Orbital-optimized coupled-cluster theory does not reproduce the full configuration-interaction limit

It is shown that due to the mixing of the usual projection approach of coupled cluster with variational orbital optimization, orbital-optimized coupled cluster (OCC) fails to reproduce the full configuration-interaction (full CI) limit when the cluster operator becomes complete. It is pointed out that the fulfillment of the projected singles equations, which define the orbital gradient in Brueckner coupled cluster (BCC), is mandatory for a correct behavior. As numerical examples we present general OCC and BCC calculations up to the full CI limit on CH(2) and an active-space model of ozone. The observed deviations of OCC from full CI are of the order of the correlation error obtained in calculations with up to quadruples excitations. Thus the failure of OCC may be considered tolerable in more approximate calculations but clearly prohibitive for any benchmark application. For applications to active-space models a hybrid approach for OCC is suggested in which for active particle-hole rotations the Brueckner orbital gradient is employed, whereas for the remaining orbital rotations the variational orbital gradient is retained.     

The role of orbital transformations in coupled-pair functionals

The replacement of single excitations by orbital transformations in coupled-pair functionals derived from a single double configuration interaction approach is discussed. It is demonstrated that this modification leads to considerably improved density matrices and better agreement with results from coupled cluster singles doubles calculations taken as a reference. A comparison between the variationally optimized orbitals and the Brueckner orbitals shows that these two sets of orbitals are different.     

Investigation of a localised second-order Brueckner correlation method

Local second-order Brueckner correlation potentials have been derived from their non-local counterparts by starting from the assumption that the orbitals generated by these potentials are the same. The structure of the local correlation potentials and its components have been analysed for the neon atom and a range of small molecules, namely HF, HCl, H(2)O, CO and ethyne. The orbitals from the local Brueckner correlation potentials yield first-order electric molecular properties which are close to those inferred from second-order M?ller-Plesset theory and Brueckner coupled cluster doubles with perturbative triples.     

The optimization of molecular orbitals for coupled cluster wavefunctions

The purpose of this work is to make the coupled cluster (CC) energy stationary with respect to molecular-orbital (MO) variations in the reference configuration. To achieve this, we have used the Z vector, the solution of a set of perturbation-independent CPHF-like equations, to rotate the MOs. A new energy and gradient calculation is carried out with these non-SCF orbitals to obtain a new Z vector. The process is repeated until the orbitals are optimized (Z = 0), i.e. the contribution to the analytic CC gradient coming from orbital relaxation (CPHF) is zero. At the CCD level the orbitals thus obtained are approximate Brueckner orbitals. At the CCSD level, convergence problems were found in the iterative procedure to optimize the orbitals. Results obtained for several molecules show that CCD wavefunctions constructed from these optimized orbitals are of CCSD quality. We conclude that the presence of exp(t₁) in the CCSD model accounts for most relaxation effects and there is not much to gain by orbital optimization in CCSD wavefunctions.     

Symmetry breaking in the cyclic C(3)C(2)H radical

We have employed high-level coupled cluster methods including connected triple excitations to study the possibility of symmetry-breaking in the (2)B(2) ground state of the c-C(3)C(2)H radical. Specifically, we find that spin-restricted open-shell Hartree-Fock (ROHF) reference orbitals yield a C(2v) structure, whereas spin-unrestricted Hartree-Fock (UHF) and Brueckner orbitals lead to a symmetry-broken C(s) minimum-energy geometry. Equation-of-motion coupled cluster singles and doubles method for ionized states yields a C(s) structure with a double-zeta basis set, but not with a triple-zeta basis set. Through a detailed analysis of the orbital instability/near-instability behavior of each type of Hartree-Fock reference, we have determined that the UHF reference wave function is more reliable than the ROHF reference in this case and that the Born-Oppenheimer potential surface for c-C(3)C(2)H exhibits a symmetry broken C(s) global minimum. This result is supported by excited-state computations, which indicate that a second-order (pseudo) Jahn-Teller interaction is responsible for the symmetry-breaking.     

Electron densities from the Brueckner Doubles method

Summary The concept of the Brueckner orbital is examined, following a resurgence of interest in wavefunctions constructed from them. The distinction between Self Consistent Field, Natural and Brueckner orbitals are discussed. Total electron densities are calculated for several examples, and correlation densities are studied. It is found that the Brueckner orbitals are more localised than SCF orbitals. The total electron density constructed from the Brueckner reference determinant with Brueckner orbitals gives qualitatively similar pictures as other correlated methods. Brueckner orbitals are found to show dissociation well.     

