Solution of the Poisson equation for two dimensional periodic structures (slabs) in an overlapping localized site density scheme

Bertaut's equivalent electric density idea (E. F. Bertaut, Journal de Physique {\bf 39}, 1331 (1978)) is applied to the case of two dimensional periodic continuous charge density distributions. The following derivation differs from what was introduced by Bertaut. The presented method solves the Poisson equation for the scheme of overlapping localized site densities with periodic boundary conditions in the ($x,y$) plane and with the general finite voltage boundary condition in the perpendicular $z$-direction.As usual the long-range potential is calculated in the Fourier space. For the $K_{||}\ne 0$ case a Fourier transformation helps to calculate the solution in a three dimensional periodic sense, while for %the $K_{||}=0$ %case the required charge neutrality is the starting point. For both cases suitable representations of the spherical harmonics are needed to arrive at expressions that are convenient for numerical implementation. In this localized density scheme an explicit relation can be derived between the finite voltage in $z$-direction and the $z$-component of the dipole density.