The 'Umbrella' Function Describing the Density of External Electrons Injected in NaCl Crystal

Electrons injected in NaCl crystal are studied using Thomas-Fermi-Dirac-von Weizsacker (TFDW) model. The external potential that contributes to the TFDW energy functional is not only due to the nuclei but also to the internal electrons. In order to obtain the potential term due to the internal electrons, the density of the internal electrons in the crystal is derived from the Fourier transformation of the structure factor equation. The external potential in the TFDW energy functional is equal to the summation of the potential owing to the internal electrons and the potential due to the nuclei. The external potential determines the 'umbrella' function that bounds the minimizer of the TFDW energy functional from above. In this present work, we assume that the density of the internal electrons is not affected by the injected external electrons.