Ll BRARY Michigan State University This is to certify that the dissertation entitled COUPLED-CLUSTER METHODS FOR OPEN-SHELL MOLECULAR AND OTHER MANY-FERMION SYSTEMS presented by JEFFREY R. GOUR has been accepted towards fulfillment of the requirements for the PhD. degree in Chemistry (MM Makfl’rofessofi Signature A o 535/39“) Date MSU is an Affirmative Action/Equal Opportunity Employer ~l_ .— .- c—-u---l-l-l-l-h-l-I-l-O-l-D-l—I- PLACE IN RETURN BOX to remove this checkout from your record. To AVOID FINES return on or before date due. MAY BE RECALLED with earlier due date if requested. DATE DUE DATE DUE DATE DUE 5/08 KIProj/Achres/CIRCIDatGDueJndd COUPLED-CLUSTER METHODS FOR OPEN-SHELL MOLECULAR AND OTHER MANY-FERMION SYSTEMS By Jeffrey R. Gour A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Chemistry 2010 Q?» one of the best methods in this category, termed CR—CC(2,3) or CR—EOMCC(2,3), in which a noniterative correction due to triple excitations is added to the CCSD or EOMCCSD energy, and its higher-order CR—CC(2,4)/CR—EOMCC(2,4) approach, in which a noniterative correction due to triple and quadruple excitations is added to the CCSD/EOMCCSD energy, to open-shell systems. In this thesis the theoretical details of all of these new methodologies as well as a sample of benchmark examples that illustrate their performance in studies of ground and excited states of open-shell molecular systems are discussed. In addition, since there is nothing in the underlying theoretical framework specific to electronic structure, the CC approaches developed in this thesis are not restricted to molecular cases and can be applied to other many- fermion systems, such as atomic nuclei. Representative examples of applications of the new CC methods developed in this thesis research in the context of quantum chemistry to studies of nuclear structure are given as well. ABSTRACT COUPLED-CLUSTER METHODS FOR OPEN-SHELL MOLECULAR AND OTHER MANY-FERMION SYSTEMS By Jeffrey R. Gour The description of the electronic structure of radicals and other open—shell molecu- lar systems represents a significant challenge for current theoretical methodologies. Since the low-lying electronic states of open-shell species often possess a manifestly multi-determinantal character, it is difficult to perform calculations for these systems that are both highly accurate and practical enough to be applied to a wide range of chemical problems of interest. To overcome these difficulties, we have developed two new classes of coupled-cluster (CC) methods, which are capable of accounting for the high-level electron correlation effects that characterize open-shell systems at a rela- tively low computational cost. The first class of methods, the active-space variants of the electron-attached (EA) and ionized (IP) equation-of—motion CC (EOMCC) theories, utilize the idea of applying a linear electron-attaching or ionizing opera- tor to the correlated, ground-state CC wave function of an N-electron closed-shell system in order to generate the ground and excited states of the related (N :l: 1)— electron radical species. Furthermore, these approaches use a physically motivated set of active orbitals to a priori select the dominant higher-order correlation effects to be included in the calculation, which significantly reduces the costs of the high-level approximations needed for obtaining accurate results for open—shell species without sacrificing accuracy. The second class consists of the size extensive, left-eigenstate completely-renormalized (CR) CC approaches based on the biorthogonal formulation of the method of moments of CC equations, in which noniterative corrections due to higher-order excitations are added to the energies obtained with the standard CC approximations, such as CCSD (CC with singles and doubles). We have extended ‘ COpyright by JEFFREY RICHARD GOUR 2010 ‘fl' ACKNOWLEDGMENT First and foremost, I wish to thank my Ph.D. advisor, Professor Piotr Piecuch for his teaching and guidance. Thanks to his constant support and his desire to see me be- come the best that I can be, I not only learned a great deal about electronic structure theory, but also about how to be a better scientist. His inspiration, expectations, and patience have been key components in my growth and success as a researcher, and I am deeply grateful for all he has done for me. I would also like to thank the rest of my guidance committee, namely Professor Katharine C. Hunt, Professor Robert Cukier, and Professor James K. McCusker, for their advice, support, and patience in overseeing my graduate study. I also owe a great deal of gratitude to Professor Marta Wloch, a former postdoc- toral research associate in the Piecuch group. When I first joined the group, it was her that took me under her wing and helped introduce me to various details regarding the research to be done. Without her I would have been lost in the beginning. Not only that, but as I grew into my own and became more independent, she turned from mentor to valuable collaborator. I was able to overcome many difficulties in my re- search thanks to discussions with her. Without her help, most of this research would not have been possible. I would also like to thank other members of the Piecuch group, both past and present, for the help they provided me, including Dr. Karol Kowalski, Dr. Maricris Lodriguito, Dr. Wei Li, Mr. Jesse Lutz, and Ms. Janelle Bradley. Finally, I would like to acknowledge the National Science foundation for providing me with a Graduate Research Fellowship that supported most of my research. Ad- ditional support was also provided through several fellowships from Michigan State University (including some from the Department of Chemistry), and from the US Department of Energy. I would also like to acknowledge the MSU High Performance Computing Center for providing resources utilized for much of this research. TABLE OF CONTENTS List of Tables ............................ viii List of Figures ........................... xi 1 Introduction ........................... 1 2 Active-Space Coupled-Cluster Methods for Open-Shell Systems . 12 2.1 Theory and Computer Implementation ................. 13 2.1.1 The Electron-Attached and Ionized Equation—of-Motion Coupled— Cluster Theories ......................... 13 2.1.2 The Active-Space EA- and IP-EOMCC Methodologies . . . . 20 2.1.3 Key Details of the Efficient Computer Implementation of the Active-Space EA-EOMCCSDt and IP-EOMCCSDt Approaches ................... 25 2.2 Applications ................................ 46 2.2.1 Excitation Energies of Diatomic Radicals: CH and SH . . . . 46 2.2.2 Potential Energy Curves of OH ................. 54 2.2.3 Excitation Energies of C2N, CNC, N3, and NCO ....... 69 Noniterative Coupled-Cluster Methods for Open-Shell Systems . 74 3.1 Theory ................................... 75 3.1.1 The Biorthogonal Formulation of the Method of Moments of Coupled-Cluster Equations .................... 75 3.1.2 The CR—CC(2,3)/CR—EOMCC(2,3) and CR—CC(2,4)/CR—EOMCC(2,4) Approaches ........... 83 3.1.3 Computer Implementation of the Open-Shell Variants of the CR-CC(2,3)/CR-EOMCC(2,3) and CR-CC(2,4)/CR—EOMCC(2,4) Approaches ........... 94 3.2 Applications ................................ 101 3.2.1 Bond Breaking in Radical Species: HgC-X and HQSI-X . . . . 102 3.2.2 Singlet-Triplet Gaps in Biradicals ................ 109 3.2.3 Excitation Energies of C2N, CNC, N3, and NCO ....... 129 Coupled-Cluster Calculations for Nuclei ............. 135 4.1 Details of Coupled-Cluster Calculations for Nuclei ........... 135 4.2 Low-lying States of 160 and the Surrounding Valence Systems . . . . 141 4.3 Ground and Excited States of 55Ni, 56Ni, and 57N i .......... 155 Summary and Future Perspectives ................ 166 vi Appendix A: Factorized Form of the EA-EOMCCSD(3p—2h) and IP-EOMCCSD(3h-2p) Equations ................. 172 Appendix B: Derivation of the Noniterative Energy Correction Defining the Biorthogonal MMCC Theory ................ 174 References ............................. 178 vii 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 LIST OF TABLES Explicit algebraic expressions for the one— and two—body matrix ele- - (CCSD) ments of H N, Op en . The various classes of restricted projections that must be considered when generating the computationally efficient form of the equations defining the EA-EOMCCSDt and IP-EOMCCSDt eigenvalue prob- lems. ................................... The ground-state energies and the adiabatic excitation energies corre— sponding to the low-lying excited states of the CH radical, as obtained with the aug-cc-pVxZ (:1:=D, T, and Q) basis sets. .......... The average time per iteration for the EA-EOMCCSD(3p—2h) and EA- EOMCCSDt calculations performed for the CH radical with the aug- cc—meZ (:r=D, T, and Q) basis sets. .................. The ground-state energies and the vertical excitation energies corre- sponding to the low-lying excited states of the SH radical, as obtained with the aug-cc-pV(z+d)Z basis set for S and the aug-cc-pVxZ basis set for H where a:=D and T. ....................... The average time per iteration for the IP-EOMCCSD(3h-2p) and IP- EOMCCSDt calculations performed for the SH radical with the aug- cc-pV(:r+d)Z basis set for S and the aug—cc-strZ basis set for H where 32D and T ................................. An analysis of the major full CI configurations for the low-lying II and A states of the OH radical for a selected set of internuclear separations RO_H. ................................... An analysis of the major full CI configurations for the low-lying 2 states of the OH radical for a selected set of internuclear separations RO_H . ................................... 50 51 52 2.9 2.10 2.11 2.12 3.1 3.2 3.3 3.4 3.5 3.6 A comparison of the total energies obtained with various IP EOMCC and SAC-CI methods for the low-lying II and A states of the OH radical with the corresponding full CI results obtained for several internuclear separations RO-H ............................. A comparison of the total energies obtained with various IP EOMCC and SAC-CI methods for the low-lying 2 states of the OH radical with the corresponding full CI results obtained for several internuclear separations RO_H .............................. Equilibrium geometries (A), adiabatic excitation energies (eV), and approximate excitation levels relative to the ground states of the cor- responding reference cations for the low-lying valence excited states of C2N and CNC. .............................. Equilibrium geometries (A), adiabatic excitation energies (eV), and approximate excitation levels relative to the ground states of the cor- responding reference anions for the low-lying valence excited states of N3 and NCO. ............................... Restricted open-shell CR-CC(2,3), unrestricted CCSD(T) and MRMP2 NPE, STD, and REE relative to MRCI(Q). .............. The adiabatic A 1A1 — X 381 splitting in CH2 obtained with full CI and various CC approaches, and the DZP basis set. ......... Comparison of the total energies (in hartree) and adiabatic excitation energies (in eV) for the low-lying states of CH2 as obtained with various CC approaches, using the aug-cc—pCVxZ (:r=T, Q, 5) basis sets and extrapolating to the CBS limit, with various QMC results. ...... The A 32,1,” — X 12; gap for the linear, Dom-symmetric (HF H)— system as a function of the H—F distance RH—F ............. A comparison of the total energies obtained with various electronic structure methods for the X 12; state of the linear, Dom-symmetric (HF H)- system as a function of the H—F distance RH—F ........ A comparison of the total energies obtained with various electronic structure methods for the A 32:; state of the linear, Dock-symmetric (HF H)‘ system as a function of the H—F distance RH—F ........ ix 66 67 70 72 107 111 114 118 121 122 3.7 3.8 3.9 4.1 4.2 4.3 4.4 4.5 Equilibrium bond lengths Re (in A) and harmonic frequencies we (in cm‘l) for the lowest triplet and singlet states of BN, as obtained with cc-pV5Z basis set, and the adiabatic singlet-triplet splittings Te (in cm—l) as obtained with the cc—pVxZ (a: = D, T, Q, and 5) basis sets, as well as the extrapolated CBS limit values. ............. Adiabatic excitation energies (eV) and reduced excitation level (REL) values of the low-lying valence excited states of CN C and C2N. . . . . Adiabatic excitation energies (eV) and reduced excitation level (REL) values of the low-lying valence excited states of N3 and NCO. A comparison of the binding energies per particle for 15O and 15N (the PR—EOMCCSD(2h—1p) values), 160 (the CCSD values), and 170 and 17F (the PA-EOMCCSD(2p—1h) values), obtained with the N 3LO, CD-Bonn, and V18 potentials, and eight major oscillator shells. . . . . A comparison of the energies of the low-lying excited states of 150, 15N, 170 and 17F, relative to the corresponding ground-state ener- gies obtained with the PR-EOMCCSD(2h-1p) (150 and 15N) and PA- EOMCCSD(2h—1p) (170 and 17F) methods, the N3L0, CD-Bonn, and Argonne V18 potentials, and eight major oscillator shells, with the ex- perimental data. ............................. Energies (in MeV) of 56Ni as functions of the shell-gap shift AG, rel- ative to the reference energy ((1)0|H |0) = —-203.800 MeV. ...... Binding energies (in MeV) of 55Ni and 57Ni relative to the correspond— ing reference energies ((1)84) (j)|H |(()A) (j)), A = 55 and 57, respec- tively, as functions of the shell gap shift AG (in MeV). ........ Excitation energies (in MeV) of the low-lying states of 57N i as functions of the shell gap shift AG ........................ 127 131 132 146 160 162 2.1 2.2 2.3 2.4 3.1 3.2 3.3 4.1 LIST OF FIGURES Images in this dissertation are presented in color Pictorial illustration of the generation of the CH and OH radicals from the closed-shell CH+ and OH— ions, respectively. ........... The key elements of the algorithm used to compute (@Ajbfil(H(CCSD)RI(LN+1))C|) in the efficient implementation of the N ,open EA-EOMCCSDt method. ........................ The key elements of the algorithm used to compute (Cbllfi|(H1(€EPS£)R£N_1))CI), in the efficient implementation of the IP-EOMCCSDt method. ........................ Potential energy curves for the ground and low—lying excited states of the OH radical. ............................. The key elements of the algorithm used to compute 6§R(2’3) in the efficient open-shell implementation of CR—CC(2,3)/CR—EOMCC(2,3). Restricted open-shell CCSD, restricted open—shell CR—CC(2,3), unre- stricted CCSD(T), and MRMP2 errors relative to MRCI(Q) for H2C—H —> 3CH2 + H with the cc-pVTZ basis set. ......... Restricted open-shell CCSD, restricted open-shell CR-CC(2,3), unre- stricted CCSD(T), and MRMP2 errors relative to MRCI(Q) for H28i—H ——+ 1Sng + H with the cc-pVTZ basis set. ......... Pictorial illustration of the nuclear shell structure. .......... xi 13 42 44 55 100 104 4.2 4.3 4.4 The coupled-cluster energies of the ground-state and first-excited 3" state as functions of the number of oscillator shells N obtained with the Idaho-A interaction. ......................... Systematic comparison of IT-CI and CC results for the ground—state energy of 160 using HF-optimized single-particle bases with emax = 4, 5, 6, and 7. (a) Comparison of IT-CI(4p—4h) (open symbols) with IT-CI(4p-4h)+MRD (filled symbols). (b) Comparison of CCSD (open symbols) with CR-CC(2,3) (filled symbols). (c) Comparison of IT- CI(4p—4h)+MRD (open symbols) with CR—CC(2,3) (filled symbols). (a) The full CI, CISDTQ, and CR-CC (2,3) energies of 56N i as functions of the shell-gap shift AG. (b) Comparison of full CI energies with the trends expected for the 1p—1h, 4p-4h, and 8p-8h configurations as functions of AG. ............................. xii 142 Chapter 1 Introduction The single-reference coupled—cluster (CC) theory [1—5] is widely regarded as the pre- eminent ab initio approach for studying chemical systems. The success of the CC methodology, and its extension to excited states through the equation-of-motion (EOM) CC formalism [6—10] or its symmetry-adapted-cluster configuration—interaction (SAC-CI) [11415] and linear response CC [16—20] analogs, lies in its ability to effi- ciently account for the many-electron correlation effects, the consideration of which is essential for obtaining an accurate description of a molecular system. As is true of other single-reference quantum theories based on the idea of expanding the many- electron wave function in a basis of molecular orbitals, the conventional CC method- ology builds correlations into the wave function through excitations out of a single reference determinant. The advantage of the CC theory over other formalisms is that it uses an exponential excitation operator to describe these correlation effects, and so it is able to account for additional excitations, not explicitly included in the calcula- tion, through the product or so—called ‘disconnected’ excitations. For instance, if one were to include only operators that create singly and doubly excited configurations out of the reference in the CC calculations, the various products of these components that result from expanding the exponential in a Taylor series lead to some triple, 1 quadruple, pentuple, etc. excitations also being accounted for without increasing the computer costs. As a result, the CC formalism provides an optimum balance between high accuracy and relatively low computer effort, making it an ideal the- ory for studying many molecular systems, and for further electronic structure theory advances. Despite the success of approaches based on the CC theory over the years, there are a number of open issues in the CC methodology, one of which is the adequate description of open-shell systems. Due to their high reactivity and importance as chemical intermediates and magnetic systems, open-shell molecular systems, such as radicals and biradicals, play a significant role in chemistry, and as such a theoret— ical understanding of such species would be invaluable for many areas of chemical research. Unfortunately, such systems still represent a major challenge for modern electronic structure theories, and the CC theory is no exception. The source of the difficulty stems from the types of many-electron correlation effects that define the electronic structure of open-shell systems, particularly those where chemical bonds are stretched or broken. In general, the many-electron correlations can be classified into two types, dynamical and nondynamical. The former refers to the correlations that result from the short-range interactions whereby electrons instantaneously avoid each other, and are mathematically included in the wave function via excitations out of a reference state. As indicated by the above explanation, the CC theory has few problems with this type of correlation effects. The nondynamical (sometimes referred to as static) correlation effects, on the other hand, are long-range effects stemming from the multi—configurational character of systems having quasidegenerate electronic states (i.e. states that are close in energy). This means that for states characterized by large nondynamic correlations, a single Slater determinant is not a good refer- ence for the many-electron wave function, and so multiple determinants must be used to create a reference function on top of which dynamical correlations can be 2 built. As it turns out, the majority of open-shell systems, particularly when they un- dergo chemical transformations or when they are electronically excited, display such a multi-reference character, and so methodologies that are capable of providing an ac- curate and balanced description of both dynamical and nondynamical many—electron correlation effects are needed to accurately describe them. Unfortunately, the basic, low-order CC approaches, including the CCSD (CC with singles and doubles) approach [21—24], and its excited-state EOMCCSD [7—9], SAC- CI-SD—R [11—15], and linear response CCSD [19,20] analogs, have difficulty balancing these types of correlation effects, and thus the accurate description of the low-lying states of open—shell systems is a major challenge for such approaches. Even the pop- ular CCSD(T) approach [25], in which a noniterative, quasiperturbative correction due to triply excited clusters is added to the energy obtained with CCSD, has prob— lems describing such systems. Though it is known to offer an excellent description of dynamical correlation effects, which provide a near perfect description of closed- shell systems near the equilibrium geometry, CCSD(T) fails to properly account for nondynamical correlation effects. The full CCSDT (CC with singles, doubles, and triples) [26,27] and EOMCCSDT [28—30] approaches, which were recently extended to open-shell systems [31], are able to better balance the dynamical and nondynamical correlation effects, and thus are capable of producing high quality results for many open-shell situations, but the computational costs of such schemes are extremely high, restricting their use to small systems with only a few light atoms (a dozen or so correlated electrons). In contrast, CCSD(T) can nowadays be routinely applied to systems with up to about 100 correlated electrons and a few hundred basis functions within a canonical formulation, and one can go to systems with hundreds of corre- lated electrons and thousands of basis functions when one uses the local correlation formulation [32—44]. Thus in order to accurately study a wide range of open-shell problems of interest, alternatives to the standard single-reference CC methods that 3 are not much more expensive than CCSD(T) are needed. One of the main reasons that properly accounting for both dynamical and nondy— namical correlations within the standard CC truncation hierarchy requires high-level, and computationally expensive, approximations is the single—reference nature of these schemes. Indeed single determinants are bad starting points for the description of manifestly multi-reference states, such as those found in open-shell systems. As a result, the only way to properly describe such systems within the standard CC trun- cations is to compensate for the bad start and account for nondynamical correlations dynamically, i.e. through the inclusion of higher-order excitation effects. Based on this analysis, an obvious solution is to simply start from a better reference state that accurately accounts for the nondynamical correlations in the system, leaving only the remaining dynamical correlations to be described by the exponential excitation operator. This is the basic idea behind the genuine multi-reference (MR) CC theories of either the valence-universal [45,46] or state-universal [47] type, for which an expo- nential excitation operator is applied to a multi-determinantal reference in order to generate the wave functions for the desired many-electron states. At first glance, this would seem like the ideal solution for open-shell systems, as it properly balances non- dynamical (through the multi-determinantal reference state) and dynamical (through the exponential ansatz) correlations without the use of high-order excitations in the wave operator that transforms the zero-order reference states into the target wave functions. Unfortunately, these formalisms are not without their own problems. In particular, the genuine MRCC approaches of the above two types face issues related to unphysical [48—50] and singular [48,49,51—54] solutions, intruder states [48,49,52], and intruder solutions [48,50], all of which can cause convergence problems as well as other complications in the MRCC calculations and the ensuing analysis of the results. Furthermore, there are potential difficulties related to the size and choice of the multi-dimensional reference space for certain types of systems. Indeed, the 4 configurations included in the complete multi-dimensional reference space are gener- ally determined via all possible rearrangements of the occupancies of a selected set of molecular orbitals composing what is referred to as the active space. Unfortunately, the size of such a reference space grows factorially with the number of active orbitals and electrons, and so choosing a proper active space for a given system can be a difficult task requiring a great deal of expertise. In fact, it is possible that for some systems, such as those containing transition metal atoms that have a large degree of quasidegeneracy due to the open f or g shells, the appropriate active space may result in the MRCC calculation being prohibitively expensive. In addition, the com- plicated formalism associated with the genuine MRCC theories makes it difficult to implement highly—efficient, general-purpose computer codes that can be applied to a wide range of open-shell problems. Recently, there has been a great deal of progress and renewed interest in overcoming the above issues and further developing both the valence-universal and state-universal MRCC approaches [48,55—67]. However, despite these developments, we feel that, due to the complications arising from a genuine MR formalism, it would be ideal to investigate the possibility of overcoming the difficulties facing the standard single-reference CC schemes in calculations involving open-shell systems within the formally simpler single-reference framework. One methodology that may provide a mechanism by which to address the chal- lenges posed by open-shell systems within a single-reference formalism is that of the electron-attached (EA) [68—70] and ionized (IP) [71—77] EOMCC theories, and the analogous and historically older EA and IP SAC-CI methods [78—84]. The basic idea behind these methodologies is to construct the ground- and excited-state wave functions of an (N :l: l)—electron system by applying a linear electron-attaching or ionizing operator to the correlated CC ground state of an N-electron closed-shell system. This wave function definition leads to a natural and con'iputational1y con- venient formalism for studying ground and excited states of open—shell systems, such 5 as radicals, that differ from the corresponding closed-shell species by one electron. Fhrthermore, the use of the closed-shell N—electron reference state in calculations for the (N :l: 1)-electron systems ensures that the resulting wave functions are auto- matically orthogonally spin-adapted, and thus EA— and IP-EOMCC approaches do not suffer from the spin contamination issues that may arise in the traditional open— shell implementations of the conventional CC or EOMCC approximations that rely on the unrestricted Hartree—Fock (UHF) or restricted open-shell HF (ROHF) refer- ence determinants. Unfortunately, as was the case for the regular CC and EOMCC methodologies, the basic, low-order EA- and IP-EOMCC approaches, which include the EA-EOMCCSD [68,69] and IP-EOMCCSD [71—74] approximations, and their EA and IP SAC-CI analogs truncated at 2-particle—1—hole (2p—1h) and 2-hole—1-particle (2h-1p) excitations [78—84], have significant difficulties with describing the excita- tion spectra of most radicals [68, 84—89]. One can address these deficiencies through the inclusion of higher-order components of the electron-attaching or ionizing oper- ators, such as the 3p—2h or 3h-2p excitations, which gives rise to schemes such as EA-EOMCCSDT [70], IP-EOMCCSDT [75,76], EA-EOMCCSD(3p—2h) [85—87], IP- EOMCCSD(3h-2p) [85—87], EA-EOMCCSDTQ [90] and IP-EOMCCSDTQ [90] as well as their less complete SAC-CI analogs [81—83, 88]. Though these schemes are all capable of providing high quality results for radical systems, the associate compu- tational costs are usually prohibitively high, restricting their use to relatively small systems. Given how well suited the EA- and IP-EOMCC methodologies, as well as their multiply-attached and multiply-ionized counterparts (e.g., the doubly electron-attached (DEA) and doubly ionized (DIP) EOMCC approaches [91-93]) are for studying rad- icals and other open-shell systems, it has been our goal to develop a new formula- tion of these schemes that maintains all the good attributes of such methods while avoiding the high computer costs associated with including higher-than 2p-1h/2h-1p 6 effects. An idea for how to develop such a formulation is provided by the so-called active-space CC [94—108] and EOMCC [28,29,109—111] approaches. In these and related schemes [112—119], the multi-reference concept of active orbitals is used to a priori select the dominant triply and other higher-than-doubly excited clusters in the standard CC/EOMCC calculations. In this way, the computational costs associ- ated with the high-order CC/EOMCC approximations are greatly reduced, since the vast majority of the higher-than—double excitations are not included in the calcula- tions, and the characteristic high accuracy of the high-order CC/EOMCC schemes is maintained at the same time. Indeed, the lowest-order active-space CC meth- ods, such as SSMRCCSD(T) (state-selective MRCC with singles doubles and active— space triples) [97—105] or CCSDt [106—108], and their excited-state EOMCCSDt ana- log [28, 29, 109], have shown promising results, even for challenging cases involving bond breaking [95,96,100,101,104—108] or excited states dominated by two-electron transitions [28, 29,109—111]. Thus, it would seem that combining the active-space CC methodology with the EA- and IP-EOMCC formalisms is a natural mechanism for developing an approach capable of performing highly accurate calculations for radicals at a relatively low computational cost. The development and benchmarking of such active-space EA- and IP-EOMCC approaches, in which higher-than 2p—1h and higher-than 2h— 1p components of the electron-attaching and ionizing operators, respectively, are selected through the use of a suitably defined set of active orbitals, is one of the primary goals of this dissertation, and is discussed in Chapter 2. Despite the initial successes and considerable promise of the active-space EA- and IP-EOMCC formalisms in studies of the excitation spectra of open-shell systems developed as part of this work, we must remember that no method is bullet-proof. Indeed, the use of the multi-reference concept of active orbitals means that these methods are not ‘black-box’; they require some a pmior‘z' analysis by the user of the nature of the electronic states of interest before performing the calculation. Though 7 this difficulty is not nearly as severe in the active-space methods as it is in the genuine MRCC theories, primarily due to the fact that the computer costs of the active-space schemes, including the EA- and IP-EOMCC approaches developed in this thesis re- search, scale polynomially with the size of the active space rather than exponentially, they are still somewhat more difficult to use than the conventional single-reference CC methods. Furthermore, the structure of these theories is such that they are not generally applicable to all types of open-shell systems. Indeed if one wants to study open-shell systems that are M -electrons away from some closed-shell system, then a different hierarchy of methods must be used for each value of M. The EA- EOMCC and IP-EOMCC schemes described in this dissertation are applicable to the case of M = 1, which includes the majority of radicals and positively or negatively charged ions of closed—shell atoms and molecules. If one wants to examine systems that are two electrons away from a closed-shell, including biradicals, schemes such as the DEA- and DIP-EOMCC methods and their active-space variants [85] must be implemented and applied, etc. Furthermore, the applicability of approaches based on adding or removing M electrons from a closed-shell species to open-shell systems where M > 2 may become questionable since as one moves further away from the closed-shell system, the electronic similarities between the two decrease. As a result of these potential complications, alternative, and perhaps even complimentary, methods for studying open-shell systems would be useful. Indeed, part of the success and pop- ularity of the CCSD(T) approach is that it is able to produce highly accurate results for systems primarily described by dynamical correlations with both reasonably low computational costs and an easy-to—use black—box nature. It is this nature that has allowed CCSD(T) to be so easily accessible to both experts and non-experts alike. Given these remarks, the development of a robust formalism that maintains the com- putational costs and ease-of—use of CCSD(T) while better balancing both dynamical and nondynamical correlation effects than CCSD(T) would be a good alternative to active-space EA- and IP-EOMCC methods for accurate studies of open-shell systems. An excellent candidate for such a methodology is presented by the completely renormalized (CR) CC and EOMCC methods [120—140], particularly the recent vari- ants based on the so-called biorthogonal method of moments of coupled-cluster (MMCC) equations [133—140]. These approaches represent a new class of CC schemes based on adding noniterative corrections to the standard CC or EOMCC energies, which are designed to improve on the performance of CCSD, CCSD(T), and EOMCCSD in situations involving larger nondynamical correlations while maintaining similar costs and ease of use. Furthermore, these methods have a natural hierarchy for construct- ing corrections for truncations besides CCSD and due to higher-than-triple excita- tions, as well as a natural extension to excited states. One of the most promising methods of this type is the CR—CC(2,3) approach [133—140], which, in analogy to CCSD(T), is based on adding a noniterative correction due to triple excitations to the ground-state CCSD energy. Various applications of this scheme have revealed that it is capable of producing high quality results in studies of single bond breaking and biradical structures on singlet potential energy surfaces [133-135,137,141—147], while offering excellent values for activation barriers in thermochemical kinetics stud- ies [148,149]. Indeed, for situations where the structure is dominated by dynamical correlations, CR—CC(2,3) is as accurate as CCSD(T), but, unlike CCSD(T), it main- tains these high accuracies as one moves onto structures characterized by stronger nondynamical correlation effects, such as biradicals or the bond-breaking regions of a potential energy surface. This success for singlet states characterized by large nondynamic correlation effects begs the question of whether CR—CC(2,3), and its excited-state CR-EOMCC(2,3) analog, would perform equally well in calculations in- volving open-shell systems, such as bond breaking and excited states of radicals, and singlet-triplet gaps in biradicals. In addition the question as to how well higher-level CR—CC schemes, such as the CR—CC(2,4) approach which corrects the CCSD energy 9 for the effects of both triples and quadruples, would perform. To address these ques- tions, the CR—CC(2,3) scheme was extended to general open-shell systems [138] and excited states [135,140], and a general purpose CR—CC(2,4) code, applicable to both closed— and open-shell systems, was implemented as part of this research. The details of these methodologies as well as the results of selected applications to open-shell systems, reported in [138—140, 150,151], are presented in Chapter 3. Up until this point, all of the discussion regarding the CC theory and its applica- tions to open-shell systems has been in the context of quantum chemistry. However, there is nothing intrinsic in the CC wave function ansatz, nor in the formulation of the various CC methods studied in this work, that restricts their use to chemical sys- tems. Indeed, the underlying physics governing any many-fermion problem, whether they be chemical, nuclear, or condensed matter to name a few, is fundamentally the same, and it is only the form of the potential in which the fermions move that varies. In fact, though the major developments of the methodology have occurred in the context of quantum chemistry, the CC theory was actually first suggested within the field of nuclear physics [1,2], and it is our belief that a reintroduction of CC ap- proaches within nuclear physics would benefit the study of the structure of nuclei. Indeed, two of the main techniques for studying nuclear structure, namely the Green’s function Monte Carlo [152] and no—core shell-model [153—156], though successful in providing highly accurate results, suffer from extremely high computational costs and so are limited to light nuclei with a dozen or so nucleons at best. In order to study medium-mass and heavy nuclei, methods that better balance accuracy and computa- tional cost are needed, and, as discussed above, such a balance is precisely the origin of the success of the CC theory in quantum chemistry. Furthermore, due to the fact that the shell structure of nuclei is very similar to that of atoms and thus displays a large amount of degeneracy, the majority of nuclei are open-shell n‘iany-fermion systems characterized by large nondynamical correlation effects. As a result, the 10 methods developed and studied in this work, which are designed specifically for prop- erly describing open-shell systems while requiring reasonably low computer costs, are particularly well-suited for studies of nuclear structure. Thus, as a final component of this dissertation, selected results of CC studies of the ground— and excited-state energies of various nuclei [157—171], with a focus on the results obtained by the author of this thesis [157,161—170], are presented and discussed in Chapter 4. 