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.     

Coupling term derivation and general implementation of state-specific multireference coupled cluster theories

Simple closed-form expressions are derived for the "same vacuum" renormalization terms that arise in state-specific multireference coupled cluster (MRCC) theories. Explicit equations are provided for these coupling terms through the triple excitation level of MRCC theory, and a general expression is included for arbitrary-order excitations. The first production-level code (PSIMRCC) for state-specific and rigorously size-extensive Mukherjee multireference coupled cluster singles and doubles (MkCCSD) computations has been written. This code is also capable of evaluating analogous Brillouin-Wigner multireference energies (BWCCSD), including a posteriori size-extensivity corrections. Using correlation-consistent basis sets (cc-pVXZ, X=D,T,Q), MkCCSD and BWCCSD were tested and compared on two classic multireference problems: (1) the dissociation potential curve of molecular fluorine (F(2)) and (2) the structure and vibrational frequencies of ozone. Comparison with experimental data shows that the Mukherjee method is generally superior to the Brillouin-Wigner theory in predicting energies, structures, and vibrational frequencies. Particularly accurate results for F(2) are obtained by applying the MkCCSD method with localized molecular orbitals. Although the MkCCSD theory greatly improves upon single-reference CCSD for the geometric parameters and a(1) vibrational frequencies of ozone, the antisymmetric stretching frequency omega(3)(b(2)) remains pathological and cannot be properly treated without the inclusion of connected triple excitations. Finally, preliminary multireference MkCCSD results are reported for the singlet-triplet splittings in ortho-, meta-, and para-benzyne, coming within 1.5 kcal mol(-1) of experiment in all cases.     

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.     

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.     

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 singlet-triplet gap in trimethylenmethane and the ring-opening of methylenecyclopropane: a multireference Brillouin-Wigner coupled cluster study

We performed an ab initio study of the singlet-triplet gap in trimethylenmethane (TMM) and of the ring-opening of methylenecyclopropane by the multireference BWCC method. Since the singlet states of TMM and intermediates between TMM and methylenecyclopropane have a strong multiconfigurational character, it is necessary to use a multireference method. The cc-pVDZ and cc-pVTZ basis sets were used. We compared our results with experiments, where available, and with previous calculations performed by MCSCF and spin-flip coupled-cluster-type methods.     

Towards the multireference Brillouin-Wigner coupled-clusters method with iterative connected triples: MR BWCCSDT-alpha approximation

We developed and implemented an approximation of the state-specific Brillouin-Wigner coupled-cluster method with singles, doubles, and triples, called MRBWCCSDT-alpha, for a general number of closed- and open-shell reference configurations. The accuracy of the method is assessed on the calculation of the oxygen molecule in the X3sigma(g-), a1delta(g), and b1sigma(g+) states and the results of this multireference treatment are compared with previous MRBWCCSD results and with those obtained by the doubly ionized similarity transformed equation-of-motion CCSD and multireference configuration interaction methods and with experimental spectroscopic data. Explicit tests of the size-extensivity of the MRBWCCSDT-alpha method with iterative size-extensivity correction are also performed.     

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.     

A state-specific partially internally contracted multireference coupled cluster approach

A state-specific partially internally contracted multireference coupled cluster approach is presented for general complete active spaces with arbitrary number of active electrons. The dominant dynamical correlation is included via an exponential parametrization of internally contracted cluster operators ( ?T) which excite electrons from a multideterminantal reference function. The remaining dynamical correlation and relaxation effects are included via a diagonalization of the transformed Hamiltonian ?? =e(- ?T)H?e( ?T) in the multireference configuration interaction singles space in an uncontracted fashion. A new set of residual equations for determining the internally contracted cluster amplitudes is proposed. The second quantized matrix elements of ?? , expressed using the extended normal ordering of Kutzelnigg and Mukherjee, are used as the residual equations without projection onto the excited configurations. These residual equations, referred to as the many-body residuals, do not have any near-singularity and thus, should allow one to solve all the amplitudes without discarding any. There are some relatively minor remaining convergence issues that may arise from an attempt to solve all the amplitudes and an initial analysis is provided in this paper. Applications to the bond-stretching potential energy surfaces for N(2), CO, and the low-lying electronic states of C(2) indicate clear improvements of the results using the many-body residuals over the conventional projected residual equations.     

