conclude the ground state might be more on the open- shell side of the diradical continuum, we also find that for 12, the contribution from configurations other than the first two dominant ones has already been above 15% with the U geometry, suggesting an emerging polyradical character. Note that this polyradical nature was also observed by natural orbital analysis [244, 248, 249], and was claimed to emerge as early as 8 [249], but based on our analysis, it does not show up until 11 and 12. We anticipate the ground state of even higher acenes will possess even more polyradical character. Before we finish up our discussion on the ground state of higher acenes, we want to discuss UDFT calculations, which currently seems to be widely accepted. If we calculate using UHF or UDFT for both H2 and acenes, at some point in the diradi- cal continuum we will observe the energy deviation from the restricted theory. The orbitals obtained become two SOMOs and their shape resembles the transformed orbitals [240, 243]. Although the energy from UHF or UDFT is usually more ac- curate than restricted Hartree-Fock (RHF) or restricted density functional theory (RDFT), the broken-spin-symmetry solution of UHF or UDFT does not lead to a correct wave function description nor to a conclusion that the ground state has al- ready become an open-shell singlet. According to the basic quantum mechanics, the spin symmetry broken under a spin-symmetric Hamiltonian is unphysical and there should not be any localized net spins without any perturbative observations. This deviation between unrestricted and restricted calculations simply raises a concern

113 Figure 5.4: Shape of delocalized frontier molecular orbitals (A and B) and localized transformed atomic-like orbitals (a and b) for H2 at equilibrium geometry (a) and stretched geometry (b). Note the decreasing spatial overlap between a and b. for those methods based on a single reference without considering static correla- tion. In this case, in order to achieve both accurate energy and meaningful wave function that describes the degree of the diradical character, one should refer to methods with a reference state maintaining proper spin symmetry, which usually means a multi-reference-based method, rather than literally interpreting the UDFT determinants. This statement should also apply to many current research projects concerning graphene fragments and spins on graphene edges.

5.2.3 Lowest Bright Singlet Excitation

We next look into the vertical excitations. For acenes longer than 2, the singlet

HOMOÑLUMO excitation is the lowest bright singlet excitation. Considering the

1 symmetry of the orbitals involved, this excitation can be noted as B2u. In addition, it is also widely known as the “p band” or “La excitation” [196, 281]. The excitation energies for this state are listed in Table 5.2 and plotted in Figure 5.7. With the increasing number of benzene rings, as expected, there is a decreasing trend for the excitation energy. Note that since acenes up to 10 are more on the closed-shell side,

114 Figure 5.5: Shape of delocalized molecular orbitals and transformed localized or- bitals for 6. All orbitals involved are π orbitals. The HOMO is in fact LUMO for the N-2 reference, and it has au symmetry. The LUMO (LUMO+1 for N-2) has b2g sym- metry. Adding or subtracting HOMO and LUMO makes two localized orbitals along the short axis. The HOMO-1 (HOMO for N-2) has b3g symmetry and LUMO+1 (LUMO+2 for N-2) has b1u symmetry and adding or subtracting them also makes two localized orbitals along the short axis. The HOMO and HOMO-1 can make localized orbital pairs along the long axis and same to LUMO and LUMO+1.

the results at R geometry should be more reliable, and those with U and T geometry may underestimate the excitation. For 11 and 12, the values at all three geometries are fairly close. There have been many studies on this excitation for acenes up to 6, both experimentally and theoretically [112, 228, 234, 250, 251, 252, 253, 254, 255, 256, 257, 258, 261, 262, 263, 282, 271, 283, 273, 274, 284, 272]. For these acenes, except for 2, in which the pp-RPA overestimates the excitation energy by 0.3 eV, our predictions match well with experiments, and the deviation is mostly within 0.05 eV. This small error is in sharp contrast with TDDFT, which underestimates the excitation energy by 0.4-0.5 eV with the same B3LYP functional [257]. For higher

115 Figure 5.6: Shape of delocalized molecular orbitals and transformed localized or- bitals for 11. All orbitals involved are π orbitals. The results are very similar to those for 6, but note the much more delocalized transformed orbitals as well as different orbital symmetry due to an odd number of benzene rings. The HOMO is in fact LUMO for the N-2 reference, and it has b3g symmetry. The LUMO (LUMO+1 for N-2) has b1u symmetry. Adding or subtracting HOMO and LUMO makes two local- ized orbitals along the short axis. The HOMO-1 (HOMO for N-2) has au symmetry and LUMO+1 (LUMO+2 for N-2) has b2g symmetry and adding or subtracting them also makes two localized orbitals along the short axis. The HOMO and HOMO-1 can make localized orbital pairs along the long axis and same to LUMO and LUMO+1. acenes, we predict the excitation energy to be 1.58 eV, 1.36 eV, 1.20 eV, 1.08 eV, 0.96-0.99 eV, and 0.93-0.94 eV for 7, 8, 9, 10, 11, and 12, respectively. We notice that some experimental data have been available for 7, 8, and 9 in polymer matrix or inert gas matrix [235, 236, 273], and there has been a fairly accurate DFT/MRCI study [255]. It seems that our predictions might be slightly too low ( 0.2 eV) for these higher acenes. However, now we are not sure of the reason, because both experimental and theoretical studies are at rather preliminary stage. Relying on our

116 1 Figure 5.7: Vertical excitation energy of B2u state for acenes with restricted singlet geometry (R), unrestricted singlet geometry (U) and triplet geometry (T). Since the ground state up to 10 are all closer to the closed-shell side, the R series provides a more reliable prediction, while for 11 and 12, all geometries give similar results. The T series provides smaller numbers than the R series. The U series first coincides with R and later approaches T. Based on exponential decay fitting, the predicted polyacene limit of this vertical excitation is around 0.85 eV. current data, we perform the exponential fitting and predict the excitation energy for this state at the polyacene limit is around 0.85 eV.

