Many-electron systems with fractional electron number and spin: exact properties above and below the equilibrium total spin value

TL;DR

This paper investigates the exact properties of many-electron systems with fractional electron number and spin, providing new theoretical insights and conditions that enhance the development of density functional approximations.

Contribution

It derives the form of ensemble ground states for fractional electron and spin cases, and generalizes key theorems and conditions in density functional theory.

Findings

Identified ambiguity in low-spin ground states and proposed entropy maximization to resolve it.

Established properties that constrain the composition of ensemble states.

Related Kohn-Sham orbital energies to total energy differences, extending the ionization potential theorem.

Abstract

In this work, we analyze the fundamental question of what is the ensemble ground state of a general, finite, many-electron system at zero temperature, with a given, possibly fractional, electron number $N_{tot}$ and a given $z$-projection of the spin, $M_{tot}$, distinguishing between low- and high-spin cases. For the low-spin case, the general form of the ensemble ground state has been rigorously derived in J. Phys. Chem. Lett. 15, 2337 (2024), finding the presence of an ambiguity in the ground state. Here we further discuss this ambiguity, and show that it can be removed via maximization of the entropy. For the high-spin case, we find that the form of the ensemble ground state strongly depends on the system in question. Furthermore, we prove three general properties which characterize the ensemble, and narrow the list of pure states it may consist of. We relate the frontier Kohn-Sham orbital energies to total energy differences, providing a generalization of the ionization potential theorem to systems with arbitrary fractional $M_{tot}$. Furthermore, we derive expressions for new derivative discontinuities, which appear as jumps in the KS potentials when crossing a boundary in the $N_{\uparrow}$-$N_{\downarrow}$ plane. Our analytical results are supported by an extensive numerical analysis of the Atomic Spectra Database of the National Institute of Standards. The new exact conditions for many-electron systems derived in this work are instrumental for development of advanced approximations in density functional theory and other many-electron methods.