where the first equality is exact but the second represents a finite-difference approximation that allows the coupling term in Eq. ( ) to be evaluated by determining the overlap (in time) of the adiabatic basis functions. While the approximation in Eq. ( ) is numerically advantageous as compared to approximating Eq. ( ) via finite differences involving coordinate displacements, even the more efficient finite-difference in time is less desirable as compared to analytic evaluation of d JK .