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).     

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).     

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).     

SCF and localized orbitals in ethylene: MBPT/CC results and comparison with one-million configuration Cl

Many-body perturbation theory (MBPT) and coupled-cluster (CC) calculations are performed on the ethylene molecule employing canonical SCF and simple bond-orbital localized orbitals (LO). Full fourth-order MBPT [i.e. SDTQ MBPT(4)], CC doubles (CCD) and CC singles and doubles (CCSD) energies are compared with the over one-million configuration ‘bench-mark” Cl calculation of Saxe et al. Though the SCF and LO reference determinant energies differ by 0.29706 hartree, the CCSD energy difference is only 1.7 mhartrees (mh). Our most extensive SCF orbital calculation, CCSD plus fourth-order triples, is found to be lower in energy than the CI result by 5.3 mh.     

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.     

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.     

Calculation of current densities using gauge-including atomic orbitals

A method for calculating the various components of the magnetically induced current-density tensor using gauge-including atomic orbitals is described. The method is formulated in the framework of analytical derivative theory, thus enabling implementation at the Hartree-Fock self-consistent-field (HF-SCF) as well as at electron-correlated levels. First-order induced current densities have been computed up to the coupled-cluster singles and doubles level (CCSD) augmented by a perturbative treatment of triple excitations [CCSD(T)] for carbon dioxide and benzene and up to the full coupled-cluster singles, doubles, and triples (CCSDT) level in the case of ozone. The applicability of the gauge including magnetically induced current method to larger molecules is demonstrated by computing first-order current densities for porphin and hexabenzocoronene at the HF-SCF and density-functional theory level. Furthermore, a scheme for obtaining quantitative values for the induced currents in a molecule via numerical integration over the current flow is presented. For benzene, a perpendicular magnetic field induces a (field dependent) ring current of 12.8 nA T(-1) at the HF-SCF level using a triple-zeta basis set augmented with polarization functions (TZP). At the CCSD(T)/TZP level the induced current was found to be 11.4 nA T(-1). Gauge invariance and its relation to charge-current conservation is discussed.     

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.     

Partially linearized, fully size-extensive, and reduced multireference coupled-cluster methods. I. Formalism and mutual relationship

We describe a fully size-extensive alternative of the reduced multireference (RMR) coupled-cluster (CC) method with singles (S) and doubles (D) that generates a subset of higher-than-pair cluster amplitudes, using linearized CC equations from the full CC chain, projected onto the corresponding higher-than-doubly excited configurations. This approach is referred to as partially linearized (pl) MR CCSD method and characterized by the acronym plMR CCSD. In contrast to a similar CCSDT-1 method [Y. S. Lee et al., J. Chem. Phys. 81, 5906 (1984)] this approach also considers higher than triples (currently up to hexuples), while focusing only on a small subset of such amplitudes, referred to as the primary ones. These amplitudes are selected using similar criteria as in RMR CCSD. An extension considering secondary triples via the standard (T)-type corrections, resulting in the plMR CCSD(T) method, is also considered. The relationship of RMR and plMR CCSD and CCSD(T) approaches is discussed, and their performance and characteristics are the subject of the subsequent Part II of this paper.     

State-of-the-art computations of dipole moments using analytic gradients of high-level density-fitted coupled-cluster methods with focal-point approximations

Using the analytic derivatives approach, dipole moments of high-level density-fitted coupled-cluster (CC) methods, such as coupled-cluster singles and doubles (CCSD), and coupled-cluster singles and doubles with perturbative triples [CCSD(T)], are presented. To obtain the high accuracy results, the computed dipole moments are extrapolated to the complete basis set (CBS) limits applying focal-point approximations. Dipole moments of the CC methods considered are compared with the experimental gas-phase values, as well as with the common DFT functionals, such as B3LYP, BP86, M06-2X, and BLYP. For all test sets considered, the CCSD(T) method provides substantial improvements over Hartree–Fock (HF), by 0.076–0.213 D, and its mean absolute errors are lower than 0.06 D. Furthermore, our results indicate that even though the performances of the common DFT functionals considered are significantly better than that of HF, their results are not comparable with the CC methods. Our results demonstrate that the CCSD(T)/CBS level of theory provides highly-accurate dipole moments, and its quality approaching the experimental results. © 2019 Wiley Periodicals, Inc.     

