  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.
