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Physical review. B, Condensed matter

Electron correlations. II. Ground-state results at low and metallic densities

R. F. Bishop, K.H. Lührmann

Published Nov 15, 1982• 142 citations
Publisher

Abstract

In this second work in a series of papers the coupled-cluster or $\mathrm{exp}(S)$ formalism is applied to the problem of ground-state correlations in the one-component Fermi plasma. The main approximation employed is the so-called SUB2 approximation for the two-body subsystem correlation operator ${S}_{2}$ which provides a measure of the two-particle---two-hole component in the true ground-state wave function, and in which the coupling to larger subsystems is neglected. The SUB2 equations for Fermi systems are brought into tractable form by a "state-averaging" procedure which we develop by analogy with the comparable Bose equations, and which is shown by comparison with earlier exact results to be accurate in the metallic density regime at about the 1% level. Our final results for metallic densities include the completely integrated and self-consistent effects of the terms which by themselves generate (i) the random-phase approximation (RPA) and its associated exact long-range screening effects, (ii) the extra random-phase approximation exchange terms necessary to keep the RPA explicitly antisymmetric, (iii) the self-consistent particle-particle ladders (LAD) that describe two-particle scattering within the many-body medium and which describe the exact short-range behavior, (iv) a class of particle-hole ladder terms, and (v) the self-consistent hole-potential terms. Particular attention is paid to the important effects caused by the interference at intermediate separations of the long-range RPA and the short-range LAD effects. By comparison with recent and essentially exact stochastic simulations of the many-body Schr\"odinger equation, our results are seen to be accurate to about 1% for metallic densities, and hence to provide what is probably the currently best microscopic description available for this system. We show also that even in the low-density limit the SUB2 approximation provides a good "translationally-invariant-solid" description of both charged Fermi and Bose systems in this exact Wigner crystal regime.

