Editing Electron density (section)

== Definition ==
The electronic density corresponding to a normalised <math>N</math>-electron [[wavefunction]] <math>\Psi</math> (with <math>\textbf r</math> and <math>s</math> denoting spatial and spin variables respectively) is defined as<ref>{{cite book|last1=Parr|first1=Robert G.|last2=Yang | first2= Weitao|title=Density-Functional Theory of Atoms and Molecules|publisher=Oxford University Press|location=New York|year=1989|isbn=978-0-19-509276-9}}</ref>

:<math>
\rho(\mathbf{r}) = \langle\Psi|\hat{\rho}(\mathbf{r})|\Psi\rangle,
</math>

where the operator corresponding to the density observable is

:<math>\hat{\rho}(\mathbf{r}) = \sum_{i=1}^{N}\ \delta(\mathbf{r}-\mathbf{r}_{i}).</math>

Computing <math>\rho(\mathbf r)</math> as defined above we can simplify the expression as follows.

<math>
\begin{align}
\rho(\mathbf{r})&= \sum_{{s}_{1}} \cdots \sum_{{s}_{N}} \int \ \mathrm{d}\mathbf{r}_1 \ \cdots \int\ \mathrm{d}\mathbf{r}_N  \ \left( \sum_{i=1}^N \delta(\mathbf{r} - \mathbf{r}_i)\right)|\Psi(\mathbf{r}_1,s_{1},\mathbf{r}_{2},s_{2},...,\mathbf{r}_{N},s_{N})|^2 \\
&= 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
\end{align}
</math>

In words: holding a single electron still in position <math>\textbf r</math> we sum over all possible arrangements of the other electrons. The factor N arises since all electrons are indistinguishable, and hence all the integrals evaluate to the same value.

In [[Hartree–Fock]] and density functional theories, the wave function is typically represented as a single [[Slater determinant]] constructed from <math>N</math> orbitals, <math>\varphi_k</math>, with corresponding occupations <math>n_k</math>. In these situations, the density simplifies to

:<math>\rho(\mathbf{r})=\sum_{k=1}^N n_{k}|\varphi_k(\mathbf{r})|^2.</math>