T-Matrix Approach to Electron Distribution in Crystals

The wave functions for valence or conduction electrons are given in the form proposed by Phillips and Kleinman. The wave equation for the smooth part of the electron wave functions is rewritten as an integral equation which is solved by using the $t$-matrix formalism. The smooth part of the wave function is then given by a plane wave plus waves arising from electron scattering by the effective potential of the crystal which might contain point imperfections. Approximate expressions for the wave functions, density and energy of the valence electrons and the self-consistent crystal potential are given. Numerical results obtained for the Fourier coefficients of the valence electron charge density in diamond exhibit covalent bonding and are in good agreement with experimental results. The presented treatment of valence or conduction electrons is expected to be particularly useful in determining the electronic structure and the formation and migration energies of point defects in valence crystals.