An alternative and efficient way to approximate the bulk properties of an infinite system is the use of periodic boundary conditions (PBC) on a finite box. These boundary conditions mean that the simulation cell is wrapped onto itself and, as an electron moves out of one side of the super-cell it immediately moves back through the opposite side (see figure 4.1). The advantage of using such boundary conditions is that there are no longer "surface electrons" and hence no surface effects. However even with PBC size effects are still present. This is due to the lack of long wavelength fluctuations in the charge density. For a simulation box of linear dimension , the periodicity will remove any correlation length greater than .

In this thesis we used a cubic simulation cell with volume with PBC, and the size effects are partially taken into account by increasing the size of the super-cell.

The general hydrogen Hamiltonian with periodic boundary condition is written as:

where are the vectors of the periodic lattice associated with the simulation box, are electron coordinates and are the proton coordinates, and is the number of electrons in the simulation cell. Infinite mass of the protons is assumed so that the kinetic term contains only the electronic contribution. Notice that the Hamiltonian 4.1 is invariant under the translation of any electron coordinate by a vector in . Moreover if the one body potential is generated by a ionic lattice, the Hamiltonian 4.1 has to be invariant also with respect to a translation given by a vector of the ionic lattice. Notice that only for neutral systems the sum of the one and the two body potential 4.1 is well defined and convergent.

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