Deriving Density Functionals

Recent developments have shown that density functionals can be derived from semiclassical quantum mechanics in a systematic fashion reminiscent of wavefunction electronic structure. This new technique is completely different from other approaches to functional design (such as the satisfaction of exact conditions or fitting empirical parameters). The Thomas-Fermi kinetic energy functional, which Lieb and Simon showed is exact in the semiclassical limit, is the lowest order term in a semiclassical expansion of the exact kinetic energy functional. We will show, at least in one dimension, how higher order corrections to this series can be obtained by calculating the sums of eigenvalues and then inverting these sums into density functionals. Thus we show how to generate the exact asymptotic expansion of the true kinetic energy functional. We will also demonstrate that boundary terms arise from the turning points/surfaces where the density cannot be slowly varying, which are missed by the traditional gradient expansion. To simplify our analysis we have worked in one dimension but we shall comment on the application of our work to real electronic systems.