DETERMINISTIC AND SEMI-STOCHASTIC CC(P;Q) APPROACHES: NEW DEVELOPMENTS AND APPLICATIONS TO SPECTROSCOPY AND PHOTOCHEMISTRY By Stephen Haniel Yuwono A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of Chemistry – Doctor of Philosophy 2022 ABSTRACT DETERMINISTIC AND SEMI-STOCHASTIC CC(P;Q) APPROACHES: NEW DEVELOPMENTS AND APPLICATIONS TO SPECTROSCOPY AND PHOTOCHEMISTRY By Stephen Haniel Yuwono The development of electronic structure methods that can accurately describe ground and excited states of molecular systems with manageable computational costs and in a system- atically improvable manner continues to be the central theme of quantum chemistry. This dissertation focuses on some of the recent developments in the coupled-cluster (CC) the- ory and its equation-of-motion (EOM) extension to excited electronic states. One of the key challenges in the development of the CC and EOMCC methodologies is the incorpora- tion of many-electron correlation effects due to higher-rank components of the cluster and EOM excitation operators without incurring significant increase in the computational costs, while avoiding failures of perturbative methods of the CCSD(T) type in multireference sit- uations, such as bond breaking and excited states dominated by two-electron transitions, and in certain weakly bound systems. Among the best ways to address these issues is the CC(P;Q) framework, which provides robust and computationally affordable noniterative en- ergy corrections to lower-order CC/EOMCC calculations. In this dissertation, we discuss the different CC(P;Q) variants relying on both the conventional and unconventional truncations in the cluster and EOM excitation operators. The advantages of the CC(P;Q) hierarchy are illustrated using a few examples ranging from small molecule spectroscopy to photo- chemistry of large organic species in solution. In particular, we discuss the computational investigations of the novel super photobase FR0-SB, which exhibits a drastic increase in basicity upon photoexcitation, including the energetics and properties of its excited states, the steric effects governing the excited-state proton transfer involving FR0-SB and alco- hols, and the enhanced photoreactivity of FR0-SB resulting from two-photon excitations, where the δ-CR-EOMCC(2,3) approach that belongs to the CC(P;Q) hierarchy played a key role. Furthermore, we demonstrate that the relatively inexpensive CC(t;3) and CC(q;4) approaches derived from the CC(P;Q) framework are as accurate in describing the challeng- ing weakly bound magnesium dimer, including its ground-state potential and vibrational levels supported by it, as the much more demanding CCSDT and CCSDTQ parent theo- ries. We also show how the highly accurate ground- and excited-state ab initio potentials obtained in the state-of-the-art CCSDT, CR-EOMCCSD(T), and full configuration inter- action (CI) computations allowed us to resolve the existing laser-induced fluorescence and photoabsorption spectra of the magnesium dimer and find the missing high-lying vibrational states of Mg2 that have eluded scientists for half a century. Last, but not least, we dis- cuss our recent extension of the semi-stochastic CC(P;Q) framework, which combines the deterministic CC(P;Q) theory with stochastic CI quantum Monte Carlo (QMC), to excited electronic states, providing rapid convergence to the parent high-level EOMCC methods, such as EOMCCSDT, out of the early stages of QMC propagations. The advantages of the semi-stochastic CC(P;Q) approach targeting EOMCCSDT are illustrated by examining vertical excitations in CH+ and adiabatic excitations in the CH and CNC species. Copyright by STEPHEN HANIEL YUWONO 2022 To Christy and Alethia. v ACKNOWLEDGMENTS First and foremost, I would like to thank my advisor, Professor Piotr Piecuch, for his guidance throughout my Ph.D. program. I am very much indebted to him for introducing me to the wonderful world of the coupled-cluster and many-body theories and encouraging me to explore new ideas in this area. I have greatly enjoyed our countless scientific discussions and tremendously benefited from his deep knowledge, contagious enthusiasm, and patient mentoring. Among the many lessons that he has taught me, I will always remember one in particular: If something is worth doing, then it is worth taking the time and effort to do it nicely; rushing to finish rarely ends well. I would also like to thank the other current and former members of my guidance com- mittee, namely, Professors Katharine Hunt, Gary Blanchard, Marcos Dantus, and Benjamin Levine for their help and guidance. I would also like to thank my initial Ph.D. advisor, Pro- fessor Angela Wilson, for recognizing my interest in computational and theoretical chemistry and recruiting me to do my graduate study in the United States. In addition, I would like to acknowledge Professors Kang Hway Chuan and Richard Wong at the National Univer- sity of Singapore, under whom I had the pleasure of working as an undergraduate student researcher, who nurtured my initial interest in theory and computations. Furthermore, I would like to acknowledge Professors Gary Blanchard, Babak Borhan, Marcos Dantus, and James Jackson and members of their groups for the opportunity to work with them on the investigation of the fascinating super photobase FR0-SB and related compounds. I have enjoyed many interesting discussions throughout the collaboration and this project has taught me how to communicate well with scientists of different backgrounds than mine. I would also like to thank Dr. Kathryn Severin and Professors John McCracken and David Weliky, under whom I have had many wonderful semesters as a teaching assistant for CEM 395 and 495. I am a beneficiary of their rich teaching experience, which at times has helped vi me remember the fundamental concepts I needed in my own research projects. In particular, I would like to thank Dr. Severin who patiently mentored and encouraged me to keep learning when I made mistakes as a teaching assistant. As a theoretical chemist, I initially felt like a fish out of the water when teaching experimental classes, but the experience has reminded me that water-dwellers did eventually evolve to walk on land and I should similarly endeavor to not be constrained to my comfort zone. I am also grateful to the current and former members of the Piecuch research group: Dr. Jun Shen, Dr. Suhita Basu Mallick, Dr. Adeayo Ajala, Dr. J. Emiliano Deustua, Dr. Ilias Magoulas, Mr. Arnab Chakraborty, Mr. Karthik Gururangan, Mr. Tiange Deng, and Ms. Swati Priyadarsini. The projects that I have been involved with during my Ph.D. program would not have been easily accomplished without their help and I have learned so much from our discussions and interactions. In particular, I would like to give a shout out to Dr. Ilias Magoulas, my “partner in crime”, for the countless hours we have worked together and for his stalwart friendship that goes beyond our Ph.D. work. I would also like to thank my friends in the Greater Lansing Area who have made my and my family’s stay here enjoyable and unforgettable. Specifically, I would like to thank the MSU Indonesian Student Association, for organizing many events that have become the remedy to our homesickness. I would like to acknowledge my friends at the MSU Graduate InterVarsity Christian Fellowship and Riverview Church for their support throughout my Ph.D. journey. Many thanks as well to Ms. Laura Castro Diaz, Ms. Eleni Lygda, and little Maria Magoulas for being such wonderful friends and family away from home. I would like to thank my parents, Budhi Sutikno Yuwono and Padmawati Sutanto, as well as my brother, Ebbe, and his family, for their support while I have been studying abroad. Many thanks to Uncle Erwien, Aunt Junita, and little Yohana, for their support and encouragement as well. Last, but not least, I would like to thank my wife, Christy Natalia, and my daughter, Alethia, for their unconditional love, patience, help, and support as I navigated my Ph.D. studies. vii TABLE OF CONTENTS LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv CHAPTER 1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 CHAPTER 2 SINGLE-REFERENCE COUPLED-CLUSTER THEORY AND ITS EQUATION-OF-MOTION EXTENSION TO EXCITED ELECTRONIC STATES . . . . . . . . . . . . . . . . . . . . . . . . 14 2.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2 Application: Resolving a Half-Century-Old Enigma: The Elusive v ′′ = 14–18 Vibrational Levels of Mg2 . . . . . . . . . . . . . . . . . . . . . . . . . 19 CHAPTER 3 THE CC(P;Q) FORMALISM . . . . . . . . . . . . . . . . . . . . . 47 3.1 Introduction to the CC(P;Q) Methodology . . . . . . . . . . . . . . . . . . . 47 3.2 The Completely Renormalized CC/EOMCC Framework . . . . . . . . . . . 51 3.2.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.2.2 Application: Photochemistry of the Novel Super Photobase FR0-SB 54 3.3 Active-Space CC/EOMCC Approaches and Their CC(P;Q) Extensions . . . 88 3.3.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 3.3.2 Application: Ground-State Potential Curve and Vibrational Term Values of Mg2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 CHAPTER 4 THE SEMI-STOCHASTIC CC(P;Q) METHODOLOGY FOR EX- CITED ELECTRONIC STATES . . . . . . . . . . . . . . . . . . . . 119 4.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 4.2 Application: Electronic Excitation Spectra of CH+ , CH, and CNC . . . . . . 127 CHAPTER 5 CONCLUDING REMARKS AND FUTURE OUTLOOK . . . . . . . 146 APPENDIX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 viii LIST OF TABLES Table 2.1: Comparison of the ab initio (Calc.) and experimentally derived (Expt) rovibrational G(v ′′ , J ′′ ) energies for selected values of J ′′ characterizing 24 Mg2 in the ground electronic state (in cm−1 ), along with the corre- sponding dissociation energies De (in cm−1 ) and equilibrium bond lengths re (in Å). The G(v ′′ , J ′′ ) energies calculated using the ab initio X 1 Σ+ g PEC defined by Eq. (2.21) are reported as errors relative to experiment, whereas De and re are the actual values of these quantities. If the exper- imental G(v ′′ , J ′′ ) energies are not available, we provide their calculated values in square brackets. Quasi-bound rovibrational levels are given in italics. Horizontal bars indicate term values not supported by the X 1 Σ+ g PEC. Adapted from Ref. [2]. . . . . . . . . . . . . . . . . . . . . . . . . . 39 Table 2.2: Comparison of the rovibrational G(v ′′ , J ′′ ) energies obtained using the ab initio X 1 Σ+ g PEC (Calc.) and its X-representation counterpart con- structed in Ref. [161] (X-rep.) for selected values of J ′′ characterizing 24 Mg2 in the ground electronic state (in cm−1 ), along with the corre- sponding dissociation energies De (in cm−1 ) and equilibrium bond lengths re (in Å). The G(v ′′ , J ′′ ) energies calculated using the ab initio X 1 Σ+ g PEC defined by Eq. (2.21) are reported as errors relative to experiment, whereas De and re are the actual values of these quantities. If the exper- imental G(v ′′ , J ′′ ) energies are not available, we provide their calculated values in square brackets. Quasi-bound rovibrational levels are given in italics. Horizontal bars indicate term values not supported by the X 1 Σ+ g PEC. Adapted from Ref. [2]. . . . . . . . . . . . . . . . . . . . . . . . . . 40 ′ ′ Table 2.3: Comparison of the theoretical line positions of the A 1 Σ+ u (v = 3, J = ′′ ′′ 11) → X 1 Σ+ g (v , J = 10, 12) fluorescence progression in the LIF spec- 24 trum of Mg2 calculated in this work with experiment. All line positions are in cm−1 . The available experimental values are the actual line posi- tions, whereas our calculated results are errors relative to experiment. If the experimentally determined line positions are not available, we pro- vide their calculated values in square brackets. Horizontal bars indicate term values not supported by the X 1 Σ+ g PEC. Adapted from Ref. [2]. . . 41 Table 3.1: Orbital character, vertical excitation energies ωn(EOMCC) (in eV and nm), oscillator strengths, and electronic dipole moment values µn (in D) of the four lowest-energy excited singlet electronic states Sn of FR0-SB as obtained in the EOMCC calculations described in the text. Adapted from Ref. [3]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 ix Table 3.2: The vertical transition energies ω10 (abs.) (in eV) and transition dipole moments µ10 (in D) corresponding to the S0 → S1 absorption, along with the µ0 and µ1 dipoles characterizing the S0 and S1 states (in D) and their ratios for FR0-SB in the gas phase and in selected alcohol solvents calculated at the respective S0 minima following the CC/EOMCC-based protocol described above. Adapted from Ref. [5]. . . . . . . . . . . . . . . 76 Table 3.3: The vertical transition energies ω10 (em.) (in eV) and transition dipole moments µ10 (in D) corresponding to the S1 → S0 emission, along with the µ0 and µ1 dipoles characterizing the S0 and S1 states (in D) and their ratios for FR0-SB in the gas phase and in selected alcohol solvents calculated at the respective S1 minima following the CC/EOMCC-based protocol described in the text. Adapted from Ref. [5]. . . . . . . . . . . . 76 Table 3.4: A comparison of the calculated S0 –S1 adiabatic transition energies with- out [ω10 (ad.)] and with [ω10 (0-0)] zero-point energy (ZPE) vibrational corrections (in eV), along with the differences and ratios of the µ0 and µ1 dipoles characterizing the S0 and S1 states at the respective minima (in D) for FR0-SB in the gas phase and in selected alcohol solvents obtained following the CC/EOMCC-based protocol described in the text with the corresponding experimentally derived data. Adapted from Ref. [5]. . . . . 77 Table 3.5: The key elements of the various deterministic CC methodologies em- ployed in this work summarized using the language of the CC(P;Q) formalism, including the P spaces H (P ) adopted in the iterative CC calculations and the Q spaces H (Q) and D0,K (P ) denominators defining the appropriate noniterative δ0 (P ; Q) corrections (if any), along with the corresponding CPU time scalings. Adapted from Ref. [1] . . . . . . . . . . 91 Table 3.6: Electronic energies of the magnesium dimer at selected internuclear sep- arations r (in Å) obtained in the various valence CC calculations with up to triply excited clusters using the A(T+d)Z basis set. Adapted from Ref. [1]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 Table 3.7: Electronic energies of the magnesium dimer at selected internuclear sep- arations r (in Å) obtained in the various subvalence CC calculations with up to triply excited clusters correlating all electrons other than the 1s shells of the Mg atoms and using the AwCTZ basis set. Adapted from Ref. [1]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 Table 3.8: Electronic energies of the magnesium dimer at selected internuclear sep- arations r (in Å) obtained in the various valence CC calculations with up to triply excited clusters using the A(Q+d)Z basis set. Adapted from Ref. [1]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 x Table 3.9: Electronic energies of the magnesium dimer at selected internuclear sep- arations r (in Å) obtained in the various subvalence CC calculations with up to triply excited clusters correlating all electrons other than the 1s shells of the Mg atoms and using the AwCQZ basis set. Adapted from Ref. [1]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 Table 3.10: Vibrational energies G(v) (in cm−1 ), dissociation energies De (in cm−1 ), and equilibrium bond lengths re (in Å) for the magnesium dimer obtained in the various valence CC calculations with up to triply excited clusters using the A(T+d)Z basis set. Adapted from Ref. [1]. . . . . . . . . . . . . 102 Table 3.11: Vibrational energies G(v) (in cm−1 ), dissociation energies De (in cm−1 ), and equilibrium bond lengths re (in Å) for the magnesium dimer obtained in the various subvalence CC calculations with up to triply excited clus- ters using the AwCTZ basis set. Adapted from Ref. [1]. . . . . . . . . . . 103 Table 3.12: Vibrational energies G(v) (in cm−1 ), dissociation energies De (in cm−1 ), and equilibrium bond lengths re (in Å) for the magnesium dimer obtained in the various valence CC calculations with up to triply excited clusters using the A(Q+d)Z basis set. Adapted from Ref. [1]. . . . . . . . . . . . . 104 Table 3.13: Vibrational energies G(v) (in cm−1 ), dissociation energies De (in cm−1 ), and equilibrium bond lengths re (in Å) for the magnesium dimer obtained in the various subvalence CC calculations with up to triply excited clus- ters using the AwCQZ basis set. Adapted from Ref. [1]. . . . . . . . . . . 105 Table 3.14: Electronic energies of the magnesium dimer at selected internuclear sep- arations r (in Å) obtained in the various valence CC calculations with up to quadruply excited clusters using the A(T+d)Z basis set. Adapted from Ref. [1]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 Table 3.15: Vibrational energies G(v) (in cm−1 ), dissociation energies De (in cm−1 ), and equilibrium bond lengths re (in Å) for the magnesium dimer obtained in the various valence CC calculations with up to quadruply excited clusters using the A(T+d)Z basis set. Adapted from Ref. [1]. . . . . . . . 110 Table 3.16: Vibrational energies G(v) (in cm−1 ), dissociation energies De (in cm−1 ), and equilibrium bond lengths re (in Å) for the magnesium dimer obtained in the subvalence CCSD(T) calculations using the AwCnZ basis sets with n = T, Q, and 5 and two CBS extrapolation schemes, designated as CBS- 1 and CBS-2, described in the text. Adapted from Ref. [1]. . . . . . . . . 117 xi Table 3.17: Vibrational energies G(v) and spacings ∆Gv+1/2 ≡ G(v + 1) − G(v) (in cm−1 ), dissociation energies De (in cm−1 ), and equilibrium bond lengths re (in Å) for the magnesium dimer, as obtained with the two compos- ite schemes discussed in the text and integrating the radial Schrödinger equation from 3.2 to 100.0 Å to capture the barely bound v = 18 state. Adapted from Ref. [1]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 Table 4.1: Convergence of the CC(P )/EOMCC(P ) and CC(P;Q) energies toward CCSDT/EOMCCSDT for CH+ , calculated using the [5s3p1d/3s1p] ba- sis set of Ref. [258], at the C–H internuclear distance R = Re = 2.13713 bohr. The P spaces used in the CC(P ) and EOMCC(P ) calculations were defined as all singles, all doubles, and subsets of triples extracted from i-FCIQMC propagations for the lowest states of the relevant sym- metries. Each i-FCIQMC run was initiated by placing 1500 walkers on the appropriate reference function [the RHF determinant for the 1 Σ+ g states, the 3σ → 1π state of the 1 B1 (C2v ) symmetry for the 1 Π states, and the 3σ 2 → 1π 2 state of the 1 A2 (C2v ) symmetry for the 1 ∆ states], set- ting the initiator parameter na at 3, and the time step ∆τ at 0.0001 a.u. The Q spaces used in constructing the CC(P;Q) corrections consisted of the triples not captured by i-FCIQMC. Adapted from Ref. [113]. . . . . . 140 Table 4.2: Same as Table 4.1 for the stretched C–H internuclear distance R = 2Re = 4.27426 bohr. Adapted from Ref. [113]. . . . . . . . . . . . . . . . . . . . 141 Table 4.3: Convergence of the CC(P )/EOMCC(P ) and CC(P;Q) energies toward CCSDT/EOMCCSDT for CH, calculated using the aug-cc-pVDZ basis set. The P spaces used in the CC(P ) and EOMCC(P ) calculations were defined as all singles, all doubles, and subsets of triples extracted from i-FCIQMC propagations for the lowest states of the relevant symme- tries. Each i-FCIQMC run was initiated by placing 1500 walkers on the appropriate reference function [the ROHF 2 B2 (C2v ) determinant for the X 2 Π state, the 1π → 4σ state of the 2 A1 (C2v ) symmetry for the A 2 ∆ and C 2 Σ+ states, and the 3σ → 1π state of the 2 A2 (C2v ) symmetry for the B 2 Σ− state], setting the initiator parameter na at 3, and the time step ∆τ at 0.0001 a.u. The Q spaces used in constructing the CC(P;Q) corrections consisted of the triples not captured by i-FCIQMC. Adapted from Ref. [113]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 xii Table 4.4: Convergence of the CC(P )/EOMCC(P ) and CC(P;Q) energies toward CCSDT/EOMCCSDT for CNC, calculated using DZP[4s2p1d] basis set. The P spaces used in the CC(P ) and EOMCC(P ) calculations were defined as all singles, all doubles, and subsets of triples extracted from i-FCIQMC propagations for the lowest states of the relevant symmetries. Each i-FCIQMC run was initiated by placing 1500 walkers on the ap- propriate reference function [the ROHF 2 B2g (D2h ) determinant for the X 2 Πg state and the 3σu → 1πg state of the 2 B1u (D2h ) symmetry for the A 2 ∆u and B 2 Σ+ u states], setting the initiator parameter na at 3, and the time step ∆τ at 0.0001 a.u. The Q spaces used in constructing the CC(P;Q) corrections consisted of the triples not captured by i-FCIQMC. Adapted from Ref. [113]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 Table A.1: Blocks of the CCSDT 1-RDM with the corresponding labels from Figs. A.2–A.5, contraction types, and possible ranks of Lµ and Rν components. 154 xiii LIST OF FIGURES Figure 2.1: The wave functions of the high-lying, purely vibrational, states of 24 Mg2 and the underlying X 1 Σ+ g potential. The last experimentally observed v ′′ = 13 level is marked in blue, the predicted v ′′ = 14 to 18 levels are marked in green, and the ab initio X 1 Σ+ g PEC obtained in this study is marked by a long-dashed black line. The inset is a Birge–Sponer plot comparing the rotationless G(v ′′ + 1) − G(v ′′ ) energy differences as functions of v ′′ + 1/2 obtained in this work (black circles) with their experimentally derived counterparts (red open squares) based on the data reported in Refs. [153] (v ′′ = 0 to 12) and [158] (v ′′ = 13; cf. also Table 2.1). The red solid line is a linear fit of the experimental points. Adapted from Ref. [2]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Figure 2.2: The VJ ′′ (r) effective potentials including centrifugal barriers character- izing the rotating 24 Mg2 molecule at selected values of J ′′ , along with the corresponding vibrational wave functions and information about the lifetimes for predissociation by rotation, τ (v ′′ ), characterizing quasi- bound states. The selected values of J ′′ used to construct the effective potentials VJ ′′ (r) (black curves) and determine the corresponding bound (blue lines) and quasi-bound (red lines) vibrational wave functions are (a) 20, (b) 40, (c) 60, and (d) 80. The black dashed line represents the rotationless, purely electronic, X 1 Σ+ g potential V (r) = VJ ′′ =0 (r) calculated using the ab initio composite scheme defined by Eq. (2.21). The horizontal black dotted line at 431.4 cm−1 marks the dissociation threshold of the ab initio X 1 Σ+ g potential. Adapted from Ref. [2]. . . . . 43 ′′ ′′ 1 + ′ ′ Figure 2.3: Schematics of the pump, X 1 Σ+ g (v = 5, J = 10) → A Σu (v = 3, J = ′ ′ 1 + ′′ ′′ 11), and fluorescence, A 1 Σ+ u (v = 3, J = 11) → X Σg (v , J = 10, 12), 24 processes resulting in the LIF spectrum for Mg2 shown in Fig. 3 of Ref. 1 + ′′ [161]. The A 1 Σ+ 1 + u and A Σu PECs and the corresponding X Σg (v = 5, J ′′ = 10) and A 1 Σ+ ′ ′ u (v = 3, J = 11) rovibrational wave functions were calculated in this work. The A 1 Σ+ u PEC was shifted to match the experimentally determined adiabatic electronic excitation energy Te of 26,068.9 cm−1 [162]. Adapted from Ref. [2]. . . . . . . . . . . . . . . . . 44 xiv ′ ′ 1 + ′′ ′′ Figure 2.4: The A 1 Σ+ u (v = 3, J = 11) → X Σg (v , J = 10, 12) LIF spectrum ′ ′ of 24 Mg2 . (a) Comparison of the experimental A 1 Σ+ u (v = 3, J = ′′ ′′ 11) → X 1 Σ+ g (v , J = 10, 12) fluorescence progression (black solid lines; adapted from Fig. 3 of Ref. [161] with the permission of AIP Publishing) with its ab initio counterpart obtained in this work (red dashed lines). The theoretical line intensities were normalized such that the tallest peaks in the calculated and experimental spectra corresponding to the v ′′ = 5 P12 line match. (b) Magnification of the low-energy region of the LIF spectrum shown in (a), with red solid lines representing the cal- culated transitions. The blue arrows originating from the v ′′ = 13 label indicate the location of the experimentally observed v ′′ = 13 P12/R10 doublet. The blue arrows originating from the v ′′ = 14 and 15 labels point to the most probable locations of the corresponding P12/R10 dou- blets. Spectral lines involving v ′′ = 16 and 17 are buried in the noise (see also Table 2.3). Adapted from Ref. [2]. . . . . . . . . . . . . . . . . 45 Figure 2.5: Comparison of the vibrational term values characterizing 24 Mg2 sup- ported by the ab initio X 1 Σ+ g potential calculated in Ref. [2] with their experimentally derived counterparts. The top and bottom pan- els show the errors and relative errors, respectively, in the G(v ′′ , J ′′ = 0) vibrational energies corresponding to the X 1 Σ+ g PECs obtained us- ing the CCSDT-based composite scheme defined by Eq. (2.21) and its CCSD(T)-based analog defined by Eq. (2.23) relative to the experimen- tally derived data [153, 158]. Adapted from Ref. [2]. . . . . . . . . . . . . 46 Figure 3.1: The molecular structure of FR0-SB with a representation of the excited- state proton transfer (ESPT) process. The bottom left panel, adapted from Ref. [150], shows the absorption (unshaded) and fluorescence (shaded) spectra of FR0-SB dissolved in acetonitrile (ACN, blue), ethanol (EtOH, black), and ethanol acidified with HClO4 (EtOH/HClO4 , red). The hν1 and hν2 labels correspond to the approximate 0–0 tran- sition wavelengths of FR0-SB and FR0-HSB+ , respectively. The bot- tom right panel provides a simplified illustration of the excitation and emission processes involving the unprotonated and protonated forms of FR0-SB, with labels corresponding to the maxima observed in the absorption and emission spectra. . . . . . . . . . . . . . . . . . . . . . . 61 Figure 3.2: The minimum-energy geometry of the ground electronic state of FR0- SB along with the EOMCCSD/6-31+G* electronic densities and dipole moment vectors characterizing the S0 (orange) and S1 (magenta) states. The S1 –S0 electronic density difference, adapted from Ref. [4], is shown in the bottom, in which areas shaded with red (blue) indicate an increase (decrease) in the electronic density upon S1 → S0 photoexcitation. The change in Mulliken charges of the amine and imine nitrogens of the FR0-SB chromophore upon photoexcitation are also indicated. . . . . . 62 xv Figure 3.3: OPE and TPE steady-state fluorescence spectra obtained for FR0-SB in (a) methanol, (b) ethanol, (c) n-propanol, and (d) i-propanol. In each of the panels, OPE (blue line) is compared with TPE (red line). The fluorescence spectra are normalized to the nonprotonated emission intensity. The ratio between the areas for FR0-HSB+ * (∼15,000 cm−1 ) and FR0-SB* (∼21,000 cm−1 ) emission following OPE and TPE is determined by fits to log-normal functions (thin black lines). Adapted from Ref. [5]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 Figure 3.4: The absorption and emission spectra of FR0-SB in various solvents to compare steric hindrance. The long wavelength emission near 630 nm (∼15,870 cm−1 ) corresponds to FR0-HSB+ *, while the short wave- length emission near 460 nm (∼21,740 cm−1 ) corresponds to FR0-SB*. Adapted from Ref. [4]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 Figure 3.5: Schematic representation of the r1 and r2 N–H internuclear distances needed to create the grid defining the ESPT reaction pathway. . . . . . . 84 Figure 3.6: Results from the reaction pathway calculations showing ground- and excited-state energy differences as a function of proton abstraction. The CAM-B3LYP/6-31+G*/SMD ground-state (S0 ) and excited-state (S1 ) reaction pathways corresponding to the proton abstraction from n-propanol (blue) and i-propanol (orange) by FR0-SB along the in- ternuclear distance between the imine nitrogen and the alcohol proton being transferred. The energies ∆E are shown relative to the ground- state minimum of the respective pathways. The dashed line in each pathway indicates the excited-state geometry relaxation following the S0 –S1 excitation of FR0-SB. Adapted from Ref. [4]. . . . . . . . . . . . . 85 Figure 3.7: Snapshots of the proton abstraction process from n-propanol. The CAM-B3LYP/6-31+G*/SMD optimized geometries of the reactant ([FR0-SB*· · · HOR]), transition state ([FR0-SB*· · · H· · · OR]), and product ([FR0-HSB+ *· · ·− OR]) of the ESPT process between FR0-SB in its S1 electronic state and three n-propanol molecules. The ∆E val- ues in kJ mol−1 are given relative to the reactant energy. The energies inside parentheses, in eV, are given relative to the [FR0-SB· · · HOR] minimum in the ground electronic state S0 , while those inside square brackets correspond to the S0 –S1 vertical transitions at each respective geometry. The rO–H and rN–H distances at each geometry represent the internuclear separations between the proton being transferred and the oxygen of n-propanol and the imine nitrogen of FR0-SB, respectively. Adapted from Ref. [4]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 xvi Figure 3.8: Snapshots of the proton abstraction process from i-propanol. The CAM- B3LYP/6-31+G*/SMD optimized geometries of the reactant ([FR0- SB*· · · HOR]), transition state ([FR0-SB*· · · H· · · OR]), and product ([FR0-HSB+ *· · ·− OR]) of the ESPT process between FR0-SB in its S1 electronic state and three i-propanol molecules. The ∆E values in kJ mol−1 are given relative to the reactant energy. The energies inside parentheses, in eV, are given relative to the [FR0-SB· · · HOR] minimum in the ground electronic state S0 , while those inside square brackets correspond to the S0 –S1 vertical transitions at each respective geometry. The rO–H and rN–H distances at each geometry represent the internuclear separations between the proton being transferred and the oxygen of i- propanol and the imine nitrogen of FR0-SB, respectively. Adapted from Ref. [4]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 Figure 3.9: The ground-state PECs of Mg2 resulting from the CCSD and various CC calculations with up to triply excited clusters using the (a) A(T+d)Z, (b) AwCTZ, (c) A(Q+d)Z, and (d) AwCQZ basis sets. All PECs have been aligned such that the corresponding electronic energies at the inter- nuclear separation r = 15 Å are identical and set at 0 hartree. Adapted from Ref. [1]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 Figure 3.10: The ground-state PECs of Mg2 resulting from the CCSD, CCSDT, and various CC calculations with up to quadruply excited clusters using the A(T+d)Z basis set. All PECs have been aligned such that the corresponding electronic energies at the internuclear separation r = 15 Å are identical and set at 0 hartree. Adapted from Ref. [1]. . . . . . . . . 111 Figure 4.1: Convergence of the EOMCC(P ) [panels (a) and (c)] and CC(P;Q) [pan- els (b) and (d)] energies toward EOMCCSDT for the three lowest-energy excited states of the 1 Σ+ symmetry, two lowest states of the 1 Π sym- metry, and two lowest 1 ∆ states of the CH+ ion, as described by the [5s3p1d/3s1p] basis set of Ref. [258], at the C–H internuclear distance R set at Re = 2.13713 bohr [panels (a) and (b)] and 2Re = 4.27426 bohr [panels (c) and (d)]. Adapted from Ref. [113]. . . . . . . . . . . . . . . . 144 (MC) Figure 4.2: The distributions of the differences between the Rµ,3 amplitudes and their EOMCCSDT counterparts resulting from the EOMCC(P ) com- putations at (a) 4000, (b) 10,000, and (c) 50,000 MC iterations using τ = 0.0001 a.u. for the 21 Σ+ state of CH+ at R = 2Re with the analogous distribution characterizing the Rµ,3 amplitudes obtained with the EOM- CCSDt approach employing the 3σ, 1πx , 1πy , and 4σ active orbitals to define the corresponding triples space [panel (d)]. All vectors Rµ needed to construct panels (a)–(d) were normalized to unity. Adapted from Ref. [113]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 xvii p(CCSDT) Figure A.1: The 17 nonoriented skeletons of Γq . . . . . . . . . . . . . . . . . 152 p(CCSDT) Figure A.2: Oriented skeletons of Γq corresponding to the oo block (γij ), or- dered according to increasing many-body rank. . . . . . . . . . . . . . . 