1 The nature of the B2u state is less controversial than the ground state. This excitation is dominated by one electron excited from HOMO to LUMO, together with a small contribution from higher single excitation and even double excitations that can be neglected for acenes up to 12. The dominant nature of this state can be noted as 1 1 ? p|AB¯y ´ |AB¯ yq “ ? p|aa¯y ´ |b¯byq, (5.3) 2 2

117 As is called by Salem [264], this state is a zwitterionic state. Because the transformed orbitals a and b localize on the two opposite edges, there is a hidden charge separation — if we look at only one particular configuration, there is charge transfer, but if we look at the two together, there is no net charge transfer, nor any spin separation. It is worth noting that this orbital rotation trick to unveil the hidden charge separation has been employed to explain the failure of TDDFT in describing this state [253].

Since we previously made an analogy to the H2 stretch process, here we just make one more comment on this resemblance. This excitation is similar to the ionic state reached as a result of charge transfer or charge separation excitation in stretched H2.

For stretched H2, DFT and TDDFT fail to describe the ground state and this excited state [21, 285, 141], while the pp-RPA is exact with a special reference named HF*

1 [176]. Therefore, there is no wonder that pp-RPA is also able to describe this B2u state for acenes.

1 5.2.4 Doubly Excited Ag State

To describe the double excitations in acenes is an even more challenging problem. In order to measure the excitation energies, experimentalists usually need to conduct non-linear optics experiments with a two-photon process involved. For theoreticians, we can no longer perform the inexpensive adiabatic TDDFT calculation because it cannot capture double excitations [77, 78, 74, 259, 260]. Fortunately, along with some expensive methods, the pp-RPA can well handle those double excitations excited

1 from HOMO [176]. The excitation energies for the lowest doubly excited state ( Ag) are listed in Table 5.3 and plotted in Figure 5.8. The molecule 5 is the one that has been studied most. Among those existing studies, considering the level of theory as well the basis set, the CASPT2/SA-CASSCF(14,14)/ANO-L-VTZP calculation should be the most accurate [261]. Our prediction (2.74eV) agrees well with its value (2.88eV) while DFT-MRCI [251, 262] (2.52eV) might systematically underestimate

118 1 this whole Ag series. The energy of this excitation also decreases with the size of

1 the acene, however, it decreases much faster than the B2u excitation. In 2 and 3,

1 1 Ag is over 1eV higher than B2u, whereas it becomes 0.4 eV lower in 10 and 0.6 eV lower in 12. From our calculation, the order switch occurs between 6 and 7, which is in accordance with a previous DFT-MRCI study [262]. For 5, the doubly excited

1 state 1Ag is about 0.5 eV higher than the singly excited B2u, therefore it is not quite likely that this doubly excited state would be involved in its singlet fission process

1 [228]. However, we do not refute the hypothesis that this Ag state might play a role in the singlet fission process for acenes longer than 6, in which it has become the lowest excited singlet state.

1 To the best of our knowledge, the nature of the Ag state has not been discussed in detail due to the relative scant studies. Here we will continue using the two-orbital diradical model for discussion.

For a species with barely any diradical character, such as H2 near its equilibrium geometry, the ground state is |AA¯y, while the doubly excited state is |BB¯y, both of which have regular covalent nature.

For a species in the diradical limit, such as the infinitely stretched H2, as is discussed earlier, the ground state is open-shell with a diradical covalent nature. In

1 contrast, the doubly excited Ag state in the diradical limit is

1 1 ? p|AA¯y ` |BB¯yq “ ? p|aa¯y ` |b¯byq, (5.4) 2 2

In the picture of molecular orbital theory, it differs from the ground state by only an opposite sign. Nonetheless, after the orbital transformation and then using the language of valence bond theory, this state becomes another zwitterionic state with hidden charge separation. Thus, it can also be seen as a closed-shell species, and this is in sharp contrast to the open-shell nature of the ground state.

119 It is also worth noting that in the diradical limit of the two-orbital model, with no overlap between the transformed orbitals a and b, the two zwitterionic states ? ? 1{ 2p|aa¯y ´ |b¯byq and 1{ 2p|aa¯y ` |b¯byq are degenerate. This happens with the stretched H2, which is dominated by two orbitals. For those species lying in the diradical continuum, the nature of their excited

1 Ag state lie in between above two extremes, and gradually shifts from a regular molecular species with more double excitation nature to a diradical species with more zwitterionic nature. The situation for acenes becomes much more complicated than this model. First,

1 for small acenes, the Ag state is not a pure doubly excited state; it has high ex- citation energy and combines with some high single excitation configurations. It is

1 similar to polyenes, whose Ag state is a mixture of HOMO,HOMOÑLUMO,LUMO, HOMO-1ÑLUMO, and HOMOÑLUMO+1 [286, 287]. However, because of the ad- ditional symmetry involved along the short axis of polyacenes, the next singly excited

1 configuration that has the Ag symmetry does not show up until fairly late compared with polyenes (mostly HOMOÑLUMO+4 and also possibly HOMO-4ÑLUMO in the B3LYP orbital lineup for the neutral polyacene systems). Note here, because of the lack of non-HOMO excitations and therefore not being able to count in the

HOMO-4ÑLUMO contribution for pp-RPA, our theoretical predictions on this state for those small acenes might be slightly overestimated. However, for these small acenes, these problems can be solved with higher accuracy wavefunction methods, and here we mainly focus on higher acenes.

