Method of moments of coupled-cluster equations: a new formalism for designing accurate electronic structure methods for ground and excited states

The method of moments of coupled-cluster equations (MMCC), which provides a systematic way of improving the results of the standard coupled-cluster (CC) and equation-of-motion CC (EOMCC) calculations for the ground- and excited-state energies of atomic and molecular systems, is described. The MMCC theory and its generalized MMCC (GMMCC) extension that enables one to use the cluster operators resulting from the standard as well as nonstandard CC calculations, including those obtained with the extended CC (ECC) approaches, are based on rigorous mathematical relationships that define the many-body structure of the differences between the full configuration interaction (CI) and CC or EOMCC energies. These relationships can be used to design the noniterative corrections to the CC/EOMCC energies that work for chemical bond breaking and potential energy surfaces of excited electronic states, including excited states dominated by double excitations, where the standard single-reference CC/EOMCC methods fail. Several MMCC and GMMCC approximations are discussed, including the renormalized and completely renormalized CC/EOMCC methods for closed- and open-shell states, the quadratic MMCC approaches, the CI-corrected MMCC methods, and the GMMCC approaches for multiple bond breaking based on the ECC cluster amplitudes.     

Full configuration interaction potential energy curves for the X (1)Sigma(g) (+), B (1)Delta(g), and B(') (1)Sigma(g) (+) states of C(2): a challenge for approximate methods

The C(2) molecule exhibits unusual bonding and several low-lying excited electronic states, making the prediction of its potential energy curves a challenging test for quantum chemical methods. We report full configuration interaction results for the X (1)Sigma(g) (+), B (1)Delta(g), and B(') (1)Sigma(g) (+) states of C(2), which exactly solve the electronic Schrodinger equation within the space spanned by a 6-31G( *) basis set. Within the D(2h) subgroup used by most electronic structure programs, these states all have the same symmetry ((1)A(g)), and all three states become energetically close for interatomic distances beyond 1.5 A. The quality of several single-reference ab initio methods is assessed by comparison to the benchmark results. Unfortunately, even coupled-cluster theory through perturbative triples using an unrestricted Hartree-Fock reference exhibits large nonparallelity errors (>20 kcal mol(-1)) for the ground state. The excited states are not accurately modeled by any commonly used single-reference method, nor by configuration interaction including full quadruple substitutions. The present benchmarks will be helpful in assessing theoretical methods designed to break bonds in ground and excited electronic states.     

Theoretical characterization of end-on and side-on peroxide coordination in ligated Cu2O2 models

The relative energetics of mu-eta1:eta1 (trans end-on) and mu-eta2:eta2 (side-on) peroxo isomers of Cu2O2 fragments supported by 0, 2, 4, and 6 ammonia ligands have been computed with various density functional, coupled-cluster, and multiconfigurational protocols. There is substantial disagreement between the different levels for most cases, although completely renormalized coupled-cluster methods appear to offer the most reliable predictions. The significant biradical character of the end-on peroxo isomer proves problematic for the density functionals, while the demands on active space size and the need to account for interactions between different states in second-order perturbation theory prove challenging for the multireference treatments. In the latter case, it proved impossible to achieve any convincing convergence.     

Balancing dynamic and nondynamic correlation for diradical and aromatic transition states: a renormalized coupled-cluster study of the cope rearrangement of 1,5-hexadiene

Single-reference coupled-cluster calculations employing the completely renormalized CCSD(T) (CR-CCSD(T)) approach have been used to examine the mechanism of the Cope rearrangement of 1,5-hexadiene. In agreement with multireference perturbation theory, the CR-CCSD(T) method favors the concerted mechanism of the Cope rearrangement involving an aromatic transition state. The CCSD(T) approach, which is often regarded as the "gold standard" of electronic structure theory, seems to fail in this case, favoring pathways through diradical structures.     