152 p(CCSDT) Figure A.3: Oriented skeletons of Γq corresponding to the uu block (γab ), or- dered according to increasing many-body rank. . . . . . . . . . . . . . . 153 p(CCSDT) Figure A.4: Oriented skeleton of Γq corresponding to the ou block (γia ). . . . . 153 p(CCSDT) Figure A.5: Oriented skeletons of Γq corresponding to the uo block (γai ), or- dered according to increasing many-body rank. . . . . . . . . . . . . . . 153 xviii CHAPTER 1 INTRODUCTION The goal of quantum chemistry is to solve the electronic Schrödinger equation for molecular systems. Although the analytical solution of this equation can only be obtained for nothing more than one-electron systems, such as the hydrogen atom or the H+ 2 molecule, numerically exact solutions can still be obtained by designing suitable basis sets and computer algorithms. Indeed, since the time the Schrödinger equation was proposed, decades of advancement in computer technologies have allowed for the calculations of increasingly complex and large molecular systems, ranging from the spectroscopically accurate ab initio description of weakly bound diatomics in the gas phase [1, 2] to the computations of excitation energies and one- electron properties of solvated organic chromophores [3–8], to name a few examples from my own work. Nevertheless, current quantum chemical calculations are still wrought with problems, especially when the target chemical systems suffer from a significant multireference (MR) character, which is a main issue in situations such as potential energy surfaces along bond breaking coordinates, electronic spectra of radicals and biradicals, and excited states dominated by two- and other many-electron transitions. The brute-force solution to this problem exists in the form of full configuration inter- action (FCI), where the electronic wave function is expressed as a linear combination of all possible Slater determinants that can be obtained from a given basis set of one-particle functions. By inserting this expansion into the electronic Schrödinger equation, one obtains an eigenvalue problem which is equivalent to diagonalizing the Hamiltonian matrix in the many-electron Hilbert space spanned by Slater determinants. In doing so, the FCI method provides the numerically exact solution of the many-electron Schrödinger equation in a given basis and all that remains to be done is to perform FCI computations in larger basis sets and extrapolate the complete basis set (CBS) limit to obtain the numerically exact solu- tion of the Schrödinger equation. While this entire procedure seems straightforward, it is 1 almost of no practical use because the dimensionality of the FCI eigenvalue problem scales factorially with respect to the system size (i.e., the number of electrons and the size of the basis set) [9, 10]. Indeed, this scaling is so steep that it can hardly be overcome by relying on the growth of computer processing speed alone, making the routine application of FCI in quantum chemistry impractical for systems with more than a few electrons. This issue motivates one of the core efforts of quantum chemistry research, namely, the development of alternative approaches that are computationally much more efficient than FCI without compromising accuracy too much. Since using all Slater determinants that a given basis set provides is generally imprac- tical, one might consider the simplest possible alternative, namely, employing only a single Slater determinant to describe the many-electron wave function. This is exactly what is done in the well-known Hartree–Fock (HF) procedure [11–15], where one applies the varia- tional principle to a single Slater determinant via the procedure which is usually called the self-consistent field (SCF) approach. While the HF method is much more affordable than FCI, it treats inter-electronic repulsion in an averaged manner and, thus, fails to capture the many-electron correlation effects that are fundamental in describing most of chemical problems of interest. For example, when we consider the potential energy curve (PEC) of the F2 molecule, the restricted HF (RHF) approach, in which each molecular orbital can be occupied by a pair of electrons with opposite spins, significantly overestimate the binding energy due to RHF overemphasizing the ionic character at the dissociation limit (i.e., RHF predicts F2 → F+ + F− ) [16]. On the other hand, the unrestricted HF (UHF) method, in which the alpha and beta spin orbitals are allowed to have different spatial components, does not bind the molecule at all [16], in addition to introducing various symmetry-broken solutions (see, e.g., Refs. [17, 18] for the classification of the various symmetry-broken UHF solutions). Furthermore, the HF approximation also fails to describe the binding interaction in systems such as the dimers of alkaline earth metal and noble gas atoms, because the un- derlying dispersion interactions require explicit treatment of the many-electron correlation 2 effects, which are neglected in HF computations, as already mentioned above. Nevertheless, despite all its shortcomings, the HF approach provides a convenient starting point for various correlated electronic structure methods. Among the numerous quantum chemistry approaches that have been developed so far to capture the many-electron correlation effects, methods based on the exponential wave func- tion ansatz of coupled-cluster (CC) [19–24] theory offer the best balance between accuracy and computational costs, thus providing an excellent alternative to FCI. Historically, the CC theory emerged as an infinite-order generalization of the finite-order many-body perturba- tion theory (MBPT), which is achieved by summing the linked wave function and connected energy diagrams to infinite order with the help of the linked [25–28] and connected [27, 28] cluster theorems, respectively. As a result of this construction, CC methods satisfy several important conditions characterizing the exact theory. First of all, CC approaches are size extensive, i.e., the energy is expressed in terms of connected diagrams only. In practice, size extensivity means that the results of CC calculations do not lose accuracy as the size of the system is increased. Furthermore, the exponential form of the CC wave function allows for separability or size consistency of the wave function in the noninteracting limit, provided the reference function is also separable, enabling CC methods to properly describe fragmentation phenomena. These properties, among others that will be discussed further in this dissertation, establish the CC theory as the de facto standard in high-accuracy ab initio quantum chemistry calculations, even those involving larger molecular systems. Within the CC framework, the seemingly natural way of dealing with MR situations is by using a genuine MRCC formalism. In the MRCC theory, one constructs a multi-dimensional model space consisting of multiple reference determinants such that a proper zeroth-order description of the problem of interest is attained when a single Slater determinant, obtained, for example, from a HF calculation, is a poor reference state. Then, the remaining, mostly dynamical, correlation effects are captured through particle–hole excitations from each ref- erence determinant included in the model space. There are various ways to achieve this, 3 leading to a number of MRCC formulations (see, e.g., Refs. [29–33] for selected reviews), which, unfortunately, also means that there is no unambiguous way of writing the exponential ansatz of the CC wave function within this framework. The situation is further complicated by the fact that the genuine MRCC methodologies cannot compete with the ease of use and implementation of their single-reference (SR) counterparts, which are capable of recovering the relevant dynamical and nondynamical correlation effects in a dynamical manner, namely, through conventional particle–hole excitations from a single reference determinant, as long as the underlying cluster operator contains the many-body components of the sufficiently high rank. Thus, this dissertation will focus on the more straightforward SRCC formalism and for the remainder of this document the term SRCC and CC will be used interchangeably, unless the explicit distinction is required for clarity. In the SRCC framework, the exact ground-state N -electron wave function |Ψ0 ⟩ is ex- pressed using the exponential ansatz, |Ψ0 ⟩ = eT |Φ⟩, where T is the cluster operator, which PN is expressed in terms of its many-body components as T = n=1 Tn , and |Φ⟩ is the Fermi vacuum, usually a HF determinant. When Tn acts on |Φ⟩, it creates all possible connected n-tuply excited components of the exact ground-state wave function |Ψ0 ⟩, while powers of Tn in eT produce the remaining, disconnected but linked, contributions to |Ψ0 ⟩. One can extend the CC formalism to excited states in a relatively straightforward manner through the use of, for example, the equation-of-motion (EOM) [31, 34–36] and linear response (LR) [37–40] formalisms, as well as their symmetry-adapted-cluster (SAC) CI counterpart [41], where one applies a linear excitation operator Rµ to the ground-state CC wave function |Ψ0 ⟩, thus pro- ducing the µ-th excited-state wave function |Ψµ ⟩ = Rµ eT |Φ⟩. The Rµ operator, in analogy PN to T , is expressed in terms of a many-body expansion, namely, Rµ = rµ,0 1 + n=1 Rµ,n , with 1 representing the unit operator and rµ,0 and Rµ,n being the zero- and n-body components of Rµ , respectively. It is worth mentioning that the CC and EOMCC formalisms described above, where the T and Rµ operators contain up to N -tuple excitations, are equivalent to FCI and, thus, they are numerically exact, albeit computationally intractable. Therefore, in 4 practice, one truncates T and Rµ at a particular excitation rank mA < N (usually mA ≪ N ). For example, by truncating the many-body expansions of T and Rµ at mA = 2, i.e., by set- ting T ≈ T1 + T2 and Rµ ≈ rµ,0 1 + Rµ,1 + Rµ,2 , one obtains the basic CC approach with singles and doubles (CCSD) [42, 43] and its excited-state EOMCCSD [34, 35, 44] counter- part. One could, of course, incorporate the higher-order components of T and Rµ to define higher-level CC and EOMCC schemes, such as the CC approach with singles, doubles, and triples (CCSDT) [45, 46] and its EOMCCSDT [47–51] extension, where mA = 3, the CC ap- proach with singles, doubles, triples, and quadruples (CCSDTQ) [52–55] and EOMCCSDTQ [49, 50, 56, 57], where mA = 4, and so on. One of the main appeals of the SRCC theory, as described above, is that the CCSD/EOMCCSD, CCSDT/EOMCCSDT, CCSDTQ/EOM- CCSDTQ, etc. hierarchy rapidly converges to the exact, FCI limit (see, e.g., Ref. [31] and references therein). As long as the number of strongly correlated electrons is not too large, this remains true even when the system of interest suffers from substantial MR character. As already alluded to above, the flexibility and ease of implementation of the CC and EOMCC theories make them attractive choices for handling quantum chemical problems with significant MR character, since one can account for MR correlation effects by the ex- plicit inclusion of higher–than–doubly excited components of T and Rµ . Unfortunately, the computational cost of CC/EOMCC methods quickly becomes prohibitively expensive as one goes from CCSD/EOMCCSD to CCSDT/EOMCCSDT, CCSDTQ/EOMCCSDTQ, and so on. For example, the CPU time scalings of CCSDT/EOMCCSDT and CCSDTQ/EOM- CCSDTQ are n3o n5u (N 8 ) and n4o n6u (N 10 ), respectively, where no (nu ) is the number of corre- lated occupied (unoccupied) orbitals and N is a measure of the system size. These scalings are much higher than the n2o n4u , or N 6 , CPU time scaling characterizing the basic CCSD/ EOMCCSD approach. Therefore, the key challenge in the development of CC/EOMCC ap- proaches is the incorporation of many-electron correlation effects brought by the Tn and Rµ,n components with n > 2 without incurring the computational costs of the parent CCSDT/ EOMCCSDT, CCSDTQ/EOMCCSDTQ, etc. methods. 5 Traditionally, one can include the correlation effects due to the higher–than–two-body components of T and Rµ through MBPT arguments, either iteratively, as in the CCSDT-n [58–61] and CCSDTQ-n [62] approaches, or through the use of noniterative corrections, re- sulting in ground-state methods such as CCSD[T] [63], CCSD(T) [64], and CCSD(TQf ) [65], and their various excited-state extensions based on EOMCC or LRCC [66–72] (see Ref. [31] for a review). Although these methods reduce the prohibitive costs of the full CCSDT/EOM- CCSDT and CCSDTQ/EOMCCSDTQ approaches, while offering high accuracies near the equilibrium geometries of molecules or for excited states dominated by one-electron transi- tions, they fail at properly describing bond breaking and doubly excited states due to the perturbative nature of the employed approximations. Furthermore, while it is well-known that perturbative corrections of the CCSD(T) type fail to properly describe the dissociation of a closed-shell system into its constituent open-shell fragments, it is worth pointing out that such methods are also far from being accurate in describing certain classes of weakly bound dimers that dissociate into closed-shell atoms, such as Be2 [73] and Mg2 [1, 2], and we will further discuss the latter example in this dissertation. Among the most successful remedies to failures of perturbative CC/EOMCC approxi- mations, such as CCSD(T) and its EOM extensions, in MR situations, within the SRCC framework is the CC(P;Q) [73–77] formalism developed by the Piecuch group, which will be the main focus of this dissertation. The key idea behind the CC(P;Q) theory is to first solve the CC/EOMCC problem in a subspace of the N -electron Hilbert space designated as the P space and correct the resulting CC/EOMCC energies using the suitably generalized form of the method of moments of CC equations (MMCC) [74, 78–93], with the help of de- terminants residing on another subspace of the N -electron Hilbert space called the Q space. The CC(P;Q) formalism is very versatile owing to the flexibility in defining the P and Q spaces. For example, if the P and Q spaces are defined following the conventional trunca- tion of the cluster and EOM excitation operators, the resulting CC(P;Q) approaches become equivalent to the left-eigenstate completely renormalized (CR) CC/EOMCC approaches and 6 related schemes [74, 89–95]. Examples of methods in those categories include the ground- state CR-CC(2,3) [89–92] and CR-CC(2,4) [89, 90, 94, 96] approaches, which correct the CCSD energy for the correlation effects due to connected triples or triples and quadruples, respectively, and their excited-state extensions, including CR-EOMCC(2,3) [91, 93] and its rigorously size-intensive modification designated as δ-CR-EOMCC(2,3) [95], to name a few examples. The CR-CC/EOMCC methods have shown considerable successes in recover- ing the correlation effects due to the higher–than–two-body components of T and Rµ at a reasonable cost, while avoiding the failures of CCSD(T)-type approaches. However, the CR- CC/EOMCC computations may fail in situations where higher–than–two-body components of T and Rµ become large and strongly coupled to their low-rank T1 , T2 , Rµ,1 , and Rµ,2 counterparts. In the CR-CC(2,3)/CR-EOMCC(2,3) approach, for example, one uses the T1 and T2 as well as Rµ,1 and Rµ,2 components obtained in CCSD/EOMCCSD calculations to determine the noniterative triples corrections, even though one should relax the T1 , T2 , Rµ,1 , and Rµ,2 amplitudes in the presence of their T3 and Rµ,3 counterparts, which become prominent in MR situations, affecting the one- and two-body components or T and Rµ . One can address the issue of coupling the lower- and higher-rank Tn and Rµ,n components by turning to the active-space CC/EOMCC ideas [47, 48, 55, 81, 97–101], where the cluster and excitation amplitudes defining the Tn and Rµ,n components with n > 2 are downse- lected by introducing a set of active orbitals relevant to the MR problem of interest, which in most cases are much fewer than the total number of orbitals, allowing the active-space CC/EOMCC schemes to capture most of the relevant nondynamical and dynamical correla- tion effects with relatively low computational costs. The resulting unconventional truncations include methods such as CCSDt (the CC approach with all singles, all doubles, and a subset of triples defined through active orbitals), CCSDtq (the CC method with all singles and dou- bles, and active-space triples and quadruples), and CCSDTq (the CC scheme with all singles, doubles, and triples and active-space quadruples), as well as their EOMCC extensions. Com- bined with the CC(P;Q) moment corrections, they yield the CC(t;3), CC(t,q;3), CC(t,q;3,4), 7 CC(q;4), etc. hierarchy, in which the energies obtained in the active-space CC/EOMCC cal- culations, such as CCSDt/EOMCCSDt, CCSDtq/EOMCCSDtq, or CCSDTq/EOMCCS- DTq, are corrected for the missing, mostly dynamical, correlations due to the remaining triples [CC(t;3), CC(t,q;3)], triples and quadruples [CC(t,q;3,4)], or quadruples [CC(q;4)] that cannot be captured with active orbitals, and the relevant lower-rank components of T and Rµ are relaxed in the presence of their higher-rank counterparts. As shown in Refs. [1, 73–77]), this leads to substantial improvements to their corresponding CR-CC results, es- pecially when the higher–than–two-body cluster components become large, and the CC(t;3), CC(t,q;3), CC(t,q;3,4), and CC(q;4) approaches accurately reproduce the parent CCSDT and CCSDTQ energetics, but performing CC(P;Q) computations in this way requires chem- ical intuition regarding the choice of the appropriate set of active orbitals, needed to select the dominant triply, triply and quadruply, or quadruply excited determinants in the wave function for the inclusion in the P space defining the CC(P;Q) expansions. Therefore, the natural next step in the development of novel CC(P;Q) approaches is to search for possible avenues for automating the selection of higher–than–doubly excited determinants entering the P space in a CC(P;Q) computation. If such an automated protocol exists, it needs to be designed in such a way that the resulting CC(P;Q) energies rapidly converge to their CCSDT/EOMCCSDT, CCSDTQ/EOMCCSDTQ, etc. parents, even when the higher–than– two-body T and Rµ,n components become substantial, at the fraction of the computational costs. To that end, the Piecuch group has recently proposed a new class of hybrid CC(P;Q) methods that can be loosely categorized into two different approaches. The first one arose from the merger of the deterministic CC(P;Q) theory with the stochastic quantum Monte Carlo (QMC) wave function propagations in the many-electron Hilbert space defining the CIQMC [102–106] and CC Monte Carlo (CCMC) [107–110] methods, culminating in the “semi-stochastic” or “QMC-driven” CC(P;Q) schemes [111–114]. The second strategy re- sulted from combining CC(P;Q) with deterministic or largely deterministic ways of sampling 8 many-electron wave functions carried out with the help of selected CI [115–131] diagonaliza- tions, which led to the selected-CI-driven CC(P;Q) framework [132]. These hybrid CC(P;Q) approaches combine the strengths of CIQMC, CCMC, and selected CI, especially their effec- tiveness in identifying the leading determinants in the many-electron wave function, with the robustness of the CC(P;Q) moment corrections. One of the main topics of this dissertation is the extension of the semi-stochastic CC(P;Q) methodology of Refs. [111, 114] to excited electronic states, but before discussing the key concepts behind it, let us take a moment to review some background information pertaining to QMC. The idea behind QMC methods dates back to the 1949 work of Metropolis and Ulam [133], which resulted in stochastic sampling procedures to numerically integrate various forms of differential equations [133–135]. This, in turn, inspired the simplest form of QMC method- ology, namely, the variational MC (VMC) method [136, 137], where one optimizes a trial wave function, while applying stochastic sampling to compute the required expectation val- ues of the Hamiltonian operator. Although VMC calculations are computationally efficient, their results depend heavily on the quality of the trial wave function. An improvement to VMC was found in the diffusion MC (DMC) approach [138–140], which works by treating the Schrödinger equation in the imaginary time as a diffusion equation and letting a trial wave function evolve to the exact wave function by adopting a projection technique (see, e.g., Refs. [141–143] for further details). Thus, as long as the trial wave function is not or- thogonal to the exact wave function, the DMC propagation is guaranteed to project out the exact solution of the Schrödinger equation in the infinite imaginary time limit. In addition, DMC (and its VMC predecessor) is appealing due to its capability to circumvent the need for finite one-particle basis sets, because one can perform the propagation in the real space of 3N electronic coordinates instead. The DMC and VMC algorithms are also easily paral- lelizable across many multi-core nodes, further increasing their popularity in the quantum chemistry community. Nevertheless, despite the advantages that DMC propagation offers, it is plagued by the fact that if the wave function propagation is run without any constraint, 9 it will produce the bosonic solution to the many-electron Schrödinger equation, which is the true ground state of the spin-free Hamiltonian violating the Pauli exclusion principle, result- ing in the so-called “boson catastrophe” or “fermion sign problem”. The most common way to circumvent this issue has been to employ the so-called fixed-node approximation [144– 147], where one imposes the nodal structure of the wave function obtained in an inexpensive quantum chemistry calculation, such as HF or multiconfigurational SCF, but in doing so the DMC propagation can no longer produce the exact solution of the Schrödinger equation. A novel way to tackle the fermion sign problem plaguing the DMC methodology is given by the full CIQMC (FCIQMC) approach and its truncated CIQMC analogs [102, 103], where one replaces the wave function propagation in the real space of 3N electronic coordinates by the propagations of CI expansions in the N -electron Hilbert space spanned by Slater determinants. Because Slater determinants are antisymmetric with respect to the exchange of any pair of electronic coordinates by construction, the many-electron wave functions pro- duced by the FCIQMC and other CIQMC propagations are guaranteed to be antisymmetric as well. FCIQMC and its truncated CIQMC analogs use stochastic walker population dy- namics in the wave function propagations, where the more important Slater determinants are populated by larger walker numbers. If all possible Slater determinants are allowed to be populated throughout the CIQMC simulation, as in FCIQMC, then at the infinite imagi- nary time limit one is guaranteed to converge the FCI solution within the employed basis set. Similar to the fully deterministic CI computations, one could also converge the truncated CISD, CISDT, CISDTQ, etc. results by limiting the space in which the propagation is being performed to be spanned by determinants of up to a certain excitation rank (e.g., including up to triply excited Slater determinants in the sampled subspace means that the CIQMC propagation will converge to CISDT). While the rate of convergence of CIQMC can be slow, there exist modifications, such as the initiator approximation [103] or its newer adaptive shift modification [105, 106], that serve to accelerate the convergence of the CIQMC prop- agations. One can also develop the CC analogs of the CIQMC methods, resulting in the 10 CCMC approaches of Refs. [107–110], by replacing the CI expansions, when propagating the wave functions in the many-electron Hilber space, by the CC ansatz. Although the FCIQMC algorithm is guaranteed to provide the exact solution to the Schrödinger equation within a given basis set and truncated CIQMC methods converge the corresponding truncated CI results, one has to deal with the stochastic noise inherent to these methodologies. To reduce the numerical noise to acceptable levels, one usually needs to run the propagation for a very long time, namely, tens or hundreds of thousands of MC imaginary time steps called “MC cycles” or “MC iterations”. Furthermore, if excited states of the same symmetry as the ground state are of interest, one has to resort to highly complex protocols by adopting, for example, a Gram–Schmidt procedure to orthogonalize higher-energy states against the lower-energy ones, so that the collapse of the dynamically propagated excited states on the lower-energy states within the same irreducible represen- tation is avoided [148, 149]. Nevertheless, FCIQMC and its truncated counterparts are able to identify the leading determinants in the many-electron wave functions in the early stages of the QMC propagation. This observation is the basis of the semi-stochastic CC(P;Q) approach, where one uses the information about the leading determinants populated in the early stages of CIQMC (or CCMC) simulations to build a P space for the CC(P;Q) considerations, while accounting for the remaining determinants using the CC(P;Q) nonit- erative corrections. In this way, one can do what is done using the aforementioned CC(t;3), CC(t,q;3), CC(t,q;3,4), etc. hierarchy, but without relying on system- and user-dependent active orbitals or any other a priori knowledge of the wave function to capture the dominant higher–than–two-body components of the cluster and excitation operators of CC/EOMCC. In other words, by combining the stochastic CIQMC and CCMC ideas with the deterministic CC(P;Q) framework, one can achieve an objective construction of the P and Q spaces for the CC(P;Q) considerations because the identification of the leading triply excited, quadruply excited, etc. determinants does not depend on any input from the user. Recent studies in the Piecuch group have demonstrated that this new paradigm shows a lot of promise in the 11 ground-state considerations [111, 114]. In this dissertation, we will discuss our efforts in ex- tending the semi-stochastic CC(P;Q) algorithm to excited electronic states [112, 113], where we do not have to rely on the complicated excited-state CIQMC propagations described in Refs. [148, 149]. The overall goal of this dissertation is to survey different variants of the CC(P;Q) theory defined using conventional and unconventional truncations in the cluster and EOM excitation operators. This includes several molecular applications of the CR-CC/EOMCC methodolo- gies and active-space CC(P;Q) approaches and our work on extending the semi-stochastic CC(P;Q) formalism to excited electronic states. The molecular examples illustrating the accuracies of the various CC(P;Q) methods include chemical problems relevant to spec- troscopy and photochemistry. We will begin with the discussion of the salient features of the CC and EOMCC theories. This will be followed by the sophisticated quantum chemistry and spectroscopic computations for the weakly bound magnesium dimer, where we utilized high-level CC and EOMCC approaches including up to triple excitations and FCI to describe the PECs of Mg2 in its ground and excited states relevant to experimental measurements, demonstrating that we can achieve spectroscopic accuracy when comparing the theoretical results with the experimentally observed spectral lines. This allowed us to provide infor- mation about the unresolved high-lying vibrational states of the magnesium dimer in the ground-state potential. We will then discuss the key concepts behind the CC(P;Q) formal- ism and examine the conventional ways of using it via the CR-CC/EOMCC formalism and the unconventional active-space CC(P;Q) approaches. The efficacy of the CR-CC/EOMCC methods will be illustrated by examining the super photobase FR0-SB, which is a molecule exhibiting a drastic increase in pKa , of about 14 units, upon photoexcitation [3–8, 150]. We will also return to the discussion of the magnesium dimer, focusing on its ground elec- tronic state, to demonstrate how the active-space-based CC(P;Q) approaches can improve the results obtained using their CR-CC predecessors. Last, but not least, we will discuss the semi-stochastic CC(P;Q) theory, especially its extension to excited electronic states that 12 resulted from this dissertation research [112, 113]. The usefulness of the semi-stochastic CC(P;Q) methodology in excited-state applications will be illustrated by examining vertical excitations in the CH+ ion and adiabatic excitations in the CH and CNC radicals. 