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Journal of Physics B Atomic Molecular and Optical Physics

On the acceleration of convergence of many-body perturbation theory. II. Benchmark checks for small systems

Klaus Dietz, C Schmidt, Markus Warken, B.A. Hess

Published Jul 14, 1993• 22 citations
Publisher

Abstract

For pt.II see ibid., vol.26, no.13, p.1885-96 (1993). The authors employ the method developed in a previous paper for small systems-Be, LiH, H 2 -where full CI calculations are available for monitoring convergence of many-body perturbation theory. It is shown that divergent series, in particular for excited states, can be transformed into fast converging ones. In essence their method consists in performing infinite subsummations of perturbation series in order to improve convergence: coupling constants are redefined such that singularities are incorporated in a non-perturbative manner and remaining correlations can be expanded in a larger domain of the complex coupling constant plane. It is in this way that the notion of 'improved convergence' has a well defined meaning.

DOI
10.1088/0953-4075/26/13/013

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For pt.II see ibid., vol.26, no.13, p.1885-96 (1993). The authors employ the method developed in a previous paper for small systems-Be, LiH, H 2 -where full CI calculations are available for monitoring convergence of many-body perturbation theory. It is shown that divergent series, in particular for excited states, can be transformed into fast converging ones. In essence their method consists in performing infinite subsummations of perturbation series in order to improve convergence: coupling constants are redefined such that singularities are incorporated in a non-perturbative manner and remaining correlations can be expanded in a larger domain of the complex coupling constant plane. It is in this way that the notion of 'improved convergence' has a well defined meaning.
The theoretical background of schemes for accelerating convergence of Moller-Plesset-Goldstone perturbation theory is discussed. Aiming at practical applications the authors consider approximate formulations of the many-body theory of atoms and molecules in finite-dimensional Hilbert spaces. A key role is played by nearby singularities of energy levels and eigenfunctions considered as functions of a suitably chosen coupling parameter. Infinite subsummations in the perturbation series are performed such that the influence of these singularities is eliminated or, at least, sufficiently attenuated. The danger of double counting of diagrammatic contributions already included in these infinite subsummations is avoided by the very construction of the authors' scheme.
Recently the possible divergence of many-body perturbation theory or many-body green function (propagator) methods with lattice summations in extended systems has been raised. The convergence of these methods with lattice summations is not only the key to establishing their correct scaling properties in inhomogeneous systems, but is also a necessity if numerical calculations are to be meaningful. In this report, it is rigorously shown that many-body perturbation theory (MBPT), coupled cluster theory (CC), and many-body green function (MBGF) methods all converge uniformly with lattice summations, although the integrand for the integration over the reciprocal lattice vector, k, could become infinite at special k values. Our proof is given not only for infinite polymers but also for crystals. In our proof, only the continuity of the zeroth-order band structure and Bloch orbitals with k is used. We show that MBPT, CC, and MBGF methods converge with the radius of the lattice summation range R at least as fast as 1/R1/n where n is the dimensionality of the system. It could be much faster if the zeroth-order band structure and Bloch orbitals have continuous first- or even higher-order derivatives with k. In practical numerical calculations, one should not pursue the convergence of the integrand, but the convergence of the correction considered with lattice summations.
The conventional many-body perturbation theory suffers from three major ills: (i) some terms in the perturbation series may be unbounded; (ii) the series as a whole may not have a limit and (iii) despite convergence,the calculated limit may be false ('bogus' convergence). In renormalizable field theories, as the name indicates, problem (i) can be bypassed. Problems (ii) and (iii) remain. Here we demonstrate that these two problems can also be resolved, by choosing an appropriate Kohn-Sham Hamiltonian as the 'unperturbed' Hamiltonian. For this to be possible, some pertinent ground-state densities must be pure-state non-interacting v-representable. We illustrate our findings by means of two examples.
The convergence problem of perturbation expansions in the many-body problem is discussed by employing some simple models. The expectation value of rhe resolvent operator is analytic around the origin of the coupling constant so long as the system is enclosed in a finite volume, but as the volume tends to infinity with the density of particles kept constant, the question arises whether this limit is allowed to be taken prior to the sum of the perturbation series. Even if this is the case, the resulting series turns out to be usually divergent, but in some models we consider we can find a function whose asymptotic expansion coincides with this series. It is shown that for the repulsive interactions the function thus obtained gives a correct answer, while for the attractive interactions which lead to a superconducting state it is far from the true one. The asymptotic behavior of the perturbation series in the latter case is examined in some detail.
A formulation of Schr\"odinger perturbation theory is developed that gives a unified treatment of non-degenerate and degenerate cases, is unique, and has a nonzero radius of convergence under very general conditions. Two alternative procedures are given for finding perturbed eigenvectors, one of which is simpler for the nondegenerate case or for small finite degeneracy, the other simpler for infinite or large finite degeneracy. The low-order terms in the perturbation expansions of quantities used in applications are given. The perturbation theory formulated in this paper has the following advantages over the conventional Schr\"odinger and Brillouin-Wigner perturbation theories: (i) When the convergence criterion is satisfied, bounds on the error made in replacing an appropriate infinite perturbation series by its first nnn terms can be obtained. (ii) For the case of degeneracy, the conventional Schr\"odinger perturbation theory can break down under conditions to which the convergence of the perturbation theory developed in this paper are insensitive. (iii) There is no implicit dependence on the eigenvalue, such as appears in the Brillouin-Wigner perturbation theory. (iv) For the case of degeneracy, statistical information about the distribution of certain eigenvalues can be obtained without finding the individual eigenvalues. (v) The theory is applicable to a wider class of problems than the conventional Schr\"odinger and Brillouin-Wigner perturbation theories.