On the connections between Brueckner–coupled-cluster,density-dependent Hartree–Fock,and density functional theory

The existence of an effective one-particle Hamiltonian in the Brueckner coupled cluster model naturally leads to the definition of an effective interaction G, which is a function of the T₂ amplitudes. Two types of approximations to G are proposed: One is purely phenomenological, while the other is based on approximations to the Brueckner T₂ equation. In both cases, the resulting effective interaction may be viewed as electron-density-dependent. Generalizing Hartree–Fock theory to accommodate density-dependent interactions (DDHF ), a method is obtained that is capable of accounting for correlation effects in an independent particle framework. The heuristic Skyrme force, successfully used in nuclear physics to model nucleon–nucleon interactions, is presented here as an example of an effective electron–electron correlation interaction. Due to the δ-function character of the Skyrme force, it is possible to express the energy in this model by an integral over an energy density, thus formally providing a connection between DDHF and density functional theory for this particular case. An approximation to the Brueckner T₂ equation is also proposed in the coordinate representation. In this model, the density-matrix dependence of T₂ is reduced to a nonlocal electron density dependence by means of an expansion which introduces terms that depend on the gradient of the density. The first term in this expansion amounts to a “local density approximation” to Brueckner coupled cluster theory. © 1995 John Wiley & Sons, Inc.     

Toward accurate thermochemical models for transition metals: G3Large basis sets for atoms Sc-Zn

An augmented valence triple-zeta basis set, referred to as G3Large, is reported for the first-row transition metal elements Sc through Zn. The basis set is constructed in a manner similar to the G3Large basis set developed previously for other elements (H-Ar, K, Ca, Ga-Kr) and used as a key component in Gaussian-3 theory. It is based on a contraction of a set of 15s13p5d Gaussian primitives to 8s7p3d, and also includes sets of f and g polarization functions, diffuse spd functions, and core df polarization functions. The basis set is evaluated with triples-augmented coupled cluster [CCSD(T)] and Brueckner orbital [BD(T)] methods for a small test set involving energies of atoms, atomic ions, and diatomic hydrides. It performs well for the low-lying s-->d excitation energies of atoms, atomic ionization energies, and the dissociation energies of the diatomic hydrides. The Brueckner orbital-based BD(T) method performs substantially better than Hartree-Fock-based CCSD(T) for molecules such as NiH, where the starting unrestricted Hartree-Fock wavefunction suffers from a high degree of spin contamination. Comparison with available data for geometries of transition metal hydrides also shows good agreement. A smaller basis set without core polarization functions, G3MP2Large, is also defined.     

On the Presumptive Similarity of Kohn–Sham and Brueckner Orbitals

For states of many-electron systems disclosing various degrees of quasidegeneracy, we have carried out comparative studies of Kohn–Sham orbitals (KSO) generated for several xc-potentials, Brueckner orbitals (BO) represented by the Brueckner-coupled cluster orbitals, and Hartree–Fock (HF) orbitals by means of criteria directly related to the orbital structure which are based on relative distance indices for various pairs of equidimensional subspaces defined by the KSO and BO basis sets. We have found that both for weak and strong quasidegeneracy there are systems for which the KSO–BO distances are larger than the BO–HF ones. For strongly quasidegenerate states it is found that the distance indices are the largest for hybrid potentials, and that the subspaces spanned by KSOs are closer to those spanned by HF orbitals than by BOs. Hence, our results do not support the recently formulated expectations concerning the similarity of Brueckner orbitals and Kohn–Sham orbitals, including those corresponding to purely local exchange-correlation potentials.     

Orbital invariance issue in multireference methods

The orbital invariance problem is analyzed from the tensor theory point of view, with an emphasis on multireference coupled cluster methods. Using the transformation properties of second‐quantized operators, we discuss the orbital invariance properties of various methods by examining the tensor properties of the residual equations. A simple self‐consistency‐checking algorithm is proposed. We first establish the orbital invariance properties for the Hartree–Fock, single reference configuration interaction, single reference coupled cluster, complete‐active‐space self‐consistent‐field, and multireference configuration interaction methods, and then discuss the invariance properties of the complete‐active‐space coupled cluster and CCSDt methods. Finally, we demonstrate theoretically the lack of orbital invariance for Jeziorski–Monkhorst ansatz based methods. It appears necessary to modify the ansatz to achieve orbital invariance, and internal contraction serves as one possible solution. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2010     

Molecular properties from coupled-cluster Brueckner orbitals

We investigate to which extent a single determinant made up from orbitals obtained by a Brueckner coupled-cluster doubles calculation is able to reproduce correlated one-electron properties. It is shown that dipole and quadrupole moments and radial expectation values compare quite well with BCCD finite-field results for a test selection of nine molecules enclosing HF, H₂O, NH₃, CO, N₂, NO⁺, HCN, CuH and CH₃OH and three rare-gas atoms He, Ne and Ar. Furthermore, we find that even second-order properties such as dipole and quadrupole polarizabilities are reproduced fairly well when determined as first derivatives of the corresponding Brueckner orbital expectation values.     

