Valence charge density and effective charges within the density-response theory.

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In many applications of the electron-response theory a decomposition of the electronic charge distribution \ensuremath{\rho}(x) into contributions ${\ensuremath{\rho}}_{\ensuremath{\alpha}}$(x) of the different sublattices (ion types \ensuremath{\alpha}) is of interest. It is shown that such a decomposition is indeed possible. Starting with the acoustic sum rule the electronic charge density can directly be expressed by the density-response function and the bare-ion pseudopotential, and this representation then allows for a decomposition in a unique way. The integrated partial densities define partial charges ${Z}_{\ensuremath{\alpha}}^{e}$, which are shown to be the electronic contribution to the longitudinal (Callen) charge. All the statements are numerically illustrated with Si as an example. Furthermore, the local-field effect in chemical binding is discussed using the partial densities ${\ensuremath{\rho}}_{\ensuremath{\alpha}}$(x). In the Appendix the relation between the acoustic and Keating's sum rule is investigated with regard to the partial charge densities and the long-wavelength limit of the acoustic-phonon modes.