A density-functional perspective on force fields

TL;DR

This paper establishes a theoretical connection between force fields and density functional theory (DFT), showing how exact force fields can be derived variationally from DFT principles.

Contribution

It presents a novel conceptual framework linking force fields, DFT, and response theory through a derivative hierarchy, emphasizing a theoretical perspective.

Findings

Force fields are variationally induced by DFT.

The Born-Oppenheimer surface is a pullback of the external-potential energy functional.

Density and response functions relate to nuclear forces and Hessians.

Abstract

Force fields are usually formulated directly in nuclear configuration space, whereas density functional theory is naturally formulated in terms of external potentials, densities, and variational duality. We show that exact force fields are variationally induced by DFT: the Born-Oppenheimer potential-energy surface is the pullback of the external-potential energy functional along the map from nuclear configurations to Coulomb potentials. In the Lieb formulation of density functional theory, the density is the first functional derivative of the energy with respect to the external potential, while the density-density response function is the second. Pulling these derivative objects back to nuclear configuration space yields the force and the nuclear Hessian, together with explicit terms induced by the nuclear-generated potential and the nuclear-nuclear repulsion. The resulting picture places force fields, density functional theory, and response theory within a single derivative hierarchy. The purpose of the present work is conceptual rather than algorithmic.