Electronic density

In quantum mechanics, and in particular in quantum chemistry, the electronic density $\rho$ corresponding to an N-electron wavefunction $\Psi ^{(N)}$ is the one-electron function given by

$\rho (x)=\int \ dx_{2}\ ...\ dx_{N}\ |\Psi ^{(N)}(x,x_{2},...,x_{N})|^{2}$

In the case $\Psi ^{(N)}$ is a Slater determinant made of N spin orbitals $\varphi _{k}$:

$\rho (x)={1 \over N}\sum _{k=1}^{N}|\varphi _{k}(x)|^{2}$

The two-electron electronic density is given by

$\rho (x,x')=\int \ dx_{3}\ ...\ dx_{N}\ |\Psi ^{(N)}(x,x',x_{3},...,x_{N})|^{2}$

Those quantities are particularly important in the context of density functional theory.

The coordinates x used here are the spin-spatial coordinates.