  We discuss a $\beta$-dependent family of electronic density scalings of the
form $n_\lambda(\R)=\lambda^{3\beta+1}\; n(\lambda^\beta \R)$ in the context of
density functional theory. In particular, we consider the following special
cases: the Thomas-Fermi scaling ($\beta=1/3$ and $\lambda \gg 1$), which is
crucial for the semiclassical theory of neutral atoms; the uniform-electron-gas
scaling ($\beta=-1/3$ and $\lambda\gg 1$), that is important in the
semiclassical theory of metallic clusters; the homogeneous density scaling
($\beta=0$) which can be related to the self-interaction problem in density
functional theory when $\lambda \leq 1$; the fractional scaling ($\beta=1$ and
$\lambda\leq 1$), that is important for atom and molecule fragmentation; and
the strong-correlation scaling ($\beta=-1$ and $\lambda \gg 1$) that is
important to describe the strong correlation limit.
  The results of our work provide evidence for the importance of this family of
scalings in semiclassical and quantum theory of electronic systems, and
indicate that these scaling properties must be considered as important
constraints in the construction of new approximate density functionals. We also
show, using the uniform-electron-gas scaling, that the curvature energy of
metallic clusters is related to the second-order gradient expansion of kinetic
and exchange-correlation energies.