13 CHAPTER 2 SINGLE-REFERENCE COUPLED-CLUSTER THEORY AND ITS EQUATION-OF-MOTION EXTENSION TO EXCITED ELECTRONIC STATES 2.1 Theory In the SRCC formalism, the exact ground-state wave function of an N -electron system is expressed as N |Ψ0 ⟩ = eT |Φ⟩ , X T = Tn , (2.1) n=1 where |Φ⟩ is an independent-particle–model (e.g., HF) reference determinant defining the Fermi vacuum and T is the cluster operator. In Eq. (2.1), Tn is the n-body component of T and is defined as a1 ...an tia11...i X Tn = ...an Ei1 ...in , n (2.2) i1 <··· 12. It did not take long to detect one of such states. In 1973, Li and Stwalley [166] identified 1 + ′′ X 1 Σ+g → A Σu transitions involving the v = 13 level in the spectra reported in Ref. [161]. They accomplished this by extending the original RKR PEC of Balfour and Douglas to the asymptotic region beyond 7.16 Å using theoretical values of C6 and C8 van der Waals coef- ficients [167, 168]. The resulting PEC supported 19 vibrational levels, i.e., five levels more than what was observed experimentally [166]. Four decades later, in an effort to character- ize states with v ′′ > 13, Knöckel et al. [169, 170] examined the A 1 Σ+ 1 + u → X Σg transition using laser-induced fluorescence (LIF), repeating and refining the earlier LIF experiment by 24 Scheingraber and Vidal [171]. They improved and expanded the original Mg2 dataset of Balfour and Douglas by reporting a total of 333 G(v ′′ , J ′′ ) and 1,351 G(v ′ , J ′ ) rovibrational term values involving v ′′ = 0–13 and v ′ = 1–46, respectively, and constructed a few experi- mentally derived analytical forms of the X 1 Σ+ g PEC, extrapolated to the asymptotic region 20 using the theoretical C6 [172], C8 [173], and C10 [173] coefficients, which support the discrete spectral data in the 3.27 to 8.33 Å range [169]. Although these refined PECs supported 19 24 Mg2 vibrational levels, reinforcing the initial prediction of Li and Stwalley [166], Knöckel ′ ′ 1 + ′′ ′′ et al. [169] were unable to identify A 1 Σ+u (v , J ) → X Σg (v , J ) transitions involving the elusive high-lying vibrational levels with v ′′ > 13 in their LIF spectra. Typically, high-lying vibrational states near dissociation constitute a small fraction of the entire vibrational manifold, but this is not the case for the weakly bound magnesium dimer, which has a shallow minimum on the ground-state PEC at re = 3.89039 Å [169] and a tiny dissociation energy De of 430.472(500) cm−1 [169, 170]. If the five extra levels, which have been speculated about, truly existed, they would represent more than a quarter of the entire vibrational manifold in the ground electronic state. Furthermore, without precise knowledge of the ground-state PEC of Mg2 , especially its long-range part that determines the positions of the high-lying vibrational states near the dissociation threshold, one can- not accurately interpret the aforementioned ultracold and collisional phenomena involving interacting magnesium atoms. It is intriguing why a seemingly docile main group diatomic continues to challenge state-of-the-art spectroscopic techniques. The experimental difficul- ties in detecting the elusive v ′′ > 13 states of the magnesium dimer originate from several factors, including small energy gaps between high-lying vibrations that are comparable to rotational spacings [161, 174], resulting in overlapping spectral lines, and unfavorable signal- to-noise ratio in the existing LIF spectra [169]. Rotational effects complicate the situation even more, since, in addition to affecting line intensities [169, 171, 174], they may render the high-lying vibrational states of Mg2 unbound. All of these and similar difficulties prompted Knöckel et al. [169, 170] to conclude that experimental work alone is insufficient and that ac- curate theoretical calculations are needed to guide further analysis of the ground-state PEC and rovibrational states of Mg2 , especially the elusive v ′′ > 13 levels near the dissociation threshold. Unfortunately, there have only been a handful of theoretical investigations attempting 21 to determine the entire vibrational manifold of the magnesium dimer. This is, at least in part, related to the intrinsic complexity of the underlying electronic structure and dif- ficulties with obtaining an accurate representation of the ground-state PEC using purely ab initio quantum-chemical means. As already mentioned above, at the HF theory level, which neglects electron correlation and dispersion interactions, Mg2 remains unbound. As demonstrated below, one needs to go to much higher theory levels, incorporate high-order many-electron correlation effects, including valence as well as inner-shell electrons, and use large, carefully calibrated, one-electron basis sets to accurately capture the relevant physics and obtain a reliable description of the X1 Σ+g potential and of the corresponding rovibrational manifold (see Ref. [1] for a detailed discussion and historical account, including references to the earlier quantum chemistry computations for the magnesium dimer). Ab initio quantum mechanical calculations for the A 1 Σ+ u PEC, the rovibrational states supported by it, and 1 + the X 1 Σ+g − A Σu electronic transition dipole moment function, needed to interpret and aid the photoabsorption and LIF experiments using purely theoretical means, are similarly challenging, and this investigation shows this too [2]. The initial theoretical estimates of the number of vibrational states supported by the X1 Σ+g potential ranged from 18 to 20 [175], while the more recent ab initio quantum chemistry computations based on the various levels CC theory, reported in Refs. [1, 176], suggested that the highest vibrational level of 24 Mg2 is v ′′ = 18. Among the previous theoretical studies, only Amaran et al. [176] considered the A 1 Σ+ u state involved in the photoabsorption and LIF experiments and included rotational effects, but they have not provided any information about the calculated rovibrational term values other than the root mean square deviations (RMSDs) relative to the experimental data of Balfour and Douglas [161]. Furthermore, as demonstrated in our earlier benchmark study [1], which will be summarized in a later chapter in this dissertation as well, where a large number of CC methods were tested using the X 1 Σ+ g 24 PEC of the magnesium dimer and the rotationless term values of Mg2 as examples, and consistent with the earlier calculations [177, 178], the popular CCSD(T) approximation 22 exploited in Ref. [176] could not possibly produce the small RMSD value reported in Ref. [176], of 1.3 cm−1 , for the rovibrational manifold of Mg2 in its ground electronic state; the value on the order of a dozen cm−1 would be more appropriate [1]. Similar remarks apply to the A 1 Σ+u state, which was treated in Ref. [176] using the LRCCSD approach, resulting in noticeable deviations from the experimentally derived A 1 Σ+ u potential shown in Fig. 4 of Ref. [170]. To simulate and properly interpret the A 1 Σ+ 1 + u → X Σg LIF spectra obtained in Ref. [169] using purely theoretical means, one needs much higher accuracy levels in the computations of line positions and robust information about line intensities, which has not been obtained in the previous quantum chemistry studies. The goal of the ab initio electronic structure calculations performed in Ref. [2] and sum- 1 + marized here is to obtain highly accurate X 1 Σ+ g and A Σu PECs of the magnesium dimer 1 + X-A and the corresponding X 1 Σ+ g − A Σu transition dipole moment function µz (r) involved in the photoabsorption and LIF experiments reported in Refs. [161, 169–171]. In the case of the ground-state PEC, we combined the numerically exact description of the valence electron correlation effects provided by FCI with the high-level description of subvalence correlations involving all electrons but the 1s shells of Mg atoms obtained using CCSDT. Thus, the X 1 Σ+ g PEC of Mg2 reported in this work was obtained by adopting the composite scheme   CCSDT/AwCQZ FCI/A(Q+d)Z CCSDT/A(Q+d)Z EX 1 Σ+g = EX 1 Σ+ + EX 1 Σ+ − EX 1 Σ+ . (2.21) g g g The first term on the right-hand side of Eq. (2.21) denotes the total electronic energy ob- tained in the full CCSDT calculations correlating all electrons other than the 1s shells of the Mg monomers and using the aug-cc-pwCVQZ basis set developed in Ref. [179], abbre- viated as AwCQZ. The second and third terms on the right-hand side of Eq. (2.21), which represent the difference between the frozen-core FCI and CCSDT energies obtained using the aug-cc-pV(Q+d)Z basis of [179], abbreviated as A(Q+d)Z, correct the nearly all-electron CCSDT/AwCQZ energy for the valence correlation effects beyond CCSDT. The A(Q+d)Z and AwCQZ basis sets were taken from the Peterson group’s website [180]. We used these bases rather than their standard aug-cc-pVnZ and aug-cc-pCVnZ counterparts, since it has 23 been demonstrated that the aug-cc-pV(n+d)Z and aug-cc-pwCVnZ basis set families, includ- ing A(Q+d)Z and AwCQZ, accelerate the convergence of bond lengths, dissociation energies, and spectroscopic properties of magnesium compounds [1, 179]. We will comment on the convergence of our computational scheme in Eq. (2.21) with respect to the size of the basis set and the level of theory employed later. In principle, one could extend the above composite scheme, given by Eq. (2.21), to the electronically excited A 1 Σ+u state by replacing CCSDT in Eq. (2.21) with its EOMCCSDT counterpart, but the nearly all-electron full EOMCCSDT calculations using the large AwCQZ basis set turned out to be prohibitively expensive for us. To address this problem, we resorted to one of the CR-EOMCCSD(T) approximations to EOMCCSDT, namely, CR- EOMCCSD(T),IA [85], which is capable of providing highly accurate excited-state PECs of near-EOMCCSDT quality at the small fraction of the cost. Thus, our composite scheme for the calculations of the A 1 Σ+u PEC was defined as CR-EOMCCSD(T),IA/AwCQZ CR-EOMCCSD(T),IA/A(Q+d)Z   FCI/A(Q+d)Z EA 1 Σ+u = EA 1 Σ+ + EA 1 Σ+ − EA 1 Σ+ , (2.22) u u u where the first term on the right-hand side of Eq. (2.22) is the total electronic energy of the A 1 Σ+ u state obtained in the CR-EOMCCSD(T),IA/AwCQZ calculations correlating all electrons other than the 1s shells of the Mg monomers and the next two terms correct the nearly all-electron CR-EOMCCSD(T),IA/AwCQZ calculations for the valence correlation effects beyond the CR-EOMCCSD(T),IA level using the difference of the FCI and CR- EOMCCSD(T),IA energies obtained with the A(Q+d)Z basis. Before deciding on the use of CR-EOMCCSD(T),IA, we tested other CR-EOMCC schemes by comparing the resulting ′ ′ A 1 Σ+ u potentials obtained using Eq. (2.22) and the corresponding rovibrational term G(v , J ) values with the available experimentally derived data reported in Refs. [169, 181]. Although all of these schemes worked well, the computational protocol defined by Eq. (2.22), with the CR-EOMCCSD(T),IA approach serving as a baseline method, turned out to produce the smallest maximum unsigned errors and RMSD values relative to experiment. 24 All electronic structure calculations for Mg2 performed in this study were based on the tightly converged RHF reference functions (the convergence criterion for the RHF density matrix was set up at 10−9 ). The valence FCI calculations for the X 1 Σ+ 1 + g and A Σu states were performed using the GAMESS package [182–184], whereas the valence and subvalence CCSDT computations for the X 1 Σ+ g state were carried out with NWChem [185]. The valence and subvalence CR-EOMCCSD(T),IA calculations for the A 1 Σ+ u state were executed using the RHF-based CR-EOMCCSD(T) routines developed in [85], which take advantage of the underlying ground-state CC codes described in [186] and which are part of GAMESS as well. The GAMESS RHF-based CC routines [186] were also used to perform the CCSD(T) calculations needed to explore the basis set convergence and the viability (or the lack thereof) of the alternative to the CCSDT-based composite scheme given by Eq. (2.21) (vide infra). The convergence thresholds used in the post-RHF steps of the CC and EOMCC computations reported in this work were set up at 10−7 for the relevant excitation amplitudes and 10−7 hartree (0.02 cm−1 ) for the corresponding electronic energies. The default GAMESS input options that were used to define our FCI calculations guaranteed energy convergence to 10−10 hartree. 1 + The grid of Mg–Mg separations r, at which the electronic energies of the X 1 Σ+ g and A Σu states determined were determined, was as follows: 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, 2.8, 2.9, 3.0, 3.1, 3.2, 3.3, 3.4, 3.6, 3.7, 3.8, 3.9, 4.0, 4.1, 4.2, 4.4, 4.6, 4.8, 5.0, 5.2, 5.4, 5.6, 5.8, 6.0, 6.4, 6.8, 7.2, 7.6, 8.0, 8.4, 8.8, 9.2, 9.6, 10.0, 11.0, 12.0, 13.0, 15.0, 20.0, 25.0, 30.0, and 100.0 Å [2]. We adopted the same set of r values to determine the electronic transition dipole moment function µX-A z (r) between the X 1 Σ+ 1 + g and A Σu electronic states, needed to calculate LIF line intensities using the Einstein coefficients. The µX-A z (r) calculations reported in this work were performed using the valence FCI approach, as implemented in GAMESS, adopting the A(Q+d)Z basis set of Ref. [179]. The rovibrational term values, including bound and quasi-bound states supported by our ab initio X 1 Σ+ 1 + g and A Σu PECs defined by Eqs. (2.21) and (2.22), were computed 25 by numerically integrating the radial Schrödinger equation from 2.2 to 100.0 Å using the Numerov–Cooley algorithm [187] available in the LEVEL16 code [188] (LEVEL16 uses the Airy function approach described in Ref. [189] to locate quasi-bound states). The widths and the tunneling lifetimes for predissociation by rotation characterizing the quasi-bound rovibrational states supported by the X 1 Σ+ g potential were calculated using LEVEL16 as well. In this case, we followed the semiclassical procedure described in Ref. [188] and im- plemented in LEVEL16, which requires numerical integrations between turning points in the classically allowed and classically forbidden regions of the relevant effective potentials including centrifugal barriers (see Ref. [188] for further details). We also used LEVEL16 to determine the rovibrational term values characterizing the experimentally derived analytical X-representation potential developed in [169], which we used to assess the accuracy of our ab initio–determined X 1 Σ+ g PEC. To be consistent with our LEVEL16 calculations for the ground-state PEC resulting from the ab initio protocol based on Eq. (2.21), we first determined the energies corresponding to the X-representation potential on the grid of 47 internuclear distances r adopted in our ab initio work. We then followed the same numerical procedure as described above for the X 1 Σ+ g PEC resulting from the ab initio quantum chemistry calculations. Last, but not least, we used LEVEL16 to compute the line positions of all allowed ′ ′ 1 + ′′ ′′ A 1 Σ+u (v , J ) → X Σg (v , J ) rovibronic transitions and, with the help of our ab initio transition dipole moment function µX-Az (r), the corresponding line intensities, as defined by the Einstein coefficients. The only adjustment that we had to make to be able to compare ′ ′ 1 + ′′ ′′ our calculated line positions and intensities for the allowed A 1 Σ+ u (v , J ) → X Σg (v , J ) transitions with the LIF data reported in Refs. [169, 170] was a uniform downward shift of −1 the entire A 1 Σ+ u PEC resulting from our ab initio computations by 1,543.2 cm , needed to match the experimentally determined adiabatic electronic gap Te of 26,068.9 cm−1 [170]. Other than that, all of the calculated spectroscopic properties, including the De , re , and rovi- brational term values corresponding to the X 1 Σ+ 1 + g and A Σu states and the line positions 26 ′ ′ 1 + ′′ ′′ and intensities characterizing the A 1 Σ+ u (v , J ) → X Σg (v , J ) transitions reported in this study, rely on the raw ab initio data combined with the LEVEL16 processing, as described above. The most essential numerical information, generated using the computational protocol described above, is summarized in Tables 2.1–2.3 and Figs. 2.1–2.5. In describing and dis- cussing our results, we begin with the PECs and rovibrational term values characterizing the X 1 Σ+ 1 + g and A Σu states of the magnesium dimer, focusing on a comparison of our ab initio calculations with the available experimental and experimentally derived data reported in Refs. [161, 169, 170]. Next, we compare the experimental LIF spectra reported in Refs. [169, 170] with those resulting from our computations and suggest potential avenues for detection of the elusive v ′′ > 13 levels of the magnesium dimer. Lastly, we also discuss fur- ther observation on the convergence of the computational protocol described in Eqs. (2.21) and (2.22). Additional information that complements the discussion in this section, includ- ing further comments on the accuracy and convergence characteristics of the computational protocol used in the present study, the effect of isotopic substitution on the calculated rovi- brational term values, the discussion of the validity of the Franck–Condon analysis adopted in Ref. [169] to examine the LIF spectra reported in Refs. [169, 170], and the lifetimes for predissociation by rotation characterizing quasi-bound rovibrational states supported by the X 1 Σ+g potential, can be found in Ref. [2] and the accompanying Supplementary Materials. As shown in Table 2.1, our ab initio X 1 Σ+ g PEC reproduces the experimentally derived dissociation energy De and equilibrium bond length re of Mg2 [169, 170] to within 0.9 cm−1 (0.2%) and 0.003 Å (0.07%), respectively. These high accuracies in describing De and re 24 are reflected in our calculated rovibrational term values of Mg2 and its isotopologs, which are in very good agreement with the available experimental information [161, 169, 170]. Indeed, the RMSDs characterizing our ab initio G(v ′′ , J ′′ ) values for 24 Mg2 relative to their experimentally determined counterparts, reported in Ref. [161] for v ′′ < 13 and Refs. [169, 170] for v ′′ < 14, are 1.1 cm−1 , when the spectroscopic data from Ref. [161] are used, and 1.5 27 cm−1 , when we rely on Refs. [169, 170] instead [2]. At the same time, the maximum unsigned errors in our calculated G(v ′′ , J ′′ ) values relative to the experiment do not exceed ∼2 cm−1 , even when the quasi-bound states above the potential asymptote arising from centrifugal barriers are considered. Although the experimental information about the G(v ′′ , J ′′ ) values 24 characterizing other Mg2 isotopologs is limited to Mg25 Mg, 24 Mg26 Mg, and 26 Mg2 and includes very few v ′′ values [169, 170], the RMSDs relative to the experiment resulting from our calculations are similarly small (see the Supplementary Materials to Ref. [2] for more details). Further insights into the quality of our ab initio calculations for the ground-state PEC can be obtained by comparing the resulting rovibrational term values with their counterparts determined using the most accurate, experimentally derived, analytical forms of the X 1 Σ+ g potential to date constructed in Ref. [169]. In the discussion below, we focus on the so- called X-representation of the ground-state PEC developed in Ref. [169], which the authors of Ref. [169] regard as a reference potential in their analyses (see Table 2.2). We recall that the X-representation of the ground-state PEC of the magnesium dimer was obtained by simultaneously fitting the X 1 Σ+ 1 + g and A Σu PECs to a large number of the experimentally ′ ′ 1 + ′′ ′′ determined A 1 Σ+ u (v , J ) → X Σg (v , J ) rovibronic transition frequencies and extrapolating the resulting X 1 Σ+g PEC to the asymptotic region using the theoretical C6 [172], C8 [173], and C10 [173] coefficients. As shown in Table 2.2, our ab initio G(v ′′ , J ′′ ) energies characterizing 24 the most abundant Mg2 isotopolog are in very good agreement with those generated using the X-representation of the ground-state PEC developed in Ref. [169]. When all of the rovibrational bound states supported by both potentials are considered, the RMSD and the maximum unsigned error characterizing our ab initio G(v ′′ , J ′′ ) values for 24 Mg2 relative to their counterparts arising from the X-representation are 1.3 and 2.0 cm−1 , respectively [2]. What is especially important in the context of the present study is that our ab initio ground-state PEC and the state-of-the-art analytical fit to the experimental data defining the X-representation, constructed in Ref. [169], bind the v ′′ = 18 level if the rotational quantum 28 number J ′′ is not too high (see the discussion below). The high quality of our calculated G(v ′′ , J ′′ ) values and spacings between them, which can also be seen in Tables 2.1 and 2.2 and Fig. 2.1, allows us to comment on the existence of the v ′′ > 13 levels that have escaped experimental detection for decades. As already alluded to above and as shown in Table 2.2 and Fig. 2.1, our ab initio X 1 Σ+ g PEC supports the same number of rotationless vibrational levels as the latest experimentally derived PEC 24 defining the X-representation [169], which for the most abundant Mg2 isotopolog is 19. 24 Table 2.1, which compares the rovibrational term values of Mg2 resulting from our ab initio calculations for the representative rotational quantum numbers ranging from 0 to 80 with the available experimental data, shows that the elusive high-lying states with v ′′ > 13 quickly become unbound as J ′′ increases, so by the time J ′′ = 20, the v ′′ = 15 to 18 levels are no longer bound. This is demonstrated in Fig. 2.2, where we show a graphical representation of the J ′′ = 20, 40, 60, and 80 effective potentials including centrifugal barriers characterizing 24 the rotating Mg2 molecule, along with the corresponding vibrational wave functions and information about the lifetimes for predissociation by rotation associated with tunneling through centrifugal barriers characterizing quasi-bound states. In fact, according to our ab initio data [2], the maximum rotational quantum number that allows for at least one bound rovibrational state decreases with v ′′ , from J ′′ = 68 for v ′′ = 0 to J ′′ = 4 for v ′′ = 18, with all states becoming quasi-bound or unbound when J ′′ ≥ 70, when the most abundant 24 Mg2 isotopolog is considered. In general, as exemplified in Fig. 2.2 (cf., also, the lifetime data compiled in the Supplementary Materials to Ref. [2]), the mean lifetimes for predissociation by rotation characterizing quasi-bound states with a given J ′′ rapidly decrease as v ′′ becomes larger. They decrease equally fast when J ′′ increases and v ′′ is fixed. These observations imply that the spectroscopic detection of the high-lying vibrational states of Mg2 can only be achieved if the molecule does not rotate too fast (cf. Table 2.1 and Fig. 2.2). As shown in Fig. 2.1, where we plot the wave functions of the high-lying, purely vibra- tional, states of 24 Mg2 , starting with the last experimentally observed v ′′ = 13 level, along 29 ′′ with the X 1 Σ+ g PEC obtained in our ab initio calculations, the v = 18 state, located only 0.2 cm−1 below the potential asymptote, is barely bound (see also Table 2.1). This makes the existence of an additional, v ′′ = 19, level for the most abundant isotopolog of the magne- sium dimer unlikely. Further insights into the number of purely vibrational bound states of 24 Mg2 supported by the X 1 Σ+ g PEC are provided by the inset in Fig. 2.1, where we plot the rotationless G(v ′′ + 1) − G(v ′′ ) energy differences, resulting from the ab initio calculations reported in this work and the experiment, as a function of v ′′ + 1/2 (the Birge–Sponer plot). Fitting the experimental data to a line, i.e., assuming a Morse potential, results in v ′′ = 16 24 being the last bound vibrational level of Mg2 . Although the deviation from the Morse potential, as predicted by our ab initio calculations, is not as severe as in the case of Be2 [73], it is large enough to result in the v ′′ = 17 and 18 states becoming bound, emphasizing the importance of properly describing the long-range part of the PEC. As shown in Table 2.1 and Fig. 2.1, the G(v ′′ + 1) − G(v ′′ ) vibrational spacings rapidly decrease with increasing v ′′ , from 47.7 cm−1 or 68.6 K for v ′′ = 0 to 11.7 cm−1 or 16.8 K for v ′′ = 12, and to 0.8 cm−1 or 1.2 K for v ′′ = 17, when 24 Mg2 is considered. This means that at regular temperatures all vibrational levels of the magnesium dimer, which is a very weakly bound system, are substantially populated, making selective probing of the closely spaced higher-energy states, including those with v ′′ > 13, virtually impossible, since practically every molecular collision (e.g., with another dimer) may result in a superposition of many rovibrational states, with some breaking the dimer apart. At room temperature, for example, the cumulative population of the v ′′ > 13 states of 24 Mg2 , determined using the normalized Boltzmann distribution involving all rotationless levels bound by the X 1 Σ+ g potential, of about 12%, is comparable to the populations of the corresponding low-lying states (16% for v ′′ = 0, 13% for v ′′ = 1, and 10% for v ′′ = 2). The situation changes in the cold/ultracold regime, where the available thermal energies, which are on the order of millikelvin or even microkelvin, are much smaller than the vibrational spacings, even when the high-lying states with v ′′ > 13 near the dissociation threshold are considered, suppressing collisional effects 30 and allowing one to probe the long-range part of the ground-state PEC, where the v ′′ > 13 states largely localize (cf. Fig. 2.1). This makes the accurate characterization of the v ′′ > 13 bound and quasi-bound states provided by the high-level ab initio calculations reported in this work relevant to the applications involving cold/ultracold Mg atoms separated by larger distances in magneto-optical traps (see, e.g., Ref. [156]). The accuracy of our ab initio description of the more strongly bound A 1 Σ+ u electronic state (De = 9414 cm−1 and re = 3.0825 Å [170]; cf. Fig. 2.3 for the corresponding PEC), which we need to consider to simulate the LIF spectra, is consistent with that obtained for the weakly bound ground state. For example, the errors relative to the experiment [170] resulting from our calculations of the dissociation energy De and equilibrium bond length re are 0.91% (86 cm−1 ) and 0.2% (0.006 Å), respectively [2]. This high accuracy of our ab initio A 1 Σ+u PEC, obtained using Eq. (2.22), is reflected in the excellent agreement between the 24 Mg2 G(v ′ , J ′ ) values obtained in Ref. [2] and their experimentally derived counterparts reported in Refs. [161, 170]. In particular, the RMSDs characterizing our rovibrational term values in the A 1 Σ+ u state relative to the data of Balfour and Douglas [161] and Knöckel et al. [170] are only 3.2 and 4.5 cm−1 , respectively, which is a major improvement over the RMSD of 30 cm−1 reported in Ref. [176]. According to our ab initio calculations using the computational protocol described above, the total number of vibrational states supported 24 by the A 1 Σ+ u potential well for the most abundant Mg2 species is 169 [2]. The most compelling evidence for the predictive power of our ab initio electronic structure and rovibrational calculations is the nearly perfect reproduction of the experimental A 1 Σ+ u → X 1 Σ+g LIF spectrum reported in Refs. [169, 170], shown in Fig. 2.4 and Table 2.3 (The theoretical line intensities shown in Fig.2.4 were normalized such that the tallest peaks in the calculated and experimental LIF spectra corresponding to the v ′′ = 5 P12 line representing ′ ′ 1 + ′′ ′′ the A 1 Σ+u (v = 3, J = 11) → X Σg (v = 5, J = 12) transition match). Figure 2.3 uses our calculated X 1 Σ+ 1 + g and A Σu PECs and the corresponding rovibrational wave functions to illustrate the photoexcitation and fluorescence processes that resulted in the experimental 31 LIF spectrum shown in Fig. 3 of Ref. [169], which is reproduced in Fig. 2.4(a). This particular ′ ′ spectrum represents the fluorescence progression from the A 1 Σ+ u (v = 3, J = 11) state of 24 ′′ ′′ Mg2 , populated by laser excitation from the X 1 Σ+ g (v = 5, J = 10) state, to all accessible ′′ ′′ X 1 Σ+g (v , J ) rovibrational levels, resulting in the P12/R10 doublets that correspond to J ′′ = 12 for the P branch and J ′′ = 10 for the R branch. Figure 2.4 and Table 2.3 compare ′ ′ 1 + ′′ ′′ the experimentally observed A 1 Σ+ u (v = 3, J = 11) → X Σg (v , J = 10, 12) transitions with the corresponding line positions (Fig. 2.4 and Table 2.3) and intensities (Fig. 2.4) resulting from our ab initio calculations. As already mentioned earlier, the only adjustment that we made to produce the theoretical LIF spectrum shown in Fig. 2.4 and Table 2.3 was a uniform shift of the entire A 1 Σ+ u PEC obtained in our ab initio computations to match the experimentally determined adiabatic electronic excitation energy Te of 26,068.9 cm−1 [170]. Other than that, the theoretical LIF spectrum in Fig. 2.4 and Table 2.3 relies on the raw ab initio electronic structure and rovibrational data. Note that in order to produce Fig. 2.4, we ′ ′ 1 + ′′ ′′ superimposed our theoretical A 1 Σ+ u (v , J ) → X Σg (v , J = 10, 12) LIF spectrum on top of the experimental one reported in Fig. 3 of Ref. [169]. The theoretical line intensities shown in Fig. 3 were normalized such that the tallest peaks in the calculated and experimental LIF ′ ′ spectra corresponding to the v ′′ = 5 P12 line representing the A 1 Σ+ u (v = 3, J = 11) → ′′ ′′ X 1 Σ+g (v = 5, J = 12) transition match. The notable agreement between the theoretical and experimental LIF spectra shown in Fig. 2.4(a) and Table 2.3, with differences in line positions not exceeding 1 to 1.5 cm−1 and with virtually identical intensity patterns, suggests that our predicted transition frequencies involving the elusive v ′′ > 13 states are very accurate, allowing us to provide guidance for their potential experimental detection in the future. Before discussing our suggestions in this regard, we note that owing to our ab initio calculations, we can now locate the previously unidentified P12/R10 doublets involving the v ′′ > 13 states within the experimental LIF spectrum reported in Fig. 3 of Ref. [169]. Indeed, as shown in Fig. 2.4 and Table 2.3, ′ ′ 1 + ′′ ′′ the LIF spectrum corresponding to the A 1 Σ+ u (v = 3, J = 11) → X Σg (v , J = 10, 12) 32 transitions contains the P12/R10 doublets involving the v ′′ = 0 to 16 states and the R10 line involving the v ′′ = 17 state. It is worth mentioning that the A 1 Σ+ ′ ′ u (v = 3, J = 11) → ′′ ′′ ′ ′ 1 + ′′ ′′ X 1 Σ+g (v = 17, J = 12) and A 1 Σ+ u (v = 3, J = 11) → X Σg (v = 18, J = 10, 12) transitions are absent, since the v ′′ = 17, J ′′ = 12 and v ′′ = 18, J ′′ = 10 and 12 states are ′ ′ unbound, but they could potentially be observed if one used different initial A 1 Σ+ u (v , J ) states (see the discussion below). As one can see by inspecting Fig. 2.4, and consistent with the remarks made by Knöckel et al. in Ref. [169], the experimental detection of the P12/R10 doublets involving v ′′ > 13, when ′ ′ transitioning from the A 1 Σ+ u (v = 3, J = 11) state, was hindered by the unfavorable signal- to-noise ratio (transitions to the v ′′ = 16 and 17 states exhibit low Einstein coefficients) and the presence of overlapping lines outside the P12/R10 progression, originating from collisional relaxation effects [169] and having similar (v ′′ = 15) or higher (v ′′ = 14) intensities. To fully appreciate this, in Fig. 2.4(b), we magnified the region of the LIF spectrum recorded in Ref. ′ ′ 1 + ′′ ′′ [169] that contains the calculated A 1 Σ+ u (v = 3, J = 11) → X Σg (v = 13 to 16, J = ′ ′ 1 + ′′ ′′ 10, 12) and A 1 Σ+ u (v = 3, J = 11) → X Σg (v = 17, J = 10) transitions. As shown in Fig. 2.4 and Table 2.3, the identification of the P12/R10 doublets corresponding to the ′ ′ 1 + ′′ ′′ A 1 Σ+u (v = 3, J = 11) → X Σg (v = 0 to 13, J = 10, 12) transitions is unambiguous. The observed and calculated line positions and intensities and line intensity ratios within every doublet match each other very closely. Figure 2.4(b) demonstrates that the identification of the remaining doublets in the P12/R10 progression is much harder. On the basis of our ab initio work and taking into account the fact that our calculated line positions may be off by about 1 cm−1 (cf. Table 2.3), the v ′′ = 14 P12/R10 doublet, marked in Fig. 2.4(b) by the blue arrows originating from the v ′′ = 14 label, is largely hidden behind the higher- intensity feature that does not belong to the P12/R10 progression and that most likely originates from collisional relaxation [169]. Because of our calculations, we can also point to the most likely location of the v ′′ = 15 P12/R10 doublet in the LIF spectrum recorded in Ref. [169] [see the blue arrows originating from the v ′′ = 15 label in Fig. 2.4(b)]. Doing this 33 without backing from the theory is virtually impossible due to the presence of other lines ′ ′ 1 + ′′ ′′ near the A 1 Σ+ u (v = 3, J = 11) → X Σg (v = 15, J = 10, 12) transitions having similar ′ ′ intensities. As shown in Fig. 2.4(b), the situation with the remaining A 1 Σ+ u (v = 3, J = ′′ ′′ 1 + ′ ′ 1 + ′′ ′′ 11) → X 1 Σ+ g (v = 16, J = 10, 12) and A Σu (v = 3, J = 11) → X Σg (v = 17, J = 10) transitions is even worse, since they have very low Einstein coefficients that hide them in the noise. In general, our ab initio calculations carried out in this work indicate that under the constraints of the LIF experiments reported in Refs. [169, 170], where the authors populated ′ ′ ′ 1 + ′′ ′′ ′′ the A 1 Σ+ u (v , J ) states with v = 1 to 46, the X Σg (v , J ) states with v = 14 to 18 cannot be realistically detected because of very small Franck–Condon factors and Einstein ′ ′ 1 + ′′ ′′ coefficients characterizing the corresponding A 1 Σ+ u (v , J ) → X Σg (v , J ) transitions [2]. As shown in Fig. 2.1, the v ′′ = 14 to 18 states are predominantly localized in the long- range r = 8 to 16 Å region. At the same time, as illustrated in Fig. 2.3, the potential well characterizing the electronically excited A 1 Σ+ u state is much deeper and shifted toward shorter internuclear separations compared to its X 1 Σ+ g counterpart. Thus, the only way ′′ ′′ ′′ to access the X 1 Σ+ g (v , J ) states with v = 14 to 18 via fluorescence from A Σu is by 1 + ′ ′ ′ populating the high-lying A 1 Σ+ u (v , J ) levels with v ≫ 46. In an effort to assist the experimental community in detecting the elusive v ′′ = 14 to 18 ′ ′ 1 + ′′ ′′ ′ vibrational levels, we searched for the A 1 Σ+ u (v , J ) → X Σg (v = 14 to 18, J = J ± 1) 24 transitions in the most abundant isotopolog of the magnesium dimer, Mg2 , that would result in spectral lines of maximum intensity based on the computed Einstein coefficients. To ensure the occurrence of allowed transitions involving the last, v ′′ = 18, level, which for 24 Mg2 becomes unbound when J ′′ > 4, we focused on the J ′′ values not exceeding 4, ′ ′ ′ i.e., the fluorescence from the A 1 Σ+ u (v , J ) states with J = 1, 3, and 5. According to our calculations, the optimum v ′ values for observing the v ′′ = 14 to 18, J ′′ ≤ 4 states via the LIF spectroscopy are in the neighborhood of v ′ = 60, 66 to 69, and 74 to 84 for v ′′ = 14; 72 to 75 and 80 to 91 for v ′′ = 15; 79 to 82 and 88 to 100 for v ′′ = 16; 88, 89, 34 and 97 to 111 for v ′′ = 17; and 109 to 129 for v ′′ = 18 [see Ref. [2] for the details of all 24 allowed rovibronic transitions in Mg2 involving the X 1 Σ+ 1 + g and A Σu states, including, in ′′ ′′ 1 + ′ ′ 1 + ′ ′ particular, the relevant X 1 Σ+ g (v , J ≤ 4) → A Σu (v , J ) pump and A Σu (v , J = 1, 3, 5) → ′′ ′′ X 1 Σ+ g (v = 14 to 18, J ≤ 4) fluorescence processes]. In determining these optimum v ′ values, we chose the cutoff value of 1.0 × 107 Hz in the Einstein coefficients, which is similar to the Einstein coefficients calculated for the most intense v ′′ = 5 P12/R10 