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The Journal of Chemical Physics

Application of projection techniques to the electron correlation problem

K. W. Becker, Peter Fulde

Published Oct 1, 1989• 46 citations
Publisher

Abstract

By applying projection techniques a unified description of weakly and strongly correlated electronic systems can be given. For that purpose the theory is formulated in Liouville instead of Hilbert space. By introducing an appropriate metric in terms of cumulants it is ensured that all energies are size consistent. An energy expression is derived which is a generalization of Goldstone’s linked cluster theorem for arbitrary Hamiltonians. It is suitable for applying the projection method of Zwanzig and Mori. The equations for the correlation energy within the Local Ansatz and within coupled electron pair approximation CEPA-2 follow straightforwardly from the present theory.

DOI
10.1063/1.456801

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By applying projection techniques a unified description of weakly and strongly correlated electronic systems can be given. For that purpose the theory is formulated in Liouville instead of Hilbert space. By introducing an appropriate metric in terms of cumulants it is ensured that all energies are size consistent. An energy expression is derived which is a generalization of Goldstone’s linked cluster theorem for arbitrary Hamiltonians. It is suitable for applying the projection method of Zwanzig and Mori. The equations for the correlation energy within the Local Ansatz and within coupled electron pair approximation CEPA-2 follow straightforwardly from the present theory.
Abstract Recently, it has been shown that the ground‐state energy of a quantum many‐body system can be written in terms of cumulants. In this article, we show that the energies of excited states can be expressed similarly. These representations are suitable for various approximation schemes, e.g., projection techniques. The explicit use of cumulants ensures size consistency in all approximations. The theory is then applied to the computation of the energy bands of a semiconductor with a diamond structure. As we focus especially on the effects of electron correlations, we consider a simplified model and rederive results obtained previously by the variational local ansatz method. © 1994 John Wiley & Sons, Inc.
Cumulants represent a natural language for expressing macroscopic properties of many-body systems. The most important property of cumulants is that of size consistency, i.e. a cumulant expression for an extensive variable scales with the size of the system, independent of possible further approximations used in the evaluation procedure. Cumulants can be considered as a generalization of linked diagrams known from diagrammatic technique of many-body theory. In this paper we outline a recently introduced method based on cumulants in order to derive expressions for zero-temperature properties of many-particle systems, i.e. the ground-state energy, static expectation values and dynamical correlation functions. This cumulant formalism allows one to describe weakly and strongly correlated systems along the same lines. We show that the coupled-cluster method known from quantum chemistry can be derived from our cumulant approach. Finally, we demonstrate the usefulness of the cumulant method by applying it to examples from solid-state physics and quantum chemistry.
Projected Hartree-Fock theory provides an accurate description of many kinds of strong correlations but does not properly describe weakly correlated systems. On the other hand, single-reference methods, such as configuration interaction or coupled cluster theory, can handle weakly correlated problems but cannot properly account for strong correlations. Ideally, we would like to combine these techniques in a symmetry-projected coupled cluster approach, but this is far from straightforward. In this work, we provide an alternative formulation to identify the so-called disentangled cluster operators, which arise when we combine these two methodological strands. Our formulation shows promising results for model systems and small molecules.
For a system consisting of two interacting subsystems, effective Hamiltonians for one subsystem are usually constructed by integrating out the degrees of freedom of the second subsystem. In contrast to usual treatments, our method for deriving effective Hamiltonians is based on the introduction of cumulants. Cumulants guarantee size consistency, a property that is not always evident in other approaches. The formalism applies for any temperature. Our method is based on a matrix formulation in the framework of a recently introduced cumulant-projection method. This method allows the treatment of strongly correlated systems, where conventional diagrammatic techniques are difficult to apply. The relation with perturbation theory is also shown. As a simple application we discuss the small polaron problem. Evaluating the one-electron dispersion relation in the strong-coupling case shows good agreement with recent results from exact Lanczos diagonalization. Our method has substantial advantages over perturbation theory.
Standard approaches to approximating solutions to the Schroedinger equation rely upon the assumption that there is a single dominant electronic configuration with a small number of low-lying excited states. As such, they are referred to as single reference approaches and are accurate only when the band gap is small in comparison with the inter-electronic repulsion of the valence electrons. Systems for which this is the case are weakly-correlated. If the inter-electronic repulsion larger than the band gap, then a single electronic reference is not an appropriate ansatz for the physical wavefunction. As the strength of the interaction is increased many Slater determinants become near-degenerate, each providing non-negligible contributions. Such systems are strongly-correlated. The goal of the thesis was to employ wavefunctions which are: non-trivial, tractable, size-consistent and extensive, and not given in an active space. We have outlined approaches based on wavefunction forms more suited to describe strong correlation. Built on the Lie algebra su(2), our first trial wavefunction was the Bethe ansatz solution to the reduced BCS Hamiltonian derived by Richardson. This wavefunction is expressed as a product of quasi-particles each parametrized in terms of quasi-momenta. Consistency among the quasi-momenta is ensured by a set of non-linear equations to be solved numerically. This approach is feasible in both variational and projected flavours. We next considered a wavefunction of the same form without requiring it to be an eigenvector of the reduced BCS Hamiltonian. A variational approach in this case is only feasible provided there are two equivalent descriptions in terms of particles and holes. In any case, a bivariational principle and a projected approach are both feasible. Also based on su(2), there is AP1roG (coupled cluster pair doubles). Open-shell singlets and triplets were constructed with the Lie algebras sp(N) and so(4N). The resulting formulae appear to be infeasible unless it is possible to efficiently compute generalizations of determinants to three-dimensional objects. Treating each spatial orbital individually, all four possible occupations close the Lie superalgebra gl(2|2).