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Molecular Physics

Electron correlation and quantum electrodynamics

Ingvar Lindgren

Published May 1, 1998• 15 citations
Publisher

Abstract

A review is given of the coupled-cluster approach for a multi-reference model space. Various schemes of normalization are discussed, particularly the hermitian formulation. Relativistic many-body schemes are analysed, starting from the no-virtual-pair approximation (NVPA). E ects beyond NVPA are discussed in the framework of QED, and in particular the QED e ects on the electron correlation for He-like ions are analysed. 1.

DOI
10.1080/00268979809482291

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A review is given of the coupled-cluster approach for a multi-reference model space. Various schemes of normalization are discussed, particularly the hermitian formulation. Relativistic many-body schemes are analysed, starting from the no-virtual-pair approximation (NVPA). E ects beyond NVPA are discussed in the framework of QED, and in particular the QED e ects on the electron correlation for He-like ions are analysed. 1.
The coupled cluster [or exp(S)] formulation of quantum many-body theory is flrst extensively reviewed. It is then applied to the general problem of pairing correlations within a many-fermion system. The method is illustrated by restricting ourselves to the exact calculations that can be performed within the various generalized ladder approximations. It is shown how the formalism provides an efficient and unified framework in which to describe all aspects of pairing at the same level of approximation. Possible extensions to deal with composite clusters of more than two particles and to incorporate the effects of the collective excitations, are sketched
In this paper we review the coupled cluster formulation of quantum many-body theory.Apart from a brief and heuristic, but essentially complete, discussion of 8 the by now rather standard tr-eatments of both ground and excited states within the 2 coupled cluster method (CeM), our main aim is to.give a rather careful critique of the various choices that have to be made in order to implement .themethod.These lS--•divide essentially into two classes.On the one hand we have the inputs to the method, 11 which amount largely to.achoice.ofstarting (or zero-order or model or uncorrelated) wayefunctions; while on the other hand we have the choice of decomposition of the ~equations into a natural hierarchy, and which amounts in practice to a particular approximation or truncati on scheme.We shall show expl tc+t ly .thatwhil e these two 13 ---choices are in principle independent of each other, in practice they are intimately 14
Many-body perturbation theories (MBPT) (both single and multireference), by design, are capable of representing electron correlation very well. However, the practical implementation of these methods for many-electron systems beyond a certain order is extremely difficult, if not impossible. For instance, there are hardly any calculations beyond third order at the multireference and fifth order at the single reference of perturbation theories. Thus, these methods are typically limited to systems where the higher-order electron correlation effect is not significant. A practical solution to this problem is the coupled cluster theory, which was first introduced by Coester [20] and Coester and Kümmel [21] in nuclear physics and was later adopted in the electron structure theory by Čížek [23, 106] and Paldus and Čížek [107]. The coupled-cluster (CC) theory is a non-perturbative all-order theory. This stateof-the-art theory has now been recognized as the basis of modern quantitative quantum chemistry and atomic physics for high-precision calculations. Its superior accuracy is rooted in the exponential ansatz for the wave function, which provides an energy expression that scales correctly with the number of electrons and under certain provisions leads to a faster convergence of the many-body expansion.
The recently introduced coupled cluster (CC) downfolding techniques for reducing the dimensionality of quantum many-body problems recast the CC formalism in the form of the renormalization procedure allowing, for the construction of effective (or downfolded) Hamiltonians in small-dimensionality sub-space, usually identified with the so-called active space, of the entire Hilbert space. The resulting downfolded Hamiltonians integrate out the external (out-of-active-space) Fermionic degrees of freedom from the internal (in-the-active-space) parameters of the wave function, which can be determined as components of the eigenvectors of the downfolded Hamiltonians in the active space. This paper will discuss the extension of non-Hermitian (associated with standard CC formulations) and Hermitian (associated with the unitary CC approaches) downfolding formulations to composite quantum systems. The non-Hermitian formulation can provide a platform for developing local CC approaches, while the Hermitian one can serve as an ideal foundation for developing various quantum computing applications based on the limited quantum resources. We also discuss the algorithm for extracting the semi-analytical form of the inter-electron interactions in the active spaces.
Some recent advances in the area of multi-reference coupled-cluster theory of the state-universal type are overviewed. An emphasis is placed on the following new developments: (i) the idea of combining the state-universal multi-reference coupled-cluster singles and doubles method (SUMRCCSD) with the multi-reference many-body perturbation theory (MRMBPT), in which cluster amplitudes of the SUMRCCSD formalism that carry only core and virtual orbital indices are replaced by their first-order MRMBPT estimates; and (ii) the idea of combining the recently proposed method of moments of coupled-cluster equations with the SUMRCC formalism. It is demonstrated that the new SUMRCCSD(1) method, obtained by approximating the SUMRCCSD cluster amplitudes carrying only core and virtual orbital indices by their first-order MRMBPT values, provides the results that are comparable to those obtained with the complete SUMRCCSD approach.