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 n_λˣ(x,y,z)={λ}n({λ}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_λˣ]{≤}λ²{T}ₛˣ$[n]+${T}ₛʸ$[n]+${T}ₛ$ $ᶻ]$, where ${T}ₛˣ$,${T}ₛʸ$, and ${T}ₛᶻ$ are the x,y, and z components of ${T}ₛ$. Surprisingly, the gradient expansion through fourth order violates the inequalities. We also observe that the Thomas-Fermi approximation for ${T}ₛ$,${T}ₛTF$, and the local-density approximation for the exchange-correlation energy, ${E}xcLDA$, do not distinguish between nonuniform scaling along different coordinates. That is, ${T}ₛTF$[${n}_{{λ}}ˣ$]=${T}ₛTF$[${n}_{{λ}}ʸ$] and ${E}xcLDA$[${n}_{{λ}}ˣ$]=${E}xcLDA$[${n}_{{λ}}ʸ$]. In contrast, for the true noninteracting kinetic energy it is proved that ${T}ₛ$[${n}_{{λ}}ˣ]≠{T}ₛ$[${n}_{{λ}}ʸ$] for a general density without special symmetry, and corresponding inequalities are conjectured to apply as well to the exact ${E}xc$. Moreover, ${T}ₛTF$ incorrectly gives the same value for its x, y, and z components.