1 The contributions to this Ag state from different configurations are plotted in Figure 5.9. It can be seen that starting from 5, this state can be roughly seen as a double excitation state, with the weight of the dominant configuration (LUMO pair) being around 75%, while the weight of its counterpart (HOMO pair) is only about 1%. If we now simply focus on these two configurations and consider the two-level

120 1 Figure 5.8: Vertical excitation energy of Ag excited state for acenes with restricted singlet geometry (R), unrestricted singlet geometry (U) and triplet geometry (T). Since the ground state up to 10 are all closer to the closed-shell side, the R series provides a more reliable prediction, while for 11 and 12, all geometries give similar results. The T series provides smaller numbers than the R series. The U series first coincides with R, later approaches T, and finally even crosses below T. Based on exponential decay fit, the predicted polyacene limit of this vertical excitation can be vanishingly small. However, considering its zwitterionic nature with respect to the short axis, it should never become the ground state, and the negative limit by R series is an artifact introduced by extrapolation.

diradical model, the higher acenes truly resemble the stretched H2. With more and more fused benzene rings, there is less and less contribution from the LUMO pair configuration, and more and more contribution from the HOMO pair configuration, which indicates a transition from the doubly excited |BB¯y state to the zwitterionic ? 1{ 2p|aa¯y ` |b¯byq state. However, this only reveals part of its nature. Unlike the ground state, in which the HOMO pair and LUMO pair configurations contribute more than 90%, the total weight of these two configurations is only around 50%- 70% for this state. Another part of contribution comes from the LUMO+1 pair,

121 1 Figure 5.9: Contributions from first few dominant configurations of the excited Ag state for acenes (5 to 12) with restricted singlet (R) geometry. The contribution from the first dominant configuration (LUMO pair) decreases with the increasing size of acenes. The increasing weight of HOMO pair indicates a diradical nature along the short axis, while the increasing weight of LUMO+1 pair indicates a diradical nature with respect to the long axis. The bonding nature of these two diradicals in the limit are zwitterionic and covalent, respectively. The decreasing weight for “HOMO and higher” suggests a decreasing contribution from a regular single excitation, which is important for shorter acenes (2 to 4). The increasing weight for “LUMO and higher” stands for an increasing weight for higher doubly excited configurations that is beyond the two orbital diradical models. which stands for a doubly excited configuration from HOMO to LUMO+1. As the weight of LUMO pair decreases, the weight of LUMO+1 pair also increases, just like the HOMO pair. Chemistry intuition tells us this correlation probably roots in an intrinsic orbital relation between LUMO and LUMO+1. We therefore plot LUMO+1 and its transformed orbital with LUMO (Figure 5.5 and Figure 5.6). Interestingly, it can be clearly seen that these two orbitals can localize along the long axis under the orbital rotation. This means that if we completely ignore previous discussions on the

122 HOMO pair and only focus on the current LUMO pair and LUMO+1 pair, we can obtain another two-level diradical model. The difference is that the previous model concerns the short axis while the new one is related to the long axis. We now denote the LUMO+1 as C and its transformed orbital with LUMO as b1 and c. (Note this b1 is different from b.) In both of the diradical limit, these two orthogonal diradical

1 pictures should be able to combine, and they can generate four Ag states, c c ?1 p|AA¯y ´ |BB¯yq ` ?1 p|BB¯y ´ |CC¯yq, (5.5a) 2 2 c c ?1 p|AA¯y ` |BB¯yq ` ?1 p|BB¯y ´ |CC¯yq, (5.5b) 2 2 c c ?1 p|AA¯y ´ |BB¯yq ` ?1 p|BB¯y ` |CC¯yq, (5.5c) 2 2 c c ?1 p|AA¯y ` |BB¯yq ` ?1 p|BB¯y ` |CC¯yq (5.5d) 2 2

where c1and c2 are combination coefficients and vary in each state. These states can be rewritten in terms of transformed orbitals c c ?1 p|a¯by ´ |ab¯ yq ` ?1 p|b1c¯y ´ |b¯1cyq, (5.6a) 2 2 c c ?1 p|aa¯y ` |b¯byq ` ?1 p|b1c¯y ´ |b¯1cyq, (5.6b) 2 2 c c ?1 p|a¯by ´ |ab¯ yq ` ?1 p|b1b¯1y ` |cc¯yq, (5.6c) 2 2 c c ?1 p|aa¯y ´ |b¯byq ` ?1 p|b1b¯1y ` |cc¯yq (5.6d) 2 2

Therefore, the first state has diradical covalent nature along both short and long axes, the second state has zwitterionic nature along the short axis and diradical covalent nature along the long axis, the third state is completely opposite to the second one, and the fourth state has zwitterionic nature on both axes.

1 Further examination on the sign of the ground state and the lowest excited Ag state of acenes suggests that the ground state is the same as the first state in nature

123 1 and the excited Ag state is the same as the second state. This energy alignment is easily understandable. First, with the help of a simple H¨uckels model (we will not show in detail here), we know that the ionic state is higher in energy than its corresponding covalent state. Therefore, the doubly covalent state should be the lowest in energy, and doubly ionic state should be the highest in energy. Secondly, intuitively speaking, the zwitterion along the short axis has less hidden charge sep- aration character than that along the long axis, and thus should have lower energy. In this sense, even though some of our quantitative extrapolation (R series in Figure

1 5.8) based on existing data suggests that the excited Ag state might become the ground state with a negative limit, it in principle should never happen based on the qualitative analysis. This raises a warning for the results obtained from extrapola- tion, especially in the close-to-zero region, where a qualitative error might occur due to a quantitative uncertainty.

