Gradient expansion of the exchange energy from second-order density response theory

The basic idea behind the present work is that an atom is not a linear perturbation of the electron gas. We have thus analyzed the exchange energy of the inhomogeneous electron gas to third order in the deviation from a constant density. We give the symmetry properties obeyed by the corresponding second-order response function Lₓ, and demonstrate how Lₓ gives rise to gradient corrections to the exchange energy. The expansion, which is taken up to sixth order in the density gradient, also includes the Laplacian of the density. In the case of a statically screened Coulomb interaction, we have calculated the coefficients of second- and fourth-order gradient terms both analytically and numerically. In analogy with the corresponding results from linear-response theory, the fourth-order coefficient is shown to diverge as the screening is made to vanish. For the bare Coulomb interaction we have not succeeded in obtaining analytical results, and, due to numerical problems at small-q vectors, our numerically obtained coefficients have an estimated uncertainty of 20%.