Partially linearized, fully size-extensive, and reduced multireference coupled-cluster methods. II. Applications and performance

The partially linearized (pl), fully size-extensive multireference (MR) coupled-cluster (CC) method, fully accounting for singles (S) and doubles (D) and approximately for a subset of primary higher than doubles, referred to as plMR CCSD, as well as its plMR CCSD(T) version corrected for secondary triples, as described in Part I of this paper [X. Li and J. Paldus, J. Chem. Phys. 128, 144118 (2008)], are applied to the problem of bond breaking in the HF, F2, H2O, and N2 molecules, as well as to the H4 model, using basis sets of a DZ or a cc-pVDZ quality that enable a comparison with the full configuration interaction (FCI) exact energies for a given ab initio model. A comparison of the performance of the plMR CCSD/CCSD(T) approaches with those of the reduced MR (RMR) CCSD/CCSD(T) methods, as well as with the standard single reference (SR) CCSD and CCSD(T) methods, is made in each case. For the H4 model and N2 we also compare our results with the completely renormalized (CR) CC(2,3) method [P. Piecuch and M. W?och, J. Chem. Phys. 123, 224105 (2005)]. An important role of a proper choice of the model space for the MR-type methods is also addressed. The advantages and shortcomings of all these methods are pointed out and discussed, as well as their size-extensivity characteristics, in which case we distinguish supersystems involving noninteracting SR and MR subsystems from those involving only MR-type subsystems. Although the plMR-type approaches render fully size-extensive results, while the RMR CCSD may slightly violate this property, the latter method yields invariably superior results to the plMR CCSD ones and is more easy to apply in highly demanding cases, such as the triple-bond breaking in the nitrogen molecule.     

A truncated version of reduced multireference coupled-cluster method with singles and doubles and noniterative triples: application to F2 and Ni(CO)n (n=1, 2, and 4)

A perturbatively truncated version of the reduced multireference coupled-cluster method with singles and doubles and noniterative triples RMR CCSD(T) is described. In the standard RMR CCSD method, the effect of all triples and quadruples that are singles or doubles relative to references spanning a chosen multireference (MR) model space is accounted for via the external corrections based on the MR CISD wave function. In the full version of RMR CCSD(T), the remaining triples are then handled via perturbative corrections as in the standard, single-reference (SR) CCSD(T) method. By using a perturbative threshold in the selection of MR CISD configuration space, we arrive at the truncated version of RMR CCSD(T), in which the dimension of the MR CISD problem is significantly reduced, thus leaving more triples to be treated perturbatively. This significantly reduces the computational cost. We illustrate this approach on the F2 molecule, in which case the computational cost of the truncated version of RMR CCSD(T) is only about 10%-20% higher than that of the standard CCSD(T), while still eliminating the failure of CCSD(T) in the bond breaking region of geometries. To demonstrate the capabilities of the method, we have also used it to examine the structure and binding energy of transition metal complexes Ni(CO)n with n=1, 2, and 4. In particular, Ni(CO)2 is shown to be bent rather than linear, as implied by some earlier studies. The RMR CCSD(T) binding energy differs from the SR CCSD(T) one by 1-2 kcal/mol, while the energy barrier separating the linear and bent structures of Ni(CO)2 is smaller than 1 kcal/mol.     

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.     

Full potential energy curve for N2 by the reduced multireference coupled-cluster method

Relying on a 56-dimensional reference space and using up to the correlation-consistent, polarized, valence-quadruple-zeta (cc-pVQZ) basis sets, the reduced multireference (RMR) coupled-cluster method with singles and doubles (CCSD), as well as its perturbatively corrected version for secondary triples [RMR CCSD(T)], is employed to generate the full potential energy curves for the nitrogen molecule. The resulting potentials are then compared to the recently published accurate analytic potential based on an extensive experimental data analysis [R. J. Le Roy et al., J. Chem. Phys. 125, 164310 (2006)], and the vibrational term values of these potentials are compared over the entire well. A comparison with single-reference CCSD and CCSD(T) results, as well as with earlier obtained eight-reference RMR CC results, is also made. Excellent performance of RMR CCSD, and its systematic improvement with the increasing dimension of the reference space employed, is demonstrated. For the first 19 vibrationally excited levels, which are based on experimentally observed bands, we find an absolute average deviation of 8 cm(-1) from the computed RMR CCSD/cc-pVQZ values. The perturbative correction for triples increases this deviation to 126 cm(-1), but only to 61 cm(-1) when extrapolated to the basis set limit. Both RMR CCSD and RMR CCSD(T) potentials perform well when compared to the experiment-based analytic potential in the entire range of internuclear separations.     