Comparison of low-order multireference many-body perturbation theories

Tests have been made to benchmark and assess the relative accuracies of low-order multireference perturbation theories as compared to coupled cluster (CC) and full configuration interaction (FCI) methods. Test calculations include the ground and some excited states of the Be, H(2), BeH(2), CH(2), and SiH(2) systems. Comparisons with FCI and CC calculations show that in most cases the effective valence shell Hamiltonian (H(v)) method is more accurate than other low-order multireference perturbation theories, although none of the perturbative methods is as accurate as the CC approximations. We also briefly discuss some of the basic differences among the multireference perturbation theories considered in this work.     

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.     

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.     

Electronic structure of spherical quantum dots using coupled cluster method

2, 6, 12, and 20 electron quantum dots have been studied using coupled cluster at singles and doubles level and extensive multireference coupled cluster (MRCC) method. A Fock-space version of MRCC (FSMRCC) containing single hole-particle excited determinants has been used to calculate low-lying excited states of the above system. The ionization potential and electron affinity are also calculated. The effect of correlation energy on excitation energy and charge density is shown by calculating them at the high density region (low value of density parameter rs) and at the low density region (high value of density parameter rs).     

Orbitally invariant internally contracted multireference unitary coupled cluster theory and its perturbative approximation: theory and test calculations of second order approximation

A unitary wave operator, exp (G), G(+) = -G, is considered to transform a multiconfigurational reference wave function Φ to the potentially exact, within basis set limit, wave function Ψ = exp (G)Φ. To obtain a useful approximation, the Hausdorff expansion of the similarity transformed effective Hamiltonian, exp (-G)Hexp (G), is truncated at second order and the excitation manifold is limited; an additional separate perturbation approximation can also be made. In the perturbation approximation, which we refer to as multireference unitary second-order perturbation theory (MRUPT2), the Hamiltonian operator in the highest order commutator is approximated by a Mo?ller-Plesset-type one-body zero-order Hamiltonian. If a complete active space self-consistent field wave function is used as reference, then the energy is invariant under orbital rotations within the inactive, active, and virtual orbital subspaces for both the second-order unitary coupled cluster method and its perturbative approximation. Furthermore, the redundancies of the excitation operators are addressed in a novel way, which is potentially more efficient compared to the usual full diagonalization of the metric of the excited configurations. Despite the loss of rigorous size-extensivity possibly due to the use of a variational approach rather than a projective one in the solution of the amplitudes, test calculations show that the size-extensivity errors are very small. Compared to other internally contracted multireference perturbation theories, MRUPT2 only needs reduced density matrices up to three-body even with a non-complete active space reference wave function when two-body excitations within the active orbital subspace are involved in the wave operator, exp (G). Both the coupled cluster and perturbation theory variants are amenable to large, incomplete model spaces. Applications to some widely studied model systems that can be problematic because of geometry dependent quasidegeneracy, H4, P4, and BeH(2), are performed in order to test the new methods on problems where full configuration interaction results are available.     

An orbital-invariant internally contracted multireference coupled cluster approach

We have formulated and implemented an internally contracted multireference coupled cluster (ic-MRCC) approach aimed at solving two of the problems encountered in methods based on the Jeziorski-Monkhorst ansatz: (i) the scaling of the computational and memory costs with respect to the number of references, and (ii) the lack of invariance of the energy with respect to rotations among active orbitals. The ic-MRCC approach is based on a straightforward generalization of the single-reference coupled cluster ansatz in which an exponential operator is applied to a multiconfigurational wave function. The ic-MRCC method truncated to single and double excitations (ic-MRCCSD) yields very accurate potential energy curves in benchmark computations on the Be + H(2) insertion reaction, the dissociation of hydrogen fluoride, and the symmetric double dissociation of water. Approximations of the ic-MRCC theory in which the Baker-Campbell-Hausdorff expansion is truncated up to a given number of commutators are found to converge quickly to the full theory. In our tests, two commutators are sufficient to recover a total energy within 0.5 mE(h) of the full ic-MRCCSD method along the entire potential energy curve. A formal analysis shows that the ic-MRCC method is invariant with respect to rotation among active orbitals, and that the orthogonalization procedure used to produce the set of linearly independent excitation operators plays a crucial role in guaranteeing the invariance properties. The orbital invariance was confirmed in numerical tests. Moreover, approximated versions of the ic-MRCC theory based on a truncated Baker-Campbell-Hausdorff expansion, preserve the orbital invariance properties of the full theory.     

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