doublet in the experimental LIF spectrum shown in Fig. 3 of Ref. [169], reproduced in Fig. 2.4(a). Our ′ ′ 1 + ′′ ′′ predicted A 1 Σ+ u (v , J = 1, 3, 5) → X Σg (v = 14to18, J ≤ 4) fluorescence frequencies resulting from the aforementioned optimum v ′ ranges, which might allow one to detect the v ′′ = 14 to 18 states of 24 Mg2 via a suitably designed LIF experiment, are estimated at about 33,360, 33,740 to 33,910, and 34,150 to 34,530 cm−1 for v ′′ = 14; 34,050 to 34,190 and 34,390 to 34,710 cm−1 for v ′′ = 15; 34,350 to 34,460 and 34,640 to 34,880 cm−1 for v ′′ = 16; 34,630 to 34,660 and 34,830 to 35,000 cm−1 for v ′′ = 17; and 34,990 to 35,100 cm−1 for v ′′ = 18 [given the 86 cm−1 error in the calculated De characterizing the A 1 Σ+ u state and the RMSD of ∼3 to 5 cm−1 in our 24 Mg2 G(v ′ , J ′ ) values relative to the spectroscopic data of Refs. [169, 170], the above frequency ranges may have to be shifted by a dozen or so cm−1 ]. As shown by the results reported in Ref. [2] and summarized above, the results of our high-level CC/EOMCC and FCI computations show an unprecedented level of accuracy relative to the available experimental data of the magnesium dimer. Thus, we are now well positioned to comment on the convergence of our computational protocol employed in Ref. [2]. In particular, let us focus on the ground X 1 Σ+ g PEC of Mg2 , where the De 24 and vibrational term values characterizing the Mg2 isotopolog are accurate to within ∼1 cm−1 relative to experimentally available data. In our discussion, we rely on the results of the auxiliary calculations reported in Ref. [2] employing the aug-cc-pV(T+d)Z, aug-cc- pwCVTZ, and aug-cc-pwCV5Z bases of Ref. [179], taken from the Peterson group’s website [180], which we abbreviate as A(T+d)Z, AwCTZ, and AwC5Z, respectively. To begin with, as shown in Ref. [2], the valence FCI correction on top of CCSDT in 35 FCI/A(Q+d)Z CCSDT/A(Q+d)Z Eq. (2.21), i.e., the EX 1 Σ+ − EX 1 Σ+ contribution to the X 1 Σ+ g energetics, are g g well converged with respect to the size of the basis set. Indeed, if we compare the valence FCI correction in Eq. (2.21) with its less saturated counterpart, in which we replace the FCI/A(T+d)Z A(Q+d)Z basis set used in Ref. [2] by its smaller A(T+d)Z counterpart, i.e., EX 1 Σ+ − g CCSDT/A(T+d)Z EX 1 Σ+ , the changes in the valence FCI correction are ∼1 cm−1 or less throughout g the entire 3.2–1.00 Å range of Mg–Mg separations considered in our computations. One could instead consider improving Eq. (2.21) by extrapolating, for example, the nearly all- electron CCSDT energetics to the CBS limit. Unfortunately, a widely used two-point CBS extrapolation [190, 191] based on the subvalence CCSDT/AwCTZ and CCSDT/AwCQZ data, which are the only CCSDT data of this type available to us, to determine the CBS counterpart of the first term on the right-hand side of Eq. (2.21) would not be reliable enough. As demonstrated in Ref. [1] and as elaborated on in the Supplementary Materials to Ref. [2], a CBS extrapolation using the AwCTZ and AwCQZ basis sets worsens, instead of improving, the De , re , and vibrational term values of the magnesium dimer compared to the unextrapolated results using the AwCQZ basis. The CBS extrapolation using the AwCQZ and AwC5Z basis sets would be accurate enough, but the CCSDT/AwC5Z calculations for the magnesium dimer correlating all electrons but the 1s shells of Mg atoms turned out to be prohibitively expensive for us. One could try to address the above concern by replacing CCSDT in Eq. (2.21) by the more affordable CCSD(T) approach, resulting in   CCSD(T)/AwCQZ FCI/A(Q+d)Z CCSD(T)/A(Q+d)Z ẼX 1 Σ+g = EX 1 Σ+ + EX 1 Σ+ − EX 1 Σ+ , (2.23) g g g but the computational protocol defined by Eq. (2.23) is not sufficiently accurate for the spectroscopic considerations reported in this work due to the inadequate treatment of triples by the baseline CCSD(T) approximation. Indeed, this is demonstrated in Fig. 2.5, where we compare the rotationless vibrational term values for the X 1 Σ+ g state obtained from our ab initio computations employing Eqs. (2.21) and (2.23) against their experimental counterparts [161, 166] (see the Supplementary Materials to Ref. [2] for further details on 36 how the experimental G(v ′′ = 0 to 13, J ′′ = 0) data were obtained). As shown in Fig. 2.5, our computational protocol based on the nearly all-electron CCSDT/AwCQZ and valence CCSDT/A(Q+d)Z and FCI/A(Q+d)Z calculations, as in Eq. (2.21), produces the G(v ′′ , J ′′ = 0) values that can hardly be distinguished from their experimentally derived counterparts, with errors not exceeding 1.4 cm−1 or 0.5%, when all experimentally observed v ′′ = 0–13 states are considered. What is especially important, errors in the G(v ′′ , J ′′ = 0) values resulting from the computational protocol based on Eq. (2.21) relative to experiment remain small for all v ′′ values. They slightly increase in the v ′′ = 0–9 region, from 0.0 cm−1 for v ′′ = 0 to 1.4 cm−1 for v ′′ = 8 and 9, but then they decrease again, to 0.7 cm−1 when the last experimentally observed v ′′ = 13 level is considered. These observations should be contrasted with the results obtained using Eq. (2.23), where full CCSDT is replaced by CCSD(T). As demonstrated in Fig. 2.5, errors in the G(v ′′ , J ′′ = 0) energies obtained for the X 1 Σ+ g potential resulting from Eq. (2.23) steadily grow with v ′′ , from 0.7 cm−1 for v ′′ = 0 to 14.7 cm−1 for v ′′ = 13, representing 3–4% of the corresponding experimentally derived G(v ′′ , J ′′ = 0) values. This clearly implies that it is not sufficient to run the CCSD(T) calculations for the purpose of capturing the bulk of many-electron correlation effects, assuming that one can incorporate the missing correlations with the help of valence FCI. If one is interested in attaining the nearly spectroscopic (1 cm−1 -type) accuracy, the bulk of the correlation effects must be captured by the more complete treatment of the connected triply excited clusters, beyond CCSD(T), which in the case of the X 1 Σ+ g PEC is represented in this study by full CCSDT, prior to applying the FCI-based correction. Otherwise, there is a significant risk of introducing substantial errors in the calculated vibrational term values. Given the nearly linear error growth characterizing the G(v ′′ , J ′′ = 0) values corresponding to the X 1 Σ+ g potential obtained using Eq. (2.23), seen in Fig. 2.5, one should not use CCSD(T) as a substitute for CCSDT in Eq. (2.21) in calculations involving higher-energy vibrational levels. In particular, the CCSD(T)-based composite approach defined by Eq. (2.23) is unsuitable for locating the elusive v ′′ = 14–18 states of the magnesium dimer. One 37 can see this by comparing, for example, the nearly 15 cm−1 error in the G(v ′′ = 13, J ′′ = 0) energy obtained using the X 1 Σ+ g PEC resulting from Eq. (2.23), which, according to Fig. 2.5, is expected to become even larger for v ′′ > 13 (cf., also, Ref. [1]), with the small spacings between the consecutive vibrational levels in the v ′′ = 14–18 region. Indeed, based on our best ab initio calculations summarized in Table 2.1, these spacings decrease from about 6 cm−1 for the G(v ′′ = 15, J ′′ = 0) − G(v ′′ = 14, J ′′ = 0) difference to ∼1 cm−1 when the gap between the rotationless v ′′ = 17 and v ′′ = 18 states is considered. For all these reasons, we have to rely on Eq. (2.21) in our calculations for the magnesium dimer, in which we use CCSDT, not CCSD(T), and finite (albeit large and carefully optimized) AwCQZ and A(Q+d)Z basis sets rather than the poor-quality CBS extrapolation from the CCSDT/AwCTZ and CCSDT/AwCQZ information. This analysis also highlights the need for robust approximations to high-level CC methods, such as CCSDT and CCSDTQ, which is exactly the main focus of the following chapters. 38 Table 2.1: Comparison of the ab initio (Calc.) and experimentally derived (Expt) rovi- brational G(v ′′ , J ′′ ) energies for selected values of J ′′ characterizing 24 Mg2 in the ground electronic state (in cm−1 ), along with the corresponding dissociation energies De (in cm−1 ) and equilibrium bond lengths re (in Å). The G(v ′′ , J ′′ ) energies calculated using the ab initio X 1 Σ+ g PEC defined by Eq. (2.21) are reported as errors relative to experiment, whereas De and re are the actual values of these quantities. If the experimental G(v ′′ , J ′′ ) energies are not available, we provide their calculated values in square brackets. Quasi-bound rovibrational levels are given in italics. Horizontal bars indicate term values not supported by the X 1 Σ+ g PEC. Adapted from Ref. [2]. G(v ′′ , J ′′ = 0) G(v ′′ , J ′′ = 20) G(v ′′ , J ′′ = 40) G(v ′′ , J ′′ = 60) G(v ′′ , J ′′ = 80) v ′′ Calc. Expta Calc. Exptb Calc. Exptb Calc. Exptb Calc. Exptb 0 0.0 25.2 −0.2 63.3 −0.4 171.2 −0.9 340.4 −1.8 552.8 1 −0.2 73.0 −0.4 109.7 −0.7 213.1 −1.2 374.6 −2.2 573.2 2 −0.5 117.8 −0.7 153.0 −1.0 252.0 −1.6 405.4 [585.0 ] 3 −0.7 159.4 −1.0 193.2 −1.3 287.7 −1.9 432.9 — 4 −0.9 198.0 −1.3 230.3 −1.6 320.3 −2.1 456.7 — 5 −1.1 233.6 −1.5 264.4 −1.8 349.7 −2.1 476.5 — 6 −1.2 266.2 −1.7 295.5 −1.9 375.9 −1.7 491.7 — 7 −1.3 295.8 −1.8 323.6 −1.9 398.8 — — 8 −1.4 322.5 −1.7 348.5 −1.7 418.1 — — 9 −1.4 346.2 −1.6 370.3 −1.4 433.9 — — 10 −1.3 366.8 −1.4 389.0 [444.5 ] — — 11 −1.2 384.4 −1.2 404.4 [451.6 ] — — 12 −0.9 398.8 −0.9 416.6 — — — 13 −0.7 410.3 −0.5 425.5 — — — 14 [418.4] [431.1] — — — 15 [424.6] — — — — 16 [428.4] — — — — 17 [430.4] — — — — 18 [431.2] — — — — De 431.4 430.472c re 3.893 3.89039c a Experimentally derived values for v ′′ = 0 to 12 taken from Ref. [161]. The v ′′ = 13 value is calculated as G(v ′′ = 13, J ′′ = 14) − 210B(v ′′ = 13, J ′′ = 14) with the information about G(v ′′ = 13, J ′′ = 14) and B(v ′′ = 13, J ′′ = 14) taken from Ref. [166]. b Experimentally derived values taken from the supplementary material of Ref. [169]. c Experimentally derived values taken from Ref. [169, 170] assuming the X-representation of the X 1 Σ+ g potential developed in Ref. [169]. 39 Table 2.2: Comparison of the rovibrational G(v ′′ , J ′′ ) energies obtained using the ab initio X 1 Σ+ g PEC (Calc.) and its X-representation counterpart constructed in Ref. [169] (X-rep.) for selected values of J ′′ characterizing 24 Mg2 in the ground electronic state (in cm−1 ), along with the corresponding dissociation energies De (in cm−1 ) and equilibrium bond lengths re (in Å). The G(v ′′ , J ′′ ) energies calculated using the ab initio X 1 Σ+ g PEC defined by Eq. (2.21) are reported as errors relative to experiment, whereas De and re are the actual values of these quantities. If the experimental G(v ′′ , J ′′ ) energies are not available, we provide their calculated values in square brackets. Quasi-bound rovibrational levels are given in italics. Horizontal bars indicate term values not supported by the X 1 Σ+ g PEC. Adapted from Ref. [2]. G(v ′′ , J ′′ = 0) G(v ′′ , J ′′ = 20) G(v ′′ , J ′′ = 40) G(v ′′ , J ′′ = 60) G(v ′′ , J ′′ = 80) v ′′ Calc. X-rep. Calc. X-rep. Calc. X-rep. Calc. X-rep. Calc. X-rep. 0 −0.1 25.2 −0.2 63.3 −0.4 171.2 −0.9 340.4 −1.8 552.8 1 −0.3 73.1 −0.4 109.7 −0.7 213.1 −1.2 374.6 −2.2 573.2 2 −0.6 117.9 −0.7 153.0 −1.0 252.0 −1.6 405.4 [585.0 ] 3 −0.9 159.6 −1.0 193.2 −1.3 287.7 −1.9 432.9 — 4 −1.1 198.2 −1.2 230.3 −1.6 320.3 −2.1 456.7 — 5 −1.4 233.9 −1.5 264.4 −1.8 349.7 −2.1 476.5 — 6 −1.5 266.5 −1.6 295.5 −1.9 375.9 −1.7 491.7 — 7 −1.7 296.2 −1.7 323.5 −1.9 398.8 — — 8 −1.7 322.8 −1.7 348.5 −1.7 418.1 — — 9 −1.6 346.4 −1.6 370.3 −1.4 433.8 — — 10 −1.5 367.0 −1.4 389.0 −1.0 [445.5 ] — — 11 −1.3 384.5 −1.2 404.4 [451.6 ] — — 12 −1.0 399.0 −0.9 416.6 — — — 13 −0.7 410.4 −0.5 425.5 — — — 14 −0.5 418.9 −0.2 431.2 — — — 15 −0.2 424.7 — — — — 16 0.2 428.3 — — — — 17 0.5 429.9 — — — — 18 0.8 430.4 — — — — De 431.4 430.472 re 3.893 3.89039 40 ′ ′ Table 2.3: Comparison of the theoretical line positions of the A 1 Σ+ u (v = 3, J = 11) → 1 + ′′ ′′ 24 X Σg (v , J = 10, 12) fluorescence progression in the LIF spectrum of Mg2 calculated in this work with experiment. All line positions are in cm−1 . The available experimental values are the actual line positions, whereas our calculated results are errors relative to experiment. If the experimentally determined line positions are not available, we provide their calculated values in square brackets. Horizontal bars indicate term values not supported by the X 1 Σ+ g PEC. Adapted from Ref. [2]. P12 R10 v ′′ Calc. Expt.a Calc. Expt.a 0 −1.5 26,701.9 −1.5 26,706.0 1 −1.2 26,654.5 −1.3 26,658.5 2 −1.0 26,610.3 −1.0 26,614.1 3 −0.7 26,569.2 −0.7 26,572.8 4 −0.4 26,531.1 −0.5 26,534.6 5 −0.2 26,496.0 −0.2 26,499.3b 6 0.0 26,463.9 [26,467.1] 7 0.1 26,434.9 0.1 26,437.9 8 0.1 26,408.8 0.1 26,411.7 9 0.0 26,385.9 0.0 26,388.5 10 −0.2 26,366.0 −0.2 26,368.4 11 −0.4 26,349.2 −0.4 26,351.4 12 −0.7 26,335.6 −0.6 26,337.5 13 −1.0 26,325.0 −0.9 26,326.7 14 [26,316.2] [26,317.7] 15 [26,311.1] [26,312.2] 16 [26,308.4] [26,309.1] 17 — [26,308.0] 18 — — a ′′ ′′ ′ ′ Differences between the experimental X 1 Σ+ g (v , J = 10, 12) and A 1 Σ+ u (v = 3, J = 11) term values reported in the supplementary material of Ref. [170] (see the Supplementary Materials of Ref. [2] for further details), unless stated otherwise. b ′′ ′′ 1 + ′ ′ The X 1 Σ+g (v = 5, J = 10) → A Σu (v = 3, J = 11) pump frequency reported in Fig. 3 of Ref. [169]. 41 v" = 18 430 v" = 17 v" = 16 1 + Mg X 2 g 425 v" = 15 50 ) 1 Experiment 40 G(v"+1) G(v") (cm ) Theory 420 1 E (cm v" = 14 30 20 415 10 0 410 v" = 13 0 2 4 6 8 10 12 14 16 18 v"+1/2 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 r (Å) Mg Mg Figure 2.1: The wave functions of the high-lying, purely vibrational, states of 24 Mg2 and ′′ the underlying X 1 Σ+g potential. The last experimentally observed v = 13 level is marked in ′′ blue, the predicted v = 14 to 18 levels are marked in green, and the ab initio X 1 Σ+ g PEC obtained in this study is marked by a long-dashed black line. The inset is a Birge–Sponer plot comparing the rotationless G(v ′′ + 1) − G(v ′′ ) energy differences as functions of v ′′ + 1/2 obtained in this work (black circles) with their experimentally derived counterparts (red open squares) based on the data reported in Refs. [161] (v ′′ = 0 to 12) and [166] (v ′′ = 13; cf. also Table 2.1). The red solid line is a linear fit of the experimental points. Adapted from Ref. [2]. 42 (a) (b) (c) (d) Figure 2.2: The VJ ′′ (r) effective potentials including centrifugal barriers characterizing the rotating 24 Mg2 molecule at selected values of J ′′ , along with the corresponding vibrational wave functions and information about the lifetimes for predissociation by rotation, τ (v ′′ ), characterizing quasi-bound states. The selected values of J ′′ used to construct the effec- tive potentials VJ ′′ (r) (black curves) and determine the corresponding bound (blue lines) and quasi-bound (red lines) vibrational wave functions are (a) 20, (b) 40, (c) 60, and (d) 80. The black dashed line represents the rotationless, purely electronic, X 1 Σ+ g potential V (r) = VJ ′′ =0 (r) calculated using the ab initio composite scheme defined by Eq. (2.21). The horizontal black dotted line at 431.4 cm−1 marks the dissociation threshold of the ab initio X 1 Σ+ g potential. Adapted from Ref. [2]. 43 37500 Mg(~3 s 3p 1 1 ; 1 P) + Mg(~3 s 2 ; 1 S) 35000 A1 + u 32500 30000 27500 1 ) v' = 3, J' = 11 25000 E (cm Laser-Induced Fluoresence 500 Mg(~3 s 2 ; 1 S) + Mg(~3 s 2 ; 1 S) 400 X1 + g 300 v" = 5, J" = 10 200 100 0 2 3 4 5 6 7 8 9 10 rMg Mg (Å) ′′ ′′ 1 + ′ ′ Figure 2.3: Schematics of the pump, X 1 Σ+ g (v = 5, J = 10) → A Σu (v = 3, J = 11), ′ ′ 1 + ′′ ′′ and fluorescence, A 1 Σ+ u (v = 3, J = 11) → X Σg (v , J = 10, 12), processes resulting in the LIF spectrum for Mg2 shown in Fig. 3 of Ref. [169]. The A 1 Σ+ 24 1 + u and A Σu PECs ′′ ′′ 1 + ′ ′ and the corresponding X 1 Σ+ g (v = 5, J = 10) and A Σu (v = 3, J = 11) rovibrational wave functions were calculated in this work. The A 1 Σ+ u PEC was shifted to match the experimentally determined adiabatic electronic excitation energy Te of 26,068.9 cm−1 [170]. Adapted from Ref. [2]. 44 (a) (b) ′ ′ 1 + ′′ ′′ Figure 2.4: The A 1 Σ+ u (v = 3, J = 11) → X Σg (v , J = 10, 12) LIF spectrum of 1 + ′ ′ ′′ ′′ 24 Mg2 . (a) Comparison of the experimental A Σu (v = 3, J = 11) → X 1 Σ+ g (v , J = 10, 12) fluorescence progression (black solid lines; adapted from Fig. 3 of Ref. [169] with the permission of AIP Publishing) with its ab initio counterpart obtained in this work (red dashed lines). The theoretical line intensities were normalized such that the tallest peaks in the calculated and experimental spectra corresponding to the v ′′ = 5 P12 line match. (b) Magnification of the low-energy region of the LIF spectrum shown in (a), with red solid lines representing the calculated transitions. The blue arrows originating from the v ′′ = 13 label indicate the location of the experimentally observed v ′′ = 13 P12/R10 doublet. The blue arrows originating from the v ′′ = 14 and 15 labels point to the most probable locations of the corresponding P12/R10 doublets. Spectral lines involving v ′′ = 16 and 17 are buried in the noise (see also Table 2.3). Adapted from Ref. [2]. 45 14 CCSDT-based scheme ) 1 12 CCSD(T)-based scheme ( ") (cm 10 G v expt 8 6 ( ") 4 G v calc 2 1 X + g 0 -2 ( ") (%) 4 CCSDT-based scheme CCSD(T)-based scheme G v G v expt 3 ( ")]/ 2 expt 1 1 + ( ") X G v g calc 0 [ -1 0 2 4 6 8 10 12 v" Figure 2.5: Comparison of the vibrational term values characterizing 24 Mg2 supported by the ab initio X 1 Σ+ g potential calculated in Ref. [2] with their experimentally derived counterparts. The top and bottom panels show the errors and relative errors, respectively, in the G(v ′′ , J ′′ = 0) vibrational energies corresponding to the X 1 Σ+ g PECs obtained using the CCSDT-based composite scheme defined by Eq. (2.21) and its CCSD(T)-based analog defined by Eq. (2.23) relative to the experimentally derived data [161, 166]. Adapted from Ref. [2]. 46 CHAPTER 3 THE CC(P;Q) FORMALISM 3.1 Introduction to the CC(P;Q) Methodology As mentioned in the Introduction, the CC(P;Q) approach provides a robust way of sys- tematically approaching the energetics obtained using high-level CCSDT/EOMCCSDT, CCSDTQ/EOMCCSDTQ, etc. parent methods, while avoiding failures of perturbative ap- proaches, without resorting to the full computational costs. The idea behind the CC(P;Q) formalism dates back to the original MMCC expansions of Refs. [78–80] (see Refs. [83] and [84] for reviews), where one applies a nonperturbative, noniterative correction to the energy obtained in a lower-order CC/EOMCC computation. While the MMCC ideas can be used in a plethora of ways, we will focus on the form of CC(P;Q) energy correction which resulted from generalization of the biorthogonal formulation of MMCC [74, 89–91]. In this section, we will summarize the key concepts and equations behind the CC(P;Q) theory relevant to the works described in this dissertation. The key component of the CC(P;Q) theory is the identification of two disjoint subspaces of the N -electron Hilbert space, H , which are designated as the P and Q spaces or H (P ) and H (Q) , respectively. The P space is spanned by excited Slater determinants |ΦK ⟩, which together with the reference determinant |Φ⟩ dominates the many-electron wave function. On the other hand, the Q space is spanned by the excited determinants that are utilized to construct the noniterative correction to the electronic energy obtained by solving the CC/EOMCC equations in the P space. Once the P and Q spaces are appropriately defined, we then proceed as follows. In the first, iterative, part of a CC(P;Q) computation, designated as CC(P )/EOMCC(P ), we begin 47 by approximating the T , Rµ , and Lµ operators, used in Eqs. (2.1), (2.3), and (2.11), as T ≈ T (P ) = X tK EK , (3.1) |ΦK ⟩∈H (P ) Rµ ≈ Rµ(P ) = rµ,0 1 + X rµ,K EK , (3.2) |ΦK ⟩∈H (P ) and Lµ ≈ L(P ) lµ,K (EK )† , X µ = δµ0 1 + (3.3) |ΦK ⟩∈H (P ) respectively, where EK is the usual particle–hole excitation operator that generates the excited Slater determinants |ΦK ⟩ = EK |Φ⟩. Note that in Eqs. (3.1)–(3.3), we explicitly restrict the many-body expansion of T , Rµ , and Lµ to the P space. We then proceed to solving the CC/EOMCC equations in the P space. In the case of E (P ) (P ) the ground state, we insert the CC(P ) wave function Ψ0 = eT |Φ⟩ into the Schrödinger equation to form the connected cluster form of the Schrödinger equation in the P space, (P ) (P ) H |Φ⟩ = E0 |Φ⟩, and projecting it onto excited determinants in the P space to obtain the system of nonlinear equations (P ) ⟨ΦK |H |Φ⟩ = 0 ∀ |ΦK ⟩ ∈ H (P ) , (3.4) (P ) (P ) (P ) (P ) where H = e−T HeT = (HeT )C is the similarity-transformed Hamiltonian in the (P ) P space and E0 is the ground-state CC(P ) energy. After solving Eq. (3.4) for the cluster amplitudes tK in the usual way, the CC(P ) energy is computed as (P ) (P ) E0 = ⟨Φ| H |Φ⟩ . (3.5) (P ) Subsequently, the excitation amplitudes rµ,K and excitation energies ωµ(P ) = Eµ(P ) − E0 E (P ) associated with the EOMCC(P ) wave function Ψ(P µ ) = Rµ(P ) eT |Φ⟩ are determined by (P ) diagonalizing the similarity-transformed Hamiltonian H in the P space (for simplicity, we assume that the excited state of interest has the same symmetry as the ground state; we 48 will discuss the necessary adjustment for excited states having different symmetries from the ground state later), which correspond to solving the eigenvalue problem (P ) (P ) ⟨ΦK |(H open Rµ,open )C |Φ⟩ = ωµ(P ) rµ,K ∀ |ΦK ⟩ ∈ H (P ) , (3.6) (P ) (P ) (P ) (P ) where H open = H − E0 1 and Rµ,open = Rµ(P ) − rµ,0 1. The zeroth-body component of Rµ(P ) is computed a posteriori in the usual way as (P ) (P ) rµ,0 = ⟨Φ|(H open Rµ,open )C |Φ⟩ /ωµ(P ) . (3.7) In addition to the above CC(P )/EOMCC(P ) steps, we also solve the corresponding left CC/EOMCC system of equations in the P space, (P ) (P ) δµ0 ⟨Φ|H open |ΦK ⟩ + ⟨Φ|L(P ) µ,open H open |ΦK ⟩ = ωµ lµ,K (P ) ∀ |ΦK ⟩ ∈ H (P ) , (3.8) D (P ) ) −T for the lµ,K amplitudes defining the EOMCC(P ) bra state Ψ̃(P µ ) = ⟨Φ| L(P µ e , which are necessary to construct the CC(P;Q) energy correction (vide infra), while enforcing the ) (P ) biorthonormality condition ⟨Φ|L(P µ Rν |Φ⟩ = δµν as described earlier. Once the iterative CC(P )/EOMCC(P ) procedure is completed, we proceed to the sec- ond key step in the CC(P;Q) calculation, namely, the noniterative determination of energy correction for the many-electron correlation effects of interest with the help of the Q-space determinants. The CC(P;Q) energy correction used in this dissertation has the form [74, 75] X δµ (P ; Q) = ℓµ,K (P )Mµ,K (P ), (3.9) |ΦK ⟩∈H (Q) where (P ) Mµ,K (P ) = ⟨ΦK |H Rµ(P ) |Φ⟩ (3.10) is the generalized moments of CC/EOMCC equations corresponding to projection of the connected cluster form of Schrödinger equation containing the CC/EOMCC wave function in the P space onto excited Slater determinants residing in the Q space. Note that in writing Eq. (3.10) we combined the ground- and excited-state cases, denoted with µ = 0 and µ > 0, 49 (P ) respectively, into a single formalism by setting Rµ=0 = 1. In defining the ℓµ,K (P ) amplitudes that multiply the Mµ,K (P ) quantities in Eq. (3.9), we adopt the quasi-perturbative formula (P ) ⟨Φ| L(P ) µ H |ΦK ⟩ ℓµ,K (P ) = , (3.11) Dµ,K (P ) where the denominator Dµ,K (P ) is defined via the Epstein–Nesbet partitioning, (P ) Dµ,K (P ) = Eµ(P ) − ⟨ΦK | H |ΦK ⟩ . (3.12) Note that one has the option to use the approximate Møller–Plesset form of Dµ,K (P ) (see below for details), but, as shown in Refs. [1, 73, 76, 77, 114], among others, and as illustrated by example calculations shown in this dissertation, the Epstein–Nesbet form is generally more accurate. The final CC(P;Q) energy is given by Eµ(P +Q) = Eµ(P ) + δµ (P ; Q). (3.13) At this point, the only thing that remains to be decided is how to define the P and Q spaces underlying the CC(P;Q) calculation. As already mentioned in the Introduction, the CC(P;Q) formalism is very versatile due to the flexibility it grants the user in defining the P and Q spaces. For example, the simplest manner of defining the P and Q spaces is by following the traditional truncation scheme in the many-body expansion of T , Rµ , and Lµ , i.e., partitioning the two subspaces based on the excitation ranks of determinants. In doing so, the resulting CC(P;Q) schemes become equivalent to their CR-CC/EOMCC precursors. One could also turn to defining the P and Q spaces based on active-space ideas, or turn to hybrid schemes relying on CIQMC wave function sampling or selected CI runs. For the remainder of this chapter, we will focus on the CR-CC/EOMCC methodology and active-space CC(P;Q) approaches. 50 3.2 The Completely Renormalized CC/EOMCC Framework 3.2.1 Theory Let us begin our survey of the many forms of CC(P;Q)-based approaches by discussing the most straightforward way of defining the P and Q spaces, namely, by relying on the traditional truncation level defining the CC/EOMCC hierarchy as done in the left-eigenstate CR-CC/EOMCC methods [82–84, 89–91, 93, 95]. Within the CR-CC/EOMCC methodology, we define the P space to be spanned by n-tuply excited Slater determinants with n = 1, . . . , mA , while the complementary Q space is spanned by m-tuply excited determinants with m = mA + 1, . . . , mB . In other words, we are correcting the conventional CC/EOMCC calculations where T and Rµ are truncated at mA , as described in Section 2.1, with the many-electron correlation effects described by the remaining Tn and Rµ,n components with n = mA + 1, . . . , mB . To illustrate how the CR-CC/EOMCC methodologies work, let us focus on the triples correction to CCSD/EOMCCSD, defining the CR-CC(2,3) and CR-EOMCC(2,3) approaches that are of interest in this dissertation. In these cases, we set mA = 2 and mB = 3, which means that the P space is spanned by all singly and doubly excited determinants and the Q space is populated by all triply excited determinants in the language of the CC(P;Q) theory. Thus, the initial CC(P )/EOMCC(P ) step in a CR-CC/EOMCC(2,3) calculation is none other than performing CCSD/EOMCCSD iterations as usual, in addition to solving for left CCSD/EOMCCSD amplitudes needed to construct the noniterative moment corrections. After the CCSD/EOMCCSD energy Eµ[(EOM)CCSD] and the cluster and EOM excitation and de-excitation amplitudes are obtained, the noniterative triples correction is determined as abc abc X δµ (2, 3) = ℓµ,ijk (2)Mµ,ijk (2), (3.14) i 0) states of the isolated FR0-SB molecule, including the ground-state geometry, the excitation energies and oscillator strengths characterizing the vertical S0 → Sn transitions, and the electronic dipole moments of the calculated states. With the exception of the molecular geometry, which was optimized using the Kohn–Sham formulation [222] of the density functional theory (DFT) [223], all of the characteristics of the calculated electronic states were obtained by using high-level ab initio methods of quantum chemistry based on the CC theory and its EOM extension to excited states. Given the relatively large size of the FR0-SB molecule, which consists of 58 atoms and 190 electrons, to use the CC and EOMCC methods as fully as possible and to make sure that the higher-order many-electron correlation effects beyond the basic EOMCCSD level are properly accounted for we used the following composite approach to determine the vertical excitation energies corresponding to the S0 → Sn transitions: h i ωn(EOMCC) = ωn(EOMCCSD/6-31+G*) + ωn(δ-CR-EOMCC(2,3)/6-31G) − ωn(EOMCCSD/6-31G) . (3.23) The first term on the right-hand side of Eq. (3.23) denotes the vertical excitation energy obtained in the EOMCCSD calculations using the 6-31+G* basis set [224–226], which was 56 the largest basis set we could afford in such computations. The next two terms on the right- hand side of Eq. (3.23), which represent the difference between the δ-CR-EOMCC(2,3) and EOMCCSD vertical excitation energies obtained by using a smaller 6-31G basis [224], correct the EOMCCSD/6-31+G* results for the higher-order many-electron correlation effects due to triple excitations. The size intensivity of the EOMCCSD and δ-CR-EOMCC(2,3) excitation energies entering our composite computational protocol defined by Eq. (3.23), combined with the size extensivity of the underlying CCSD and CR-CC(2,3) approaches, is important, too, since without reinforcing these formal theory features one risks losing accuracy with growing molecular size. The CCSD/6-31+G* and EOMCCSD/6-31+G* calculations were also used to determine the dipole moments in the ground and excited states and the oscillator strengths character- izing the vertical S0 → Sn (in general, Sm → Sn ) transitions. As usual, this was done by solving both the right and left EOMCCSD eigenvalue problems and constructing the relevant one-electron reduced density and transition density matrices. While triples corrections, such as those of CR-CC(2,3) and δ-CR-EOMCC(2,3), are important to improve the energetics, the description of one-electron properties, such as dipole moments and oscillator strengths characterizing one-electron transitions examined in this study, by the CCSD and EOMCCSD approaches is generally quite accurate. All single-point CC and EOMCC calculations reported in our initial investigation [3] and summarized here relied on the ground-state geometry of FR0-SB, which we optimized using the analytic gradients of the CAM-B3LYP [227] DFT approach employing the 6-31+G* basis set. We chose the CAM-B3LYP functional because the extension of this functional to excited states using the time-dependent (TD) DFT formalism [228] provided vertical excitation energies closest to those obtained with EOMCC. Furthermore, the ground-state geometry of the FR0-SB molecule resulting from the CAMB3LYP/6-31+G* calculations turned out to be virtually identical (to within 0.004 Å on average and not exceeding 0.02 Å for the bond lengths) to that obtained with the second-order Møller–Plesset perturbation 57 theory (MP2) approach using the same basis. All of the electronic structure calculations for the FR0-SB molecule reported in Ref. [3] and summarized here, including the CAM- B3LYP and MP2 geometry optimizations and the CC/EOMCC single-point calculations, were performed by using the GAMESS package [182–184]. The relevant CCSD, EOMCCSD, and δ-CR-EOMCC(2,3) computations using the RHF determinant as a reference and the corresponding left-eigenstate CCSD and EOMCCSD calculations, which were needed to determine the triples corrections of δ-CR-EOMCC(2,3) and the one-electron properties of interest, including the dipole moments and oscillator strengths, were performed by using the CC/EOMCC routines developed by the Piecuch group [183], which form part of the GAMESS code. In all of the post-RHF calculations, the core orbitals associated with the 1s shells of C and N atoms were kept frozen; i.e., we correlated 138 electrons. In the calculations employing the 6-31+G* basis set, we used spherical d-type polarization functions. The visualization of the optimized structure of FR0-SB in its ground and excited electronic states shown in Fig. 3.2 was accomplished by using VMD software [229]. We show in Table 3.1 some of the key properties of the low-lying singlet electronic states of the FR0-SB molecule obtained in the ab initio EOMCC calculations described above. They include the excitation energies and oscillator strengths characterizing the vertical S0 → Sn (n = 1–4) transitions and the electronic dipole moments of the calculated ground and excited states. To verify the reliability of our EOMCC-based computational protocol defined by Eq. (3.23), we compared our theoretical gas-phase value of the S0 → S1 vertical excitation en- ergy of FR0-SB with the corresponding experimental photoabsorption energy characterizing FR0-SB dissolved in hexane, which is the least polar solvent considered in our experiments that will be reported in a future publication. Our best ab initio EOMCC value based on Eq. (3.23), of 3.70 eV, matches closely the experimentally derived S0 → S1 transition energy corresponding to the maximum of the photoabsorption band characterizing FR0-SB dis- solved in hexane, which is 3.52 eV. If we did not correct the EOMCCSD/6-31+G* excitation energy for the triples using Eq. (3.23), we would obtain 4.10 eV, which shows that high-order 58 many-electron correlation effects beyond the EOMCCSD level, estimated in this study with the help of the δ-CR-EOMCC(2,3) approach, are significant. We should also mention that our best TD-DFT result for the S0 → S1 vertical excitation energy of FR0-SB, obtained using the CAM-B3LYP functional, of 3.92 eV, is not as good as the EOMCC value shown in Table 3.1. Moving to our main theoretical findings summarized in Table Eq. (3.23), we can see that of the four lowest-energy singlet excited states of FR0-SB calculated in this study, two, namely, S1 and S2 , can be accessed by photoabsorption. The remaining two states, S3 and S4 , are characterized by negligible oscillator strengths. What is most important for this study are the observations that both S1 and S2 have similar peak positions and intensities on the same order, resulting in broadening of the FR0-SB → FR0-SB* photoabsorption band, and that the electronic dipole moment of FR0-SB increases significantly upon pho- toexcitation, from 2.6 D in the ground electronic state to 8.6 D for S1 and to 6.7 D for S2 . This more-or-less 3-fold increase in the dipole moment as a result of the S0 → S1 and S0 → S2 optical transitions in FR0-SB, observed in our EOMCC calculations and shown in Fig. 3.2 for S1 , demonstrates that the deprotonation of protic solvent molecules by the photoactivated FR0-SB species is indeed possible, since there is an accumulation of the net negative charge on the imine nitrogen that is accompanied by a decrease of electron density on the amine nitrogen. This very small charge transfer, of about 0.1e, occurs over a very large distance of >10 Å, which is made possible by the central fluorene linker, and this is ex- actly what is causing the very large increase in the dipole moment of FR0-SB upon S0 → S1 photoexcitation. 