Singular analysis of RPA diagrams in coupled cluster theory

TL;DR

This paper offers a rigorous mathematical analysis of RPA diagrams in coupled cluster theory, connecting them to pseudo-differential operators and proposing efficient approximation schemes for their computation.

Contribution

It introduces a novel singular analysis approach to RPA diagrams, linking them to pseudo-differential operators and developing optimal approximation strategies.

Findings

Established a connection between RPA diagrams and pseudo-differential operators

Analyzed the asymptotic behavior of RPA diagrams using singular analysis

Proposed an efficient N-term approximation scheme with proven convergence rates

Abstract

Coupled cluster theory provides hierarchical many-particle models and is presently considered as the ultimate benchmark in quantum chemistry. Despite is practical significance, a rigorous mathematical analysis of its properties is still in its infancy. The present work focuses on nonlinear models within the random phase approximation (RPA). Solutions of these models are commonly represented by series of a particular class of Goldstone diagrams so-called RPA diagrams. We present a detailed asymptotic analysis of these RPA diagrams using techniques from singular analysis and discuss their computational complexity within adaptive approximation schemes. In particular, we provide a connection between RPA diagrams and classical pseudo-differential operators which enables an efficient treatment of the linear and nonlinear interactions in these models. Finally, we discuss a best $N$-term approximation scheme for RPA-diagrams and provide the corresponding convergence rates.