1 In spite of the above long discussion, the nature of this Ag state is still far from being fully elaborated. Another orbital, which is the HOMO-1, can also play a role in this state. We previously only investigated the HOMO pair configuration and doubly excited LUMO pair (HOMO, HOMOÑLUMO, LUMO) and LUMO+1 pair (HOMO, HOMOÑLUMO+1, LUMO+1) configurations, but in fact, a fourth configuration, the doubly excited (HOMO-1, HOMO-1ÑLUMO,LUMO) may also be important. Unfortunately, because of the restrictions of the pp-RPA, which now is unable to describe non-HOMO excitations, we cannot provide detailed data. Nevertheless, we here provide some qualitative analysis. The (HOMO-1, HOMO-1ÑLUMO, LUMO) configuration is another counterpart to (HOMO, HOMOÑLUMO, LUMO) config- uration. The transformed orbitals concern HOMO and HOMO-1, and they localize along the long axis (see Figure 5.5 and Figure 5.6). It is also without doubt that there are many more configurations that can play a role, for example, (HOMO-1,

HOMO-1ÑLUMO+1, LUMO+1) is a counterpart of the Aufbau configuration and

124 it correlates HOMO-1 and LUMO+1 along the short axis, but as a result of the large orbital energy difference, their contribution should be significantly smaller.

5.2.5 Singlet Fission for Higher Acenes

How likely are the higher acenes to be good candidates for singlet fission that can be used in photovoltaics? Three thermodynamic criteria are often used in an analysis

[224, 288]. The first criterion is EpS1q ą 2EpT1q. When it is satisfied, the singlet fission process is exoergic and hence is energetically favorable, whereas the reverting annihilation process is unfavorable. This is a desirable criterion for designing new molecules for singlet fission, although it may not be strictly satisfied because many organic molecules that are known to undergo singlet fission do not need to satisfy this thanks to possible thermal or artificial activations. According to our calculation, 4 is the smallest acene that satisfies the EpS1q ą 2EpT1q condition. This also agrees with experiments that the singlet fission observed in 4 and 5 does not need much energy activation [224, 289]. The second condition is EpT2q ą 2EpT1q. It simply keeps the two triplet states from annihilating and forming the second lowest triplet state. This condition is also easy to satisfy for acenes. From our calculation, we observe that it is always satisfied starting from 4. The third criterion is EpS1q « 2eV and

EpT1q « 1eV . Strictly speaking, this is not a criterion for singlet fission, but a condition that favors high photovoltaic efficiency for practical applications. We find 5 to be the best molecule that satisfies this condition. For higher acenes that are longer than 5, even though singlet fission is more energetically favorable, both the singlet and triplet excitations are much too low in energy and hence not good for high photovoltaic efficiency. However, even though not yet observed, the singlet fission does not necessarily only generate two triplets and these higher acenes might be good candidates for generating multiple triplets, which could benefit the efficiency if achievable [224]. Purely from the energy perspective, 7 and 8 favor a singlet fission

125 Figure 5.10: Excitation energy of T2 and its relative position to S1 into three triplets, while 9 and 10 are potential to generate four. Therefore, after taking into account all these three criteria, we anticipate that higher acenes are very likely to undergo singlet fission, but the energy conversion efficiency might be low because of a large system size and low excitation energies. If a singlet fission into multiple triplets can be achieved, then higher acenes might be promising materials for photovoltaic applications. However, it should be noted that because of the increasing diradical character and the high instability of higher acenes, before converting them into devices, one of the most urgent task now might be to find out a condition to stabilize these species. This points out the importance of looking for more stable derivatives [198].

5.3 Conclusion

In summary, we qualitatively and quantitatively described the nature of the ground and lowest few important excited states of higher acenes. With the increasing number

126 of fused benzene rings, their ground state slide in the diradical continuum with more and more open-shell character. For acenes up to 10, the ground state is on the closed-shell side, while for 11 and 12, the ground state tilts more to the open-shell side. The polyradical character also emerges after 10. The ground state always has

1 Ag symmetry and it has covalent nature with respect to both the short and long axes. The ST gap is predicted to be always positive but can be vanishingly small for

1 the polyacene limit based on an exponential decay fitting. The B2u singlet excited state is a zwitterionic state with respect to the short axis, and its energy decreases

1 slowly with the acene size. The excited Ag state also has a decreasing trend in

1 excitation energy but it decreases faster than that of B2u, rendering it the lowest

1 singlet excited state starting from 7. The nature of this Ag state gradually switches from a regular doubly excited state to another zwitterionic state on the short axis with the increasing size of acenes, but always keeps its covalent nature with respect to the long axis. It in principle cannot become the ground state since the ground state should be covalent for both axes. Further consideration on the relative energies

1 of B2u and the lowest two triplet excitations implies that higher acenes are likely to undergo singlet fission but with a relatively low photovoltaic efficiency. The efficiency might be improved if a singlet fission into multiple triplets can finally be achieved.

127 6

Application to Conical Intersections

This chapter is mainly adapted from the following journal article,

• Yang Yang, Lin Shen, Du Zhang, and Weitao Yang, “Conical intersections from particle-particle random phase and Tamm-Dancoff approximations” J. Phys. Chem. Lett. 7, 2407 (2016)

6.1 Introduction

The conical intersection [290, 291] is an important concept in nonadiabatic dynamics. They occur at a set of molecular geometries where two electronic states are degenerate and the potential energy surfaces intersect. Conical intersections are ubiquitous and play a vital role in many nonadiabitic processes related to energy transfer, charge transfer, and ultrafast photochemical transitions [291, 292]. Multireference wave function methods are most frequently used to describe con- ical intersections. Among these methods are CASSCF, CASPT2, and MRCI. These methods are able to provide accurate results for conical intersections with a proper choice of the active space, however, their computational cost also quickly grows with

128 the size of the active space, which is usually not affordable for complex systems. DFT and TDDFT are widely used for ground state and excited states calculations thanks to their good balance of accuracy and efficiency [74, 293]. However, they completely fail to describe those conical intersections involving the ground state and an excited state [74, 294]. This is because the excited states, which are described by TDDFT, have no state interaction with the ground state described by DFT.

