Density functional theory: Non-Born–Oppenheimer Legendre transforms and Maxwell relations, equilibrium and stability conditions

Abstract

The Legendre-transformed representations of the non-Born–Oppenheimer (NBO) density functional theory are analyzed and the corresponding Maxwell relations are derived. These relations exhibit various couplings between parameters of the NBO ground-state energy hypersurfaces: ε[{Ni, Zi, Mi}] and εφ[{Ni, Zi, Mi},φ], where Ni, Zi, and Mi denote, respectively, the number of particles i, their charge and mass, while φ stands for an external electric field. The criteria for intrinsic equilibrium and stability of molecular systems are formulated and discussed within both the BO and NBO approximations. The physical content of stability criteria is interpreted in terms of the Le Châtelier and the Le Châtelier–Braun principles. The classical nature of these criteria is revealed through the introduction of internal partial scalar pressures of the system components (groups of identical particles) within the local formulation of the theory. It is then shown that the criteria of equilibrium and stability in isolated molecular systems become the classical criteria of the ’’mechanical’’–electrostatic equilibrium and stability.