11 Chapter 2 Active-Space Coupled-Cluster Methods for Open-Shell Systems In this chapter, the extension of the active-space CC methodology to the EA- and IP-EOMCC formalisms, which results in new classes of low—cost, highly accurate ab initio approaches for ground- and excited states of open-shell systems, is discussed. Section 2.1 provides the theoretical details of these approaches, including an overview of the original EA— and IP-EOMCC theories and the details of the new active—space extensions of these methods, as described in [85—87], as well as a description of our recently developed highly efficient computer implementations of the most basic active- Space EA- and IP-EOMCC schemes [87]. Section 2.2 provides examples of several benchmark calculations, taken from {85—89, 140], in order to illustrate the performance of the active-space EA- and IP-EOMCC approximations developed in this work. 12 Active : Space : 50' 50' o . 40' : 40'— : \§ ........ 17:, Irrx—-.-o(§/Creating Creatlngg """" my 5 In,““" 5 OH CH §+o— 30' 30-0—0—i —H— 20' ZJ—C—C— —H— 10' [0—0—0— Figure 2.1: Pictorial illustration of the generation of the CH and OH radicals from the closed-shell CH+ and OH’ ions, respectively. 2.1 Theory and Computer Implementation 2.1.1 The Electron-Attached and Ionized Equation-of—Motion Coupled-Cluster Theories The key idea behind the EA- and IP-EOMCC methodologies is that, rather than treating the open-shell molecular (N :t 1)-electron system of interest directly, one instead generates the ground and excited states of it by adding an electron to or removing an electron from the related N -electron closed—shell system. This idea is illustrated in Figure 2.1, which gives a schematic representation of generating the CH and OH radicals from the closed-shell CH+ and OH“ ions, respectively. This idea can be expressed more rigorously by using the f(:)llowing form for the, electronic wave function of the nth state of an (N + 1)- or (N - 1)-electron system (where p = 0 corresponds to the ground state and p > 0 corresponds to the excited 13 states): (Ni1)>_ R(Ni1) )I pp x110.) (2.1) In this equation, RLNH) and RSV-1) are the electron-attaching and ionizing oper- ators, respectively, whereas [\IIO) is the correlated ground-state wave function of the N -electron closed-shell system, which is defined through the exponential ansatz of the single-reference CC theory, we = eTl>. (2.2) In the above equation, |) is a closed—shell N -electron reference determinant (e.g., the restricted Hartree Fock (RHF) reference) and T is the cluster operator of the standard single-reference CC theory, 2 1 ' T = E Tn, Tn = (a) 15311,,Zinaal ”cane,” ...a,-1, (2.3) where in the exact case ll/IT = N while in the approximate approaches MT < N. For instance, in the basic CCSD approach MT 2 2, so the cluster operator T (CCSD) is given by T = T1 + T2 = tgaaai + at zjbaaabajai. (2.4) Throughout this paper we employ the usual notation where z', j, . . . (a, b, . . .) refer to the spin-orbitals occupied (unoccupied) in the reference determinant |), up (up) are the creation (annihilation) operators associated with the spin-orbital basis set {|p)}, in -an entering Eq. (2.3) are the usual cluster amplitudes. In and the coeffic1ents tall" addition, whenever possible, we make use of the Einstein summation notation over repeated upper and lower indices. The electron-attaching and ionizing operators introduced in Eq. (2.1), HP +1) 14 (N-l) and R], , respectively, are defined as AIR N+1 Rig ) = Z Rp,(n+1)p—nh (2'5) 11:0 and AIR N—l Rf! ) = Z Rp,(n+1)h-np (2'6) =0 where the ((n + 1)1)-nh) component of RLNH) and the ( (n + 1)h—np) component of RLN—l) are given by 1 2' i 1--- n a a] an . . and 1 ulna, a _— — 1 an. . . . R#,(n+1)h_np ”Kn 1M7“ almana ...a am . . . (1,10,, (2.8) and where M R = N in the exact case and A! R < N in the approximate schemes. Equation (2.7) reveals that the ((n + l)p—nh)-components of RLNH) can be viewed as operators which create a particle in an unoccupied spin-orbital and, for n > 0, simultaneously cause an excitation of n electrons from occupied spin-orbitals into unoccupied spin-orbitals. Similarly, from Eq. (2.8) we see that the ((n. + l)h-np)- components of RSV—1) can be viewed as operators which remove one of the electrons from an occupied spin-orbital and, for n > 0, simultaneously excite n of the remaining electrons. By substituting the EA- and IP-EOMCC wave function ansatze, Eqs. (2.1) and (2.2) in which M R S MT, into the time-independent Schriidinger equation, one ob- tains the following non-Hermitian eigenvalue problem: - N N N (HN,openR;(L i1)lCiq» :w/(L inRit inl‘Pi (2-9) 15 fiere HN,open = (HNeTlepen = e_THN€T ‘_ (HNeTlC,elosed (210) is the similarity-transformed Hamiltonian of the CC theory in the normal-ordered form relative to the Fermi vacuum |), the subscripts “open”, “closed”, and C refer to the open (i.e., having external lines), closed (i.e. having no external lines), and connected parts of a given operator expression, and the eigenvalues obtained from solving Eq. (2.9) are the energy differences given by wLNil) = ELNi1)— EéN), (2.“) (Nil) . where Eu IS the total energy of the pth state of the (N :l: 1)-electron system and ESN) is the ground-state energy of the N -electron reference system. Thus in the EA- and IP-EOMCC formalisms, the energies of the ground- and excited-state wave functions of an (N :t 1)-electron system are obtained by diagonalizing the similarity- transformed Hamiltonian HNppen, calculated using the cluster operator of the related N -electron closed-shell system, in the subspace of .9? (N +1) spanned by the determi- nants |a) = aa|<1>) and leaflfii") = aaaal Had-nah, . . . (1,1 |) (n = 1,... ,MR) in the EA-EOMCC case, and in the subspace of if (N "1) spanned by the determi- nants [(1%) = a,|) and [(szllzfl") = aal ...aa"a,-n...a,°1a,-|k,Aj>kpj>k,Ak>l,AJ>k>l,BJ>k>l,cJ>K>l>m,C13> I‘I’ubfi) a . : :2: :52: :22: 4 I285 >2“;- 7> 72,3» IIJK> J Ijk satisfies the active-space requirements of Eq. (2.14) and so no further constraints on the indices defining this term have to be imposed. Similarly, if we consider the projection on IQA?§) (projection type 2 in Table 2.2), we obtain ABC k D jk C2( )= —2h"77nATB7Z:I€ , (2.31) which also requires no additional restrictions on the spin-orbital indices, since the 3p—2h amplitude TBC: has at least one active unoccupied index, as required by Eq. (2.14). Furthermore, comparing Eqs. (2.30) and (2.31) reveals that the only difference between these two terms is in the restriction on the index c, which in Eq. (2.30) is restricted to active unoccupied spin-orbitals and in Eq. (2.31) to inactive virtual spin- orbitals. This straightforward relationship between Eqs. (2.30) and (2.31) allows us to recombine these two contributions into one, somewhat more general term of the form BM}: (2 )= —gh{,fArB’gek, (2.32) where the unoccupied index c can be active or inactive. When we consider projection type 3 from Table 2.2 (the projection on IQAEE», we obtain AbC _ 1 je k ‘ . D jk (2) _ 2hmArCTble’ (2.33) 31 where we have made use of the antisymmetric properties of the 3p—2h amplitudes to maintain the ordering of the spin-orbital labels employed in Eq. (2.14) (i.e., the index restricted to active spin-orbitals is the leftmost of the unoccupied spin-orbital labels). We have moved the index C in rail: to the leftmost position because it is more convenient to have a consistent placement of the active spin-orbital labels when implementing the EA—EOMCCSDt and IP-EOMCCSDt schemes. As with the previous two cases, no further restrictions on the spin-orbital indices defining this term are required. Finally, we consider the projection on ICDAIJ’E) (projection type 4 in Table 2.2). The resulting expression for this contribution is b _ . DA.7<2>=—2h:5..rb:k, (2.34) mk where, unlike in the previous cases, the Tbce amplitude which enters this term does not automatically have at least one unoccupied index constrained to active unoccupied spin-orbitals. In order to impose such a condition, we must restrict the summation over all particle spin-orbitals e in Eq. (2.34) to active spin-orbital labels only since b and c are virtual (i.e., inactive) indices. The resulting expression is given by DA77<2> = —%5§A7‘E7§§, (2.35) where the overtilde denotes the fact that we imposed restrictions on the summations appearing in DA?g(2) and where once again we have made use of the antisymmetric nature of the 3p-2h amplitudes to maintain the order of unoccupied indices used in Eq. (2.14) (active indices precede the generic or inactive ones). Equations (2.32), (2.33), and (2.35) represent all contributions to the EA-EOMCCSDt working equations which result from the single term in the EA-EOMCCSD(3p—2h) equations projected on 3])- 2h excited determinants given by Eq. (2.28). By applying the above procedure to each term which enters the explicit form of 32 the equations defining the EA-EOMCCSD(3p—2h) and IP-EOMCCSD(3h-2p) meth- ods (Eqs. (A.1),(A.2), (A.3),(A.7),(A.8) and (A.9) in Appendix A) and by making use of the projection types given in Table 2.2, one can derive the explicit form of all contributions to the EA- and IP-EOMCCSDt equations. After combining all of these contributions together, we obtain the final form of the fully factorized, computation- ally efficient EA-EOMCCSDt and IP-EOMCCSDt equations, which are presented below. We begin with the expressions for the EA-EOMCCSDt scheme. The factor- ized equations defining the projections of the EA-EOMCCSDt eigenvalue problem on the 1p determinants |a) are (CCSD N+1) N+1 («WK Nopen)R( )0) 12>) = ‘Ia = of, )ra, (2.36) where TA = XA + ivSfirfié’} (2.37) and I. = x. + 7 2) 222.277}: (2.38) E2) are (CCSD N— 1 ' (<1) I( Nopen’R(N"1’C) |)= T=w£ ’72, (2.65) where 19f 177777 (:1 2X1 'i'2111’77777"~ ef (2.66) and — —x +5 2 viil My}, (2.67) M>77 with Xi ___ —}—7:nr 777 + hfnre 7777_ 1’76. n777T 7.7767277. (2.68) The IP-EOMCCSDt equations defining the projections on the 2p-1h determinants [Cpl-5’) are given by (5. 5’7 Hfiffpi‘filRLN ’)c|<1>)= mi =77)”- 1’ ’5} (2.69) where 111]): XI b + 0113+ ghnmrlgén, (2.70) zig = A” — 57",}, (2.71) and 2‘}, = A)“, (2.72) 36 with Xzb = —§h7nbrm _I_"i777’m] + 517572.23 +ZI177777Tmn — high“)? + 216%] 67,7 (2'73) Ij __ 1-eTIj777 efrljm 07 b — QIW + Ihbm ef’ (2'74) A’J— _ xi, J+ 2 7,7; ”M333, (2.75) M>77 and TMni ij 16f M'u j+ Z thT (2+ QhMTM be + ZhbMT 61" (2'76) M>77 Finally, the IP-EOMCCSDt equations defining the projections on the selected 317-27) determinants |IIJ’%) have the following computationally efficient form: (CCSD N—1 k Ik N—l I"k (2 37H HNopen’Rf. ’)cI>= «94ch 730,: w}. ’r 3,, (2.77) where 5177:777—757-(575 (2.78) I'K A'I K K' I 23C: J +6 J (2.79) and I'k I'k T&=A&, am) with I k k ’7 gc— — I177777771757én, (2.81) 613': = 37.3, 713;" + 272 2333*, ”:3", (2.82) 37 Ijk _1jk 1777 U km k_ hke 1] X be _ thbrc + h InbrC _ 2Ihbc Ger] 1hbc'r e + 2’13? 135+ She: [31:]; + ZIjkthZ ilkI n b(' [C kc 71 k I ’6 UK _ 15K 1 - Mnj Abc —Xbc_QZhM77 bc’ M>77 and Ijk_1jk_1 Ik M777 I Mjk_1-Ie Mjk A —X be 2:: thT _ZlthT be H?’ Mc 2M>77 7‘ be. (2.83) (2.84) (2.85) As was the case with the EA-EOMCCSDt scheme, the IP-EOMCCSDt equations utilize several additional recursively generated intermediates which are as follows: 6 8f 77777 I ‘20777777' f 7 11 f ~1 f 1ef 177777 C —Q 1’77777'r 60 7 and if _ ~if 8f M i 16 —IC— 2 anr 2.2, M>77 where i7f_ _ ‘hiildm + ghnfmrmn— I‘lgfnr7g7, and Jk ~Jk J k I 77 = I 77 +%1’767{77T g} 7 K K 77 [J — _1j77 + $02577 e]; 7 and jk ~jk 1 ef Mjk I 1" _§IM77T ef’ (2.86) (2.87) (2.88) (2.89) (2.90) (2.91) where [~37]: : —;L‘77,lfn7'7n — udjkiI/Sf7l7jgn. (2.93) The antisymmetrizer 527;” = .Q/pq, which enters the above equations is defined as ”m E 5237”" = 1 - (77(1), (2.94) reSpectively, with (pq) representing a transposition of indices 7) and q. We now consider the most important details of the efficient computer implemen- tation of the EA-EOMCCSDt and IP-EOMCCSDt eigenvalue equations, Eqs. (2.36)- (2.93). Figures 2.2 and 2.3 give the key elements of the algorithms that are used to compute the projections of the EA- and IP-EOMCCSDt eigenvalue problems on the selected 3p—2h and 3h-2p determinants, Ex?) and “DIE-fig), respectively, which are the most expensive and difficult parts of the EA-EOMCCSDt and IP-EOMCCSDt cal- culations (see Eqs. (2.48)—(2.56) and (2.58)-(2.64) for the EA—EOMCCSDt case and Eqs. (2.77)—(2.85) and (2.87)-(2.93) for the IP-EOMCCSDt case). The algorithms for calculating the remaining 1p, 2p-1h, lb, and 2h-1p projections are similar and are not discussed here. One of the key features of our algorithm is the fact that the explicit loops that are used to construct the EA-EOMCCSDt and IP-EOMCCSDt equations projected on |Ajblf) and IQIIfiI’ Eqs. (2.48) and (2.77), respectively, range over active indices only, as indicated in Figures 2.2 and 2.3 by the use of bold, uppercase letters for the looping variables. Within these usually short loops, thanks to the use of the one- and — (CCSD) two-body matrix elements of H "'q ‘rs . J - ‘. ’ N,open 7 hp and hpqa reSPGCtIde, and the rccursn ely generated intermediates defined above, our code possesses a high degree of vectoriza- tion, allowing us to exploit highly efficient, fast matrix multiplication routines from the BLAS library to perform the necessary computations. To avoid confusion with the summations performed by the explicit loops over active indices, all of the remaining 39 summations which are performed by the fast matrix multiplication routines are explic- itly labeled in Figures 2.2 and 2.3 using the traditional summation symbol 2, rather than relying on the Einstein summation convention used in the rest of this disserta- tion. By incorporating the short loops over one or at most two active indices at a time, we are able to make the full use of the computational benefits offered by the fast matrix multiplication routines while simultaneously ensuring that any unnecessary overcom- putation of terms of the EA-EOMCCSD(3p—2h) and IP-EOMCCSD(3h—2p) meth- ods that vanish in the active space EA-EOMCCSDt and IP-EOMCCSDt schemes is avoided. As a result, the EA-EOMCCSDt and IP-EOMCCSDt codes described in this work take full advantage of the low Nungnfi and Nongnfi CPU operation counts characterizing these approaches. In addition, although our current implementation is a serial code, the explicit loops over active indices used in our algorithm can easily be parallelized without altering the encompassed fast matrix multiplications, further improving the efficiency. Finally, it should be noted that the explicit loops over ac- tive indices make it possible to avoid storing the Nungng (the EA-EOMCCSDt case) and Nongnfi (the IP—EOMCCSDt case) objects in memory. As a result, the memory requirements for the present implementations of the EA— and IP-EOMCCSDt meth- 3 ods are ~ 277077,, words, i.e., similar to the memory requirements of the conventional CCSD or EOMCCSD approaches. These memory requirements are solely defined by H(ccso) the construction of the matrix elements of N, Open (see Table 2.1) and are the same as the memory requirements characterizing the low-order EA—EOMCCSD(2p—1h) and IP-EOMCCSD(2h—1p) schemes. The highly efficient computer programs based on the above algorithms for the EA-EOMCCSDt and IP-EOMCCSDt methods and the cor- responding EA-EOMCCSD(2p—1h), IP-EOMCCSD(2h-1p), EA-EOMCCSD(3p—2h), and IP-EOMCCSD(3h—2p) codes were interfaced with the RHF/ROHF and integral routines available in GAMESS. The EA—EOMCCSD(2p—1h), IP-EOlV’ICCSD(2h—1p), EA-EOMCCSD(3p—2h), and EA-EOMCCSDt GAMESS options will be released to 40 the world within the next few weeks. The release of the IP-EOMCCSD(3h-2p) and IP-EOMCCSDt options will follow in the not-too—distant future. 41 Figure 2.2: The key elements of the algorithm used to compute ((DAJI-IEI(HICCSDIRLN+1))C|), Eq. (2.48), in the efficient implementation of N ,open the EA-EOMCCSDt method. 42 Calculate Ia; for all values of a,b,f, Eq.(2.64) Calculate 7,7,, for all values of. a, m, j, Eq.(2.60) SetI Iabf_ — Ia bf for all values of a, b, f Set Iam- — am for all values of a, m ,j LOOP OVER D Calculate 115;: IDE —% Z US$763 for all values of b,f, Eq.(2.61) e,,mn Calculate 1&2: ”mfnTDae for all values of a, f, Eq. (2. 62) 26,,77’27717 Calculate Ia f=I :Df+12 vaf "I'Dab for all values of a, b, f, Eq. (2.63) Calculate IDJm- — ID] m+ 217%: vfnnrDje f for all values of m ,j, Eq. (2.58) e ,,f77 Calculate 1am: a777+Z Z vmnrDIfI; for all values of a, 7n,j, Eq.(2.59) " f(>D) END OF LOOP oven D LOOP OVER A jk Calculate ’IAbc’ fiAbc’ and XAJcb for all values of b, c ,j,k, Eqs.(2.52)-(2.54) Set A jk— jk for all values of b c 'k Abc — XAbc 7 73’ LOOP OVER E Calculate AAbkC— — AAfllg-é Z 1713 CTEfb for all values of b, C ,j,k, 2f()>E Eq. (2.55) . Jk JE 7771: Calculate AAbcz AAbc 22 hAc E fb + ZihArEbc 2 thATEbc f)(>E In for all values of b, c,j,k, Eq. (2.56) END OF LOOP OVER E Set IAbcz AAbkc’ Eq.(2.51) END OF LOOP OVER A LOOP OVER A LOOP OVER D Calculate TAD]; for all values of c ,j,k, Eq. (2. 49) Calculate IAbD for all values of b,j,k, Eq. (2.50) END OF LOOP OVER D A — CCSD Calculate (‘1’ Jbgl(H1(\/',open) RLN+1))C|) by antisymmetrizing ‘IA‘IbIz, Eq.(2.48) END OF LOOP OVER A 43 Figure 2. 3: The key elements of the algorithm used to compute ((1)1? -k|(H 1113553)}? BIN—1))CIQ) Eq. (2.77), in the efficient implementation of the JI-P EOMCCSDt method. 44 Calculate I]: for all values of j,k,77, Eq.(2.93) Calculate Iiéf for all values of i,c,f, Eq.(2.89) Set Ijk =Ijk for all values of j,k, 77 Set Ic sz II; for all values of 7, c, f LOOP OVER L Calculate ILA: = ILk +1 Z v7,{ang;k for all values of 173,77, Eq.(2.90) m ,e ,f Calculate Ijfi=1515+§2 777,7,an ef mfor all values of j,77, Eq. (2.91) m,e ,f Calculate 151;: Ijk- 122%UL717L‘Ifkfor all values of j,k, 77, Eq. (2.92) Calculate IIéf = IIgf— 18 thZ 7)$,,,7‘an for all values of c,f, Eq.(2.87) Qflmne Calculate [10 = 115—: X 71317ng for all values of i, c,f, Eq.(2.88) ‘9 n<(L) END OF LOOP OVER L LOOP OVER I Ijk Ijk Ijk . Calculate 7bc’ [3 bc’ and x bc for all values of 1,17, b,c, Eqs.(2.81)-(2.83) Set A195: X12119 for all values of j,k,b,c LOOP OVER M Calculate AIjK— — AI‘IK— % Z hlVIKn 7M1? for all values of j,K,b,c, 77(Q-oo-w:w am: 88.” 83 888.8 88.8 38.8 28.... 8.8 88.8 +m8 D :3 mm...“ 38.8 3m.” we...” 83 38.8 as 38 (mm m Km: 58 :8.“ 58.8 was 8.8.8 82.... 88.8 was <8. #8 as: 83 23 82.8 38.8 .883 88.8 88.8 8.3 (m8 8 :8 28.8”- 28 88.8”- ”8 $8.88. Ea 28.8...- mm. 88.8”- .38 28.8”- can 88.8...- :8 x NH>m-oo-w:m $8: 88..” was... 88.8 $3. 83. Sam .8... 88.... +5 b :8: was was 33.. 888.8 km.” 82.8 a...” :88 (mm m aw: Sm so.” 88.” 888.8 88.... 888.8 No.8 8.8.8. 48 8. am: 83 @888 2.8.8 8.88.8 888.8 883 88.8 38 (N8 8 88 £8.88..- was 888.8”- 88... 88.8. 8: 888.8- so £3”. .88 $38- mow 88.88”- :8 8.. ND>Q-oo-w=.m ea ..Aozomz 858.00 8580 388.8580 $23800 8.880208... ammoozom 8.8m 28-5 >580 580 49 .amaob mes» H.338 88 3.88 $832 23 3033538 @338on :e 5 .3mew2 Soc 5x3 98 82853813 OUEOmiOU 2%. E cam: mmfluoaomw .8205: was 8.2328 cosfiwoxm Runwaflomxm .8meme 8833688 as... Sm >m was .3388 magmccsew one Hem @898: was 8.33 .SwTwsa 3% Emma— AG use .8 .QHHV NH>Q-oo-w=w 23 fig 8:830 mm. 4868 m0 was. m0 889% @3858 wEmTBB map 3 wcwcaoammhoo mmwwaoao :oSefioxo 252.868.. 23 was 8.2326 mpeuméqsgm 23. “MN @385 Table 2.4: The average time per iteration for the EA-EOMCCSD(3p—2h) and EA- EOMCCSDt calculations“ performed for the CH radical with the aug—cc-meZ (:1:=D, T, and Q) basis sets [178—180]. The average times T are reported as T / T2p_1 h: where T2p_1h is the average time per iteration for the corresponding EA-EOMCCSD(2p—1h.) calculation. In all correlated calculations the lowest energy core orbital was kept frozen. Method aug-cc—pVDZ aug-cc-pVTZ aug—cc—pVQZ EA-EOMCCSD(3p—2h) 68.67 61.42 56.42 EA-EOMCCSDtb 7.39 4.18 3.00 EA-EOMCCSDtc 9.44 5.57 3.45 CCSDd 8.56 5.86 4.68 EOMCCSDe 1.72 1.00 1.19 3‘ For consistency purposes, the average iteration time is computed using the first 10 iterations of the calculation for the ground state of the CH radical. The excited state calculations show essentially identical timings. b The active space consisted of the 17m; and lvry orbitals of CH+. C The active space consisted of the 1am, lay, and 40 orbitals of CH+. d ROHF-based CCSD calculation for the ground state using codes described in [132]. e ROHF-based EOMCCSD calculation for the a 42“ state using codes described in [132]. of the vertical excitation energies corresponding to six low—lying excited states, as well as the CASSCF-based MRCI(Q) calculations for comparison purposes (see [184] for further information). The equilibrium geometry for SH used in these calculations was taken from [192] and the basis sets considered consisted of the aug-cc-pV(:r+d)Z basis [180,193] for the S atom and the aug-cc—pVxZ basis [178,180] for the H atom (a: = D and T). The RHF determinant of the closed—shell SH’ ion was used as a reference in all IP-EOMCC calculations and the active—space for the IP-EOMCCSDt calculations consisted of the 27m; and 27ry orbitals of SH‘. The lowest-energy core orbital (which correlates with the 13 orbital of S) was kept frozen, and the spherical components of the d and (where present in the basis set) f functions were utilized. In analogy to CH, the 02v symmetry was exploited. As was the case for the CH radical, the SH system is very difficult to describe with the standard, low-order CC / EOMCC methods, such as the basic IP-EOMCCSD(2h- 1p) 50 Table 2.5: The ground-state energies and the vertical excitation energies correspond- ing to the low-lying excited states of the SH radical, as obtained with the aug-cc- pV(:r+d)Z basis set for S [180,193] and the aug-cc-pVxZ basis set for H [178,180], where :c=D and T. Units are hartree for the ground-state energy and eV for the excitation energies.a IP-EOM State CCSD(2h-lp) CCSD(3h-2p) CCSDtb aug—cc—pV(D+d) Z / aug-cc—pVDZ X 2H -398.287 215 -398.294 398 -398.294 167 —398.299 262 MRCI(Q)C A 223+ 4.002 3.936 3.926 3.962 1 42:- 9.168 5.830 5.824 5.437 1 22— 10.575 6.567 6.561 6.034 1 2A 10.713 7.383 7.378 7.001 B 22+ 10.341 8.191 8.210 7.862 1 411 11.738 8.425 8.420 8.079 aug-cc-pV(T+d)Z/aug-cc—pVTZ X 211 -398.378 344 -398.388 242 -398.387 932 —398.395 274 A 22+ 3.987 3.926 3.912 3.949 1 42- 11.252 5.994 5.986 5.527 1 22:— 10.878 6.766 6.758 6.175 1 2A 12.565 7.405 7.398 6.992 B 22+ 12.197 8.265 8.284 7.863 1 4H 12.028 8.561 8.554 8.103 3‘ All calculations were performed at the experimental equilibrium geometry, re 2 1.3409 A, taken from [192]. In all correlated calculations the lowest energy core orbital was kept frozen. b The active space consisted of the 277$ and 2771, orbitals of SH’. c The active space consisted of 10 orbitals; see [184] for further information. 51 Table 2.6: The average time per iteration for the IP—EOMCCSD(3h-2p) and IP- EOMCCSDt calculationsa performed for the SH radical with the aug—cc-pV(:1:+d)Z basis set for S [180,193] and the aug-cc-pVxZ basis set for H [178,180] where n=D and T. The average times T are reported as T/T2h_1p, where T2h_1p is the average time per iteration for the corresponding IP-EOMCCSD(2h-1p) calculation. In all correlated calculations the lowest energy core orbital was kept frozen. Method aug—cc—pV(D+d)Z aug-cc-pV(T+d)Z IP-EOMCCSD(3h—2p) 332 1144 IP—EOMCCSDtb 118 280 CCSDc 44 342 EOMCCSDd 17 71 a For consistency purposes, the average iteration time is computed using the first 10 iterations of the calculation for the ground state of the SH radical. The excited state calculations show essentially identical timings. b The active space consisted of the 27p; and 27ry orbitals of SH“. 0 ROHF-based CCSD calculation for the ground state using codes described in [132]. d ROHF-based EOMCCSD calculation for the A 22+ state using codes described in [132]. approach. Table 2.5 reveals that, with the exception of the A 22+ state, the IP- EOMCCSD(2h-1p) approximation completely fails for the low-lying excited states studied in this work, producing errors relative to MRCI(Q) of 3.93 — 5.73 eV when the aug-cc—pV(T+d)Z basis set is employed. This dramatic failure illustrates how essential the higher-order correlation effects, such as the 3h-2p component of the ion- ) , are when describing the excited states of SH. By explicitly izing operator RSV—1 including the 3h—2p contributions in the IP-EOMCC formalism, we can significantly improve these poor results. As shown in Table 2.5, the IP-EOMCCSD(3h-2p) method reduces the huge errors relative to MRCI(Q) produced by the IP-EOMCCSD(2h-1p) calculations to ~ 0.40 — 0.59 eV. These errors are still larger than one would like, pos- sibly indicating a need for incorporating 4h—3p effects in the calculation, but clearly the IP-EOMCCSD(3h-2p) results represent a major improvement over the poor per- formance of the IP-EOMCCSD(2h-1p) method. Again, as was the case for CH, this improvement comes with a significant increase in the computational costs compared 52 to the IP-EOMCCSD(2h—1p) calculations. As shown in Table 2.6, the average time per iteration required by the IP-EOMCCSD(3h-2p) calculations with the aug-cc- pV(T+d)Z basis set is approximately 1100 times longer than that characterizing the IP-EOMCCSD(2h-1p) calculations. In order to reduce these high costs of the IP-EOMCCSD(3h-2p) calculations we turn to the active-space IP-EOMCCSDt method. An analysis of the excitation e11- ergies in Table 2.5 shows that the IP-EOMCCSDt results are practically identical to those generated by the full IP-EOMCCSD(3h-2p) method. In fact, the differences between the excitation energies produced by these two approaches do not exceed 0.02 eV for all states listed in Table 2.5. Most importantly, the IP-EOMCCSDt ap- proach offers this excellent performance at a fraction of the high costs characterizing the IP-EOMCCSD(3h—2p) calculations. This is shown in Table 2.6, where one can see that the average time per iteration required by the IP-EOMCCSDt approach is about four times smaller than the average time per iteration characterizing the IP— EOMCCSD(3h-2p) calculation, when the aug—cc—pV(T+d)Z basis set is employed. The smaller degree of the savings in the computer effort in this case, compared to the EA-EOMCCSDt calculations for CH, is related to the fact that the ratio of the number of all occupied orbitals to active occupied orbitals in the SH / SH" system is not as large as the ratio of the number of all unoccupied orbitals to active unoccupied orbitals in the CH/CH+ system. The computational savings offered by the active- space IP-EOMCC schemes would be much more dramatic for larger systems, as well as for the higher-order active-space approximations, such as the IP-EOMCCSth method discussed in Section 2.1.2, where both active occupied and unoccupied or- bitals are considered in selecting the dominant 4h-3p excitations (see Eq. (2.22)). On the other hand, it is encouraging to observe that a factor of four speed-up compared to the IP—EOMCCSD(3h—2p) approach offered by the IP-EOMCCSDt calculations for SH is not achieved at the expense of losing the relatively high accuracy of the 53 IP-EOMCCSD(3h—2p) results. 2.2.2 Potential Energy Curves of OH In this section, we discuss IP-EOMCC and SAC-CI calculations of the potential energy curves for the low-lying states of the OH radical [86,88], which were performed using the 6-31G** [180,194,195] basis set and compared to the full CI results that were obtained with GAMESS (Figure 2.4 and Tables 2.7 — 2.10). The equilibrium geometry of OH was taken from [192]. In all calculations the lowest core orbital was kept frozen, the spherical components of the (1 functions were employed, and the 02v symmetry was exploited. The SAC-CI calculations were carried out using the development version of the Gaussian suite of programs that was made available to my advisor by Professor Masahiro Ehara. 54 Figure 2.4: Potential energy curves for the ground and low—lying excited states of the OH radical. Energies are in hartree and the O-H distance RO-H is in A. (a) The full CI results. (b) The IP-EOMCCSD(2h—1p) results. (c) The IP-EOMCCSD(3h-2p) results. (d) The IP-EOMCCSDt results. (6) The SAC-CI(4h-3p) results (doublet states only). (f) The SAC-CI(4h-3p){3,1} results (doublet states only). 55 -743. [Ar—A X 2[1 ]<1—< 1411 Vv——v 122' 8 0 [54—8 142’ g '7580’ G 0A22+ m 1H 1 2A v 0—0 2 2H - . ~+——+ + g 75 2 0(‘S)+H(28), B 22 5 «(1D)+H(2S)'l -75.4~ . ,. ____ 0(3P)+H(ZS 75 [- I 7 ..... - - (a) ] '35 1.0 7 1.5 2.0 2.5 3.0 RGH(Angstroms) -743» .. l A—A X211 \ 8—4 1411 A \ ‘, ~v———v 122‘ 8 ...—48 142 g -75.05 5.x ,‘i\.. - ‘G—OA 22+ m {‘2 Fri“, ”7 // g, f— [[8 8 12A v _£ .4 + a! ’ r t H 2 Zn 5.3 -75.2[ . +_, 322* «‘3 ' 8 8 8 :1 _ _ : ~ 111 . => ~ _ -75.45 .- - - i (b) ’ -75.6L—— ---- ___ __- _1__. __, ---- J 0.5 1.0 1.5 2.0 2.5 3.0 R0H(Angstroms) 56 [ ————‘—I'— -74.8— - A—AXZII <1—<114r1 A . «v—V 122' 8 ' D —- 142' g '75-0’ ‘e—eAzz“ a ‘a—a 12A 0—02211 :6 '75-2 J 1+—+ 322+ E . .1. fl m .. -75.4~ 5 '2 (c) -758 -__+, _,-,__ ___- . J .5 1.0 1.5 2.0 2.5 3.0 RGH(Angstroms) -74.8- A 8 g -75.0~ is -75.2— ’2 LL] -75.4~ -75.6- ‘ . ‘ -..__- . _.4_ __ .— - , 0.5 1.0 1.5 2.0 2.5 3.0 RGH(Angstroms) Figure 2.4 Continued 57 -74.8 1HX :11 EVL——s7 1 22+ A o {G—GA 2 8 121—8 12A g '75-“ 1<>——<> 22H l" 2 + I - i+——+ B 2 v 1 >5 _ , . an 75.2 . . i a v = ‘3\ fi 1 m n V ; g——. a -75.4; . = = . = - . 2, g 1 Mr (9) 1 -75. ...- _ ._ l _ _._L_._—. -44—4 _ _i. 8.5 1.0 1.5 2.0 2.5 3.0 R0_H(Angstroms) -74.8 1HX:H_ {V——V 1 22+ A . iG—CA 2 8 1 H 1 2A S -75.0 10—0221'1 iJr—Jr 2 + a , . B Z >~. _ - an 75.21 . ‘ i 0) ' - = ‘ j 1 c: E\ _ 1 m o ' . ; 1.3———; o -75.4~ . = ' — 7 3 1 1 Kan/$7 (f) i -75.6 . . 