Extensive generalization of renormalized coupled-cluster methods

The recently developed completely renormalized (CR) coupled-cluster (CC) methods with singles, doubles, and noniterative triples or triples and quadruples [CR-CCSD(T) or CR-CCSD(TQ), respectively], which are based on the method of moments of CC equations (MMCC) [K. Kowalski and P. Piecuch, J. Chem. Phys. 113, 18 (2000)], eliminate the failures of the standard CCSD(T) and CCSD(TQ) methods at larger internuclear separations, but they are not rigorously size extensive. Although the departure from strict size extensivity of the CR-CCSD(T) and CR-CCSD(TQ) methods is small, it is important to examine the possibility of formulating the improved CR-CC methods, which are as effective in breaking chemical bonds as the existing CR-CCSD(T) and CR-CCSD(TQ) approaches, which are as easy to use as the CR-CCSD(T) and CR-CCSD(TQ) methods, and which can be made rigorously size extensive. This may be particularly useful for the applications of CR-CC methods and other MMCC approaches in calculations of potential energy surfaces of large many-electron systems and van der Waals molecules, where the additive separability of energies in the noninteracting limit is very important. In this paper, we propose different types of CR-CC approximations, termed the locally renormalized (LR) CCSD(T) and CCSD(TQ) methods, which become rigorously size extensive if the orbitals are localized on nointeracting fragments. The LR-CCSD(T) and LR-CCSD(TQ) methods rely on the form of the energy expression in terms of the generalized moments of CC equations, derived in this work, termed the numerator-denominator-connected MMCC expansion. The size extensivity and excellent performance of the LR-CCSD(T) and LR-CCSD(TQ) methods are illustrated numerically by showing the results for the dimers of stretched HF and LiH molecules and bond breaking in HF and H2O.     

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.     

Breaking bonds with the left eigenstate completely renormalized coupled-cluster method

The recently developed [P. Piecuch and M. Wloch, J. Chem. Phys. 123, 224105 (2005)] 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 CCL, is compared with the full configuration interaction (FCI) method for all possible types of single bond-breaking reactions between C, H, Si, and Cl (except H2) and the H2Si[Double Bond]SiH2 double bond-breaking reaction. The CCL method is in excellent agreement with FCI in the entire region R=1-3Re for all of the studied single bond-breaking reactions, where R and Re are the bond distance and the equilibrium bond length, respectively. The CCL method recovers the FCI results to within approximately 1 mhartree in the region R=1-3Re of the H-SiH3, H-Cl, H3Si-SiH3, Cl-CH3, H-CH3, and H3C-SiH3 bonds. The maximum errors are -2.1, 1.6, and 1.6 mhartree in the R=1-3Re region of the H3C-CH3, Cl-Cl, and H3Si-Cl bonds, respectively, while the discrepancy for the H2Si[Double Bond]SiH2 double bond-breaking reaction is 6.6 (8.5) mhartree at R=2(3)Re. CCL also predicts more accurate relative energies than the conventional CCSD and CCSD(T) approaches, and the predecessor of CR-CC(2,3) termed CR-CCSD(T).     

Extension of the renormalized coupled-cluster methods exploiting left eigenstates of the similarity-transformed hamiltonian to open-shell systems: a benchmark study

The recently formulated completely renormalized coupled-cluster method with singles, doubles, and noniterative triples, exploiting the biorthogonal form of the method of moments of coupled-cluster equations (Piecuch, P.; W?och, M. J. Chem. Phys. 2005, 123, 224105; Piecuch, P.; W?och, M.; Gour, J. R.; Kinal, A. Chem. Phys. Lett. 2006, 418, 467), termed CR-CC(2,3), is extended to open-shell systems. Test calculations for bond breaking in the OH radical and the F2+ ion and singlet-triplet gaps in the CH2, HHeH, and (HFH)- biradical systems indicate that the CR-CC(2,3) approach employing the restricted open-shell Hartree--Fock (ROHF) reference is significantly more accurate than the widely used CCSD(T) method and other noniterative triples coupled-cluster approximations without making the calculations substantially more expensive. A few molecular examples, including the activation energies of the C2H4 + H --> C2H5 forward and reverse reactions and the triplet states of the CH2 and H2Si2O2 biradicals, are used to show that the dependence of the ROHF-based CR-CC(2,3) energies on the method of canonicalization of the ROHF orbitals is, for all practical purposes, negligible.     

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.     

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).     

