Energy of a many-electron system in an ensemble ground-state, versus electron number and spin: piecewise-linearity and flat plane condition generalized

TL;DR

This paper provides an exact description of the ground state energy and spin-density dependence on fractional electron number and spin, generalizing key principles and revealing new degeneracies and discontinuities relevant for density functional theory.

Contribution

It generalizes the piecewise-linearity and flat-plane conditions for many-electron systems with fractional electrons and spins, and uncovers new degeneracies and derivative discontinuities.

Findings

Generalized piecewise-linearity and flat-plane conditions.

Identified a new derivative discontinuity in the Kohn-Sham potential.

Discovered a degeneracy where energy and density are unique but spin densities are not.

Abstract

Description of many-electron systems with a fractional electron number $N_\textrm{tot}$ and fractional spin $M_\textrm{tot}$ is of great importance in physical chemistry, solid state physics and materials science. In this Letter, we provide an exact description of the zero-temperature ensemble ground state of a general, finite, many-electron system, and characterize the dependence of the energy and the spin-densities on both $N_\textrm{tot}$ and $M_\textrm{tot}$, when the total spin is at its equilibrium value. We generalize the piecewise-linearity principle and the flat-plane condition and determine which pure states contribute to the ground-state ensemble. We find a new derivative discontinuity, which manifests for spin variation at constant $N_\textrm{tot}$, as a jump in the Kohn-Sham potential. We identify a previously unknown degeneracy of the ground state, such that the total energy and density are unique, but the spin-densities are not. Our findings serve as a basis for development of advanced approximations in density functional theory and other many-electron methods.