Exchange and correlation in silicon

A combination of the coupling constant integration technique and the quantum Monte Carlo method is used to investigate the most relevant quantities in Kohn-Sham density-functional theory. Variational quantum Monte Carlo is used to construct realistic many-body wave functions for diamond-structure silicon at different values of the Coulomb coupling constant. The exchange-correlation energy density along with the coupling constant dependence and the coupling-constant-integrated form of the pair-correlation function, the exchange-correlation hole, and the exchange-correlation energy are presented. Comparisons of these functions are made with results obtained from the local-density approximation, the average density approximation, the weighted density approximation, and the generalized gradient approximation. We discuss reasons for the success of the local-density approximation. The insights provided by this approach will make it possible to carry out stringent tests of the effectiveness of exchange-correlation functionals and in the long term aid in the search for better functionals.