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Physical review. B, Condensed matter

Linked-cluster perturbative expansion in correlated-basis theory

Stefano Fantoni

Published Mar 1, 1984• 15 citations
Publisher

Abstract

We give a brief outline of the perturbative expansion and the cluster properties of nondiagonal matrix elements in correlated-basis theory. A structural study of the perturbative series shows the presence of systematic cancellations among the terms of the series. It is proved by virtue of these cancellations that the expansion is linked and closely resembles the perturbative expansion of Brueckner-Goldstone theory. It is found that the ladder diagrams can be summed up by solving a Bethe-Goldstone---type equation which has recently been derived in correlated coupled-cluster theory. A brief discussion of some properties of ring diagrams is also given.

DOI
10.1103/physrevb.29.2544

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We give a brief outline of the perturbative expansion and the cluster properties of nondiagonal matrix elements in correlated-basis theory. A structural study of the perturbative series shows the presence of systematic cancellations among the terms of the series. It is proved by virtue of these cancellations that the expansion is linked and closely resembles the perturbative expansion of Brueckner-Goldstone theory. It is found that the ladder diagrams can be summed up by solving a Bethe-Goldstone---type equation which has recently been derived in correlated coupled-cluster theory. A brief discussion of some properties of ring diagrams is also given.
The correlated-basis-functions method of Feenberg and the coupled-cluster formalism of Coester and K\"ummel are joined to form a new ground-state many-body method combining the advantages of both older methods and avoiding their disadvantages. From the point of view of the correlated-basis-functions method, coupled-cluster theory is used to sum the perturbation series partially to arbitrary order. From the point of view of the coupled-cluster method, correlated basis functions are used to take out the repulsive core of the two-body interaction in order to allow more efficient truncation schemes. It is found that powerful renormalizations are possible. Explicit equations are given for the two-body subsystems embodying generalized Bethe-Goldstone and random-phase equations summing, in the correlated basis, ladder and ring diagrams to arbitrary order.
Connections between the Brueckner and Goldstone forms of the linked-cluster theorem are clarified in an elementary fashion employing a finite many-body system of electrons interacting via two-body forces as an illustrative example. It is shown that the order-by-order cancellations of Brueckner unlinked terms in the many-body wave function and energy occur in the Goldstone development upon the additions of the different time orderings of Feynman-Goldstone diagrams having disconnected vacuum parts. The factoring of the vacuum amplitude from the wave function, upon which customary presentations of the theorem focus attention, is seen to be irrelevant to the cancellations of Brueckner unlinked terms, and has to do, rather, with the presence of secular and normalization terms in the time-dependent perturbation functions for an adiabatically switched static perturbation. Similarly, the equivalence of the vacuum amplitude with the exponential of all connected vacuum diagrams, originally demonstrated by Feynman in the case of hole theory, is seen to be irrelevant to the cancellations of Brueckner unlinked terms in the energy, which occur in the Goldstone development upon the summations over all time orderings of disconnected vacuum diagrams. The distinction between Brueckner unlinked terms on the one hand, and secular and normalization terms on the other, is confused in customary presentations of the linked-cluster theorem by the use of exponential switching, following Gell-Mann and Low, and is clarified in the present development by considering the Friedrichs limit of ideal adiabatic switching.
The cluster expansion for the energy for a system with Bijl-Dingle-Jastrow (BDJ) correlation structure is derived in a new form. It is shown how the theory may be related to the Brueckner-Bethe-Goldstone theory with a particular choice for the correlation operator (or wave matrix) appearing in the BDJ wave function. The three-body cluster terms are discussed. It is shown that these clusters contain a limited class of dispersive terms which may be included in the analysis of the two-body clusters. The three-body clusters are also shown to contain terms which correct for violations of the Pauli principle in the correlation operator.
A linked-cluster expansion for the free energy in the petit canonical ensemble is presented and proved. It has the advantage of avoiding the introduction of the unknown chemical potential in the perturbation series. As a consequence of correlations among the population numbers n(K)n(\mathrm{K})n(K), additional linkages representing these correlations appear. The result is used to find the ground-state energy of a many-body fermion system. This expression reduces to the Brueckner-Goldstone expansion only in the case of central forces in an isotropic system, a theorem due to Kohn, Luttinger, and Ward. It is also shown that in the random phase approximation, correlation bonds do not contribute. Finally the relation of our formalism to the Bloch-De Dominicis expansion for the grand partition function is discussed.
The Goldstone-Brueckner perturbation theory is extended to incorporate in a simple way correlations associated with large-amplitude collective motions in nuclei. The new energy expansion making use of nonorthogonal vacua still allows to remove the divergences originating from the hard core of the bare interaction. This is done through the definition of a new Brueckner matrix summing generalized Brueckner ladders. At the lowest order, this formalism motivates variational calculations beyond the mean field, such as the generator coordinate method (GCM) and the projected mean-field method, from a perturbative point of view for the first time. Going to higher orders amounts to incorporate diabatic effects in the GCM and to extend the projection technique from product states to well-defined correlated states.
A linked-cluster expansion for the expectation values of operators between correlated wave functions of the Jastrow type is described. The expansion is based on the logarithm of the correlation function and leads to terms that are explicitly proportional to the particle number for infinite systems. The lowest-order terms for a weakly interacting system of spin-1/2 particles are calculated explicitly. From these, the Euler-Lagrange equations for the correlation functions have been calculated. The equation for the correlation function between particles of antiparallel spin can be solved by Fourier transform, and it is shown that for $r\ensuremath{\rightarrow}\ensuremath{\infty}$, the correlation function behaves like 1+Ar21+\frac{A}{{r}^{2}}1+r2A​. In the case of a system of particles interacting by hard-core repulsions, the logarithm of the correlation function does not exist inside the hard-core radius. It is then necessary to sum over ladder diagrams as in the derivation of the reaction matrix in many-body perturbation theory. This is carried out to provide a low-density approximation to the ground-state energy of the system and the Euler-Lagrange equation for the correlation function. It is argued that the correlation function again behaves like 1+Ar21+\frac{A}{{r}^{2}}1+r2A​ for large rrr. The expansion is applied to the problem of calculating the correlation energy of the electron gas. It is shown that a variational calculation based on the lowest-order terms in an expansion of the wave function leads to a divergent result (unbounded from below), but that a convergent result is obtained by summing over ring diagrams, which correspond to long-range polarization of the electron cloud. The result is similar to that of Gaskell obtained by a somewhat different method. It is shown analytically that at high densities the result behaves logarithmically, specifically as $0.0570\mathrm{ln}{r}_{s}\ensuremath{-}0.1324$ Ry. The lowest-order exchange contributions have also been calculated and found to contribute 0.040 Ry to the energy at densities of physical interest.