Self-consistent Madelung Potential Method

1 Self-Consistent Madelung Potential (SCMP) Method 

The whole crystal can be constructed by elementary translations (and eventually by appropriate space-group operations) of a basic unit (unit cell or asymmetric unit). We shall call this basic unit the motif . It is choosen as a closed shell molecule or molecular complex or even a neutral assembly of ions, containing n electrons and M nuclei. Let N be the number of unit cells within the Born-Karman boundaries and denote by Q the elementary translation vectors pointing from the origin to the L-th cell. (For the sake of notational simplicity we shall assume that each unit cell contains only one motif. Generalization to several motifs per unit cell is straightforward.) 

The total Hamiltonian of the crystal is constituted of the Hamiltonians of free motifs (Hl) and interaction terms  

The interaction Hamiltonian can be written in the Longuet-Higgins notation as  

Here we introduced the operator of the total (nuclear and electronic) charge density of the L-th subsystem  

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In very thin films, new effects may take place, because the finite thickness is responsible for further modifications of the Madelung potential. This shows up, for example, when one considers unsupported MgO films, with thicknesses n ranging from 1 to 6 atomic planes and several orientations ((100), (110) and (211)). As a first gross approximation, the atoms may be assumed to remain at their bulk positions. The application of the self-consistent tight-binding method yields the gap width A and the ionic charges Qs borne by the surface atoms, as a function of their coordination number Zg. The results are as follows ... 

Densities of states, band widths and, more generally, all quantities related to the electron delocalization are not very sensitive to the way electron-electron interactions are treated. On the contrary, charge densities, band positions and gap widths require a precise estimation of the Madelung potentials. This is particularly important in the case of polar surfaces but also on non-polar surfaces, when one wishes to compare the surface to the bulk, or several surfaces of different orientations. This is the reason why, in the following, we will only discuss the results obtained by self-consistent methods. 

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