Improving the accuracy of ab initio methods with summation approximants and singularity analysis

Abstract

ABSTRACT:The accuracy of Møller-Plesset (MP) perturbation theory and coupledcluster (CC) theory can be significantly improved, at essentially no increase in computational cost, by using summation approximants that model the way in which these theories converge to the full configuration interaction limit. Approximants for MP4 and CCSD(T) are presented, their size scaling is analyzed, and the functional analysis of the MP energy, on which the MP4 approximant is based, is discussed. The MP approximants are shown to have a form that is appropriate for describing resonance energies.

Computing singularities of perturbation series

“…One cannot hope to be able to compute the whole set of singularities for a very large manybody system or even to determine the exact ROC for a stateof-the-art MPn calculation. However, obtaining insight into the distribution of singularities for typical quantum systems, such as the examples considered in this article and others [17], makes the construction and analysis of general resummation schemes much easier [10,27]. Also, it is clear that the study of multireference perturbation theory can benefit from such knowledge.…”

“…Indeed the now extremely popular coupled cluster method, which has to a large extent supplanted low-order MBPT as the most effective method for ab initio structure calculations, can be described in terms of summations of selected classes of diagrams (i.e., selected terms in the series) to infinite order [7]. Numerous other ways of resumming the series give improvements of the convergence such as Padé or algebraic approximants [6,[8][9][10][11]. Especially in nuclear physics, summations of classes of diagrams to infinite order, such as the random-phase approximation [12], have widespread use.…”

Appendix to “Studies in Perturbation Theory”: The Problem of Partitioning

“…It was noted [107][108][109]] that Feenberg's procedure is equivalent to a Padé approximation 2 scheme [116], as the above equation indeed suggests. An important question is whether the extensivity is maintained by the above modification, for which affirmative answers were already published [117,118]. However, a closer look into the problem reveals that Eq.…”

“…In the recent studies by Goodson [118,120,121], connection between Padé approximants and Feenberg scaling has further been exploited. Goodson has used quadratic Padé approximants [121], the denominators of which may exhibit not only poles, but also branching points.…”

Singularities of Møller-Plesset energy functions

“…The fourth-order perturbation theory was formerly considered the method of choice for high-accuracy ab initio computations, but on account of concerns raised about the convergence of the perturbation series [1][2][3][4][5][6][7] it has, in recent years, been supplanted by the popular CCSD͑T͒ coupled-cluster theory. [10][11][12][13][14][15] In principle, there exist two classes of singularities: 14,15 Class ␣ singularities are isolated complexconjugate pairs of square-root branch points, which correspond to the avoided crossings between the ground state and an excited state on a path along the real axis, while class ␤ singularities lie on or near the real axis and correspond to critical points at which one or more electrons dissociate from the nuclei. The MP energy can be thought of as an asymptotic series of a function of a perturbation parameter, and as such can be analyzed using methods of functional analysis.…”