Therefore, although the conical intersections should occur in a subspace of N int ´ 2 dimension, with N int the internal degree of freedom [291], the dimension of this intersection space is incorrectly expanded by 1 with TDDFT [74, 294]. In the past few years, many DFT-based methods have been developed to cir- cumvent this problem by introducing the state interaction between the ground state and excited states [295]. Li et al. directly introduced a configuration interaction coupling element to the matrix of TDDFT within the Tamm-Dancoff approximation to break the unphysical degeneracy [296]. Van Voorhis’s group developed configu- ration interaction based constrained density functional theory (CDFT-CI) [297] and well described the conical intersections with some prior knowledge on the target system [298]. Based on a spin-restricted ensemble construction of the Kohn-Sham DFT (REKS) and a state-averaged (SA) energy functional, Huix-Rotllant et al. sug- gested a state interaction element in the SI-SA-REKS method, which also properly describes the conical intersections and yields the correct dimensionality [299, 300]. A natural introduction of the interaction is from the SF-TDDFT [99, 149]. It starts from a high-spin state and targets the low-spin ground and excited states by a spin- flip operator. The state interaction element between the ground and excited states are naturally built in the SF-TDDFT matrix and therefore the dimensionality of intersection space is also correct. Furthermore, the SF-TDDFT has been applied to locate the geometries of conical intersections [300, 301]. Here we will apply the pp-RPA and pp-TDA to the description of conical in-

129 tersections. The pp-RPA starts from a two-electron deficient (N-2) reference, and targets the neutral (N) states by adding two electrons, thus describing the ground and excited states on the same footing. Similar to SF-TDDFT, the interaction be- tween ground and excited states are naturally taken into consideration. Therefore, it is expected that the pp methods can also describe conical intersections with a relatively low (O(N 4)) computational cost [162]. However, it should be noted that methods with non-Hermitian Hamiltonian in principle cannot describe all conical intersections with possible complex eigenvalues.[302, 303] As a result, methods based on coupled cluster theory may face challenges for accidental same-symmetry conical intersections.[302, 303] Here before we go into the detailed performances of pp methods, we briefly comment on the Hermiticity of the pp methods. The matrices A and C in pp-RPA and pp-TDA equations are both Hermitian. Therefore, mathematically speaking, the pp-TDA eigenvalue problem is guaranteed to be stable with only real eigenvalues. In contrast, because of the spe- cial metric rI, 0; 0, ´Is in the pp-RPA equation, the pp-RPA in principle can have imaginary eigenvalue problems. However, till now we have never encountered this problem in regular systems thanks to the physical fact that the two-electron addition and two-electron removal processes are well decoupled. In other words, the pp-RPA solutions preserve the stability of molecules with respect to the spontaneous dispro- portionation reactions into two-electron addition and removal products, leading to real eigenvalues in the pp-RPA solutions. But we cannot guarantee it to also per- form well for challenging accidental same-symmetry conical intersections. Therefore, although in later sections we present results both from pp-RPA and pp-TDA, we tentatively recommend pp-TDA for general conical intersection problems.

130 6.2 Method

Starting from an N-2 reference, the energy of the neutral ground and excited states can be obtained by [304]

N N´2 `2 E0 “E0 ` ω0 , (6.1a)

N N´2 `2 En “E0 ` ωn . (6.1b) where 0 and n denote the ground and excited states, respectively. The pp-RPA calculations are performed with QM4D [45]. TDDFT and CASSCF calculations are performed with Gaussian09 [109]. The B3LYP functional is used in pp-RPA and TDDFT calculations. Trihydrogen (H3) and amonia (NH3) are em- ployed as test systems. The 6-31G basis set is used for H3 and aug-cc-pVDZ basis set is used for NH3. The active space in CASSCF is chosen to be (3,6) for H3 and (6,6) for NH3. To overcome the convergence problem for regular CASSCF, SA-CASSCF is used in NH3, which averages over the lowest two states. Because H3 is an open- shell doublet species, the pp-RPA based on unrestricted N-2 calculations generates two set of spin-contaminated data, and we average the two series to reduce the spin contamination [176, 120].

6.3 Results

6.3.1 D3h H3

Previously, H3 has been used to demonstrate the problem of TDDFT as well as the success of CDFT-CI in describing conical intersections [294, 298]. For this open- shell doublet system, a seam of symmetry-required conical intersections between the lowest two doublet states is located at equilateral geometries with degenerate SOMO and LUMO. However, if we fix two atoms and only move the third one, the higher

D3h symmetry will reduce to lower C2v or Cs symmetries and hence break the orbital and state degeneracy. Therefore, as was done in Ref. [294], we fix two atoms on the

131 0.3 0.3 0.3 0.2 0.2 0.2 0.1 0.1 0.1 0 0 0 −0.1 −0.1 −0.1 Energy (eV) Energy (eV) Energy (eV) −0.2 −0.2 −0.2 −0.3 −0.3

