Analytic gradient for the multireference Brillouin-Wigner coupled cluster method and for the state-universal multireference coupled cluster method

We present the analytic gradient theory and its pilot implementation for the multireference Brillouin-Wigner coupled cluster (BWCC) method and for the state-universal multireference coupled cluster method. The analytic gradient has been derived for three cases: (i) BWCC method without a size-extensivity correction, (ii) BWCC method with the iterative size-extensivity correction, and (iii) for the rigorously size-extensive state-universal method. The pilot implementation is based on full-configuration interaction expansions and is presently limited to single and double excitation levels; however, the resulting equations are general. For BWCC methods, they also do not contain terms explicitly mixing amplitudes of different reference configurations and can thus be implemented in an efficient way. The analytic gradients have been verified with respect to numerically computed ones on the example of CH2 molecule, and geometry optimizations of CH2 and SiH2 have been carried out.     

Alternative single-reference coupled cluster approaches for multireference problems: the simpler, the better

We report a general implementation of alternative formulations of single-reference coupled cluster theory (extended, unitary, and variational) with arbitrary-order truncation of the cluster operator. These methods are applied to compute the energy of Ne and the equilibrium properties of HF and C(2). Potential energy curves for the dissociation of HF and the BeH(2) model computed with the extended, variational, and unitary coupled cluster approaches are compared to those obtained from the multireference coupled cluster approach of Mukherjee et al. [J. Chem. Phys. 110, 6171 (1999)] and the internally contracted multireference coupled cluster approach [F. A. Evangelista and J. Gauss, J. Chem. Phys. 134, 114102 (2011)]. In the case of Ne, HF, and C(2), the alternative coupled cluster approaches yield almost identical bond length, harmonic vibrational frequency, and anharmonic constant, which are more accurate than those from traditional coupled cluster theory. For potential energy curves, the alternative coupled cluster methods are found to be more accurate than traditional coupled cluster theory, but are three to ten times less accurate than multireference coupled cluster approaches. The most challenging benchmark, the BeH(2) model, highlights the strong dependence of the alternative coupled cluster theories on the choice of the Fermi vacuum. When evaluated by the accuracy to cost ratio, the alternative coupled cluster methods are not competitive with respect to traditional CC theory, in other words, the simplest theory is found to be the most effective one.     

The coupled cluster approach with a hybrid treatment of connected triple excitations based on the restricted Hartree-Fock reference

A generalization of the coupled cluster (CC) singles, doubles, and a hybrid treatment of connected triples [denoted as CCSD(T)-h] [Shen et al., J. Chem. Phys. 132, 114115 (2010)] to the restricted Hartree-Fock (RHF) reference is presented. In this approach, active (or pseudoactive) RHF orbitals are constructed automatically by performing unitary transformations of canonical RHF orbitals so that they spatially mimic the natural orbitals of the unrestricted Hartree-Fock reference. The present RHF-based CCSD(T)-h approach has been applied to study the potential energy surfaces in several typical bond breaking processes and the singlet-triplet gaps in a diradical (HFH)(-1). For all systems under study, the overall performance of CCSD(T)-h is very close to that of the corresponding CCSD(T) (CC singles, doubles, and triples), and much better than that of CCSD(T) (CC singles, doubles, and perturbative triples).     

Restricted Hartree-Fock and unrestricted Hartree-Fock as reference states in many-body perturbation theory: a critical comparison of the two approaches

Summary The paper deals with two topics related to the problem which reference state is better for many-body perturbation theory: restricted Hartree-Fock (RHF) or unrestricted Hartree-Fock (UHF)? The first topic concerns the potential surfaces. Several examples are presented to show shortcomings of the two approaches and a simple way is presented which seems to give a useful potential curve in the whole range of interatomic distances by a composition of RHF and UHF potential curves. The second topic concerns the many-body perturbation theory for open-shell systems in the RHF formalism. The method is critically examined and compared with the ordinary many-body perturbation theory using UHF as the reference. This examination of many-body techniques provides also some insight into the problems inherent of the SCF theory: spin contamination from higher multiplets, localization of orbitals, and self-consistency effects.     

