Integral Hellmann—Feynman Theorem

In the Born—Oppenheimer approximation, it is shown that for any molecular isoelectronic process the total energy change, excluding nuclear kinetic energy, is given by the formula $${\mathrm{\Delta W=\Delta V}}_{\mathrm{nn}}+{\displaystyle \int }\text{ρ(1)}{\mathcal{H}}^{\prime }\text{(1)dτ(1).}$$ Here ΔV_nn is the nuclear—nuclear repulsion‐energy change, and H′ is the change in the one‐electron nuclear—electron attraction operator. The function ρ(1) is the normalized transition density for the change, the integral over Electrons 2 through N of the product of the initial and final electronic wavefunctions, multiplied by N and divided by the overlap integral between the initial and final wavefunctions. For infinitesimal changes, it is demonstrated that this theorem reduces to the conventional Hellmann—Feynman theorem. A corresponding differential equation and its equivalent variational principle are derived. Use of approximate wavefunctions in the several formulas is discussed, and various possible applications of the theorem are indicated. The theorem is shown to provide a rigorous basis for a pseudoclassical theory of molecular vibrations, and shown to indicate why electron‐correlation effects ordinarily do not control molecular deformation.