Integrated and Integral Hellmann—Feynman Formulas

For an isoelectronic molecular process X→Y, the energy change ΔW = WY—WX may be computed from any one of three equivalent formulas if the exact molecular wavefunctions, ψX and ψY, are known: the expectation-value difference, the integrated Hellmann—Feynman formula, and the integral Hellmann—Feynman formula. Should only approximate wavefunctions be available, Ψ̃X and Ψ̃Y, these formulas give different estimates of the energy change. If the Hamiltonians for X and Y differ in values of some parameter or parameters λ, say λ = 0 for X and λ = 1 for Y, one has HX=H(0), HY=H(1), H=H(λ), Ψ̃X=Ψ̃(0), Ψ̃Y=Ψ̃(1), Ψ̃=Ψ̃(λ), and the three formulas are as follows: ΔWed=〈Ψ̃(1) |H(1) |Ψ̃(1)〉/〈Ψ̃(1) |Ψ̃(1)〉−〈Ψ̃(0) |H(0) |Ψ̃(0)〉/〈Ψ̃(0) |Ψ̃(0)〉,ΔW̃d= ∫ 01dλ〈Ψ̃(λ) |∂H(λ)/∂λ |Ψ̃(λ)〉〈Ψ̃(λ) |Ψ̃(λ)〉,ΔWl=〈Ψ̃(0) |H(1)−H(0) |Ψ̃(1)〉/〈Ψ̃(0) |Ψ̃(1)〉.Relative advantages and disadvantages of these formulas are discussed, and illustrations are given of their use. Conditions for the equivalence of the formulas are established. It is shown that if Ψ̃X and Ψ̃Y are selected by the linear variational method from a fixed basis set, the three formulas give the same ΔW̃. If each of Ψ̃X and Ψ̃Y is selected variationally from a given class of functions, as is the case when each is an exact Hartree—Fock function, ΔW̃ed and ΔW̃d are equal, but possibly different from ΔW̃l. Examples are included which show that ΔW̃l sometimes gives a better estimate of an actual energy change than does ΔW̃ed. Implications for the energy of interaction between two ions or ionic fragments in a molecule are discussed.