Breaking bonds of open-shell species with the restricted open-shell size extensive left eigenstate completely renormalized coupled-cluster method

The recently developed restricted open-shell, size extensive, left eigenstate, completely renormalized (CR), coupled-cluster (CC) singles (S), doubles (D), and noniterative triples (T) approach, termed CR-CC(2,3) and abbreviated in this paper as ROCCL, is compared with the unrestricted CCSD(T) [UCCSD(T)] and multireference second-order perturbation theory (MRMP2) methods to assess the accuracy of the calculated potential energy surfaces (PESs) of eight single bond-breaking reactions of open-shell species that consist of C, H, Si, and Cl; these types of reactions are interesting because they account for part of the gas-phase chemistry in the silicon carbide chemical vapor deposition. The full configuration interaction (FCI) and multireference configuration interaction with Davidson quadruples correction [MRCI(Q)] methods are used as benchmark methods to evaluate the accuracy of the ROCCL, UCCSD(T), and MRMP2 PESs. The ROCCL PESs are found to be in reasonable agreement with the corresponding FCI or MRCI(Q) PESs in the entire region R = 1-3Re for all of the studied bond-breaking reactions. The ROCCL PESs have smaller nonparallelity error (NPE) than the UCCSD(T) ones and are comparable to those obtained with MRMP2. Both the ROCCL and UCCSD(T) PESs have significantly smaller reaction energy errors (REE) than the MRMP2 ones. Finally, an efficient strategy is proposed to estimate the ROCCL/cc-pVTZ PESs using an additivity approximation for basis set effects and correlation corrections.     

Excited states in the multireference state-specific coupled-cluster theory with the complete active space reference

The recently proposed multireference state-specific coupled-cluster theory with the complete active space reference has been used to study electronically excited states with different spatial and spin symmetries. The algorithm for the method has been obtained using the computerized approach for automatic generation of coupled-cluster diagrams with an arbitrary level of the electronic excitation from a formal reference determinant. The formal reference is also used to generate the genuine reference state in the form of a linear combination of determinants contracted to a configuration with the spin and spatial symmetries of the target state. The natural-orbital expansions of the one-electron configuration inferaction density matrix allowed us to obtain the most compact orbital space for the expansion of the reference function. We applied our approach in the calculations of singlet and triplet states of different spatial symmetries of the water molecule. The comparisons of the results with values obtained using other many-particle methods and with the full configuration interaction results demonstrate good ability of the approach to deal with electronic excited states.     

On the electronically excited states of uracil

Vertical excitation energies in uracil in the gas phase and in water solution are investigated by the equation-of-motion coupled-cluster and multireference configuration interaction methods. Basis set effects are found to be important for converged results. The analysis of electronic wave functions reveals that the lowest singlet states are predominantly of a singly excited character and are therefore well described by single-reference equation-of-motion methods augmented by a perturbative triples correction to account for dynamical correlation.Our best estimates for the vertical excitation energies for the lowest singlet n --> pi* and pi --> pi* are 5.0 +/- 0.1 eV and 5.3 +/- 0.1 eV, respectively. The solvent effects for these states are estimated to be +0.5 eV and +/- 0.1 eV, respectively. We attribute the difference between the computed vertical excitations and the maximum of the experimental absorption to strong vibronic interaction between the lowest A" and A' states leading to intensity borrowing by the forbidden transition.     

Efficient strategies for accurate calculations of electronic excitation and ionization energies: theory and application to the dehydro-m-xylylene anion

In the dehydro-m-xylylene (DMX) anion [Munsch; et al. J. Org. Chem. 2004, 69, 5735], three nearly degenerate orbitals host four electrons, which results in a large number of nearly-degenerate electronic states. By using this challenging example, we assess the performance of the multireference "brute force" approach and the two-step schemes based on single-reference methods for calculating accurate energy differences. Different schemes for calculating adiabatic ionization potential (IP) of DMX- are also investigated. IP calculated by single-reference based schemes is in excellent agreement with experiment.     

High-order excitations in state-universal and state-specific multireference coupled cluster theories: model systems

For the first time high-order excitations (n>2) have been studied in three multireference couple cluster (MRCC) theories built on the wave operator formalism: (1) the state-universal (SU) method of Jeziorski and Monkhorst (JM) (2) the state-specific Brillouin-Wigner (BW) coupled cluster method, and (3) the state-specific MRCC approach of Mukherjee (Mk). For the H4, P4, BeH(2), and H8 models, multireference coupled cluster wave functions, with complete excitations ranging from doubles to hextuples, have been computed with a new arbitrary-order string-based code. Comparison is then made to corresponding single-reference coupled cluster and full configuration interaction (FCI) results. For the ground states the BW and Mk methods are found, in general, to provide more accurate results than the SU approach at all levels of truncation of the cluster operator. The inclusion of connected triple excitations reduces the nonparallelism error in singles and doubles MRCC energies by a factor of 2-10. In the BeH(2) and H8 models, the inclusion of all quadruple excitations yields absolute energies within 1 kcal mol(-1) of the FCI limit. While the MRCC methods are very effective in multireference regions of the potential energy surfaces, they are outperformed by single-reference CC when one electronic configuration dominates.     