Reduced multireference coupled-cluster method: barrier heights for heavy atom transfer, nucleophilic substitution, association, and unimolecular reactions

The recently developed reduced multireference coupled-cluster method with singles and doubles (RMR CCSD) that is perturtatively corrected for triples [RMR CCSD(T)] is employed to compute the forward and reverse barrier heights for 19 non-hydrogen-transfer reactions. The method represents an extension of the conventional single-reference (SR) CCSD(T) method to multireference situations. The results are compared with a benchmark database, which is essentially based on the SR CCSD(T) results. With the exception of seven cases, the RMR CCSD(T) results are almost identical with those based on SR CCSD(T), implying the abatement of MR effects at the SD(T) level relative to the SD level. Using the differences between the RMR CCSD(T) and CCSD(T) barrier heights as a measure of MR effects, modified values for barrier heights of studied reactions are given.     

Multireference coupled-cluster study of the symmetry breaking in the C2B radical

The potential energy surfaces (PESs) for both the ground and the excited electronic states of the C(2)B radical are investigated using various multireference (MR) coupled-cluster (CC) approaches. In the ground state case we employ the reduced MR (RMR) CC approach with singles (S) and doubles (D), the RMR CCSD method, as well as its RMR CCSD(T) version corrected for secondary triples, relying on various model spaces and basis sets. The reliability of this approach is also tested against the benchmark full configuration interaction results obtained for a small Dunning-Hay (DH) basis set. The results imply a clear preference for a cyclic structure which, however, breaks the C(2v) symmetry. This symmetry breaking manifests itself strongly at the level of the independent particle model, as represented by the restricted open-shell Hartree-Fock approximation, but the tendency toward symmetry breaking diminishes with the increasing size of the basis set employed as well as with the enhanced account of the correlation effects. It is likely to disappear in the complete basis set limit. The general model space CCSD method is then used to compute vertical excitation energies for a number of excited states as well as the cuts of the PES as the boron atom moves around the C(2) fragment. These results also explain why no symmetry breaking is found when relying on a spin contaminated unrestricted Hartree-Fock reference, as in the UMP2 method.     

Hybrid coupled cluster methods: combining active space coupled cluster methods with coupled cluster singles, doubles, and perturbative triples

Based on the coupled-cluster singles, doubles, and a hybrid treatment of triples (CCSD(T)-h) method developed by us [J. Shen, E. Xu, Z. Kou, and S. Li, J. Chem. Phys. 132, 114115 (2010); and ibid. 133, 234106 (2010); and ibid. 134, 044134 (2011)], we developed and implemented a new hybrid coupled cluster (CC) method, named CCSD(T)q-h, by combining CC singles and doubles, and active triples and quadruples (CCSDtq) with CCSD(T) to deal with the electronic structures of molecules with significant multireference character. These two hybrid CC methods can be solved with non-canonical and canonical MOs. With canonical MOs, the CCSD(T)-like equations in these two methods can be solved directly without iteration so that the storage of all triple excitation amplitudes can be avoided. A practical procedure to divide canonical MOs into active and inactive subsets is proposed. Numerical calculations demonstrated that CCSD(T)-h with canonical MOs can well reproduce the corresponding results obtained with non-canonical MOs. For three atom exchange reactions, we found that CCSD(T)-h can offer a significant improvement over the popular CCSD(T) method in describing the reaction barriers. For the bond-breaking processes in F(2) and H(2)O, our calculations demonstrated that CCSD(T)q-h is a good approximation to CCSDTQ over the entire bond dissociation processes.     