Potential energy curve for isomerization of N2H2 and C2H4 using the improved virtual orbital multireference Møller-Plesset perturbation theory

Multireference M?ller-Plesset (MRMP) perturbation theory [K. Hirao, Chem. Phys. Lett. 190, 374 (1992)] is modified to use improved virtual orbitals (IVOs) and is applied to study ground state potential energy curves for isomerization and dissociation of the N2H2 and C2H4 molecules. In contrast to traditional MRMP or multistate multiconfiguration quasidegenerate perturbation theory where the reference functions are obtained from (often difficult to converge) state averaged multiconfiguration self-consistent field methods, our reference functions are represented in terms of computationally efficient IVOs. For convenience in comparisons with other methods, a first order complete active space configuration interaction (CASCI) calculation with the IVOs is followed by the use of the IVOs in MRMP to incorporate residual electron correlation effects. The potential energy curves calculated from the IVO-MRMP method are compared with computations using state-of-the-art coupled cluster singles and doubles (CCSD) methods and variants thereof to assess the efficacy of the IVO-MRMP scheme. The present study clearly demonstrates that unlike the CCSD and its variants, the IVO-MRMP approach provides smooth and reliable ground state potential energy curves for isomerization of these systems. Although the rigorously size-extensive completely renormalized CC theory with noniterative triples corrections (CR-CC(2,3)) likewise provides relatively smooth curves, the CR-CC(2,3) calculations overestimate the cis-trans barrier height for N2H2. The ground state spectroscopic constants predicted by the IVO-CASCI method agree well with experiment and with other highly correlated ab initio methods.     

Second- and third-order triples and quadruples corrections to coupled-cluster singles and doubles in the ground and excited states

Second- and third-order perturbation corrections to equation-of-motion coupled-cluster singles and doubles (EOM-CCSD) incorporating excited configurations in the space of triples [EOM-CCSD(2)T and (3)T] or in the space of triples and quadruples [EOM-CCSD(2)TQ] have been implemented. Their ground-state counterparts--third-order corrections to coupled-cluster singles and doubles (CCSD) in the space of triples [CCSD(3)T] or in the space of triples and quadruples [CCSD(3)TQ]--have also been implemented and assessed. It has been shown that a straightforward application of the Rayleigh-Schrodinger perturbation theory leads to perturbation corrections to total energies of excited states that lack the correct size dependence. Approximations have been introduced to the perturbation corrections to arrive at EOM-CCSD(2)T, (3)T, and (2)TQ that provide size-intensive excitation energies at a noniterative O(n(7)), O(n(8)), and O(n(9)) cost (n is the number of orbitals) and CCSD(3)T and (3)TQ size-extensive total energies at a noniterative O(n(8)) and O(n(10)) cost. All the implementations are parallel executable, applicable to open and closed shells, and take into account spin and real Abelian point-group symmetries. For excited states, they form a systematically more accurate series, CCSD1 eV) and the ground-state wave function has single-determinant character. In other cases, however, the corrections tend to overestimate the triples and quadruples effects, the origin of which is discussed. For ground states, the third-order corrections lead to a rather small improvement over the highly effective second-order corrections [CCSD(2)T and (2)TQ], which is a manifestation of the staircase convergence of perturbation series.     

Coupled-cluster methods including noniterative corrections for quadruple excitations

A new method is presented for treating the effects of quadruple excitations in coupled-cluster theory. In the approach, quadruple excitation contributions are computed from a formula based on a non-Hermitian perturbation theory analogous to that used previously to justify the usual noniterative triples correction used in the coupled cluster singles and doubles method with a perturbative treatment of the triple excitations (CCSD(T)). The method discussed in this paper plays a parallel role in improving energies obtained with the full coupled-cluster singles, doubles, and triples method (CCSDT) by adding a perturbative treatment of the quadruple excitations (CCSDT(Q)). The method is tested for an extensive set of examples, and is shown to provide total energies that compare favorably with those obtained with the full singles, doubles, triples, and quadruples (CCSDTQ) method.     

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).     

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.     

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.     

Linear response coupled cluster singles and doubles approach with modified spectral resolution of the similarity transformed Hamiltonian

This paper discusses practical scheme for correcting the linear response coupled cluster with singles and doubles (CCSD) equations by shifting the poles corresponding to the equation-of-motion CCSD excitation energies by adding noniterative corrections due to triples. A simple criterion is derived for the excited states to be corrected in the spectral resolution of similarity transformed Hamiltonian on the CCSD level. Benchmark calculations were performed to compare the accuracies of static and dynamic polarizabilities obtained in this way with the CC3 and CCSDT counterparts.     

Coupled-cluster dynamic polarizabilities including triple excitations

Dynamic polarizabilities for open- and closed-shell molecules were obtained by using coupled-cluster (CC) linear response theory with full treatment of singles, doubles, and triples (CCSDT-LR) with large basis sets utilizing the NWChem software suite. By using four approximate CC methods in conjunction with augmented cc-pVNZ basis sets, we are able to evaluate the convergence in both many-electron and one-electron spaces. For systems with primarily dynamic correlation, the results for CC3 and CCSDT are almost indistinguishable. For systems with significant static correlation, the CC3 tends to overestimate the triples contribution, while the PS(T) approximation [J. Chem. Phys. 127, 164105 (2007)] produces mixed results that are heavily dependent on the accuracies provided by noniterative approaches used to correct the equation-of-motion CCSD excitation energies. Our results for open-shell systems show that the choice of reference (restricted open-shell Hartree-Fock versus unrestricted Hartree-Fock) can have a significant impact on the accuracy of polarizabilities. A simple extrapolation based on pentuple-zeta CCSD calculations and triple-zeta CCSDT calculations reproduces experimental results with good precision in most cases.     

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.     

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).