Adiabatic wave functions beyond the Born–Oppenheimer approximation: Phase linking between electrons and nuclei

In order to improve the Born–Oppenheimer treatment of adiabaticity we impose to the electronic and nuclear wave functions to have a well defined phase difference. It enables us to correctly evaluate matrix elements of such quantities as momenta which are poorly evaluated in the Born–Oppenheimer description. The phase factor obeys an equation which we solve by successive approximations and leads to simple expressions for this phase factor. Taking the coupling of two harmonic oscillators as an exactly soluble example which may also be treated within this adiabatic approximation we show that the proposed formalism is rather easy to use and leads to significant improvements with respect to Born–Oppenheimer treatments. We suggest that this framework might be useful when treating systems which are weakly coupled but are not so different in their time evolutions as are electrons and nuclei.