New coupled cluster approaches based on the unrestricted Hartree-Fock reference for treating molecules with multireference character

Here we review the basic formalism, implementation details, and performance of two newly developed coupled cluster (CC) methods based on the unrestricted Hartree-Fock (UHF) reference for treating molecules with multireference character. These two approaches can be considered to be approximations to the CC singles, doubles, and triples (CCSDT) method. The key concept of these two approaches is the corresponding orbitals, which are unitary transformations of canonical UHF molecular orbitals so that all spin orbitals are grouped into unique orbital pairs. In one approach called CCSDT(5P), a subset of triple excitations involving up to five-pair indices is included. In another approach called CCSD(T)-h, the contribution of connected triple excitations is treated in a hybrid way. With the concept of active corresponding orbitals, triple excitations can be automatically partitioned into two subsets, and the amplitudes of these two subsets are determined via solving different equations. Both CCSD(T)-h and CCSDT(5P) computationally scale as the seventh power of the system size. A survey of a number of applications demonstrates that CCSD(T)-h is an excellent approximation to the full CCSDT method, and CCSDT(5P) provides a good approximation to CCSDT for single-bond breaking processes. The overall performance of CCSDT(5P) is less accurate than that of CCSD(T)-h, but significantly better than that of the widely used CCSD(T).     

Binding in transition metal complexes: Reduced multireference coupled-cluster study of the MCH2+ (M=Sc to Cu) compounds

The recently developed reduced multireference coupled-cluster method with singles and doubles (RMR CCSD), which is perturbatively corrected for triples [RMR CCSD(T)], is employed to compute binding energies of nine transition metal ions with CH2. Unlike analogous compounds involving main-group elements, the MCH2+ (M=Sc to Cu) transition metal complexes often exhibit a non-negligible multireference character. The authors thus employ the RMR CCSD(T) method, which represents an extension of the standard single-reference (SR) CCSD(T) method and can account for multireference effects, while employing only small reference spaces. In this way the role of quasidegeneracy effects on the binding energies of these complexes can be assessed at a higher SD(T) level than is possible with the widely used ab initio methods, namely, with the standard SR CCSD(T) approach, and provide a new benchmark for these quantities. The difference between the RMR and the standard CCSD(T) methods becomes particularly evident when considering nonequilibrium geometries.     

A coupled cluster approach with a hybrid treatment of connected triple excitations: implementation and applications for open-shell systems

An implementation of the coupled cluster (CC) singles, doubles, and a hybrid treatment of connected triples [denoted as CCSD(T)-h], based on the unrestricted Hartree-Fock (UHF) reference, is presented. Based on the spin-integrated formulation, we have developed a computer program to achieve the automatic derivation and implementation of the CCSD(T)-h approach. The CCSD(T)-h approach computationally scales as the seventh power of the system size, and is affordable for many medium-sized systems. The present approach has been applied to study the equilibrium geometries and harmonic vibrational frequencies in a number of open-shell diatomic molecules and bond breaking potential energy profiles in several open-shell molecules, including CH(3), NH(2), and SiH(2). For all systems under study, the overall performance of the UHF-based CCSD(T)-h approach is very close to that of the corresponding CCSDT (CC singles, doubles, and triples), and much better than that of the UHF-based CCSD(T) (CC singles, doubles, and perturbative triples).     

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.     

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).     

The coupled cluster singles, doubles, and a hybrid treatment of connected triples based on the split virtual orbitals

We have proposed a simple strategy for splitting the virtual orbitals with a large basis set into two subgroups (active and inactive) by taking a smaller basis set as an auxiliary basis set. With the split virtual orbitals (SVOs), triple or higher excitations can be partitioned into active and inactive subgroups (according to the number of active virtual orbitals involved), which can be treated with different electron correlation methods. In this work, the coupled cluster (CC) singles, doubles, and a hybrid treatment of connected triples based on the SVO [denoted as SVO-CCSD(T)-h], has been implemented. The present approach has been applied to study the bond breaking potential energy surfaces in three molecules (HF, F(2), and N(2)), and the equilibrium properties in a number of open-shell diatomic molecules. For all systems under study, the SVO-CCSD(T)-h method based on the unrestricted Hartree-Fock (UHF) reference is an excellent approximation to the corresponding CCSDT (CC singles, doubles, and triples), and much better than the UHF-based CCSD(T) (CC singles, doubles, and perturbative triples). On the other hand, the SVO-CCSD(T)-h method based on the restricted HF (RHF) reference can also provide considerable improvement over the RHF-based CCSD(T).     

