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 ${\mathit{n}}_{\lambda }^{\mathit{x}}$(x,y,z)=λn(λx,y,z) is an example of a nonuniformly scaled electron density. Inequalities are derived for the exact noninteracting kinetic energy ${\mathit{T}}_{\mathit{s}}$[n]. For example, ${\mathit{T}}_{\mathit{s}}$[${\mathit{n}}_{\lambda }^{\mathit{x}}$]≤${\lambda }^2$ ${\mathit{T}}_{\mathit{s}}^{\mathit{x}}$[n]+${\mathit{T}}_{\mathit{s}}^{\mathit{y}}$[n]+${\mathit{T}}_{\mathit{s}}$ ${}_{}{}^{\mathit{z}}{}_{}{}^{}]$, where ${\mathit{T}}_{\mathit{s}}^{\mathit{x}}$,${\mathit{T}}_{\mathit{s}}^{\mathit{y}}$, and ${\mathit{T}}_{\mathit{s}}^{\mathit{z}}$ are the x,y, and z components of ${\mathit{T}}_{\mathit{s}}$. Surprisingly, the gradient expansion through fourth order violates the inequalities. We also observe that the Thomas-Fermi approximation for ${\mathit{T}}_{\mathit{s}}$,${\mathit{T}}_{\mathit{s}}^{\mathrm{TF}}$, and the local-density approximation for the exchange-correlation energy, ${\mathit{E}}_{\mathrm{xc}}^{\mathrm{LDA}}$, do not distinguish between nonuniform scaling along different coordinates. That is, ${\mathit{T}}_{\mathit{s}}^{\mathrm{TF}}$[${\mathit{n}}_{\lambda }^{\mathit{x}}$]=${\mathit{T}}_{\mathit{s}}^{\mathrm{TF}}$[${\mathit{n}}_{\lambda }^{\mathit{y}}$] and ${\mathit{E}}_{\mathrm{xc}}^{\mathrm{LDA}}$[${\mathit{n}}_{\lambda }^{\mathit{x}}$]=${\mathit{E}}_{\mathrm{xc}}^{\mathrm{LDA}}$[${\mathit{n}}_{\lambda }^{\mathit{y}}$]. In contrast, for the true noninteracting kinetic energy it is proved that ${\mathit{T}}_{\mathit{s}}$[${\mathit{n}}_{\lambda }^{\mathit{x}}$]≠${\mathit{T}}_{\mathit{s}}$[${\mathit{n}}_{\lambda }^{\mathit{y}}$] for a general density without special symmetry, and corresponding inequalities are conjectured to apply as well to the exact ${\mathit{E}}_{\mathrm{xc}}$. Moreover, ${\mathit{T}}_{\mathit{s}}^{\mathrm{TF}}$ incorrectly gives the same value for its x, y, and z components.