1 “A-“ 5 . .—. 0.5 1.0 1.5 2.0 2.5 3.0 RO_H(Angstr0ms) Figure 2.4 Continued 58 As was the case for the CH and SH radicals discussed in Section 2.2.1, many of the low-lying excited states of OH show a strong contribution from higher-order excitations or a significant multi-reference character (cf., e.g., [82,196]). As a result, the OH radical is a challenging system to describe, especially for the low-order IP- EOMCCSD(2h—1p) approach. In fact, if we look at the leading full CI configurations shown in Tables 2.7 and 2.8 we see that, with the exception of the X 2H ground state and the A 22+ excited state, the leading configuration for each state studied, at all four geometries analyzed in these tables, is at least a 2h— 1p excitation relative to the ground-state reference configuration of OH". Just as the standard EOMCCSD approach fails for excited states dominated by doubles, the major contributions of 2h-lp excitations in the excited states of interest should cause a failure of the IP- EOMCCSD(2h-1p) method, and, in fact, this is exactly what we observe. Tables 2.9 and 2.10 show that except for the X 2H and A 22+ states, the errors in the IP- EOMCCSD(2h-1p) energies relative to full CI are huge, ranging from N 70 to ~ 450 millihartree. A comparison of Figures 2.4(a) and 2.4(b) reveals that these errors are not merely a quantitative concern, as this low-order approximation produces a qualita- tively incorrect representation of the excited states of OH. The IP-EOMCCSD(2h- 1p) scheme shows some limited success only for the X 2II and A 22+ states, producing errors in the range of 0.6—4.3 millihartree for the O—H distances of 0.77—1.27 A . How- ever, it is important to notice that even in these two cases the relatively high accuracy of the IP-EOMCCSD(2h— 1p) results rapidly deteriorates as soon as the O—H distances become larger, with the errors in the IP-EOMCCSD(2h-1p) results for the X 2H and A 22+ states steadily increasing to 73.5 and 31.6 millihartree, respectively, at the O—H separation of 3.0 A. As is the case for the other states, this behavior is perfectly in sync with the nature of the leading configurations in the wave functions defining the X 2H and A 223+ states. Tables 2.7 and 2.8 show that at internuclear separations of 0.77 and 0.96966 A, both states are predominantly single-reference states defined 59 by a 1h excitation, and as a result, the IP-EOMCCSD(2h-1p) approach describes them well. As one moves out of the spectroscopic region, however, we see that the multi—reference character of the X 211 and A 22+ states increases, and 211-119 (and even 3h-2p) contributions become significant. In fact, for the X 211 state, which of the two states analyzed here is the one that the IP-EOMCCSD(2h-1p) approach has more difficulty with, a 2h-1p excitation actually becomes the dominant contribution. The significant contributions of the 2h—1p excitations and the failure of the IP- EOMCCSD(2h-1p) approach illustrate the importance of considering the 3h-2p com- ponents of the ionizing operator RLN—l) in calculations of the excited states of OH. Indeed, as shown in Tables 2.9 and 2.10, the inclusion of these effects through the IP-EOMCCSD(3h-2p) approach does offer considerable improvements over the huge errors produced by IP-EOMCCSD(2h-1p). The IP-EOMCCSD(3h-2p) approach does a particularly good job of describing the 1 411 state, reducing the 225.443 millihartree maximum unsigned error (MUE) and 154.788 millihartree non-parallelity error (NPE value; defined as the difference between the most positive and most negative signed errors relative to full C1 along a given potential energy curve) down to 2.674 and 2.374 millihartree, respectively. Since this state is strongly dominated by 2h- 1p excitations and shows relatively small contributions from 3h-2p excitations, as shown in Table 2.7, it is not surprising that it is described so well at the 3h-2p level of theory. Though it does show somewhat larger contributions from the 3h-2p excitations than the 1 4H state, the 2 211 state is also described reasonably well by the IP-EOMCCSD(3h-2p) method, producing the reasonable MUE and N PE values of 7.793 and 6.658 milli- hartree, respectively. The X 2II and A 22+ states are also well described by the IP-EOMCCSD(3h-2p) scheme, especially in the spectroscopic region, where the er- rors relative to full CI are roughly 1.5 millihartree. Unfortunately the growing role of the 3h-2p excitations as the internuclear separation is increased results in errors that become as large as 11.777 and 6.272 millihartree, respectively, as the O—H distance 60 .5888 82.88580 A858 8: 8 80588 @8898 8:: 2: mo 88m 8: wAAAAAAAAoU 89c :38 AAA: .88? 8% 85:58.58 AAAoAmmAU AA 3 mAAAcAAoAAmmAAoQ some £80588 33m 8038595280 528on 025 2: A8 28 :39? 3858580 88%8 95 A58 A5 ._m§m§mbmmbmmb: 83am IE0 22668vo 2: .3 80588950 88888 38m 88ko 2: on. 3358 888530 5 .Amm: 80¢ :33 8&8: 853 885885 3 .8832: m8 3.0% me 8:13 @8028 2: mo .80 $82 AAA A3 3.0 $52 AAA .6 820558 AA AAA; 8058:38vo A2. A Tom .8 835/ 82:5 .8“ 8820580 AAAAAA 2:5. :55. AAAAV- AA; _A A55VA AAAAVA AAAAVA A5AVA A5AV_ AAA5A AAAVAV AVAAAV- AAAAV AAAVAV- _AA5AVA AAAAAVA A55AVA A5AVA AAA5AV AA5 AV_ AAAAAA AAAAV AAA. AAA 58.? AAAAVAAVAAAV- A5AA .AAAA- 5_AA55VA AAAAAAVM AAAAVA AA5AV MA5AVMA5AV_ AAA 8AA 3AA- AAAAV AVAAVAV- _AAAAAVMAAAAVAA5AVAA5AVA A5AV_ AAAA AAAAAA ASA- AAAVAV- AVAAAV- A5AA- _A AAAAVA A55VA A55AVA A5AVA A5A AVAA5AV_ AAAAAA AAAVAV- A5AA- AAAAV- AAAAV- _A AAAAVA A55VA AAAAAVA AAA55AV AAA5AV A5AA A5VAA AV_ AAAAAA ASA AAAAV- AAAAV AAAAAV _ A5A VA AAAAAAVA AAAAVA AA5A AVAA5A VAA5 AV_ AAA 83 A23 AAAAV AAAAV _AA5 VAAAAAAVA A55AV AA5AVAA5A VAA5 AV_ 85A 5AA; AVAAAV- AAAAV- 3AA ASA _AA5A5VAAA AAEVI AA 5AVVAA5AVAA5AVAA5AV_ 515A AAAA AAAA AAAAV AAAAV- _AA5A5VAAA55AV AA 5AVVAA5AVAA5AVAA5AV_ 4AA 5A-AAA 8AA A5AA- A5AAVAV- AAVAVv _AA5A5VA AAAAV AAAAAVAA5AVAA5AV_ AAAIAA AAA .83 AAA .83 A5A.A.AAA.A AVAAVAV .AAVAVv 5_A A55V AAAAAAVAAEAVA A5AVAA5AVA A5AV_ AAA AAAtAV- AAAAV AAAAV A5AA _AAAAAVAA55AVAA5AVAA5AVAA5AV_ AAAAA. 525. 5555558 .A. 8A AA AAA AAA. 5853 AA A; 555880 A5535 85558550 ssm +70% 8058.598 8228.888 mo new @3028 AA .8“ 8068 m0 2: mo mgfim Q 88 E wAAAAAAéoA 58, A8 58288388 HO :8 8.38 23 .8 $9388 8< ”Nam £an 61 080800 803088000 A3580 AAA? 8 805020 008.8888 0083 05 A0 88% 0A: w8AA0800 80¢ AAAAAAAA EA: .880 8% 0AAAAA008A0A8A 820:6 0 3 8680800800 8000 $803088 08% 803083800 65808 03A 08A 80A 08 8396 38065000 820:8 03A 085 0 ._mkAm§m0mm0mm0A_ 806% I30 :05. 80A0A0 05 .A0 80A9088wc800 00808308 08%. 855% 0A: 0A 0>AAAAA0A 8088860 0 4mm: 80¢ 8045 AAAwAA0A U808 88AAAAAAAAAA0m 8 808308 08.0 m-O- «0 00883 8080—00 0A: A0 080 802 A0 80A 26 $02 AAA A0 8205000 0 AAAAB 0805883800 :< A AAAAAA AAAAAV AAAAAV- AAAAA ASA- _AA5AAVAAAAAAV+A AAAAVAA A5 AAV A5 AVAA5AV_ AAAAAA AAAAV- AAAAAV AAAVAV AAAVAV _A A5AAVA AAAAAV AAAAAAVAA5 AVAA5AV_ AAAAAA AAAAV AAAAV- AAAAA- AAAAAV- _A AAA5AAV AAAAAV AA5:AVAA AVAA5AV_ AAAAAA AAAAV AAAAV AAAAV AAAAV _A A5AAVAAAAAAV+AAA AAA VAAAA5A VAA5 AV5AA AV_ 5 AAAAAV BAA- AAAAAV AAAVAV _ AAAAAAVAAAAAV A5AVAA5 AVAA5A AV_ +wAm AAAAAA AAAAVAAAAAAAV- AAAAAAAAAAV- AAAAVAAVAAA- AAAVAVAAAVAV- AA_AA5AAVAAAAAVAAAAAVAA5AVAA5AVAA5AV_ AAAAAA AAAAALAAAAV AAAAV ,AAAAV- AAAAV .AAVAAv AA; .AAAAAV- AA_AA5AAVAAAAAVAAAAAVAA5AVAA5AVAA5AV_ IAAAA AAAAAA AAAAAA AAAAAV AAAAV AAAVAAV _AA5AAVAAAAAVAAAAAVAAA5AVAA5 AVAA5AV_ SA b as AA AAA; AAAAAA AAAAVAV- AA5A. AAAAAV AAAAA _AAA AVAA AAVAA V AA AVA A5 AAVAA VA VAAA5AVA A5A AAV5AA AV_ AAAAAA AAAAV- AAVAAV AAAAV AAAAV _AA5AA AAVA AAAAVA AAAAxVA AAA5AV A5 V5AAA AV_ AAAAAA AAAVAV- AAAAAV- AAAAV AAAVAV _AA5A VA AAAAAAV AAAAAVAA5AVA A5AA A5AVAA AV_ IAAAAA AAAAAA AAAAA- AAAVAA- AAAVAV AAVAAv _ AA5AAVAAAAAAV+ AAAAAAAVAAA5AV A5 AAV A5 V: AAAAAA AVAAAA- AAAAV AAAAAv AAVAAv _ AA5AAVAAAAAAV+ AAAAAAVAAA5AV A5 AVA AA5AV_ AAAAAA AAAAAV AAAAVv AAVAAV AVAA; _AA5AAVA AAAAAAV AAAAVA A5 AVAA5AV_ AA AA; AAAAAAV AAAAV 8AA _AAAAAVAAAAAVAA5AVAA5AVAA5AV_ +8AAAA 525A 8:098 A 8A A AVAA AAA AAAAAAV AAA AAA A8888 A355 8:853:50 BEA mémw A0 008A0> 080A8> 80A 38065000 8-0% 8030888 8208538 A0 90m 830200 .0 80A A0088 $0 0A3 A0 00880 N w8AAAA4$0A 08A 80A Am80AA08mA5800 AD :5 80.38 08. A0 mAmmAAAAAAA 8< ”MAN 0305 62 approaches 3.0 A. The remaining 1 2A, 1 423—, 1 22—, and B 22+ states pose more of a challenge to the IP-EOMCCSD(3h—2p) approach. An analysis of Tables 2.7 and 2.8 reveals that the role of the 3h-2p contributions in these states is very large, partic— ularly at the significantly stretched geometries where several of the electronic states of OH become dominated by such excitations. This suggests a need for the explicit inclusion of the 4h—3p component in the ionizing operator RELN—l) in order to obtain an accurate description, as discussed below. Despite this, the IP-EOMCCSD(3h-2p) method describes the 1 2A, 1 423—, 1 22—, and B 223+ states in the region around the ground-state equilibrium geometry reasonably well, producing errors for these four states of 4203—12392 millihartree at the equilibrium O—H distance of 0.96966 A. As we move to larger distances, where the 3h-2p contributions begin to dominate the 1 2A, 1 42-, 1 22-, and B 22+ states, we see that the errors steadily increase toward 26164—49731 millihartree. It is worth noting that even though these errors are relatively large, they represent a significant improvement of the 335.385—450.090 millihartree errors obtained with the IP-EOMCCSD(2h-1p) approach at 3.0 A for the same four states. We can make similar qualitative observations by comparing Figures 2.4(a) and 2.4(c). This comparison shows that if we focus our attention on the spectroscopic (Franck-Condon) region, the IP-EOMCCSD(3h—2p) method does a reasonable job of faithfully reproducing the full CI curves for all of the states of OH studied in this work. As we shift our attention to the larger O—H distances, we notice that while the success of the IP-EOMCCSD(3h—2p) approach continues for the X 2H, A 22+, 1 4H, and 2 2H states, the results for the remaining states of OH Show increasingly large deviations from the corresponding full CI data. Based on the above analysis it appears that one needs to incorporate the 4h-3p excitations in the IP-EOMCC calculations in order to obtain a better description of the bond-breaking region of the potential energy curves of OH. As discussed below, this is indeed the case. 63 Having analyzed the performance of the IP-EOMCCSD(3h-2p) approach in detail, we now turn our attention to its active-space IP-EOMCCSDt variant. A comparison of Figures 2.4(c) and 2.4(d) reveals that, with the exception of the B 22+ state, the IP-EOMCCSDt potential energy curves are almost a perfect match to the full IP-EOMCCSD(3h—2p) curves. An analysis of Tables 2.9 and 2.10 confirms this obser- vation. If we compare the errors in the two approaches, we find that for all states, ex- cept B 22+, the differences between the IP-EOMCCSDt and IP-EOMCCSD(3h—2p) results are less than 2.410 millihartree for all geometries considered. Furthermore, the corresponding NPE values differ by less than 1.570 millihartree for all of these states (in many cases, less than 0.01 millihartree). Unfortunately, the B 22+ state is a problem for the IP-EOMCCSDt approach at larger O—H separations, where the IP-EOMCCSDt and IP-EOMCCSD(3h-2p) results differ by as much as 25.770 mil- lihartree at the O—H distance of 3.0 A (for the O—H separations in the equilibrium region, the differences between the IP—EOMCCSDt and IP-EOMCCSD(3h—2p) ener- gies of the B 22+ state are only 2—3 millihartree). This large discrepancy between the IP-EOMCCSDt and IP-EOMCCSD(3h—2p) energies of the B 22+ state at larger O—H separations is easily explainable by analyzing the data presented in Table 2.8. The second to the last configuration contributing to this state, shown in Table 2.8, is a 3h-2p excitation in which no electron is removed from the 1m; or 17ry orbitals. Since the active space used in the IP-EOMCCSDt calculations consisted of only those two orbitals, the amplitude that represents this particular 3h-2p configuration is not present in the IP-EOMCCSDt calculation, and since this is a relatively significant configuration for the proper description of the B 223+ state at larger O—H distances, the accuracy of the IP-EOMCCSDt results for this state is hurt. One of the advan- tages of the active-space approaches, however, is that these kinds of problems can be dealt with by expanding the active space. The only problem with that strategy in the particular example of the OH radical is that with the core electrons frozen, expand- 64 ing the active space to include the 30 orbital results in the IP-EOMCCSDt method becoming the full IP-EOMCCSD(3h-2p) approach since no 3h-2p amplitudes are ig- nored when there is only one inactive orbital. In the end, these results show that with an appropriate choice of the active space the IP-EOMCCSDt approach is capable of accurately reproducing the results of the full IP—EOMCCSD(3h-2p) calculations at the cost of the standard CCSD/EOMCCSD schemes. We now return to the important issue of improving on the IP-EOMCCSD(3h- 2p) results. As mentioned above, the analysis of the dominant contributions to the full CI wave functions, along with the IP-EOMCCSD(3h—2p) results, lead one to the ) conclusion that the inclusion of 4h—3p components of the RSV—1 operator in the IP- EOMCC calculations is necessary to accurately describe the entire potential energy curves of OH. In order to test this hypothesis, my advisor and I, in collaboration with Dr. Yuhki Ohtsuka, Professor Masahiro Ehara, and Professor Hiroshi Nakatsuji, per- formed SAC—CI(4h-3p) calculations for the low-lying doublet states of OH [88]. As explained in Section 2.1.1, the IP SAC-CI methodology is equivalent, up to some implementational differences and unimportant terms, to IP-EOMCC and so can pro- vide similar insight into the role of 411-31) excitations as the corresponding IP-EOMCC method. In order to verify the equivalence of the IP EOMCC and SAC—CI methodologies, we compare the SAC-CI(3h—2p) results with those obtained with IP-EOMCCSD(3h- 2p). Examination of Tables 2.9 and 2.10 reveals that the two methods do in fact provide very similar results. This is particularly true in the spectroscopic region, where the differences between these two methods range from 0.022 — 4.755 milli- hartree. Past this range, particularly for states that become dominated by 3h-2p excitations (see Tables 2.7 and 2.8), the differences can become larger, but as these are the regions and states for which the 3h-2p methods have difficulties, this is not a major concern. 65 .IEO mo ESE-8 bv «:8 $5 5.5 Sn 2: mo US$980 8me 950d 25. v .IEO mo £355 B; 98 a: 93 mo vmummmaoU 8me 950d 23; o .32. so: 5&3 gang 28 833235 a .amuob mm? 1355 3 commsooo $832 2: .mcosflsofio wouflotonv 2w E .EE E 958 on :8 £023 603$, lamb-"Em HO :3 wamvaommotoo 23 3 332m: vegans—MS E 958%: m3 momwcoco H0-0 0) wave functions can then be expressed as: A . 95:40 = 1,535,374), == Riflefl ’I<1>>. (3.1) 75 where the excitation operator RLA) which transforms the CC ground state Nam) into the excited-state wave function lily”) is given by A "1A A A R] > = Rio) + 3,23,,“ 2 mo 1 + 2: RM, (3.2) 1121 with 1 2 7,1...i'n a an Rflfll = E? Tu,a1...a7l 0’ 1 ° . ' 0’ ain . . I ail? (3'3) 1 representing the identity operator, and $1,532,,” designating the corresponding exci- tation amplitudes of EOMCC. One may notice that the structure of the wave function ansatz given by Eq. (3.1) is very similar to the EA/IP—EOMCC wave function ansatz of Eq. (2.1), except that instead of an electron-attaching or ionizing operator, RELA) is a particle-conserving excitation operator. It should also be noted that although Eq. (3.1) is formally applicable to excited states, one can easily extend it to include the ground-state (11 = 0) case if we adopt a convention in which REA) is the identity operator. We use this convention throughout this work. The REA) = 1 condition for the 11 = 0 RLA) operator is equivalent to defining 700 = 1 and 1’31 £111.70” = 0 for n 2 1. With the CC / EOMCC approximation A defined as above, the noniterative energy correction recovering the full CI energy from the CC/EOMCC energy, 6,874), which defines the biorthogonal formulation of the MMCC theory, is given by the following 76 compact expression [133—137,139, 140] A) _ A) NflaA = Z CPL-gum A’Ip,'n(mA)lq)) n=mA+1 N 18A n=m +1 A i1<'-- = <?,1,,,‘:,"IRf. 01¢», (36) where n > mA, and HM) is the similarity-transformed Hamiltonian of CC method A given by Eq. (2.10) with T = TM). Since it was established that RéA) = 1, Eq. (3.6) is general and defines both the ground- and excited-state moments. If one specifically examines the ground-state case, then Eq. (3.6) reduces to the generalized moments 77 of the ground-state CC equations defining method A, 911“”fo (m1) 2 92121111112340: <‘-‘1"=“"IHI>. (3.7) 21...'Ln At this point it should be noted that for a given CC / EOMCC method A, not all of the moments EmulfllfiaAm/j) With n > m A are non-zero. Indeed, for a given approxnna- tion, there is generally a value of n above which all moments 932232,, (m A) are zero. This is the source of the upper summation limit N u, A in Eq. (3.4), which is equal to this value of n for the CC / EOMCC method A. For instance, in the case of CCSD (the m A = 2 case), only the triply, quadruply, pentuply, and hextuply excited moments, i.e. moments with n = 3 — 6, are nonzero when the Hamiltonian contains pairwise interactions only, and thus No, A = 6 (the CCSD equations are solved by zeroing the singly and doubly excited moments, 2118,00) and 9313], ab(2), respectively, hence the triply excited moments are the first to be nonzero). Similarly, in the EOMCCSD case with ,u > 0, N11,A = 8. The second fundamental contribution to the equation for the noniterative energy correction defining the biorthogonal MMCC approach, Eq. (3.4), are the operators a]...an «544,22 and the corresponding deexcitation amplitudes Eu 2.1-”in . The $14.72 operators are the n-body components of the deexcitation operator .2”), which is designed to give the following parameterization for the exact full CI “bra” wave function: m A that contribute to the energy corrections 6&4). As a result, it is useful to deconstruct the exact operator .2], in the following way: 2,, = .25") + 5.2,)“, (3.10) where the A part of .2], is given by A 771A of") )= 22’1””, (3.11) n=0 and the remainder corresponding to n-tuply deexcited contributions with n > m A by N A 6.21) l = 2 21,3. (3.12) n=mA+1 This helps in designing the approximate schemes based on Eq. (3.4), such as the CR—CC(2,3) or CR-EOMCC(2,3) approaches discussed in the next section. It is also important to note that the validity of Eq. (3.4) depends on the normal- ization of 3)). Indeed, Eq. (3.4) only gives the exact difference between the full CI and CC / EOMCC energies if the following normalization condition is used: (6)3308?qu = 1. (3.13) Once again, we note that because of the convention in which REA) = 1 this equation is general, and in the ground-state case, reduces to (<1>|$(]A)|) = 1. (3.14) Inspection of Eq. (3.13) reveals that it is very similar to the biorthonormality 79 condition defining the CC/EOMCC bra states (\11 M], |, namely, ~ A A A A M. >wa )> = «PILL >125. ’I9) = 61.... (315) Here, the CC/EOMCC bra state (\IIH \IIAH | corresponding to the ket state [‘11], A)) defined by Eq. (3.1) is given by ~ .4 (effll = (QILLA)e—T( ) (3.16) (cf. [9,10]), Where LLA) is a deexcitation operator defined as A with Lffol = 6],,0 1 (3.18) and 1 2 alman 2'1 in Ln, Aopen — _2 141,71, [111,71 2 a l/l'vil-"i‘n a -- - a Gan - ° - aal. . (3.19) Examination of the above equations reveals obvious similarities between the standard EOMCC deexcitation operator LISA) Eq. (3.17) and the A part 25A) of the operator .3?” Eq. (3.11), but it is important to stress that these are two different objects. Indeed, the bra states (ML/(LA) are the left eigenstates of the non-Hermitian similarity- transformed Hamiltonian HM), which are obtained by solving the left eigenvalue problem for the CC/EOMCC method A, a .a a a 614,0 <¢|Hopenl¢i11 in n) + ((DlLumopenHoffen[(1)2113in,” _ (A) 01- -an _wfl ly.,i1...in 21 < ‘ ’ ' <1", (11 < ' ° ' < an, (3.20) 80 where wISA ) =E(A)-— EéA) is the corresponding vertical excitation energy obtained with method A (wéA )— — 0), in the subspace of the N -e1ectron Hilbert space spanned by the excited determinants [(Dglm-an) with n = 1, . . . ,mA. These left eigenstates are 1...’£n the bra counterparts of the ket eigenstates RLA)|) of BM) which are obtained by solving the EOMCC right eigenvalue problem, —- A A ((1)91 :1|(H(()pan[1,())pen)C [(13): WE), )Tzlalinan, (3.21) 11...! in the space spanned by the excited determinants Wild-~17!) with n=1,. mA, followed ".207: by the determination of the zero—body amplitude TM) using the equation A A 14,0ZN](HopenRh,dpen)C[q>>/w/(A )- (3.22) On the other hand, the bra state ((IJIEIEA) is only one of two contributions to the exact state (CHEM, which is obtained by solving the eigenvalue problem (44.8,, HM) = E, ((512,, (3.23) which is equivalent to the adjoint form of the Schrbdinger equation, (\Ilu [H = E), (\IIM, for the exact bra wave function (\Ilpl and exact energy Eu in the entire N —e1ectron A) Hilbert space. As a result, the two operators, LISA) and 3]) , are identical only in the special case of m A = N, where N is the number of correlated electrons. However, despite the fact that the two operators are formally different, the observed similarities between them are useful in formulating approximate CC / EOMCC methods based on the biorthogonal MMCC theory, which is the subject of the next section. As a final point, it is worth noting that the formula for the noniterative energy correction 6,8 ) , Eq. (3.4), origlnates from a cons1derat10n of the followmg asymmetric 81 energy functional [120, 122, 127, 129,130,133, 134] _ (\II|HR),A)eT(A) (<5) A[\II] _ . (WIRELA)8T(A) IQ) (3.24) The unique feature of this expression is that when ‘1! is the exact, full CI state \IJM, then the value of the A[\II] functional is the full CI energy EM (i.e. A[‘Il#] = Eu), regardless of the choice of Rf)!” or eT(A) for the CC/EOMCC method A. This is a consequence of the Hermitian nature of the Hamiltonian, which results in the equation being unchanged if one applies the Hamiltonian to the bra state (‘I’ul in Eq. (3.24) rather than to the ket state RLA)eT(A) |). It is the invariance of this expression with respect to the CC / EOMCC truncation m A that makes it such a good starting point for the derivation of the noniterative correction 6&4), Eq. (3.4). For a derivation of the formula for (5)14), Eq. (3.4), see Appendix B. 82 3.1.2 The CR-CC(2,3)/CR—EOMCC(2,3) and CR—CC(2,4) /CR—EOMCC(2,4) Approaches Equation (3.4) provides one with a well-defined procedure for correcting the CC / EOMCC energy for the ground or excited state of a given molecular system in order to obtain a good approximation to the exact energy. By applying a series of systematic approxi- mations to Eq. (3.4), it is possible to generate a class of practical and highly accurate CC / EOMCC methods, referred to as the completely renormalized (CR) CC / EOMCC approaches. This section discusses the systematic approximations that generate the so-called CR-CC(mA,mB)/CR—EOMCC(mA,m3) schemes, with a particular focus on the CR—CC(2,3)/CR—EOMCC(2,3) and CR—CC(2,4)/CR-EOMCC(2,4) approxi- mations utilized in this work. When attempting to generate practical computational approaches from Eq. (3.4) the first issue faced is that of the summation over 72 and the upper summation limit N H, A- With the exception of small molecular systems consisting of only a few cor- related electrons, it is not feasible to consider all the terms up to N u, A in electronic structure calculations. Indeed, as discussed in Section 3.1.1, if one starts from the basic CCSD/EOMCCSD approach, then No, A = 6 and N”, A = 8 for p > 0, which means that terms involving up to hextuply excited moments of the CCSD equa- tions and up to octuply excited moments of the EOMCCSD equations would have to be included. Such high-order moments are computationally too expensive to be included in calculations for all but the smallest systems. In order to overcome this difficulty, one can truncate the summation over 72 in Eq. (3.4) at some value 271. B, where m A < m B < N114 This leads to a systematic hierarchy of approximate CC methods, referred to as the MMCC(mA, m5)? schemes, for which the total electronic energy is calculated as [133—137, 139,140] E),(mA,mB) = Eff) + 6),(mA,mB), (3.25) 83 where the noniterative correction 6])(mA, m3) is given by: 7723 514(mAamB) = Z (©l32.n1‘42,n(m.4)l<1>) n=mA+l m8 = z: 2 221.3. 22:22:72me <22» n=mA+1 i1<.'. 0) energy are given by (511(213) = (‘I’Ii’ps Mp,3(2)|‘1’) _ ijk — Z 2” .),. 222,. .142) (322) i + (¢|$#,4M,,,4(2)I) m2 k = 2 [abc p,2'jk E”2110569) i+ +<¢|£p 3 H(CCSD)[(I)ajbC) = Eu<<1>|$13l<1>§3bf>= E11 Babe (3.33) 12,2'jk By replacing the exact energy Eu by the CCSD/EOMCCSD energy E(CCSD), ap— proximating the triples-triples block of H (CCSD) by its diagonal, and solving for the amplitudes defining E], 3, one obtains the quasiperturbative expression sz'jk’ (CCSD - k 22%,, _ —,(,(|L )H(CCSD)|<1>abg)/D:jabc, (3.34) 86 Where the Epstein-Nesbet-like denominator ijfle is defined as thjfzbc ELCCSD)... (@ijc|H( CCSD) [Qajb0) wijCSD)— (QajbclH(CCSDl |¢abr> C(CSD) b FEES-E) ccsn , -<?jf|H§ 293% (3.30) 4692311; Here wLCCSD) represents the EOMCCSD vertical excitation energy, wLCCSD) (CCSD) = E” (CCSD) d BESCSD) represents the m-body component of the CCSD similarity-transformed Hamiltonian. By substituting ($1331.: defined through Eq. (3.34), into the equations defining the MMCC(2,3)_g2 approach, Eqs. (3.27) and (3.29), we obtain the formula for the total electronic energy defining the CR-CC(2,3) (12 = 0) or CR-EOMCC(2,3) (12 > 0) approach, namely, CR(2,3) _ (CCSD) CR(2,3) _ (CCSD) be k _ E], + Z 6:142ch 91230,":(2), (3.37) i 0) case, the zero-, one-, and two—body components RM), R1212 and Rm? obtained by solving the EOMCCSD eigenvalue problem. It is important to mention that the above expression for the gzbf'jk amplitudes, Eq. (3.34), may 87 have to be modified if one of the indices 2', j, k, a, b, c corresponds to an orbital which is degenerate with some other orbitals. In that case, to make sure that the CR- CC(2,3)/CR~EOMCC(2,3) energies remain invariant with respect to rotations among degenerate orbitals, one should replace Eq. (3.34) by a more elaborate expression in which, instead of using the diagonal matrix elements (<1)?ng (CCSD) @273?) that enter ijk 11,abc’ Eq. (3.35), one solves a small system of the Epstein-Nesbet-like denominator D linear equations, similar to Eq. (3.33), where all amplitudes [721%. 1: involving indices of degenerate spin-orbitals are coupled together through the off-diagonal matrix ele- dc ments ((1)1 m n|H(CCSD) |gjbfl involving the triply excited determinants that carry the indices of degenerate spin-orbitals [138]. Without taking care of this issue, the CR- EOMCC(2,3) energy correction 6§R(2’3) is not strictly invariant with respect to the rotations among degenerate orbitals, although the dependence of the 65R(2’3) correc- tion employing Eq. (3.34) to determine the (72132-31: amplitudes on the rotations among degenerate orbitals is minimal. Indeed, as shown in [138], changes in the values of triples corrections due to the rotations among degenerate orbitals do not exceed 0.1 millihartree when one uses Eq. (3.34) to determine amplitudes Zfibfjk. Thus, the issue of the lack of invariance of the CR—EOMCC(2,3) correction 6ER(2’3) employing Eq. (3.34) with respect to the rotations among degenerate orbitals is more of a formal problem than the practical one as long as one does not calculate energy derivatives for systems with orbital degeneracies. If the molecule has at most an Abelian symmetry or if the orbitals employed break the non-Abelian symmetry (for example, due to the use of symmetry-broken reference determinant or external fields), so that there are as is. no orbital degeneracies, one can apply Eq. (3.34) to all amplitudes (72%,; The computational costs of the CR—CC(2,3)/CR—EOMCC(2,3) schemes are char- acterized by the iterative 7137121, steps of the underlying CCSD/EOMCCSD approach plus a single noniterative step that scales as 71271:. These scalings are similar to those that characterize CCSD(T), and so, like CCSD(T), CR-CC(2,3)/CR-EOMCC(2,3) 88 can be applied to systems with up to about 100 correlated electrons and a few hun- dred basis functions. Furthermore, the overall structure of the CR—CC(2,3)/CR— EOMCC(2,3) formulas for the energy given by Eqs. (3.34) — (3.37) is the same as that of CCSD(T). Indeed, the formula for the CCSD(T) energy can be obtained from Eq. (3.37) by approximating 2823-]: and 9313542) in the following manner: in the definition of Egg-0%, Eq. (3.34), we replace the denominator Dbfclibc’ Eq. (3.35), by the spin-orbital energy difference (6,- +ej+ek—ea—eb—ec), neglect the (L021? (00813)) DC contribution to the (|L£,CCSD)H(CCSD)|§‘JI?§) numerator of Eq. (3.34), which is at least a fourth-order term in many-body perturbation theory (MBPT) if the HF ref- erence is used, replace LOJ, Log, and HéCCSD) by T1;r , T21, and VN in the remaining contributions to (@lLELCCSDUTmCSD)@355), where VN is the two-body part of the Hamiltonian in the normal ordered form relative to the Fermi vacuum and TI and T; are the adjoints of the T1 and T2 cluster operators determined by solving the CCSD equations, and replace the triply excited moments 9113,3369), Eq. (3.7) with n = 3 and mA = 2, by the lead term (ngf’ngVNTflchb). As a result, the CR-CC(2,3) and CR-EOMCC(2,3) approaches maintain the same “black-box” nature and ease- of-use that CCSD(T) is known for. There are, however, many advantages of the CR—CC(2,3)/CR—EOMCC(2,3) methodology over CCSD(T). First of all, as shown in Section 3.2 and {138—140, 150] for the open-shell systems and [133—135,137, 141—149] for singlet ground states, CR—CC(2,3) eliminates the failures of CCSD(T) in bond breaking and biradical situations, while being as accurate as CCSD(T) when the multi-reference character of the calculated state is small. Furthermore, CR-CC(2,3) has a natural extension to excited states in the CR—EOMCC(2,3) scheme, as described above, and there is no natural formal extension of CCSD(T) to excited states. The earlier CR-CCSD(T) approach of [127], which relies on the original MMCC formal- ism of [127,130] that does not utilize the left eigenstates of the similarity-transformed Hamiltonian, has an extension to excited states as well [131,132], but unlike CR- 89 CC(2,3), the earlier CR—CCSD(T) is not strictly size extensive. The CR-CC(2,3) approach and its higher-order CR-CC(2,4) analog are both size extensive. A procedure similar to that used to derive the CR—CC(2,3) and CR—EOMCC(2,3) equations can be used to develop methods based on the MMCC (2,4) g approximation. In this case, the approximate form of if“, given by 6% ~ L(ccsr)) + 23%;), + 2,,4, (3.38) is substituted into the adjoint form of the Schrbdinger equation, Eq.(3.23), and then right-projected on the triply excited determinants @3323) and quadruply excited de- terminants @3123?) In the CR—CC(2,4)/CR—EOMCC(2,4) approximation studied in this work, one then replaces the exact energy Eu in the resulting system of equations for 2,1,3 and .37)“; by the CCSD/EOMCCSD energy ELCCSD) and approximates the triples-triples and quadruples-quadruples blocks of E(CCSD) by their diagonals, while ignoring any coupling between triples and quadruples. As a result of these manipulations, one obtains the same mathematical expressions for E 11 I J k as in the CR—CC(2, 3)/CR-EOMCC(2,3) case, Eqs. (3.34) and (3.35), and the following for- mula for gzbfdjkl: CCSD kl eabicjdkl— ((PIL( )H(CCSD) l¢gjbcfi>/D:jabcd’ (339) where the denominator D322“! is defined as DgaCd = ELCCSD) _ (@qu:|fi((CCSD)l¢qucd) CCSD —(< 2j2zIH§ 3222» (3.40) 90 By substituting [szcjlw Eqs. (3.34) and (3.35), and (fibffa, Eqs. (3.39) and (3.40), into the equations defining the MMCC(2,4)$ approach, Eqs. (3.28) and (3.30), we obtain the formula for the total energy that defines the CR-CC(2 ,4) / CR-EOMCC(2,4) approach, CR@,4) EERQA) +6 Efixsn) (CCSD) abc 2J'k E” + Z €12,2ch Emu abc(2 ) ifiljblffi) term in the definition of Dfigécd. Finally, variant A is ob— tained by replacing the Epstein-Nesbet-like denominators D222“ and D3226 d by the corresponding Moller-Plesset-like denominators, [wEJCCSm - (ea, + Eb + cc — e,- — ej — ek)] for triple excitations, and [4.2/(£00813) — (Q; + 6b + cc + 6d — e,- — ej —— ék — 61)] for quadruple excitations. Finally, there are also relationships between the CR-CC(2,3)/CR—EOMCC(2,3) and CR-CC(2,4)/CR-EOMCC(2,4) approaches and other noniterative CC/EOMCC methods. For instance, the CR—EOMCC(2,4),A method is equivalent to the EOM- 92 CC(2)PT(2) of [203,204] when the canonical Hartree-Fock orbitals are used. A simi- lar equivalency exists between the CR—EOMCC(2,3),A approach and the triples cor- rection of EOM-CC(2)PT(2). The ground-state CR—CC(2,3),A and CR—CC(2,4),A sChemes are equivalent to the CCSD(2)T and CCSD(2)TQ methods of [205] when the canonical Hartree-Fock orbitals are used, and the analogous relationships exist between the CR—CC(2,3),B and CR—CC(2,4),B schemes and the CCSD(2) approach of [206-209]. For example, the CR-CC(2,3),B approach is equivalent, up to small details, to the triples correction of CCSD(2). As mentioned earlier, there is also a straightforward relationship between CR—CC(2,3) and CCSD(T). 