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Explaining the apparent arbitrariness of the LDA-1/2 self-energy correction method applied to purely covalent systems

The LDA-1/2 method expands Slater's half occupation technique to infinite solid state materials by introducing a self-energy potential centered at the anions to cancel the energy associated with electron-hole self-interaction. To avoid an infinite summation of long-ranged self-energy potentials they must be trimmed at a variationally-defined cutoff radius. The method has been successful in predicting accurate band gaps for a large number of elementary and binary semiconductors. Nevertheless, there has been some confusion regarding carbon and silicon, both in the cubic diamond structure, which require different ionizations of the valence charge, 1/2 for carbon and 1/4 for silicon respectively, to yield band gaps in agreement with experimental data. We here analyze the spatial distribution of the valence electrons of these two materials to conclude that in silicon and in carbon LDA-1/4 and LDA-1/2, respectively, must be adopted for the proper cancellation of the self-energies. Such analysis should be applied to other covalent semiconductors in order to decide which ionization to adopt for the proper correction of the self-energy.