Approximate noninteracting kinetic energy functionals from a nonuniform scaling requirement

We have recently proven an inequality for the exact noninteracting kinetic energy density functional {article}{empty}{document}$ T_s [n]: {lim }x → ∞ T_s [n_λ ^x] ≤ T_s [n_s^y [n] + T_S^2 [n] < ∞,where{ }n_λ ^x (x,{ }y,{ }z) = λ n(λ x,{ }λ { y, }λ z) ${document} . It is known that the gradient expansion through fourth order, T [ n ], violates this inequality. Toward improving T s GE [ n ], we have constructed two new functionals, T s 1 [ n ] and T s 2 [ n ], by keeping the zeroth and second orders in T s GE [ n ] and replacing the fourth order with two simple terms, respectively, so that these new functionals satisfy the inequality. Numerical tests are presented for T s 1 [ n ], T s 2 [ n ], and T s GE [ n ] and for the gradient expansion through second order. Hartree–Fock and hydrogenic atomic densities are employed.