Uncoupled multireference state-specific Mukherjee's coupled cluster method with triexcitations

We have developed the uncoupled version of multireference Mukherjee's coupled cluster method with connected triexcitations. The method has been implemented in ACES II program package. The agreement between the uncoupled and the standard version of Mukherjee's multireference coupled cluster method has been reported previously at the singles and doubles level by Das et al. [J. Mol. Struct.: THEOCHEM 79, 771 (2006); Chem. Phys. 349, 115 (2008)]. The aim of this article is to investigate this method further, in order to establish how its performance changes with the size of the basis set, size of the model space, multireference character of different molecules, and inclusion of connected triple excitations. Assessment of the new method has been performed on the singlet methylene, potential energy curve of fluorine molecule, and third b?(1)Σ(g)(+) electronic state of oxygen molecule.     

Reduced multireference coupled cluster method with singles and doubles: Perturbative corrections for triples

The reduced multireference coupled-cluster method with singles and doubles (RMR CCSD) that employs multireference configuration interaction wave function as an external source for a small subset of approximate connected triples and quadruples, is perturbatively corrected for the remaining triples along the same lines as in the standard CCSD(T) method. The performance of the resulting RMR CCSD(T) method is tested on four molecular systems, namely, the HF and F(2) molecules, the NO radical, and the F(2) (+) cation, representing distinct types of molecular structure, using up to and including a cc-pVQZ basis set. The results are compared with those obtained with the standard CCSD(T), UCCSD(T), CCSD(2), and CR CCSD(T) methods, wherever applicable or available. An emphasis is made on the quality of the computed potentials in a broad range of internuclear separations and on the computed equilibrium spectroscopic properties, in particular, harmonic frequencies omega(e). It is shown that RMR CCSD(T) outperforms other triply corrected methods and is widely applicable.     

Fock space multireference coupled cluster method with full inclusion of connected triples for excitation energies

We report the initial Fock space multireference coupled cluster method with the full inclusion of single, double, and triple excitations (FS-CCSDT) for the (1,1) sector. We present pilot applications for calculating excitation energies for the N(2) molecule and the Ne atom. The performance of the current model, along with the FS-CCSD one, has been studied in comparison with the equation-of-motion coupled-cluster and the similarity transformed methods.     

A sequential transformation approach to the internally contracted multireference coupled cluster method

The internally contracted multireference coupled cluster (ic-MRCC) approach is formulated using a new wave function ansatz based on a sequential transformation of the reference function (sqic-MRCC). This alternative wave function simplifies the formulation of computationally viable methods while preserving the accuracy of the ic-MRCC approach. The structure of the sqic-MRCC wave function allows folding the effect of the single excitations into a similarity-transformed Hamiltonian whose particle rank is equal to the one of the Hamiltonian. Consequently, we formulate an approximation to the sqic-MRCC method with singles and doubles (included respectively up to fourfold and twofold commutators, sqic-MRCCSD[2]) that contains all terms present in the corresponding single-reference coupled cluster scheme. Computations of the potential energy curves for the dissociation of BeH(2) show that the untruncated sqic-MRCCSD scheme yields results that are almost indistinguishable from the ordinary ic-MRCCSD method. The energy obtained from the computationally less expensive sqic-MRCCSD[2] approximation is found to deviate from the full ic-MRCCSD method by less than 0.2 mE(h) for BeH(2), while, in the case of water, the harmonic vibrational frequencies of ozone, the singlet-triplet splitting of p-benzyne, and the dissociation curve of N(2), sqic-MRCCSD[2] faithfully reproduces the results obtained via the ic-MRCCSD scheme truncated to two commutators. A formal proof is given of the equivalence of the ic-MRCC and sqic-MRCC methods with the internally contracted and full configuration interaction approaches.     

Fock space multireference coupled cluster calculations based on an underlying bivariational self-consistent field on Auger and shape resonances

The Fock space multireference coupled cluster based on an underlying bivariational self-consistent field is applied to the problem of computing complex energy associated with Auger and shape resonances in e-atom scattering. It is concluded that the Fock space multireference coupled cluster based on a bivariational self-consistent field provides a useful and practical approach to calculation of resonance parameters. Numerical results are presented for the 2P shape resonance of Mg and Auger 1 s(-1) hole of Be.     