DOI
10.1103/physrevb.26.5523

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In this second work in a series of papers the coupled-cluster or exp(S)\mathrm{exp}(S)exp(S) formalism is applied to the problem of ground-state correlations in the one-component Fermi plasma. The main approximation employed is the so-called SUB2 approximation for the two-body subsystem correlation operator S2{S}_{2}S2​ which provides a measure of the two-particle---two-hole component in the true ground-state wave function, and in which the coupling to larger subsystems is neglected. The SUB2 equations for Fermi systems are brought into tractable form by a "state-averaging" procedure which we develop by analogy with the comparable Bose equations, and which is shown by comparison with earlier exact results to be accurate in the metallic density regime at about the 1% level. Our final results for metallic densities include the completely integrated and self-consistent effects of the terms which by themselves generate (i) the random-phase approximation (RPA) and its associated exact long-range screening effects, (ii) the extra random-phase approximation exchange terms necessary to keep the RPA explicitly antisymmetric, (iii) the self-consistent particle-particle ladders (LAD) that describe two-particle scattering within the many-body medium and which describe the exact short-range behavior, (iv) a class of particle-hole ladder terms, and (v) the self-consistent hole-potential terms. Particular attention is paid to the important effects caused by the interference at intermediate separations of the long-range RPA and the short-range LAD effects. By comparison with recent and essentially exact stochastic simulations of the many-body Schr"odinger equation, our results are seen to be accurate to about 1% for metallic densities, and hence to provide what is probably the currently best microscopic description available for this system. We show also that even in the low-density limit the SUB2 approximation provides a good "translationally-invariant-solid" description of both charged Fermi and Bose systems in this exact Wigner crystal regime.
The coupled-cluster or exp(S)\mathrm{exp}(S)exp(S) formalism is applied to the problem of correlations in a many-electron system. In this first work in a series of papers we concentrate on exact results and restrict ourselves to various ring approximations applicable essentially to the high-density regime. It is shown that the well-known random-phase approximation (RPA) can be formulated as a nonlinear three-dimensional integral equation for the four-point function S2{S}_{2}S2​ which provides a measure of the two-particle---hole pair component in the true ground-state wave function. Exact analytic solutions are presented for both S2{S}_{2}S2​ and the three-point particle-hole vertex function in the RPA, and their properties are discussed. The Tamm-Dancoff approximation to the ring summation is also formulated, and the analagous exact solutions are presented for the first time. It is demonstrated that both solutions are unique, and a comparison of the two approximations is presented in terms of the correlation energy.
In an accompanying paper, we have described a new approach to the computation of ground-state energies and of correlated particle-hole excitations in terms of a phenomenological pseudo-Hamiltonian containing two- and more-body smooth effective potentials which sum the results of short-range correlations and of density fluctuation amplitudes which describe long-range correlations. In this paper we study the problem of relating the pseudopotentials to a (possibly singular) microscopic interaction with the aid of tools developed within the framework of the coupled cluster theory of K\"ummel and his collaborators, the results containing both familiar and unfamiliar elements. For example, the formulas derived, which depend on Bethe-Goldstone and Bethe-Faddeev amplitudes, include new definitions of particle-hole scattering matrix elements. An important consistency check is satisfied, in that the problem of defining the phenomenological potentials in terms of the microscopic ones must yield two separate but equivalent solutions, once within the framework of the theory of the ground-state energy and a second time within the framework of the theory of excitations. The entire package is studied with the aid of a modified version of coupled cluster theory, and shown to form a self-consistent entity. Among the desirable features of the formalism is that the large gap in the single-particle energy spectrum often utilized in existing formalisms is naturally absent from the current one.
We develop a general variational theory for inhomogeneous Fermi systems such as the electron gas in a metal surface, the surface of liquid 3He^{3}\mathrm{He}3He, or simple models of heavy nuclei. The ground-state wave function is expressed in terms of two-body correlations, a one-body attenuation factor, and a model-system Slater determinant. Massive partial summations of cluster expansions are performed by means of Born-Green-Yvon and hypernetted-chain techniques. An optimal single-particle basis is generated by a generalized Hartree-Fock equation in which the two-body correlations screen the bare interparticle interaction. The optimization of the pair correlations leads to a state-averaged random-phase-approximation equation and a strictly microscopic determination of the particle-hole interaction.
A general theory of long range correlations in the ground state of a closed shell nucleus is outlined, using a number of ideas introduced or adumbrated, but not always successfully developed, in previous work of the authors and others. These include the following: (i) Any operator can be formally expressed as a series in particle-hole (p-h) excitation and destruction operators which should converge rapidly for a suitable choice of single-particle basis at the closed shells. By means of these relations and with the help of sum rules all observables can be computed without the explicit use of wave functions. (ii) For a fixed choice of single-particle basis, the ground state energy can be computed from a variational principle in which in lowest nontrivial approximation, in consequence of (i), the parameters are amplitudes of the same type as occur in the random phase approximation (RPA). The precise relation to RPA is fully detailed. (iii) The single particle basis chosen is the natural orbital basis of Lowdin, which presents itself as the obvious choice both for formulation of the dynamics and for the algorithm of solution. (iv) Though the theory deals with pairs of fermion operators, particular attention is paid that all Pauli principle restrictions are satisfied. (v) It is verified that in the perturbation limit, known results are reproduced. The method is generally applicable to any system for which the correlations can be viewed as particle-hole excitations with respect to a reference Slater determinant.
This paper is the first of two in which the coupled cluster method (CCM) or exp(S) formalism is applied to two-component Fermi systems, the aim being to describe real metals and superconductivity. In this paper the authors concentrate on exact results and restrict ourselves to a ring approximation, applicable essentially to a high-density regime. They show that in the ground-state formalism the random phase approximation (RPA) can be formulated as a system of coupled, bilinear integral equations satisfied by functions associated with the so-called four-point functions of the system which provide a measure of the two-particle two-hole components in the true ground-state wavefunction. These equations are analysed in the dimensionless parameter formed by the ratio of the species masses and exact analytic solutions obtained. For Coulombic potentials (V 11 V 22 =V 2 12 ) they show that the exact analytic solution is unique and obtain an expression for the correlation energy. For non-Coulombic potentials (V 11 V 22 not=V 2 12 ) they indicate how to obtain a possible analytic solution. An RPA-like treatment of the one- and two-body equations in the excited-state formalism is provided for completeness.