The theoretical charge density of silicon: Experimental testing of exchange and correlation potentials

31 highly accurate, experimentally measured, structure factors of silicon for reflections from (111) to (880) are used to test various approximations for exchange and correlation potentals. Specifically, the Hartree-Fock method and some of its extensions, and some of its extensions, and the density fuctional theory in the local density approximation (LDA), and two newly developed refinements, namely the generalized gradient approximations (GGAs) of Perdew and Wang (Perdew J P. Chevary J A. Vosko S H. Jackson K A. Pederson M R. Singh D J and Fiolhais C 1992 Phys. Rev. B. 46 6671) and Engel and Vosko (Engel E and Vosko S H 1993 Phys. Rev. B 47 13 164) are used. The multi-configuration hartree-Fock method includfing relativistic effects gives the best description of the core electron densities. The charge density calculated with the GGA of Perdew and Wang leads to structure factors which deviate from experimental data only half as much as previously reported LDA results. The improvement comes mainly from the description of the core charge density. The GGA of Engel and Vosko is even better for the core electrons, but the valence electrons are not described as well. The experimental and theoretical description of the covalent bonding in silicon is studied by means of difference maps and multipole expansions. The limitation of the multipole model is investigated by fitting both experimental and theoretical charge densities.