59 Table 3.1: Orbital character, vertical excitation energies ωn(EOMCC) (in eV and nm), oscillator strengths, and electronic dipole moment values µn (in D) of the four lowest-energy excited singlet electronic states Sn of FR0-SB as obtained in the EOMCC calculations described in the text. Adapted from Ref. [3]. ωn(EOMCC) State Orbital character Oscillator strength µn a (eV) (nm) ∗ S1 π→π 3.70 335 0.74 8.6 S2 π → π∗ 3.96 313 0.35 6.7 ∗ S3 π→π 4.23 293 0.02 5.4 S4 π → π∗ 4.45 279 0.03 4.3 a The CCSD value of the dipole moment in the ground electronic state S0 is 2.6 D. 60 Figure 3.1: The molecular structure of FR0-SB with a representation of the excited-state proton transfer (ESPT) process. The bottom left panel, adapted from Ref. [150], shows the absorption (unshaded) and fluorescence (shaded) spectra of FR0-SB dissolved in acetonitrile (ACN, blue), ethanol (EtOH, black), and ethanol acidified with HClO4 (EtOH/HClO4 , red). The hν1 and hν2 labels correspond to the approximate 0–0 transition wavelengths of FR0- SB and FR0-HSB+ , respectively. The bottom right panel provides a simplified illustration of the excitation and emission processes involving the unprotonated and protonated forms of FR0-SB, with labels corresponding to the maxima observed in the absorption and emission spectra. 61 Figure 3.2: The minimum-energy geometry of the ground electronic state of FR0-SB along with the EOMCCSD/6-31+G* electronic densities and dipole moment vectors characterizing the S0 (orange) and S1 (magenta) states. The S1 –S0 electronic density difference, adapted from Ref. [4], is shown in the bottom, in which areas shaded with red (blue) indicate an increase (decrease) in the electronic density upon S1 → S0 photoexcitation. The change in Mulliken charges of the amine and imine nitrogens of the FR0-SB chromophore upon photoexcitation are also indicated. 62 Once we understood the origin of photobasicity of FR0-SB, our experimental collabora- tors performed experiments where chromophore is excited via TPE instead of the conven- tional one-photon excitation (OPE) process and observed an interesting phenomenon. This is demonstrated in Fig. 3.3, adapted from Fig. 2 of Ref. [5], where we show the steady-state fluorescence spectra following OPE and TPE of FR0-SB dissolved in methanol, ethanol, n-propanol, and i-propanol. Focusing on the spectra in methanol shown in Fig. 3.3(a), it is clear that the emission peak area corresponding to the protonated FR0-SB species is larger for the TPE-induced fluorescence than in the OPE-induced fluorescence after normalizing for the unprotonated emission peak area. In fact, ESPT between FR0-SB and methanol is enchanced by a factor of ∼62%. Interestingly, we also observe a solvent-dependent behav- ior in this TPE enhancement of ESPT of FR0-SB, where the enhancement decreases from methanol to ethanol to n-propanol and there is no observed enhancement at all in i-propanol. To investigate this phenomenon, we begin by looking at the OPE and TPE absorption cross sections arising from the first- and second-order time-dependent perturbation theory, respectively (see, e.g., Ref. [230]). The 0 → f OPE absorption cross section, with 0 and f denoting the initial and final electronic states, respectively, is [230] (1) σf 0 (ω) = A |µf 0 |2 gM1 (ω), (3.24) where ω is the frequency of the exciting photon (in our case, the frequency of a 400 nm laser), A is a constant, µf 0 denotes the magnitude of the transition dipole moment between the ground and excited electronic states, and gM1 (ω) is the OPE distribution function or linewidth associated with the molecular systenm of interest. In presenting Eq. (3.24), we have assumed an isotropic averaging over the directions of the transition dipole moment vector µf 0 . To arrive at an expression for the absorption cross section associated with the isoenergetic one-color TPE, where the laser frequency is half of its OPE counterpart, we take advantage of the fact that no resonance at 800 nm is observed in our experiments, in agreement with our electronic structure calculations. Under these conditions, the absorption 63 cross section for TPE becomes [231] 2 (2) X µf ν µν0 σf 0 (ω) =B gM2 (ω), (3.25) ν ων0 − ω/2 + iΓν (ω/2) where B is a constant, ων0 is the frequency needed to reach the intermediate state ν from the ground state 0, iΓν (ω/2) is a damping factor that is inversely proportional to the lifetime of a given intermediate state ν, and gM2 (ω) is the TPE line shape function. In analogy to the OPE absorption cross section, we have performed an isotropic averaging over the directions of the transition dipole moment vectors µf ν and and µν0 . Equation (3.25) is useful, but in this work we are interested in relating the TPE absrption cross section with the change in the dipole moment upon 0 → f photoexcitation. One can derive such a relationship if we perform the following mathematical manipulations [232]. First, we separate the ν = 0 and ν = f terms from the sum over states in Eq. (3.25). Next, we take advantage that, in our case, 0 and f correspond to the elctronically bound S0 and S1 states of FR0-SB, respectively. This allows us to eliminate the iΓν (ω/2) term in the ν = 0 and ν = f denominators in Eq. (3.25). In the final step, we replace ωf 0 in the ν = f denominator by ω and combine the ν = 0 and ν = f contributions to obtain [232] 2 (2) X µf ν µν0 µf 0 ∆µf 0 σf 0 (ω) =B + gM2 (ω). (3.26) ν̸=0,f ων0 − ω/2 + iΓν (ω/2) ω/2 It is customary to refer to the first term in Eq. (3.26) as the “virtual” pathway and to the second one, which relies on the transition dipole moment µf 0 and the difference between the permanent ground- and excited-state dipoles ∆µf 0 ≡ µf f − µ00 , as the “dipole” pathway [233]. Equation (3.26) shows that for centrosymmetric molecules, for which ∆µf 0 vanishes identically, the virtual pathway is the only contributing term to the TPE absorption cross section. However, FR0-SB is not centrosymmetric and, thus, it is interesting to examine the extent to which each pathway contributes to the S0 → S1 one-color TPE considered here. For the first term in Eq. (3.26) to be large, the following three conditions would have to be satisfied: (i) the ων0 frequency characterizing the 0 → ν transition would have to be close to the frequency ω/2 of each of the two photons associated with TPE, (ii) 64 the iΓν (ω/2) damping factor would have to be very small, i.e., the intermediate state ν would have to be sufficiently long-lived, and (iii) the 0 → ν and ν → f transitions would have to be allowed, giving rise to larger µν0 and µf ν transition dipole moments. In the case of the TPE experiments performed in Ref. [5], it is unlikely that conditions (i)–(iii) can be simultaneously satisfied. Indeed, since there are no dipole-allowed electronic states between S0 and S1 , the intermediate state ν satisfying condition (i) would have to be a rovibrational resonance supported by the ground-state electronic potential. It is unlikely that such resonances are long-lived and characterized by large 0 → ν and ν → f transition dipole moments. It is possible that the intermediate states ν characterized by larger µν0 and µf ν values exist, but those would have to be electronic states higher in energy than S1 , which cannot satisfy the resonant condition (i). Furthermore, as demonstrated in Ref. [3] and summarized above, the low-lying electronically excited states above S1 are characterized by small or even negligible transition dipole moments from the ground state. In other words, while the virtual pathway contributes to the TPE cross section, the probability that it dominates it seems low, especially when we realize that there are reasons for the dipole pathway to play a substantial role in the case of the molecular systems considered in this work. Indeed, as shown in Refs. [3, 4] and as further elaborated on below, the S0 → S1 excitations in the isolated and solvated FR0-SB are characterized by large transition dipole moments and significant changes in the permanent dipoles. This suggests that the second term in Eq. (3.26) plays a major role, which is consistent with the well-established fact that the dipole pathway becomes critical when TPE involves charge transfer associated with substantial change in the permanent dipole upon photoexcitation [234–240]. Although the S0 → S1 transition in FR0-SB is accompanied by a migration of a small amount of charge [3, 4], this migration happens over a very large distance, giving rise to more than a threefold increase in dipole moment and a substantial enhancement of the second term in Eq. (3.26). Given the above analysis, from this point on, we focus on the dipole pathway and assume that we can approximate the TPE absorption cross section by the second term in Eq. (3.26), 65 i.e. [238], (2) σf 0 (ω) ≈ B ′ |µf 0 |2 |∆µf 0 |2 gM2 (ω), (3.27) where B ′ = 4B/ω 2 . Furthermore, by forming the ratio of Eqs. (3.24) and (3.27), we can obtain a new expression that summarizes the difference between OPE and TPE, in which the change in permanent dipole moment acts as an amplification factor, (2) σf 0 (ω/2) B ′ |µf 0 |2 |∆µf 0 |2 gM2 (ω) |∆µf 0 |2 gM2 (ω) = = C , (3.28) (1) σf 0 (ω) A |µf 0 |2 gM1 (ω) gM1 (ω) where C = B ′ /A. As illustrated in the mathematical derivation above, the absorption cross sections for both one- and two-photon excitation processes depend on the square of the absolute value of the transition dipole moment µf 0 characterizing the 0 → f vertical electronic excitation, which, in our case, is the transition dipole µ10 corresponding to the S0 → S1 photoabsorption for the FR0-SB system in various solvents. However, in the case of TPE, the absorption cross section also depends on the difference between the electronic dipole moments of the f and 0 states, ∆µf 0 , which, in our case, is the difference ∆µ10 ≡ µ1 − µ0 between the dipole moment µ1 characterizing the first excited singlet S1 state of FR0-SB and its S0 counterpart µ0 . Consequently, ∆µ10 and its dependence on the solvent environment hold the key to understanding the enhancement of the ESPT reactions between the FR0-SB photobase and alcohol solvents observed in the case of TPE. To provide insights into the effect of solvation on ∆µ10 and other properties characterizing the S0 and S1 states of the solvated FR0-SB chromophore and transitions between them, we performed electronic structure calculations using the following CC/EOMCC-based composite approach. In the computations reported in Ref. [5] and summarized here, we focus on analyzing the role of solvation effects on the S0 –S1 vertical and adiabatic transition energies and vertical transition dipole moments, along with the electronic dipoles characterizing the individual S0 and S1 states of FR0-SB, which are key quantities in comparing the one- and two-photon S0 → S1 absorption cross sections. The solvent dependence of the enhancement nessecitates 66 the incorporation of solvation effects in the CC/EOMCC computations, which our protocol described in Ref. [3] and summarized above does not take into account. As explained in Ref. [5], in modeling the solvated FR0-SB chromophore, we considered the complex of FR0-SB hydrogen-bonded to a cluster of three alcohol solvent molecules, designated as [FR0-SB· · · HOR], which, according to our earlier investigation of the steric effects on the ESPT process involving FR0-SB and n- and i-propanol, is the minimum number of explicit solvent molecules required for the proton transfer to occur [4]. Following Ref. [4], we used the “branched” arrangement of the three alcohol solvent molecules treated in our modeling explicitly, with one of them hydrogen-bonded to FR0-SB and the other two solvating it, since such an arrangement leads to the lowest energy barriers for the ESPT reactions involving FR0-SB (see Ref. [4] and the discussion below for further details). The remaining, bulk, solvation effects were described using the continuum solvation model based on the solute electron density (SMD) approach [241]. The alcohol solvents considered in our computations were methanol, ethanol, n-propanol, and i-propanol. For each of the alcohol solvents considered in our calculations, the geometry optimiza- tion of the [FR0-SB· · · HOR] complex in its S0 state, used in the subsequent CC/EOMCC calculations, was performed employing the Kohn–Sham formulation of DFT. To obtain the corresponding minimum-energy structures of the [FR0-SB· · · HOR] species in the S1 state, we used the TD-DFT extension to excited electronic states. Following Refs. [3, 4], in car- rying out these geometry optimizations we used the CAM-B3LYP functional. All geometry optimizations of the [FR0-SB· · · HOR] complex employed the 6-31+G* basis and accounted for the bulk solvation effects using the aforementioned SMD model. To provide accurate information about the transition energies and transition dipole mo- ments characterizing the absorption (S0 → S1 ) and emission (S1 → S0 ) processes involving the solvated FR0-SB species and the corresponding dipoles in the S0 and S1 states, which are all needed to model the one- and two-photon cross sections for each of the alcohol sol- vents considered in our calculations, we performed the following series of single-point CC 67 and EOMCC computations at the aforementioned CAM-B3LYP/6-31+G*/SMD optimized geometries. First, we determined the S0 –S1 electronic transition energies, (EOMCC) (EOMCC) (CC) ω10 = ES1 − E S0 , (3.29) corresponding to the [FR0-SB· · · HOR] complex in the absence of the SMD continuum solvation, where the total electronic energies of the S0 and S1 states entering Eq. (3.29) were computed as h i (CC) (CCSD/6-31+G*) (CR-CC(2,3)/6-31G) (CCSD/6-31G) ES0 = ES0 + ES0 − ES0 (3.30) for the ground state and h i (EOMCC) (EOMCCSD/6-31+G*) (δ-CR-EOMCC(2,3)/6-31G) (EOMCCSD/6-31G) ES1 = ES1 + ES1 − ES1 (3.31) for the first excited singlet state. The first term on the right-hand side of Eq. (3.31) denotes the total electronic energy of the S0 state computed at the CCSD level utilizing the largest basis set considered in this study, namely, 6-31+G*. The term in the square brackets on the right-hand side of Eq. (3.31) corrects the CCSD/6-31+G* energy for the many-electron correlation effects due to triply excited clusters obtained in the CR-CC(2,3) calculations employing the smaller and more affordable 6-31G basis. Similarly, the first term on the right-hand side of Eq. (3.31) designates the EOMCCSD/6-31+G* energy of the S1 state and the expression in the square brackets represents the triples correction to EOMCCSD obtained in the δ-CR-EOMCC(2,3)/6-31G calculations. Ideally, one would like to use basis sets larger than 6-31+G* and, in particular, incorporate polarization and diffuse functions on hydrogen atoms, but such calculations at the CC/EOMCC levels used in this work turned out to be prohibitively expensive. Nevertheless, we tested the significance of the polarization and diffuse functions on hydrogen atoms by performing the CAM-B3LYP/6-31++G**/SMD calculations for the [FR0-SB· · · HOR] complexes that show that neither the excitation ener- gies nor the dipole and transition dipole moment values change by more than 1% compared to the CAM-B3LYP/6-31+G*/SMD results. 68 Before describing the remaining elements of our computational protocol, it is important to emphasize that the composite approach defined by Eqs. (3.29)–(3.31) is more general than the analogous expressions shown in Eq. (3.23) [3], where we focused on the vertical excitation processes only. Equations (3.29)–(3.31) encompass both the vertical and adiabatic transition (CC) (EOMCC) energies. Indeed, if ES0 and ES1 are calculated at the minimum on the S0 potential (EOMCC) energy surface, ω10 given by Eqs. (3.29)–(3.31) becomes the vertical excitation energy (EOMCC) (CC) ω10 (abs.) characterizing the S0 → S1 absorption defined by Eq. (3.23). If ES0 and (EOMCC) ES1 are determined at the minimum characterizing the [FR0-SB· · · HOR] complex (EOMCC) in the S1 state, we obtain the vertical transition energy ω10 (em.) corresponding to (EOMCC) the S0 → S1 emission. The ω10 energy defined by Eq. (3.29) becomes the adiabatic (EOMCC) (CC) (EOMCC) transition energy, abbreviated as ω10 (ad.), when ES0 and ES1 are computed at their respective minima. As far as the transition dipole moments characterizing the vertical absorption and emission processes involving the solvated FR0-SB species are concerned, they were calculated from the one-electron transition density matrices obtained at the EOMCCSD level of theory employing the 6-31+G* basis set. Similarly, we used the CCSD/6-31+G* and EOMCCSD/6-31+G* one-electron reduced density matrices to determine the dipole moments of the S0 and S1 states at each of the two potential minima. Given the large computational costs associated with the EOMCCSD and δ-CR-EOMCC(2,3) calculations for the [FR0-SB· · · HOR] system, which consists of three alcohol molecules bound to the FR0-SB chromophore and requires correlating as many as 216 electrons and 758 molecular orbitals in the case of the n- or i-propanol solvents when the 6-31+G* basis set is employed, we replaced the three explicit alcohol molecules with the cor- responding effective fragment potentials (EFPs) [242]. We were able to do this because, based on our CAM-B3LYP/6-31+G*/SMD calculations for the [FR0-SB· · · HOR] complexes, the S0 –S1 electronic transition does not involve charge transfer between the photobase and its solvent environment. Indeed, the S0 –S1 transition in the bare and solvated FR0-SB species has a predominantly π–π* character with the π and π* orbitals localized on the FR0-SB 69 chromophore, i.e., the alcohol solvent molecules are mere spectators to this excitation process (see the supplementary material of Ref. [5] for further details). The use of EFPs to represent the cluster of three alcohol molecules bonded to FR0-SB in our CC/EOMCC computa- tions allowed us to reduce the system size to that of the bare FR0-SB species embedded in the external potential providing a highly accurate description of the intermolecular interac- tions between FR0-SB and solvent molecules in the [FR0-SB· · · HOR] complex, including electrostatic, polarization, dispersion, and exchange repulsion effects [242]. Once the electronic transition energies and the corresponding one-electron properties of the [FR0-SB· · · HOR] complex were determined, we proceeded to the second stage of our modeling protocol, which was the incorporation of the remaining bulk solvation effects that turned out to be nonnegligible as well. As in the case of the aforementioned geometry opti- mizations, the bulk solvation effects were calculated with the help of the implicit solvation SMD approach. Due to the limitations of the computer codes available to us, we could not perform the CC/EOMCC computations in conjunction with the SMD model, so we estimated (SMD) the SMD effects using the a posteriori corrections δX to the various CC/EOMCC prop- erties X of the [FR0-SB· · · HOR] complex, such as transition energies and dipole moments, using DFT and TD-DFT. These corrections were constructed in the following way. First, for each of the four alcohol solvents considered in our calculations, we performed single-point DFT/TD-DFT calculations for the [FR0-SB· · · HOR] complex at the previously optimized S0 and S1 geometries accounting for the bulk solvation effects using SMD. As in the case of the geometry optimizations, we used the CAM-B3LYP functionals and the 6-31+G* basis set and, in analogy to the CC/EOMCC computations, replaced the cluster of three explicit alcohol solvent molecules bound to FR0-SB by the corresponding EFPs. We then repeated (SMD) the analogous calculations without SMD. This allowed us to determine the desired δX corrections using the formula (SMD) δX = X (CAM-B3LYP/6-31+G*/SMD) − X (CAM-B3LYP/6-31+G*) , (3.32) where the first and second terms on the right-hand side of Eq. (3.32) designate property X 70 obtained in the CAM-B3LYP/6-31+G* calculations with and without SMD, respectively. The final SMD-corrected EOMCC electronic transition energies were computed as (EOMCC) ω10 = ω10 + δω(SMD) 10 , (3.33) (EOMCC) where ω10 is the transition energy for the [FR0-SB· · · HOR] complex defined by Eqs. (3.29)–(3.31), whereas the SMD-corrected one-electron properties were determined using the formula (SMD) X = X [(EOM)CCSD/6−31+G∗] + δX , (3.34) with X [(EOM)CCSD/6−31+G∗] denoting the value of property X calculated at the (EOM)CCSD/6- 31+G* level. If the property of interest was a vector, such as dipole or transition dipole moment, we used Eq. (3.34) for each of the Cartesian components of the vector. Finally, to gauge the effects of solvation on the various calculated properties, including transition energies and dipole and transition dipole moments, we also performed single-point CC/EOMCC calculations for the bare super photobase, i.e., FR0-SB without the presence of explicit solvent molecules or equivalent EFPs and SMD implicit solvation, at the gas- phase geometry of the S1 state optimized using CAM-B3LYP/6-31+G*. In the case of the S0 minimum-energy structure, we relied on our previous gas-phase CC/EOMCC results reported in Ref. [3] and summarized earlier. All of the electronic structure calculations reported in Ref. [5] and summarized here, including the CAM-B3LYP geometry optimizations with and without the SMD continuum solvation, the CC and EOMCC single-point calculations without implicit SMD solvation, and the CAM-B3LYP single-point calculations with and without SMD, needed to estimate the SMD corrections to CC/EOMCC properties, were performed using the GAMESS package (specifically, we used the 2019 R2 version of GAMESS for our works in Ref. [5] and discussed here). In the case of the S0 → S1 absorption process, whenever the SMD implicit solvation model was utilized, we incorporated the nonequilibrium solvation effects associated with the solvent relaxation delay, as implemented in GAMESS [243]. In all of our CC/EOMCC 71 calculations, the core orbitals associated with the 1s shells of C and N atoms of FR0-SB were kept frozen. The EFPs that were used to replace the cluster of three explicit alcohol solvent molecules bound to FR0-SB in the CC/EOMCC single-point calculations and the CAM- B3LYP computations aimed at determining the SMD solvation effects were generated using the RHF approach and the 6-31+G* basis set. Thanks to the use of EFPs, our frozen-core CC/EOMCC calculations for the [FR0-SB· · · HOR] complex correlated only 138 electrons of the FR0-SB system. In all of the calculations employing the 6-31+G* basis set, we used spherical components of d orbitals. In Table 3.2, we report the vertical transition energies ω10 (abs.) and transition dipole moments µ10 characterizing the S0 → S1 photoabsorption process, along with the dipoles corresponding to the S0 and S1 states, µ0 and µ1 , respectively, and their ratios resulting from our calculations for FR0-SB in the gas phase and in the aforementioned four solvents determined at the minima on the respective S0 potential energy surfaces. The analogous information for the S1 → S0 emission and the dipole moment values of the S0 and S1 states determined at the S1 minima characterizing the isolated and solvated FR0-SB is presented in Table 3.3. We begin our discussion of computational results by comparing the vertical absorption and emission energies characterizing the solvated FR0-SB species obtained with the CC/EOMCC-based protocol adopted in this work against their experimental counter- parts. The vertical excitation energies for the [FR0-SB· · · HOR] complexes calculated at the respective S0 minima, shown in Table 3.2, are essentially identical to the locations of the peak maxima in the corresponding experimental photoabsorption spectra reported in Ref. [4], which are 3.32 eV, 3.33 eV, 3.32 eV, and 3.34 eV for methanol, ethanol, n-propanol, and i-propanol, respectively. The same accuracies are also seen in the case of the vertical emission energies calculated at the S1 minima of the [FR0-SB· · · HOR] species reported in Table 3.3, which can hardly be distinguished from the maxima in the experimental emission peaks for FR0-SB in methanol, ethanol, n-propanol, and i-propanol of 2.57 eV, 2.61 eV, 2.62 eV, and 2.65 eV, respectively [4]. These observations corroborate the accuracy of the com- 72 putational protocol used in this study to model the interactions of the FR0-SB photobase with the various alcohol solvents. The observed good agreement between the theoretical ver- tical transition energies reported in Tables 3.2 and 3.3 and the corresponding experimental data can largely be attributed to the use of high-level ab initio CC/EOMCC approaches in describing the [FR0-SB· · · HOR] complexes. This becomes apparent when one considers the errors relative to the experiment characterizing the vertical transition energies obtained in the single-point CAM-B3LYP/6-31+G*/SMD computations, which are about 0.2 eV–0.3 eV (9%–11%). Having established the accuracy of our quantum chemistry protocol, we proceed to the discussion of our computational findings regarding the dipole moments of the S0 and S1 states and the transition dipoles between them, which are the key quantities for the one- and two-photon absorption cross sections given by Eqs. (3.24) and (3.27), respectively. In the absence of direct experimental information, our computations provide insights into the effects of solvation on these quantities. To begin with, as reported in our earlier work for the bare FR0-SB species [3], and as shown in Table 3.2, there is a large, by a factor of more than 3, increase in the electronic dipole moment following S0 → S1 photoabsorption, giving rise to the superbase character of FR0-SB*. Upon solvation, both S0 and S1 dipole moments of the FR0-SB chromophore are significantly enhanced, becoming approximately twice as large as their gas-phase counterparts. This can be attributed to the polarization of the electron cloud of the FR0-SB photobase by the alcohol molecules surrounding it. Furthermore, the fact that the electronic dipole moment characterizing the S1 state is much larger than its S0 counterpart translates into a stronger stabilization of the S1 state relative to S0 , leading to lower S0 → S1 vertical excitation energies in the case of FR0-SB in alcohol solvents when compared to the bare FR0-SB system. The transition dipole moment characterizing the S0 → S1 photoabsorption process is amplified by solvation as well (by about 40%), which results in larger OPE and TPE absorption cross sections for the solvated FR0-SB species relative to their gas-phase values. Similar trends are observed when we examine the dipoles 73 and transition dipoles shown in Table 3.3. It is also interesting to note that the dipole moments characterizing the S0 and S1 states and the corresponding transition dipoles increase upon geometrical relaxation from the S0 to S1 minima, with a concomitant red shift in the vertical transition energies. This bathochromic shift is more pronounced in the case of the solvated FR0-SB species as a consequence of µ1 being much larger than µ0 , implying a stronger stabilization of the S1 state due to the polar solvent environment compared to the S0 state. As already alluded to above, the transition dipole moments characterizing the S0 –S1 ab- sorption and emission processes and the S0 and S1 dipoles at the respective potential minima could not be determined from our experiments. However, by analyzing the solvatochromic shift of the absorption and fluorescence bands in 16 different solvents as a function of solvent dielectric constant and index of refraction, we could estimate the magnitude of the transition dipole moment µ10 and the change in the dipole moment, ∆µ10 , associated with the S0 → S1 adiabatic excitation [244, 245]. Based on our analysis, we found ∆µ10 of FR0-SB in the alcohol solvents considered in our experiments to be ∼15 D, a magnitude usually associated with substantial charge transfer, and µ10 to be about 10 D. The procedure outlining how the experimental values of µ10 and ∆µ10 were derived is given in the supplementary material of Ref. [5]. Having access to the dipole moments characterizing the S0 and S1 states at their respective minimum-energy structures and the vertical transition dipole moments associated with the S0 –S1 transitions resulting from our quantum chemistry computations (see Tables 3.2 and 3.3) allowed us to assess the quality of our experimentally derived values of µ10 and ∆µ10 . As shown in Tables 3.2 and 3.3, the vertical transition dipole moments µ10 charac- terizing the FR0-SB chromophore in the alcohol solvents included in our calculations range from 9.4 D to 11.8 D, in very good agreement with the experimentally derived value of about 10 D. According to the data collected in Table 3.4, the calculated and experimentally derived changes in the dipole moment associated with the S0 → S1 adiabatic transition, which are about 15 D in both cases, are virtually identical. Given that both theory and experiment 74 point to the large values of µ10 and ∆µ10 as a result of solvation and that the dipole pathway defined by the second term in Eq. (3.26) is anticipated to be the dominant TPE pathway, as discussed above, we can conclude that using Eq. (3.27) in approximating the TPE absorption cross section of FR0-SB in alcohol solvents is justified. We are now well-situated to rely on our computational data to help interpret the exper- imental findings of Ref. [5]. Indeed, as shown in Tables 3.2–3.4, there is a massive increase in permanent dipole moment as FR0-SB undergoes S0 → S1 excitation, which is further en- hanced by the presence of polar solvent environment enveloping the FR0-SB chromophore. We also know from Eq. (3.28) that the TPE absorption cross section is enhanced by a signif- icant ∆µ10 value, while OPE absorption cross section does not take such contribution into account. Therefore, in Ref. [5], we hypothesized that by using TPE, one can selectively excite [FR0-SB· · · H-OR] configurations which would result in considerable increase in permanent dipole moment. Such configurations may then be characterized by energetically favorable ESPT reaction pathways toward the formation of protonated FR0-SB in the excited state, especially considering the magnitude of the dipole moment in the excited state. This could be confirmed by performing, for example, molecular dynamics simulations of FR0-SB in various alcohol solvent environment, taking snapshots of different solvent configurations around the FR0-SB chromophore, and performing the same solvation-corrected EOMCC calculations described above, which we will consider in our future works. 