Correction to kinetic energy density using exactly solvable model

An accurate non-gradient-expansion based correction to Thomas--Fermi is developed using solvable model. The used model is a system of $N$ non-interacting electrons moving independently in the Coulomb field of the nuclear charge. The presented correction is applicable for atoms and should be extendable beyond that. The method exploits the fact that the difference between the Thomas--Fermi approximation and the non-interacting kinetic energy is comparable to the difference between the same values inside the proposed solvable model. The numerical experiments show that by adding this correction factor, the precision of Thomas--Fermi approximation is enhanced by an order of magnitude.