Semiclassical Approximation to Sums of Eigenvalues with Application to DFT

I present a new mathematical framework developed by Kieron Burke and Michael V. Berry for estimating the sum of the lowest N eigenenergies of a one-dimensional potential. I apply this method to several model onedimensional systems (harmonic oscillator, Poschl-Teller well, quartic oscillator, linear well, and exponential well). This new method gives the sum of the energies as a semiclassical series, which can be shown to reproduce the DFT gradient expansion for slowly varying densities, and also produces a correction to the gradient expansion for finite systems with a discrete spectrum. Explicit corrections to the gradient expansion of the kinetic energy are derived which in simple cases greatly improve accuracy. All work is done assuming a system of non-interacting identical fermions. This research is funded by the NSF (CHE 1856165).