93 3.1.3 Computer Implementation of the Open-Shell Variants of the CR—CC(2,3)/CR—EOMCC(2,3) and CR—CC(2,4) /CR—EOMCC(2,4) Approaches Unlike the active-space EA/IP-EOMCC methods discussed in Chapter 2, which are built from the ground up specifically for open-shell valence systems, there is noth- ing about the CR—CC(2,3)/CR-EOMCC(2,3) or CR—CC(2,4)/CR—EOMCC(2,4) ap- proaches that specifically depends on the open- or closed-shell nature of the molec- ular system of interest other than the choice of the reference determinant |). In- deed, the first computer implementation of the CR—CC(2,3)/CR—EOMCC(2,3) ap- proach, developed by Professors Marta Wloch and Piotr Piecuch [133] and available in GAMESS, was written specifically for closed-shell references. As discussed in the Introduction, the early applications of this approach to singlet states of molecular systems [133—135,137, 141—149] revealed its ability to provide highly accurate results at a reasonable low computer cost. In particular, CR—CC(2,3) proved to be very effective in calculations of the singlet states of biradical systems and single bond breaking on singlet potential energy surfaces, which are characterized by a significant multi-reference character. Though restricted to open-shell singlet states, these results offered the first glimpse of the potential applicability of the CR-CC(2,3) approach to general open-shell molecular systems. As the overarching goal of this dissertation is the development of practical methods for studying open-shell systems, the original closed-shell CR—CC(2,3) work of [133] was followed up by the development of the general—purpose open-shell computer imple- mentation of the CR—CC(2,3) and CR—EOMCC (2,3) approaches that are applicable to singlet as well as non-singlet states [138,140]. Furthermore, in order to study the per- formance of higher-order CR—CC approximations based on the biorthogonal MMCC theory of [133,134], the CR-CC(2,4)/CR—EOMCC(2,4) method was implemented by 94 the author of this dissertation as well (with the help of routines for quadruply excited moments provided by Dr. Maricris Lodriguito, reviewed in [210]), using a general for- mulation applicable to both singlet and high-spin non-singlet reference determinants. The resulting computer codes. are highly efficient, fully vectorized codes which make use of recursively generated intermediates and fast matrix multiplication routines. The programs are general in the sense that they can work with any type of high- spin reference determinant, though currently they are used with the RHF and ROHF references generated by GAMESS. These codes utilize the standard spin-orbital ba- sis of oz and fl spin-orbitals in performing calculations, and the most recent version of the open-shell ground-state CR—CC(2,3) program, which is specifically designed for non-relativistic molecules, is written in a spin-integrated form which avoids ex- plicit calculations of terms that go to zero because of spin symmetry, making it as fast as possible [138]. This version of the CR—CC(2,3) code is available as part of the GAMESS package. Currently, the open-shell CR-EOMCC (2,3) and CR—CC(2,4)/CR- EOMCC(2,4) codes are stand-alone pilot programs that are only loosely interfaced with the integral transformation and RHF/ROHF routines of GAMESS, but work is under way to develop more efficient, spin-integrated implementations that will be fully included in GAMESS and thus available to the public. As already mentioned, the CR—CC(2,4)/CR—EOMCC(2,4) computer codes developed as part of this research benefitted from the thesis project of Dr. Maricris Lodriguito, who provided the au- thor of this thesis with the routines for calculating the quadruply excited moments mfifigcda) [210,211]. As an illustration of the algorithmic details that are associated with the above method and code development efforts, we present the key details of the computer im- plementation of the CR-CC(2,3)/CR—EOMCC(2,3) method. The CR-CC(2,3)/CR- EOMCC(2,3) computer codes implemented as part of this work consist of four main parts. In the first part, as was the case in the EA-EOMCC and IP-EOMCC codes 95 discussed in Section 2.1.3, we solve the usual CCSD equations for the ground state in order to obtain the singly and doubly excited cluster amplitudes, t2, and t3), respec— tively. In the next step, again in analogy to the EA—EOMCC and IP-EOMCC codes, we use these amplitudes to construct the one— and two—body matrix elements of the CCSD similarity-transformed Hamiltonian, 72;], and 72%, respectively. As a reminder, the explicit equations for these matrix elements in terms of the one— and two-electron integrals, fg and 22%, respectively, and the cluster amplitudes t2, and t3) can be found in Table 2.1. In the third step, we construct and solve the EOMCCSD equations. In executing this step, we must distinguish between two situations. If our interest is in the noniterative corrections 65R(2’3) for the ground ()2 = 0) and excited (,u > 0) states, we must construct and solve both the right and left EOMCCSD equations, Eq. (3.21), augmented by Eq. (3.22), and Eq. (3.20), respectively, in which A = CCSD and m A = 2. The right eigenstates RLCCSDMCD) needed to construct the EOMCCSD 321,42) With 1” > 0, EQ- (3.6) in which mA = 2 and n = 3, whereas the left eigenstates LEJCCSD) |) are needed to construct the deexcitation amplitudes Eq. (3.34). If our interest is in the ground-state correction 6gR(2’3) only, we moments rm ~ bc €72,222, must solve the CCSD left eigenvalue problem obtained by setting a = 0, A = CCSD, and m A = 2 in Eq. (3.20). The right EOMCCSD eigenvalue problem for the excited- state (a > 0) case is solved using the Hirao—Nakatsuji generalization [176] of the Davidson diagonalization algorithm [177] to non-Hermitian eigenvalue problems and is generally solved for first. Once the excitation energy wLCCSD) is known (in the ground-state case, wéCCSD) = 0), it is substituted into the left (EOM)CCSD eigen- value problem, Eq. (3.20), which converts it into a system of linear equations for the If” and [$132 j amplitudes, which we can solve with the same DIIS solver [212—214] that we normally use to solve the ground-state CC equations. As the focus of this work is the CR—CC(2,3)/CR—EOMCC(2,3) approach, we will not go into further details of the above steps in this dissertation and rather focus on the final step, which is the 96 CR(2 3) determination of the 6M triples correction. CR(2,3) From Eq. (3.37), we know that in order to calculate 6,) we must first express gabe the deexcitation amplitudes jk and the triply excited moments 931 jk Nab (2 ) in terms of the molecular integrals fp and 213,3 ,CCSD cluster amplitudes t3 and tij , right EOMCCSD excitation amplitudes r), 0,r aand r27 ab(when a > 0 only), and left EOMCCSD deexcitation amplitudes lg and Mlzsz (pa = 0 and a > 0). To that end gabe we begin by reexpressing the fin amplitudes as follows: I) ijk (Babe aijk_ —NS,z‘cjk/Du,abc' (3.43) Here the denominator DIE/Sim is still defined by Eq. (3.35) and the numerator is given by (cf. Eq. (3.34)) (CCSD - Nfigpjk = (©[L( )H(CCSD) [(1)an0> = dabcrz’jfjk. (3.44) abck is defined as where the partially antisymmetric quantity F” i] be _ k1 vbc ‘ b b P3,, ,jk _g/C/J [2232113 + 2133123 +2 133h3— 313,333.73 (3.45) (we dropped the symbol a representing the electronic state of interest from I“ z- and [abj- for clarity reasons). Inspection of Eq. (3.35) reveals that the determination of the DLjabc denominators requires the calculation of the diagonal matrix elements <¢an€ H(mCCSD) using the following expressions: (1)3331?) with m = 1, 2, and 3. These three quantities can be computed <¢abc H ijk @313) = 433 _ a; _ 13.3 +233, + 13.3 + 713, (3.46) 97 abc ‘ (CCSD) abc _ ‘ai ‘b2' “02' <‘I’2jk H2 ¢ijk> - _hai — hbi — hcz' -aj -bj -cj _th ‘ hbj ‘ hcj ‘ak ’blc ‘ck _hak — hbk — hck ‘ij "ik -J'k - b - ’b. +th + 133 + hbg, (3.47) and abc “ (CCSD) abc _ *2'aJ' ‘2'ak ‘jak < z'J'k H3 ijk> — _ha'ij ‘ haik “ hajk -z'bj ab}: 17% —hb2'j — hbik " hbjk —iCj ‘2'ck 'jCk —hc2'j - hcik — hcjk +1133]; + I?“ + WC aic bz'c +5332 + W“ + 729"” aJ'c bJ'c +2333 + 5.333 + 5333, (3.48) where the one— and two-body matrix elements 12% and 1.2;?) are defined in Table 2.1 and the specialized three—body matrix elements entering Eq. (3.48) are defined as 5332 = - 2 1232153113 (3.49) m and fi?§fi$$ em 6 For the triply excited moments 9,113,123,0(2) that multiply the amplitudes £21)ka to R(2,3) ( produce the correction 65 cf. Eq. (3.37 )), we can write (2) =dabciijk (2), (3.51) iJ’k {m u,abc p,abc 98 where the partially antisymmetric Tijk 2 quantities are defined by #,abc "k k k $33542) = :i/jklbhzifl‘ég— $717M; Z71?" 2177wtab +Iczzetjc) 7M ebtéf—Im 315:". 53)] (3.52) (again, we dropped the symbol u from 7'), 0,r u a and r for the sake of clarity). (tab The intermediates that enter the above expression for ‘35“? ab 6(2) are defined as 3% = 2’1337‘2 - Sin 3'? + zihircmab "m hincbra , (3-53) 'k k k k k k 153 = 1,643+ 2}}:sz re}— hfmra m +31] (53%,, 2+ +53: re), (3.54) k k 'k- 13,—:th —t{.ah,¢, (3.55) and If: 433%“; (3.56) while the antisymmetrizers that enter several of the above equations are given by dpqr E 427”" = 1 — (pq) - (pr) - ((1?) + (W) + (pm), (3-57) flip/qr s W ‘1’“ = 1 — (pq) — (pr), (3.58) and Win 5 37”" = 1-(pq), (3.59) with (pq) and (pqr) representing the cyclic permutations of two and three indices, respectively. As in the earlier equations, the one— and two-body matrix elements of E(ccsp) hq and hrs pq, respectively, that enter Eqs. (3.52) — (3.56) can be found in Ta- ble 2.1. If the object of the calculation IS the ground-state ([1: 0) correction 6CR(2 3), we modify Eqs. (3.52) — (3.56) by replacing TO by 1 while zeroing all amplitudes r}, 99 SUM = 0.0 LOOP OVER i H28i + X (X = H, Cl, CH3 and SiH3), were studied using CR—CC(2,3) [150]. One of the interesting features of these radical reactions is that they are of importance in high temperature chemical vapor deposition (CVD) processes, such as the silicon carbide CVD process [215,216]. In order to evaluate the performance of the ROHF-based CR-CC(2,3) method, either the full CI method or the internally contracted MRCI(Q) [181,182] scheme (depending on the Size of the basis) was used to provide benchmark potential energy surfaces. Phrthermore, the results were compared with those obtained with the UHF-based CCSD(T) approach, which is often used in high accuracy studies of radical species, and the multi-reference second-order perturbation theory (MRMP2) [217, 218]. Calculations of the potential surfaces for all eight of the reactions described above were performed. To save space, two of these reactions, namely HgC—H -———+ 3CH2 + H and H2Si—H —-1 1Sng + H, will be described in detail and the remaining ones will only be summarized. Five different basis sets were utilized in the calculations, 102 namely, MINI [219], 6-31G [194,195], 6—31g(d) [195], cc—pVDZ [178,220] and cc- pVTZ [178,220]. Full CI is used as the benchmark for calculations utilizing the MINI basis set, while the remaining calculations use MRCI(Q) to provide the reference results. All five basis sets were used for each reaction, except for the CHg—Cl and Sng—Cl reactions, for which the MINI basis set was replaced by a mixed basis set, abbreviated as MIX, in which MINI is used for C, Si, and Cl and 6—311G is used for H, and for the H2C(Si)—C(Si)H3 reactions, for which the 6-31G(d), cc-pVDZ, and cc- pVTZ basis sets could not be applied due to the excessive computational cost of the MRCI(Q) calculations. The potential energy surfaces were constructed by sampling the breaking bond distances from Slightly shorter than the equilibrium bond length (Re) to roughly 3R3 in 0.2 A increments. In order to ensure that the two fragments on the product Side of each reaction have negligible interaction, so that the bond can be considered to be ‘broken’, if the last two structures at 311.; had an energy difference of more than 0.16 millihartree (0.1 kcal/mol), then more structures with longer bond lengths would be computed until the energy difference was below the 0.16 millihartree threshold. For all eight reactions with the MINI (or MIX as the case may be) basis set, as well as for the H2C(Si)—H reaction with the 6—31G and 6- 31G(d) basis sets, full CI was used to optimize the structures on the one-dimensional bond-breaking reaction surface. In all other calculations, the full-valence CASSCF was used for the structure optimizations. Finally, it Should be noted that the frozen core approximation was used in all calculations, and the full valence CASSCF was used as the zero-order wave function for all MRCI and MRMP2 calculations. Figure 3.2 shows the CCSD, CR—CC(2,3), CCSD(T), and MRMP2 error curves relative to MRCI(Q) for the HgC—H ——-> 3CH2 + H reaction as computed with the cc-pVTZ basis set. Inspection of these error curves reveals that the CR—CC(2,3) results are either as accurate as or more accurate than CCSD(T) for all geometries considered. Indeed we can observe that the error in CCSD(T) begins to increase 103 - CCSD OCR-CC(2,3) 16 xccsncr) 14 .. ‘ AMRMP2-4mh § 12 *‘TA A ' . E 10 A A " - 8 ' A - A 6 _ - A A A A A A A A A i " . 4 X X X 0 - - 2 X V x 0 y X ~ 0 ~ 3 owzaéo?°. . fl*6§ 1 1.5 2 2.5 3 3.5 4 R(angstrom) Figure 3.2: Restricted open-shell CCSD, restricted open-shell CR—CC(2,3), unre- stricted CCSD(T), and MRMP2 errors relative to MRCI(Q) for HgC—H ——> 3CH2 + H with the cc-pVTZ basis set. around 1.5 A while a similar increase does not occur for CR—CC(2,3) until about 2.5 A, indicating that CR—CC(2,3) remains accurate at longer bond distances than CCSD(T). Interestingly, both CR—CC(2,3) and CCSD(T) produce smaller errors than MRMP2 for all geometries studied. A similar picture arises for the Hgsi—H ——> 1Sng + H potential energy curve, which is shown for the cc-pVTZ basis set in Figure 3.3. In this case, CR-CC(2,3) is more accurate than CCSD(T) for all points examined, though for the very short distances and very long distances the discrepancy between the two is relatively small. Once again, a bump in the error curve of CCSD(T) is observed, with the rise in the error beginning at roughly 2 A. As was the case for HgC—H, a similar rise in the error of the CR-CC(2,3) result does not occur until about 2.6 A, once again indicating that CR—CC(2,3) remains accurate at longer bond distances than CCSD(T). Furthermore, the bump in the CR—CC(2,3) error curve at about 2.6 A is considerably smaller 104 - CCSD 10 0CR-CC(2,3) " xCCSD(T) A81 - AMRMP2-15mh .2 AA . " E 6___A_._____A__A_A_A__A_A_A,_A_A_A v -A --------- I- ..-- A 2 4 5a I. III xxx 2 xxx“ oxxxxxxxxx 33000000 000000000 0 l T l T l l 1.4 1.9 2.4 2.9 3.4 3.9 4.4 R(angstrom) Figure 3.3: Restricted open-shell CCSD, restricted open-shell CR-CC(2,3), unre- stricted CCSD(T), and MRMP2 errors relative to MRCI(Q) for HQSI—‘H —-> 1Sng + H with the cc-pVTZ basis set. than that of CCSD(T), leading to a smoother and overall consistently more accurate potential energy surface. Once again, both methods are significantly more accurate than MRMP2 at all geometries considered. Rather than analyzing all of the remaining potential energy surfaces generated in a similar fashion, the rest of this discussion will focus on summarizing the result through the use of three fundamental quantities that serve as valuable indicators of the accuracy of the potential energy surfaces, namely the nonparallelity error (NPE), the standard deviation error (STD), and the reaction energy error (REE). As described earlier in this thesis, NPE is defined as the difference between the most positive and the most negative signed errors along the surface, and thus is a measure of how closely the shape of the potential energy surface matches that of the benchmark surface. As the name implies, REE measures the error in the reaction energy and is defined as the difference between the error at the longest bond distance (i.e. the ‘broken bond’ 105 distance) and the error at the equilibrium geometry. Finally, STD is defined as 2 1 N N STD: N Z[Err(R,-)]2— ZlErr(R,-)| , (3.60) 1221 i=1 where N is the number of geometries, and Err(R,,-) is the error at the bond length R]. Table 3.1 gives the N PE, STD, and REE values for all eight potential energy surfaces determined in this study. Analysis of Table 3.1 reveals that for all reactions other than HgC—Cl, the CR- CC(2,3) NPE values for the various basis set calculations are less than (in many cases, much less than), or in a few cases virtually identical to, the corresponding CCSD(T) value. Similarly, the average CR-CC(2,3) NPE as computed with 6-31G (the largest basis set that all reactions were computed with) is 0.4 millihartree less than that of CCSD(T), while the average CR—CC(2,3) NPE for the cc-pVTZ basis (the largest basis set considered) is 0.3 millihartree less than CCSD(T). Thus it would seem that, at least in terms of NPE, the ROHF-based CR—CC(2,3) scheme is capable of outperforming the frequently used UHF-based CCSD(T) approach in these reactions. Interestingly, the situation is a little different when CR—CC(2,3) and MRMP2 are compared, as neither is clearly superior with regards to NPE. Indeed, for half of the reactions, namely HgC—H, HgSi—H, HgSi—Cl, and HQSI_CH3, the CR-CC(2,3) NPE values are smaller than the corresponding MRMP2 values, while for the remaining reactions it is the other way around. Looking at the average NPE values, we see that for 6-31G, the CR—CC(2,3) average NPE is higher than that of MRMP2 by roughly 0.5 millihartree, whereas for cc-pVTZ the CR-CC(2,3) value is actually lower by 0.4 millihartree. Moving on, Table 3.1 reveals that the STD patterns observed are quite similar to those observed for the NPE values. Indeed, it turns out that with the exception of 106 Table 3.1: Restricted open-shell CR—CC(2,3), unrestricted CCSD(T) and MRMP2 NPE, STD, and REE values relative to MRCI(Q). Units are millihartree.a NPE STD REE C R- MR- C R— MR— CR— M R- Basis CC(2,3) CCSD(T) MP2 CC(2,3) CCSD(T) MP2 CC(2,3) CCSD(T) MP2 H2C—H MINI 0.366 1.093 N.A. 0.146 0.431 N.A. -0.313 0.851 N.A. 6—31G 1.725 2.081 1.883 0.565 0.766 0.627 0.104 -0.035 -1.507 6—31G(d) 3.279 3.339 3.447 1.072 1.160 1.226 0.152 -0.052 -3.114 cc—pVDZ 3.621 3.699 4.807 1.162 1.286 1.832 0.170 -0.040 -4.529 cc—pVTZ 4.256 4.223 5.613 1.365 1.455 2.181 0.181 -0.062 -5.276 HgC-Cl MIX 2.022 1.965 0.436 0.624 0.754 0.125 -0.840 -1.118 -0.114 6—31G 3.772 2.674 0.832 1.175 0.957 0.293 0.111 -0.421 0.123 6—31G(d) 6.787 6.146 2.694 1.919 1.956 0.777 0.583 0.248 -1.519 cc-pVDZ 7.112 6.705 2.875 2.029 2.072 0.855 0.643 0.423 -1.653 cc-pVTZ 8.367 7.794 4.646 2.398 2.327 1.410 0.582 1.165 -2.879 H2C—CH3 MINI 0.301 2.543 N.A. 0.197 0.849 N.A. —0.326 0.300 N.A. 6—31G 3.126 4.023 1.176 0.981 1.401 0.401 0.160 -0.039 —O.753 HQC—SiH3 MINI 0.412 1.814 N.A. 0.127 0.666 N.A. 0.064 0.017 N.A. 6—31G 2.681 3.142 1.553 0.821 1.069 0.539 -0.136 -0.236 -1.272 HQSi-H MINI 0.237 0.443 N.A. 0.063 0.142 N.A. 0.049 0.063 N.A. 6—31G 0.526 0.801 0.697 0.151 0.277 0.228 0.041 0.035 -0.059 6—31G(d) 1.061 1.450 2.271 0.239 0.366 0.767 0.300 0.509 0.585 cc-pVDZ 0.923 1.100 3.128 0.212 0.315 0.847 0.245 0.236 -0.822 cc-pVTZ 1.132 2.092 3.310 0.295 0.547 0.873 0.584 0.577 -0.842 HQSi-Cl MIX 0.744 1.377 0.478 0.225 0.406 0.163 -0.114 -0.342 0.294 6—31G 0.682 2.094 1.917 0.175 0.592 0.565 0.463 0.346 -1.870 6-31G(d) 1.917 3.616 3.302 0.583 0.842 1.308 0.054 0.474 2.034 cc—pVDZ 1.833 3.661 4.170 0.543 0.814 1.688 0.068 0.499 2.823 cc-pVTZ 2.705 3.600 4.472 0.801 0.764 1.604 —1.013 0.221 2.006 H2Si—CH3 MINI 0.392 0.696 N.A. 0.127 0.235 N.A. -0.040 -0.110 N.A. 6—31G 0.434 0.765 1.288 0.114 0.226 0.321 -0.054 -0.031 -0.740 H2Si-SiH3 MINI 0.485 1.991 N.A. 0.158 0.571 N.A. -0.046 0.000 N.A. 6—31G 1.510 2.048 0.797 0.357 0.545 0.184 -0.077 -0.115 0.103 Unsigned Average 6—31G 1.807 2.204 1.268 0.542 0.729 0.395 0.143 0.157 0.803 cc-pVTZ 4.115 4.427 4.510 1.215 1.273 1.517 0.590 0.506 2.751 a Full CI is the benchmark method with the MINI basis set. 107 the CR—CC(2,3) MINI surface for HgC—CHg, the STD values are roughly 20—30% of the corresponding NPE values. As result, much of the previous NPE discussion also applies in the STD case. However, in the REE case, such patterns are not observed, and a more detailed analysis is useful. Upon inspection of Table 3.1, it becomes clear that both CR-CC(2,3) and CCSD(T) significantly outperform MRMP2 in terms of REE. Indeed, in some cases the MRMP2 REE values exceed those of CR—CC(2,3) and CCSD(T) by as much as 5 millihartree. This is further illustrated by the average REE values, for which the MRMP2 result exceeds those of CR—CC(2,3) and CCSD(T) by roughly 0.7 and 2.2 millihartree for the 6—31G and cc—pVTZ basis sets, respectively. A more direct comparison of just CR-CC(2,3) and CCSD(T) reveals that these two methods perform very similarly to each other with respect to average REE values. In fact, the average CR—CC(2,3) REE value is less than that of CCSD(T) by only 0.01 millihartree for the 6-31G basis set, while it is greater than that of CCSD(T) by only 0.09 millihartree with the cc-pVTZ basis. Based on these results, it is clear that the ROHF-based CR—CC(2,3) approach is a promising alternative for performing high accuracy calculations of single-bond break- ing processes in radicals. Indeed, these results indicate that for HgC—X and HgSi-X reactions, CR-CC(2,3) outperforms the popular UHF-based CCSD(T) and MRMP2 approaches, which are traditionally considered the recommended options for high ac- curacy, low-cost calculations of bond breaking in radicals. In particular, we see that CR—CC(2,3) produces NPE values that are notably smaller than those of CCSD(T) and which are comparable to those of MRMP2, while producing REE values that are comparable to the small values produced by CCSD(T) and significantly smaller than those of MRMP2. Thus, the CR—CC(2,3) approach provides a more balanced description of the radical potential energy surfaces than CCSD(T) or MRMP2 in spite of the use of the ROHF reference, which does not dissociate as well as the UHF reference used in the CCSD(T) calculations, and in spite of being a single-reference 108 black-box approach, which is much easier to use than MRMP2. Furthermore, the ROHF-based CR—CC(2,3) approach is able to provide such accurate results while avoiding the pitfalls associated with unrestricted or multi—reference formalisms, such as errors in relative energies resulting from spin contamination of UHF in the former case or the intruder state problem and the need for carefully selecting active orbitals in the latter. 3.2.2 Singlet-Triplet Gaps in Biradicals The description of the singlet-triplet gaps in biradical systems represents a significant challenge for many electronic structure methods. The difficulty stems from the fact that the singlet and the triplet states, which result from the parallel or antiparallel coupling of the two unpaired electrons forming the radical centers, are often primarily characterized by consistently different correlation effects, particularly when the radical centers are separated by some distance. Indeed, the non-degenerate, high-spin triplet state is dominated by dynamical correlations, while the singlet state often shows a strong contribution from nondynamical correlations, and thus has a more multi- reference nature. As a result, the accurate description of the gap between these states requires an electronic structure method capable of accurately balancing both types of correlations. It has generally been thought that the genuine multi-reference methods of the MRCI or MRCC type are needed to produce accurate results under such conditions, but, as it turns out, the single-reference CR—CC(2,3) scheme employing the RHF reference for the singlet state and ROHF reference for the triplet state is also capable of accurately describing such systems, being very competitive with genuine multi-reference methods. To illustrate this, we consider several systems which are characterized by varying degrees of a biradical nature. We begin with methylene, which is well known for being the subject of controver- sies between theory and experiment (cf., e.g., [221—226], and references therein). As 109 a first test, we performed calculations of the adiabatic singlet-triplet gap using the same DZP-type basis sets and geometries that were used in the well-known bench- mark calculations of Bauschlicher and Taylor [227]. The results are presented in Table 3.2, along with the exact full CI results of Bauschlicher and Taylor. As can clearly be seen, the most complete CR—CC(2,3),D approach provides the best description of the singlet-triplet gap out of all the triples-type CC methods considered, producing an error relative to full CI of only 0.21 kcal/mol. This is a notable improvement over the 0.32, 0.44, and 0.41 kcal/mol errors obtained with CCSD(T), CR—CCSD(T), and CCSD(2)T (=CR—CC(2,3),A), respectively. Furthermore, only the more com- plete variants C and D of CR—CC(2,3) are capable of providing a singlet-triplet gap that is more accurate than that of CCSD(T). Variants A and B, which are equivalent to the CCSD(2)T methods of [205—209] are not as effective. It also important to note that the high accuracy of the CR-CC(2,3),D singlet-triplet gap value is not the result of a fortuitous cancellation of errors. Looking at the total energies for the sin- glet and triplet states, we see that the CR—CC(2,3) values are in excellent agreement with those of full CI. Indeed, the errors in the singlet and triplet energies are only 0.333 and -0.001 millihartree, respectively, which represent substantial improvements over the CCSD(T), CR—CCSD(T), and CCSD(2)T triplet—state errors of 0.367, 0.516, and 0.469 millihartree and singlet-state errors of 0.873, 1.213, and 1.125 millihartree, respectively. In order to further explore methylene, we performed a sequence of calculations using the aug—cc-pCVmZ (93=T, Q, and 5) basis sets [178,179,228] and compared the results with the Quantum Monte Carlo (QMC) [229—231] calculations of Umrigar [232]. Two different variants of QMC were used in Umrigar’s calculations, namely variational Monte Carlo (VMC) and diffusion Monte Carlo (DMC). The trial functions for both variants consisted of Jastrow-Slater multi-determinant CAS wave functions which were optimized using the linear optimization method [233—235]. Three different 110 Table 3.2: The adiabatic A 1A1 various CC approaches, and the DZP basis set.8L — X 331 splitting in CH2 obtained with full CI and E(X 331) E(A 1A1) E(A 1.41) — E(X 381) Method (hartree) (hartree) (kcal /mol) Full CIa 39.046 260 -39.027 183 11.97 CCSD -39.044 111 -39.023 639 12.85 CCSD(T) 439.045 893 -39.026 310 12.29 CR—CCSD(T -39.045 744 -39.025 970 12.41 CR-CC(2, 3), -39045 791 -39.026 058 12.38 CR—CC(2, 3,B) -39.045 743 -39.025 960 12.41 CR-CC(2, 3,C) -39.046 267 -39.026 864 12.18 CR—CC(2, 3),D -39.046 261 -39.026 850 12.18 3‘ The basis sets, geometries, and full CI energies were taken from [227]. As in [227], in all correlated calculations, the lowest occupied orbital was kept frozen and Cartesian components of the carbon d orbital were employed. b Equivalent to the CCSD(2)T approach of [205]. C Equivalent to the triples part of the (2) correction of the CCSD(2) method of [206—209] active spaces were used to generate CAS trial functions, specifically the (2,2), (4,4) and (6,6) active spaces (recall that (n, m) denotes an active space of n electrons and m orbitals). Rather than restricting ourselves to the ground X 331 and first excited A 1A1 states, as was done in the DZP calculations discussed above, we follow the calculations of Umrigar and also compute the higher-energy B 181 and C 1A1 states. It should be noted that since the X 331 and A 1A1 states are the lowest- energy states of their respective symmetries, they can be computed using the ground- state CR—CC(2,3) formalism, which is how both the DZP and this set of calculations were performed. The B 131 and C 1A1 states were calculated using the excited- state CR—EOMCC(2,3) approach of Section 3.1.2, with the A 1A1 state acting as the correlated ground state for the EOMCC ansatz (see Eq. (3.1)). The C 1A1 state is particularly important here, since this is a strongly multi-reference state of the same symmetry as the A 1A1 state. In both the CC and QMC calculations, the geometries for each state were taken from [236], where they were generated using the full CI 111 calculations with the [5s3p/2s] triple zeta basis set of Dunning [237] augmented with two sets of polarization functions (TZ2P). Finally, since QMC is derived using the first quantization formalism, and so does not utilize the concept of a basis set, a proper comparison between QMC and CC requires the determination of the complete basis set (CBS) limit for the CC results. This can be done through extrapolation from the aug—cc-pCVxZ data. In order to verify the stability of the CBS results, two different extrapolation schemes were utilized in this work. In the first scheme, the CBS total energy of the X 381 state was determined by first extrapolating the correlation energy using the formula AE(:1:) = AE00 + Ans—3, (3.61) with :1: = 3,4,5. Here a: is the cardinal number of the aug-cc-pCVmZ basis set, AE(:I:) is the correlation energy obtained with the aug-cc-pCVJIZ basis, and AEOO is the correlation energy in the CBS limit. The resulting extrapolated correlation energy was then added to the aug-cc-pCV5Z reference energy, which, due to the fast (exponential) convergence of the RHF/ROHF energy with respect to the basis set, is equivalent to the CBS reference energy to an extremely high (0.1 millihartree) accuracy. The CBS total energies of the remaining states were subsequently obtained by adding the aug-cc-pCV5Z excitation energy to the extrapolated total energy of the X 381 state. This approach was designed based on the assumption that with aug-cc-pCV5Z the excitation energies were essentially converged with respect to the basis set. The validity of this assumption will be discussed below. In the second extrapolation scheme, the CBS total energy of each state was extrapolated using the formula E(2) = E00 + Bea—(H) + Ce‘(‘”—1)2, (3.62) with a: = 3, 4, 5. Here a: is again the cardinal number of the aug-cc—pCVrZ basis set, 112 E (2:) is the total energy of the state as computed with the aug—cc—pCVzZ basis, and E00 is the total energy of the state in the CBS limit. Throughout the rest of this discussion, we will refer to the results obtained using the first extrapolation scheme as CBS-A and those obtained using the second as CBS-B. It should also be noted that since the CR—CC(2,3)/CR—EOMCC(2,3) programs are part of the GAMESS package, as mentioned in Section 3.1.3, and since the integral codes in GAMESS are currently restricted to g-functions, the h—functions of the aug-cc-pCV5Z basis set were omitted in these calculations. Furthermore, all electrons were correlated and the spherical components of the d, f, and g orbitals were employed in all CC calculations. Table 3.3 shows the results of the methylene CC and QMC calculations. We begin by analyzing the stability of the CBS extrapolations. As mentioned above, the CBS-A extrapolation scheme was based on an assumption that although the total CC/EOMCC energies are not converged to the CBS limit with the aug—cc-pCV5Z basis, the excitation energies are. An analysis of Table 3.3 reveals that for this system this is indeed a valid assumption, as the EOMCCSD and CR—EOMCC(2,3) excitation energies do not significantly change when moving from the aug-cc—pCVQZ to aug-cc-pCV5Z basis sets, with the largest change of 0.02 eV occurring for the C 1A1 state. Moving on to a direct comparison of the two types of CBS CR—CC(2,3)/CR— EOMCC(2,3) total energies for each state, we see that the two extrapolation schemes produce results that are in reasonably good agreement. Indeed, the discrepancies between the two sets of results do not exceed 2.8 millihartree. The situation for the CCSD/EOMCCSD total energies is similar. The agreement between the two different CBS schemes for the adiabatic excitation energies is even better. For the A 1A1 and B 1B1 excitation energies, the two CBS CR—EOMCC(2,3) values differ by 0.001 eV or less, while in the case of the C 1A1 state, the discrepancy in the excitation energies is 0.013 eV. Again, similar observations apply to the EOMCCSD approach. We can conclude that the CBS CCSD/EOMCCSD and CR—CC(2,3)/CR—EOMCC(2,3) 113 .Ammfi 80.: 5wa 8.0 315.2 02> 0:0 035 =< 2:02.003 AA 2: 30030: 00.6.: pom Ewen Nm>0909w§w 0A: 0m USA 62333 mmmE Agatha A823 AVVAVAAAV Am VA. m8. AVAA- AmeEAVAVAV- A... VAV AV.~..AA AVAV- AmVEmAsAV- A A.Vm<0 02> AwVAVmAV.m AwVAAE wVAAVEV Am VAAAVAV. an- A8289”- Am V NAAAVAA- ANVENAAVAV- Am 330 02> A323 AA.VAVAA..A A.VAVAVA..AV A VAAVmAV. an- AAVAWAVAVAVAVAV- AA VAVRA a..- AAVAANEAVAV- AAV 8.2.0 020 AA.VAVAVm.N AA.VAWAVA..A eVmAAAV AA VmAVAAV. am- AAVAAVAWAVAVAV- AA V22. AVAA- AAVwAVEaAV- AA. A.Vm<0 .020 AA.VAVAVAV.m AA.VA..AVA..A eVmAeAV AAVAmAAV. AVAA- AAVAVAVAWAVAVAV- AA V22 AVA..- AAVAVAVAAaAV- Am AVm<0 .020 2.3 ”AVA; omA..AV AVAVA. AVAAV. AVA..- AAVAV NAVAVAVAA- 2a AVAHAAVAV- AVE AVA.A.AVAV- m-mm0 2.6a A.AVA..A AVmA..AV AVAVAV 232. AR mAVAVAVAV- mg was”- as 22.3.. <20 2.6a A.AVA..A AVAVA..AV AVAA. 2.3m- mam AAVAVAVAV- AVAVAV AVNAAAAV- AVAVA 22am- mus was AVAVA..A AVAA..AV A22 AVAAVAVAA- 26 25.3. AVAA AVNAAVAV- 22 «2.3. an... 23 23 221V 8A. 29%. AAVAV EAVAVAA- 8A. A.AA.AAA..- 3A. 22%. an... 800028.20 A.AVAV.A. AAVA..A A23 AA.AV 302... m2. AVAVAVAVAV- ...AVAV EAAVA..