Solving one-body ensemble N-representability problems with spin

TL;DR

This paper characterizes the set of possible one-body reduced density matrices for N-electron systems with spin, refining the N-representability problem and providing a comprehensive solution using mathematical tools, crucial for ensemble density functional theory.

Contribution

It introduces a complete characterization of the one-body N-representability problem considering spin symmetries and mixedness, using convex polytopes and spectral constraints.

Findings

Set of admissible orbitals described by linear spectral constraints

Constraints are independent of the number of orbitals and M

Dependence on N and S is linear, enabling calculations for various systems

Abstract

The Pauli exclusion principle is fundamental to understanding electronic quantum systems. It namely constrains the expected occupancies $n_i$ of orbitals $\varphi_i$ according to $0 \leq n_i \leq 2$. In this work, we first refine the underlying one-body $N$-representability problem by taking into account simultaneously spin symmetries and a potential degree of mixedness $\boldsymbol w$ of the $N$-electron quantum state. We then derive a comprehensive solution to this problem by using basic tools from representation theory, convex analysis and discrete geometry. Specifically, we show that the set of admissible orbital one-body reduced density matrices is fully characterized by linear spectral constraints on the natural orbital occupation numbers, defining a convex polytope $\Sigma_{N,S}(\boldsymbol w) \subset [0,2]^d$. These constraints are independent of $M$ and the number $d$ of orbitals, while their dependence on $N, S$ is linear, and we can thus calculate them for arbitrary system sizes and spin quantum numbers. Our results provide a crucial missing cornerstone for ensemble density (matrix) functional theory.