75 Table 3.2: The vertical transition energies ω10 (abs.) (in eV) and transition dipole moments µ10 (in D) corresponding to the S0 → S1 absorption, along with the µ0 and µ1 dipoles characterizing the S0 and S1 states (in D) and their ratios for FR0-SB in the gas phase and in selected alcohol solvents calculated at the respective S0 minima following the CC/EOMCC- based protocol described in the text. Adapted from Ref. [5]. Solventa ω10 (abs.) µ10 µ0 µ1 µ1 /µ0 None (gas phase)b 3.70 6.9 2.6 8.6 3.3 MeOH 3.30 9.6 4.4 16.4 3.8 EtOH 3.32 9.5 4.3 15.9 3.7 n-PrOH 3.32 9.5 4.2 15.9 3.7 i-PrOH 3.33 9.4 4.2 15.7 3.7 a Abbreviations: MeOH = methanol, EtOH = ethanol, n-PrOH = n-propanol, and i-PrOH = i-propanol. b Taken from our previous gas-phase CC/EOMCC calculations reported in Ref. [3]. Table 3.3: The vertical transition energies ω10 (em.) (in eV) and transition dipole moments µ10 (in D) corresponding to the S1 → S0 emission, along with the µ0 and µ1 dipoles char- acterizing the S0 and S1 states (in D) and their ratios for FR0-SB in the gas phase and in selected alcohol solvents calculated at the respective S1 minima following the CC/EOMCC- based protocol described in the text. Adapted from Ref. [5]. Solventa ω10 (em.) µ10 µ0 µ1 µ1 /µ0 None (gas phase) 3.26 8.9 3.4 10.9 3.2 MeOH 2.68 11.8 6.6 20.0 3.0 EtOH 2.69 11.8 6.5 19.7 3.0 n-PrOH 2.70 11.8 6.5 19.6 3.0 i-PrOH 2.72 11.7 6.4 19.1 3.0 a Abbreviations: MeOH = methanol, EtOH = ethanol, n-PrOH = n-propanol, and i-PrOH = i-propanol. 76 Table 3.4: A comparison of the calculated S0 –S1 adiabatic transition energies without [ω10 (ad.)] and with [ω10 (0-0)] zero-point energy (ZPE) vibrational corrections (in eV), along with the differences and ratios of the µ0 and µ1 dipoles characterizing the S0 and S1 states at the respective minima (in D) for FR0-SB in the gas phase and in selected alcohol sol- vents obtained following the CC/EOMCC-based protocol described in the text with the corresponding experimentally derived data. Adapted from Ref. [5]. Theory Experiment Solventa ω10 (ad.) ω10 (0-0)b ∆µ10 c µ1 /µ0 c Solventa ω10 (0-0) ∆µ10 d µ1 /µ0 d None (gas phase) 3.42 3.33 8.3 4.2 c-Hexane 3.4 — — MeOH 2.88 2.80 15.6 4.6 MeOH 2.9 15.2 ± 0.2 4.4 ± 0.1 EtOH 2.89 2.80 15.4 4.6 EtOH 3.0 15.3 ± 0.3 4.6 ± 0.1 n-PrOH 2.89 2.81 15.3 4.6 n-PrOH 3.0 15.3 ± 0.3 4.6 ± 0.1 i-PrOH 2.88 2.80 14.9 4.5 i-PrOH 3.0 15.5 ± 0.5 4.7 ± 0.1 a Abbreviations: MeOH = methanol, EtOH = ethanol, n-PrOH = n-propanol, i-PrOH = i-propanol, and c-Hexane = cyclohexane. b Calculated as ω10 (ad.) + ∆ZPE, where ∆ZPE is the difference between the zero-point vibrational energies characterizing the S1 and S0 electronic states of the bare FR0-SB molecule in the gas phase computed at the CAM-B3LYP/6-31+G∗ level of theory. Our calculations with and without solvent indicate that the effect of solvation on ∆ZPE is negligible (less than 0.01 eV). c Calculated using the µ0 values reported in Table 3.2 and the µ1 values reported in Table 3.3. d Calculated using the theoretical values of µ0 reported in Table 3.2 and the procedure based on the analysis of the experimental solvatochromic shifts described in the supplementary material of Ref. [5]. 77 Figure 3.3: OPE and TPE steady-state fluorescence spectra obtained for FR0-SB in (a) methanol, (b) ethanol, (c) n-propanol, and (d) i-propanol. In each of the panels, OPE (blue line) is compared with TPE (red line). The fluorescence spectra are normalized to the nonprotonated emission intensity. The ratio between the areas for FR0-HSB+ * (∼15,000 cm−1 ) and FR0-SB* (∼21,000 cm−1 ) emission following OPE and TPE is determined by fits to log-normal functions (thin black lines). Adapted from Ref. [5]. 78 As already alluded to above and as shown in the experimental data reported in Refs. [3–5], there are various effects that govern the extent of ESPT between the alcohol solvent environment and the FR0-SB chromophore. Among these effects, one that could significantly affect the ESPT reaction is steric effects due to the structure of the alcohol solvent molecules. We have seen this behavior in the OPE- and TPE-induced fluorescence spectra of FR0- SB in i-propanol, which do not differ from each other. Furthermore, when steady-state OPE-induced fluorescence measurements were performed on FR0-SB dissolved in various alcohol solvents, including primary, secondary, and tertiary alcohols, one observe varying degrees of proton transfer occurring. As shown in Fig. 3.4, adapted from Fig. 2 of Ref. [4], where each fluorescence spectrum is normalized with respect to the tallest peak in the spectrum, the extent of ESPT reaction, represented by the ratio between the emission peak areas of the protonated and unprotonated FR0-SB*, is considerably lower in secondary alcohols (represented by i-propanol and cyclopentanol in the figure) than in primary alcohol (represented by n-propanol in the figure). In fact, the ESPT reaction is not observed at all in tertiary alcohols, such as t-amyl alcohol, which exhibits very similar spectrum to the aprotic acetonitrile solvent. Thus, we set out to provide insights into the effect of solvent steric factors on the ESPT process. Specifically, we report the details of the ESPT reaction pathways between FR0-SB and representative primary and secondary alcohols predicted by quantum chemistry calculations. To address this issue and to provide deeper insights into the role of steric effects in the proton transfer reactions between the excited FR0-SB* chromophore and alcohol solvent molecules, we augmented the experimental effort by performing electronic structure calcula- tions focusing on the ground, S0 , and first-excited singlet, S1 , electronic states of the solvated FR0-SB system. In the calculations reported in this work, we focused on the reactions of FR0-SB* with n- and i-propanol. The n- and i-propanol molecules are the smallest alcohol species in the primary and secondary categories considered in our experiments that permit structural isomerism. 79 In modeling the ESPT process, we considered the interaction between FR0-SB* and a cluster of three alcohol molecules, which, according to our computations, is the minimum number of explicit solvent molecules necessary for the proton transfer to occur. In trying to use complexes consisting of FR0-SB* bound to fewer alcohol molecules, our calculations could not detect the presence of the second minimum corresponding to ESPT. The remaining, i.e., bulk, solvation effects were incorporated using the SMD continuum solvation model. In constructing the reaction pathways characterizing the proton transfer between FR0- SB* and n- and i-propanol, the following protocol was adopted. For each of the two alcohols, the geometries of the electronically excited reactant and product complexes were optimized. The reactant complex is the FR0-SB* chromophore hydrogen-bonded to the cluster of three solvent molecules, i.e., the [FR0-SB*· · · HOR] species with two ROH molecules attached to the alcohol bonded to FR0-SB*. The product of the proton transfer reaction is the [FR0-HSB+ *· · · − OR] complex with two ROH molecules attached to it. Having established the internuclear distances between the proton being transferred and the imine nitrogen of FR0-SB* in the reactant and the product complexes, designated in Fig. 3.5 as r1 and r2 , respectively, we probed the [FR0-SB*· · · HOR]→[FR0-HSB+ *· · · − OR] reaction pathway by introducing an equidistant grid of N–H separations using the step size defined as (r1 −r2 )/10. The molecular structure at each point along the above ESPT reaction pathway was obtained by freezing the N–H distance at the respective grid value and reoptimizing the remaining geometrical parameters. We also optimized the geometry of FR0-SB hydrogen-bonded to the cluster of three alcohol molecules in the ground electronic state, needed to calculate the S0 → S1 vertical excitation energy. All of the geometry optimizations relied on the Kohn–Sham formulation of DFT using, in the case of the structures along the ESPT reaction pathway, the TD-DFT extension to excited states combined with the SMD continuum solvation model to account for the bulk solvation effects (the analogous protocol employing SMD was used in the DFT ground- state optimizations). All of the calculations reported for this work [4] employed the 6- 80 31+G* basis set using spherical components of d functions and the CAM-B3LYP functional. All of the electronic structure calculations reported in this study were performed using the GAMESS package. As in the case of the OPE vs TPE study discussed above, when considering the optimized ground-state geometries, the S0 → S1 vertical excitation energies, which, from the fundamental physics perspective, correspond to a very fast process resulting in an abrupt change in the solute electron density, were computed by taking advantage of the nonequilibrium solvation effects associated with the solvent relaxation delay, incorporating a fast component of the solvent dielectric constant in addition to its bulk value, as implemented in GAMESS. The results of our quantum chemistry computations, shown in Figs. 3.6–3.8, reveal the intricacies of the excited-state proton abstraction process initiated by the formation of the [FR0-SB*· · · HOR] complex. In Fig. 3.6, we present the calculated minimum-energy path- ways characterizing the ESPT reactions involving FR0-SB in its first-excited singlet S1 state and the n- and i-propanol molecules along the internuclear distance between the imine nitrogen of FR0-SB and the proton being transferred. For completeness, the energetics characterizing the corresponding S0 ground states as well as the S0 and S1 energies obtained at the optimized ground-state structures of the relevant [FR0-SB*· · · HOR] complexes are also provided (the leftmost points in Fig. 3.6). As shown in Fig. 3.6, the ground-state energy monotonically increases as the alcohol proton approaches the imine nitrogen of FR0-SB, indicating that the proton abstraction occurs in the excited state of FR0-SB, not in the ground state, in agreement with the experimental observations. As elaborated on above, in the experiments reported in this work, the excited state of FR0-SB is populated by pho- toabsorption from the ground electronic state. Our calculated S0 → S1 excitation energies of FR0-SB in n- and i-propanol of ∼3.6 eV agree quite well with their corresponding ex- perimental values of ∼3.3 eV (see Fig. 3.4 and 3.6–3.8). Upon relaxing the excited-state geometries (see the dashed lines in Fig. 3.6), the difference in the behavior of the bulkier i-propanol species in the [FR0-SB*· · · HOR] complex relative to its n-propanol counterpart 81 becomes apparent already in the early stages of the deprotonation process. In particular, the internuclear distance between the imine nitrogen of FR0-SB and the alcohol proton that is hydrogen-bonded to it is ∼0.1 Å larger in i-propanol than in n-propanol (cf. Fig. 3.6–3.8). Furthermore, Fig. 3.6 reveals that even though the ESPT process takes place in both n- and i-propanol, the barrier height characterizing the reaction involving the secondary alco- hol i-propanol species is ∼50% higher than the analogous barrier associated with its primary alcohol n-propanol counterpart, consistent with the larger distance between the proton being transferred and the oxygen of the alcohol in i-propanol relative to that in n-propanol in the corresponding transition states (see Figs. 3.7 and 3.8). At the same time, the barrier for the reverse process, i.e., deprotonation of FR0-HSB+ *, in i-propanol is about 35% lower than that characterizing the analogous process in n-propanol. Our calculations summarized in Figs. 3.7 and 3.8 imply that there is a need for a complex with two hydrogen bonds to the –OH group of the alcohol that transfers the proton. This “branched” arrangement is unusual; X-ray diffraction structures of the n-alkanols ethanol and butanol, congeners of n-propanol, show only linear structures of –OH moieties, in which each oxygen accepts only one hydrogen bond [246, 247]. However, the “structure” of n- propanol in the liquid phase has been studied and consists of chains of various lengths with modest amounts (a few percent) of branching [248–250]. For i-propanol, which has a stronger preference for cyclic clusters, such configurations are unlikely and again, are not observed in the crystal structure of the pure solvent [251]. Indeed, for both n- and i-propanol, our computations predict the linear alcohol clusters to be about 8–12 kJ mol−1 lower in energy compared to the branched arrangements, not only for the ground-state [FR0-SB· · · HOR] species, but also in the case of the [FR0-SB*· · · HOR] ESPT reactant. Nevertheless, the situation changes dramatically, in favor of the branched alcohol conformations, when one considers the [FR0-HSB+ *· · · − OR] product of the ESPT reaction. In the case of n-propanol, for example, the branched [FR0-HSB+ *· · · − OR] struc- ture is lower in energy than the linear one by about 2 kJ mol−1 . This is related to the fact that 82 the branched alcohol arrangement solvates the RO− species more effectively. Consequently, the Eproduct − Ereactant energy difference in the case of the linear n-propanol configuration, of 14.3 kJ mol−1 , is higher than the 13.1 kJ mol−1 activation barrier characterizing the branched conformation (see Fig. 3.7), implying that the activation energy characterizing the linear ar- rangement is even larger. The difference between the branched and linear conformations is pronounced even more when one considers i-propanol. In this case, the Eproduct − Ereactant energy difference in the linear cluster is about 8 kJ mol−1 higher than the activation barrier characterizing the branched arrangement (cf. Fig. 3.8). Based on our calculations we can conclude that the branched structures adopted in modeling of the ESPT reactions, while unusual in the case of the pure solvents, are a more realistic representation of the [FR0- SB*· · · HOR]→[FR0-HSB+ *· · · − OR] process, since they lead to smaller activation energies compared to the linear arrangements of alcohol molecules bound to FR0-SB*. Last, but not least, the difficulty in achieving the configurations shown in Fig. 3.8 is consistent with the greatly diminished protonation yield observed for i-propanol and the lack of protonation observed for tertiary alcohols. 83 Figure 3.4: The absorption and emission spectra of FR0-SB in various solvents to compare steric hindrance. The long wavelength emission near 630 nm (∼15,870 cm−1 ) corresponds to FR0-HSB+ *, while the short wavelength emission near 460 nm (∼21,740 cm−1 ) corresponds to FR0-SB*. Adapted from Ref. [4]. Figure 3.5: Schematic representation of the r1 and r2 N–H internuclear distances needed to create the grid defining the ESPT reaction pathway. 84 Figure 3.6: Results from the reaction pathway calculations showing ground- and excited- state energy differences as a function of proton abstraction. The CAM-B3LYP/6- 31+G*/SMD ground-state (S0 ) and excited-state (S1 ) reaction pathways corresponding to the proton abstraction from n-propanol (blue) and i-propanol (orange) by FR0-SB along the internuclear distance between the imine nitrogen and the alcohol proton being trans- ferred. The energies ∆E are shown relative to the ground-state minimum of the respective pathways. The dashed line in each pathway indicates the excited-state geometry relaxation following the S0 –S1 excitation of FR0-SB. Adapted from Ref. [4]. 85 E 13.1 kJ/mol (3.36 eV) [2.80 eV] 3.4 kJ/mol 0.0 kJ/mol rO–H = 1.25 Å (3.26 eV) (3.22 eV) rN–H = 1.26 Å [2.66 eV] [2.94 eV] rO–H = 1.71 Å rO–H = 1.01 Å rN–H = 1.06 Å rN–H = 1.71 Å [FR0-SB*⋯HOR] [FR0-SB*⋯H⋯OR] [FR0-HSB+*⋯−OR] Figure 3.7: Snapshots of the proton abstraction process from n-propanol. The CAM- B3LYP/6-31+G*/SMD optimized geometries of the reactant ([FR0-SB*· · · HOR]), transi- tion state ([FR0-SB*· · · H· · · OR]), and product ([FR0-HSB+ *· · ·− OR]) of the ESPT process between FR0-SB in its S1 electronic state and three n-propanol molecules. The ∆E values in kJ mol−1 are given relative to the reactant energy. The energies inside parentheses, in eV, are given relative to the [FR0-SB· · · HOR] minimum in the ground electronic state S0 , while those inside square brackets correspond to the S0 –S1 vertical transitions at each respective geometry. The rO–H and rN–H distances at each geometry represent the internuclear sepa- rations between the proton being transferred and the oxygen of n-propanol and the imine nitrogen of FR0-SB, respectively. Adapted from Ref. [4]. 86 E 19.0 kJ/mol (3.44 eV) 12.9 kJ/mol [2.80 eV] (3.38 eV) rO–H = 1.33 Å [2.68 eV] 0.0 kJ/mol rN–H = 1.21 Å rO–H = 1.66 Å (3.25 eV) rN–H = 1.07 Å [2.97 eV] rO–H = 1.01 Å rN–H = 1.76 Å [FR0-SB*⋯HOR] [FR0-SB*⋯H⋯OR] [FR0-HSB+*⋯−OR] Figure 3.8: Snapshots of the proton abstraction process from i-propanol. The CAM- B3LYP/6-31+G*/SMD optimized geometries of the reactant ([FR0-SB*· · · HOR]), tran- sition state ([FR0-SB*· · · H· · · OR]), and product ([FR0-HSB+ *· · ·− OR]) of the ESPT pro- cess between FR0-SB in its S1 electronic state and three i-propanol molecules. The ∆E values in kJ mol−1 are given relative to the reactant energy. The energies inside parentheses, in eV, are given relative to the [FR0-SB· · · HOR] minimum in the ground electronic state S0 , while those inside square brackets correspond to the S0 –S1 vertical transitions at each respective geometry. The rO–H and rN–H distances at each geometry represent the internu- clear separations between the proton being transferred and the oxygen of i-propanol and the imine nitrogen of FR0-SB, respectively. Adapted from Ref. [4]. 87 3.3 Active-Space CC/EOMCC Approaches and Their CC(P;Q) Extensions 3.3.1 Theory In the previous section, we discussed the simplest forms of CC(P;Q) formalism, namely, the CR-CC and CR-EOMCC schemes. As shown by applying the CR-CC(2,3) and δ-CR- EOMCC(2,3) methods to the FR0-SB photobase, the CR-CC/EOMCC approaches can be very accurate. However, as mentioned in the Introduction, there are certain problems where CR-CC/EOMCC methods fail. In such situation, there usually exists a large coupling between the lower-order components of the cluster and EOM excitation operators, such as T1 and T2 in the former case and Rµ,1 and Rµ,2 in the latter case, and their higher-order counterparts, e.g., T3 (Rµ,3 ), T4 (Rµ,4 ), etc. Within the CR-CC/EOMCC way of performing CC(P;Q) calculations, the P space is defined by relying on traditional truncations of the cluster and EOM excitation operators and, thus, the aforementioned coupling is neglected. This becomes an issue when the higher-order Tn and Rµ,n components with n > 2 become large. However, one can rely on “unconventional” partitioning of the P and Q spaces to allow for the relaxation of T1 and T2 , as well as Rµ,1 and Rµ,2 if EOMCC computations are concerned, in the presence of their dominant higher-order counterparts. One way to do this is via the use of active-space CC and EOMCC methodology, where one downselects T3 , T4 , etc. and their Rµ,n counterparts using a set of predefined active orbitals. Here, we focus on the ground-state approaches relevant to this dissertation work, namely, active-space CC methods with up to quadruply excited clusters and their CC(P;Q) extension. In the active-space CC/EOMCC approaches, we divide the spin-orbitals used in the CC/EOMCC calculations into four separate categories, namely core, active occupied, active unoccupied, and virtual spin-orbitals, and approximate the higher-order cluster and exci- tation operators with the help of active orbitals, while treating the lower-order ones fully. This provides us with a mechanism to relax the T1 , T2 , Rµ,1 , and Rµ,2 amplitudes in the 88 presence of the dominant, higher-order cluster and excitation amplitudes, which is absent in the CR-CC/EOMCC framework, while avoiding the steep computational costs of the full CCSDT/EOMCCSDT and CCSDTQ/EOMCCSDTQ approaches. The active-space methods employed in this dissertation are CCSDt and CCSDTq, where we follow a general recipe how to design the active-space CC methods at any level of trun- cation in the cluster operator laid down in Refs. [99, 101]. Thus, in CCSDt, we approximate the cluster operator T as T ≈ T (CCSDt) = T1 + T2 + t3 , (3.35) where T1 and T2 are the standard one- and two-body components of T , treated fully, and tijK Abc X t3 = Abc EijK (3.36) i E0 , (4.1) τ →∞       0 for S < E0   119 i.e., knowing that the long time limit of such a propagation accompanied by S approaching E0 projects out the desired ground state |Ψ0 ⟩, one propagates the CI (e.g., FCI) state |Ψ(τ )⟩ = c0 (τ ) |Φ0 ⟩ + cK (τ ) |ΦK ⟩ in the Slater determinant space, where the τ -dependent P K CI coefficients satisfy the system of equations [102] ∂cK (τ ) X = −(HKK − S)cK (τ ) − HKL cL (τ ), (4.2) ∂τ L(̸=K) which in the τ → ∞ limit and for S → E0 becomes equivalent to the conventional CI eigenvalue problem X HKL cL (∞) = E0 cK (∞). (4.3) L A direct numerical integration of the system given by Eq. (4.2) would require the determi- nation of the full set of cK (τ ) coefficients at each time step, which is prohibitive. Instead, in the spirit of DMC, in the FCIQMC and truncated CIQMC schemes introduced in Ref. [102], one considers a population of walkers, denoted by α, which can carry positive or negative signs, sα = ±1, and defines the cK (τ ) amplitudes associated with determinants |ΦK ⟩ to be proportional to the signed sums of walkers, i.e., X cK (τ ) ∼ NK = sα δK,Kα , (4.4) α where Kα designates the determinant on which walker α is located. In other words, the walkers inhabit the Slater determinant space, arriving at various determinants with positive or negative signs and evolving according to simple rules that include, in every time step, spawning and diagonal birth or death processes, which reflect on the content of Eq. (4.2), and annihilation. Spawning walkers at different (“child”) determinants is associated with the second term on the right-hand side of Eq. (4.2), which translates into X cK (τ + ∆τ ) = cK (τ ) − ∆τ HKL cL (τ, ) (4.5) L(̸=K) in which ∆τ is a propagation time step. The diagonal birth or death, which is a creation or a destruction of a walker at a given determinant, is driven by the first term on the right 120 hand side of Eq. (4.2), which represent the amplitude change cK (τ + ∆τ ) = [1 − (HKK − S)∆τ ]cK (τ ). (4.6) The annihilation step eliminates pairs of oppositely signed walkers at a given determinant. The pattern of walker growth displays a characteristic plateau once a critical or sufficiently large number of walkers is reached, at which point one begins to stabilize the correlation energy and walker population using suitable energy shifts S [which in the long time limit ap- proach the ground-state energy E0 ; cf. Eq. (4.1)]. Upon convergence, the FCIQMC propaga- tion, where walkers are allowed to explore the entire Hilbet space, produces a FCI-level state and the corresponding energy without any a priori knowledge of the nodal structure of the wavefunction needed in traditional DMC considerations [140, 144, 145, 147, 252, 253], since the population of walkers evolves in the space of Slater determinants, which have the proper fermionic symmetry. Similarly, the truncated CIQMC approximations, such as CISDT-MC, CISDTQ-MC, etc., where spawning walkers at determinants beyond the specified truncation (i.e., determinants with higher than triples in the CISDT-MC case, determinants higher than quadruples in the case of CISDTQ-MC, etc.) is not allowed, converge to the corresponding truncated CI (CISDT, CISDTQ, etc.) states. Several ideas have been explored to improve the original CIQMC methodology and accel- erate its convergence [103, 254–257], including the initiator CIQMC (i-CIQMC) approach, where only those determinants that acquire walker population exceeding a preset value (na ) are allowed to spawn new walkers onto empty determinants [103]. These determinants, called the initiator determinants, are dynamically adjusted, i.e., they remain initiators as long as their walker population exceeds na . One can begin the i-FCIQMC or truncated i-CIQMC simulations using a fixed set of initiator determinants, following, for example, MR ideas, or start from a single determinant, e.g., a RHF state, placing a certain, sufficiently large, number of walkers on it, allowing the corresponding i-CIQMC algorithm to grow the walker population capturing other determinants. The CIQMC ideas can be extended to other many-body schemes [107, 258], including high-level CC (CCSDT, CCSDTQ, etc.) theories, 121 resulting in the corresponding CCMC (CCSDT-MC, CCSDTQ-MC, etc.) methodologies, in which instead of sampling determinants, one samples the space of excitation amplitudes (am- plitudes of “excitors”) by “excips,” whose population dynamics converges to the desired CC solution [107–109]. As in the case of CIQMC, one can use the initiator CCMC (i-CCMC) algorithm, adopted in Ref. [111], to accelerate convergence [109]. Last, but not least, as with other stochastic approaches, it is rather straightforward to parallelize the CIQMC and CCMC techniques through partitioning of the relevant many-fermion Hilbert space across multiple processors [255] and, to reduce the amount of communication among processors, by running independent simulations on different processors and combining statistics gathered in each calculation [257]. Although one may need longer propagation times τ to stabilize walker (CIQMC) or excip (CCMC) populations to achieve the desired wavefunction and energy convergence using purely stochastic means, the most important determinants or cluster amplitude types, which significantly contribute to the wavefunction in the end, are captured already in the early propagation stages, which require small computational effort relative to the target CC calculation. In other words, it may take longer propagation times τ to come up with the reasonably stable numbers of walkers/excips at the individual determinants/excitors, but the leading determinants and cluster amplitude types are identified much sooner. The usefulness of this idea has been demonstrated in Ref. [111], where one could use the information about the leading determinants or excitations captured during the early stages of i-CIQMC or i- CCMC propagations to create lists of determinants defining P spaces for CC(P ) calculations and then use the CC(P;Q) correction δ0 (P ; Q) to capture the remaining correlation effects missing in the P -space CC calculations. In essence, the T operator in the P space is defined (MC) (MC) (MC) (MC) as T (P ) = T1 + T2 + T3 + T4 + · · · , where T3 , T4 , etc., are the cluster operator components defined using the lists of selected triples, quadruples, etc., identified via full or truncated i-CIQMC or i-CCMC runs at a given imaginary time. In the initial study of the semi-stochastic CC(P;Q) methodology [111], we demonstrated 122 that the CIQMC methodology of Refs. [102, 103] is very good in identifying the leading determinants and generating meaningful P spaces for the deterministic CC(P )/EOMCC(P ) calculations already in the early stages of QMC propagations without any a priori knowledge of the states being calculated. We show in this work that the excited-state CC(P;Q) correc- tions δµ (P ; Q), defined by Eq. (3.9), similarly to their µ = 0 ground-state counterparts exam- ined in Ref. [111], are highly effective in accounting for the many-electron correlation effects outside the stochastically determined P spaces. While the specific computations reported in this work, which aim at recovering the EOMCCSDT energetics, rely on the FCIQMC propagations to identify the dominant triply excited determinants for defining the relevant P spaces, the algorithm summarized below is quite general, permitting the use of truncated CIQMC and CCMC approaches and extensions to higher EOMCC levels than EOMCCSDT, such as EOMCCSDTQ (not implemented yet). In the stochastic part of the excited-state CC(P;Q) algorithm proposed in this work, we rely on the initiator CIQMC (i-CIQMC) approach developed in Ref. [103], but we could certainly take advantage of improvements in the original i-CIQMC and i-CCMC algorithms, such as those recently reported in Refs. [104, 105, 259]. It is also worth pointing out that by combining the stochastic CIQMC and deterministic EOMCC ideas via the CC(P;Q) methodology, we can extract highly accurate excited-state information on the basis of relatively short CIQMC propagations for the ground state or the lowest-energy state of a given symmetry, without having to resort to the more complex excited-state CIQMC framework proposed in Refs. [148, 149], although exploring the utility of the latter framework would be an interesting direction to pursue. The key steps of the semi-stochastic CC(P;Q) algorithm proposed in Ref. [113], which builds upon the semi-stochastic CC(P )/EOMCC(P ) framework suggested in Ref. [112] (steps 1–3 below) and which extends the previously developed merger of the ground-state CC(P;Q) methodology with CIQMC or CCMC to excited states, are as follows: 1. Initiate a CIQMC (or CCMC) run for the ground state and, if the system of interest has spin, spatial, or other symmetries, the analogous QMC propagation for the lowest 123 state of each irreducible representation (irrep) to be considered in the CC(P;Q) cal- culations by placing a certain number of walkers (in CCMC, “excips” [108, 109]) on the appropriate reference function(s) |Φ⟩ [e.g. the RHF or restricted open-shell HF (ROHF) determinants]. 2. At some propagation time τ > 0, i.e. after a certain number of CIQMC (or CCMC) time steps, called MC iterations, extract a list or, if states belonging to multiple irreps are targeted, lists of determinants relevant to the desired CC(P;Q) computations from the QMC propagation(s) initiated in step 1 to determine the P space or spaces needed to set up the ground-state CC(P ) and excited-state EOMCC(P ) calculations. If the goal is to converge the CCSDT/EOMCCSDT-level energetics, the P space for the CC(P ) calculations and the EOMCC(P ) calculations for excited states belonging to the same irrep as the ground state is defined as all singly and doubly excited determinants and a subset of triply excited determinants, where each triply excited determinant in the subset is populated by a minimum of nP positive or negative walkers/excips (in this work, nP = 1). For the excited states belonging to other irreps, the P space defining the CC(P ) problem is the same as that used in the case of the ground state, but the lists of triply excited determinants defining the EOMCC(P ) diagonalizations are provided by the CIQMC (or CCMC) propagations for the lowest-energy states of these irreps. One proceeds in a similar way when the goal is to converge other types of high-level CC/EOMCC energetics. For example, if we want to obtain the results of the CCSDTQ/EOMCCSDTQ quality, we also have to extract the lists of quadruples, in addition to the triples, from the CIQMC (or CCMC) runs to define the corresponding P spaces. 3. Solve the CC(P ) and EOMCC(P ) equations in the P space or spaces obtained in the previous step. If we are targeting the CCSDT/EOMCCSDT-level energetics and the excited states of interest belong to the same irrep as the ground state, we define T (P ) = 124 (MC) (MC) (MC) T1 +T2 +T3 , Rµ(P ) = rµ,0 1+Rµ,1 +Rµ,2 +Rµ,3 , and L(P ) µ = δµ0 1+Lµ,1 +Lµ,2 +Lµ,3 , (MC) (MC) (MC) where the list of triples in T3 , Rµ,3 , and Lµ,3 is extracted from the ground-state CIQMC (or CCMC) propagation at time τ . For the excited states belonging to other (P ) irreps, we construct the similarity-transformed Hamiltonian H , to be diagonalized in the EOMCC steps, in the same way as in the ground-state computations, but then use the CIQMC (or CCMC) propagations for the lowest states of these irreps to define the (MC) (MC) lists of triples in Rµ,3 and Lµ,3 . We follow a similar procedure when targeting the (MC) (MC) CCSDTQ/EOMCCSDTQ-level energetics, in which case T (P ) = T1 +T2 +T3 +T4 , (MC) (MC) (MC) (MC) Rµ(P ) = rµ,0 1+Rµ,1 +Rµ,2 +Rµ,3 +Rµ,4 , and L(P ) µ = δµ0 1+Lµ,1 +Lµ,2 +Lµ,3 +Lµ,4 . 