Coulomb Matrix and Electric Fields

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 

The particular choice of a Gaussian charge distribution is made for mathematical convenience and cp is, of course, independent of the parameter n In Appendix K, we derive an explicit expression for C(q) using the ft-space representation and EWALD s transformation. The result is 

The sum in (4.38) extends over the reciprocal lattice points t and originates from On the other hand, (, ) is expressed as a sum in crys- 

We are now in a position to introduce the Coulomb field and the polarization. Instead of (4.30) we write 

The form (4.42) illustrates that the Coulomb fiel d is a dipole field  

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