Methods for electronic-structure calculations - an overview from a reduced-density-matrix point of view

TL;DR

This paper reviews quantum chemistry and solid state methods for many-body problems, emphasizing reduced density matrices, their properties, and recent extensions like density matrix functional theory and dynamical mean field theory.

Contribution

It provides a comprehensive overview of reduced density matrix approaches and recent advancements in quantum chemical and solid state methods for many-body systems.

Findings

Highlights the importance of the cumulant two-body reduced density matrix.

Discusses spectral resolution and geminals as key concepts.

Proposes using contraction sum rules for variational methods.

Abstract

The methods of quantum chemistry and solid state theory to solve the many-body problem are reviewed. We start with the definitions of reduced density matrices, their properties (contraction sum rules, spectral resolutions, cumulant expansion, $N$-representability), and their determining equations (contracted Schr\"odinger equations) and we summarize recent extensions and generalizations of the traditional quantum chemical methods, of the density functional theory, and of the quasi-particle theory: from finite to extended systems (incremental method), from density to density matrix (density matrix functional theory), from weak to strong correlation (dynamical mean field theory), from homogeneous (Kimball-Overhauser approach) to inhomogeneous and finite systems. Measures of the correlation strength are discussed. The cumulant two-body reduced density matrix proves to be a key quantity. Its spectral resolution contains geminals, being possibly the solutions of an approximate effective two-body equation, and the idea is sketched of how its contraction sum rule can be used for a variational treatment.