Correlated one-particle method: numerical results

In a previous paper a correlated one-particle method was formulated, where the effective Hamiltonian was composed of the Fock operator and a correlation potential. The objective was to define a correlated one-particle theory that would give all properties that can be obtained from a one-particle theory. The Fock-space coupled-cluster method was used to construct the infinite-order correlation potential, which yields correct ionization potentials (IP's) and electron affinities (EA's) as the negative of the eigenvalues. The model, however, was largely independent of orbital choice. To exploit the degree of freedom of improving the orbitals, the Brillouin-Brueckner condition is imposed, which leads to an effective Brueckner Hamiltonian. To assess its numerical properties, the effective Brueckner Hamiltonian is approximated through second order in perturbation. Its eigenvalues are the negative of IP's and EA's correct through second order, and its eigenfunctions are second-order Brueckner orbitals. We also give expressions for its energy and density matrix. Different partitioning schemes of the Hamiltonian are used and the intruder state problem is discussed. The results for ionization potentials, electron affinities, dipole moments, energies, and potential curves are given for some sample molecules.     

Localized optimized orbitals, coupled cluster theory, and chiroptical response properties

The impact of orbital localization on the efficiency and accuracy of the optimized-orbital coupled cluster model is examined for the prediction of chiroptical properties, in particular optical rotation. The specific rotations of several test cases-(P)-[4]triangulane, (S)-1-phenylethanol, and chiral conformers of 1-fluoropentane, heptane, and nonane-were computed using an approach in which localization is enforced throughout the orbital optimization and subsequent linear response computation. This method provides a robust local-correlation scheme for future production-level implementation. Although the cross-over point between the canonical and localized coupled cluster approach lies at larger molecules than for ground-state energies, the scheme presented should still provide reduced scaling sufficient to investigate much larger molecules than are presently accessible.     

Approximate variational coupled cluster theory

We show that it is possible to construct an accurate approximation to the variational coupled cluster method, limited to double substitutions, from the minimization of a functional that is rigorously extensive, exact for isolated two-electron subsystems and invariant to transformations of the underlying orbital basis. This approximate variational coupled cluster theory is a modification and enhancement of our earlier linked pair functional theory. It is first motivated by the constraint that the inverse square root of the matrix that transforms the cluster amplitudes must exist. Low-order corrections are then included to enhance the accuracy of the approximation of variational coupled cluster, while ensuring that the computational complexity of the method never exceeds that of the standard traditional coupled cluster method. The effects of single excitations are included by energy minimization with respect to the orbitals defining the reference wavefunction. The resulting quantum chemical method is demonstrated to be a robust approach to the calculation of molecular electronic structure and performs well when static correlation effects are strong.     

Brueckner doubles coupled cluster method with the polarizable continuum model of solvation

We present the theory and implementation for computing the (free) energy and its analytical gradients with the Brueckner doubles (BD) coupled cluster method in solution, in combination with the polarizable continuum model of solvation (PCM). The complete model, called PTED, and an efficient approximation, called PTE, are introduced and tested with numerical examples. Implementation details are also discussed. A comparison with the coupled-cluster singles and doubles CCSD-PCM-PTED and CCSD-PCM-PTE schemes, which use Hartree-Fock (HF) orbitals, is presented. The results show that the two PTED approaches are mostly equivalent, while BD-PCM-PTE is shown to be superior to the corresponding CCSD scheme when the HF reference wave function is unstable. The BD-PCM-PTE scheme, whose computational cost is equivalent to gas phase BD, is therefore a promising approach to study molecular systems with complicated electronic structure in solution.     

Application of double ionization state-specific equation of motion coupled cluster method to organic diradicals

The state-specific equation of motion coupled cluster method is applied to three systems of diradical character: automerization of cyclobutadiene, singlet-triplet gaps of trimethylmethylene, and Bergman reaction. The aim of the paper is to assess the performance of the method and test numerically the importance of orbital optimization, three-body terms in transformed Hamiltonian, and the choice of cluster equations.     

