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This paper establishes a formal connection between force fields (FFs) and density functional theory (DFT) by showing that exact FFs are variationally induced by DFT through a pullback operation. Specifically, the Born-Oppenheimer potential-energy surface is shown to be the pullback of the external-potential energy functional along the map from nuclear configurations to Coulomb potentials. By pulling back DFT derivative objects (density and density-density response function) to nuclear configuration space, the authors derive the force and nuclear Hessian, placing FFs, DFT, and response theory within a unified derivative hierarchy.
Force fields are revealed as the natural consequence of applying density functional theory to nuclear configurations, bridging two traditionally distinct approaches to molecular simulation.
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.
Stanford HAI