General harmonic approximation for time-dependent molecular dynamics

The linear approximation to TDVP equations for the complete molecular system is used to derive equations of motion for the electrons and the nuclei in the random phase approximation (RPA). Definitions of the relevant matrix elements in terms of nuclear gradients and hessians of the energy are given. The linear approximation to the evolution equations corresponds to a Gaussian approximation to the overlap and a quadratic approximation to the overall phase of the wave function.