Density Functional Theory for Fractional Particle Number: Derivative Discontinuity of the Energy at the Maximum Number of Bound Electrons

TL;DR

This paper explores the derivative discontinuity in ensemble DFT at the maximum number of bound electrons, revealing real and imaginary jumps in the exchange-correlation potential related to electron binding and metastability.

Contribution

It extends complex-scaled ensemble DFT to fractional particles, uncovering the nature of potential jumps at the maximum bound electron number, including the imaginary component at $J_{max}$.

Findings

Discontinuous jumps in the exchange-correlation potential at integer particle numbers.

Real jumps below $J_{max}$ due to chemical potential shifts.

Imaginary component of the jump at $J_{max}$ indicating finite lifetime of the next state.

Abstract

The derivative discontinuity in the exact exchange-correlation potential of ensemble Density Functional Theory (DFT) is investigated at the specific integer number that corresponds to the maximum number of bound electrons, $J_{max}$. A recently developed complex-scaled analog of DFT is extended to fractional particle numbers and used to study ensembles of both bound and metastable states. It is found that the exact exchange-correlation potential experiences discontinuous jumps at integer particle numbers including $J_{max}$. For integers below $J_{max}$ the jump is purely real because of the real shift in the chemical potential. At $J_{max}$, the jump has a non-zero imaginary component reflecting the finite lifetime of the $(J_{max}+1)$ state.