  We model the exchange-correlation (XC) energy density of the Si crystal and
atom as calculated by variational Monte Carlo (VMC) methods with a gradient
analysis beyond the local density approximation (LDA). We find the Laplacian of
the density to be an excellent predictor of the discrepancy between VMC and LDA
energy densities in each system. A simple Laplacian-based correction to the LDA
energy density is developed by means of a least square fit to the VMC XC energy
density for the crystal, which fits the homogeneous electron gas and Si atom
without further effort.
