Harmonic phonons in a periodic lattice

In the previous section, the correction to the electronic energy, due to a perturbing potential, was evaluated to second order. If this perturbation results from ion displacements, there is an additional correction to the total energy of the lattice, arising from the interaction between the displaced ions. In the adiabatic approximation, the previous derivation of Eq. 88 for the electronic part of the energy remains valid. If the ions do not overlap, the ion-ion interaction energy for ions of valence Z at distance R is just the 

Coulomb interaction jR. Their interaction energy is then  

Since also the electron-ion potential V is a sum of terms from each ion  

The correction to the total energy, including the ion-ion interaction and the electronic contribution, thus becomes  

Since the energy has to be stationary with respect to the displacements to first order, we first examine the first order correction. Denoting the ion displacements by  

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