Extension of many-electron theory and approximate density functionals to fractional charges and fractional spins

TL;DR

This paper develops a general framework to extend approximate density functionals and many-electron theories to fractional charges and spins, enabling better analysis of their accuracy and violations of exact conditions.

Contribution

It introduces a universal method for extending various approximate theories to fractional systems based on ensemble averaging of the Green's function.

Findings

Framework applicable to LDA, GGA, hybrid functionals, and many-body theories.

Allows analysis of functional violations against exact conditions.

Facilitates calculation of chemical potentials and band gaps.

Abstract

The exact conditions for density functionals and density matrix functionals in terms of fractional charges and fractional spins are known, and their violation in commonly used functionals has been shown to be the root of many major failures in practical applications. However, approximate functionals are not normally expressed in terms of the fractional variables. Here we develop a general framework for extending approximate density functionals and many-electron theory to fractional-charge and fractional-spin systems. Our development allows for the fractional extension of any approximate theory that is a functional of $G^{0}$, the one-electron Green's function of the non-interacting reference system. The extension to fractional charge and fractional spin systems is based on the ensemble average of the basic variable, $G^{0}$. We demonstrate the fractional extension for the following theories: (1) any explicit functional of the one-electron density, such as the LDA and GGA; (2) any explicit functional of the one-electron density matrix of the non-interacting reference system, such as the exact exchange functional and hybrid functionals; (3) many-body perturbation theory; and (4) random-phase approximations. A general rule for such an extension has also been derived through scaling the orbitals and should be useful for functionals where the link to the Green's function is not obvious. The development thus enables the examination of approximate theories against known exact conditions on the fractional variables and the analysis of their failures in chemical and physical applications in terms of violations of exact conditions of the energy functionals. The present work should facilitate the calculation of chemical potentials and fundamental band gaps with approximate functionals and many-electron theories through the energy derivatives with respect to the fractional charge.