Variational formulation of perturbative explicitly-correlated coupled-cluster methods

We present a variational formulation of the recently-proposed CCSD(2)(R12) method [Valeev, Phys. Chem. Chem. Phys., 2008, 10, 106]. The centerpiece of this approach is the CCSD(2)(R12) Lagrangian obtained via L?wdin partitioning of the coupled-cluster singles and doubles (CCSD) Hamiltonian. Extremization of the Lagrangian yields the second-order basis set incompleteness correction for the CCSD energy. We also developed a simpler Hylleraas-type functional that only depends on one set of geminal amplitudes by applying screening approximations. This functional is used to develop a diagonal orbital-invariant version of the method in which the geminal amplitudes are fixed at the values determined by the first-order cusp conditions. Extension of the variational method to include perturbatively the effect of connected triples produces the method that approximates the complete basis-set limit of the standard CCSD plus perturbative triples [CCSD(T)] method. For a set of 20 small closed-shell molecules, the method recovered at least 94.5/97.3% of the CBS CCSD(T) correlation energy with the aug-cc-pVDZ/aug-cc-pVTZ orbital basis set. For 12 isogyric reactions involving these molecules, combining the aug-cc-pVTZ correlation energies with the aug-cc-pVQZ Hartree-Fock energies produces the electronic reaction energies with a mean absolute deviation of 1.4 kJ mol(-1) from the experimental values. The method has the same number of optimized parameters as the corresponding CCSD(T) model, does not require any modification of the coupled-cluster computer program, and only needs a small triple-zeta basis to match the precision of the considerably more expensive standard quintuple-zeta CCSD(T) computation.     

A comparative assessment of the perturbative and renormalized coupled cluster theories with a noniterative treatment of triple excitations for thermochemical kinetics, including a study of basis set and core correlation effects

The CCSD, CCSD(T), and CR-CC(2,3) coupled cluster methods, combined with five triple-zeta basis sets, namely, MG3S, aug-cc-pVTZ, aug-cc-pV(T+d)Z, aug-cc-pCVTZ, and aug-cc-pCV(T+d)Z, are tested against the DBH24 database of diverse reaction barrier heights. The calculations confirm that the inclusion of connected triple excitations is essential to achieving high accuracy for thermochemical kinetics. They show that various noniterative ways of incorporating connected triple excitations in coupled cluster theory, including the CCSD(T) approach, the full CR-CC(2,3) method, and approximate variants of CR-CC(2,3) similar to the triples corrections of the CCSD(2) approaches, are all about equally accurate for describing the effects of connected triply excited clusters in studies of activation barriers. The effect of freezing core electrons on the results of the CCSD, CCSD(T), and CR-CC(2,3) calculations for barrier heights is also examined. It is demonstrated that to include core correlation most reliably, a basis set including functions that correlate the core and that can treat core-valence correlation is required. On the other hand, the frozen-core approximation using valence-optimized basis sets that lead to relatively small computational costs of CCSD(T) and CR-CC(2,3) calculations can achieve almost as high accuracy as the analogous fully correlated calculations.     

Spin-Adapted restricted Hartree–Fock reference coupled-cluster theory for open-shell systems: Noniterative triples for noncanonical orbitals

The coupled clusters singles and doubles (CCSD ) method for calculations of open-shell systems with the single restricted Hartree–Fock (ROHF ) reference determinant is extended by the noniterative triples to give CCSD(T) . Our approach profits from the fact that (a) single- and double-excitation amplitudes are spin-adapted, which directly leads to a computationally less demanding algorithm than are nonadapted procedures and produces the spin-adapted CCSD wave function and (b) triple excitations calculated from converged spin-adapted (SA ) CCSD amplitudes are also obtained more effectively. Altogether, computer demands of our SA CCSD(T) approach, applicable to high-spin open-shell cases which are well represented by a single-determinant reference is comparable to that for closed-shell systems. Our approach is not based on semicanonical orbitals, applied by Bartlett's group. However, we compare some other possible choices of ROHF orbitals to this “standard.” Numerical results for a series of atoms and molecules demonstrate little sensitivity to this selection. © 1995 John Wiley & Sons, Inc.     

