On the evaluation of nonadiabatic coupling matrix elements for MCSCF/CI wave functions using analytic derivative methods. III. Second derivative terms

A method for the efficient evaluation of nonadiabatic coupling matrix elements of the form 〈Ψ J(r;R)‖(∂2/∂R2α) Ψ I(r;R)〉r is presented. The electronic wave functions Ψ J and Ψ I are assumed to be MCSCF/CI wave functions whose common molecular orbital basis is determined within the state averaged MCSCF (SA-MCSCF) approximation. The method derives its efficiency by exploiting analogies with analytic CI second derivative techniques and from the first and second derivative coupled perturbed SA-MCSCF equations. This method is compared with an existing finite difference procedure which is reformulated to take maximal advantage of analytic gradient methods.