Implications of the unitary invariance and symmetry restrictions on the development of proper approximate one-body reduced density matrix functionals

TL;DR

This paper investigates how symmetry restrictions and unitary invariance influence the development of approximate 1RDM functionals, highlighting the importance of symmetry considerations in accurately modeling the two-body reduced density matrix.

Contribution

It clarifies the role of symmetry restrictions versus universal functionals and explores the impact of unitary invariance on the 2RDM in approximate theories.

Findings

Symmetry restrictions are more accurately described as variants of the universal functional.

Degenerate natural orbitals exhibit large deviations in 2RDM elements due to unitary invariance.

H4 geometries serve as a guide for constructing symmetry-restricted 1RDM functionals.

Abstract

In many of the approximate functionals in one-body reduced density matrix (1RDM) functional theory, the approximate two-body reduced density matrix (2RDM) in the natural orbital representation only depends on the natural occupation numbers. In Phys. Rev. A 92, 012520 (2015) Wang and Knowles initialised a discussion of to what extent this simplification is valid, by introducing two different H$_4$ geometries with identical natural occupation numbers, but different 2RDMs. Gritsenko has argued that this feature is due symmetry [Phys. Rev. A 97, 026501 (2018)]. This work aims to contribute to the discussion on the following points: 1) one should rather speak of symmetry-restricted variants of the universal functional, than saying that the universal functional is symmetry dependent; 2) the unitary invariance of degenerate NOs can lead to large deviations in the 2RDM elements, especially the phase of the NOs; 3) symmetry-restricted functionals are constructed for the H$_4$ geometries considered by Wang and Knowles, whose structure could serve as guide in the construction of approximate 1RDM functionals.