4. Correct the CC(P ) and EOMCC(P ) energies for the missing correlations of interest that were not captured by the CIQMC (or CCMC) propagations at the time τ the lists of the P -space excitations were created (the remaining triples if the goal is to recover the CCSDT/EOMCCSDT energetics, the remaining triples and quadruples if one targets CCSDTQ/EOMCCSDTQ, etc.) using the CC(P;Q) corrections δµ (P ; Q) defined by Eq. (3.9). 5. Check the convergence of the resulting Eµ(P +Q) energies calculated using Eq. 3.13 by repeating steps 2–4 at some later CIQMC (or CCMC) propagation time τ ′ > τ . If the Eµ(P +Q) energies do not change within a given convergence threshold, we can stop the calculations. One can also stop them if τ in steps 2–4 is chosen such that the stochastically determined P space(s) contain sufficiently large fraction(s) of higher– than–doubly excited determinants relevant to the target CC/EOMCC level. Our un- published tests using the CC(t;3) corrections to the EOMCCSDt energies, the ground- state semi-stochastic CC(P;Q) calculations reported in Ref. [113], and the excited-state CC(P;Q) calculations using i-FCIQMC to generate the underlying P spaces performed in this work indicate that one should be able to reach millihartree or sub-millihartree accuracies relative to the parent CC/EOMCC computations, when the stochastically 125 determined P spaces contain as little as ∼5–10% and no more than ∼30–40% of higher– than–double excitations of interest, although this may need further study. Similarly to the semi-stochastic form of the ground-state CC(P;Q) methodology intro- duced in Ref. [111], the above algorithm offers significant savings in the computational effort compared to the fully deterministic, high-level, EOMCC approaches it targets. These savings originate from three factors. First, the computational times associated with the early stages of the i-CIQMC or i-CCMC walker/excip propagations are very short compared to the corre- sponding converged runs. Second, the CC(P ) calculations and the subsequent EOMCC(P ) diagonalizations offer significant speedups compared to their CC/EOMCC parents, when the corresponding excitation manifolds contain small fractions of higher–than–doubly ex- cited determinants. For example, as pointed out in Refs. [111, 112], when the most expensive (2) (2) D E D E Φabc ijk [H, T3 ] Φ (or Φabc ijk [H , T3 ] Φ , where H = e−T1 −T2 HeT1 +T2 ) and (P ) D E Φabc ijk [H , Rµ,3 ] Φ terms in the CCSDT and EOMCCSDT equations are isolated and reprogrammed using techniques similar to implementing selected CI approaches, combined with sparse matrix multiplication and index rearrangement routines (rather than conven- tional many-body diagrams that assume continuous excitation manifolds labelled by occupied and unoccupied orbitals from the respective ranges of indices; generally, the stochastically determined lists of excitations do not form continuous manifolds that could be a priori iden- tified), one can speed up their determination by a factor of up to (D/d)2 , where D and d denote the number of all triples and stochastically determined triples in the P space, respec- (P ) D E D E D E tively. Other terms, such as Φabc ijk [H, T2 ] Φ and Φabc ijk [H , Rµ,2 ] Φ or Φab ij [H, T3 ] Φ (P ) D E and Φab ij [H , Rµ,3 ] Φ , when treated in a similar manner, may offer additional speedups, on the order of (D/d)2 , too. Our current CC(P ) and EOMCC(P ) routines are not as efficient D E yet, but the speedups that scale linearly with (D/d) in the most expensive Φabc ijk [H, T3 ] Φ (P ) D E and Φabc ijk [H , Rµ,3 ] Φ contributions are attainable. The third factor contributing to ma- jor savings in the computational effort offered by the semi-stochastic CC(P;Q) approach is the observation that the determination of the noniterative correction δµ (P ; Q) for a given 126 electronic state µ is much less expensive than the time required to complete a single iteration of the target CC/EOMCC calculation (in the case of the calculations aimed at the CCS- DT/EOMCCSDT energetics, the computational time associated with each δµ (P ; Q) scales no worse than ∼ 2n3o n4u , as opposed to the n3o n5u scaling of every CCSDT and EOMCCSDT iteration). There exists another interesting aspect of the semi-stochastic CC(P;Q) algorithm as out- lined above if we examine how the CC(P )/EOMCC(P ) and CC(P;Q) runs behave as τ is varied. For example, if we focus on the CCSDT and EOMCCSDT schemes that are of the main interest in this chapter, at τ = 0, where the P space contains only singly and doubly ex- cited determinants, the CC(P )/EOMCC(P ) run becomes equivalent to CCSD/EOMCCSD, whereas the τ = 0 CC(P;Q) step corresponds to CR-CC(2,3) and CR-EOMCC(2,3). The target CCSDT/EOMCCSDT method is equivalent to CC(P )/EOMCC(P ) at τ = ∞, where all triply excited determinants have been captured by the QMC propagation and included in the P space. Consequently, CCSDT/EOMCCSDT are also equivalent to the τ = ∞ CC(P;Q) runs, because at this point the Q space is empty and the δµ (P ; Q) correction be- comes zero by definition. Thus, the variable τ serves to connect the τ = 0 and τ = ∞ limits of our CC(P )/EOMCC(P ) and CC(P;Q) computations, which is useful for checking the correctness of our calculations. 4.2 Application: Electronic Excitation Spectra of CH+ , CH, and CNC In order to assess the performance of the semi-stochastic CC(P;Q) approach to excited described above and examine, in particular, the ability of the noniterative δµ (P ; Q) correc- tions to accelerate the convergence of the CIQMC-driven EOMCC(P ) calculations toward the desired EOMCC energetics, represented here by EOMCCSDT, we carried out bench- mark calculations for the frequently studied vertical excitations in the CH+ molecule at the equilibrium [Table 4.1 and Fig. 4.1(a) and (b)] and stretched [Table 4.2 and Fig. 4.1(c) 127 and (d)] geometries and the adiabatic excitations in the challenging open-shell CH (Table 4.3) and CNC (Table 4.4) systems, which have low-lying excited states dominated by two- electron transitions that require the EOMCCSDT theory level to obtain a reliable description [56, 86, 93, 101, 260–263]. The CH+ ion was described by the [5s3p1d/3s1p] basis set of Ref. [264] and we used the aug-cc-pVDZ [265, 266] and DZP[4s2p1d] [267, 268] bases for the CH and CNC species, respectively. Following Refs. [111, 113] (cf., also, Ref. [269]), we used the HANDE software package [270, 271] to execute the stochastic i-FCIQMC runs, needed to generate the lists of triply excited determinants included in the CC(P ) and EOMCC(P ) calculations. Our standalone CC/EOMCC codes, interfaced with the RHF, ROHF, and integral routines in the GAMESS package, were used to carry out the required CC(P ), EOMCC(P ), CC(P;Q), and fully deterministic (CCSD/EOMCCSD and CCSDT/EOM- CCSDT) computations [the Q spaces used to construct the CC(P;Q) corrections to the CC(P ) and EOMCC(P ) energies consisted of the triples not captured by the i-FCIQMC runs at the corresponding propagation times τ ]. It should be noted that the CC(P ) and EOMCC(P ) energies at τ = 0 CC(P;Q) corrections are equivalent to those of CR-CC(2,3) (the ground state) and CR-EOMCC(2,3) (excited states). It should also be noted that the CC(P ) and EOMCC(P ) energies at τ = ∞ are identical to the energies obtained in the full CCSDT and EOMCCSDT calculations. The semi-stochastic CC(P;Q) calculations recover the CCSDT and EOMCCSDT energetics in this limit, too, although the τ = ∞ values of the δµ (P ; Q) corrections are zero in this case, since the τ = ∞ P spaces contain all the triples, i.e., the corresponding Q-space triples lists are empty. These relationships between the semi-stochastic CC(P ), EOMCC(P ), and CC(P;Q) approaches and the fully deterministic CCSD/EOMCCSD, CR-CC(2,3)/CR-EOMCC(2,3), and CCSDT/EOMCCSDT methodolo- gies were helpful in examining the correctness of our codes. They also point to the ability of the CC(P ), EOMCC(P ), and CC(P;Q) calculations driven by the information extracted from CIQMC to offer a systematically improvable description as τ approaches ∞. Each i-FCIQMC run was initiated by placing 1500 walkers on the relevant reference function (see 128 Tables 4.1–4.4 for the details) and we set the initiator parameter na at 3. All of the i- FCIQMC propagations used the time step τ of 0.0001 a.u. In the post-ROHF computations for the CH and CNC species, the core electrons corresponding to the 1s shells of the carbon and nitrogen atoms were kept frozen. In the case of CH+ , we correlated all electrons. We begin our discussion of the numerical results with the CH+ ion, where we investigated the three lowest excited states of the 1 Σ+ symmetry (labeled as 2 1 Σ+ , 3 1 Σ+ , and 4 1 Σ+ ; the ground state is designated as 1 1 Σ+ ), two lowest states of the 1 Π+ symmetry (1 1 Π+ and 2 1 Π+ ), and two lowest 1 ∆+ states (1 1 ∆+ and 1 1 ∆+ ). Two C–H internuclear separations were considered, the equilibrium distance R = Re = 2.13713 bohr [Table 4.1 and Fig. 4.1(a) and (b)] and the stretched R = 2Re geometry [Table 4.2 and Fig. 4.1(c) and (d)]. Following the semi-stochastic CC(P;Q) algorithm, as described above, and our interest in converging the CCSDT/EOMCCSDT energetics, the cluster and right and left EOM operators used (MC) in the calculations for the states were approximated by T (P ) = T1 + T2 + T3 , Rµ(P ) = (MC) (MC) rµ,0 1 + Rµ,1 + Rµ,2 + Rµ,3 , and L(P µ ) = δµ0 1 + Lµ,1 + Lµ,2 + Lµ,3 , respectively, where (MC) (MC) (MC) the list of triples defining the three-body components T3 , Rµ,3 , and Lµ,3 at a given time τ was obtained from the ground-state i-FCIQMC propagation at the same value of (MC) τ . The T3 component of T (P ) used in the CC(P;Q) computations of the 1 Π and 1 ∆ (P ) states, needed to determine the similarity-transformed Hamiltonian H to be diagonalized in the subsequent EOMCC steps, was defined in the same way as in the case of the states, (MC) (MC) but the lists of triples entering the Rµ,3 and component of Rµ(P ) and the Lµ,3 component of L(Pµ ) were obtained differently. They were extracted from the i-FCIQMC runs for the lowest states within the irreps of C2v relevant to the symmetries of interest, meaning the 1 B1 (C2v ) component of 1 1 Π for the 1 Π states and the 1 A2 (C2v ) component of 1 1 ∆ for the 1 ∆ states (C2v is the largest Abelian subgroup of the true point group of CH+ , C∞v ; our codes cannot handle non-Abelian symmetries). As implied by Eq. (3.9), the corrections to the CC(P ) and EOMCC(P ) energies at a given time τ were computed using the Mµ,K (P ) and ℓµ,K (P ) amplitudes corresponding to the triply excited determinants |ΦK ⟩ not captured 129 by i-FCIQMC at the same τ . As pointed out in Refs. [47, 48, 112], the 2 1 Σ+ , 2 1 Π, 1 1 ∆, and 2 1 ∆ states of CH+ at R = Re and all of the excited states of the stretched CH+ /R = 2Re system, which we calculated in this work, are characterized by substantial MR correlations that originate from large two-electron excitation contributions (the 2 1 ∆ state at R = 2Re also has significant triple excitations [47, 48, 112]). It is, therefore, not surprising that the basic EOMCCSD level, equivalent to the EOMCC(P ) calculations at τ = 0, performs poorly for all of these states, producing very large errors relative to EOMCCSDT that are about 12, 20, and 34–35 millihartree for the 2 1 Σ+ , 2 1 Π, and both 1 ∆ states, respectively, at R = Re and ∼14–144 millihartree when the excited states at R = 2Re are considered (see Tables 4.1 and 4.2). The EOMCCSD energies for the 3 1 Σ+ , 4 1 Σ+ , and 1 1 Π, states at the equilibrium geometry, which are dominated by one-electron transitions, are more accurate, but errors on the order of 3–6 millihartree still remain. As shown in Tables 4.1 and 4.2, the CR-EOMCC(2,3) triples correction to EOMCCSD, equivalent to the CC(P;Q) calculations at τ = 0, offers substantial improvements, as exemplified by the small errors, on the order of 1–3 millihartree, for the majority of excited states of CH+ considered in this article, but there are cases, especially the and states at R = 2Re , where the differences between the CR-EOMCC(2,3) and parent EOMCCSDT energies, which are about 13 millihartree in the former case and more than 63 millihartree in the case of the latter state, remain very large. This is related to the substantial coupling of the one- and two-body components of the cluster and EOM excitation and de-excitation operators with their three-body counterparts, which the CR-EOMCC(2,3) corrections to EOMCCSD neglect. Our older active-space EOMCCSDt calculations for CH+ reported in Refs. [47, 48] and the more recent semi-stochastic EOMCC(P ) calculations for the same system described in Ref. [112] are telling us that the incorporation of the leading triples in the relevant P spaces, which allows the one- and two-body components of T , Rµ , and Lµ to relax in the presence of their three-body counterparts, is the key to improve the results of the CR-EOMCC(2,3) calculations. 130 This is exactly what we observe in Tables 4.1 and 4.2 and Fig. 4.1. In agreement with Ref. [112], by enriching the P spaces used in the CC(P ) and EOMCC(P ) computations with the subsets of triples captured during i-FCIQMC propagations, the results greatly improve, allowing us to reach the millihartree or sub-millihartree accuracy levels for all the calculated excited states of CH+ at both nuclear geometries considered in this work when the stochastically determined P spaces contain about 20–30% of all triples. The CC(P;Q) corrections to the EOMCC(P ) energies based on Eq. (3.9) accelerate the convergence toward EOMCCSDT even further. As shown in Tables 4.1 and 4.2 and Fig. 4.1, these corrections are so effective that we reach the millihartree or sub-millihartree accuracy levels relative to the parent EOMCCSDT energetics almost instantaneously, i.e., out of the early stages of the i-FCIQMC propagations, when no more than 5–10% of all triples are included in the relevant P spaces. This is true even when the highly complex 4 1 Σ+ and 2 1 ∆ states at R = 2Re, for which the EOMCCSD calculations produce the massive, ∼33 and ∼144 millihartree, errors, which remain large (about 13 and 63 millihartree, respectively) at the CR-EOMCC(2,3) level. As shown in Table 4.2, the CC(P;Q) corrections to the EOMCC(P ) energies, which account for the missing triples that the i-FCIQMC propagations at a given time τ did not capture, allow us to reach the sub-millihartree accuracy levels with less than 5% (the 2 1 ∆ state) or ∼10% (the 4 1 Σ+ state) of triples in the relevant P spaces. The uncorrected EOMCC(P ) calculations display the relatively fast convergence toward EOMCCSDT as well, but they reach similar accuracies at later propagation times τ , when about 15% (the 2 1 ∆ state) or 25% (the 4 1 Σ+ state) of triples are captured by i-FCIQMC. Obviously, the details of the rate of convergence of the semi-stochastic CC(P;Q) calculations toward EOMCCSDT, especially when one wants to tighten it, depend on the specific excited state being calculated, but, as shown in Tables 4.1 and 4.2, once about 20% of triples are captured by the i-FCIQMC propagations, we recover the EOMCCSDT energetics for all the calculated excited states of CH+ at both geometries examined in this study to within 0.1 millihartree or better. Interestingly, there is a great deal of consistency between the behaviour of the uncorrected 131 semi-stochastic EOMCC(P ) approach, in which the lists of triples defining the relevant P spaces are extracted from i-FCIQMC propagations, and the fully deterministic EOMCCSDt calculations for CH+ reported in Refs. [47, 48], in which the leading triples were identified using active orbitals. Indeed, once the stochastically determined P spaces extracted from i- FCIQMC capture about 20–30% of all triples, which in the case of the CH+ system examined here is achieved after 50,000 or fewer ∆τ = 0.0001 a.u. MC iterations, the energies resulting from the EOMCC(P ) computations become very similar to those obtained with the EOM- CCSDt method using the active space that consists of the highest-energy occupied (3σ) and three lowest-energy unoccupied (1πx , 1πy , and 4σ) orbitals. Following the definitions of the ‘little t’ T3 and Rµ,3 operators adopted in EOMCCSDt, for the state symmetries considered in this work, the active space consisting of the 3σ, 1πx , 1πy , and 4σ valence orbitals amounts to about 26–29% of all triples included in the respective EOMCC diagonalization spaces [47, 48]. This suggests that the types and values of the triply excited amplitudes defining the components of the EOM operators , which characterize the EOMCCSDt computations reported in Refs. [47, 48], and those that define the components obtained in the i-FCIQMC- driven EOMCC(P ) calculations performed after 50,000 MC iterations using ∆τ = 0.0001 a.u. should be similar too. This is illustrated in Fig. 4.2, where we compare the distributions (MC) of the differences between the Rµ,3 amplitudes and their full EOMCCSDT counterparts resulting from the EOMCC(P ) computations at 4000 [Fig. 4.2(a)], 10,000 [Fig. 4.2(b)], and 50,000 [Fig. 4.2(c)] MC iterations for the 2 1 Σ+ state of CH+ at R = 2Re with the anal- ogous distribution characterizing the amplitudes obtained with the EOMCCSDt approach using the 3σ, 1πx , 1πy , and 4σ active orbitals to define the corresponding triples space [Fig. 4.2(d); all EOM vectors Rµ needed to construct Fig. 4.2, corresponding to the EOMCC(P ), EOMCCSDt, and EOMCCSDT calculations, were normalized to unity]. As shown in Fig. (P ) 4.2 [cf. Fig. 4.2(c) and Fig. 4.2(d)], the small differences between the Rµ,3 amplitudes result- ing from the EOMCC(P ) calculations performed after 50,000 MC iterations and the Rµ,3 amplitudes obtained with EOMCCSDT, including their numerical values and distribution, 132 closely resemble those characterizing the active-space EOMCCSDt computations reported in Refs. [47, 48]. This is in perfect agreement with the small errors relative to EOMCCSDT characterizing the two calculations, which are 0.302 millihartree in the former case (cf. Table 4.2) and 0.576 millihartree in the case of EOMCCSDt [47, 48]. When we start using con- siderably smaller fractions of triples and, as a consequence, significantly smaller P spaces in the EOMCC(P ) calculations, which is what happens when the underlying i-FCIQMC prop- agation is terminated too soon, the differences between the amplitudes resulting from the EOMCC(P ) calculations and their EOMCCSDT counterparts, including their values and distribution, and the errors in the EOMCC(P ) energies relative to EOMCCSDT increase. This can be seen in Fig. 4.2, especially when one compares panel (a), which corresponds to the EOMCC(P ) calculations performed after 4000 MC iterations that use only 7% of triples, with panel (d) representing EOMCCSDt, which uses a much larger fraction of triple excitations (∼30%), and in Table 4.2, where the error in the EOMCC(P ) energy of the state of CH+ at R = 2Re relative to EOMCCSDT obtained after 4000 MC iterations, of 4.263 millihartree, is ∼14 times larger than the analogous error obtained after 50,000 MC steps. The above analysis, which could be repeated for the remaining states of CH+ , reach- ing similar conclusions, has several interesting implications for the semi-stochastic CC(P;Q) methodology pursued in this study, which will be examined by us in the future. It sug- gests, for example, that the CC(P )/EOMCC(P ) and CC(P;Q) approaches using CIQMC propagations to determine the lists of higher–than–double excitations in the correspond- ing P spaces can be regarded as natural alternatives to the fully deterministic active-space EOMCC methods, such as EOMCCSDt, and their CC(P;Q)-corrected counterparts, such as CC(t;3), whose performance in excited-state calculations will be reported by us in a separate study. It also suggests that the fractions of higher–than–double excitations used to define the stochastically determined P spaces, needed to achieve high accuracies observed in the semi-stochastic CC(P;Q) calculations discussed in this work, should decrease with the basis set. We have already observed this in the previous ground-state work [111], and we anticipate 133 that the same will remain true in the CIQMC-driven excited-state CC(P;Q) calculations. While this remark requires a separate thorough study, beyond the scope of this initial work on the excited-state CC(P;Q), we can rationalize it by referring to the analogies between the semi-stochastic CC(P )/EOMCC(P ) and CC(P;Q) approaches and their deterministic CCSDt/EOMCCSDt and CC(t;3) counterparts. Indeed, the aforementioned (D/d) ratio that controls the speedups offered by the CC(P )/EOMCC(P ) and CC(P;Q) calculations becomes (no /No )(nu /Nu ) when the active-space CCSDt/EOMCCSDt and CC(t;3) calcula- tions, based on the ideas laid down in Refs. [47, 48, 74, 75, 81, 101], are considered, where No and Nu are the numbers of active occupied and active unoccupied orbitals, respectively, which either do not grow with the basis set or grow with it very slowly compared to no and nu . Finally, before moving to the next molecular example, we would like to point out that, in analogy to the CC(P;Q)-based CC(t;3), CC(t,q;3), and CC(t,q;3,4) calculations using active orbitals to define the underlying P spaces (see, e.g., Ref. [77]), one is better off by using smaller P spaces in the semi-stochastic CC(P )/EOMCC(P ) considerations, which can be extracted out of the early stages of CIQMC propagations, and capturing the remain- ing correlations using noniterative CC(P;Q) corrections, than by running long-time CIQMC simulations to generate larger P spaces for the uncorrected CC(P )/EOMCC(P ) calcula- tions. This can be seen in Tables 4.1 and 4.2 for CH+ and in the remaining Tables 4.3 and 4.4 discussed in the next two subsections. We illustrate this remark by inspecting the EOMCC(P ) and CC(P;Q) calculations for the state of CH+ . As shown in Table 4.1, one needs to capture about 50% of triples in the P space to reach a 0.1 millihartree accuracy relative to EOMCCSDT at R = Re using the uncorrected EOMCC(P ) approach. When the CC(P;Q) correction is employed, only 15% of triples are needed to reach the same accuracy level. At the more challenging R = 2Re geometry (Table 4.2), one reaches a ∼0.1 milli- hartree accuracy level with about 40% of triples in the P space when using the uncorrected EOMCC(P ) approach. This fraction reduces to about 20%, without any accuracy loss, when 134 the CC(P;Q) correction is added to the EOMCC(P ) energy. Based on the information pro- vided in above, running the EOMCC(P ) calculations with a smaller fraction of triples in the P space offers much larger savings in the computational effort than the additional time spent on determining the CC(P;Q) correction, which is, as pointed out above, considerably less expensive than a single EOMCCSDT iteration. For example, in the pilot implementation of the excited-state EOMCC(P ) and CC(P;Q) approaches aimed at recovering EOMCCSDT energetics, employed in this work, the uncorrected EOMCC(P ) run using 50% of triples in the P space, needed to reach a ∼0.1 millihartree accuracy relative to EOMCCSDT for the state of CH+ at R = Re , is about twice as fast as the corresponding EOMCCSDT calcu- lation. The EOMCC(P ) diagonalization that forms part of the analogous CC(P;Q) run, which needs only 15% of triples in the P space to reach the same accuracy level, is about 6 times faster than EOMCCSDT. The noniterative CC(P;Q) correction is so inexpensive here that one can largely ignore the computational costs associated with its determination in this context [cf. Ref. [73] for the analogous comments made in the context of comparing costs of the CCSDt computations with those of CC(t;3)]. Similar convergence patterns in the semi-stochastic EOMCC(P ) and CC(P;Q) calcula- tions are observed for the CH radical (see Table 4.3). In this case, following the earlier deterministic EOMCC work, including the CR-EOMCC [86, 93] and electron-attachment (EA) EOMCC [93, 260, 261] approaches, and a wide range of EOMCC computations, in- cluding the high EOMCCSDT and EOMCCSDTQ levels, published by Hirata [56], along with the X 2 Π ground state, we examined the three low-lying doublet excited states, des- ignated as A 2 ∆, B 2 Σ− , and C 2 Σ+ , which belong to different irreducible representations than that of the ground state. In analogy to the aforementioned EOMCC studies of CH [56, 86, 93, 260, 261], the relevant CC(P ) (the X 2 Π state) and EOMCC(P ) (excited states) electronic energies and their CC(P;Q) counterparts were determined at the corresponding experimentally derived equilibrium C–H distances, which are 1.1197868 Å for the X 2 Π state [272], 1.1031 Å for the A 2 ∆ state [272], 1.1640 Å for the B 2 Σ− state [273], and 1.1143 Å 135 for the C 2 Σ+ state [274] (cf. Table 4.3). Since all of our CC(P )/EOMCC(P ) and CC(P;Q) calculations, starting from the τ = 0 CCSD/EOMCCSD and CR-EOMCC(2,3) levels and ending up with the larger values of τ needed to examine the convergence toward the parent CCSDT/EOMCCSDT energetics, were performed using the ROHF reference determinant, we also computed the ROHF-based CCSDT and EOMCCSDT energies, which formally cor- respond to the τ = ∞ CC(P )/EOMCC(P ) and CC(P;Q) results. We had to do it, since the previously published CCSDT/EOMCCSDT results [56] relied on the UHF rather than (MC) the ROHF reference. In analogy to CH+ , the lists of triples defining the T3 component of (MC) (MC) the cluster operator T (P ) and the Rµ,3 and Lµ,3 components of the EOM excitation and de-excitation operators, Rµ(P ) and L(P ) µ , respectively, used in the CC(P ), EOMCC(P ), and CC(P;Q) calculations for the CH radical, were extracted from the i-FCIQMC propagations for the lowest-energy states of the relevant irreps of C2v , namely, the 2 B2 (C2v ) component of the X 2 Π state, the lowest state of the 2 A1 (C2v ) symmetry in the case of the A 2 ∆ and C 2 Σ+ states, and the lowest 2 A2 (C2v ) state when considering the B 2 Σ− state (again, we used C2v , which is the largest Abelian subgroup of the true point group of CH, C∞v ). As explained in Refs. [86, 93, 260, 261] and as shown in Ref. [56], all three excited states of the CH radical considered here, especially B 2 Σ− and C 2 Σ+ , which are dominated by two-electron excitations (cf. the reduced excitation level (REL) diagnostic values in Tables II and III of Ref. [93] or Table II of Ref. [86]), constitute a significant challenge, requiring the full EOMCCSDT treatment to obtain a reliable adiabatic excitation spectrum. This can be seen by inspecting the τ = 0 EOMCC(P ), i.e., EOMCCSD, energies for the A 2 ∆, B 2 Σ− , and C 2 Σ+ states of CH shown in Table 4.3, which are characterized by the ∼13, ∼39, and ∼44 millihartree errors relative to EOMCCSDT, respectively. The CR-EOMCC(2,3) triples corrections to EOMCCSD, represented in Table 4.3 by the τ = 0 CC(P;Q) values, help, especially in the case of the C 2 Σ+ state, but the situation is far from ideal, since errors on the order of 8 and 5 millihartree for the A 2 ∆ and B 2 Σ− states, respectively, remain. The situation considerably improves when we turn to the semi-stochastic CC(P;Q) 136 calculations, which incorporate the leading triples in the relevant P spaces by extracting them from the corresponding i-FCIQMC propagations and correct the resulting energies for the remaining triple excitations that were not captured by i-FCIQMC at a given time τ . As shown in Table 4.3, in the case of the A 2 ∆ and B 2 Σ− states, which are not only challenging to EOMCCSD, but also to CR-EOMCC(2,3), we can reach comfortable 1–2 millihartree errors relative to EOMCCSDT using the semi-stochastic CC(P;Q) corrections developed in this work once the relevant P spaces contain about 20–40% of all triples. With ∼50% triples in the same P spaces, the CC(P;Q) energies of the A 2 ∆ and B 2 Σ− states are within fractions of a millihartree from EOMCCSDT. These are considerable improvements relative to the purely deterministic EOMCCSD and CR-EOMCC(2,3) computations, which give ∼13–39 and ∼5–8 millihartree errors, respectively, for the same two states, and the semi- stochastic EOMCC(P ) calculations that reach 1–2 millihartree accuracy levels with about 70–80% triples in the respective P spaces. In the case of the C 2 Σ+ state, which is a major challenge to EOMCCSD, but not to CR-EOMCC(2,3), the behaviour of the EOMCC(P ) and CC(P;Q) approaches is different, since the CC(P;Q) corrections obtained with the help of some triples in the P space captured by i-FCIQMC are no longer needed to obtain the well-converged energetics, i.e., the τ = 0 CC(P;Q) result, where the P space is spanned by singles and doubles only, is sufficiently accurate, but it is still interesting to observe that one can tighten the convergence further, reaching stable <0.1 millihartree errors relative to EOMCCSDT with about 50% of all triples in the P space. In analogy to the A 2 ∆ and B 2 Σ− states, it is also interesting to observe a reasonably smooth convergence of the uncorrected EOMCC(P ) energies toward EOMCCSDT. It is clear from the results presented in Table 4.3 that the CC(P;Q) corrections to the semi-stochastic CC(P ) and EOMCC(P ) energies offer considerable speedups compared to the uncorrected CC(P )/EOMCC(P ) calculations, not only for the closed-shell molecules, such as CH+ , but also when examining open-shell species. Our last example, which is also the largest many-electron system considered in the present 137 study, is the linear, D∞h symmetric, CNC molecule. Following the earlier CR-CC(2,3)/CR- EOMCC(2,3) and electron-attached (EA) EOMCC calculations for this challenging open- shell molecular species [93, 262, 263], we considered the X 2 Πg ground state and two low-lying doublet excited states, A 2 ∆u and B 2 Σ+ u . The i-FCIQMC-driven CC(P ) ground-state and EOMCC(P ) excited-state energies and the corresponding CC(P;Q) corrections, along with their deterministic EOMCCSD, CR-EOMCC(2,3), and EOMCCSDT counterparts, were cal- culated using the equilibrium C–N distances optimized in Ref. [262] with EA-SAC-CI. They are 1.253 Å for the X 2 Πg state, 1.256 Å for the A 2 ∆u state, and 1.259 Å for the B 2 Σ+ u state. As in the case of the CH radical, we used the ROHF reference determinant. Fol- lowing the computational protocol adopted in this study, and in analogy to the CH+ and (MC) (MC) (MC) CH species, the lists of triples defining the T3 , Rµ,3 , and Lµ,3 components used in the semi-stochastic CC(P ), EOMCC(P ), and CC(P;Q) calculations for CNC were obtained using the i-FCIQMC propagations for the lowest-energy states of the relevant irreps of the largest Abelian subgroup of D∞h , i.e., D2h , meaning the 2 B2g (D2h ) component of the X 2 Πg state and the lowest state of the 2 B1u (D2h ) symmetry in the case of the A 2 ∆u and B 2 Σ+ u states. As shown in Table 4.4 and in agreement with one of our previous studies [93], all three states of CNC considered in this work, especially A 2 ∆u and B 2 Σ+ u , are poorly described by CCSD and EOMCCSD, which produce more than 18, 31, and 111 millihartree errors, respectively, relative to the target EOMCCSDT energetics [see the τ = 0 CC(P ) and EOMCC(P ) energies in Table 4.4]. The excessively large, >111 millihartree, error in the EOMCCSD energy of the B 2 Σ+ u state is related to its strongly MR character dominated by two-electron excitations (cf. the REL values characterizing the excited states of CNC in Table IV of Ref. [93]). In the case of the ground state and the B 2 Σ+ u excited state, the CR-CC(2,3) and CR-EOMCC(2,3) corrections to CCSD and EOMCCSD seem to be quite effective, reducing the large errors relative to CCSDT/EOMCCSDT observed in the CCSD and EOMCCSD calculations to a sub-millihartree level, but the ∼16 millihartree error 138 resulting from the CR-EOMCC(2,3) calculations for the A2 ∆u state, while considerably lower than the >31 millihartree error obtained with EOMCCSD, is still rather large (see the τ = 0 CC(P;Q) energies in Table 4.4). By incorporating the dominant triply excited determinants captured by the i-FCIQMC propagations in the respective P spaces, the semi-stochastic CC(P ) and EOMCC(P ) approaches help, allowing us to reach stable ∼1–2 millihartree accuracy levels for the and states relative to the target CCSDT/EOMCCSDT energetics with about 50–60% triples, but the CC(P;Q) corrections that account for the remaining triples, missing in the i-FCIQMC wave functions, are considerably more effective. In the case of the state, which poses problems to both EOMCCSD and CR-EOMCC(2,3), which give about 31 and 16 millihartree errors relative to EOMCCSDT, respectively, we reach a stable ∼1–2 millihartree accuracy level with about 30–40% triples in the corresponding P space, as opposed to the aforementioned 50–60% needed in the uncorrected EOMCC(P ) run. The benefits of using the semi-stochastic CC(P;Q) versus deterministic CR-EOMCC(2,3) corrections for the X 2 Πg and A 2 ∆u states are less obvious, but it is encouraging to observe the rapid convergence toward the target CCSDT and EOMCCSDT energetics in the former calculations. In particular, they allow us to lower the 0.4–0.5 millihartree errors obtained with CR-EOMCC(2,3) to a 0.1 millihartree level with about 10% of all triples, identified by i-FCIQMC, in the case of the state and with ∼30–40% triples in the P space when the state is considered. Once again, the CC(P;Q) corrections to the energies resulting from the semi-stochastic CC(P ) and EOMCC(P ) calculations speed up the uncorrected CC(P )/EOMCC(P ) computations, while allowing us to improve the CR-CC(2,3) and CR- EOMCC(2,3) energetics by bringing them very close to the CCSDT and EOMCCSDT levels at the fraction of the cost. 139 Table 4.1: Convergence of the CC(P )/EOMCC(P ) and CC(P;Q) energies toward CCSDT/EOMCCSDT for CH+ , calculated using the [5s3p1d/3s1p] basis set of Ref. [264], at the C–H internuclear distance R = Re = 2.13713 bohr. The P spaces used in the CC(P ) and EOMCC(P ) calculations were defined as all singles, all doubles, and subsets of triples extracted from i-FCIQMC propagations for the lowest states of the relevant symmetries. Each i-FCIQMC run was initiated by placing 1500 walkers on 1 the appropriate reference function [the RHF determinant for the 1 Σ+ g states, the 3σ → 1π state of the B1 (C2v ) symmetry for the 1 Π states, and the 3σ 2 → 1π 2 state of the 1 A2 (C2v ) symmetry for the 1 ∆ states], setting the initiator parameter na at 3, and the time step ∆τ at 0.0001 a.u. The Q spaces used in constructing the CC(P;Q) corrections consisted of the triples not captured by i-FCIQMC. Adapted from Ref. [113]. 