Exchange and correlation energy of an inhomogeneous electron gas at metallic densities

A convenient expression is derived for the coefficient, $B_{\mathrm{xc}}\mathrm{}\left(n\right)\mathrm{}$, which determines the first gradient corrections to the exchange and correlation energy of an inhomogeneous electron gas. The result is exact to all orders in $e^2$ and is expressed in terms of a single-particle propagator. The lowest-order contributions to $B_{\mathrm{xc}}$ are evaluated without the simplifying high-density approximation. The formal structure of the derived expression for $B_{\mathrm{xc}}$ is then used to develop an approximation method for including the contribution of all higher-order graphs within the random-phase approximation. Numerical results are given for $B_{\mathrm{xc}}$ in the metallic density range. Application to surface energy of metals show an improvement over any previous approximations for $B_{\mathrm{xc}}$.