Ewald procedure potential

The hypersphere method, as discussed below, is a very efficient way to sample the configurations of charged or dipolar particles, in particular, because the pair potentials have a closed analytical expression. It was demonstrated to give results identical to those obtained by using the Ewald procedure. The method is certainly an eloquent illustration of the equivalence... 

Ewald summation is one of the procedures developed to solve the problems just mentioned. While VDW has rapid potential drop across certain interatomic distances due to its 6-12 exponential function, the electrostatic interaction s convergence over the interatomic distance variation is very slow due to its 1/r dependency. The use of a two-step summation (one in real space and one in reciprocal space) for the periodic system will give a more accurate value for the electrostatic interactions (63). One summation is carried out in reciprocal space the other is carried out in real space. Based on Ewald s formulation, the simple lattice sum can be reformulated to give absolutely convergent summations that define the principal value of the electrostatic potential, called the intrinsic potential. Given the periodicity present in both crystal calculations and in dynamics simulations using periodic boundary conditions, the Ewald formulation becomes well suited for the calculation of electrostatic energy and force. 

It is not feasible to evaluate the series (4.32) directly in fact, it does not tend to a unique value as q 0, but to a value which depends on the relative directions of q and the electrical polarization associated with the mode concerned. Fortunately, there is a very elegant procedure, known as EWALD s method [4.32], by which C(q) can be expressed as the sum of two rapidly convergent series, one involving a summation over a limited region in the reciprocal lattice and the other over a limited region in the direct lattice. Let be the potential energy of a point charge z e at a... 

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