Wide-range shell correction to the Thomas--Fermi theory and equation of   state for electrons
  Shell effects reflects irregularities of physical quantities caused by a discrete energy spectrum. The theory of the shell effects by Kirzhnits and Shpatakovskaya is valid only at relatively low densities providing for oscillations of thermodynamic functions. Similar effects for the electronic binding energy of a neutral atom were considered by Englert and Schwinger. In this work we propose a method of calculation of shell effects applicable in a wide range of density and temperature. The model is based on the finite-temperature Thomas-Fermi theory. Shell corrections to thermodynamic functions are obtained by special accounting of semiclassical states of bound electrons in the Thomas-Fermi potential. The results are in good correspondence with the precise Saha approach for the low density plasma and density functional theory simulation at high density. 
Sergey Dyachkov and Pavel Levashov
physics.plasm-ph