Electronic Density - Definition

Definition

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

\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

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

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