One-dimensional kinetic energy density functionals compatible with the differential virial theorem

For a system, of one-dimensional fermions moving in a common effective potential V(x), a method is described to find the most general kinetic energy functional epsilon k which satisfies the requirements that epsilon k (i) is a sufficiently differentiable function of the density rho and of its first n derivatives rho , rho ', . . . rho (n); (ii) is non-negative for arbitrary density distributions >or=0; (iii) obeys the differential virial theorem. The cases n=0, 1, 2 have been worked out yielding the results that epsilon k=k rho 3+ lambda W rho '2/ rho is the only solution compatible with these conditions, where k>or=0 is an indetermined coefficient to the Thomas-Fermi term and lambda W=h(cross)2/8m is the full Weizsacker coefficient.