Editing Electron density (section)

== General properties ==
From its definition, the electron density is a non-negative function integrating to the total number of electrons. Further, for a system with kinetic energy ''T'', the density satisfies the inequalities<ref name="lieb83">{{cite journal|last=Lieb|first=Elliott H.|year=1983|journal=International Journal of Quantum Chemistry|volume=24|issue=3|pages=243–277|title=Density functionals for coulomb systems|doi=10.1002/qua.560240302}}</ref>

:<math>\frac{1}{2}\int\mathrm{d}\mathbf{r}\ \big(\nabla\sqrt{\rho(\mathbf{r})}\big)^{2} \leq T.</math>

:<math>\frac{3}{2}\left(\frac{\pi}{2}\right)^{4/3}\left(\int\mathrm{d}\mathbf{r}\ \rho^{3}(\mathbf{r})\right)^{1/3}  \leq T.</math>

For finite kinetic energies, the first (stronger) inequality places the square root of the density in the [[Sobolev space]] <math>H^1(\mathbb{R}^3)</math>. Together with the normalization and non-negativity this defines a space containing physically acceptable densities as

:<math>
\mathcal{J}_{N} =
\left\{ \rho \left| \rho(\mathbf{r})\geq 0,\ 
\rho^{1/2}(\mathbf{r})\in H^{1}(\mathbf{R}^{3}),\ 
\int\mathrm{d}\mathbf{r}\ \rho(\mathbf{r}) = N
\right.\right\}.
</math>

The second inequality places the density in the [[Lp space|''L''<sup>3</sup> space]]. Together with the normalization property places acceptable densities within the intersection of ''L''<sup>1</sup> and ''L''<sup>3</sup>&nbsp;– a superset of <math>\mathcal{J}_{N}</math>.