Perturbative determination of nonadiabatic coupling matrix elements

Equations for the first- and second-derivative coupling matrix elements between the adiabatic electronic wave functions of a canonical Van Vleck approach are derived for the case of a single nuclear coordinate. Expressions for their order-by-order evaluation, analogous to the multidimensional perturbative expansion of the effective Hamiltonian (given elsewhere) are presented. Diagrammatic notations useful for enumerating certain components of the derivatives are introduced, and the diagrams required through first order are shown