0.72 0.72 0.72

0.71 0.71 0.71

0.7 0.7 0.7 0.02 0.03 0.02 0.03 0.02 0.03 0 0.01 0 0.01 0 0.01 0.69 −0.01 0.69 −0.01 0.69 −0.01 x (angstrom) −0.03 −0.02 x (angstrom) −0.03 −0.02 x (angstrom) −0.03 −0.02 y (angstrom) y (angstrom) y (angstrom) (a) CASSCF(3,6) (b) TDDFT-B3LYP (c) pp-RPA-B3LYP (Same as pp-TDA-B3LYP here)

Figure 6.1: Potential energy surfaces of H3 around a conical intersection. Two atoms are are fixed on the y axis at (0, 0.409) and (0, -0.409), respectively, while the position of the third atom is varied along the x and y axes. TDDFT gives an unphysical butterfly shape for the excited state surface, while pp-RPA correctly describes the conical intersection with potential energy surfaces closely resembling CASSCF.

y axis with coordinates being (0, 0.409) and (0, -0.409), and place the third atom

initially at (0.708, 0) with the D3h symmetry and then move it along the x and y directions for scanning purpose. The results are shown in Figure 6.1a-c. The reference potential energy surfaces are obtained from CASSCF(3,6)/6-31G, which is essentially a full active-space cal- culation within the selected basis set. The surfaces display a double-cone feature. TDDFT on top of a ground state DFT calculation provides a butterfly-like surface.

The two “wings” have unphysical basins that occur at Cs symmetry. The “body” originates from the instability problem, which yields imaginary TDDFT eigenvalues that are plotted as negative ones in the figure. In contrast to TDDFT, the pp meth- ods yield results that closely resemble the CASSCF calculations in terms of both the potential energy surface shape and the transition energy amplitude. Note that the pp-RPA reduces to the pp-TDA for this three-electron system since the dimension of B and C are both 0. The accurate double-cone surfaces indicate that the pp methods can also yield good numerical results in the vicinity of conical intersections.

132 0.25

0.3 0.2 0.3 0.15 0.2 0.2 0.1 2 0.1 0.05 0.1 1 0 0 0 Energy (eV) Energy (eV) Energy (eV) −0.05 Energy (eV) 0 1.85 −0.1 −0.1 −0.1 1.8 86 −0.2 −0.15 89 88 1.75 89 −0.2 89

1.48 90 1.39 1.385 1.485 90 1.7 1.395 90 1.39 90 1.49 92 1.4 1.395 1.495 1.405 1.4 1.5 91 94 1.65 radius (angstrom) 91 1.405 91 β (degree) β (degree) β (degree) β (degree) radius (angstrom) radius (angstrom) radius (angstrom) (a) CASSCF(6,6) (b) TDDFT-B3LYP (c) pp-RPA-B3LYP (d) pp-TDA-B3LYP

Figure 6.2: Potential energy surfaces of NH3 around the conical intersection with D3h symmetry. Radius denotes the bond length of the three N-H bonds, and β is the angle between any N-H bond and the vector that trisects the solid angle spanned by the three N-H bonds. TDDFT completely fails and predicts a line of intersections. In contrast, the pp methods correctly predict the double-cone feature, although the bond length of the conical intersection is shorter compared to the CASSCF result.

6.3.2 D3h NH3

NH3 is another well studied model system [296, 305, 306, 307]. Many conical in- tersections can be located at planar geometries [305]. We first study the one with the D3h symmetry. Unlike H3, whose D3h geometries are all in the seam of coni-

cal intersections, there is only one point of degeneracy for NH3. At this point, the

1 1 1 2 closed-shell A1 state and the open-shell A2 state are degenerate. We locate this point simply by scanning over symmetric bond stretching. Afterwards, the potential energy surfaces are scanned by extending the degree of freedom to β, which is the angle between any N-H bond and the vector that trisects the solid angle spanned by

the three N-H bonds. The D3h symmetry is essentially reduced to C3v and the two

1 ˝ states involved are both reduced A1 when β deviates from 90 . The results are plotted in Figure 6.2. CASSCF, pp-RPA and pp-TDA all predict a sloped conical intersection, whereas TDDFT in combination with ground state DFT predicts a curved line of intersections due to the absence of state interaction. TDDFT also suffers from instability issues with imaginary eigenvalues when the bond length gets longer, and displays an unphysical valley because the imaginary values are plotted as negative values. For the pp methods, although their potential energy

133 surfaces look similar to the CASSCF results, the locations of conical intersections differ from CASSCF. With the aug-cc-pVDZ basis set, CASSCF predicts a conical intersection with bond length equal to 1.487 A,˚ while pp-RPA and pp-TDA predicts 1.397 Aand˚ 1.391 A,˚ respectively. The bond length predicted by pp methods is smaller than that predicted by CASSCF, and in fact this is true with any given basis set. While the reason could partially lie in the fact that CASSCF lacks some dynamic correlation, it is also probable that the pp methods have some errors. It has been known that pp-RPA and pp-TDA tend to underestimate the equilibrium bond lengths for ground and excited states [304]. Therefore, it seems that the pp methods could often predict the occurrence of chemical phenomena with a short bond length. The reason might be that the pp methods start with an N-2 SCF calculation, whose low-lying molecular orbitals are overly contracted due to the less screening of the electron-nuclei attraction compared to the neutral species. Nonetheless, it is possible that this problem may be overcome with a better choice of reference orbitals, such as using the orbitals from the neutral reference system (e.g. the HF* reference in Ref. [176]).

