NONUNIFORM COORDINATE SCALING REQUIREMENTS IN DENSITY-FUNCTIONAL THEORY

For the purpose of improving upon present approximate functionals, nonuniform coordinate scaling is introduced into density-functional theory, where x(x,y,z)=n(x,y,z) is an example of a nonuniformly scaled electron density. Inequalities are derived for the exact noninteracting kinetic energy Ts[n]. For example, Ts[n x] 2Tsx[n]+Tsy[n]+Ts] z, where Tsx,Tsy, and Tsz are the x,y, and z components of Ts. Surprisingly, the gradient expansion through fourth order violates the inequalities. We also observe that the Thomas-Fermi approximation for Ts,TsTF, and the local-density approximation for the exchange-correlation energy, ExcLDA, do not distinguish between nonuniform scaling along different coordinates. That is, TsTF[n x]=TsTF[n y] and ExcLDA[n x]=ExcLDA[n y]. In contrast, for the true noninteracting kinetic energy it is proved that Ts[n x] Ts[n y] for a general density without special symmetry, and corresponding inequalities are conjectured to apply as well to the exact Exc. Moreover, TsTF incorrectly gives the same value for its x, y, and z components. © 1990 The American Physical Society.