Dynamical matrix by Ewald-Fuchs method

In this appendix, the Ewald-Fuchs method for the calculation of the dynamical matrix is outlined in quite some detail, provided the structure dependent contribution to the energy per ion is given by  

The appearance of the energy-wave number characteristic is not essential for the derivation, but is quite natural from perturbation theory with pseudopotentials. In the more elaborate treatments, as e.g. the dielectric formulation, the electronic contribution is more complicated, as is outlined in the related section. This contribution has then to be treated in the appropriate way. The main purpose of the Ewald-Fuchs method is to handle the ionic contribution, which is written as in the previous appendix, where q is an arbitrary constant, chosen to obtain optimum convergence for the direct and the reciprocal lattice sum. Denoting the equilibrium positions of the ions by 

For the calculation of the phonons in the harmonic approximation, this equation of motion has to be considered to first order in the displacements (which corresponds to a second order calculation in the potential). Expanding the exponential function and the structure factor, the sum in reciprocal space becomes  

Inserting the explicit expression for the equilibrium structure factor, this term becomes  

Since the zero-order terms have to be zero in the equilibrium configuration, 

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