Derivative Discontinuity in Many-Body Perturbation Theory and Chemical Potentials in Random Phase Approximation

TL;DR

This paper derives analytical expressions for chemical potentials within RPA, revealing a fundamental discontinuity at integer particle numbers that explains discrepancies in $GW$ quasiparticle energies and total energies.

Contribution

It demonstrates the existence of a derivative discontinuity in the $GW$ correlation energy, addressing a key inconsistency in many-body perturbation theory.

Findings

Discontinuity in the $GW$ correlation self-energy at integer particle numbers.

Validation of chemical potential calculations via two equivalent methods.

Insight into the nonanalytic behavior of correlation energy functionals.

Abstract

We derive analytical expressions for chemical potentials within the random phase approximation (RPA), equivalently the $GW$ energy functional evaluated using non interacting Green's functions ($G_s$). The chemical potential is obtained using two formally equivalent approaches: a direct derivative of the total energy with respect to particle number, and a functional derivative via the chain rule through $G_s$, both validated with finite difference benchmarks. We show that the functional derivative of the $GW$ correlation energy$\unicode{x2013}$i.e., the $GW$ correlation self energy$\unicode{x2013}$exhibits a discontinuity at integer particle numbers with finite jumps. This resolves the apparent inconsistency between accurate $GW$ quasiparticle energies and the large delocalization errors observed in RPA total energies, as standard $GW$ self energies neglect this nonanalytic behavior. Our results suggest that derivative discontinuities are a fundamental feature of correlation energy functionals, analogous to the known discontinuity in the exact exchange correlation energy.