Electronic Density - General Properties

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

For finite kinetic energies, the first (stronger) inequality places the square root of the density in the Sobolev space H1(R3). Together with the normalization and non-negativity this defines a space containing physically acceptable densities as

\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\}.

The second inequality places the density in the L3 space. Together with the normalization property places acceptable densities within the intersection of L1 and L3 – a superset of .

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