Efficient elimination of response parameters in molecular property calculations for variational and nonvariational energies.

A general method is presented for the efficient elimination of response parameters in molecular property calculations for variational and nonvariational energies. For variational energies, Wigner's 2n+1 rule is obtained as a special case of the more general k(2n+1) rule, which states that for a subset of k perturbations within a total set of z>or=k perturbations, response parameters may be eliminated according to the 2n+1 rule (normally applied to the full set of perturbations). Nonvariational energies may be treated by introducing Lagrange multipliers that satisfy the stronger 2n+2 rule for the k perturbations, while the wave-function parameters still satisfy the 2n+1 rule for the k perturbations. The corresponding rule for nonvariational energies is referred to as the k(2n+1,2n+2) rule. For k=z, the well-known 2n+2 rule for the multipliers is reproduced, while the wave-function parameters satisfy the 2n+1 rule. The application of the k(2n+1) and k(2n+1,2n+2) rules minimizes the total number of response equations to be solved when the molecular property contains k extensive perturbations (e.g., geometrical derivatives) and z-k intensive perturbations (e.g., electric fields).