where is the periodic distance between the nucleus and the electron :

(4.18) |

Notice that is a periodic function because we are using a periodic trial wave-function (see section 4.1.2). This auxiliary function removes the divergence in the bare force and it is consistent with the periodicity of the system. At variance of the case without periodic boundary conditions does not cancel exactly the term coming from the derivative of the ion-electron potential. Therefore we have to included both the Laplacian of the and the derivatives of the ion-electron potential in the calculation of the forces. More precisely the expression we used for the force is:

(4.19) |

where: