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

Extending spin-symmetry projected coupled-cluster to large model spaces using an iterative null-space projection technique

Recently, we introduced an orbital-invariant approximate coupled-cluster (CC) method in the spin-projection manifold. The multi-determinantal property of spin-projection means that the parametrization in the spin-extended CC (ECC) ansatz is nonorthogonal and overcomplete. Therefore, the linear dependencies must be removed by an orthogonalization procedure to obtain meaningful solutions. Multi-reference methods often achieve this by diagonalizing a metric of the equation system, but this is not feasible with ECC because of the enormous size of the metric, a consequence of the incomplete active space of the spin-projected Hartree-Fock reference. As a result, the applicability of ECC has been limited to small benchmark systems, for which the ansatz was shown to be superior to the configuration interaction and linearized approximations. In this article, we provide a solution to this problem that completely avoids the metric diagonalization by iteratively projecting out its null-space from the working equations. As the additional computational cost required for this iterative projection is only marginal, it greatly expands the application range of ECC. We demonstrate the potential of approximate ECC by studying the complete basis set limit of F₂ and transition metal complexes such as NiO, Mn₂ , and [Cu₂ O₂ ]²⁺ , which have all been hindered by the prohibitively large metric size. We also identify the potential inadequacy of the molecular orbitals given by spin-projected Hartree-Fock in some cases, and propose possible solutions. © 2018 Wiley Periodicals, Inc.

Orbital-invariant spin-extended approximate coupled-cluster for multi-reference systems

We present an approximate treatment of spin-extended coupled-cluster (ECC) based on the spin-projection of the broken-symmetry coupled-cluster (CC) ansatz. ECC completely eliminates the spin-contamination of unrestricted CC and is therefore expected to provide better descriptions of dynamical and static correlation effects, but introduces two distinct problems. The first issue is the emergence of non-terminating amplitude equations, which are caused by the de-excitation effects inherent in symmetry projection operators. In this study, we take a minimalist approach and truncate the Taylor series of the exponential ansatz at a certain order such that the approximation safely recovers the traditional CC without spin-projection. The second issue is that the nonlinear equations of ECC become underdetermined, although consistent, yielding an infinitude of solutions. This problem arises because of the redundancies in the excitation manifold, as is common in other multi-reference approaches. We remove the linear dependencies in ECC by employing an orthogonal projection manifold. We also propose an efficient solver for our method, in which the components are usually sparse but not diagonal-dominant. It is shown that our approach is rigorously orbital-invariant and provides more accurate results than its configuration interaction and linearized CC analogues for chemical systems.

Iterative universal state selective correction for the Brillouin-Wigner multireference coupled-cluster theory

As a further development of the previously introduced a posteriori Universal State-Selective (USS) corrections [K. Kowalski, J. Chem. Phys. 134, 194107 (2011); J. Brabec et al., ibid. 136, 124102 (2012)], we suggest an iterative form of the USS correction by means of correcting effective Hamiltonian matrix elements. We also formulate USS corrections via the left Bloch equations. The convergence of the USS corrections with excitation level towards the full configuration interaction (FCI) limit is also investigated. Various forms of the USS and simplified diagonal USS corrections at the singles and doubles and perturbative triple levels are numerically assessed on several model systems and on the ozone and tetramethyleneethane molecules. It is shown that the iterative USS correction can successfully replace the previously developed a posteriori Brillouin-Wigner coupled cluster size-extensivity correction, while it is not sensitive to intruder states and performs well also in other cases when the a posteriori one fails, like, e.g., for the asymmetric vibration mode of ozone.

REDUCED MULTIREFERENCE COUPLED-CLUSTER METHOD AND ITS APPLICATION TO THE PYRIDYNE DIRADICALS

The reduced multireference (RMR) coupled-cluster (CC) method with singles and doubles (RMR CCSD) that employs a modest-size MR CISD wave function as an external source for the most important (primary) triples and quadruples in order to account for the nondynamic correlation effects in the presence of quasidegeneracy, and which is further perturbatively corrected for the remaining (secondary) triples, RMR CCSD(T), is employed to compute the molecular geometry and the energy of the lowest-lying singlet and triplet states, as well as the corresponding singlet–triplet splitting, for all possible isomers of the m, n-pyridyne diradicals. A comparison is made with earlier results that were obtained by other authors, and the role of the multireference effects for both the geometry and the spin multiplicity of the lowest state, as described by the RMR-type methods, is demonstrated on the example of 2,6- and 3,5-pyridynes.

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.

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.

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.

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.

Multireference Brillouin-Wigner coupled clusters method with noniterative perturbative connected triples

We developed and implemented the state-specific Brillouin-Wigner coupled cluster method with singles, doubles, and noniterative perturbative triples, called MR BWCCSD(T), for a general number of closed- and open-shell reference configurations. To assess the accuracy of the method, we performed calculations of the three lowest electronic states of the oxygen molecule and of the automerization barrier of cyclobutadiene. For the oxygen molecule, the results were in a good agreement in comparison with those of the iterative MR BWCCSDTalpha method. For cyclobutadiene, the effect of connected triples was found to be minor, which is in agreement with the previous study by and Balková and Bartlett [J. Chem. Phys. 101, 8972 (1994)].

A Multireference Coupled-Cluster Potential Energy Surface of Diazomethane, CH2N2

The intrinsically multireference dissociation of the C-N bond in ground-state diazomethane (CH(2)N(2)) at different angles has been studied with the multireference Brillouin-Wigner coupled-cluster singles and doubles (MRBWCCSD) method. The morphology of the calculated potential energy surface (PES) in C(s)() symmetry is similar to a multireference perturbational (CASPT3) PES. The MRBWCCSD/cc-pVTZ H(2)C-N(2) dissociation energy with respect to the asymptotic CH(2)(ã(1)A(1)) + N(2)(X(1)Sigma(g)(+)) products is D(e) = 35.9 kcal/mol, or a zero-point corrected D(0) = 21.4 kcal/mol with respect to the ground-state CH(2)(X(3)B(1)) + N(2)(X(1)Sigma(g)(+)) fragments.