First-degree homogeneous N-particle noninteracting kinetic-energy density functionals

It is known in density-functional theory that the noninteracting kinetic-energy density functional T_s[ρ] is not first-degree homogeneous in density scaling. However, it is shown here that, for every particle number N, there is an N-particle noninteracting kinetic-energy density functional T_N[ρ], that is, a density functional that gives the noninteracting kinetic energy for N-particle densities, which is of first-degree homogeneity in the density ρ(r⇀). This gives a powerful tool, a strong requirement, for constructing such functionals. A systematic procedure to obtain the real part of T_N[ρ], the full T_N[ρ] in one-dimension, for each N is also proposed. It is pointed out, further, that in the Euler-Lagrange equations that determine the one-particle orbitals that define T_s[ρ], the Lagrange multiplier that forces the orbitals to yield ρ(r⇀) is not other than the first derivative of T_s[ρ], δT_s[ρ]/δρ(r⇀), which yields a natural derivation of the Kohn-Sham equations. Utilizing the same idea, it is shown for ground states how the Schrödinger equation can be derived from the basics of density-functional theory as well.