6.3.3 C2v NH3

We also studied the conical intersections for NH3 with a C2v symmetry. One bond

(r1) can be freely stretched while the other two are always kept equal (r2 “ r3). The bond angles are all fixed to be 120˝. The contour plots in figure 6.3 show the energy splitting between the two potential energy surfaces. The seam of conical intersections lies in between the two lines marked with 0.3 eV. Again, we can see that CASSCF and the pp methods predict similar chemical phenomena within different bond length scales. The results for pp-RPA and pp-TDA with bond length 1-1.6 Ahave˚ a roughly similar landscape as CASSCF with bond length 1-2 A.˚

134 2 1 0.3 1 2 1 2 1.6 1 2 1.6 2 1.9 1 2 0.3 0.3 0.3 1.8 0.3 0.3 0.3 0.3 1.5 1.5 1 0.3 1.7 1 1 0.3 1 0.3 1 0.3 1 1.4 1 1.4 1.6 2 2 2 1 1 2 2 2 1.5 0.3 1.3 3 1.3 3 3 0.3

(angstrom) 1.4 2 (angstrom) 2 (angstrom) 1 1 1 2

r 1 r r

0.3 0.3 4 0.3 1.3 1.2 1.2 4 4 1 3 3 1.2 3 1 1 1 1 0.3 2 1.1 5 0.3 1.1 5 1.1 2

2 2 1 1 1 1 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 1 1.1 1.2 1.3 1.4 1.5 1.6 1 1.1 1.2 1.3 1.4 1.5 1.6 r =r (angstrom) r =r (angstrom) r =r (angstrom) 2 3 2 3 2 3 (a) CASSCF(6,6) (b) pp-RPA-B3LYP (c) pp-TDA-B3LYP Figure 6.3: Contour plot of potential energy surface splitting (in eV) near the seam ˝ of conical intersections for NH3 with C2v symmetry. Bond angles are fixed at 120 . The plot is with respect to r1 and r2(“ r3). The seam of conical intersections lies in between the pair of lines marked with 0.3 eV. The results from the pp methods are very similar, and with bond length 1-1.6 A,˚ they have a roughly similar landscape as the CASSCF results with bond length 1-2 A.˚

6.4 Conclusion

The pp methods can correctly describe the dimensionality of the conical intersection space. The potential energy surfaces in the vicinity of conical intersections are also well depicted. This is due to the fact that the pp methods are essentially multi- configuration approaches that describe the ground and excited states on the same footing, and naturally take into account the interstate interaction without any ad hoc correction elements. In this sense, it is similar to SF-TDDFT. From a quantitative perspective, the pp-RPA and pp-TDA may underestimate the bond lengths for con- ical intersections as a result of the contracted N-2 molecular orbitals, a shortcoming that might be overcome with different reference orbitals. Nonetheless, as an effi- cient DFT-based approach, the pp methods, especially the pp-TDA that in principle will never suffer from instability problems, are promising in possible applications to conical intersections and non-adiabatic dynamics.

135 7

Outlook and Future Directions

7.1 Future Developments within Pairing Matrix Fluctuation

The pairing matrix fluctuation in combination with the adiabatic connection has been shown to be an elegant theory of describing both ground state correlation and electronic excitation. Because the exact pairing matrix fluctuation, which should in principle lead to exact correlation energies and excitation energies, is unknown, the simplest approximations — pp-RPA and pp-TDA — have been adopted for pactical calculations. As is shown in Chapters2 and3, the pp-RPA is capable of describing correlation and excitation energies reasonably well. However, there still remains some problems and challenges. In the context of correlation energy calculation, when the HF reference is adopted, the pp-RPA is the same as the ladder-CCD. This fact on the one hand suggests that the pp-RPA with HF references can be systematically improved simply by going to higher levels of CC theory; however, on the other hand, it also makes the development of the pp-RPA not so attractive since the CC theory has been so mature that the

136 ladder-CCD is not a significant progress. The employment of DFT references is just the opposite: while its development is considered interesting, the systematical improvement on the approximation is not easily foreseen. Currently, it seems that more practical improvements on pp-RPA correlation energy calculation is to lower the computational cost rather than to improve the accuracy. The pp-RPA has been developed by Shenvi et al. [308] to achieve a scaling of OpN 4q. However, it is not very exciting as a result of a large prefactor. More attempts to approximate the contributions from numerous virtual orbitals may be worthwhile. The pp-RPA for excitation energy calculations seems to be a more promising direction. As is shown in Chapter3, it has already been practical for double exci- tations, CT excitations, diradical systems, and conical intersections. However, the greatest challenge is that currently the pp-RPA is still only limited to excitations excited from the HOMO. Therefore, the most urgent task is to break this limit. The HF* reference could be one solution. Similarly, it is also possible to use a DFT* reference, in which the N-electron DFT calculation is first performed and then two electrons are manually removed. Another way out is to adopt a non-Aufbau reference. For example, with an N- 2 reference whose HOMO is doubly occupied and HOMO-1 is empty, a series of excitations excited from HOMO-1 can be obtained. It is also possible to employ a high-spin triplet N-2 reference, although it may lead to sever spin contamination and spin incompleteness. It may also be worth trying a multi-determinant N-2 reference, or ensemble ref- erence. However, the derivation can be tricky, and we have not been able to achieve any meaningful formula. The other great challenge for pp-RPA excitations is the underestimated bond-

137 length issue in geometry optimization. It happens to ground states, excited states, and conical intersections. As is mentioned in Chapter6, this may be due to the overly contracted molecular orbitals from the N-2 reference. Therefore, a reference that is specially designed for the regular N reference is highly desirable. The HF* reference is the simplest case for this kind of N-electron reference. It is also possible to adopt a DFT* reference. Nonetheless, a reference that is able to minimize the total energy for a specific target state may be a better choice. This kind of reference is under investigation in our group.

