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 .

Read more about this topic: Electronic Density

Famous quotes containing the words general and/or properties:

“They make a great ado nowadays about hard times; but I think that ... this general failure, both private and public, is rather occasion for rejoicing, as reminding us whom we have at the helm,—that justice is always done. If our merchants did not most of them fail, and the banks too, my faith in the old laws of the world would be staggered.”
—Henry David Thoreau (1817–1862)

“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)