Fock space multireference coupled cluster theory: noniterative inclusion of triples for excitation energies

In this paper we propose and numerically implement a specific scheme for calculating the excitation energies (EEs) within the Fock space multireference coupled cluster framework, which includes the contributions from noniterative triples cluster amplitudes. These contribute to the EEs at the third order. We present results for CH⁺ and N2, and study the effects of these noniterative triples on EEs. Received: 28 July 1997 / Accepted: 8 December 1997     

Potential energy curve for ring-opening reactions: comparison between broken-symmetry and multireference coupled cluster methods

The spin-unrestricted Hartree-Fock (UHF)-based coupled cluster singles and doubles (UHF-CCSD) and Mukherjee's state-specific multireference CCSD (MkCCSD) methods are applied to four ring-opening reactions. The spin-restricted Hartree-Fock (RHF)-based CCSD (RHF-CCSD) calculations are also performed for comparison. In the case of the UHF-CCSD method, an approximate spin-projection (AP) method is applied to the broken-symmetry (BS) singlet solution to remove the spin contamination effect. For potential energy curves (PECs) of all reactions presented in this study, the results of RHF-CCSD and UHF-CCSD are substantially different from those of MkCCSD, while the results after the AP method (AP-UCCSD) reproduce the MkCCSD results well. It strongly suggests that the spin contamination effect should be removed by the AP correction even at the UHF-CCSD level to predict reliable energetics of these reactions.     

Implementation of the multireference Brillouin-Wigner and Mukherjee's coupled cluster methods with non-iterative triple excitations utilizing reference-level parallelism

In this paper we discuss the performance of the non-iterative state-specific multireference coupled cluster (SS-MRCC) methods accounting for the effect of triply excited cluster amplitudes. The corrections to the Brillouin-Wigner and Mukherjee's MRCC models based on the manifold of singly and doubly excited cluster amplitudes (BW-MRCCSD and Mk-MRCCSD, respectively) are tested and compared with exact full configuration interaction results for small systems (H(2)O, N(2), and Be(3)). For the larger systems (naphthyne isomers) the BW-MRCC and Mk-MRCC methods with iterative singles, doubles, and non-iterative triples (BW-MRCCSD(T) and Mk-MRCCSD(T)) are compared against the results obtained with single reference coupled cluster methods. We also report on the parallel performance of the non-iterative implementations based on the use of processor groups.     

Real or artifactual symmetry breaking in the BNB radical: a multireference coupled cluster viewpoint

The ground state of the linear BNB radical has been examined via the recently developed reduced multireference coupled cluster method with singles and doubles that is perturbatively corrected for triples [RMR CCSD(T)] using the correlation consistent basis sets (cc-pVXZ, X=D, T, and Q). Similar to earlier results that were based on the single reference CCSD(T) and BD(T) approaches, the RMR CCSD(T) method also predicts an asymmetric structure with two BN bonds of unequal length, even though the MR effects significantly reduce the barrier height. The computed frequencies for the symmetric and antisymmetric stretching modes agree reasonably well with the experimental data.     

Correction to constrained coupled cluster doubles models based on the second coupled cluster central moment

We develop a correction for the coupled cluster version of the perfect pairing (PP) model. The correction is based on finding modified values of the PP amplitudes such that the second coupled cluster central moment defined in the space of all valence single and double substitutions vanishes and, subject to this constraint, minimizing the deviation between the modified and unmodified PP amplitudes with respect to a chosen metric. We discuss how this correction can be generalized to other constrained doubles models, such as local correlation and active-space models. While the correction is not strictly size consistent and retains some of the deficiencies of the PP model, numerical results indicate that much of the missing active-space coupled cluster singles and doubles correlation energy is recovered.     

Local polarizabilities in molecules,based on ab initio Hartree-Fock calculations

A method to separate the total molecular polarizability, calculated in the uncoupled Hartree-Fock approximations, into local contributions is proposed. The method is tested for H₂, H₂O, H₂CO and C₆H₆ and the results are discussed. It is found that the ratio of the polarizability contributions for two atoms in a molecule almost only depends on the type of atoms and is almost independant of molecule.     

A cluster expansion theory with multireference functions using the unitary ansatz

A cluster expansion theory with multireference functions is developed. The method is applied to the ground state of H₂, the lowest two ¹∑⁺ states of LiH, and the lowest three ¹A₁ states of H₂O. Well-balanced potential curves are obtained for H₂ and LiH. Excitation energies are in good agreement with experiment.     

Locally correlated equation-of-motion coupled cluster theory for the excited states of large molecules

We report an extension of the local correlation concept to electronically excited states via the equation-of-motion coupled cluster singles and doubles (EOM-CCSD) method. We apply the same orbital domain structure used successfully for ground-state CCSD by Werner and co-workers and find that the resulting localized excitation energies are in error generally by less than 0.2 eV relative to their canonical EOM-CCSD counterparts, provided the basis set is flexible and includes Rydberg-like functions. In addition, we account for weak-pair contributions efficiently using a correction to local-EOM-CCSD transition energies based on the perturbative (D) correction used with configuration interaction singles (CIS).