Formal derivation of nonadiabatic couplings from time-dependent density functional theory and the extension within modified linear response

Most of modern ab initio theories rely on the Born-Oppenheimer approximation to separate the electronic and nuclear motion, due to the fact that the smallness of the mass ratio between electrons and nuclei drastically scales down the nonadiabatic coupling (NAC) terms, which can thus be assumed to be negligible. However, this approximation will break down if there exists degenerate states, i.e. the crossing of potential energy surfaces, which are involved in numerous and rather diverse phenomena. NACs become infinite at the degeneracy points and needs to be seriously considered in the vicinity of these points.