Electronic density

In quantum mechanics, and in particular quantum chemistry, the electronic density is a probabilistic measure of the number of electrons occupying a given element of space. It is a scalar quantity depending upon three spatial variables and is typically denoted as either "ρ"(r) or "n"(r). The density is determined, through definition, by the normalized "N"-electron wave function which itself depends upon 4"N" variables (3"N" spatial and "N" spin coordinates). Conversely, the density determines the wave function modulo a phase factor, providing the formal foundation of density functional theory.

Definition

The electronic density corresponding to a normalized "N"-electron wave function (with r and "s" denoting spatial and spin variables respectively) is defined as [cite book|last=Parr|first=Robert G.|coauthors=Yang, Weitao|title=Density-Functional Theory of Atoms and Molecules|publisher=Oxford University Press|location=New York|date=1989|isbn=0-19-509276-7]

:$ho(mathbf\{r\})=Nsum\_$s}_{1 cdots sum_s}_{N int mathrm{d}mathbf{r}_2 cdots int mathrm{d}mathbf{r}_N |Psi(mathbf{r},s_{1},mathbf{r}_{2},s_{2},...,mathbf{r}_{N},s_{N})|^2.

From this definition, the electron density is a non-negative function integrating to the total number of electrons.

In Hartree-Fock and density functional theories the wave function is typically represented as a single Slater determinant constructed from "N" orbitals, "φ"_"k", with corresponding occupations "n"_"k". In these situations the density simplifies to

:$ho(mathbf\{r\})=sum\_\{k=1\}^N\; n\_\{k\}|varphi\_k(mathbf\{r\})|^2.$

General Properties

The ground state electronic density of an atom is conjectured to be a monotonically decaying function of the distance from the nucleus. [cite journal|last=Ayers|first=Paul W.|coauthors=Parr, Robert G.|date=2003|title=Sufficient condition for monotonic electron density decay in many-electron systems|journal=International Journal of Quantum Chemistry|volume=95|issue=6|pages=877 - 881|doi=10.1002/qua.10622]

Nuclear cusp condition

The electronic density displays cusps at each nucleus in a molecule as a result of the unbounded electron-nucleus Coulomb potential. This behavior is quantified by the Kato cusp condition formulated in terms of the spherically averaged density, , about any given nucleus as [cite journal|last=Kato|first=Tosio |date=1957|title=On the eigenfunctions of many-particle systems in quantum mechanics|journal=Communications on Pure and Applied Mathematics|volume=10|issue=2|pages=151 - 177|doi=10.1002/cpa.3160100201]

:

That is, the radial derivative of the spherically averaged density, evaluated at any nucleus, is equal to twice the density at that nucleus multiplied by the negative of the atomic number ("Z").

Asymptotic behavior

The nuclear cusp condition provides the near-nuclear (small "r") density behavior as

:$ho(r)\; sim\; e^\{-2Z\_\{alpha\}r\},.$

The long-range (large "r") behavior of the density is also known, taking the form [cite journal|last=Morrell|first=Marilyn M.|coauthors=Parr, Robert. G.; Levy, Mel|date=1975|title=Calculation of ionization potentials from density matrices and natural functions, and the long-range behavior of natural orbitals and electron density|journal=Journal of Chemical Physics|volume=62|pages=549-554|doi=10.1063/1.430509]

:$ho(r)\; sim\; e^\{-2sqrt\{2mathrm\{Ir\},.$

where I is the ionization energy of the system.that's all

References