Can a single-reference approach provide a balanced description of ground and excited states? A comparison of the completely renormalized equation-of-motion coupled-cluster method with multireference quasidegenerate perturbation theory near a conical intersection and along a photodissociation coordinate in ammonia

We calculated the two lowest electronically adiabatic potential energy surfaces of ammonia in the region of the conical intersection and at a sequence of geometries along which one of the N-H bonds is broken. We employed both a multireference (MR) method and a single-reference (SR) method. The MR calculations are based on multiconfiguration quasidegenerate perturbation theory (MC-QDPT) with a 6-311+G(3df,3pd) basis set. The SR calculations, carried out with the same basis, employ the completely renormalized equation-of-motion coupled-cluster method with singles and doubles, and a noniterative treatment of triples, denoted CR-EOMCCSD(T). At 91 geometries used for comparison, including geometries near a conical intersection, the surfaces agree to 7% on average.     

Strongly correlated mechanisms of a photoexcited radical reaction from the anti-Hermitian contracted Schrödinger equation

Photoexcited radical reactions are critical to processes in both nature and materials, and yet they can be challenging for electronic structure methods due to the presence of strong electron correlation. Reduced-density-matrix (RDM) methods, based on solving the anti-Hermitian contracted Schro?dinger equation (ACSE) for the two-electron RDM (2-RDM), are examined for studying the strongly correlated mechanisms of these reactions with application to the electrocyclic interconversion of allyl and cyclopropyl radicals. We combine recent extensions of the ACSE to excited states [G. Gidofalvi and D. A. Mazziotti, Phys. Rev. A 80, 022507 (2009)] and arbitrary spin states [A. E. Rothman, J. J. Foley IV, and D. A. Mazziotti, Phys. Rev. A 80, 052508 (2009)]. The ACSE predicts that the ground-state ring closure of the allyl radical has a high 52.5 kcal/mol activation energy that is consistent with experimental data, while the closure of an excited allyl radical can occur by disrotatory and conrotatory pathways whose transition states are essentially barrierless. Comparisons are made with multireference second- and third-order perturbation theories and multireference configuration interaction. While predicted energy differences do not vary greatly between methods, the ACSE appears to improve these differences when they involve a strongly and a weakly correlated radical by capturing a greater share of single-reference correlation that increases the stability of the weakly correlated radicals. For example, the ACSE predicts a -39.6 kcal/mol conversion of the excited allyl radical to the ground-state cyclopropyl radical in comparison to the -32.6 to -37.3 kcal/mol conversions predicted by multireference methods. In addition, the ACSE reduces the computational scaling with the number of strongly correlated orbitals from exponential (traditional multireference methods) to quadratic. Computed ground- and excited-state 2-RDMs are nearly N-representable.     

Linear-response theory for Mukherjee's multireference coupled-cluster method: Excitation energies

The recently presented linear-response function for Mukherjee's multireference coupled-cluster method (Mk-MRCC) [T.-C. Jagau and J. Gauss, J. Chem. Phys. 137, 044115 (2012)] is employed to determine vertical excitation energies within the singles and doubles approximation (Mk-MRCCSD-LR) for ozone as well as for o-benzyne, m-benzyne, and p-benzyne, which display increasing multireference character in their ground states. In order to assess the impact of a multireference ground-state wavefunction on excitation energies, we compare all our results to those obtained at the single-reference coupled-cluster level of theory within the singles and doubles as well as within the singles, doubles, and triples approximation. Special attention is paid to the artificial splitting of certain excited states which arises from the redundancy intrinsic to Mk-MRCC theory and hinders the straightforward application of the Mk-MRCC-LR method.     

