The forces on both electrons and nuclei due to their electric charge are of the same order of magnitude, and so the changes which occur in their momenta as a result of these forces must also be the same. One might, therefore, assume that the actual momenta of the electrons and nuclei were of similar magnitude. In this case, since the nuclei are so much more massive than the electrons, they must accordingly have much smaller velocities. Thus it is plausible that on the typical time-scale of the nuclear motion, the electrons will very rapidly relax to the instantaneous ground-state configuration, so that in solving the time-independent Schrödinger equation resulting from the Hamiltonian in equation 2.12, we can assume that the nuclei are stationary and solve for the electronic ground-state first, and then calculate the energy of the system in that configuration and solve for the nuclear motion. This separation of electronic and nuclear motion is known as the Born-Oppenheimer approximation .
Following Ziman , we assume the following form of an
eigenfunction for the Hamiltonian (2.12):
The second non-adiabatic term in the final term of equation 2.20
will be largest when the electrons labelled are tightly bound to the nuclei
labelled in which case
first order correction from this term is
We therefore neglect the non-adiabatic terms and note that equation 2.20 is
Schrödinger equation of the form
This adiabatic principle is crucial because it allows us to separate the nuclear and electronic motion, leaving a residual electron-phonon interaction. From this point on it is assumed that the electrons respond instantaneously to the nuclear motion and always occupy the ground-state of that nuclear configuration. Varying the nuclear positions maps out a multi-dimensional ground-state potential energy surface, and the motion of the nuclei in this potential can then be solved. In practice Newtonian mechanics generally suffices for this part of the problem2.7, and relaxation of the nuclear positions to the minimum-energy configuration or molecular dynamics [11,12] can be performed. These aspects go beyond the scope of this dissertation so that from now on it is assumed that a system with a fixed nuclear configuration is to be treated, so that the electronic energy is a constant and the electronic wave-function obeys the Schrödinger equation 2.19. The dependence of the electronic wave-function on the nuclear positions is now suppressed.