- NAVAV 2.2V- m-mm0 $3. AVAA..A AVAVAAV 22 422%- owe AAVAVAVAV- SA SAAVAV- man 22%- <20 23. m2..A was 22 AVAVAVAWAV- 2AA AAVAVAVAA- 3m ANAAVAV- gm AVA.A.AVAV- mu... 2.3. AVAA..A was AAVm 22.2- AV: ...wAVAVAV- AVAK AVNAAVAV- NAVAV 223.”. an... AVAVAVA. ASA 33V So was»: as EAVAVAV- AVAWA was”- NANA AVNAAVAA- Bus 800200 Am” x - AA..A 0 Am” A. - AmA m Amm x - A3 A.. A5 0 Am: m AA..A Ax Am... x am 230 A6262 A>0V QAmAAm 2053606. ofianfitaq. AmmAfiamV 2325 A308 6.3368 020 95:9 £2, .28: mmO 0A: 3 222222208 0A8 33 £me 3 .0 .BHHV NRA/Oméomsw 2: mafia 60500390 OD 90¢? :23 002630 mm NED mo mopepm wAAAbéoA 05 8% A>0 EV mmmonAAm ASSSASB 03.33% was 3053: :5 $2908 233 0A: .8 28300800 And 03er. values are accurate to the level of approximately 2 millihartree in total energies, and approximately 0.01 eV in excitation energies. We now turn our attention to the comparison of the CC results with those of QMC. Beginning with the basic CCSD/EOMCCSD calculations, we see that the results for the A 1A1 — X 3B1 and B 131 — X 3B1 gaps are in reasonably good agreement with the various QMC results. Furthermore, accounting for the effects of triples through CR-CC(2,3)/CR—EOMCC(2,3) does not significantly alter the values for these gaps, changing the excitation energies for the A 1A1 and B 1B1 states in the CBS limit by only about 0.03—0.04 eV. The situation is quite a bit different for the C 1A1 — X 381 excitation energy. For this gap, the EOMCCSD excitation energies differ from those generated by CR—EOMCC (2,3) and the various QMC approaches by roughly 1.4 — 1.5 eV, thus indicating that this is the most challenging state considered here, which is characterized by a significant multi-reference or two-electron excitation nature. Looking at the total energies of each state, we see that for the X 3B1, A 1A1, and B 1B1 states, which correspond to the states for which the EOMCCSD excitation energies were reasonable, CCSD/EOMCCSD produces errors relative to CR—CC(2,3)/CR—EOMCC(2,3) on the order of 3.5 to 6.1 millihartree, i.e. errors which are relatively small. The EOMCCSD calculation for the C 1A1 state, however, generates a huge 58 millihartree error relative to CR—EOMCC(2,3), illustrating the much larger role of triply excited clusters in the description of this multi-reference state. Moving on, it can be seen from Table 3.3 that the CR—CC(2,3)/CR-EOMCC(2,3) results for the adiabatic excitation energies are in very good agreement with the var- ious QMC results. Indeed, depending on the size of the CAS for the trial function, the discrepancies between the DMC and CR-CC(2,3)/CR-EOMCC(2,3) excitation energies range from 0018—0024, 0047—0059, and 0033—0122 eV for the A 1A1, B 1B1, and C 1A1 states, respectively. Interestingly, the agreement of the CR- 115 CC(2,3)/CR—EOMCC(2,3) excitation energies with the corresponding VMC data is even better, with the discrepancy ranges reducing to 0.000—0.023, 0003—0053, and 0.013—0.096 eV for the A 1A1, B 1B1, and C 1A1 states, respectively. It is worth noting, however, that the improvements in the excitation energy when going from DMC to VMC are not dramatic, and so we can conclude that the two QMC variants produce comparable results in this regard. The total energies, however, paint a some- what different picture. The discrepancies between the VMC and DMC results are on the order of 10 millihartree, which is a rather substantial disagreement. Looking at the CR-CC(2,3)/CR-EOMCC(2,3) total energies, it is clear that they agree much more strongly with the DMC results than with the VMC results. If we focus on the largest CAS(6,6) QMC calculations, then the CBS-A CR—CC(2,3)/CR-EOMCC(2,3) energies differ from the DMC results by 2.2—6.3 millihartree while the CBS-B energies differ by only 06—39 millihartree, which represents an excellent level of agreement. The discrepancy with the VMC energies, on the other hand, is an order of magni- tude worse, ranging from 13.4 to 16.4 millihartree for the CBS-A results and 10.6 to 14.0 millihartree for the CBS-B values. Given the agreement between the inde- pendent CR-CC(2,3)/CR—EOMCC(2,3) and DMC results, it is safe to conclude that the VMC results are the ones in larger error, producing total energies that are too high. However, given the fact that the increase in the VMC energies relative to the CR—CC(2,3)/CR-EOMCC(2,3) and DMC approaches is nearly constant for all four states, the resulting adiabatic excitation energies are still highly accurate and in very good agreement with those of the latter two methods. In summary, we see that by using the aug—cc-pCVzZ series of basis sets to extrap- olate the CBS limit of the CR-CC(2,3)/CR—EOMCC(2,3) energies for the low-lying states of methylene, the results are on the level of highly accurate, yet computa- tionally expensive, QMC calculations, which implicitly produce CBS results. This is a very encouraging result that illustrates the large potential of CR-CC(2,3)/CR- 116 EOMCC(2,3) in describing the low-lying states of biradical species. The methylene system, though a challenging case, is a relatively weak biradical. In order to investigate the performance of the CR—CC(2,3) scheme in describing the singlet-triplet gaps in systems characterized by a larger degree of biradical charac- ter, we consider the (HFH)‘ species. This Bowl-symmetric linear molecule, which consists of two unpaired spins, represented by the hydrogen atoms, separated by a polarizable, diamagnetic bridge, namely F‘, is not only a strong biradical, but also a simple, yet informative, model for many magnetic systems. The singlet-triplet gap in such magnetic systems is particularly important as it provides information about the magnetic exchange coupling constant. In the case of (HFH)—, this corresponds to the gap between the ground X 123" and the first-excited A 32,] states. To test the performance of CR—CC(2,3) as a function of the degree of biradical character, calculations for the A 32;] — X 123‘ gap of the Dooh-symmetric linear (HFH)_ system were performed for various values of the H—F distance RH—F [138, 139]. All calculations were performed using the 6—31G(d,p) basis set [194,195], employing the spherical components of the d orbitals, and the lowest-energy molecular orbital cor- relating to the 13 orbital of F was kept frozen. To gauge the performance of the RHF/ROHF—based CR—CC(2,3) calculations, performed using the codes discussed in Section 3.1.3, the results are compared with full CI data obtained with GAMESS [173] and MOLPRO [183], and with the UHF-based CCSD and CCSD(T) results obtained with Gaussian 98 [238]. The (HFH)' system is a strong biradical, particularly for larger H-F distances. This can be seen quantitatively in Table 3.4, which gives the absolute value of the ratio of the full CI coefficients at the doubly excited (HOMO)2 ——> (LUMO)2 determinant (62) and the RHF ground-state determiant (co) for the ground X 12; state. As is the case for the H2 molecule, the symmetries of the HOMO and LUMO in (HFH)_ are 09 and an, respectively, and the ratio c2 /cO is equivalent to the full CI value of the 117 .0030 .mwA N 05 mo :oAmAAdAAVS HO :3 0A: 2202083000 Aouv 2850:0200 02300::ko him 0% 0:0 33 2855000 A0353 T AOEOEV 002020 23:00 05 0 08066000 0A: 00 033 0A: 00 030». 0020000 008 0 .3808 00 090.08 AmmeOO 05 .8 530028 A8 05. 00 200 0072.3 0A: 3 0:300 :25. o... a: .AAA0EZAAUH AA 0020350 03 235-8 200.”.— 0058m 00080800 00» :00? .fioa 00 5.003% HANVQmOO 0A: 8. 000A0>Asdm AA 00:0-A0m0m .mED 00:0A0m0m him 5AA AV AV AV mm. m... NAV- A.AV- 2wa AVAA- 8%...- AV AVAVAVA. EAAVA AVA A A.- n. AAVAV 22- 22. AVE 8...“- AVAVA.AVA.- AA AVAVAVAA News EA mm Aw- ...AVA. AVAVAA w:- AA.A.- .32. 82V- «2%. E 83 02.3 as NANA AAA. 0% 9.2 A2. A...- EAA Emm- AaAsAA- 22. E? NAVEAV 00A. AVAA AVE- AVAAV AAV: mom- 0AA 3% $2. AVAVAS- RA. Sew maAVAVAV AVE. 2.6 A2- AVAA-A 80A 2: mAAV ”AAV... AWAVAVA- AVSAVN- $2 .23 mwAVAVAV ASA «AAA AAV- $8 2.2 02.. 9.2 NANA SAVA- mamm- SAA AVAVAVA A225 2% AAA: AVwA £2 82 ”AVAA ~02 8% EA- ...mAAm- 2me 8.3 2.2.5 AAVAVAA AVmAAV AVAA 2:. as... am” 2:. EA... we: 82:. 20.. S: 2.23 AVE AAAAVA. AAA: 3.2V AAVAVAV AWAVAAV AVRAV 8A: fin. AVAVAVmA- AWAVAVA 3: wEAVAV 0AA... max .28 22V Asa A22 3% was. AVNAK $3. .28 AVAVAA 60623 ABA-.500 800 A00 0.880000 0.380000 20.830020 628300-00 A5800 200 000 A022 0.00 AAV-A. EV mlmm 0000020 mum 0A: 00 203023 0 00 .33 .3: 000 £003 3.302-0 2: .3 82660 A Tso 2V Assam (AEAAV 656880-280 62.: 9AA 6A .60 FA N 1 ....m... A4 2AA. 2..” 62.0. 118 T2 cluster amplitude corresponding to the (HOMO)2 —> (LUMO)2 double excitation. The magnitude of this T2 amplitude is an indicator of the biradical character of the system, with values around zero representing essentially no biradical character and values around 1 representing a virtually pure biradical. From Table 3.4 we see that even for small values of RH—F, the (HFH)- system shows a large degree of biradical character. Furthermore, as the H-F distance is increased the degree of biradical nature steadily increases until the molecule becomes an essentially pure biradical at the largest distances. As a result of this strong biradical character, we expect the A 32:; - X 12; gap to be relatively small, and thus very sensitive to the electron correlation treatment, even for the relatively small values of RH—F, and steadily approach zero as the H—F distance is increased. As can be seen in Table 3.4, this behavior is displayed by the exact full CI results. Analyzing the performance of the various CC approximations based on adding a noniterative correction due to triples to the CCSD energy, we see from Table 3.4 that the only method to provide a uniformly accurate description of the singlet-triplet gap across all values of RH—F is CR-CC(2,3) (with the complete variant D of CR—CC(2,3) offering the best results). Indeed, if we focus specifically on the CCSD(T) scheme, it is clear that it has severe difficulties in uniformly describing the (HFH)— system regardless of which type of reference determinant is utitlized. With the RHF/ROHF reference, CCSD(T) does provide reasonably accurate results for the singlet—triplet gap at RH—F = 1.5 and 1.625 A, generating errors relative to full CI of approximately 60 cm‘l. However, this accuracy quickly breaks down as the H-F distance, and thus the biradical character, increases. The RHF/ROHF-based CCSD(T) approach does not produce the correct dependence of the gap on RH—Fa with the energy actually increasing after RH—F = 2.25 A rather than systematically approaching zero as the H-F distance is increased. The resulting errors relative to full CI are significant, growing from 268 cm-1 at RH_F = 1.75 A to 8900 cm—1 at RH_F = 4.00 A. This 119 behavior is easily understandable if we analyze the behavior of CCSD(T) with re- spect to each state (see Tables 3.5 and 3.6). Table 3.6 reveals that CCSD(T) provides accurate results for the A 323,”: state, producing errors relative to full CI of less than 0.7 millihartree for all geometries considered. This excellent performance makes sense since theA 32,4," state is a nondegenerate high-spin triplet state and so it is character- ized primarily by dynamical correlations, which CCSD(T) describes well. However, as shown in Table 3.5, the situation is considerably different for the X 12; state. Because this state is characterized by strong nondynamical correlation effects, which CCSD(T) has difficulty describing, the results are not nearly so accurate. Indeed, though CCSD(T) performs well for small H-F distances where the biradical character of the state is not as large, producing errors relative to full CI that are less than 0.8 millihartree, the performance of CCSD(T) quickly breaks down as RHVF increases, with the errors relative to full CI growing from 2 millihartree at RH—F = 1.875 A to 40 millihartree at RH—F = 4.000 A. Thus, we see that the problems in the RHF/ROHF based CCSD(T) description of the singlet-triplet gap stem from the un- balanced treatment of the two states, where the triplet is described very well but the description of the singlet is problematic. Switching to an unrestricted UHF ref- erence for the CCSD(T) calculations improves the situation, but the description of the singlet-triplet gap is still problematic. Though the gap approaches zero as the H-F distance is increased when the UHF-based CCSD(T) approach is used, it decays too fast as well as a result of the considerable mixing of singlet and triplet contribu- tions in the spin-contaminated CCSD(T) / UHF calculation. Furthermore, the results for shorter H-F distances, for which the restricted CCSD(T) calculations were quite 1 errors in the successful, are considerably worse now. Indeed, the 57 and 58 cm“ restricted CCSD(T) results relative to full CI for RH—F = 1.5 and 1.625 A increase to 1107 and 1232 cm‘l, respectively, when the UHF-based CCSD(T) approach is employed. 120 .Eomlwofi 00 000008 AvamOO 000 .00 003000.80 Amv 000 mo 0000 00—0000 000 00 .000000 00000 3 a: .00000200m 0 00.8380 000 0000000 000m 000000m 00000000 000 0003 .fiofi 00 00000000 BANK—moo 000 0... 0020205" 0 4|. I I 4 hiflflull Q... |- _ h .n F P; 3.0 as; AVEAV: AAAVAV awAVAV- A000 A03 22V...- 030 08.00. 2.08002- AVAVAV... AWAV...AV A02 A000: AVAVAVAV 00.0- mm...” .03 0.00%- 9.02 0.3% 0023.2:- AVAVAV.AV ...-0A .03 A500 «SAV- vwflV- AVAAV0 800 0.8.3- 02: A008... $8387 A53 AVA“: 08.” 9.0.2: was- $3- 33 3.3 ”8.3- A008 0000” 8832:- $0 09.0 AVAWAV... 9.002 83. A000- .EVa an...” ......NAVA- 9.....8 5.0.0 $23.03- Sam «AVA... EA... $0.02 AVEAV AVAA:- 2000 A5.” 0.50- 50.8 ”3A0” 0000.00.02- 000 0...... A03 2080 AAV.A.AV 0.3- $2 £3. E...- wAVmAVA 9.00:. 08832. AVAVAV.A 02.... 00...... 03.08 AWEAV 09.0- 0.0.0 32. :3. maAVAwA A028 ABAVBAVAVA- .50 AVAAAV 22: 3020 A30 AVmAVAV 3A.... .03. 32V- A02: 0.23 A0352:- 0...: 3A0 20.: 8;: A80 $0.0 A05. A00.” 82V 8...: 3200 3302A:- 0A.: 8...... 9.02 82me AVAAVAV A03 23. E... 300 3.0.2 AV...AV.E $800.02- AVAVE 30.0.00 0000 00m 0.0000000 000800.00 20.00.000-00 200000-00 3.0000 0000 00m 00 0.0 0-0.0 000000m0m mmD 000000m0m mmm 0030., HO :3 w0000oam00-80 000 00 0300000 00000000000 00 003m 000 0003000 30000800 000 0003 .0000000 00 003m 000 0009000 HO :3 008 .Q. 00 mlmm 00000000 him 000 00 003003 0 00 .33 .3: 000 00000 3.3050 000 .3 000000000 00000.? LEE-S 23088.3-08Q .0000: 000 00 00000 mmfi N 000 000 00000000 000000000 0000000000 00000., 003 00000000 0003000 00000 000 00 0000000800 < And 0308 121 .Sealeeae e0 00508 AaVQmOO 05 mo 080000.80 av 05 00 0000 00085 000 00 020000 2080 00. 0: .80_0>80m 0 00002080 000 00005.8 800m 005005 00080000 000 0083 .Eeafi 00 80000000 HAaVQmOO 000 00 8203030 0 vmvd m5; eahd: dead vefid emwd Hmvd vmvd at; aaudfi mfimeamdea- eeegV wand new; vmwéwfi de 05d umvd emvd nmmd dz; aweéwfi nmafimmdefl- deed vmed 5:4 35de mefid dead uwvd ~1de aued new: aeedwfi eehevmdefi- eema weed anA meaSwH aead dead mend mpvd aeed aaaé emedwfi eeemvmdefi- Ema emed 3d; @92me mHad dead wand aevd deed vawé evvdefi mfimpvmdefi- emaa eHed Sea mmedwfi eHad A“Had Hmmd aBd nweed eve.“ ewaaefi eamemmdefi- mafia mend :Ha emndefi wHad mHad nEnd Heme ted eee.a 03.03 Eammmdefi- eee.a ewmd eHaa mewéefi eHad :ad Mvemd evmd awed eeaa weeddfi neemmmdefi- mum; vemd eama mEdeH aead edfid deed ummd nvaed evma wmesefi Sammmdefi- emu.“ evmd emva weeds de mead Sed Hemd mend aeva wfimdefi mtammdefi- maeA nmmd nvma memddfi mid ede eaed emmd eemd waea veedea meemvmdefi- dew; 60000 0000 000 000000.000 000000.00 00.00.000.00 00.00.000.00 E0000 0000 000 000.0 0-00 000000m0m 0033 000000030 mmm .0080> HO :8. w80000000000 000 00 0300800 000000838 8 003m 000 0080000 w880800 000 0:83 0000000 8 003m 000 0080000 HO :3 08H. .300 8V 01mm 0000006 him 08. 00 080008 0 00 .33 .3: 000 00000 3.3036 080. .3 00800000 8000.00 Iflmmmv 800088z0é8Q .0000: 08. 00 00000 fiwm «x 080 0090 00000.08 0850850 080000000 0080000 083 008030 0080000 _0000 05 00 000300800 < “ed 0308 122 Moving on to variants A and B of CR—CC(2,3), which, as discussed in Section 3.1.2, are equivalent, up to small details, to the CCSD(2)T approach of [205] and the triples part of the CCSD(2) method of [206—209], respectively, it is clear that they too have some difficulties describing this system, although they are not nearly as big as in the CCSD(T) case. The singlet-triplet gaps generated by the CR—CC(2,3),A and CR-CC(2,3),B methods decrease too fast, and, as a result, end up passing zero and becoming negative. Thus, for the larger geometries both approaches predict the wrong ordering of states. Eventually, a turnover occurs and the gap begins to increase again back towards zero, but this does not happen until the value of the gap reaches -230 and -718 cm—1 for the A and B variants, respectively. Furthermore, the discrepancies in the singlet-triplet gap energies relative to full CI remain substantial for all values of the H-F distance besides RH—F = 4.0 A, with errors ranging from 247—822 cm"1 in the CR—CC(2,3),A case and 651—1164 cm—1 in the CR—CC(2,3),B case . The full variant D of CR—CC(2,3), which, from now on, will simply be referred to as CR—CC (2,3), does not suffer from such difficulties in describing the A 322: —X 12; gap. Indeed, we see clearly from Table 3.4 that CR—CC(2,3) reproduces the systematic decay of the energy gap accurately. Though it does overshoot zero slightly at RH_ F = 4.0 A, it is only by the very small value of 33 cm‘l. Quantitatively, the errors in the CR—CC(2,3) results relative to full CI never exceed the relatively small value of 174 cm—1 regardless of the value of the H-F distance. Looking at Tables 3.5 and 3.6, it is clear that this excellent description in the singlet-triplet gap is not a result of a fortuitous cancellation of errors. In fact, CR~CC(2,3) provides a highly accurate description of the total energies of both the A 323;“ and X 12; states, with errors relative to full CI ranging from 0.143 to 0.219 millihartree for the former, and from 0.060 to 0.967 millihartree in the latter. Thus, we see that of the various approximate triples methods that have similar computational costs and ease of use, only CR- 123 CC(2,3) (specifically the complete variant D of it) is capable of properly balancing the dynamical and nondynamical correlation effects in a way necessary to accurately describe the singlet-triplet gap in (HFH)— as a function of the H—F distance. As a final example in this section, we consider the BN molecule. This is an extremely challenging biradical system that is seemingly characterized by a large contribution from connected quadruply excited clusters (i.e T4 clusters). Indeed, a calculation [239] of the singlet-triplet separation using the full CCSDT approach with the cc-pVQZ basis set [178,220] generated a value that was significantly larger than experiment. Given the challenging nature of this biradical and the large T4 effects, the CR—CC(2,3) approach, which only accounts for the effects of up to T3 clusters, is not expected to accurately describe the singlet-triplet gap in this system, and so a higher-order method such as CR-CC(2,4), which accounts for the combined effect of T3 and T4 clusters, is likely to be necessary to produce high quality results. In order to study the role of T4 clusters in the description of the adiabatic singlet- triplet gap of BN, T e, as well as to examine the performance of the CR-CC(2,4) approach in an application involving a biradical system, CR—CC and reduced multi- reference (RMR) CC calculations for T g were performed in collaboration with Dr. Xiangzhu Li and Professor Josef Paldus. The RMR CCSD approach [240] is based on solving the so—called externally corrected CCSD equations, which include all terms of the standard CCSD equations plus correction terms involving the most important (i.e. primary) T3 and T4 clusters [241,242]. In the RMR CCSD scheme, the T3 and T4 amplitudes that enter the T3 plus T4 corrected CCSD equations are obtained from the cluster analysis of a modest-size MRCISD wave function. As a result of this procedure, the T3 and T4 clusters that are accounted for in this scheme primarily describe nondynamical correlations. In order to provide a more balanced description of the correlation effects, a standard noniterative perturbative correction due to the remaining triply excited clusters can be added to the RMR CCSD energy to account 124 for those T3 components that describe the missing dynamical correlation effects (this correction has the same form as that of the standard CCSD(T) approach, except that one uses it only for the dynamical T3 contributions, i.e. those T3 contributions that are not captured by MRCISD). This gives rise to the RMR CCSD(T) scheme [243]. In this study, all calculations were performed using the cc-szrZ (a: = D, T, Q, 5) basis sets. As was the case for the methylene study discussed above, the h-functions were omitted from the cc-pV5Z basis set due to the restriction of the integral routines of GAMESS to g-functions. These results were then used to extrapolate the CBS value of the singlet-triplet gap, Te(CBS), using the following formula (cf. Eq. (3.62)): Te(a:) = Te(CBS) + Be—(x-ll + Ce-2(a; = 2, 3, 4, 5), (3.63) where Te(:r:) is the value of the singlet—triplet gap as computed with the cc-meZ basis set. In addition to the adiabatic singlet-triplet gap Te, the equilibrium bond lengths Re and harmonic vibrational frequencies we, as obtained with the cc-pV5Z basis, were computed as well. Due to the pilot nature of the CR—CC(2,4) code that we have developed, as mentioned in Section 3.1.3, only variant A of CR-CC(2,4) was used to perform the calculations. As discussed in Section 3.1.2, CR—CC(2,4),A is equivalent to the CCSD(2) approach of [205] when the canonical RHF orbitals are used. As a further consequence of the pilot implementation, the cc—pV5Z CR—CC(2,4) calculations could not be performed, and thus Eq. (3.63) was used to extrapolate this value from the cc-pVxZ, x = D, T, and Q, data. For the RMR CC calculations, two different model (or reference) spaces were considered for the underlying MRCI calculations, which lead to what will be referred to here as the A- and B-type RMR CC calculations. The a 12+ state, which is the more multi-reference, and thus more difficult, state to calculate, is described by a model space spanned by four symn'ietry- adapted determinants in the A-type and B—type RMR CC calculations. The ground 125 X 3H state uses a single reference determinant in the A-type RMR CC calculations while a two determinant model space is used in the B—type RMR CC calculations. Table 3.7 gives the results of the various calculations. We begin by considering the results for Re and we. Both types of RMR CCSD calculations produce a bond length that is too short and a harmonic frequency that is too large relative to the experimental values for the X 3H state. On the other hand, the CCSD(T) aproach, which neglects T4 completely and primarily describes only the dynamical T3 effects, produces Re and we values that are in very good agreement with experiment. Fur- thermore, the results of the CR—CC(2,3) and RMR CCSD(T),B calculations do not significantly differ from those of CCSD(T) in this case (recall that for the triplet state, RMR CCSD(T),A uses a single reference and thus is identical to CCSD(T)). These data suggest that the X 311 state is primarly characterized by dynamical correlation effects with little contribution from connected quadruples, which makes sense given the single-reference nature of this state. The a 12+ state, on the other hand, has a strongly multi-reference nature, and thus the emerging picture is quite a bit different. CCSD(T) produces Re and we values that are somewhat too short and too large, respectively, relative to experiment. The CR—CC(2,3) approach, which has already been shown to better balance the dynamical and nondynamical correlation effects, performs better but still has difficulty, producing a bond length that is slightly too long and an oscillator frequency that is somewhat too small. This indicates that accounting for both the dynamical and nondynamical triples is not enough to accu- rately describe the spectroscopic properties of the a 12+ state, and the T4 effects must also be included. This is supported by the RMR CCSD(T) results, which show good agreement with experiment for both the bond length and harmonic frequency. Moving on to the a 12+ —X 3 II gap, Te, we see that the standard CCSD approach dramatically fails, producing errors relative to experiment of more 4000 cm‘l. Using the RMR CCSD scheme, which incorporates the primary T3 and T4 correlation effects 126 Table 3.7: Equilibrium bond lengths Re (in A) and harmonic frequencies we (in cm‘l) for the lowest triplet and singlet states of BN, as obtained with cc-pV5Z basis set, and the adiabatic singlet-triplet splittings Te (in cm”1) as obtained with the cc-pVxZ (a: = D, T, Q, and 5) basis sets, as well as the extrapolated CBS limit values. X311 612 Te Method Re we Re we x=D x=T x=Q x=5 CBS CCSD 1.317 1586 1.272 1705 4196 43914459 4471 448846 RMR CCSD,Aa 1.317 1586 1.273 1727 125913571450 1490 151241 RMR CCSD,Bb 1.321 1559 1.273 1727 167717561827 1863 187843 CCSD(T) 1.329 1510 1.269 1739 -34 -92 —94 -87 —8843 CCSDTC 1.330 1512 1.277 1702 — — 844 — — CR—CC(2,3) 1.329 1518 1.281 1686 404 666 817 919 946419 CR—CC(2,4),Ad — — — — 253 289 323 (337) (345) RMRCCSD(T),A31.329 1510 1.277 1691 361 264 267 269 27141 RMR CCSD(T),Bb 1.330 1501 1.277 1691 548 422 408 406 40441 Experiment 1.329e 1519.2f 1.275% 170545 15—182f 1496e 1.274h 1700.9f 153In a A-Type RMR CC calculations were based on the (2,2) active space, which leads to a four-dimensional model space for the singlet and a one-dimensional model space for the triplet [244]. b B-Type RMR CC calculations utilized a four-dimensional model space for the singlet and a two-dimensional model space for the triplet [244]. C From [239] using the cc-pVQZ basis set. d Due to the pilot nature of the CR—CC(2,4),A code used in this work, gradient calculations were not possible. As a result, the CR—CC(2,3) optimized geometries were employed. The cc—pV5Z and CBS values for the singlet-triplet gap are extrapolated values obtained using Eq. (3.63). 9 1'0 and wo values from [245]. f From [246]. 3 From [247]. h From [248]. 127 ‘7 captured by MRCISD, does significantly improve the results, reducing the error in the calculated gap by a factor of more than two. Unfortunately, the error relative to experiment is still sizable, pointing to the importance of dynamical connected triples in the description of the singlet-triplet gap of BN missing in the RMR CCSD calculations. CCSD(T), on the other hand, produces negative values for Te, meaning that it gives the wrong ordering of states. This results from CCSD(T) accurately treating the single-reference triplet state, while overshooting the energy of the multi- reference singlet state. The CR-CC(2,3) method produces the proper ordering of states as well an error reduction compared to RMR CCSD. In fact, for the cc-pVDZ basis set, the results look promising and are in reasonable agreement with the CR— CC(2,4) and RMR CCSD(T) results. Unfortunately, the error in the calculated gap increases as the size of the basis set increases, and so both the larger basis set and CBS CR-CC(2,3) values Show sizable errors relative to experiment. It is important to note, however, that for the cc-pVQZ basis set, the CR—CC(2,3) value of Te is in excellent agreement with the full CCSDT value of [239], differing by less than 30 cm—l. This suggests that CR—CC(2,3) is accurately and properly describing the T3 effects, and is unable to provide a reasonable description of the gap due to missing quadruply excited clusters. The above data all suggests that both a properly balanced description of dynamical and nondynamical triples as well as the inclusion of connected quadruply excited clusters is necessary to accurately describe the singlet-triplet gap of BN. Table 3.7 reveals that this is indeed the case, as both the CR—CC(2,4) and RMR CCSD(T) approaches produce significantly better values for Te, which are in good, although not perfect, agreement with the available experimental data, which are not well- established either, due to the smallness of the a 12+ — X 3H gap. Additionally it is encouraging to see that these two completely different methods of including the effects of T3 and T4 clusters in the CC calculations are in very good agreement with 128 each other. 3.2.3 Excitation Energies of C2N, CNC, N3, and NCO As a final example of the performance of the left-eigenstate CR—CC/CR—EOMCC schemes in studies of open-shell systems, we once again examine the low-lying states of C2N, CN C, N3, and NCO, which were studied using the active—space EA-EOMCCSDt and IP-EOMCCSDt approaches in Section 2.2.3. Not only do these representative radical systems provide an interesting and informative test case for the open-shell CR-EOMCC(2,3) implementation, but they also allow us to directly compare the performance of both of the new approaches for studying open-shell systems developed in this research. To that end, the computational details, including basis set and equilibrium geometries used, are the same as those defining the calculations of Section 2.2.3 and will not be repeated here. Table 3.8 shows the results of the calculations for CN C and CgN. The EA-EOMCC results discussed in detail in Section 2.2.3 are also included to facilitate comparisons with the CR—EOMCC(2,3) data. One of the first things apparent from these results is that the basic EOMCCSD approach performs reasonably well for the A 2131, state of CNC and the A 2A state of C2N, producing results that are in reasonable agreement with both experiment and the high quality EA-EOMCCSD(3p—2h) results. Indeed, the difference between the EOMCCSD and EA—EOMCCSD(3p—2h) results are only 0.186 and 0.136 eV for the A 2Au state of CNC and the A 2A state of CQN, respec- tively. Furthermore, it is clear that including the effects of triples has little impact on these states, as all four variants of CR—EOMCC (2,3) provide similar values for the excitation energies that do not differ substantially from those of EOMCCSD. This is not too surprising given the relatively simple structure of these two states. Indeed, the so-called reduced excitation level (REL) values are 1.099 and 1.090 for the A 2A.” state of CNC and the A 2A state of CQN, respectively. The REL diagnostic, intro- 129 duced in [132], is a quantitative measure of the nature of an excited state relative to the corresponding ground state, with values close to 1.0 indicating the state is dom- inated by single excitations and values close to 2.0 indicate domination by double excitations. It is well known that EOMCCSD performs well for states dominated by single excitations, and so it is not surprising to see the reasonable performance of this scheme for these two states. The situation is different for the remaining states of CN C and C2N, however. Indeed, the REL values characterizing the B 22,] state of CNC, and the B 22“ and C 22+ states of C2N are 1.979, 1.856, and 1.897, respectively, indicating that all of these states are dominated by two—electron transistions. Not surprisingly, given this observation, EOMCCSD fails to accurately describe the excitation energies, producing errors of 2.7 — 3.0 eV relative to experiment. When variant A of CR—EOMCC(2,3), which, as mentioned in Section 3.1.2, is equivalent to the EOM-CC(2)PT(2) scheme of [203, 204] when the canonical RHF orbitals are employed, is used to calculate the excitation energies for the above three states, the errors are reduced but are still substantial. Indeed, the discrepancies with experiment are 1.117, 1.239, and 1.435 eV for the B 22,] state of ON C, and the B 223— and C 22+ states of C2N, respectively. Variant B, which, as a reminder, replaces the orbital energies of the Meller-Plesset- like denominators of CR—EOMCC(2,3),A with the one-body diagonal elements of the similarity—transformed Hamiltonian (the latter essentially corresponding to ‘dressed’ orbital energies), shows essentially equivalent performance. In contrast to variants A and B, the full variant D of CR—EOMCC(2,3) pro- duces accurate results for the B 22,“: state of CNC, and the B 22‘ and C 22+ states of C2N. Indeed, the errors relative to experiment for the B 22.: state of CNC, and the B 22" and C 253"" states of C2N are 0.284, 0.331, and 0.518 eV, respectively, which is a substantial improvement over the results of variants A and B (not to mention, EOMCCSD). Furthermore, the CR-EOMCC(2,3),D excitation 130 48$ 86¢ :83 5 . +ZmO ho +020 mo $358 53838 $298002: $952 .58 as... mo Umummmaoo 8me @388 951 3 .:.m 233. E @85me $93 We madam 23 2d 939:: 983 $338 .885an Smfimeéeam «ad 2: mo manomgfiw 282.28 2: fits vofiwfio 239:: 989 8&8“on HEB wow: v.8 moEuoEoww 32:4 8 8m.” a?“ 8.2 SE. 3.2. £3 £3 8w.” 9% 5: EN Q at.“ 22.. $3.. 83. 23. :3 £8.” Ex” 83 as: .5 m was $3 $3 as...” E...” a; was 34.4 88 8.3 4a w 28 22. 8.3 $3. 4%.... News ME 35. SS. 443 83 ”mm m a; 82. 33. 34.3 83. 32. £3. 8:. 83 83 54a «4 czo 6356:8334 a o m < amoozom ammoo $24580 €35.80 4mm 88m 2826: 33828-5 284m 3.26 25 020 a6 magnum @3658 928?». 