Electronic density.html

In quantum mechanics, and in particular quantum chemistry, the electronic density is a measure of the probability of an electron occupying an infinitesimal element of space surrounding any given point. 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 wavefunction which itself depends upon 4N variables (3N spatial and N spin coordinates). Conversely, the density determines the wave function modulo a phase factor, providing the formal foundation of density functional theory.

1 Definition
2 General Properties
3 Topology
- 3.1 Nuclear cusp condition
- 3.2 Asymptotic behavior
4 See also
5 References

Definition

The electronic density corresponding to a normalized N-electron wavefunction (with r and s denoting spatial and spin variables respectively) is defined as¹

$\begin{align} \rho(\mathbf{r})&=N\sum_{{s}_{1}} \cdots \sum_{{s}_{N}} \int \ \mathrm{d}\mathbf{r}_2 \ \cdots \int\ \mathrm{d}\mathbf{r}_N \ |\Psi(\mathbf{r},s_{1},\mathbf{r}_{2},s_{2},...,\mathbf{r}_{N},s_{N})|^2, \\ &= \langle\Psi|\hat{\rho}(\mathbf{r})|\Psi\rangle, \end{align}$

where the operator corresponding to the density observable is

$\hat{\rho}(\mathbf{r}) = \sum_{i=1}^{N}\sum_{s_{i}}\ \delta(\mathbf{r}-\mathbf{r}_{i}).$

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

$\rho(\mathbf{r})=\sum_{k=1}^N n_{k}|\varphi_k(\mathbf{r})|^2.$

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 inequalities²

$\frac{1}{2}\int\mathrm{d}\mathbf{r}\ \big(\nabla\sqrt{\rho(\mathbf{r})}\big)^{2} \leq T,$

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

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 $\mathcal{J}_{N}$ .

Topology

The ground state electronic density of an atom is conjectured to be a monotonically decaying function of the distance from the nucleus.³

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, $\bar{\rho}$ , about any given nucleus as⁴

$\left.\frac{\partial}{\partial r_{\alpha}}\bar{\rho}(r_{\alpha})\right|_{r_{\alpha}=0} = -2Z_{\alpha}\bar{\rho}(0).$

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

$\rho(r) \sim e^{-2Z_{\alpha}r}\,.$

The long-range (large r) behavior of the density is also known, taking the form⁵

$\rho(r) \sim e^{-2\sqrt{2\mathrm{I}}r}\,.$

where I is the ionization energy of the system.

References

^ Parr, Robert G.; Yang, Weitao (1989). Density-Functional Theory of Atoms and Molecules. New York: Oxford University Press. ISBN 0-19-509276-7.
^ Lieb, E. H. (1983). "Density functionals for coulomb systems". International Journal of Quantum Chemistry 24: 243-277. doi:10.1002/qua.560240302.
^ Ayers, Paul W.; Parr, Robert G. (2003). "Sufficient condition for monotonic electron density decay in many-electron systems". International Journal of Quantum Chemistry 95 (6): 877 - 881. doi:10.1002/qua.10622.
^ Kato, Tosio (1957). "On the eigenfunctions of many-particle systems in quantum mechanics". Communications on Pure and Applied Mathematics 10 (2): 151 - 177. doi:10.1002/cpa.3160100201.
^ Morrell, Marilyn M.; Parr, Robert. G.; Levy, Mel (1975). "Calculation of ionization potentials from density matrices and natural functions, and the long-range behavior of natural orbitals and electron density". Journal of Chemical Physics 62: 549-554. doi:10.1063/1.430509.

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