Development of a relaxation-inducing cluster expansion formalism for treating strong relaxation and correlation effects

We present in this paper a comprehensive account of an explicitly spin-free coupled cluster theory for treating energy differences of open-shell states relative to a closed-shell ground state, where the open-shell states of interest are dominated by a few simple configuration state functions. We develop a valence-universal coupled cluster formalism to achieve this via a novel cluster expansion ansatz for the valence part of the wave operator, where the orbital relaxation and the correlation relaxation accompanying ionization/excitation from the ground state are taken care of to all orders in compact, efficient, and explicitly spin-free manner. The essential difference of our proposed ansatz from the ordinary and the normal-ordered cluster ansatz in vogue is that (a) we allow the valence cluster operators to be connected among themselves with spectator valence lines only and (b) we use suitable combinatoric factors accompanying powers of cluster operators thus connected, which are equal to the number of ways the operators can be joined, leading to the same excitation (the automorphic factor). We emphasize that such an ansatz does not generate terms (diagrams) with chains of cluster operators joined among themselves via spectator lines only. Barring only a few, almost all the terms in the working equations determining the cluster amplitudes involve contraction of the Hamiltonian with the cluster operators via at least one nonspectator line, leading to what we call a "strongly connected" series. The structure of the working equation is remarkably similar to the single-reference closed-shell equation, with a few additional terms. The presence of contractions among cluster operators via spectator lines introduces the additional physical effects of orbital and correlation relaxation using low-body cluster operators. As an illustrative application of the new multireference coupled cluster (CC) theory, we consider in this paper computation of ionization potentials (IPs) of one-valence problem with only one active orbital. The numerical applications are made for both the core- and the inner- and outer-valence IPs for several molecular systems. The numerical values demonstrate the superiority of the relaxation-inducing CC theory, as compared to the normal-ordered ansatz.     

The structures of m-benzyne and tetrafluoro-m-benzyne

The structures of m-benzyne and its fluorinated derivative, tetrafluoro-m-benzyne, were investigated using coupled cluster methods including triple excitations [CCSD(T) and CCSDT], different reference wave functions (spin-restricted Hartree-Fock, spin-unrestricted Hartree-Fock, and Brueckner), and different basis sets [6-31G(d,p) and correlation-consistent valence triple-zeta (cc-pVTZ)]. The inclusion of triple excitations in conjunction with d- and f-type polarization functions is paramount to correctly describe through-bond delocalization of the monocyclic form. At the highest level of theory, the C1-C3 distance of the minimum energy form of m-benzyne is 2.0 A and the profile of the potential energy surface along the C1-C3 distance is that of an asymmetric, single well, in agreement with previous density-functional theory and coupled cluster studies. In addition, the calculated CCSD(T) fundamental frequencies are in excellent agreement with the measured infrared frequencies, thus confirming the monocyclic form of m-benzyne. For tetrafluoro-m-benzyne, however, the increased eclipsing strain between the ring-external C-X bonds stabilizes the bicyclo[3.1.0]hexatriene form: the C1-C3 distance is calculated at the CCSD(T)/cc-pVTZ level to be approximately 1.75 A, which is in the range of elongated CC bonds. Computed harmonic vibrational frequencies compare reasonably well with the experimental neon-matrix difference spectrum and provide further evidence for the existence of a bicyclic form.     

An orbital-invariant and strictly size extensive post-Hartree-Fock correlation functional

A strictly size extensive post-Hartree-Fock correlation functional being invariant with respect to orbital transformations within the occupied and virtual subspaces is presented. While avoiding the necessity to solve additional Z vector equations for the calculation of properties and energy gradients, this functional reproduces almost exactly the results of coupled-cluster singles doubles (CCSD) calculations. In particular, it is demonstrated that the method is rigorous in the sense that it can be systematically improved by the perturbative inclusion of triple excitations in the same way as CCSD. As to the computational cost, the presented approach is somewhat more expensive than the CCSD if the energy is variationally optimized with respect to both the orbitals and the excitation amplitudes. Replacement of orbital optimization by the Brueckner condition reduces the computational cost by a factor of two, thus making the method less expensive than CCSD.     

The electronic spectrum of SiH4: Jahn-Teller Rydberg series

The aim of the present theoretical work is to provide data necessary for a better understanding of the electronic spectrum of the silane molecule, which is affected by the Jahn-Teller effect. By selecting an adequate distorted C(2v) geometry of SiH(4), the three lower Koopmans ionization potentials are evaluated with the equation of motion coupled cluster of singles and doubles method. Vertical excitation energies for the different Rydberg series converging to the three Jahn-Teller components are inferred from ab initio coupled cluster linear response calculations. Absorption oscillator strengths for dipole-allowed electronic transitions are also determined with the molecular-adapted quantum defect orbital methodology. Predictions of new spectroscopic data on SiH(4) are reported.