Integrated and Integral Hellmann—Feynman Formulas

For an isoelectronic molecular process X→Y, the energy change ΔW = W_Y—W_X may be computed from any one of three equivalent formulas if the exact molecular wavefunctions, ψ_X and ψ_Y, are known: the expectation‐value difference, the integrated Hellmann—Feynman formula, and the integral Hellmann—Feynman formula. Should only approximate wavefunctions be available, ${\mathrm{\Psi ̃}}_X$ and ${\mathrm{\Psi ̃}}_Y$ , 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 ${\mathcal{H}}_X=\mathcal{H}\text{(0)}$ , ${\mathcal{H}}_Y=\mathcal{H}\text{(1)}$ , $\mathcal{H}=\mathcal{H}\mathrm{(\lambda )}$ , ${\mathrm{\Psi ̃}}_X\text{=Ψ̃(0)}$ , ${\mathrm{\Psi ̃}}_Y\text{=Ψ̃(1)}$ , $\mathrm{\Psi ̃=\Psi ̃(\lambda )}$ , and the three formulas are as follows: $$\begin{array}{c}{\mathrm{\Delta W}}_{\mathrm{ed}}\text{=〈Ψ̃(1)\hspace{0.17em}|}\mathcal{H}\text{(1)\hspace{0.17em}|Ψ̃(1)〉/〈Ψ̃(1)\hspace{0.17em}|Ψ̃(1)〉−〈Ψ̃(0)\hspace{0.17em}|}\mathcal{H}\text{(0)\hspace{0.17em}|Ψ̃(0)〉/〈Ψ̃(0)\hspace{0.17em}|Ψ̃(0)〉,}\\ {\mathrm{\Delta W̃}}_d={\displaystyle {\int }_0^1}\mathrm{d\lambda }\frac{\mathrm{〈\Psi ̃(\lambda )\hspace{0.17em}|\partial }\mathcal{H}\mathrm{(\lambda )/\partial \lambda \hspace{0.17em}|\Psi ̃(\lambda )〉}}{\mathrm{〈\Psi ̃(\lambda )\hspace{0.17em}|\Psi ̃(\lambda )〉}},\\ {\mathrm{\Delta W}}_l\text{=〈Ψ̃(0)\hspace{0.17em}|}\mathcal{H}\text{(1)−}\mathcal{H}\text{(0)\hspace{0.17em}|Ψ̃(1)〉/〈Ψ̃(0)\hspace{0.17em}|Ψ̃(1)〉.}\end{array}$$ 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 ${\mathrm{\Psi ̃}}_X$ and ${\mathrm{\Psi ̃}}_Y$ are selected by the linear variational method from a fixed basis set, the three formulas give the same ΔW̃. If each of ${\mathrm{\Psi ̃}}_X$ and ${\mathrm{\Psi ̃}}_Y$ is selected variationally from a given class of functions, as is the case when each is an exact Hartree—Fock function, ΔW̃_ed and ΔW̃_d are equal, but possibly different from ΔW̃_l. Examples are included which show that ΔW̃_l sometimes gives a better estimate of an actual energy change than does ΔW̃_ed. Implications for the energy of interaction between two ions or ionic fragments in a molecule are discussed.