Electronic density

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.

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

where the operator corresponding to the density observable is

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

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

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

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