Analytic computation of energy derivatives. Relationships among partial derivatives of a variationally determined function

This paper considers three functions of several variables, W(r,x), λ(r), and E(r), related by E(r) = W[r,λ(r)] and the condition that W(r,x) be stationary with respect to variations of x when x=λ. Formulas are presented which relate coefficients in the Taylor series expansions of these three functions. We call λ the response function. Partial derivatives of the response function are obtained by solution of a recursive system of linear equations. Solution through order n yields derivatives of E through order 2n+1. This analysis extends Pulay's demonstration of the applicability of Wigner's 2n+1 rule to partial derivatives in coupled perturbation theory. A four-term second derivative formula is shown to be numerically more stable than the usual two-term formula. We refute previous claims in the literature that energy derivatives are stationary properties of the wave function.