9:43-32 2: mo 83.? Saw: 6?: cofimfioxm @83on was A>mv mmwwawco :oEwfioxo 05.8334. ”mam @3er 131 .fiofi 8on ~8me o .IOU Z .8 (m2 mo 3.358 826208 wofisoooas $23: 025 23 mo @3238 85% 6383 Eat n .:.m 33am. 5 @3583 305 mm 22.3 map v.8 magmafl @203 323. £0on95 EmémvHOAV/wm mv 8&2me 2033623 938.3344 “ad @3er 132 energies agree very well with the high-level EA-EOMCCSD(3p—2h) results. Indeed, for the B 223: state of CNC and the C 22+ state of C2N, the discrepancies be— tween the CR—EOMCC(2,3),D and EA-EOMCCSD(3p-2h) data are only 0.119 and 0.015 eV, respectively. For the B 2E“ state, which poses some difliculty for the EA- EOMCCSD(3p—2h) scheme and likely requires 4p—3h excitations to properly describe it, the discrepancy between CR-EOMCC(2,3),D and EA—EOMCCSD(3p—2h) is some- what larger, with CR—EOMCC(2,3),D improving the result by 0.567 eV, which is a remarkable improvement considering the black-box nature of the CR—EOMCC(2,3) theory. Turning to the low-lying states of N3 and NCO, Table 3.9 reveals that all three states included in it are dominated by single excitations. Thus, not surprisingly, the basic EOMCCSD approach provides accurate results for all of these states, with errors relative to experiment ranging from 0.110 to 0.207 eV. As observed earlier for the states of CNC and C2N dominated by singles, all four variants of CR—EOMCC(2,3) provide essentially equivalent results, with differences among the results of less than 0.05 eV, and with small changes in excitation energies when going from EOMCCSD to CR—EOMCC(2,3). Finally, it should be noted that the CR-EOMCC(2,3) results are in good agreement with those of the high-level IP-EOMCCSD(3h-2p) calculations, with discrepancies of approximately 0.1 - 0.4 eV between the CR—EOMCC(2,3) and IP-EOMCCSD(3h-2p) data. Thus, it is clear that the excited-state CR—EOMCC(2,3) approach performs very well for the low-lying states of open-shell systems, with the capability to produce results of similar quality as that provided by the high-level EA-EOMCCSD(3p—2h) and IP-EOMCCSD(3h-2p) approaches discussed in Chapter 2. More specifically, for states dominated by single-excitations, all variants of CR-EOMCC(2,3) perform equally well, and offer only minor improvements over the already satisfactory EOM- CCSD results. For the more challenging states dominated by double excitations, only 133 the more complete variants C and D are able to successfully describe the excitation energies, providing results of the same quality as those of EA-EOMCCSD(3p—2h) and IP—EOMCCSD(3h—2p). From these results it is clear that the full CR—EOMCC(2,3) scheme, which has costs on the order of the popular ground-state CCSD(T) scheme, is capable of providing high-quality results for both the simple singly excited and more complicated doubly excited low-lying states of open-shell systems, and thus is a promising approach for studying the excitation spectra of radicals. 134 Chapter 4 Coupled-Cluster Calculations for Nuclei In this chapter, the application of various CC approaches, in particular those detailed in Chapters 2 and 3, to the study of nuclear structure is discussed. In Section 4.1, a description of the main elements associated with performing CC calculations for nuclei, particularly those related to how nuclear CC calculations differ from molec- ular CC calculations, are presented. The rest of the chapter presents and discusses the results of some representative nuclear CC calculations performed as part of this research. 4.1 Details of Coupled-Cluster Calculations for Nu- clei Despite the many differences between nuclei and molecules, it turns out that the un- derlying physics describing the motion of the nucleons and electrons, all of which are spin 1 / 2 fermions, Within the nuclear and molecular potentials, respectively, is essen- tially identical. Indeed, the time-independent, many—particle Schrodinger equations 135 for these two types of physical systems only differ in the form of the potential. Since the formulation of various CC methods in quantum chemistry makes virtually no assumptions about the form of the potential (the major exception will be discussed in detail below), they can be applied to calculations of nuclear structure, at least for Hamiltonians with pairwise nucleon-nucleon interactions, without any theoretical reformulation. For example, the explicit equations defining the EDA-EOMCCSDt, IP- EOMCCSDt, and CR-CC(2,3)/CR—EOMCC(2,3) approaches, presented in Sections 2.1.3 and 3.1.3, utilize the integrals 1153 = (pq|v|rs) — (pq|v|sr), but, besides the re- quirement that integrals are real and not complex, make no assumptions regarding the functional form of 12 nor the values of v53. Thus, the CC methods developed in quantum chemistry require no substantial theoretical reformulation in order to be ap- plied to nuclear physics (though some changes in the details of the implementation of the methods may be necessary). The fundamental difference between running nuclear CC calculations and molecular CC calculations is the form of the operator 7), and thus the resulting values of the matrix elements v73, defining the second-quantized form of the Hamiltonian, that enter the equations defining the CC approximation of choice. Having stated all of the above, there are some complications in using the nucleon- nucleon potentials in CC calculations that may need out attention. We discuss them DOW. Since it is the electrostatic force that governs the interaction between electrons, as well as their motion in the field of the nuclei, Coulomb’s law defines the potential in nonrelativistic molecular calculations. The interactions between nucleons, on the other hand, are described primarily by the strong force, with small contribution from the Coulomb force through the mutual repulsion of protons. Unfortunately, there is no Coulomb’s law analog known for the strong force, and as a result generating the Hamiltonian for nuclear calculations is not as simple or straightforward as in quantum chemistry. Because of this, a great deal of research has gone toward the development 136 of realistic nucleon—nucleon interactions, and, as a result, there are a variety of poten- tials to choose from, such as the well-known Argonne V18 [249] and CD-Bonn [250] potentials or the more recent interactions generated within the framework of chiral effective field theory [251, 252], such as Idaho-A or N3LO [253, 254] to name a few. Because of this choice of interactions, the nuclear CC calculations performed as part of this work, and discussed in this dissertation, were based on several different inter- actions, both semi-empirical and ab initio, in order to gauge how the performance of the CC methods varies with regard to the choice of potential. Besides the lack of an exact analytical expression for the nucleon—nucleon interac- tion, there is yet another complication that arises with this potential. Whereas elec- trons are point particles, and so the resulting interaction between them is two-body at most, nucleons have an underlying structure in terms of quarks. As a result, the interactions between nucleons are effective and not fundamental, and as such can be characterized by three—body and higher many-body components (to find an analogy in chemistry, the description of intermolecular forces that originate from the underlying electron-electron, electron-nucleus, and nucleus-nucleus interactions is characterized by a similar situation). This issue is important when attempting to apply the CC methods developed in quantum chemistry to nuclear structure theory because, al- though the quantum chemistry CC formulations make virtually no assumptions on the form of the potential, they do in fact assume that it is at most two-body in na- ture. There is nothing intrinsic in the CC theory that requires this assumption, and indeed a formulation of the basic CCSD approach based on three-body Hamiltonians has recently been derived and implemented [171]. However, the resulting calculations are considerably more expensive than those based on a two—body Hamiltonian. Phr- thermore, if one wants to explore more advanced methods, such as those described in this dissertation that have already been derived and implemented for two-body Hamiltonians, they would have to be rederived and re—implemcnted, with the result- 137 1st MajorShe” — ...................... _131/2 Figure 4.1: Pictorial illustration of the nuclear shell structure. ing equations being considerably more complicated than their two—body Hamiltonian counterparts, before they could be used to study nuclei. Of course, it is possible to avoid these issues by simply restricting the calculations to two—body Hamiltonians only, which allows for a direct application of the quantum chemistry CC formulations to nuclear calculations, which is the route chosen for this research. In fact, two of the goals of this work are to explore the effect of neglecting three-body components of the Hamiltonian as well as the possibility of generating an accurate description of these effects within an effective two-body interaction. The latter makes a lot of sense since, as shown in [171], once the Hamiltonian is written in the normal-ordered form, as is usually done in CC considerations any way, the “true” three-body interactions play a negligible role. The main role of the three-body interactions is to modify the two- body interactions through the effective density-dependent terms that originate from the normal-ordered form of the Hamiltonian in a natural way, providing additional justification for focusing on effective pairwise potentials. 138 A final element of performing nuclear CC calculations that should be discussed before considering some examples is that of the basis set. The structure of the nuclear Hamiltonian written in internal coordinates is, in part, reminiscent of a harmonic os- cillator potential and, as such, single-particle, three-dimensional harmonic oscillator functions are generally used as the basic one-nucleon basis functions. A key param- eter for defining such a basis is the energy gap between adjacent oscillator energy levels, hw, which is generally chosen such that it minimizes the ground-state energy of interest. These functions can be used directly in the CC or other correlated cal- culations, or a mean-field calculation, such as Hartree—Fock, can be performed first to optimize the nuclear orbitals represented as linear combinations of harmonic oscil- lator basis functions. As is the case for atoms and molecules, these nuclear orbitals form a shell structure when arranged according to energy. The first four shells within this structure are illustrated in Figure 4.1. As can be seen, the shell structure is closer to that of atoms than of molecules. However, unlike atoms, the magnitude of spin-orbit splitting is very large, and so even for light nuclei it must be accounted for. Thus the energy level splittings on the right-hand side of Figure 4.1 are the ones that should be used in calculations. Both protons and neutrons each have an identical energy level diagram, and so the number of basis functions associated with a given energy level is 2 * (23' + 1), where j is the total angular momentum for the spin-orbit coupled function. Generally, when building a basis set for use in a CC or other type of correlated calculation, basis functions are added entire shells at a time, and so the number of major shells is used to indicate the size of the basis. Unfortunately, due to the hard core of the nucleon-nucleon potential, the convergence of quantum calcu- lations with respect to the number of major shells is very slow if the bare interaction is used, and so extremely large basis sets are needed to produce accurate results in such calculations. In order to improve the convergence with the size of the basis set, and thus reduce the corresponding computational cost, the Hamiltonian is generally 139 renormalized in order to soften the potential. Several of the calculations performed in this and related work [157—167] made use of the no—core G—matrix procedure described in [255,256], whereas our most recent calculations for 160 [169,170] made use of the renormalized two—body interaction that accounts, to some extent, for three-body in- teractions, called VUCOM [257—260]. The effective semi-empirical Hamiltonians, such as those used in our CC calculations for heavy nuclei [167,168], are generally soft and need no renormalization. Last, but not least, we must remember that unlike atoms or molecules, that consist of electrons moving in a fairly rigid framework of nuclei creating an external potential, nuclei are self-bound systems, which introduces a new layer of complexity absent in electronic structure calculations. The exact nuclear wave function factorizes into the intrinsic part and the center-of-mass part, but this is no longer true for approximate methods, such as those based on CC theory. This issue becomes particularly dramatic for light nuclei where one can easily produce center-of-mass contaminated results, although our recent study shows that the issue does not automatically disappear when heavier nuclei are examined [170]. The issue of center-of-mass contamination is not discussed here and we refer the reader to the literature, including some of our recent work [169,170]. Needless to say, we will mention our ways of addressing the issue, as appropriate, in the next sections, without going into details. It is virtually impossible to describe all of our nuclear CC applications to date [157—171]. Thus, in the following section we focus on a few representative examples only. 140 4.2 Low-lying States of 16O and the Surrounding Valence Systems One of this project’s first applications of CC approaches to nuclear physics was the calculation of the ground and first excited J = 3‘ states of 160 [157,161—164]. Two different Hamiltonians, both derived from chiral effective field theory, were used in these calculations, namely Idaho-A and N3LO [253,254]. The former includes up to chiral-order three diagrams, while the latter includes up to chiral-order four diagrams as well as charge-symmetry and charge-independence breaking terms. Additionally, the proton-proton Coulomb repulsion is included in the N3LO interaction [253,254]. In order to soften the potential, the Idaho-A and N3LO Hamiltonians were renor- malized using the no—core G-matrix approach [255, 256]. Each of the resulting renor- malized effective two-body Hamiltonians was subsequently corrected by adding the fiHCM term to it, where HCM is the center-of-mass Hamiltonian and fl is the La- grange multiplier that are optimized to minimize that center-of-mass contaminations. Finally, once the final form of the renormalized, center-of-mass-corrected Hamiltonian was generated, we determined the relevant one— and two—body matrix elements, f3 and 12%, in the basis set consisting of the standard harmonic oscillator single—particle functions (i.e. no mean-field optimization of orbitals was performed). We refer the reader to [160—164] for the details of the procedure used to define the final form of the Hamiltonian from the Idaho-A and N3LO interactions that was used in the CC calculations 160. The CC methods used in the calculations for 160 reported in [157,161—164] in- cluded the basic CCSD/EOMCCSD approach, for which basis sets with up to 8 major oscillator shells (480 single-particle functions) were used for the ground state and with up to 7 major shells (336 single-particle functions) for the 3“ state, and the completely renormalized CR-CCSD(T)/CR-EOMCCSD(T) scheme [120, 122, 123, 126—128, 131, 141 -100 . CCSD . Cit-CCSD(T) ' o / E(3')-Eu=12.0 MeV -105 2;ng -110 - - % E—ns —- _ m f -120 - = f _ _ E‘s-420.5 MeV -125 - - - . 1 4 m m 1 1304 6 8 . Number of oscillator shells, N Figure 4.2: The CC energies of the ground-state (g.s.) and first-excited 3" state of 160 as functions of the number of major oscillator shells N obtained with the Idaho—A interaction. 132], for which basis sets with up to 7 oscillator shells were used for both states. The latter scheme is essentially an older variant of CR—CC(2,3)/CR-EOMCC(2,3) discussed in Chapter 3 based on the original formulation of the MMCC theory [120—123, 126—129] (as opposed to the biorthogonal formulation CR—CC(2,3)/CR- EOMCC(2,3) is based on). For the calculations with 7 and 8 major oscillator shells, ho.) was set at 11 MeV, to minimize the CCSD energy. It should be noted, however, that the results were virtually independent of the value of flu, when the larger basis sets were employed, which is a typical behavior since the dependence of the results on the basis set parameter hw disappears in the CBS limit. Figure 4.2 shows the CCSD/EOMCCSD and CR—CCSD(T)/CR—EOMCCSD(T) energies as functions of the number of major oscillator shells, N, for both the ground and excited 3‘ states. The symbols correspond to the actual CC energies, while the 142 lines were generated by fitting the energies to the following formula: E(N) = E00 + ae(_bN), (4.1) where the extrapolated infinite basis set (quantum chemists’ CBS) energy, E00, as well as the parameters a and b are determined by the fit. It is clear from Figure 4.2 that this is an excellent fit to the calculated energies, and as such the extrapolated CC energies should be an accurate representation of the infinite basis set results. For the ground-state case, the CCSD scheme generates an extrapolated binding energy of -7.46 MeV / nucleon with the Idaho-A interaction, whereas the extrapolated CR—CCSD(T) binding energy is -7.53 MeV/ nucleon. It is immediately clear that the inclusion of the effects of triply excited clusters has very little effect on the result, changing the binding energy by only 0.9 ‘70. It is also clear from Figure 4.2 that the discrepancy between the two results is consistently small for all basis sets for which both calculations were performed. This small effect due to connected triply excited clusters would suggest that all of the important correlation effects are already included at the CCSD and CCSD(T) levels, and that the inclusion of higher-order clusters would have little impact on the resulting binding energies. This is not too surprising since 160 is a strongly closed-shell nucleus, or “doubly magic” system as it is called in nuclear physics, and so the role of T3 and higher—order clusters cannot be significant. Before comparing the above CCSD and CR—CCSD(T) results to experiment, however, we must remember that, as mentioned above, the Idaho-A interaction used in these calculations does not include the effects of Coulomb repulsion of the protons. The effect of Coulomb repulsion is known to add approximately 0.7 MeV/nucleon to the binding energy, and so adding this value to the converged CR—CCSD(T) binding energy obtained with Idaho—A gives an approximate final CR- CC'SD(T) binding energy of -6.8 MeV/nucleon. As shown in [164], this value is in 143 excellent agreement with the extrapolated N3LO CR—CCSD(T) binding energy of — 7.0 MeV / nucleon (the N3 LO interaction includes the Coulomb repulsion). Comparing these results with the experimental value of -8.0 MeV/ nucleon, we see that the CC calculations underbind 160 by about 1.0 MeV/ nucleon. This result immediately begs the question of where this discrepancy comes from. Indeed, as illustrated by Figure 4.2, the above results appear to be fully converged with the basis set. Furthermore, based on the above discussion, it is safe to conclude that the CR—CCSD(T) method has saturated the particle correlations describing this system. Thus, it would appear, at least at first glance, that our CC calculations for 16O not missing any fundamental physics, and should reproduce the experimental result with much better accuracy. However, we must not forget that the Hamiltonians used in our calculations included only up to two-body interactions, and all three-body and higher interactions and their effect on pairwise interactions were neglected. Thus, we conclude that the three- body interactions in the Hamiltonian neglected in our calculations should provide an additional 1.0 MeV/ nucleon in the binding for the 1GO nucleus. We will return to this interesting aspect of our calculations in a later part of this section. Turning to the 3" state, the basic EOMCCSD approach provides an extrapo- lated excitation energy of 11.3 MeV with Idaho-A, while the extrapolated Idaho- A CR—EOMCCSD(T) excitation energy is ~ 12.0 MeV. N3LO produces essentially equivalent results. This 3“ state is thought to be dominated by single (i.e. 1p—1h) excitations [261] (specifically, a single excitation from the 1p1/2 orbital to the 1d5/2 orbital; see Figure 4.1). As discussed in Chapter 3, it is the experience of quantum chemistry that even the basic EOMCCSD approach is capable of providing an accu- rate description of states with such a structure, and the incorporation of T3 effects has little impact on the overall result if the Hamiltonian contains two-body interactions only (as would be the case in quantum chemistry). It is clear from Figure 4.2 that the inclusion of triply excited clusters through CR—EOMCCSD(T) causes only a slight 144 change in the excitation energy of the 3“ state, in agreement with quantum-chemist intuition. The experience of quantum chemistry is telling us then that it is safe to conclude that CR-EOMCCSD(T) provides an essentially exact solution (from the point-of—view of many-body correlation effects) for the 3' state of 160. The creates a real puzzle, since our converged EOMCCSD and CR—EOMCCSD(T) results for the excitation energy of the 3’ state significantly differ from the experimental excitation energy of 6.12 MeV. Once again, we believe that the discrepancy between CC and experiment is due to missing three-body forces in the Hamiltonian. In order to expand the above study to include open-shell nuclei, a number of CC calculations were performed for the valence systems surrounding 16O, namely 150, 15N, 17O, and 17F [165]. Given that these systems are all one-nucleon away from 160, which is a closed—shell system, they are excellent candidates for EA—type and IP-type EOMCC calculations. Of course, the names ‘electron-attached’ and ‘ionized’ make no sense in the context of nuclear physics, so we will describe the nuclear analogs of the EA- and IP-EOMCC methods as the particle-attached (PA) and particle- removed (PR) EOMCC schemes. Thus, we performed the basic PA-EOMCCSD(2p— 1h) calculations for 17O and 17F and the PR—EOMCCSD(2h-1p) calculations for 15O and 15N. We used the chiral N 3LO interaction from the above 160 study as well the phenomenological Argonne V18 and CD-Bonn potential mentioned in Section 4.1. For 150 and 15N, the ground and first—excited (3/2); states were computed, while for 17O and 17F, the ground, (3/2)'1f and (1/2);L states were calculated. As in the 1GO case, a basis of harmonic oscillator states was utilized. Although the calculations reported in [165] include basis sets with various numbers of oscillator shells, in this discussion we focus on the largest N = 8 basis set only. Furthermore, because there is almost no dependence of the results on the value of fin) when the large N = 8 basis set is employed, we focus on the results with ha) 2 11 MeV for N3LO and CD-Bonn, and fiw = 10 MeV for Argonne V18. The discussion of the results for other flu) values 145 Table 4.1: A comparison of the binding energies per particle for 15O and 15N (the PR—EOMCCSD(2h—1p) values), 160 (the ccsu values), and 17o and 17 F (the PA- EOMCCSD(2p—1h) values), obtained with the N3LO [253, 254], CD-Bonn [250], and V18 [249] potentials, and eight major oscillator shells, with the experimental data taken from [262]. All entries are in MeV. For the CD-Bonn and N3LO interactions, we used hw = 11 MeV. For V18, we used he.) 2 10 MeV. Interaction Nucleus N3LO CD-Bonn V18 Expt 150 6.643 7.584 5.246 7.464 15N 6.824 7.751 5.414 7.699 160 7.406 8.327 5.897 7.976 170 7.150 8.032 5.617 7.751 ”P 6.987 7.879 5.462 7.542 and basis sets with N S 7 can be found in [165]. Table 4.1 gives PA-EOMCCSD(2p—1h) and PR-EOMCCSD(2h-2p) results for the binding energies per particle for 150, 15N, 170 and 17F. Focusing on the results obtained with the N3LO interaction, we see that the PA—EOMCCSD(2p-lh) and PR- EOMCCSD(2h—1p) energies differ from the experimental values by roughly 0.5 to 0.8 MeV/ nucleon. One potential source for this discrepancy could be that the most basic PA—EOMCCSD(2p—1h) and PR-EOMCCSD(2h—1p) schemes do not include all of the relevant correlation effects necessary to describe the ground states of these nuclei. In particular, they lack the 3p—2h/3h-2p components of the particle-attaching and particle-removing operators, which, as discussed in Chapter 2, are in some cases essential for obtaining accurate results for the low-lying states of open-shell systems. However, it is known that the ground states of all four of these valence nuclei are dominated by 11) (170, 17F) or 1h (150, 15N) transitions relative to the ground- state of 160. As illustrated in Section 2.2.3 for molecular applications, the basic PA- EOMCCSD(2p—1h) and PR—EOMCCSD(2h-1p) schemes perform quite well for states dominated by 1p or 1h transitions, and improvements resulting from using the higher- order PA-EOMCCSD(3p—2h) and PR-EOMCCSD(3h-2p) approaches was minimal 146 in those situations. Thus, it is safe to conclude that the higher-order components of the cluster, particle-attaching, and particle-removing operators play a relatively small role in the structure of the ground states of 15O, 15N, 170 and 17F, and so PA-EOMCCSD(2p—1h) and PR-EOMCCSD(2h-1p) accurately describe the relevant many—body correlations. As a result of this analysis, we once again conclude that the discrepancy with experiment is likely due to the inadequate treatment of the nucleon- nucleon interaction resulting from neglecting three-body forces. In fact, the 0.5 to 0.8 MeV / nucleon differences between the PA-EOMCCSD(2p—1h) /PR—EOMCCSD(2h-1p) results and experiment for the valence nuclei are in good agreement with the 0.6 or 1.0 MeV/ nucleon differences between the 8-shell N3 LO CCSD (Table 4.1) or infinite-basis set CR-CCSD(T) binding energies, respectively, and experiment, observed earlier. This relatively consistent discrepancy with experiment across all five nuclei around 1GO indicates that the role of three-body forces is approximately the same in each (3888. Turning to a comparison of the results obtained with different potentials, Table 4.1 shows that the binding energies obtained with the N3LO and CD-Bonn interactions, both of which are based on a nonlocal model for the interaction defined in momentum space, are in reasonable agreement with each other, although the binding is greater with the CD-Bonn potential. On the other hand, the results obtained with the Argonne V18 potential, which is based on a local parameterization in coordinate space, shows a much larger deviation, underbinding the nuclei by approximately 1.4 to 1.5 MeV/ nucleon relative to N3LO. Since the above arguments regarding the convergence of the CC calculations and the potential role three-body interactions apply to a lesser or larger extent, to any interaction, our calculations strongly suggest that the role of three-body forces depends on what interaction model is used, and so every potential needs its own unique three-body interaction. This is an important finding, since it has been a tendency of the nuclear physics community for quite some time to 147 add some form of three-body interaction to some form of two—body interaction in the Hamiltonian without realizing the possible inconsistency in such an ad hoc treatment. Before moving on to an analysis of the CC results for the low-lying excited states of the 15O, 15N, 17O and 17F nuclei, it is useful at this point to return to the discussion of the potential role of three-body interactions in the case of the lowest excited state of 160 of the 3‘ symmetry. As mentioned above, a zero-order description of this state is that of a 1p—1h excitation from the 1p1/2 orbital to the 1d5/2 orbital. The approximate energy associated with such an excitation, relative to the ground state of 160, can easily be estimated using the formulas A67; = 677(1d5/2) — 64(1171/2) = [BE(160) — BE(17F)] + [813(160) — BE(15N)] (4.2) and A61! = 611(1d5/2) ‘ €u(1P1/2) = [BE(16O) — 313070)] + [88(160) — BE(15O)] (4.3) for the proton (77) and neutron (V) excitations, respectively, where BE represents the relevant total binding energy for the labeled nucleus. Using the results presented in Table 4.1, Eqs. (4.2) and (4.3) can be used to calculate both CC and experimental values for A67;- and Ac”, which give a zeroth—order estimate for the excitation energy of the 3‘ state of 160. The resulting zeroth—order CC values are A6“ = 15.846 MeV and A6,, = 15.789 MeV, whereas the corresponding experimentally derived values are A6” = 11.526 MeV and A6,, = 11.521 MeV. Thus, regardless of whether a proton or a neutron is excited from the 1101/2 orbital to the 1d5/2 orbital, there is a discrepancy between the zeroth-order CC and experimental estimates of the 3‘ excitation energy of about 4.3 MeV, which is a large fraction of the missing 6 MeV 148 Table 4.2: A comparison of the energies of the low-lying excited states of 15O, 15N , 17O and 17F, relative to the corresponding ground-state energies (the (1 / 2)1‘ states of 15O and 15N and the (5/2)?‘ states of 17O and 17F) obtained with the PR-EOMCCSD(2h- 1p) (150 and 15N) and PA-EOMCCSD(2p—1h) (170 and 17F) methods, the N3LO [253,254], CD—Bonn [250], and Argonne V18 [249] potentials, and eight major oscillator shells, with the experimental data taken from [263]. All entries are in MeV. For the CD-Bonn and N3LO interactions, we used hw = 11 MeV. For V18, we used ’17..) = 10 MeV. Interaction Excited state N3LO CD-Bonn V18 Expt 150(3/2); 6.264 7.351 4.452 6.176 15N(3/2)‘ 6.318 7.443 4.499 6.323 170(3/2)!’r 5.675 6.406 3.946 5.084 170(1/2)’r -0025 0.311 -0390 0.870 ll 5.891 6.677 4.163 5.000 1 0.428 0.805 0.062 0.495 in the CR—EOMCCSD(T) excitation energy for this state. Based on this analysis, it appears that the discrepancy between the CC and experimental results for the 3‘ state is primarily due to a relatively poor reproduction of the shell structure of 160, in particular, an incorrect energy gap between the 1p and 231d shells (see Figure 4.1), which is known to be affected by three-nucleon forces. This is a strong evidence that it is the lack of three-body forces in the Hamiltonian, and not any intrinsic deficiencies in our CC/EOMCC calculations for 160, that is largely responsible for the ~ 6 MeV discrepancy between the converged EOMCCSD or CR—EOMCCSD(T) and experimental excitation energies for the 3‘ state of 160. Moving on to the excited states of the 150, 15N, 17O and 17F systems, Table 4.2 reveals that the N3LO PA-EOMCCSD(2p—1h.) and PR—EOMCCSD(2h-1p) results for the lowest (3/2)1‘ states of 150 and 15N and the (U2)? state of 17F are in good agreement with the experimental values, differing by less than 0.1 MeV. The description for the remaining states is not as good, but it is still reasonable, with the errors relative to experiment increasing to 0.6 to 0.9 MeV. It is important to note 149 that the (3/2)? states of 17C and 17F are known to be resonances, which cannot be treated properly by the CC codes utilized in this work since they are designed for the bound states only. This fact is likely one of the main contributions to the discrepancy between the CC and experimental results for these particular states. Finally, it is worth noting that once again there are significant differences between the results obtained with different potentials. Indeed, the CD—Bonn interaction produces excitation energies that are notably higher than the experimental values in most cases, while Argonne V18 consistently underestimated the excitation energies in the 150, 15N, 17O and 17F systems. Of the potentials used in our calculation, it appears that N 3L0 provides the best overall results. Based on the above analysis, it seems that the three-body contributions to the nucleon—nucleon interaction may have a significant effect on calculated nuclear prop— erties. However, as already mentioned above, the explicit inclusion of three-body interactions in the nuclear Hamiltonian is not the best way to proceed, as it in- creases the computational costs of the many-body nuclear structure calculations, and requires the rederivation of the equations defining various correlated methodologies that are typically formulated for pairwise potentials. This certainly applies to all quantum-chemistry-inspired CC theories discussed in this thesis. Furthermore, as shown in [171], Where the CCSD calculations for the two- and three-body interac- tions were presented, the main contribution of the three-body force is in the density- dependent zero—, one-, and two-body terms that result from using the normal—ordered form of the Hamiltonian. The “true” three-body terms of the normal-ordered Hamil- tonian can safely be neglected with minimum impact on the accuracy of the CC results. In other words, the most important three-body interactions can be expressed in the form of effective two—body matrix elements. This is an important finding as it points to the possibility of incorporating the effects of three-body forces within an ef- fective two-body Hamiltonian, allowing for a straightforward application of quantum 150 chemistry inspired CC methods such as the ones utilized above. An example of a mod- ern interaction that attempts to effectively account for three-body and higher many- body interactions within a two-body Hamiltonian is the VUCOM potential [257—260]. This is a pure two—body interaction derived from Argonne V18 through a clever uni- tary transformation to account for short-range central and tensor correlations, which is adjusted to reproduce the experimental binding energies of three— and four—particle nuclei. In order to gauge whether VUCOM offers any improvements over the standard two- body interactions considered above when used in CC calculations of nuclei, as well as to further test the performance of the CC theory in studies of nuclear structure, we performed extensive CR—CC(2,3) calculations for the ground state of 160 using the VUCOM potential [169]. In addition, importance-truncated configuration interaction (IT -CI) [264] calculations were performed in order to further examine the strengths and weaknesses of both the CC methodology and the IT-CI formalism through di- rect comparisons [169]. The IT-CI approach is based on the idea of truncating the CI model space in which the Hamiltonian is diagonalized through the use of an im- portance measure for the individual configurations derived from perturbation theory followed by the numerical extrapolation to the limit corresponding to all configura— tions of a given CI scheme. For instance, the IT—CI(4p—4h) method, which is the IT—CI scheme used in our study [169], is an approximation to CISDTQ (CI with sin- gles, doubles, triples, and quadruples), in which configurations with an importance measure less than a given threshold are neglected from the calculation. This reduces the large computational costs of CISDTQ without sacrificing accuracy. We then re— peat the IT—CI(4p—4h.) calculations with increasingly smaller thresholds for selecting configurations and extrapolate the results to the zero—threshold limit (which is equiv- alent to full CISDTQ). For the calculations summarized in this dissertation, all taken from [169], a Hartree-Fock basis was used. This decision is based 011 comparisons 151 of the IT—CI and CC results with both the the harmonic oscillator and Hartree-Fock (HF) bases, which show that, unlike the CC results which are almost insensitive to the choice of the basis set type, the IT-CI results for the harmonic oscillator basis become poor for larger hw values (for the complete details of this analysis see [169]). Basis sets consisting of 5, 6, 7, and 8 major oscillator shells, which can also be identified by the quantum numbers emax = 4, 5, 6, and 7, respectively, were utilized. Figure 4.3 displays the results of the CCSD, CR—CC(2,3), and IT—CI calcula- tions for the binding energy of 160 using VUCOM~ In addition to considering basic IT-CI(4p-4h) results, calculations were also performed in which a multi-reference Davidson correction [265—269] was added to the IT-CI(4p—4h) energies (which will be referred to as IT-CI(4p-4h)+MRD). This correction is meant to estimate the effect of higher—than-4p—4h configurations as well as to approximately restore size extensivity (unlike CC, truncated CI methods, including IT-CI, are not size extensive). As can be seen from panel (a) of Figure 4.3, the effect of the Davidson correction is fairly small regardless of the size of the basis set or the value of the basis set parameter hw, maintaining a value of ~ 1 MeV. This implies that with the HF basis, the contribu— tions of higher-than-4p—4h configurations to the ground-state wave function are small. Panel (b) shows a comparison of the CCSD and CR—CC(2,3) results. It is clear from these results that the effect of connected triply excited clusters is quite significant, particularly for larger values of M. Quantitatively, the shift in energy when moving from CCSD to CR-CC(2,3) can be as large as 6 MeV, indicating that the inclusion of T3 effects is important for obtaining an accurate description of the binding energies of 16O with the VUCOM interaction. Turning to a direct comparison of the CR—CC (2,3) and IT-CI(4p-4h) results, Figure 4.3 (c) reveals that the agreement between the two methodologies is remarkable. Indeed, for all but the largest basis set, the two plots are virtually on top of each other. Though the agreement is slightly worse for the largest 6mm; = 7 basis, it is still very good, with discrepancies between CR—CC(2,3) and IT- 152 (a) IT-Cl(4p4h) . _130 _ IT—CI(4p4h)+MRD . E 8, [MeV] . CCSD . CR-CC(2,3) W 4 4 L - l A l A I ' I ' ' .1. W O I (C) .1. N O I IT-CI(4p4h)+MRD _130 _ CR-CC(2,3) 15.20-25‘30‘35'40‘ h!) [MeV] Figure 4.3: Systematic comparison of IT-CI and CC results for the ground-state en- ergy of 160 using HF-optimized single-particle bases with 6mm: = 4, 5, 6, and 7. (a) Comparison of IT-CI(4p—4h) (open symbols) with IT-CI(4p—4h)+MRD (filled sym- bols). (b) Comparison of CCSD (open symbols) with CR—CC(2,3) (filled symbols). (c) Comparison of IT-CI(4p—4h)+MRD (open symbols) with CR-CC(2,3) (filled sym- bols). 