$N$-representability in non-collinear spin-polarized density functional theory

TL;DR

This paper addresses the N-representability problem in non-collinear spin-polarized density functional theory, revealing differences between pure and mixed states and providing practical conditions for their characterization.

Contribution

It demonstrates that pure and mixed state N-representable densities differ in non-collinear cases and offers a simple, checkable characterization for mixed states.

Findings

Pure and mixed state N-representable densities are generally different.

Provided necessary and sufficient conditions for mixed state densities.

Clarified the N-representability problem in non-collinear spin DFT.

Abstract

The $N$-representability problem for non-collinear spin-polarized densities was left open in the pioneering work of von Barth and Hedin setting up the Kohn-Sham density functional theory for magnetic compounds. In this letter, we demonstrate that, contrarily to the non-polarized case, the sets of pure and mixed state $N$-representable densities are different in general. We provide a simple characterization of the latter by means of easily checkable necessary and sufficient conditions on the components $\rho^{\alpha \beta} (\br)$ of the spin-polarized density.