Quantum chemical study of the electronic structure of NiCH2 + in its ground state and low-lying electronic excited states

The electronic structure of NiCH(2) (+), representative of transition metal carbene ions, is investigated by means of several methods of quantum chemistry. The relative stabilities of the four low-lying doublet electronic states ((2)A(1), (2)A(2), (2)B(1), and (2)B(2)) are determined at the coupled cluster singles and doubles level (CCSD) and triples level [CCSD(T) and CCSDT-3] with both a Hartree-Fock and density functional theory (Kohn-Sham) reference. The equation-of-motion coupled cluster for treatment of excited states in singles and doubles approximation (EOM-CCSD) is used to characterize the transition energies from the (2)A(1) electronic ground state to the low-lying doublet excited states. The (2)A(2) and (2)B(1) states are nearly degenerate, found to be separated by 940 cm(-1) at the EOM-CCSD level, in agreement with the CASSCF energy ordering. The (2)B(2) state is calculated to be higher in energy by more than 1.0 eV. The spin purity of the low-lying doublet and quadruplet states described by CCSD calculations based on the unrestricted open-shell Hartree-Fock reference is discussed.     

Calculated electronic transitions of the water ammonia complex

We have calculated vertical excitation energies and oscillator strengths of the low lying electronic transitions in H2O, NH3, and H2ONH3 using a hierarchy of coupled cluster response functions [coupled cluster singles (CCS), second order approximate coupled cluster singles and doubles (CC2), coupled cluster singles and doubles (CCSD), and third order approximate coupled cluster singles, doubles, and triples (CC3)] and correlation consistent basis functions (n-aug-cc-pVXZ, where n=s,d,t and X=D,T,Q). Our calculations indicate that significant changes in the absorption spectra of the photodissociative states of H2O and NH3 monomers occur upon complexation. In particular, we find that the electronic transitions originating from NH3 are blueshifted, whereas the electronic transitions originating from H2O are redshifted.     

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.     

Closed-shell coupled-cluster theory with spin-orbit coupling

A two-component closed-shell coupled-cluster (CC) approach using relativistic effective core potentials with spin-orbit coupling included in the post-Hartree-Fock treatment is proposed and implemented at the CC singles and doubles (CCSD) level as well as at the CCSD level augmented by a perturbative treatment of triple excitations [CCSD(T)]. The latter invokes as an additional approximation the neglect of the occupied-occupied and virtual-virtual blocks of the spin-orbit coupling matrix in order to avoid the iterative N(7) steps in the treatment of triple excitations. The computational effort of the implemented two-component CC methods is about 10-15 times that of its corresponding nonrelativistic counterpart, which needs to be compared to the by a factor of 32 higher cost for fully relativistic schemes and schemes with spin-orbit coupling included already at the Hartree-Fock self-consistent field (HF-SCF) level. This substantial computational saving is due to the use of real molecular orbitals and real two-electron integrals. Results on 5p-, 6p-, and 7p-block element compounds show that the bond lengths and harmonic frequencies obtained with the present two-component CCSD method agree well with those computed with the CCSD approach including spin-orbit coupling at the HF-SCF level even for the 7p-block element compounds. As for the CCSD(T) approach, high accuracy for 5p- and 6p-block element compounds is retained. However, the difference in bond lengths and harmonic frequencies becomes somewhat more pronounced for the 7p-block element compounds.     

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.     

The permanent dipole moment of gas-phase para-amino benzoic acid revisited

The permanent dipole moment of para-amino benzoic acid has been calculated at various theoretical levels, including Hartree-Fock, second-order M?ller-Plesset perturbation (MP2), coupled cluster singles and doubles (CCSD), and triples corrections CCSD(T), and hybrid density functional theory at B3LYP level. It is found that the B3LYP method fails to provide correct results for the geometry and the permanent dipole moment. These results are significantly improved by MP2 calculations. Our best estimated dipole moments obtained at CCSD and CCSD(T) levels are in good agreement with experiment.     

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.     

Benchmarking NMR indirect nuclear spin-spin coupling constants: SOPPA, SOPPA(CC2), and SOPPA(CCSD) versus CCSD

Accurate calculations of NMR indirect nuclear spin-spin coupling constants require especially optimized basis sets and correlated wave function methods such as CCSD or SOPPA(CCSD). Both methods scale as N(6), where N is the number of orbitals, which prevents routine applications to molecules with more than 10-15 nonhydrogen atoms. We have therefore developed a modification of the SOPPA(CCSD) method in which the CCSD singles and doubles amplitudes are replaced by CC2 singles and doubles amplitudes. This new method, called SOPPA(CC2), scales only as N(5), like the original SOPPA-method. The performance of the SOPPA(CC2) method for the calculation of indirect nuclear spin-spin coupling constants is compared to SOPPA and SOPPA(CCSD) employing a set of benchmark molecules. We also investigate the basis set dependence by employing three different basis sets optimized for spin-spin coupling constants, namely the HuzIV-su4, ccJ-pVTZ, and ccJ-pVQZ basis sets. The results of the corresponding CCSD calculations are used as a theoretical reference.     

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.