Explicitly correlated coupled-cluster theory using cusp conditions. II. Treatment of connected triple excitations

The coupled-cluster singles and doubles method augmented with single Slater-type correlation factors (CCSD-F12) determined by the cusp conditions (also denoted as SP ansatz) yields results close to the basis set limit with only small overhead compared to conventional CCSD. Quantitative calculations on many-electron systems, however, require to include the effect of connected triple excitations at least. In this contribution, the recently proposed [A. Ko?hn, J. Chem. Phys. 130, 131101 (2009)] extended SP ansatz and its application to the noniterative triples correction CCSD(T) is reviewed. The approach allows to include explicit correlation into connected triple excitations without introducing additional unknown parameters. The explicit expressions are presented and analyzed, and possible simplifications to arrive at a computationally efficient scheme are suggested. Numerical tests based on an implementation obtained by an automated approach are presented. Using a partial wave expansion for the neon atom, we can show that the proposed ansatz indeed leads to the expected (L(max)+1)(-7) convergence of the noniterative triples correction, where L(max) is the maximum angular momentum in the orbital expansion. Further results are reported for a test set of 29 molecules, employing Peterson's F12-optimized basis sets. We find that the customary approach of using the conventional noniterative triples correction on top of a CCSD-F12 calculation leads to significant basis set errors. This, however, is not always directly visible for total CCSD(T) energies due to fortuitous error compensation. The new approach offers a thoroughly explicitly correlated CCSD(T)-F12 method with improved basis set convergence of the triples contributions to both total and relative energies.     

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.     

Symmetric and asymmetric triple excitation corrections for the orbital-optimized coupled-cluster doubles method: improving upon CCSD(T) and CCSD(T)(Λ): preliminary application

Symmetric and asymmetric triple excitation corrections for the orbital-optimized coupled-cluster doubles (OO-CCD or simply "OD" for short) method are investigated. The conventional symmetric and asymmetric perturbative triples corrections [(T) and (T)(Λ)] are implemented, the latter one for the first time. Additionally, two new triples corrections, denoted as OD(Λ) and OD(Λ)(T), are introduced. We applied the new methods to potential energy surfaces of the BH, HF, C(2), N(2), and CH(4) molecules, and compare the errors in total energies, with respect to full configuration interaction, with those from the standard coupled-cluster singles and doubles (CCSD), with perturbative triples [CCSD(T)], and asymmetric triples correction (CCSD(T)(Λ)) methods. The CCSD(T) method fails badly at stretched geometries, the corresponding nonparallelity error is 7-281 kcal mol(-1), although it gives reliable results near equilibrium geometries. The new symmetric triples correction, CCSD(Λ), noticeably improves upon CCSD(T) (by 4-14 kcal mol(-1)) for BH, HF, and CH(4); however, its performance is worse than CCSD(T) (by 1.6-4.2 kcal mol(-1)) for C(2) and N(2). The asymmetric triples corrections, CCSD(T)(Λ) and CCSD(Λ)(T), perform remarkably better than CCSD(T) (by 5-18 kcal mol(-1)) for the BH, HF, and CH(4) molecules, while for C(2) and N(2) their results are similar to those of CCSD(T). Although the performance of CCSD and OD is similar, the situation is significantly different in the case of triples corrections, especially at stretched geometries. The OD(T) method improves upon CCSD(T) by 1-279 kcal mol(-1). The new symmetric triples correction, OD(Λ), enhances the OD(T) results (by 0.01-2.0 kcal mol(-1)) for BH, HF, and CH(4); however, its performance is worse than OD(T) (by 1.9-2.3 kcal mol(-1)) for C(2) and N(2). The asymmetric triples corrections, OD(T)(Λ) and OD(Λ)(T), perform better than OD(T) (by 2.0-6.2 kcal mol(-1)). The latter method is slightly better for the BH, HF, and CH(4) molecules. However, for C(2) and N(2) the new results are similar to those of OD(T). For the BH, HF, and CH(4) molecules, OD(Λ)(T) provides the best potential energy curves among the considered methods, while for C(2) and N(2) the OD(T) method prevails. Hence, for single-bond breaking the OD(Λ)(T) method appears to be superior, whereas for multiple-bond breaking the OD(T) method is better.