Quantum Monte Carlo investigations of density functional theory of the strongly inhomogeneous electron gas

We use a variational quantum Monte Carlo realization of the adiabatic connection technique to calculate the most relevant quantities in Hohenberg-Kohn-Sham density functional theory for several strongly inhomogeneous electron-gas systems. Results for the coupling-constant dependence of the exchange-correlation energy, the pair-correlation function, the exchange-correlation hole, and the exchange and correlation energy densities are presented. Comparisons are made with the interaction strength interpolation (ISI) approximation, the local density approximation (LDA), the gradient expansion approximation (GEA), the generalized gradient approximation (GGA), and the weighted density approximation (WDA). The coupling-constant dependence of the exchange-correlation energy is accurately described by an ISI model that incorporates information on the strong-interaction limit. Unlike either the LDA or GEA, the WDA is successful in describing the nonlocal structure of the exchange-correlation hole. The LDA errors in the exchange-correlation energy density show a remarkable correlation with the Laplacian of the density. The GGA worsens the error in the integrated exchange-correlation energy as the inhomogeneity of the systems increases. This failure is shared by current meta-GGA functionals and is shown to be caused by the inability of these functionals to describe the LDA overestimation (in absolute value) of the exchange energy density around density maxima. It is suggested that this effect could be taken into account by including Laplacian terms in semilocal density functionals.