7.2 Future of Electronic Structure Theory

The pursuit of better approximations in quantum chemistry continues, and the pp- RPA and pp-TDA are just part of the pursuit. However, it should be aware that at current stage, to get a better approximation is becoming more and more difficult. Some people turn to computer sciences and mathematics in order to develop better algorithm for practical calculations. Some people start to focus more on chemical systems and do application work. Those staying in the field are struggling their way to develop new approximations. Currently, wave function methods are limited by the computational cost and DFT is still at the frontier of this scientific field. For the DFT community, one future direction is to learn from the wave function method, and develop more and more so-called higher-level functionals, including double hybrid, RPAs, and those functionals with orbital manipulations. However, it should be careful and not go too far on this road because after all, the ultimate “functional” is in fact the full-CI. The other direction is to employ more statistical techniques, and the most important of which are machine learning and artificial neural networks. However, there is also a problem in this area. Since the statistical tools are all limited by the completeness of the database and it is hard to obtain a relative complete set of so many kinds of molecules, their general application to

138 quantum chemistry is still questionable even though it is a super hot area recently. All in all, the building of quantum chemistry is still under construction. We have been able to put a few bricks onto it by introducing the pairing matrix fluctuation and the pp-RPA. However, this is far from enough. We need to work hard and think more, meanwhile, more talented people and bright ideas are needed in this field.

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159 Biography

Personal Information: • Full name: Yang Yang • Date of birth: Nov 2, 1989 • Place of birth: Tai’an Shandong, China

Education Background: • Ph.D. in Chemistry, Duke University, 2016 • B.S. in Physics, Peking University, 2011 • B.S. in Chemistry, Peking University, 2011

Honors and Awards: • Paul M. Gross Fellowship, 2013-2014

Publications:

1. Yang Yang, Adriel Dominguez, Du Zhang, and Weitao Yang, “Charge transfer excitations from particle-particle random phase approximation — Opportunities and challenges arising from two-electron deficient systems”, To be submitted

2. Yang Yang, Ernest Davidson, and Weitao Yang, “Nature of ground and elec- tronic excited states of higher acenes”, Submitted

3. Yang Yang, Lin Shen, Du Zhang, and Weitao Yang, “Conical intersections from particle-particle random phase and Tamm-Dancoff approximations”, J. Phys. Chem. Lett. 7, 2407 (2016)

160 4. Yang Yang, Kieron Burke, and Weitao Yang, “Accurate atomic quantum defects from particle-particle random phase approximation”, Mol. Phys. 114, 1189 (2016)

5. Yang Yang, Degao Peng, Ernest Davidson, and Weitao Yang, “Singlet-triplet energy gaps for diradicals from particle-particle random phase approximation”, J. Phys. Chem. A 119, 4923 (2015)

6. Degao Peng, Yang Yang, Peng Zhang, and Weitao Yang, “Restricted second random phase approximations and Tamm-Dancoff approximations for electronic excitation energy calculations”, J. Chem. Phys. 141, 214104 (2014)

7. Yang Yang, Degao Peng, Jianfeng Lu, and Weitao Yang, “Excitation energies from particle-particle random phase approximation: Davidson algorithm and benchmark studies”, J. Chem. Phys. 141, 124104 (2014)

8. Neil Shenvi, Helen van Aggelen, Yang Yang, and Weitao Yang, “Tensor hy- percontracted ppRPA: Reducing the cost of the particle-particle random phase approximation from O(r6) to O(r4)”, J. Chem. Phys. 141, 024119 (2014)

9. Degao Peng, Helen van Aggelen, Yang Yang, and Weitao Yang, “Linear-response time-dependent density-functional theory with pairing fields”, J. Chem. Phys. 140, 18A522 (2014)

10. Helen van Aggelen, Yang Yang, and Weitao Yang, “Exchange-correlation en- ergy from pairing matrix fluctuation and the particle-particle random phase approximation”, J.Chem. Phys. 140, 18A511 (2014)

11. Yang Yang, Helen van Aggelen, and Weitao Yang, “Double, Rydberg and charge transfer excitations from pairing matrix fluctuation and particle-particle ran- dom phase approximation”, J. Chem. Phys. 139, 224105 (2013)

12. Yachao Zhang, Yang Yang and Hong Jiang, “3d-4f magnetic interaction with

161 density functional theory plus U approach: Local coulomb correlation and ex- change pathways”, J. Phys. Chem. A 117, 13194 (2013)

13. Yang Yang, Helen van Aggelen, Stephan N. Steinmann, Degao Peng, and Weitao Yang, “Benchmark tests and spin adaptation for the particle-particle random phase approximation”, J. Chem. Phys. 139, 174110 (2013)

14. Neil Shenvi, Helen van Aggelen, Yang Yang, Weitao Yang, Christine Schw- erdtfeger, and David Mazziotti, “The tensor hypercontracted parametric re- duced density matrix algorithm: coupled-cluster accuracy with O(r4) scaling”, J. Chem. Phys. 139, 054110 (2013)

15. Helen van Aggelen, Yang Yang, and Weitao Yang, Exchange-correlation energy from pairing matrix fluctuation and the particle-particle random phase approx- imation”, Phys. Rev. A 88, 030501 (2013)

16. Jian Peng, Kuo-Chun Tang, Kaitlin McLoughlin, Yang Yang, Danika Forgach, and Roseanne J. Sension, “Ultrafast excited-state dynamics and photolysis in base-off B12 coenzymes and analogues: Absence of the trans-nitrogenous ligand opens a channel for rapid nonradiative decay.”, J. Phys. Chem. B 114, 12398 (2010)

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