Real versus artifactual symmetry-breaking effects in Hartree-Fock, density-functional, and coupled-cluster methods

We have examined the relative abilities of Hartree-Fock, density-functional theory (DFT), and coupled-cluster theory in describing second-order (pseudo) Jahn-Teller (SOJT) effects, perhaps the most commonly encountered form of symmetry breaking in polyatomic molecules. As test cases, we have considered two prototypical systems: the 2Sigmau+ states of D( infinity h) BNB and C3+ for which interaction with a low-lying 2Sigmag+ excited state leads to symmetry breaking of the nuclear framework. We find that the Hartree-Fock and B3LYP methods correctly reproduce the pole structure of quadratic force constants expected from exact SOJT theory, but that both methods appear to underestimate the strength of the coupling between the electronic states. Although the Tamm-Dancoff (CIS) approximation gives excitation energies with no relationship to the SOJT interaction, the random-phase-approximation (RPA) approach to Hartree-Fock and time-dependent DFT excitation energies predicts state crossings coinciding nearly perfectly with the positions of the force constant poles. On the other hand, the RPA excited-state energies exhibit unphysical curvature near their crossings with the ground (reference) state, a problem arising directly from the mathematical structure of the RPA equations. Coupled-cluster methods appear to accurately predict the strength of the SOJT interactions between the 2Sigmau+ and 2Sigmag+ states, assuming that the inclusion of full triple excitations provides a suitable approximation to the exact wave function, and are the only methods examined here which predict symmetry breaking in BNB. However, coupled-cluster methods are plagued by artifactual force constant poles arising from the response of the underlying reference molecular orbitals to the geometric perturbation. Furthermore, the structure of the "true" SOJT force constant poles predicted by coupled-cluster methods, although correctly positioned, has the wrong structure.     

An exponential multireference wave-function Ansatz

An exponential multireference wave-function Ansatz is formulated. In accordance with the state universal coupled-cluster Ansatz of Jeziorski and Monkhorst [Phys. Rev. A 24, 1668 (1981)] the approach uses a reference specific cluster operator. In order to achieve state selectiveness the excitation- and reference-related amplitude indexing of the state universal Ansatz is replaced by an indexing which is based on excited determinants. There is no reference determinant playing a particular role. The approach is size consistent, coincides with traditional single-reference coupled cluster if applied to a single-reference, and converges to full configuration interaction with an increasing cluster operator excitation level. Initial applications on BeH2, CH2, Li2, and nH2 are reported.     

Photodissociation dynamics of the NO dimer. I. Theoretical overview of the ultraviolet singlet excited states

Molecular orbital theory and calculations are used to describe the ultraviolet singlet excited states of NO dimer. Qualitatively, we derive and catalog the dimer states by correlating them with monomer states, and provide illustrative complete active space self-consistent field calculations. Quantitatively, we provide computational estimates of vertical transition energies and absorption intensities with multireference configuration interaction and equations-of-motion coupled-cluster methods, and examine an important avoided crossing between a Rydberg and a valence state along the intermonomer and intramonomer stretching coordinates. The calculations are challenging, due to the high density of electronic states of various types (valence and Rydberg, excimer and charge transfer) in the 6-8 eV region, and the multiconfigurational nature of the ground state. We have identified a bright charge-transfer (charge-resonance) state as responsible for the broadband seen in UV absorption experiments. We also use our results to facilitate the interpretation of UV photodissociation experiments, including the time-resolved 6 eV photodissociation experiments to be presented in the next two papers of this series.     

Inelastic scattering matrix elements for the nonadiabatic collision B(2P1/2)+H2(1Sigmag+,j)<-->B(2P3/2)+H2(1Sigmag+,j')

Inelastic scattering matrix elements for the nonadiabatic collision B(2P1/2)+H2(1Sigmag+,j)<-->B(2P3/2)+H2(1Sigmag+,j') are calculated using the time dependent channel packet method (CPM). The calculation employs 1 2A', 2 2A', and 1 2A" adiabatic electronic potential energy surfaces determined by numerical computation at the multireference configuration-interaction level [M. H. Alexander, J. Chem. Phys. 99, 6041 (1993)]. The 1 2A' and 2 2A', adiabatic electronic potential energy surfaces are transformed to yield diabatic electronic potential energy surfaces that, when combined with the total B+H2 rotational kinetic energy, yield a set of effective potential energy surfaces [M. H. Alexander et al., J. Chem. Phys. 103, 7956 (1995)]. Within the framework of the CPM, the number of effective potential energy surfaces used for the scattering matrix calculation is then determined by the size of the angular momentum basis used as a representation. Twenty basis vectors are employed for these calculations, and the corresponding effective potential energy surfaces are identified in the asymptotic limit by the H2 rotor quantum numbers j=0, 2, 4, 6 and B electronic states 2Pja, ja=1/2, 3/2. Scattering matrix elements are obtained from the Fourier transform of the correlation function between channel packets evolving in time on these effective potential energy surfaces. For these calculations the H2 bond length is constrained to a constant value of req=1.402 a.u. and state to state scattering matrix elements corresponding to a total angular momentum of J=1/2 are discussed for j=0<-->j'=0,2,4 and 2P1/2<-->2P1/2, 2P3/2 over a range of total energy between 0.0 and 0.01 a.u.     

