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

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.

Figures (2)

• Figure 1: Behavior of the RPA energy of H2O as a function of the electron number. Arrows represent ionization potentials and electron affinities calculated by $GW$ quasiparticle energies, $\Delta$RPA and derivatives of the RPA energy to the particle number. HF was used as the mean-field reference. cc-pVDZ basis set was used.
• Figure 2: Mean signed errors of different methods for predicting ionization potentials and electron affinities of molecules compared to $\Delta$CCSD(T) references. $\mu_\text{RPA}$ and $\mu_\text{RPAE}$ results were obtained from the finite difference approach for chemical potentials.