153 CI(4p-4h)+MRD on the order of 1-2 MeV only. This is an important finding, which demonstrates that the relatively inexpensive CC theory with singles, doubles, and noniterative triples, represented here by CR-CC(2,3), is as accurate as the consider- ably more expensive CISDTQ-like IT—CI(4p—4h)+MRD calculations. We do not have to worry about the Davidson extensivity corrections since the CCSD, CR—CC(2,3), and most of the other CC approximations are automatically size extensive and, as mentioned earlier, the CC results are almost insensitive to the choice of the basis set type (harmonic oscillator or HF) due to the presence of the 8T1 component in the CC wave function that makes CC calculations approximately invariant with respect to orbital rotations (Thouless’ theorem). As demonstrated in [169], the IT—CI results for the harmonic oscillator basis are considerably worse than their HF counterparts shown in Figure 4.3, which is another strong argument in favor of using CC methods in nuclear applications (in addition to the computer costs which are lower in the CC case compared to CI aimed at similar accuracies). We refer the reader to [169] for further analysis. As a final element of the 16O/VUQQM study, we extrapolate the CR—CC(2,3) re— sults to the infinite basis set limit in order to compare the resulting binding energies with experiment. Unfortunately, there are a couple of issues that complicate such an extrapolation. First, the VUCOM interaction has a harder core than in an interac- tion transformed by the G-matrix approach used in our earlier work [164], and as a result a basis set including up to 8 shells is still relatively far from convergence. In addition, the HF and, in consequence, correlation energies do not change. uniformly when increasing the size of the basis set through the addition of the next major os- cillator shell. As a consequence of these issues and due to the limited amount of data obtained in our study, we could only conclude that the extrapolated infinite basis set CR—CC(2,3) energies fall within the range of -131 to -141 MeV. Compar- ing this range of values with the experimental binding energy of -127.6 MeV reveals 154 that using the VUCOM interaction in the CC calculations does improve the results over the purely two-body N3LO interaction [164]. Indeed, the error in the binding energy per nucleon for the very roughly extrapolated CR—CC(2,3) results is 0.21 to 0.84 MeV/ nucleon (-0.5 MeV/ nucleon on average), which represents an improvement over the ~ 1.0 MeV/ nucleon error obtained in the N3LO CR—CCSD(T) calculations for the 160 ground state [164] or ~ 2 MeV/ nucleon error obtained with Argonne V18 and CCSD [165] (see Table 4.1). The latter observation is important since VUCOM is based on transforming Argonne V18. Given that the CR-CCSD(T) and CR-CC(2,3) approaches should yield similar accuracies for small or medium size closed-shell sys- tems, we can conclude that the observed improvement is the result of the improved interaction. These results are certainly promising, though further calculation to help improve the extrapolation would be useful. The remaining potential errors in the CR—CC(2,3) VUCOM binding energy is likely the result of contamination from the center-of-mass motion. Indeed, unlike in quantum chemistry, where thanks to the Born-Oppenheimer approximation there is a rigid nuclear framework in which the electrons move, the self-bound nucleus has translational degrees of freedom, and in truncated CC or Cl calculations there may be spurious contributions induced by a coupling between the translational and intrinsic degrees of freedom. Such contami- nation can introduce errors into the CC results for the intrinsic energy of the nuclear state of interest. For a more detailed analysis of the center-of—mass problem in CC calculations of nuclei, see [169,170]. 4.3 Ground and Excited States of 55Ni, 56Ni, and 57Ni In order to analyze the performance of the CC theory in studies of heavy nuclei, we performed CR—CC(2,3)/CR—EOMCC(2,3) and CR-CC(2,4)/CR—EOMCC(2,4) calcu- 155 lations for the ground and low-lying excited states of 56N i [167]. Besides being an example of a heavier nucleus for which all non-CC high—level methodologies, such as the Green’s function Monte Carlo [152] and no-core shell-model [153—156] approaches, are prohibitively expensive, it is also an example of What can be described as a ‘semi- closed shell’ nucleus. Indeed, the lowest-energy configuration for the ground state of 56Ni completely fills all energy levels up to and including the 1f7/2 level (see Fig- ure 4.1). Although there is a notable energy gap between this level and the lowest unoccupied 2193/2 level, the 1f7/2 — 2p3/2 gap in 56Ni is not as large as what would be found in a perfectly closed-shell system and not as small as what would be found in a typical quasidegenerate open-shell system. This means that the degree of non— dynamical correlations in 56Ni is larger than in a typical closed-shell system in spite of the completely filled highest occupied 1f7/2 subshell. As a result, this is a very interesting test case for the CR—CC approaches discussed in this work, which have been shown to perform well for molecular systems with significant nondynamical cor- relations, and smaller HOMO-LUMO gaps. In order to extend these considerations and further explore the performance of the CR—CC(2,3) and CR-CC(2,4) schemes in studies of nuclei characterized by varying degrees of nondynamical character, we performed calculations in which the value of the 1f7/2 — 2193/2 energy gap was varied through the shifting of the energies of the 2123/2, 1f5/2, and 2191/2 levels relative to the 1 f7 /2 level by an amount AG. Both negative values of AG, for which the gap is reduced and thus the degree of nondynamical correlation is increased, and positive values, for which the gap is increased and the nucleus becomes more closed-shell, were considered [167]. AG = 0 is the original gap in a realistic description of 56Ni. The 56Ni calculations in this study utilized the semi-empirical GXPF 1A effec- tive Hamiltonian [270], which is derived from a microscopic calculation based on the renormalized G—Matrix theory with the Bonn-C interaction [255], and then param- eterized to fit experimental data for the low-lying states in nuclei from A = 47 to 156 C Energy (Mev) -5 P - — Full on - - «- CR—CC(2,3) - - - - CISDTQ - [ . -10 l l I l l I l l L l -3-2-1012 -3-2-1012 3 Gap shift (MeV) Gap shift (MeV) Figure 4.4: (a) The full CI, CISDT Q, and CR—CC(2,3) energies of 56Ni as functions of the shell-gap shift AG. (b) Comparison of full CI energies with the trends expected for the lp—lh, 4p—4h, and 8p-8h configurations as functions of AG. A = 66 [270—272] (A is the mass number). In order to accurately gauge the accuracy of the CR—CC(2,3)/CR—EOMCC(2,3) and CR—CC(2,4)/CR—EOMCC(2,4) schemes in describing the correlations within this system, we wanted to perform full CI calcula- tions as well as a sequence of truncated CI calculations for benchmarking purposes. Since 56N1 is too large for no—core full CI calculations, the full CI and all CC and other CI calculations were performed within a valence model space consisting of only the pf shell (the highest shell depicted in Figure 4.1), with the nucleons occupying the lower shells being accounted for in an effective manner through the GXPFIA effective Hamiltonian. 157 Figure 4.4 depicts the results of our CR—CC(2,3)/CR—EOMCC(2,3) calculations and their comparison with the CISDTQ and full CI data. Focusing on the results for the ground 0+ state, panel (a) shows that the CR—CC(2,3) energies agree very well with those of full CI for positive values of AG. This is confirmed quantitatively in Ta— ble 4.3, which shows that for this AG region, the difference between CR-CC(2,3) and full CI ranges from 0.07 to 0.09 MeV. This excellent agreement is not surprising given the predominantly closed-shell nature of the nucleus for AG < 0. Indeed, as seen in Table 4.3, the value of So = |(OI\Ilgml—CI)|, which measures the overlap of the refer- ]‘Ilgml_CI), is over 0.9 the ence determinant |0) with the exact, full CI wave function AG > 0 region. This indicates that the wave function is dominated by the reference configuration, and, as such, is mostly single reference in nature. For the physical value of the gap (i.e., AG = 0), the SO value drops to 0.825, indicating a somewhat stronger multi-reference character, and thus a more open-shell nature. However, even in this case, CR—CC(2,3) performs very well, producing an error relative to full CI of 0.22 MeV which is almost nothing on the scale of binding energies of heavier nuclei. It is when AG becomes negative that CR—CC(2,3) begins to have more trouble, as can be clearly seen from Figure 4.4. Quantitatively, one finds discrepancies between CR- CC(2,3) and full CI of 1.43 and 5.84 MeV for AG = -1 and —2 MeV, respectively. This is certainly a result of the much more strongly open-shell or quasidegenerate nature of the nucleus for this AG region. Indeed, SO = 0.332 and 0.022 for AG = —1 and —2 MeV, respectively. Such small overlaps between the reference determinant and the exact wave function indicate that the ground state is characterized very strong nondynamical correlations, similar to those found in metallic-like systems, that even CR—CC(2,3) has difficulty with. Interestingly, even in the strongly correlated multi-reference AG < 0 region, where CR-CC(2,3) has difficulty reproducing the full CI results, the CR—CC(2,3) energies are in excellent agreement with those of the much more expensive CISDTQ approach. 158 Indeed, the two differ by less than 0.34 MeV for all values of AG. This is a very en- couraging result given that the most expensive computational steps of CR—CC(2,3) scale as 71371.3 in the iterative CCSD part and 733713 in the noniterative triples correc- tion part (see Section 3.1.2), whereas CISDTQ is characterized by iterative steps that scale as 723,713. As in the case of 160, the CR—CC(2,3) approach offers substantial sav- ings in the computer effort compared to CI with minimal loss in accuracy. This would not be surprising if 56Ni was a closed-shell system. What is surprising and certainly very promising here is the fact that the inexpensive CR—CC(2,3) approach accurately describes all correlations among nucleons up to quadruple excitation independent of the 1f7/2 — 2p3/2 gap, i.e., independent of the degree of quasidegenerate character of our 56Ni model system and independent of the strength of the correlations. The fact that the agreement between CR—CC(2,3) and CISDTQ holds in a metallic-like region where the reference determinant |0) contributes only a few percent of the wave function is unheard of, demonstrating the great utility of CR—CC(2,3). It is also interesting to observe that quadruple excitations are largely dominated by dis— connected %T22 contributions. This is verified by the CR-CC(2,4) results which only differ from the CR—CC(2,3) binding energies in Table 4.3 by 0.001-0.102 MeV. Such small changes when moving from CR—CC(2,3) to CR—CC(2,4) indicates that there is very little contribution from the connected quadruply excited clusters T4. Turning to the excited 2+ and 4+ states of our 56Ni model, Figure 4.4 and Table 4.3 reveal that the trends displayed by the results for the ground state apply to the excited states as well, with CR-EOMCC(2,3) mimicking full CI for AG = 0, 1 and 2 MeV, but diverging from the full CI results for AG = -—2 and —1 MeV. Furthermore, as was true for the ground state, the CR-EOMCC(2,3) results are practically identical to those of CISDTQ, and the role of T4 and RIM, as indicated by the CR—EOMCC(2,4) energies, is small regardless of the AG value. One interesting observation, however, is that the REL values for these two states are less than 1.33 independent of AG, 159 WE. A.— Table 4.3: Energies (in MeV) of 56Ni as functions of the shell—gap shift AG (also in MeV), relative to the reference energy (O|H [(D0) = —203.800 MeV. SO is defined as K901953110». AG -2 —1 0 1 2 State so 0.022 0.332 0.825 0.917 0.949 0+ CCSD -3.218 -2.048 -1.509 -1202 -1002 CR—CC(2,3) -4355 —2.437 -1.686 -1.298 -1.060 CR-CC(2,4) -4253 -2415 -1.679 -1295 -1059 CISD -2.148 -1.652 -1327 -1104 -0943 CISDT -2.706 -1.946 -1.488 -1.199 -1004 CISDTQ -4013 -2.548 -1.758 -1334 -1079 Full CI -10.198 -3.868 -1909 -1370 -1.091 2+ CCSD -2440 -0.065 1.595 2.983 4.241 CR—CC(2,3) -2.695 -O.218 1.496 2.915 4.192 CR—CC(2,4) —2.700 -0222 1.493 2.913 4.190 CISD 0.864 2.000 3.093 4.162 5.215 CISDT -1227 0.359 1.771 3.066 4.283 CISDTQ —2.426 -0.335 1.378 2.833 4.137 Full CI -9.728 -3054 0.689 2.594 4.027 REL 1.309 1.178 1.114 1.080 1.060 4* CCSD -1373 0.910 2.551 3.942 5.211 CR—CC(2,3) -1.667 0.720 2.420 3.848 5.141 CR-CC(2,4) -1.626 0.736 2.428 3.852 5.144 CISD 1.554 2.743 3.884 4.994 6.082 CISDT —0.271 1.301 2.713 4.017 5.248 CISDTQ -1.465 0.606 2.308 3.769 5.087 Full CI -8.700 -1974 1.778 3.581 4.999 REL 1.333 1.215 1.152 1.115 1.090 which indicates that regardless of the quasidegenerate nature, the excited states are dominated by single excitations relative to the ground state. As discussed in Section 3.2.3, the CR—EOMCC(2,3) approach describes such states accurately. Thus, it is likely that the excitation process that generates the 2+ and 4+ excited states of 56Ni is described correctly regardless of the choice of AG, and that the discrepancies between CR—EOMCC(2,3) and full CI in the negative AG region are due to the propagation of errors from the ground-state calculations. Given the above analysis, the remaining issue is the physical nature of the source 160 of the discrepancy between CR—CC(2,3)/CR—EOMCC(2,3) and full CI in the negative AG region. Panel (b) of Figure 4.4 gives some insights into this issue. In this figure, the full CI results for the three states of 56Ni, as a function of AG, are plotted and compared to the trends expected for the 1p—1h (single), 4p—4h (quadruple), and 8p—8h (octuple) configurations. By comparing the slopes of the full CI curves against the slopes of the mp-mh lines, it is clear that as AG approaches zero from the positive side, the role of 4p—4h configurations increases, but there is little contribution from higher- than 4p—4h configurations. This explains why CR—CC(2,3)/CR—EOMCC(2,3) and CISDTQ perform so well in this region. They both capture all nucleon correlations up to 4p—4h excitation almost perfectly. However, as the gap is decreased further, it is clear that 8p—8h configurations begin to dominate the wave function. Now since the CR—CC methods used in this study neglect the effect of Tn clusters with n > 4, they are incapable of describing the entire set of 8p-8h excitations. The exponential nature of CC means that some 8p—8h correlations are accounted for via disconnected product terms, such as 21-4T24. However, because of the noniterative nature of the CR- CC schemes used here, some important 8p-8h disconnected product terms involving T3, such as %T§T2 are more or less neglected, and so only product terms involving T1 and T2 are included in CR—CC(2,3). Clearly, given our results, these are not enough to describe the 8p—8h correlations dominating the low-lying states of 56N i when AG is negative and the 1f7/2 — 2193/2 gap becomes small. As a further extension of the 56Ni study, we performed PA—EOMCCSD(3p—2h.) and PR—EOMCCSD(3h-2p) calculations for the ground g— state of 55Ni and the ground g.- and excited g. and %_ states of 57Ni, using 56Ni as the “closed-shell” reference system. As this was a continuation of the above 56Ni study, these calcu- lations were again performed within the pf model space, using the same GXPF 1A effective Hamiltonian as that used in the 56Ni calculations. It should only be noted that the two-body matrix elements of the GXPFIA interaction have a smooth mass 161 Table 4.4: Binding energies (in MeV) of 55N i and 57N i relative to the corresponding reference energies ((()A)( j )IH [(PS4) ( j )), A = 55 and 57, respectively, as functions of the shell gap shift AG (in MeV). 3A)(j) is defined as [($81) (j)|‘115‘“f]l’CI(]))| AG -2 -1 0 1 2 55Nl PR—EOMCC(2h-1p) -3.649 —2.459 -1.884 -1542 -1313 PR-EOMCC(3h-2p) -3.844 —2.567 —1.951 -1.587 —1.344 01(29211) -2505 -2013 -1.672 -1427 -1244 CI(3p—3h) -3295 -2449 -1922 -1.580 -1344 (31(4545) -4457 -2.967 -2150 -1.693 -1.406 CI(6p—6h) -6.397 -3519 -2.262 —1.723 -1417 Full-CI -9.091 -3920 -2279 —1.725 -1417 5355>(g) 0.0362 0.4023 0.8015 0.8919 0.9287 57Ni PA-EOMCC(2p—1h) -3.868 -2.671 -2.080 -1.721 -1.476 PA-EOMCC(3p—2h) -4.295 -2.871 -2.186 -1.783 -l.516 01(29212) ~2.692 -2192 -1.840 -1.584 -l.389 01(39311) -3.622 -2717 -2.146 -1.772 -1513 (31(4545) -4.697 -3217 2370 -1.884 -1.575 CI(6p-6h) -6.534 -3.768 -2493 -1.918 -1.588 Full-CI -9391 -4.151 -2511 -1921 -1.588 5559(3) 0.0335 0.4062 0.7802 0.8774 0.9182 162 dependence that scales as (42 /A)1/ 3, and so we had to scale them to the mass number of the nucleus of interest (i.e. A = 55 or A = 57). As was done in the 56Ni study, the 1f7/2 — 2p3/2 shell gap is varied by an amount AG in order to study the perfor- mance of the PA: and PR—EOMCC methods for differing degrees of nondynamical correlation effects. Table 4.4 gives the results for the binding energies of 55Ni and 57N i. As was the case for the ground state of 56Ni, it is clear that the PA-EOMCCSD(3p—2h) and PR- EOMCCSD(3h-2p) schemes perform very well in the positive AG region, producing discrepancies relative to full CI for both states of about 0.07 MeV for AG = 2 MeV and 0.14 MeV for AG = 1 MeV. This result is not unexpected as the ground states of both 55Ni and 57Ni show a predominantly single-reference nature, with an overlap between the CI reference determinant and the full CI wave function, SSA) (j), of about 0.9 (please note that the Cl calculations treat the nuclei of interest, 55Ni or 57Ni, directly rather than attempting to build them out of 56N i through particle attachment or particle removal as is the case in PA/PR—EOMCC). The more interesting question is how the PA- and PR-EOMCC schemes perform when the multi-reference character of a nucleus increases. For the physical energy gap corresponding to AG = 0 MeV, for which 935%) z 0.80 and 8557)(g) :9 0.78, the PA-EOMCCSD(3p-2h) and PR- EOMCCSD(3h-2p) approaches still produce accurate results for the ground states of 55Ni and 57'Ni, differing from full CI by only 0.33 MeV in both cases. However, similar to what was observed for 56Ni, the agreement begins to break down for negative AG values. Indeed, for AG 2 —1 MeV, for which the CI reference determinant makes up less than half of the full CI wave function, the CC and full CI energies differ by 1.35 MeV and 1.28 MeV for 55Ni and 57Ni, respectively. For AG = -—2 MeV, where the CI reference [$84) (j )) is virtually orthogonal to the corresponding full CI wave function, there is a breakdown in accuracy, with the PA-EOMCCSD(3p-2h) and PR—EOMCCSD(3h-2p) errors of 5.10 and 5.24 MeV respectively. 163 Despite the difficulties the 57Ni and 55Ni model systems pose to the PA- and PR—EOMCC methods when AG < 0 MeV, we once again see that there is a nice correlation between the CC and CISDTQ (labelled by the equivalent CI(4p—4h) des- ignation in Table 4.4) results regardless of the value of AG. Indeed, although CISDTQ is slightly more accurate than PA-EOMCCSD(3p—2h) and PR—EOMCCSD(3h-2p) in describing the ground states of 57Ni and 55Ni, there is a reasonable agreement be- tween these CI and CC results, with discrepancies that are only about 0.06 MeV for the largest AG = 2 MeV gap, and ~ 0.4 — 0.6 MeV for the smallest AG = —2 MeV energy gap. Furthermore, the PA—EOMCCSD(3p—2h) and PR-EOMCCSD(3h-2p) re- sults are consistently more accurate than those of CISDT. This result is encouraging because these PA— and PR—EOMCC schemes with up to 3p-2h and 3h-2p excita- tions are characterized by iterative steps that scale as 71.3713 and 71371;], respectively, which are both considerably less expensive than the the 77.3713 and 71372.3 steps that characterize CISDT and CISDTQ. Furthermore, if one were to consider the active- space variants of the PA- and PR-EOMCC schemes discussed above, the associated computer costs would be even lower (see Section 2.1.2). Looking at the excited (5/2)’ and (1/2)’ states of 57Ni, Table 4.5 reveals that the observed accuracy patterns are similar to those seen for the ground-state calcu- lations, with the accuracy of the PA-EOMCCSD(3p—2h) approach declining as the energy gap is decreased. Furthermore, there is again a good agreement between the PA-EOMCCSD(3p—2h) and CISDTQ results. In fact, the agreement observed for the excitation energies of the (5/2)‘ and (1 / 2)‘ states of 57Ni is superior to what was observed for the binding energies, with the PA-EOMCCSD(3p—2h) and CISDTQ ex- citation energies differing by less than 0.1 MeV for all values of AG. Again, this is an excellent result given the large savings in computational effort provided by the PA-EOMCCSD(3p—2h) approach when compared to CISDTQ. 164 Table 4.5: Excitation energies (in MeV) of the low-lying states of 57Ni as functions of the shell gap shift AG (in MeV). 59%;“) is defined as [($84) (j)|\115]J/]1‘CI(]))| AG -2 -1 O 1 2 (5/2)‘ PA-EOMCC(2p-1h) 0.658 0.819 0.895 0.937 0.961 PA-EOMCC(3p—2h) 0.625 0.771 0.856 0.908 0.939 CI(2p—2h) 0.812 0.856 0.897 0.927 0.948 01(393/2) 0.781 0.827 0.878 0.917 0.944 01(4545) 0.692 0.776 0.852 0.904 0.937 CI(6p—6h) 0.360 0.658 0.832 0.900 0.936 Full CI ~0.118 0.402 0.825 0.900 0.936 33574;) 0.0193 0.2640 0.7443 0.8596 0.9077 (U2)— PA-EOMCC(2p—1h) 1.259 1.494 1.639 1.739 1.813 PA-EOMCC(3p—2h) 0.669 1.071 1.366 1.562 1.694 01(2525) 1.279 1.451 1.592 1.699 1.781 01(39312) 1.009 1.218 1.426 1.588 1.706 CI(4p-4h) 0.763 1.021 1.312 1.530 1.676 CI(6p—6h) 0.395 0.739 1.211 1.499 1.665 Full CI 0.050 0.434 1.184 1.496 1.665 3359(1) 0.0293 0.2561 0.6577 0.8049 0.8701 165 Chapter 5 Summary and Eiture Perspectives In this dissertation, two new ab initio methodologies designed specifically to address the challenges posed by open—shell many-fermion systems within a single-reference formalism have been developed and tested. The active-space EA- and IP-EOMCC theories (and their PA- and PR—EOMCC extensions to non-electronic systems) are based on the idea of building an open-shell system, such as a radical, through the direct addition or removal of a particle to or from the related closed-shell species. Additionally, these new schemes are characterized by the use of active orbitals to a priori select the dominant higher-than 2p-1h and higher-than 2h-1p components of the electron-attaching and ionizing operators, respectively, which greatly reduces the computational cost of higher-level EA- and IP—EOMCC approximations without sacrificing any of the associated accuracy. The performance of these schemes was il- lustrated through benchmark calculations for adiabatic or vertical excitation energies of the low-lying states of CH, SH, C2N, CN C, N3, and NCO. All of these molecular examples revealed that the most basic active-space EA- and IP-EOMCC approaches, the EA-EOMCCSDt and IP-EOMCCSDt methods which include up to 3p-2h and 3h—2p components in the electron-attaching and ionizing operators, respectively, were able to provide a highly accurate description of the electronic excitation spectra while 166 requiring computational costs that are only a small prefactor times those characteriz- ing the basic and widely applicable CCSD method. Additionally, calculations for the potential energy curves of the low-lying states of the OH radical revealed that while the IP-EOMCCSDt approach performs well in the spectroscopic Franck-Condon re- gion, the inclusion of the higher-order 4h-3p effects is needed in order to accurately break chemical bonds in excited states. It was then demonstrated that the active- space SAC-CI(4h-3p) scheme, which includes the dominant 3h-2p and 4h-3p effects selected via a small subset of active orbitals and which is essentially equivalent to the active-space IP-EOMCC method with up to 4h-3p “excitations” produces essentially identical accuracies as its substantially more expensive parent approach while requir- ing a significantly less computational effort, while enabling us to obtain a perfect description of the entire ground- and excited-state potential energy surfaces of OH, including the Franck-Condon and asymptotic regions. Similar statements apply to the active-space EA-EOMCC and EA SAC-CI schemes. The second methodology discussed in this dissertation was the CR—CC/CR—EOMCC formalism based on the biorthogonal MMCC theory. In particular, the CR—CC(2,3) /CR- EOMCC(2,3) and CR—CC(2,4)/CR—EOMCC(2,4) approaches, in which noniterative corrections due to triply or triply and quadruply excited clusters are added to the en- ergies obtained with the basic CCSD/EOMCCSD scheme, were extended to general open-shell references of the high-spin ROHF type, implemented, and tested. In order to gauge the performance of CR—CC (2,3) in a variety of scenarios involving open-shell molecular systems, calculations for single-bond breaking reactions in radicals and for singlet-triplet gaps in biradical systems were performed. These calculations revealed that CR-CC(2,3) is able to provide highly accurate results for these types of prob- lems, producing values that are as good as or, particularly for states or geometries characterized by a large degree of nondynamical correlation, superior to those of both the restricted and unrestricted CCSD(T) methods that are often referred to as the 167 ‘gold standard’ of quantum chemistry (a somewhat problematic classification in view of our findings). Furthermore, the CR—CC(2,3) approach produces these excellent results while maintaining the relatively low computational costs and the ease-of—use that have made CCSD(T) so popular. Through calculations for the unusually small singlet-triplet gap in the BN molecule, which is an extreme case characterized by strong T4 effects, it was shown that the CR-CC(2,4) approach is capable of accu- rately describing the effects of quadruply excited clusters without requiring the high computational costs characterizing the full CCSDT Q method. Finally, calculations for the low-lying excited states of radicals revealed that the extension of CR—CC(2,3) to excited states, CPI-EOMCC(2,3), offers same balance of high accuracy and low computational cost demonstrated for the ground-state CR—CC(2,3) scheme when the excitation spectra of radicals and other open-shell species are examined. Although a great deal of progress was made in regards to the development and implementation of the above quantum chemistry methodologies, there remains a large amount of future development work to be done. Indeed, in the case of the active-space EA- and IP-EOMCC schemes, higher-order approximations, such as the EA—EOMCCSth and IP-EOMCCSth schemes, which include up to triples in the cluster operator defining the closed-shell reference system, and up to 4p-3h or 4h—3p components in the electron-attaching and ionizing operators, respectively, should be implemented. This is especially important given the conclusion that 4p-3h and 4h-3p effects are important in accurately describing bond—breaking processes, particularly in the excited states of radicals. Furthermore, the development and computer im- plementation of the active-space DEA- and DIP-EOMCC schemes, which build the wave function for an open-shell system by adding two particles to, or removing two particles from the related closed shell, would be a valuable development as it would make it possible to study biradicals within this formalism. In terms of the CR—CC schemes, further development of the CR—CC(2,4) approach with regards to the cou- 168 pling between triples and quadruples, which was neglected in the CR-CC (2,4) scheme described in this dissertation but may be important for obtaining accurate results for some systems, is needed. Additionally the development of higher-order CR—CC ap- proximations, such as the CR-CC(3,4) method, which corrects the CCSDT energy for the effect of connected quadruply excited clusters, would be useful. Clearly it would be great to develop the analogous CR-EOMCC(3,4) scheme that corrects the EOMCCSDT results for excited states for the quadruple excitations. In addition to the above work, an important development that would be use- ful with respect to the active-space CC/EOMCC and CR-CC/CR—EOMCC method- ologies discussed in this thesis is the extension of these approaches to very large systems through an appropriate local correlation approximation. Indeed, there has already been significant progress in this area as local variants of the CR—CC(2,3) scheme and its CCSD and CCSD(T) counterparts, based on the so—called “cluster- in-molecule” [38, 39] framework, have already been developed [40—44]. This is an important development as it extends the applicability of these types of CC methods to much larger systems that generally are the regime of inexpensive, low-order ap— proximations or semi-empirical schemes by replacing the W6 — W7 scalings of the CPU time with the system size JV by a linear scaling. As a final component of this dissertation, the CC methods developed in this work, as well as other quantum chemistry inspired CC approaches, have been used in cal- culations of the structure of nuclei. The basis of this work is that the fundamental many-body physics underlying the structures of both molecular and nuclear systems is the same, with essentially only the form of the interaction changing, and thus the CC approaches developed in quantum chemistry can also work within the context of nuclear physics. The methodologies developed in this dissertation are of partic- ular value in nuclear structure theory as, thanks to the atom—like shell structure of nuclei characterized by a large amount of degeneracy, the majority of nuclei are in 169 fact open-shell systems. Indeed, in a study of 55Ni, 56Ni, and 57'Ni, the size of the energr gap between the highest occupied and lowest unoccupied levels was varied in order to test the performance of the CC methods as a function of the degree of the open-shell character of the nucleus of interest. This study revealed that re- gardless of the degree of open-shell character and strength of the correlation effects, the CR-CC(2,3)/CR—EOMCC(2,3) (56Ni), PA-EOMCCSD(3p—2h) (57Ni), and PR- EOMCCSD(3h-2p) (55Ni) methods were capable of producing results of essentially the same quality as those of the much more expensive CISDTQ scheme. In a sep- arate set of calculations involving 160 and the surrounding valence systems, it was found that the three-body components of the nucleon-nucleon interaction, which are not present in the Coulomb interaction between electrons, plays a significant role, and cannot be ignored if one wants to obtain results in agreement with experiment. However, it was also demonstrated that it is possible to reasonably accurately include the effects of three-body forces through an effective two-body interaction. Finally, as was the case for 56Ni, the CR-CC(2,3) results for 16O are in excellent agreement with those of CI schemes including up to quadruple excitations without the need to worry about orbital optimization or size extensivity. Some potential further developments that can be pursued as part of this work include the further refinement and testing of two—body Hamiltonians that effectively account for three-body forces and the exten- sion of genuine MRCC methods to nuclear calculations as midshell nuclei and some excited states in closed- and open-shell nuclei may require such a treatment in order to be properly described. Finally, there is still an open issue of how to deal with contaminations from the translational degrees of freedom in the CC calculations for nuclei. Indeed such a problem does not exist in quantum chemistry as, in the Born- Oppenheimer approximation, the nuclei create a fixed framework within which the electrons move. Although some investigation of the degree to which such contamina- tion is an issue in CC and CI calculations has already been performed [170], further 170 ‘r’ study as well as the development of ways to overcome this issue in CC calculations is needed. 