1 1 Σ+ g 2 1 Σ+ g 3 1 Σ+g 4 1 Σ+g 1 1Π 2 1Π 1 1∆ 2 1∆ MC iter. (103 ) a b c a b a b a b a b c a b a b c a P (P ;Q) %T P (P ;Q) P (P ;Q) P (P ;Q) P (P ;Q) %T P (P ;Q) P (P ;Q) %T P (P ;Q)b d 0 1.845 0.063 0 19.694 1.373 3.856 0.787 5.537 0.954 3.080 0.792 0 11.656 2.805 34.304 −0.499 0 34.685 0.350 2 1.071 0.024 7 11.004 0.909 3.248 0.587 4.826 −4.469 0.772 0.179 13 3.746 0.530 1.492 0.151 10 5.951 0.432 4 0.423 0.009 15 5.474 0.090 1.893 0.047 1.980 0.100 0.513 0.102 20 1.852 0.128 0.525 0.051 16 2.542 0.128 6 0.249 0.003 20 4.712 0.111 1.268 0.046 1.077 0.068 0.213 0.054 25 0.957 0.073 0.471 0.028 18 1.892 0.094 8 0.181 0.003 23 1.371 0.112 0.643 0.067 0.702 0.075 0.170 0.058 27 0.743 0.060 0.240 0.021 22 0.940 0.057 10 0.172 0.004 24 1.572 0.061 0.295 0.044 0.385 0.026 0.118 0.046 29 0.411 0.047 0.198 0.017 24 0.877 0.041 50 0.077 0.001 37 0.755 0.026 0.139 0.037 0.208 0.032 0.053 0.027 43 0.157 0.027 0.039 0.008 42 0.133 0.011 100 0.044 0.000 48 0.277 0.009 0.007 0.013 0.155 0.017 0.021 0.013 57 0.063 0.012 0.014 0.005 56 0.043 0.005 150 0.015 0.000 59 0.085 0.005 0.058 0.006 0.041 0.007 0.008 0.005 71 0.020 0.004 0.004 0.002 71 0.008 0.003 200 0.006 0.000 69 0.024 0.002 0.014 0.002 0.002 0.003 0.004 0.003 82 0.008 −0.001 0.003 0.002 82 0.003 0.002 ∞e −38.019516 −37.702621 −37.522457 −37.386872 −37.900921 −37.498143 −37.762113 −37.402308 a Errors in the CC(P ) (the 1 1 Σ+ g ground state) and EOMCC(P ) (excited states) energies relative to the corresponding CCSDT and EOMCCSDT data, in millihartree. b Errors in the CC(P;Q) energies relative to the corresponding CCSDT and EOMCCSDT data, in millihartree. c The %T values are the percentages of triples captured during the i-FCIQMC propagations for the lowest state of a given symmetry [the 1 1 Σ+ g = 1 1 A1 (C2v ) ground state for the 1 Σ+ 1 1 1 1 g states, the B1 (C2v ) component of the 1 Π state for the Π states, and the A2 (C2v ) component of the 1 ∆ 1 1 state for the ∆ states]. d The CC(P ) and EOMCC(P ) energies at τ = 0.0 a.u. are identical to the energies obtained in the CCSD and EOMCCSD calculations. The τ = 0.0 a.u. CC(P;Q) energies are equivalent to the CR-CC(2,3) (the ground state) and the CR-EOMCC(2,3) (excited states) energies. e The CC(P ) and EOMCC(P ) energies at τ = ∞ a.u. are identical to the energies obtained in the CCSDT and EOMCCSDT calculations. 140 Table 4.2: Same as Table 4.1 for the stretched C–H internuclear distance R = 2Re = 4.27426 bohr. Adapted from Ref. [113]. 1 1 Σ+ g 2 1 Σ+ g 3 1 Σ+ g 4 1 Σ+ g 1 1Π 2 1Π 1 1∆ 2 1∆ MC iter. (103 ) P (P ;Q) %T P (P ;Q) P (P ;Q) P (P ;Q) P (P ;Q) %T P (P ;Q) P (P ;Q) %T P (P ;Q) 0 5.002 0.012 0 17.140 1.646 19.929 −2.871 32.639 12.657 13.552 2.303 0 21.200 −1.429 44.495 −4.526 0 144.414 −63.405 2 1.588 0.031 3 5.209 0.478 12.524 −2.079 33.400 14.297 1.398 0.306 7 1.644 −0.060 1.372 0.046 6 13.363 0.368 4 0.504 0.015 7 4.263 −1.741 6.383 −0.760 12.671 2.178 0.712 0.058 12 0.724 0.050 0.451 0.014 9 3.338 0.130 6 0.275 0.002 11 1.405 0.047 1.352 0.051 5.870 0.593 0.409 0.033 14 0.612 0.031 0.422 0.022 12 2.340 0.063 8 0.263 0.004 12 1.543 0.065 1.173 0.020 4.406 0.699 0.436 0.050 16 0.457 −0.002 0.253 0.007 13 2.088 0.021 10 0.148 0.003 14 0.792 0.094 0.613 0.047 2.331 0.342 0.227 0.039 17 0.220 0.014 0.122 −0.001 14 0.862 0.038 50 0.030 0.000 26 0.302 0.002 0.339 0.007 0.457 0.013 0.061 0.007 30 0.079 0.060 0.047 0.005 26 0.288 0.005 100 0.009 0.000 39 0.103 0.003 0.119 0.006 0.110 0.011 0.013 0.002 41 0.016 0.004 0.013 0.004 36 0.038 0.000 150 0.004 0.000 52 0.031 0.000 0.035 0.003 0.076 0.006 0.005 0.002 52 0.007 0.002 0.005 0.001 47 0.014 0.000 200 0.001 0.000 63 0.024 0.000 0.019 0.000 −0.006 0.001 0.002 0.001 65 0.001 0.000 0.001 0.000 57 0.003 0.000 ∞ −37.900394 −37.704834 −37.650242 −37.495275 −37.879532 −37.702345 −37.714180 −37.494031 141 Table 4.3: Convergence of the CC(P )/EOMCC(P ) and CC(P;Q) energies toward CCS- DT/EOMCCSDT for CH, calculated using the aug-cc-pVDZ basis set. The P spaces used in the CC(P ) and EOMCC(P ) calculations were defined as all singles, all doubles, and sub- sets of triples extracted from i-FCIQMC propagations for the lowest states of the relevant symmetries. Each i-FCIQMC run was initiated by placing 1500 walkers on the appropriate reference function [the ROHF 2 B2 (C2v ) determinant for the X 2 Π state, the 1π → 4σ state of the 2 A1 (C2v ) symmetry for the A 2 ∆ and C 2 Σ+ states, and the 3σ → 1π state of the 2 A2 (C2v ) symmetry for the B 2 Σ− state], setting the initiator parameter na at 3, and the time step ∆τ at 0.0001 a.u. The Q spaces used in constructing the CC(P;Q) corrections consisted of the triples not captured by i-FCIQMC. Adapted from Ref. [113]. X 2Π A 2∆ B 2 Σ− C 2 Σ+ MC iter. (103 ) a b c a b c a b c a P (P ;Q) %T P (P ;Q) %T P (P ;Q) %T P (P ;Q)b %Tc 0 2.987 0.231 0.0 13.474 7.727 0.0 38.620 −4.954 0.0 43.992 0.087 0.0 2 2.405 0.170 13.8 13.009 7.395 9.8 10.602 −1.848 18.5 40.700 −0.689 9.8 4 1.413 0.086 41.7 10.907 5.288 19.3 7.066 −1.259 38.9 31.017 −0.319 19.7 6 0.883 0.035 58.9 10.119 4.577 27.2 3.452 −0.371 53.2 26.364 −0.508 28.8 8 0.603 0.022 66.8 7.764 2.436 34.6 2.309 −0.149 61.4 20.545 −0.412 34.3 10 0.495 0.019 72.6 6.987 2.170 38.1 1.965 −0.024 64.8 17.180 0.435 38.3 12 0.445 0.015 76.5 6.640 1.981 42.3 1.832 −0.081 69.5 16.929 0.029 42.5 14 0.389 0.013 77.5 7.040 1.887 45.7 1.180 0.030 72.2 13.114 0.253 45.1 16 0.309 0.008 79.2 6.047 1.667 48.3 1.303 0.012 75.6 7.646 −0.041 48.7 18 0.292 0.008 80.3 4.646 0.875 49.8 1.349 −0.062 77.5 5.312 0.011 50.1 20 0.243 0.006 82.2 3.809 0.754 52.6 0.796 0.038 79.5 4.691 0.108 52.2 50 0.150 0.002 89.1 1.367 0.112 74.1 0.298 0.038 91.6 1.436 0.070 74.0 100 0.055 0.002 95.3 0.177 0.017 91.7 0.144 0.014 98.3 0.204 0.013 91.3 150 0.025 0.000 98.1 0.042 −0.003 98.0 0.010 0.007 99.6 0.063 0.010 98.2 200 0.010 0.000 99.2 0.007 0.001 99.7 −0.001 −0.001 99.9 0.010 0.001 99.7 ∞e −38.387749 −38.276770 −38.267544 −38.238205 a Errors in the CC(P ) (the X 2 Π ground state) and EOMCC(P ) (excited states) energies relative to the corresponding CCSDT and EOMCCSDT data, in millihartree, calculataed at the experimentally obtained equilibrium C–H distances used in Refs. [56, 86, 93], which are 1.1197868 Å for the X 2 Π state [272], 1.1031 Å for the A 2 ∆ state [272], 1.1640 Å for the B 2 Σ− state [273], and 1.1143 Å for the C 2 Σ+ state [274]. The lowest-energy core orbital was frozen in all correlated calculations. b Errors in the CC(P;Q) energies relative to the corresponding CCSDT and EOMCCSDT data, in milli- hartree, calculated at the experimentally determined equilibrium C–H distances as used in Refs. [56, 86, 93] (see footnote a for the C–H distances). c The %T values are the percentages of triples captured during the i-FCIQMC propagations for the lowest state of a given symmetry [the 2 B2 (C2v ) component of the X 2 Π ground state, the lowest 2 A1 (C2v ) state for the A 2 ∆ and C 2 Σ+ states, and the lowest 2 A2 (C2v ) state for the B 2 Σ− state]. d The CC(P ) and EOMCC(P ) energies at τ = 0.0 a.u. are identical to the energies obtained in the CCSD and EOMCCSD calculations. The τ = 0.0 a.u. CC(P;Q) energies are equivalent to the CR-CC(2,3) (the ground state) and the CR-EOMCC(2,3) (excited states) energies. e The CC(P ) and EOMCC(P ) energies at τ = ∞ a.u. are identical to the energies obtained in the ROHF- based CCSDT and EOMCCSDT calculations. 142 Table 4.4: Convergence of the CC(P )/EOMCC(P ) and CC(P;Q) energies toward CCS- DT/EOMCCSDT for CNC, calculated using DZP[4s2p1d] basis set. The P spaces used in the CC(P ) and EOMCC(P ) calculations were defined as all singles, all doubles, and sub- sets of triples extracted from i-FCIQMC propagations for the lowest states of the relevant symmetries. Each i-FCIQMC run was initiated by placing 1500 walkers on the appropriate reference function [the ROHF 2 B2g (D2h ) determinant for the X 2 Πg state and the 3σu → 1πg state of the 2 B1u (D2h ) symmetry for the A 2 ∆u and B 2 Σ+ u states], setting the initiator pa- rameter na at 3, and the time step ∆τ at 0.0001 a.u. The Q spaces used in constructing the CC(P;Q) corrections consisted of the triples not captured by i-FCIQMC. Adapted from Ref. [113]. X 2 Πg A 2 ∆u B 2 Σ+ u MC iter. (103 ) Pa (P ;Q)b %Tc Pa (P ;Q)b %Tc Pa (P ;Q)b %Tc 0 18.458 −0.495 0.0 31.157 16.017 0.0 111.307 −0.433 0.0 2 10.331 −0.043 13.2 18.835 9.114 6.5 81.493 −2.496 6.5 4 4.424 −0.029 33.2 10.637 5.717 16.1 53.677 −2.526 16.0 6 2.824 −0.011 44.1 7.555 4.199 22.7 35.539 −1.254 22.8 8 1.818 −0.013 49.9 6.181 3.090 27.5 26.767 −0.864 27.9 10 1.306 −0.006 53.3 5.187 2.441 30.8 21.337 −0.284 31.5 12 1.092 −0.003 56.5 4.162 1.778 34.0 17.056 0.196 34.3 14 0.911 −0.005 58.7 3.529 1.418 37.0 12.843 0.046 37.5 16 0.820 −0.003 60.6 3.106 1.149 39.5 9.197 0.134 39.9 18 0.651 −0.003 62.5 2.510 0.811 41.7 8.879 −0.034 42.4 20 0.610 −0.001 63.9 2.395 0.785 44.4 7.548 0.151 44.7 50 0.077 0.000 79.7 0.172 0.058 70.9 0.732 0.055 70.7 100 0.002 0.000 94.5 0.002 0.001 92.3 0.005 0.003 91.9 150 0.000 0.000 99.3 0.000 0.000 99.1 0.000 0.000 99.1 e ∞ −130.421932 −130.276946 −130.252999 a Errors in the CC(P ) (X 2 Πg state) and EOMCC(P ) (the remaining states) energies relative to the cor- responding CCSDT and EOMCCSDT data, in millihartree, calculataed at the experimentally obtained equilibrium C–N distances optimized in Ref. [262], which are 1.253 Å for the X 2 Πg state, 1.256 Å for the A 2 ∆u state, and 1.259 Å for the B 2 Σ+u state. The three lowest-energy core orbital was frozen in all correlated calculations. b Errors in the CC(P;Q) energies relative to the corresponding CCSDT and EOMCCSDT data, in milli- hartree, calculated at the equilibrium C–N distances optimized in Ref. [262] (see footnote a for these C–N distances). c The %T values are the percentages of triples captured during the i-FCIQMC propagations for the lowest state of a given symmetry [the 2 B2g (D2h ) component of the X 2 Πg ground state and the lowest 2 B1u (D2h ) state for the A 2 ∆ and B 2 Σ+ u states]. d The CC(P ) and EOMCC(P ) energies at τ = 0.0 a.u. are identical to the energies obtained in the CCSD and EOMCCSD calculations. The τ = 0.0 a.u. CC(P;Q) energies are equivalent to the CR-CC(2,3) (the ground state) and the CR-EOMCC(2,3) (excited states) energies. e The CC(P ) and EOMCC(P ) energies at τ = ∞ a.u. are identical to the energies obtained in the ROHF- based CCSDT and EOMCCSDT calculations. 143 Figure 4.1: Convergence of the EOMCC(P ) [panels (a) and (c)] and CC(P;Q) [panels (b) and (d)] energies toward EOMCCSDT for the three lowest-energy excited states of the 1 Σ+ symmetry, two lowest states of the 1 Π symmetry, and two lowest 1 ∆ states of the CH+ ion, as described by the [5s3p1d/3s1p] basis set of Ref. [264], at the C–H internuclear distance R set at Re = 2.13713 bohr [panels (a) and (b)] and 2Re = 4.27426 bohr [panels (c) and (d)]. Adapted from Ref. [113]. 144 (MC) Figure 4.2: The distributions of the differences between the Rµ,3 amplitudes and their EOMCCSDT counterparts resulting from the EOMCC(P ) computations at (a) 4000, (b) 10,000, and (c) 50,000 MC iterations using τ = 0.0001 a.u. for the 2 1 Σ+ state of CH+ at R = 2Re with the analogous distribution characterizing the Rµ,3 amplitudes obtained with the EOMCCSDt approach employing the 3σ, 1πx , 1πy , and 4σ active orbitals to define the corresponding triples space [panel (d)]. All vectors Rµ needed to construct panels (a)–(d) were normalized to unity. Adapted from Ref. [113]. 145 CHAPTER 5 CONCLUDING REMARKS AND FUTURE OUTLOOK This dissertation surveys the recent developments in the realms of CC and EOMCC method- ologies. In particular, we have discussed the various CC and EOMCC approaches belonging to the CC(P;Q) framework, in which one have the flexibility of defining the underlying P and Q spaces to achieve the desired level of accuracy suitable to the system of interest. In the first part of the dissertation, we have discussed the goal of quantum chemistry and the issue of computational cost scaling that plagues the brute-force FCI approach to it. We highlighted the CC theory as one of the best approximation to FCI, especially the CC(P;Q) methodology when we contain the discussion to the realm of SRCC methodology. We have also provided a brief summary of SRCC theory and its EOM extension to excited electronic states and shown examples of high-level CC/EOMCC computations for Mg2 dimer, where one has to include nearly all-electron correlation effects at the triples level and correct the results for valence FCI correlation effects to obtain spectroscopic (∼1 cm−1 ) accuracy relative to experiment. Furthermore, the case study of Mg2 also gives us the opportunity to see how the CCSD(T) approximation to CCSDT fail to be quantitative when weakly bound diatomic that dissociates into closed-shell fragments are examined. This gives us the motivation to look into more robust alternatives to high-level CC and EOMCC schemes, which brings us to the next part of this dissertation. In the second part of this work, we have discussed the CC(P;Q) formalism, beginning with a short summary of the key concepts and equations behind the CC(P;Q) methodology. We proceeded by looking at the CR-CC and CR-EOMCC approaches, which is equivalent to CC(P;Q) where the underlying P and Q spaces are defined by following the traditional truncation schemes defining the cluster and EOM excitation operators. The usefulness of the CR-CC/EOMCC method has been demonstrated by examining the performance of the δ-CR-EOMCC(2,3) triples correction to EOMCCSD in the investigation of the super photo- 146 base FR0-SB. In particular, we have shown that δ-CR-EOMCC(2,3) significantly improves the results obtained from EOMCCSD. Our calculations for FR0-SB have also demonstrates that δ-CR-EOMCC(2,3) has been applied to systems with >50 atoms and hundreds of elec- trons, including solvation effects, resulting in quantitative accuracy relative to experimental data (∼0.1–0.2 eV in terms of excitation energies). We have also examined the CC(P;Q) approaches that rely on active orbitals for defining the P and Q spaces. We also demon- strated the improvement that the resulting CC(t;3) and CC(q;4) schemes, where one corrects CCSDt and CCSDTq energetics for the missing T3 [CC(t;3)] and T4 [CC(q;4)] correlation effects not captured by active orbitals, respectively, offer compared to conventional CR-CC schemes, by returning to the example of Mg2 , where the results obtained in CC(t;3) and CC(q;4) calculations are shown to faithfully reproduce the parent CCSDT and CCSDTQ data, respectively. In the third and last part of this dissertation, we discussed the novel way of performing CC(P;Q) computations that results from the merger of the deterministic CC(P;Q) frame- work with the stochastic CIQMC wave function samplings developed in the Piecuch group. In particular, we have discussed my contribution to the extension of the semi-stochastic CC(P;Q) methodology to excited electronic states, where we targeted EOMCCSDT ener- getics. The benefit of the semi-stochastic CC(P;Q) scheme for excited state applications were demonstrated using several test cases, including the closed-shell CH+ ion and the open- shell CH and CNC radicals. In the future, it will be interesting to see how the EOMCC(P ) methodology underlying the semi-stochastic CC(P;Q) approaches can be used to compute properties other than energy by constructing the appropriate RDM in the P space. For ex- ample, one could also investigate how one-electron properties evolve when the EOMCCSD P space is enriched with subsets of triply excited determinants captured in QMC propagations. To that end, I have derived the equations for the EOMCCSDT 1-RDM and implemented a (MC) (MC) pilot code for it, which is flexible enough to accept stochastically constructed T3 , Rµ,3 , (MC) and Lµ,3 amplitudes or their active-space counterparts. The derivation of the EOMCCSDT 147 1-RDM equations using diagrammatic method, the resulting algebraic expressions, and the spin-integrated equations can be found in the Appendix. While this dissertation has explored a plethora of CC(P;Q) schemes relying on different ways of defining the P and Q spaces, there exist more avenues that are still being explored. Recent works employing the active-space [77] and QMC-driven [114] CC(P;Q) approaches including up to quadruples excitation have shown promising results in the ground-state case, so it would be interesting to extend them to excited states to converge EOMCCSDTQ ener- getics. Furthermore, as already mentioned in the Introduction, one could turn to methods belonging to the selected CI category, where one performs a sequence (or sequences) of Hamil- tonian diagonalizations in a growing space of Slater determinants, to construct the P space in an automated manner but without the risk of introducing stochasticity into the CC(P;Q) computations. As shown in Ref. [132], the resulting selected-CI-driven CC(P;Q) compu- tations produces energetics with comparable accuracy when compared to the QMC-driven CC(P;Q) analogs in recovering the parent high-level CC energetics. Another appealing aspect of the Hamiltonian diagonalizations in selected CI procedure is their capability of naturally producing state-specific P spaces for use in excited-state CC(P;Q) calculations, and, thus, it will be interesting to see how the results obtained in such computations would compare with those of our current semi-stochastic CC(P;Q) methodology for excited states, which relies on P spaces constructed for the lowest state of a given symmetry. Last, but not least, it would also be interesting to extend the hybrid CC(P;Q) ideas to particle non- conserving approaches, such as the electron-attachment (EA) and ionization-potential (IP) EOMCC formalisms, or their double EA (DEA) and double IP (DIP) counterparts. In these types of approaches, one uses EOM operators that formally add (EA and DEA) or remove (IP and DIP) electron(s) from a closed-shell reference function. These methodolo- gies are well-suited for studying open-shell systems, such as radicals and biradicals, because they produce properly spin- and symmetry-adapted wave functions, unlike the conventional CC/EOMCC calculations performed using open-shell reference functions. Earlier works in 148 our group have utilized active-space ideas to downselect the higher-rank components of the many-body expansions underlying the EA/IP- and DEA/DIP-EOMCC approaches, obtain- ing computationally efficient approximations that are capable of faithfully reproducing the results obtained with their parent methods [260, 261, 275–279]. Therefore, it would be in- teresting to see if one could achieve the same by relying on CIQMC, CCMC, or selected CI to construct the appropriate P space within the particle nonconserving methodologies. 149 APPENDIX 150 APPENDIX DIAGRAMMATIC DERIVATION OF EOMCCSDT 1-BODY REDUCED DENSITY MATRIX This appendix provides the derivation of the 1-RDM at the CCSDT/EOMCCSDT level of theory. The general formulation for γqp(CCSDT) (µ, ν) is given and specific cases subsequently examined. For simplicity, we refer to both γqp(CCSDT) (µ), which describes the density of a state µ, and γqp(CCSDT) (µ, ν), which corresponds to the transition between to states µ and ν, as 1-RDM regardless of the states µ and ν describing the CC bra and ket states, respectively. We begin the derivation by defining the CCSDT/EOMCCSDT wavefunction, namely, E (CCSDT) Ψ(CCSDT) µ = Rµ(CCSDT) eT |Φ⟩ , (A.1) where T (CCSDT) = T1 + T2 + T3 and Rµ(CCSDT) = rµ,0 1 + Rµ,1 + Rµ,2 + Rµ,3 (Rµ(CCSDT) = 1 for the µ = 0 ground-state case). The corresponding left CCSDT/EOMCCSDT wavefunction is defined as D (CCSDT) Ψ̃(CCSDT) µ = ⟨Φ| L(CCSDT) µ e−T , (A.2) where L(CCSDT) µ = δµ0 1 + Lµ,1 + Lµ,2 + Lµ,3 . We proceed to the definition of the CCSDT/EOMCCSDT 1-RDM. By inserting Eqs. (A.1) and (A.2) into Eq. (2.17), we obtain the following expression: p(CCSDT) γqp(CCSDT) (µ, ν) = ⟨Φ|L(CCSDT) µ Γq Rν(CCSDT) |Φ⟩ , (A.3) p(CCSDT) (CCSDT) where Γq = (ap aq eT )C . Let us begin our derivation by analyzing terms that will p(CCSDT) contribute to Eq. (A.3). We first focus on expanding the Γq in terms of the allowed nonoriented skeletons, which are presented in Fig. A.1. In constructing these diagrams, we consider several factors that simplify the derivation. First, only up to two T vertices may p(CCSDT) connect to ap aq . Second, because Γq can only contract with L(CCSDT) µ from the left- p(CCSDT) hand side, the Γq diagrams can have a maximum of 6 external lines going to the left. 151 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 p(CCSDT) Figure A.1: The 17 nonoriented skeletons of Γq At this point, it is useful to start generating oriented skeletons and categorizing them p(CCSDT) based on the spin orbital type associated with the indices p and q of Γq . We know a priori that the resulting CCSDT 1-RDM is a (no + nu ) by (no + nu ) matrix and, thus, we can recognize 4 distinct blocks within the matrix, namely, the occupied-occupied (oo), occupied- unoccupied (ou), unoccupied-occupied (uo), and unoccupied-unoccupied (uu) blocks. Based on the orientation of the fermion lines of ap aq , we can easily construct the oriented versions of the 17 skeletons in Fig. A.1 and categorize them based on the occupancy character of the indices, which are shown in Figs. A.2–A.5. i i i j j j i i j A B C D E p(CCSDT) Figure A.2: Oriented skeletons of Γq corresponding to the oo block (γij ), ordered according to increasing many-body rank. 152 b b b a a a b a F G H I p(CCSDT) Figure A.3: Oriented skeletons of Γq corresponding to the uu block (γab ), ordered according to increasing many-body rank. a i J p(CCSDT) Figure A.4: Oriented skeleton of Γq corresponding to the ou block (γia ). a a i a i a a a i i i i K L M N O P a a i a i i i a a i i a Q R S T U V i a a a i i i a W X Y Z p(CCSDT) Figure A.5: Oriented skeletons of Γq corresponding to the uo block (γai ), ordered according to increasing many-body rank. 153 p(CCSDT) Using the oriented skeletons of Γq shown in Figs. A.2–A.5 and relying on the fact that Eq. (A.3) corresponds to fully contracted diagrams, we can now proceed to analyze p(CCSDT) the ranks of Lµ and Rν that may form contraction with Γq . This information is summarized in Table A.1. It is interesting to note that the skeletons labeled as A and K in the oo and uo blocks, respectively, are closed diagrams, i.e., they do not have external fermion lines. Thus, if we examine Eq. (A.3) for these closed diagrams, we end up with terms p(CCSDT) p(CCSDT) of the form Γq,closed ⟨Φ| L(CCSDT) µ Rν(CCSDT) |Φ⟩ = Γq,closed δµν in the oo and uo blocks of γqp(CCSDT) (µ, ν). Table A.1: Blocks of the CCSDT 1-RDM with the corresponding labels from Figs. A.2–A.5, contraction types, and possible ranks of Lµ and Rν components. Block Label Contraction type n a p(CCSDT) A Lµ,n Γq Rν,n 0, 1, 2, 3 p(CCSDT) B Lµ,n Γq Rν,n 1, 2, 3 p(CCSDT) oo C Lµ,n+1 Γq Rν,n 0, 1, 2 p(CCSDT) D Lµ,n+2 Γq Rν,n 0, 1 p(CCSDT) E Lµ,n+3 Γq Rν,n 0 p(CCSDT) F Lµ,n Γq Rν,n 1, 2, 3 p(CCSDT) G Lµ,n+1 Γq Rν,n 0, 1, 2 uu p(CCSDT) H Lµ,n+2 Γq Rν,n 0, 1 p(CCSDT) I Lµ,n+3 Γq Rν,n 0 p(CCSDT) ou J Lµ,n+1 Γq Rν,n 0, 1, 2 p(CCSDT) Ka Lµ,n Γq Rν,n 0, 1, 2, 3 p(CCSDT) L Lµ,n Γq Rν,n+1 0, 1, 2 p(CCSDT) M, N Lµ,n Γq Rν,n 1, 2, 3 p(CCSDT) O, P Lµ,n+1 Γq Rν,n 0, 1, 2 uo p(CCSDT) Q, R Lµ,n+1 Γq Rν,n 1, 2 p(CCSDT) S, T, U Lµ,n+2 Γq Rν,n 0, 1 p(CCSDT) V, W Lµ,n+2 Γq Rν,n 1 p(CCSDT) X, Y, Z Lµ,n+3 Γq Rν,n 0 a Corresponds to a closed diagram; contraction with Lµ and Rν will produce δµν (see text). We have now reached the stage where we can derive γqp(CCSDT) (µ, ν) in the spin orbital 154 basis. We do so by performing the contractions listed in Table A.1, obtaining j(CCSDT) γi (µ, ν) = δµν δij − (lµ,ie tje + 21 lµ,in ef jn tef + 1 l ef g tjno )r 12 µ,ino ef g ν,0 ef jn − lµ,ie rν,ej − 21 lµ,in rν,ef − 1 l ef g r jno 12 µ,ino ν,ef g (A.4) − lµ,in te rν,fn − 14 lµ,ino ef j te rν,fnog − 12 lµ,ino ef g j ef g jn tef rν,go , γab(CCSDT) (µ, ν) = (lµ,m b m ta + 21 lµ,mn bf mn taf + 1 l bf g tmno )rν,0 12 µ,mno af g mn + lµ,m b rν,am + 12 lµ,mn bf rν,af + 1 l bf g r mno 12 µ,mno ν,af g (A.5) + lµ,mnbf m ta rν,fn + 14 lµ,mno ta rν,fnog + 12 lµ,mno bf g m bf g mn taf rν,go , a(CCSDT) γi (µ, ν) = lµ,ia rν,0 + lµ,in ef rν,fn + 14 lµ,ino ef g rν,fnog , (A.6) and γai(CCSDT) (µ, ν) = δµν tia + (lµ,m e im tae − lµ,m e i m te ta + 41 lµ,mn ef imn taef ef mn i ef in m ef g mo in − 12 lµ,mn taf te − 21 lµ,mn tef ta − 14 lµ,mno tag tef − 1 l ef g tmno ti 12 µ,mno af g e − 1 l ef g tino tm )rν,0 12 µ,mno ef g a in ino + δµ0 rν,ai + lµ,nf rν,af + 14 lµ,no fg rν,af e i g − lµ,m te rν,a m e m mn in − lµ,m ta rν,ei − 12 lµ,mn ef i te rν,af ef m − 21 lµ,mn ta rν,ef (A.7) − 1 l ef g ti r mno 12 µ,mno e ν,af g − 1 l ef g tm r ino 12 µ,mno a ν,ef g + lµ,mnef im tae rν,fn tae rν,fnog − lµ,mn ef g im + 14 lµ,mno te ta rν,fn − 14 lµ,mno ef i m te ta rν,fnog ef g i m ef in − 12 lµ,mn tef rν,am − 12 lµ,mnef mn taf rν,ei − 14 lµ,mno ef g in tef rν,ag mo io ef g mn − 14 lµ,mno taf rν,eg + 14 lµ,mno taef rν,go − 12 lµ,mno ef g imn ef g mn i taf te rν,go ef g in a − 12 lµ,mno tef tm rν,go − 1 l ef g tino r m 12 µ,mno ef g ν,a − 1 l ef g tmno r i . 12 µ,mno af g ν,e In writing the above expressions for γqp(CCSDT) (µ, ν), we follow the convention for spin orbitals explained in Section 2.1 when dealing with fixed labels. In the case of free labels, we use m, n, o, . . . and e, f, g, . . . to indicate occupied and unoccupied spin orbitals, respectively. 155 One can do an extra step by performing spin integration to obtain the 1-RDM in the P (CCSDT) molecular orbital basis, γQ (µ, ν), where we use capital letters to label spin orbitals following the convention described above, and molecular orbitals that correspond to the β electron spin are labeled with a tilde. Thus, we obtain, for the α spin case, J(CCSDT) EF JN γI (µ, ν) = δµν δIJ − (lµ,IE tJE + 12 lµ,IN tEF + lµ,IEÑF̃ tE J Ñ F̃ + 1 l EF G tJN O 12 µ,IN O EF G + 12 lµ,INEF G̃ JN Õ t Õ EF G̃ + 41 lµ,IEÑF̃ÕG̃ tE J Ñ Õ F̃ G̃ )rν,0 − lµ,IE rν,EJ − 12 lµ,IN EF rν,EFJN − lµ,IEÑF̃ rν,EJ ÑF̃ − 1 l EF G r JN O 12 µ,IN O ν,EF G − 12 lµ,INEF G̃ r JN Õ − 41 lµ,IEÑF̃ÕG̃ rν,EJ ÑF̃ G̃ Õ ν,EF G̃ Õ (A.8) EF J − lµ,IN tE rν,FN − lµ,IEÑF̃ tJE rν,F̃Ñ − 14 lµ,IN EF G J O tE rν,F G NO EF G̃ J − lµ,IN t r N Õ − 41 lµ,IEÑF̃ÕG̃ tJE rν,F̃ÑG̃Õ − 21 lµ,IN Õ E ν,F G̃ EF G JN O tEF rν,G O EF G̃ JN E F̃ G̃ J Ñ Õ − 12 lµ,IN t r Õ − lµ,IEÑF̃OG tE Õ EF ν,G̃ J Ñ O F̃ rν,G − lµ,I Ñ Õ tE F̃ rν,G̃ , B(CCSDT) B M BF M N B F̃ M Ñ γA (µ, ν) = (lµ,M tA + 12 lµ,M N tAF + lµ,M Ñ tAF̃ + 1 l BF G tM N O 12 µ,M N O AF G BF G̃ M N Õ + 12 lµ,M t N Õ AF G̃ + 41 lµ,MB F̃ G̃ M Ñ Õ t Ñ Õ AF̃ G̃ )rν,0 B M BF MN B F̃ M Ñ + lµ,M rν,A + 12 lµ,M N rν,AF + lµ,M Ñ rν,AF̃ + 1 l BF G r M N O 12 µ,M N O ν,AF G + 12 lµ,MBF G̃ r M N Õ + 41 lµ,M N Õ ν,AF G̃ B F̃ G̃ r mÑ Õ Ñ Õ ν,AF̃ G̃ (A.9) BF M N B F̃ M Ñ 1 BF G M NO + lµ,M N tA rν,F + lµ,M Ñ tA rν,F̃ + 4 lµ,M N O tA rν,F G BF G̃ M + lµ,M t r N Õ + 14 lµ,M N Õ A ν,F G̃ B F̃ G̃ M t r Ñ Õ + 21 lµ,M Ñ Õ A ν,F̃ G̃ BF G M N N O tAF rν,G O BF G̃ M N + 21 lµ,M t r Õ + lµ,M N Õ AF ν,G̃ B F̃ G M Ñ t r O + lµ,M Ñ O AF̃ ν,G B F̃ G̃ M Ñ t r Õ , Ñ Õ AF̃ ν,G̃ A(CCSDT) γI (µ, ν) = lµ,IA rν,0 + lµ,IN AF rν,FN + lµ,IAÑF̃ rν,F̃Ñ (A.10) + 1 l AF G r N O 4 µ,IN O ν,F G + AF G̃ lµ,IN r N Õ Õ ν,F G̃ + 1 l AF̃ G̃ r Ñ Õ , 4 µ,I Ñ Õ ν,F̃ G̃ and 156 I(CCSDT) E IM Ẽ I M̃ E I M γA (µ, ν) = δµν tIA + (lµ,M tAE + lµ,M̃ tAẼ − lµ,M tE tA EF IM N E F̃ IM Ñ Ẽ F̃ I M̃ Ñ + 14 lµ,M 1 N tAEF + lµ,M Ñ tAE F̃ + 4 lµ,M̃ Ñ tAẼ F̃ EF M N I E F̃ M Ñ I 1 EF IN M − 12 lµ,M N tAF tE − lµ,M Ñ tAF̃ tE − 2 lµ,M N tEF tA E F̃ I Ñ M EF G M O IN EF G̃ M Õ IN − lµ,M t t − 14 lµ,M Ñ E F̃ A 1 N O tAG tEF − 2 lµ,M N Õ tAG̃ tEF E F̃ G M O I Ñ E F̃ G̃ M Õ I Ñ − 12 lµ,M t t − lµ,M Ñ O AG E F̃ t t − Ñ Õ AG̃ E F̃ 1 l EF G tM N O tI 12 µ,M N O AF G E EF G̃ M N Õ I E F̃ G̃ M Ñ Õ I − 12 lµ,M t N Õ AF G̃ E t − 14 lµ,M t Ñ Õ AF̃ G̃ E t − 1 l EF G tIN O tM 12 µ,M N O EF G A EF G̃ IN Õ M E F̃ G̃ I Ñ Õ M − 12 lµ,M t t − 14 lµ,M N Õ EF G̃ A t Ñ Õ E F̃ G̃ A t )rν,0 I F IN F̃ I Ñ FG IN O F G̃ IN Õ F̃ G̃ I Ñ Õ + δµ0 rν,A + lµ,N rν,AF + lµ,Ñ rν,A F̃ + 14 lµ,N 1 O rν,AF G + lµ,N Õ rν,AF G̃ + 4 lµ,Ñ Õ rν,AF̃ G̃ E I M E M − lµ,M tE rν,A − lµ,M tA rν,EI − 21 lµ,M EF I N tE rν,AF MN E F̃ I − lµ,M t r M Ñ − 12 lµ,M Ñ E ν,AF̃ EF M IN E F̃ M N tA rν,EF − lµ,M Ñ tA rν,E F̃ I Ñ − 1 l EF G tI r M N O 12 µ,M N O E ν,AF G − 12 lµ,M EF G̃ I t r M N Õ − 41 lµ,M N Õ E ν,AF G̃ E F̃ G̃ I t r M Ñ Õ Ñ Õ E ν,AF̃ G̃ − 1 l EF G tM r IN O 12 µ,M N O A ν,EF G − 12 lµ,M EF G̃ M t r IN Õ − 14 lµ,M N Õ A ν,EF G̃ E F̃ G̃ M t r I Ñ Õ Ñ Õ A ν,E F̃ G̃ EF IM N E F̃ IM Ñ ẼF I M̃ N Ẽ F̃ I M̃ Ñ + lµ,M N tAE rν,F + lµ,M Ñ tAE rν,F̃ + lµ,M̃ N tAẼ rν,F + lµ,M̃ Ñ tAẼ rν,F̃ EF G IM NO ẼF G I M̃ NO EF G̃ IM N Õ + 14 lµ,M 1 N O tAE rν,F G + 4 lµ,M̃ N O tAẼ rν,F G + lµ,M N Õ tAE rν,F G̃ t r Ñ Õ + lµ,M̃ E F̃ G̃ IM + 14 lµ,M Ñ Õ AE ν,F̃ G̃ t r Ñ O + 14 lµ,M̃ Ẽ F̃ G I M̃ Ñ O AẼ ν,F̃ G t r Ñ Õ Ẽ F̃ G̃ I M̃ Ñ Õ AẼ ν,F̃ G̃ EF I M N E F̃ I M Ñ − lµ,M N tE tA rν,F − lµ,M Ñ tE tA rν,F̃ EF G I M NO EF G̃ I M N Õ 1 E F̃ G̃ I M Ñ Õ − 41 lµ,M N O tE tA rν,F G − lµ,M N Õ tE tA rν,F G̃ − 4 lµ,M Ñ Õ tE tA rν,F̃ G̃ EF IN M E F̃ I Ñ M EF M N I E F̃ M Ñ I − 21 lµ,M 1 N tEF rν,A − lµ,M Ñ tE F̃ rν,A − 2 lµ,M N tAF rν,E − lµ,M Ñ tAF̃ rν,E EF G IN MO EF G̃ IN M Õ 1 E F̃ G I Ñ MO E F̃ G̃ I Ñ M Õ − 14 lµ,M 1 N O tEF rν,AG − 2 lµ,M N Õ tEF rν,AG̃ − 2 lµ,M Ñ O tE F̃ rν,AG − lµ,M Ñ Õ tE F̃ rν,AG̃ EF G M N IO EF G̃ M N I Õ E F̃ G M Ñ IO E F̃ G̃ M Ñ I Õ − 14 lµ,M 1 1 N O tAF rν,EG − 2 lµ,M N Õ tAF rν,E G̃ − 2 lµ,M Ñ O tAF̃ rν,EG − lµ,M Ñ Õ tAF̃ rν,E G̃ EF G IM N O EF G̃ IM N Õ ẼF G I M̃ N O + 14 lµ,M 1 N O tAEF rν,G + 4 lµ,M N Õ tAEF rν,G̃ + lµ,M̃ N O tAẼF rν,G E F̃ G̃ IM Ñ + lµ,M t r Õ + 14 lµ,M̃ Ñ Õ AE F̃ ν,G̃ Ẽ F̃ G I M̃ Ñ t Ñ O AẼ F̃ ν,G r O + 14 lµ,M̃ Ẽ F̃ G̃ I M̃ Ñ t Ñ Õ AẼ F̃ ν,G̃ r Õ EF G M N I O EF G̃ M N I Õ E F̃ G M Ñ I O E F̃ G̃ M Ñ I Õ − 21 lµ,M 1 N O tAF tE rν,G − 2 lµ,M N Õ tAF tE rν,G̃ − lµ,M Ñ O tAF̃ tE rν,G − lµ,M Ñ Õ tAF̃ tE rν,G̃ EF G IN M O EF G̃ IN M Õ E F̃ G I Ñ M O E F̃ G̃ I Ñ M Õ − 12 lµ,M 1 N O tEF tA rν,G − 2 lµ,M N Õ tEF tA rν,G̃ − lµ,M Ñ O tE F̃ tA rν,G − lµ,M Ñ Õ tE F̃ tA rν,G̃ − 1 l EF G tIN O r M 12 µ,M N O EF G ν,A − 12 lµ,M EF G̃ IN Õ t N Õ EF G̃ ν,A r M − 41 lµ,M E F̃ G̃ I Ñ Õ t r M Ñ Õ E F̃ G̃ ν,A − 1 l EF G tM N O r I 12 µ,M N O AF G ν,E − 12 lµ,M EF G̃ M N Õ t N Õ AF G̃ ν,E r I − 41 lµ,M E F̃ G̃ M Ñ Õ t Ñ Õ AF̃ G̃ ν,E r I. 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