The closed‐shell coupled cluster theory restricted to single and double excitation operators (CCSD) is formulated in a basis of nonorthogonal local correlation functions. Excitations are made from localized molecular orbitals into subspaces (domains) of the local basis, which strongly reduces the number of amplitudes to be optimized. Furthermore, the correlation of distant electrons can be treated in a simplified way (e.g., by MP2) or entirely neglected. It is demonstrated for 20 molecules that the local correlation treatment recovers 98%–99% of the correlation energy obtained in the corresponding full CCSD calculation. Singles‐doubles configuration interac‐ tion (CISD), quadratic configuration interaction (QCISD), and Mo/ller–Plesset perturbation theory [MP2, MP3, MP4(SDQ)] are treated as special cases.

Local treatment of electron correlation in coupled cluster theory

A new implementation of local coupled-cluster theory with single and double excitations (LCCSD) is presented for which asymptotically all computational resources (CPU, memory, and disk) scale only linearly with the molecular size. This is achieved by: (i) restricting the correlation space for each electron pair to domains that are independent of molecular size; (ii) classifying the pairs according to a distance criterion and treating only strong pairs at the highest level; (iii) using efficient prescreening algorithms in the integral transformation and other integral-direct procedures; and (iv) neglect of small couplings of electron pairs that are far apart from each other. The errors caused by the various approximations are negligible. LCCSD calculations on molecules including up to 300 correlated electrons and over 1000 basis functions in C1 symmetry are reported, all carried out on a workstation.

Low-order scaling local electron correlation methods. IV. Linear scaling local coupled-cluster (LCCSD)

Conjugated Polymers and Aromaticity

This paper is the first in a series of papers on the new approach to the many-electron correlation problem, termed the method of moments of coupled-cluster equations (MMCC). A hierarchy of MMCC approximations, including the renormalized and completely renormalized CCSD[T], CCSD(T), CCSD(TQ), and CCSDT(Q) methods, which can be viewed as generalizations of the well-known perturbative coupled-cluster CCSD[T], CCSD(T), CCSD(TQf), and CCSDT(Qf) schemes, is introduced. In this initial study, an emphasis is placed on the ability of the MMCC approach to describe bond breaking and large effects due to connected triples and quadruples by modifying the standard noniterative CC approaches, such as the popular CCSD(T) method. The performance of selected MMCC approaches, including the renormalized and completely renormalized CCSD[T], CCSD(T), and CCSD(TQ) schemes, is illustrated by the results of pilot calculations for the HF and H2O molecules.

The method of moments of coupled-cluster equations and the renormalized CCSD[T], CCSD(T), CCSD(TQ), and CCSDT(Q) approaches

A critical assessment of coupled cluster method in quantum chemistry

In nondegenerate systems, the tetraexcited clusters are well approximated by products of disconnected pair clusters and the connected quadruply excited component is negligible. In contrast, when the reference state becomes quasidegenerate with the lowest biexcited configuration(s), the connected quadruply excited clusters become very important. To extend the applicability of the coupled-pair many-electron theory to such situations, we approximate the connected tetraexcited contribution in the form suggested by the unrestricted Hartree-Fock-type wave function, or one of its projected versions, such as the alternant molecular-orbital method. We show that the incorporation of the connected quadruply excited clusters into the coupled-pair equations effectively cancels certain nonlinear terms, originating from disconnected quadruple excitations, so that the resulting equations are very similar (up to a numerical factor) to the approximate coupled-pair theory, in which only those nonlinear terms which factorize with respect to the hole-electron pairs are considered. This fact shows in turn why various approximate coupled-pair approaches can often provide better results than the full coupled-pair many-electron theory.

Approximate account of the connected quadruply excited clusters in the coupled-pair many-electron theory

The role of electron correlation effects on the bond-length alternation in linear metalliclike systems, as modeled by cyclic polyenes CNHN, N = 2n = 4v + 2, v = 1,2,…, is examined using the coupled cluster approach in the localized Wannier basis formalism. A recently developed approximate coupled pair approach which accounts for connected quadruply excited clusters is employed together with various truncation schemes for the localized doubly-excited cluster components. It is found that for the physical value of the coupling constant, the electron correlation has only a very slight effect on the potential energy curves, yielding almost the same values for both the magnitude of the bond-length alternation and for the stabilization energy relative to the symmetric equidistant structures as the restricted Hartree-Fock theory. This is in contrast to a strongly correlated region where the correlation effects stabilize the undistorted non-alternating structures. Different mechanisms of the bond-length alternation or Peierl's distortion as implied by a simple Huckel Hamiltonian and by the Pariser-Parr-Pople Hamiltonian models are also pointed out.

Bond length alternation in cyclic polyenes. VI. Coupled cluster approach with wannier orbital basis

Problems which arise in the application of closed-shell coupled-cluster approaches to quasidegenerate or almost degenerate situations are discussed and the basic classification of quasidegeneracy types is outlined. Recent coupled-cluster results obtained for the cyclic polyene model, particularly in the strongly correlated limit, are briefly discussed and the unexpected features of approximate and localized coupled-pair approaches are pointed out.

Degeneracy and coupled-cluster approaches

The problem of bond length alternation in linear extended systems with conjugated double bonds is examined on a simple cyclic polyene model using finite-order many-body perturbation theory. Three different partitionings of the model Hamiltonian are employed, namely the Huckel, Moller–Plesset, and Epstein–Nesbet partitionings. The dependence of correlation energy on bond length alternation is examined in each case, showing an almost constant behavior of Moller–Plesset and Epstein–Nesbet perturbation energies in contrast to a strong dependence on distortion, favoring undistorted structures, for the Huckel perturbation and UHF correlation energies. The origin of an unphysical character of the Huckel perturbation theory and the inappropriateness of the UHF approach for the problem considered are pointed out. The second- and third-order Moller–Plesset and also the second-order Epstein–Nesbet perturbation theories yield results which are similar to those obtained with the RHF method and which clearly favor the bond length alternating structures, leading to the bond length distortion of about 0.045 A and to the stabilization energy per site (relative to the equidistant geometry) of about 0.03 eV.

Bond length alternation in cyclic polyenes. V: Local finite-order perturbation theory approach

The applicability of the finite-order many-body perturbation theory to the electron correlation problem in extended one-dimensional systems is examined. The cyclic polyenes CNHN, N = 4ν + 2, ν = 1, 2, …, with the DNh geometry as described by both the Pariser–Parr–Pople and Hubbard Hamiltonians, are employed to model the metallic-like one-dimensional systems. The second-order perturbation theory contributions to the correlation energy are obtained with three different partitionings of the Hamiltonian (Huckel, M⊘ller–Plesset, and Epstein–Nesbet). The third- and fourth-order contributions are also calculated in special cases. A comparison with other methods is given wherever available. For the Hubbard Hamiltonian the asymptotic behavior of the perturbation theory expansion is examined numerically. It is shown that the finite-order perturbation expansion can provide reliable results for the correlation energy of one-dimensional systems even in the correlation region which corresponds to the spectroscopically determined physical value of the coupling constant.

M. Takahashi

Papers

Approximate account of the connected quadruply excited clusters in the coupled-pair many-electron theory

Bond length alternation in cyclic polyenes. VI. Coupled cluster approach with wannier orbital basis

Degeneracy and coupled-cluster approaches

Bond length alternation in cyclic polyenes. V: Local finite-order perturbation theory approach

Perturbation theory and electron correlation in extended systems: Cyclic polyene model