171 Appendix A Factorized Form'of the EA-EOMCCSD(3p—2h) and IP-EOMCCSD(3h—2p) Equations In this appendix, we present the fully factorized form of the equations defining the EA-EOMCCSD(3p—2h) and IP-EOMCCSD(3h—2p) eigenvalue problems, exploited in this study, in terms of the one- and two-electron molecular integrals, f3 = (p] f lq) (f is the Fock operator) and 0,23 = (pqlvlrs) — (pqlvlsr), respectively, defining the Hamiltonian, T1 and T2 cluster amplitudes defining the underlying N -electron ground- state CCSD problem, and the Rim!» Ru,2p-1h, and Rm3p-2h and Ru.1h’ Rfl’2h_1p, and Rfl,3h_2p amplitudes defining the electron attaching and electron removing operators, RLNH) and RSV—1), respectively. The EA-EOMCCSD(3p—2h) equations can be given the following form: (CCSD N+1 — ‘ ' ((DGKHN ,open)R/(1 ))Clq)) = ’237‘8 + hfnralg + 2h“{”r€f N 1 +Yl'v7e7171Targfl— — w], + )Ta’ (A'l) 172 CCSD N+1) -' 7 ' <9“,’-’I k k k k = 7546” 7%] [— th cerae‘l' h aeTbe— ih‘znarbzl_ %hflnlcrab e 3’6 1‘4? l'Tef 3k fiha +§hc Tabe+2hmrabcm +8hbcr 06f Grebe k k _4h3£ ()3 f + 3h’77nn1‘ Margie” 2hmarbcek— —hmcralifak ‘2Iamtm mlbmtgék ZIbftjf + 219669] _ w(N+1) — wl‘ Tmmv (A.3) where, in addition to one— and two-body matrix elements of the similarity-transformed . . (CCSD) Hamlltonian H N 0p en been defined in Table 2.1 we define the following intermediates: _ 1 6 n [m — 27177777764, 1am: Bfnea're + ghafnrej f + hmnrae ‘1‘ 207757437}, [bfz hb£Te_ .flbchgjfiq’rrbel— ernfylrbzlcn. The IP-EOMCCSD(3h-2p) equations used in this work are: (CCSD N— -' -, ' ($11“ HNopen)R(1 ))Clq)> : "h‘inrm'i'hfnrzgl 76 77777 1 6 777777 (N_ 1) 7' —1h77mr +21v771777 cf :00}; 7" 173 of the CCSD approach, 77% and 77%, respectively, which have (CCSD N— _ . . ((1)9.17?](HN,open)R( 1))C](I)> = fl2J[_ 1h:nbm — (13:77]? +2h57‘i6 '1' Zh777nT17gn— h]ne(,7' 7773' + 11675:]; 1‘63 7jm+ 1 cf 7'j777 +2h7nr be 4727977777??? '1‘ 47'th cf ] (JV—1) 7.77 C(CSD) N— <9 .,-7| Recall that TM) and BLA) are the cluster and linear excitation operators that define the wave function in the truncated CC/EOMCC method A. We replace the exact bra state (\I/M in Eq. (B.1) by the ansatz given by Eq. (3.8) and use the fact that TM) and RLA) commute. We obtain, -(A) (A) E77: {6'3”}; R" [(1)], (13.2) 61.77.191.019) 175 where P104) is the similarity-transformed Hamiltonian of CC method A defined by Eq. (2.10). By imposing the normalization condition given in Eq. (3.13), the denominator in Eq. (B.2) goes to one, leaving the following expression for the full CI energy of state 77: E], = (91.76,, Emmy) |<1>). (13.3) At this point, we insert the resolution of the identity in the N -electron Hilbert space, where and P + Q”) + 62(3) = 1, (8.4) P = I‘I’X‘PL (B-5) mA 0. a. a a. ”—1 i} < ° ' ° < in a1<---<‘1’7,....-.,"|» (B7) n=mA+1 71<---<7'n al<"') + (917.27,)A)QH= Hf.A >19. W9) (39) we can simplify Eq. (8.8) to E,, = ELA)(|,Z[A)R],A)|) + (enzfimowmmlafimle). (13.10) Substituting the normalization condition given by Eq. (3.13) and the explicit form of Q(R) given by Eq. (B.7) into Eq. (B.10) yields A) 77 — A E,,=E,(,A +2 2 (|6.Z( 79311”? ><¢311...7?:‘|H(A)Rlz )|<1>). 77:777 +1 , . A zl<"'<2n a1<--- 777A. Furthermore, comparison of Eq. (B.11) with Eq. (3.6) reveals the pres— ence of the generalized moments of the CC/EOMCC equations mflgf’fianmA) = (4)311” :1an (A)R(A)|) By taking advantage of these observations, along with the fact that the moments are zero for 77 > N17,Aa Eq. (B.11) can be rewritten as N MA .77 2 6,94 l— _E,, — E],A l: E: 2 7:1,; 371977,,1,"fm‘",,,(m,,) (B12) 77=777A+1 , < <, 21 ... Zn (1.1 < - ~ - < (7,, which completes the derivation. 177 A REFERENCES [1] F. Coester, Nucl. Phys. 7, 421 (1958). [2] F. Coester and H. Kiimmel, Nucl. Phys. 17, 477 (1960). [3] J. CiZek, J. Chem. Phys. 45, 4256 (1966). [4] J. C1’Zek, Adv. Chem. Phys. 14, 35 (1969). [5] J. C12ek and J. Paldus, Int. J. Quantum Chem. 5, 359 (1971). [6] K. Emrich, Nucl. Phys. A 351, 379 (1981). [7] J. Geertsen, Mi. Rittby, and R. J. Bartlett, Chem. Phys. Lett. 164, 57 (1989). [8] D. C. Comeau and R. J. Bartlett, Chem. Phys. Lett. 207, 414 (1993). [9] J. F. Stanton and R. J. Bartlett, J. Chem. Phys. 98, 7029 (1993). [10] P. Piecuch and R. J. Bartlett, Adv. Quantum Chem. 34, 295 (1999). [11] H. Nakatsuji and K. Hirao, Chem. Phys. Lett. 47, 569 (1977). [12] H. Nakatsuji and K. Hirao, J. Chem. Phys. 68, 4279 (1978). [13] H. Nakatsuji and K. Hirao, J. Chem. Phys. 68, 4279 (1978). [14] H. Nakatsuji, In Computational Chemistry: Reviews of Current Trends, edited by J. Leszczyriski, vol. 2, pages 62—124 (World Scientific, Singapore, 1997), and references therein. [15] H. Nakatsuji, Bull. Chem. Soc. Jpn. 78, 1705 (2005), and references therein. [16] H. Monkhorst, Int. J. Quantum Chem. Symp. 11, 421 (1977). [17] E. Dalgaard and H. Monkhorst, Phys. Rev. A 28, 1217 (1983). [18] M. Takahashi and J: Paldus, J. Chem. Phys. 85, 1486 (1986). 178 [19] H. Koch and P. Jergensen, J. Chem. Phys. 93, 3333 (1990). [20] H. Koch, H. J. A. Jensen, P. Jorgensen, and T. Helgaker, J. Chem. Phys. 93, 3345 (1990). [21] G. D. Purvis, III and R. J. Bartlett, J. Chem. Phys. 76, 1910 (1982). [22] J. M. Cullen and M. C. Zerner, J. Chem. Phys. 77, 4088 (1982). [23] G. E. Scuseria, A. C. Scheiner, T. J. Lee, J. E. Rice, and H. F. Schaefer, 111, J. Chem. Phys. 86, 2881 (1987). [24] P. Piecuch and J. Paldus, Int. J. Quantum Chem. 36, 429 (1989). [25] K. Raghavachari, G. W. Trucks, J. A. Pople, and M. Head-Gordon, Chem. Phys. Lett. 102, 479 (1989). '- [26] J. Noga and R. J. Bartlett, J. Chem. Phys. 86, 7041 (1987), 89, 3401 (1988) l [Erratum]. [27] G. E. Scuseria and H. F. Schaefer, III, Chem. Phys. Lett. 152, 382 (1988). [28] K. Kowalski and P. Piecuch, J. Chem. Phys. 115, 643 (2001). [29] K. Kowalski and P. Piecuch, Chem. Phys. Lett. 347, 237 (2001). [30] S. A. Kucharski, M. Wloch, M. Musial, and R. J. Bartlett, J. Chem. Phys. 115, 8263 (2001). [31] S. Hirata, J. Chem. Phys. 121, 51 (2004). [32] C. Hampel and H.-J. Werner, J. Chem. Phys. 104, 6286 (1996). [33] M. Schiitz and H.-J. Werner, J. Chem. Phys. 114, 661 (2001). [34] M. Schiitz, J. Chem. Phys. 113, 9986 (2000). [35] M. Schiitz and H.—J. Werner, Chem. Phys. Lett. 318, 370 (2000). [36] M. Schiitz, J. Chem. Phys. 116, 8772 (2002). [37] M. Schiitz, Phys. Chem. Chem. Phys. 4, 3941 (2002). [38] S. Li, J. Ma, and Y. Jian, J. Comput. Chem. 23, 237 (2002). [39] S. Li, J. Shen, W. Li, and Y. Jiang, J. Chem. Phys. 125, 074109 (2006). [40] W. Li, P. Piecuch, and J. R. Gour, In Theory and Applications of Computational Chemistry - 2008, edited by D.-Q. Wei and X.-J. Wang, vol. 1102 of AIP Conference Proceedings, pages 68—113 (American Institute of Physics, Melville, NY, 2009). 179 [41] W. Li, P. Piecuch, and J. R. Gour, In Advances in the Theory of Atomic and Molecular Systems: Chemistry, edited by P. Piecuch, J. Maruani, G. Delgado- Barrio, and S. Wilson, vol. 19 of Progress in Theoretical Chemistry and Physics, pages 131—195 (Springer, Dordrecht, 2009). [42] W. Li, J. R. Gour, P. Piecuch, and S. Li, J. Chem. Phys. 131, 114109 (2009). [43] W. Li and P. Piecuch, J. Chem. Phys. In press; Articles ASAP; Publication Date (Web): April 7, 2010; DOI:.1021/jp100782u. [44] W. Li and P. Piecuch, J. Phys. Chem. A Submitted. [45] I. Lindgren and D. Mukherjee, Phys. Rep. 151, 93 (1987). [46] D. Mukherjee and S. Pal, Adv. Quantum Chem. 20, 291 (1989). [47] B. Jeziorski and H. J. Monkhorst, Phys. Rev. A 24, 1668 (1995). [48] K. Kowalski and P. Piecuch, Phys. Rev. A 61, 052506 (2000). [49] J. Paldus, P. Piecuch, L. Pylypow, and B. Jeziorski, Phys. Rev. A 47, 2738 (1993) [50] K. Kowalski and P. Piecuch, Int. J. Quantum Chem. 80, 757 (2000). [51] P. Piecuch, R. Tobola, and J. Paldus, Chem. Phys. Lett. 210, 243 (1993). [52] P. Piecuch and J. Paldus, Phys. Rev. A 49, 3479 (1994). [53] K. Jankowski, J. Paldus, I. Grabowski, and K. Kowalski, J. Chem. Phys. 97, 7600 (1992). [54] K. Jankowski, J. Paldus, I. Grabowski, and K. Kowalski, J. Chem. Phys. 101, 3085 (1994). [55] P. Piecuch and K. Kowalski, Int. J. Mol. Sci. 3, 676 (2002). [56] K. R. Shamsundar and S. Pal, J. Chem. Phys. 114, 1981 (2001), 115, 1979 (2001). [57] P. Piecuch and J. I. Landman, Parallel Comp. 26, 913 (2000). [58] K. Kowalski and P. Piecuch, Chem. Phys. Lett. 334, 89 (2001). [59] K. Kowalski and P. Piecuch, J. Molec. Struct.: Theochem 547, 191 (2001). [60] K. Kowalski and P. Piecuch, Mol. Phys. 102, 2425 (2004). [61] X. Li and J. Paldus, J. Chem. Phys. 119, 5320 (2003). [62] X. Li and J. Paldus, J. Chem. Phys. 119, 5334 (2003). 180 [63] X. Li and J. Paldus, J. Chem. Phys. 110, 5346 (2003). [64] X. Li and J. Paldus, J. Chem. Phys. 120, 5890 (2004). [65] M. Musial and R. J. Bartlett, J. Chem. Phys. 121, 1670 (2004). [66] M. Musial, L. Meissner, S. A. Kucharski, and R. J. Bartlett, J. Chem. Phys. 122, 224110 (2005). [67] M. Hanrath, J. Chem. Phys. 123, 084102 (2005). [68] M. Nooijen and R. J. Bartlett, J. Chem. Phys. 102, 3629 (1995). [69] M. Nooijen and R. J. Bartlett, J. Chem. Phys. 102, 6735 (1995). [70] M. Musial and R. J. Bartlett, J. Chem. Phys. 119, 1901 (2003). [71] R. J. Bartlett and J. F. Stanton, In Reviews in Computational Chemistry, edited by K. B. Lipkowitz and D. B. Boyd, vol. 5, pages 65—169 (VCH Publishers, New York, 1994). [72] M. Nooijen and J. G. Snijders, Int. J. Quantum Chem. Symp. 26, 55 (1992). [73] M. Nooijen and J. G. Snijders, Int. J. Quantum Chem. 48, 15 (1993). [74] J. F. Stanton and J. Gauss, J. Chem. Phys. 101, 8938 (1994). [75] M. Musial, S. A. Kucharski, and R. J. Bartlett, J. Chem. Phys. 118, 1128 (2003) [76] M. Musial and R. J. Bartlett, Chem. Phys. Lett. 384, 210 (2004). [77] Y. J. Bomble, J. C. Saeh, J. F. Stanton, P. G. Szalay, and M. Kallay, J. Chem. Phys. 122, 154107 (2005). [78] H. Nakatsuji and K. Hirao, Int. J. Quantum Chem. 20, 1301 (1981). [79] H. Nakatsuji, K. Ohta, and K. Hirao, J. Chem. Phys. 75, 2952 (1981). [80] H. Nakatsuji, K. Ohta, and T. Yonezawa, J. Phys. Chem. 87, 3068 (1983). [81] H. Nakatsuji, Chem. Phys. Lett. 177, 331 (1991). [82] H. Nakatsuji and M. Ehara, J. Chem. Phys. 98, 7179 (1993). [83] H. Nakatsuji, M. Ehara, and T. Momose, J. Chem. Phys. 100, 5821 (1994). [84] M. Ishida, K. Toyota, M. Ehara, H. Nakatsuji, and M. J. Frisch, J. Chem. Phys. 120, 2593 (2004). [85] J. R. Gour, P. Piecuch, and M. VVloch, J. Chem. Phys. 123, 134113 (2005). 181 [86] J. R. Gour, P. Piecuch, and M. Wloch, Int. J. Quantum Chem. 106, 2854 (2006). [87] J. R. Gour and P. Piecuch, J. Chem. Phys. 125, 234107 (2006). [88] Y. Ohtsuka, P. Piecuch, J. R. Gour, M. Ehara, and H. Nakatsuji, J. Chem. Phys. 126, 164111 (2007). [89] M. Ehara, J. R. Gour, and P. Piecuch, Mol. Phys. 107, 871 (2009). [90] M. Kamiya and S. Hirata, J. Chem. Phys. 125, 074111 (2006). . [91] M. Nooijen and R. J. Bartlett, J. Chem. Phys. 106, 6441 (1997). [92] M. Wladyslawski and M. Nooijen, In Low-Lying Potential Energy Surfaces, edited by M. R. Hoffmann and K. G. Dyall, vol. 828 of ACS Symposium Series, pages 65—92 (American Chemical Society, Washington, DC, 2002). [93] M. Nooijen, Int. J. Mol. Sci. 3, 656 (2002). [94] N. Oliphant and L. Adamowicz, J. Chem. Phys. 94, 1229 (1991). [95] N. Oliphant and L. Adamowicz, J. Chem. Phys. 96, 3739 (1992). [96] N. Oliphant and L. Adamowicz, Int. Rev. Phys. Chem. 12, 339 (1993). [97] P. Piecuch, N. Oliphant, and L. Adamowicz, J. Chem. Phys. 99, 1875 (1993). [98] P. Piecuch and L. Adamowicz, J. Chem. Phys. 100, 5792 (1994). [99] P. Piecuch and L. Adamowicz, Chem. Phys. Lett. 221, 121 (1993). [100] P. Piecuch and L. Adamowicz, J. Chem. Phys. 102, 898 (1995). [101] K. B. Ghose, P. Piecuch, and L. Adamowicz, J. Chem. Phys. 103, 9331 (1995). [102] K. B. Ghose and L. Adamowicz, J. Chem. Phys. 103, 9324 (1995). [103] V. Alexandrov, P. Piecuch, and L. Adamowicz, J. Chem. Phys. 102, 3301 (1995) [104] K. B. Ghose, P. Piecuch, S. Pal, and L. Adamowicz, J. Chem. Phys. 104, 6582 (1996). [105] L. Adamowicz, P. Piecuch, and K. B. Ghose, Mol. Phys. 94, 225 (1998). [106] P. Piecuch, S. A. Kucharski, and R. J. Bartlett, J. Chem. Phys. 110, 6103 (1999). [107] P. Piecuch, S. A. Kucharski, and V. Spirko, J. Chem. Phys. 111, 6679 (1999). [108] K. Kowalski and P. Piecuch, Chem. Phys. Lett. 344, 165 (2001). 182 [109] K. Kowalski and P. Piecuch, J. Chem. Phys. 113, 8490 (2000). [110] K. Kowalski, S. Hirata, M. Wloch, P. Piecuch, and T. L. Windus, J. Chem. Phys. 123, 074319 (2005). [111] P. Piecuch, S. Hirata, K. Kowalski, P.-D. Fan, and T. L. Windus, Int. J. Quan- tum Chem. 106, 79 (2006). [112] M. Kallay and J. Gauss, J. Chem. Phys. 121, 9257 (2004). [113] L. Adamowicz, J.-P. Malrieu, and V. V. Ivanov, J. Chem. Phys. 112, 10075 (2000). [114] V. V. Ivanov and L. Adamowicz, J. Chem. Phys. 112, 9258 (2000). [115] D. I. Lyakh, V. V. Ivanov, and L. Adamowciz, J. Chem. Phys. 122, 024108 (2005). [116] J. Olsen, J. Chem. Phys. 113, 7140 (2000). [117] J. W. Krogh and J. Olsen, Chem. Phys. Lett. 344, 578 (2001). [118] M. Kallay, P. G. Szalay, and P. G. Surjan, J. Chem. Phys. 117, 980 (2002). [119] A. K6hn and J. Olsen, J. Chem. Phys. 125, 174110 (2006). [120] P. Piecuch, K. Kowalski, I. S. O. Pimienta, and M. J. McGuire, Int. Rev. Phys. Chem. 21, 527 (2002). [121] P. Piecuch, K. Kowalski, I. S. O. Pimienta, and S. A. Kucharski, In Low-Lying Potential Energy Surfaces, edited by M. R. Hoffman and K. G. Dyall, vol. 828 of ACS Symposium Series, pages 31—64 (American Chemical Society, Washington DC, 2002). [122] P. Piecuch, K. Kowalski, P.-D. Fan, and I. S. O. Pimienta, In Advanced Top- ics in Theoretical Chemical Physics, edited by J. Maruani, R. Lefebvre, and E. Brandas, vol. 12 of Progress in Theoretical Chemistry and Physics, pages 119—206 (Kluwer, Dordrecht, 2003). [123] P. Piecuch, K. Kowalski, I. S. O. Pimienta, P.-D. Fan, M. Lodriguito, M. J. McGuire, S. A. Kucharski, T. Kus, and M. Musial, Theor. Chem. Ace. 112, 349 (2004). [124] P.-D. Fan, K. Kowalski, and P. Piecuch, Mol. Phys. 103, 2191 (2005). [125] P.-D. Fan and P. Piecuch, Adv. Quantum Chem. 51, 1 (2006). [126] P. Piecuch and K. Kowalski, In Computational Chemistry: Reviews of Cur- rent Trends, edited by J. Leszczyriski, vol. 5, pages 1-104 (World Scientific, Singapore, 2000). 183 [127] K. Kowalski and P. Piecuch, J. Chem. Phys. 113, 18 (2000). [128] K. Kowalski and P. Piecuch, J. Chem. Phys. 113, 5644 (2000). [129] K. Kowalski and P. Piecuch, J. Chem. Phys. 122, 074107 (2005). [130] K. Kowalski and P. Piecuch, J. Chem. Phys. 115, 2966 (2001). [131] K. Kowalski and P. Piecuch, J. Chem. Phys. 120, 1715 (2004). [132] M. Wloch, J. R. Gour, K. Kowalski, and P. Piecuch, J. Chem. Phys. 122, 214107 (2005). [133] P. Piecuch and M. Wloch, J. Chem. Phys. 123, 224105 (2005). [134] P. Piecuch, M. Wloch, J. R. Gour, and A. Kinal, Chem. Phys. Lett. 418, 463 (2005) [135] M. Wloch, M. D. Lodriguito, P. Piecuch, and J. R. Gour, Mol. Phys. 104, 2149 (2006) [136] P. Piecuch, M. Wloch, M. Lodriguito, and J. R. Gour, In Recent Advances in the Theory of Chemical and Physical Systems, edited by S. Wilson, J.-P. Julien, J. Maruani, E. Brandas, and G. Delgado-Barrio, vol. 15 of Progress in Theoretical Chemistry and Physics, pages 45—106 (Springer, Berlin, 2006). [137] P. Piecuch, M. Wloch, and A. J. C. Varandas, In Topics in the Theory of Chem- ical and Physical Systems, edited by S. Lahmar, J. Maruani, S. Wilson, and G. Delgado-Barrio, vol. 16 of Progress in Theoretical Chemistry and Physics, pages 63—121 (Springer, Dordrecht, 2007). [138] M. Wloch, J. R. Gour, and P. Piecuch, J. Phys. Chem. A 111, 11359 (2007). [139] P. Piecuch, J. R. Gour, and M. Wloch, Int. J. Quantum Chem. 108, 2128 (2008). [140] P. Piecuch, J. R. Gour, and M. Wloch, Int. J. Quantum Chem. 109, 3268 (2009). [141] A. Kinal and P. Piecuch, J. Phys. Chem. A 111, 734 (2007). [142] C. J. Cramer, M. Wloch, P. Piecuch, C. Puzzarini, and L. Gagliardi, J. Phys. Chem. A 110, 1991 (2006). [143] C. J. Cramer, A. Kinal, M. Wloch, P. Piecuch, and L. Gagliardi, J. Phys. Chem. A 110, 11557 (2006). [144] C. J. Cramer, J. R. Gour, A. Kinal, M. Wloch, P. Piecuch, A. R. M. Shahi, and L. Gagliardi, J. Phys. Chem. A 112, 3754 (2008). 184 [145] P. Piecuch, M. Wloch, and A. J. C. Varandas, Theor. Chem. Ace. 120, 59 (2008). [146] Y. Z. Song, A. Kinal, P. J. S. B. Caridade, A. J. C. Varandas, and P. Piecuch, [147] [148] (149] J. Mol. Struc.: Theochem 859, 22 (2008). Y. B. Ge, M. S. Gordon, and P. Piecuch, J. Chem. Phys. 127, 174106 (2007). J. J. Zheng, J. R. Gour, J. J. Lutz, M. Wloch, P. Piecuch, and D. G. Truhlar, J. Chem. Phys. 128, 044108 (2008). Y. Zhao, O. Tishchenko, J. R. Gour, W. Li, J. J. Lutz, P. Piecuch, and D. G. T1uhlar, J. Phys. Chem. A 113, 5786 (2009). [150] Y. B. Ge, M. S. Gordon, P. Piecuch, M. Wloch, and J. R. Gour, J. Phys. Chem. [151] [152] [153] (154] [155] [156] (157] [158] [159) [160] [161] A 112, 11873 (2008). X. Li, J. R. Gour, J. Paldus, and P. Piecuch, Chem. Phys. Lett. 461, 321 (2008). R. B. Wiringa and S. C. Pieper, Phys. Rev. Lett. 89, 182501 (2002). P. Navratil and W. E. Ormand, Phys. Rev. C 68, 034305 (2003). P. Navratil and W. E. Ormand, Phys. Rev. Lett. 88, 152502 (2002). P. Navratil and E. Caurier, Phys. Rev. C 69, 014311 (2004). S. Aroua, P. Navratil, L. Zamick, M. S. Fayache, B. R. Barrett, J. P. Vary, and K. Heyde, Nucl. Phys. A 720, 71 (2004). P. Piecuch, M. Wloch, J. R. Gour, D. J. Dean, M. Hjorth-Jensen, and T. Papen- brock, In Nuclei and Mesoscopic Physics: Workshop on Nuclei and Mescopic Physics WNMP 2004, edited by V. Zelevinsky, vol. 777 of AIP Conference Proceedings, pages 28—45 (American Institute of Physics, Melville, New York, 2005) D. J. Dean, M. Hjorth-Jensen, K. Kowalski, T. Papenbrock, M. Wloch, and P. Piecuch, In Key Topics in Nuclear Structure, edited by A. Covello, Proceed- ings of the 8th International Spring Seminar on Nuclear Physics, pages 147—157 (World Scientific, Singapore, 2005). D. J. Dean, M. Hjorth-Jensen, K. Kowalski, P. Piecuch, and M. Wloch, 111 Condensed Matter Theories, edited by J. W. Clark and R. M. Panoff, vol. 20 (Nova Science Publishers, 2006). K. Kowalski, D. J. Dean, M. Hjorth-Jensen, T. Papenbrock, and P. Piecuch, Phys. Rev. Lett. 92, 132501 (2004). D. J. Dean, J. R. Gour, G. Hagen, M. Hjorth—Jensen, K. Kowalski, T. Papen- brock, P. Piecuch, and M. Wloch, Nucl. Phys. A 752, 299 (2005). 185 [162] M. Wloch, D. J. Dean, J. R. Gour, P. Piecuch, M. Hjorth-Jensen, T. Papen- brock, and K. Kowalski, Eur. Phys. J. A 25, 485 (2005). [163] M. Wloch, J. R. Gour, P. Piecuch, D. J. Dean, M. Hjorth-Jensen, and T. Pa- penbrock, J. Phys. G: Nucl. Part. Phys. 31, S1291 (2005). [164] M. Wloch, D. J. Dean, J. R. Gour, M. Hjorth—Jensen, K. Kowalski, T. Papen- brock, and P. Piecuch, Phys. Rev. Lett. 94, 212501 (2005). [165] J. R. Gour, P. Piecuch, M. Hjorth-Jensen, M. Wloch, and D. J. Dean, Phys. Rev. C 74, 024310 (2006). [166] T. Papenbrock, D. J. Dean, J. R. Gour, G. Hagen, M. Hjorth-Jensen, P. Piecuch, and M. Wloch, Int. J. Mod. Phys. B 20, 5338 (2006). [167] M. Horoi, J. R. Gour, M. Wloch, M. D. Lodriguito, B. A. Brown, and P. Piecuch, Phys. Rev. Lett. 98, 112501 (2007). [168] J. R. Gour, M. Horoi, P. Piecuch, and B. A. Brown, Phys. Rev. Lett. 101, 052501 (2008). [169] R. Roth, J. R. Gour, and P. Piecuch, Phys. Rev. C 79, 054325 (2009). [170] R. Roth, J. R. Gour, and P. Piecuch, Phys. Lett. B 679, 334 (2009). [171] G. Hagen, T. Papenbrock, D. J. Dean, A. Shwenk, A. Nogga, M. Wloch, and P. Piecuch, Phys. Rev. C 76, 034302 (2007). [172] S. Hirata, M. Nooijen, and R. J. Bartlett, Chem. Phys. Lett. 328, 459 (2000). [173] M. W. Schmidt, K. K. Baldridge, J. A. Boatz, S. T. Elbert, M. S. Gordon, J. H. Jensen, S. Koseki, N. Matsunaga, K. A. Nguyen, S. J. Su, T. L. Windus, M. Dupuis, and J. A. Montgomery, J. Comput. Chem. 14, 1347 (1993). [174] P. Piecuch, S. A. Kucharski, K. Kowalski, and M. Musial, Comp. Phys. Com- mun. 149, 71 (2002). [175] J. Gauss, In Encyclopedia of Computational Chemistry, edited by P. v. R. Schleyer, N. L. Allinger, J. G. T. Clark, P. A. Kollman, H. F. Schaefer, III, and P. R. Schreiner, vol. 1, pages 615%36 (Wiley, Chichester, U.K., 1998). [176] K. Hirao and H. Nakatsuji, J. Comput. Phys. 45, 246 (1982). [177] E. R. Davidson, J. Comput. Phys. 17, 87 (1975). [178] T. H. Dunning, Jr., J. Chem. Phys. 90, 1007 (1989). [179] R. A. Kendall, T. H. Dunning, Jr., and R. J. Harrison, J. Chem. Phys. 96, 6769 (1992). 186 [180] Basis sets were obtained from the Extensible Computational Chemistry Envi- ronment Basis Set Database, Version 02 / 25 / 04, as developed and distributed by the Molecular Science Computing Facility, Environmental and Molecular Sci- ences Laboratory which is part of the Pacific Northwest Laboratory, PO. Box 999, Richland, Washington 99352, USA, and funded by the US. Department of Energy. The Pacific Northwest Laboratory is a multi-program laboratory oper— ated by Battelle Memorial Institute for the US. Department of Energy under contract DE—AC06-76RLO 1830. Contact David Feller or Karen Schuchardt for further information. [181] H.-J. Werner and P. J. Knowles, J. Chem. Phys. 89, 5803 (1988). [182] P. J. Knowles and H.-J. Werner, Chem. Phys. Lett. 145, 514 (1988). [183] H.—J. Werner, P. J. Knowles, R. Lindh, F. R. Manby, M. Schiitz, P. Celani, T. Korona, G. Rauhut, R. D. Amos, A. Bernhardsson, A. Berning, D. L. Cooper, M. J. O. Deegan, A. J. Dobbyn, F. Eckert, C. Hampel, G. Hetzer, A. W. Lloyd, S. J. McNicholas, W. Meyer, M. E. Mura, A. Nicklass, P. Palmieri, R. Pitzer, U. Schumann, H. Stoll, A. J. Stone, R. Tarroni, and T. Thorsteinsson. MOLPRO, version 2006.1, a package of ab initio programs. [184] The active-space used in the CASSCF—based MRCI(Q) calculations for CH consisted of 4 electrons and 14 orbitals (the 20 - 7o orbitals and the orbitals constituting the 177 - 377 and 16 shells of CH were used). The active-space used in the CASSCF-based MRCI(Q) calculations for SH included 7 electrons and 10 orbitals (the 40 - 9o orbitals and the orbitals constituting the 277 - 377 shells of SH were used). In the MRCI(Q) calculations the lowest-energy core orbitals (~ 1s orbitals of C and S) were kept frozen. [185] M. Zachwieja, J. Mol. Spectrosc. 170, 285 (1995). [186] T. Nelis, J. M. Brown, and K. M. Evenson, J. Chem. Phys. 92, 4067 (1990). [187] R. Kepa, A. Para, M. Rytel, and M. Zachwieja, J. Mol. Spectrosc. 178, 189 (1996) [188] K. P. Huber and G. Herzberg. Molecular Spectra and Molecular Structure: Constants of Diatomic Molecules (Van Nostrand Reinhold, New York), 1979). [189] A. Kasdan, E. Herbst, and W. C. Lineberger, Chem. Phys. Lett. 31, 78 (1975). [190] C. E. Smith, R. A. King, and T. D. Crawford, J. Chem. Phys. 122, 054110 (2005) [191] A. Kalemos, A. Mavridis, and A. Metropoulos, J. Chem. Phys. 111, 9536 (1999). 187 [192] K. P. Huber and G. Herzberg. Constants of Diatomic Molecules. NIST Chem- istry WebBook, NIST Standard Reference Database Number 69, June 2005. National Institute of Standards and Technology, Gaithersburg MD, 20899, http: / / webbook.nist. gov. [193] T. H. Dunning, Jr., K. A. Peterson, and A. K. Wilson, J. Chem. Phys. 114, 9244 (2001). [194] W. J. Hehre, R. Ditchfield, and J. A. Pople, J. Chem. Phys. 56, 2257 (1972). [195] P. C. Hariharan and J. A. Pople, Theor. Chim. Acta 28, 213 (1973). [196] A. V. Nemukhin and B. L. Grigorenko, Chem. Phys. Lett. 276, 171 (1997). [197] T. H. Dunning, Jr., J. Chem. Phys. 53, 2823 (1970). [198] T. H. Dunning, Jr. and P. J. Hay, In Methods of Electronic Structure Theory, edited by H. F. Schaefer, III, vol. 2, pages 1—28 (Plenum, New York, 1977). [199] T. Nakajima and H. Nakatsuji, Chem. Phys. Lett. 79, 280 (1997). [200] M. Ishida, K. Toyota, M. Ehara, and H. Nakatsuji, Chem. Phys. Lett. 347, 493 (2001) [201] H. Nakatsuji, Chem. Phys. 75, 425 (1983). [202] G. Herzberg. Molecular Spectra and Molecular Structure 1: Spectra of Diatomic Molecules (Van Nostrand, London, 1967). [203] S. Hirata, M. Nooijen, I. Grabowski, and R. J. Bartlett, J. Chem. Phys. 114, 3919 (2001). [204] T. Shiozaki, K. Hirao, and S. Hirata, J. Chem. Phys. 126, 224106 (2007). [205] S. Hirata, P.-D. Fan, A. A. Auer, M. Nooijen, and P. Piecuch, J. Chem. Phys. 121, 12197 (2004). [206] S. R. Gwaltney and M. Head-Gordon, Chem. Phys. Lett. 323, 21 (2000). [207] S. R. Gwaltney, C. D. Sherrill, M. Head-Gordon, and A. I. Krylov, J. Chem. Phys. 113, 3548 (2000). [208] S. R. Gwaltney and M. Head-Gordon, J. Chem. Phys. 115, 2014 (2001). [209] S. R. Gwaltney, E. F. C. Byrd, T. V. Voorhis, and M. Head-Gordon, Chem. Phys. Lett. 353, 359 (2002). [210] M. Lodriguito and P. Piecuch, In Frontiers in Quantum Systems in Chemistry and Physics, edited by S. Wilson, P. Grout, J. Maruani, G. Delgado-Barrio, and P. Piecuch, vol. 18 of Progress in Theoretical Chemistry and Physics, pages 67—174 (Springer, Dordrecht, 2008). 188 [211] M. D. Lodriguito. Single-Reference Coupled— Cluster Methods Employing Multi- Reference Perturbation Theory. Ph.D. thesis, Michigan State University (2007). [212] P. Pulay, Chem. Phys. Lett. 73, 393 (1980). [213] P. Pulay, J Comput. Chem. 3, 556 (1982). [214] T. P. Hamilton and P. Pulay, J. Chem. Phys. 84, 5728 (1986). [215] Y. B. Ge, M. S. Gordon, F. Battaglia, and R. O. Fox, J. Phys. Chem. A 111, 1475 (2007). [216] Y. B. Ge, M. S. Gordon, F. Battaglia, and R. O. Fox, J. Phys. Chem. A 111, 1462 (2007). . [217] K. Hirao, Chem. Phys. Lett. 190, 374 (1992). [218] K. Hirao, Chem. Phys. Lett. 196, 397 (1992). [219] S. Huzinaga, J. Andzelm, M. Klobukowski, E. Radzio—Andzelm, Y. Sakai, and H. Tatewaki. Guassian Basis Sets for Molecular Calculations (Elsevier, Ams- terdam, 1984). [220] D. E. Woon and T. H. Dunning, Jr., J. Chem. Phys. 98, 1358 (1993). [221] J. F. Harrison, Accts of Chem. Res. 7, 378 (1974). [222] I. Shavitt, Tetrahedron 41, 1531 (1985). [223] W. A. Goddard, III, Science 227, 917 (1985). [224] H. F. Schaefer, III, Science 231, 1100 (1986). [225] P. R. Bunker, In Comparison of Ab Initio Quantum Chemistry with Experiment for Small Molecules, edited by R. J. Bartlett, page 141 (Reidel, Dordrecht, The Netherlands, 1985). [226] P. Jensen and P. R. Bunker, J. Chem. Phys. 89, 1327 (1988). [227] C. W. Bauschlicher and P. R. Taylor, J. Chem. Phys. 85, 6510 (1986). [228] D. E. Woon and T. H. Dunning, Jr., J. Chem. Phys. 103, 4572 (1995). [229] W. M. C. Foulkes, L. Mitas, R. J. Needs, and G. Rajagopal, Rev. Mod. Phys. 73,33 (2001). [230] M. P. Nightingale and C. J. Umrigar, editors. Quantum Monte Carlo Methods in Physics and Chemistry, vol. 525 of NATO ASI Ser. C (Kluwer, Dordrecht, 1999). [231] S. Zhang and H. Krakauer, Phys. Rev. Lett. 90, 136401 (2003). 189 [232] P. M. Zimmerman, J. Toulouse, Z. Zhang, C. B. Musgrave, and C. J. Umrigar, J. Chem. Phys. 131, 124103 (2009). [233] J. Toulouse and C. J. Umrigar, J. Chem. Phys. 126, 084102 (2007). [234] C. J. Umrigar, J. Toulouse, C. Filippi, S. Sorella, and R. G. Hennig, Phys. Rev. Lett. 98, 110201 (2007). [235] J. Toulouse and C. J. Umrigar, J. Chem. Phys. 128, 174101 (2008). [236] C. D. Sherrill, M. L. Leininger, T. J. V. Huis, and H. F. Schaefer, III, J. Chem. Phys. 108, 1040 (1998). [237] T. H. Dunning, Jr., J. Chem. Phys. 55, 716 (1971). [238] M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, V. G. Zakrzewki, J. A. Montgomery, Jr., R. E. Stratmann, J. C. Burant, S. Dapprich, J. M. Millam, A. D. Daniels, K. N. Kudin, M. C. Strain, O. Farkas, J. Tomasi, V. Barone, M. Cossi, , R. Cammi, B. Mennucci, C. Pomelli, C. Adamo, S. Clifford, J. Ochterski, G. A. Petersson, P. Y. Ayala, Q. Cui, K. Morokuma, K. Malick, A. D. Rabuck, K. Raghavachari, J. B. Fores- man, J. Cioslowski, J. V. Ortiz, B. B. Stefanov, G. Liu, A. Liashenko, P. Piskorz, I. Komaromi, , R. Gomperts, R. L. Martin, D. J. Fox, T. Keith, M. A. A1- Laham, C. Peng, C. Y. Peng, A. Nanayakkara, C. Gonazalez, M. Challacombe, P. M. W. Gill, B. Johnson, W. Chen, M. W. W. J. L. Andres, M. Head-Gordon, E. S. Replogle, and J. A. Pople. GAUSSIAN98, revision A.5 (Gaussian Inc, Pittsburgh, PA, 1998). [239] J. D. Watts, In Computational Chemistry: Reviews of Current Trends, edited by J. Leszczynski, vol. 7, pages 65—169 (World Scientific, Singapore, 2002). [240] X. Li and J. Paldus, J. Chem. Phys. 107, 6257 (1997). [241] J. Paldus and J. Planelles, Theor. Chem. Acta 89, 13 (1994). [242] X. Li, G. Peris, J. Planelles, F. Rajadell, and J. Paldus, J. Chem. Phys. 107, 90 (1997). [243] X. Li and J. Paldus, J. Chem. Phys. 124, 174101 (2006). [244] X. Li and J. Paldus, Chem. Phys. Lett. 431, 179 (2006). [245] H. Bredohl, I. Dubois, Y. Houbrechts, and P. Nzohabonayo, J. Mol. Spectrosc. 112,430 (1985). [246] M. Lorenz, J. Agreiter, A. M. Smith, and V. E. Bondybey, J. Chem. Phys. 104, 3143 (1996). [247] R. S. Ram and P. F. Bernath, J. Mol. Spectrosc. 180, 414 (1996). 190 [248] K. R. Asmis, T. R. Taylor, and D. M. Neurnark, Chem. Phys. Lett. 295, 75 (1998). [249] R. B. Wiringa, V. G. J. Stoks, and R. Schiavilla, Phys. Rev. C 51, 38 (1995). [250] R. Machleidt, Phys. Rev. C 63, 024001 (2001). [251] S. Weinberg, Phys. Lett. B251, 288 (1990). [252] U. van Kolck, Prog. Part. Nucl. Phys. 43, 337 (1999). [253] D. R. Entem and R. Machleidt, Phys. Rev. Lett. B 524, 93 (2002). [254] D. R. Entem and R. Machleidt, Phys. Rev. C 68, 41001 (2003). [255] M. Hjorth—Jensen, T. T. S. Kuo, and E. Osnes, Phys. Rep. 261, 125 (1995). [256] D. J. Dean and M. Hjorth-Jensen, Phys. Rev. C 69, 054320 (2004). [257] R. Roth, H. Hergert, P. Papakonstantinou, T. Neff, and H. Feldmeier, Phys. Rev. C 72, 034002 (2005). [258] R. Roth, T. Neff, H. Hergert, and H. Feldmeier, Nucl. Phys. A 745, 3 (2004). [259] H. Feldmeier, T. Neff, R. Roth, and J. Schnack, Nucl. Phys. A 632, 61 (1998). [260] T. Neff and H. Feldmeier, Nucl. Phys. A 713, 311 (2002). [261] E. K. Warburton and B. A. Brown, Phys. Rev. C 46, 923 (1992). [262] G. Audi, A. H. Wapstra, and C. Thibault, Nucl. Phys. A 729, 337 (2003). [263] R. B. Firestone, V. S. Shirley, C. M. Baglin, S. Y. F. Chu, and J. Zipkin. Table of Isotopes, 8th Ed. (Wiley Interscience, New York, 1996). [264] R. Roth and P. Navratil, Phys. Rev. Lett. 99, 092501 (2007). [265] P. J. Bruna and S. D. Peyerimhoff, Adv. Chem. Phys. 67, 1 (1987). [266] J. Paldus, In New Horizons in Quantum Chemistry, edited by P.-O. Léwdin and B. Pullman, pages 31—60 (Reidel, Dordrecht, 1983). [267] R. Buenker and S. Peyerimhoff, In New Horizons in Quantum Chemistry, edited by P.-O. wadin and B. Pullman, pages 183—220 (Reidel, Dordrecht, 1983). [268] P. J. Bruna, S. D. Peyerimhoff, and R. J. Buenker, Chem. Phys. Lett. 7 2, 278 (1980). [269] K. Jankowski, L. Meissner, and J. Wasilewksi, Int. J. Quantum Chem. 28, 931 (1985). 191 [270] M. Honma, T. Otsuka, B. A. Brown, and T. Mizusaki, Eur. Phys. Jour. A 25, 499 (2005). [271] M. Honma, T. Otsuka, B. A. Brown, and T. Mizusaki, Phys. Rev. C 65, 061301 (2002). [272] M. Honma, T. Otsuka, B. A. Brown, and T. Mizusaki, Phys. Rev. C 69, 034335 (2004). 192 MICHlIGAN STATE UNIVERSITY LIB II] I III I 6|l||]|]|[|llll||lms 21930