The ground-state energy and external potential as functionals of the electron density and their derivatives

TL;DR

This paper investigates the mathematical properties of the ground-state energy functional in density functional theory, revealing issues with its derivatives and the implications for the external potential as a functional of electron density.

Contribution

It demonstrates that the ground-state energy functional's derivative cannot be properly determined due to non-differentiability and domain restrictions, clarifying fundamental theoretical issues.

Findings

The derivative of the energy functional is determined by its asymptotic behavior.

Traditional assumptions about non-invertibility lead to paradoxes in derivative calculations.

Resolving the paradox involves recognizing domain restrictions of the potential.

Abstract

It is shown that the ground-state energy as a functional solely of the electron density is determined by the asymptotic value of the derivative of the degree-one homogeneous extension of the universal density functional F[n] at the given electron number. This has the consequence that its derivative cannot be properly determined. Carrying out the derivative of E[n[N,v]] with respect to v(r) leads to a paradox, which is resolved by the non-differentiability of E[n] if one follows traditional wisdom regarding the non-invertibility of the linear response function. However, considering the derivative of v[n[v]] through the one-electron case shows that this paradox has a more elementary origin, namely, an unaccounted restriction of the v(r) domain.