Corrections to Thomas-Fermi densities at turning points and beyond
  Uniform semiclassical approximations for the number and kinetic-energy densities are derived for many non-interacting fermions in one-dimensional potentials with two turning points. The resulting simple, closed-form expressions contain the leading corrections to Thomas-Fermi theory, involve neither sums nor derivatives, are spatially uniform approximations, and are exceedingly accurate. 
Raphael F. Ribeiro, Donghyung Lee, Attila Cangi, Peter Elliott, Kieron   Burke
quant-ph