The Interaction Between Two Drude Molecules

In the system comprising two Dmde molecules (see Section 4.9.1), an additional term must be included in the Hamiltonian [Rigby et al. 1986]. This additional term arises from the interactions between the two dipoles. The instantaneous dipole of each molecule is qz(t), where z(t) is the separation of the charges. Thus, if we label the molecules 1 and 2, we can write the dipole-dipole interaction energy as  

This equation can be solved by making the following substitutions  

This is the Schrodinger equation for two independent (i.e. non-interacting) oscillators with frequencies given as follows  

If we now substitute for u i and and expand the square roots using the binomial theorem, 

The interaction energy of the two oscillators is the difference between this zero-point energy and the energy of the system when the oscillators are infinitely separated and so  

---