Application of renormalized coupled-cluster methods to potential function of water

The goal of this paper is to examine the performance of the conventional and renormalized single-reference coupled-cluster (CC) methods in calculations of the potential energy surface of the water molecule. A comparison with the results of the internally contracted multi-reference configuration interaction calculations including the quasi-degenerate Davidson correction (MRCI(Q)) and the spectroscopically accurate potential energy surface of water resulting from the use of the energy switching (ES) approach indicates that the relatively inexpensive completely renormalized (CR) CC methods with singles (S), doubles (D), and a non-iterative treatment of triples (T) or triples and quadruples (TQ), such as CR-CCSD(T), CR-CCSD(TQ), and the recently developed rigorously size extensive extension of CR-CCSD(T), termed CR-CC(2,3), provide substantial improvements in the results of conventional CCSD(T) and CCSD(TQ) calculations at larger internuclear separations. It is shown that the CR-CC(2,3) results corrected for the effect of quadruply excited clusters through the CR-CC(2,3)+Q approach can compete with the highly accurate MRCI(Q) data. The excellent agreement between the CR-CC(2,3)+Q and MRCI(Q) results suggests ways of improving the global potential energy surface of water resulting from the use of the ES approach in the regions of intermediate bond stretches and intermediate energies connecting the region of the global minimum with the asymptotic regions. Contribution to the Mark S. Gordon 65th Birthday Festschrift Issue.     

Spin-orbit coupling in O2(upsilon)+O2 collisions: I. Electronic structure calculations on dimer states involving the X 3Sigmag-, a 1Deltag, and b 1Sigmag+ states of O2

The importance of vibrational-to-electronic (V-E) energy transfer mediated by spin-orbit coupling in the collisional removal of O2(X 3Sigmag-,upsilon>or=26) by O2 has been reported in a recent communication [F. Dayou, J. Campos-Martinez, M. I. Hernandez, and R. Hernandez-Lamoneda, J. Chem. Phys. 120, 10355 (2004)]. The present work provides details on the electronic properties of the dimer (O2)2 relevant to the self-relaxation of O2(X 3Sigmag-,upsilon>0) where V-E energy transfer involving the O2(a 1Deltag) and O2(b 1Sigmag+) states is incorporated. Two-dimensional electronic structure calculations based on highly correlated ab initio methods have been carried out for the potential-energy and spin-orbit coupling surfaces associated with the ground singlet and two low-lying excited triplet states of the dimer dissociating into O2(X 3Sigmag-)+O2(X 3Sigmag-), O2(a 1Deltag)+O2(X 3Sigmag-), and O2(b 1Sigmag+)+O2(X 3Sigmag-). The resulting interaction potentials for the two excited triplet states display very similar features along the intermolecular separation, whereas differences arise with the ground singlet state for which the spin-exchange interaction produces a shorter equilibrium distance and higher binding energy. The vibrational dependence is qualitatively similar for the three studied interaction potentials. The spin-orbit coupling between the ground and second excited states is already nonzero in the O2+O2 dissociation limit and keeps its asymptotic value up to relatively short intermolecular separations, where the coupling increases for intramolecular distances close to the equilibrium of the isolated diatom. On the other hand, state mixing between the two excited triplet states leads to a noticeable collision-induced spin-orbit coupling between the ground and first excited states. The results are discussed in terms of specific features of the dimer electronic structure (including a simple four-electron model) and compared with existing theoretical and experimental data. This work gives theoretical insight into the origin of electronic energy-transfer mechanisms in O2+O2 collisions.