Density Matrices

Before introducing the various density functionals, it is useful to examine density matrices and the exchange correlation-hole. The N-th order density matrix is defined as
$$\begin{align}\gamma_{N}({\bf x}_{1}'{\bf x}_{2}'\cdots{\bf x}_{N}',{\bf x}_{1}{\... ...s{\bf x}_{N}')\Psi^{*}_{N}({\bf x}_{1}{\bf x}_{2}\cdots{\bf x}_{N}). \end{align}$$
From this the first- and second-order reduced density matrices can be defined:
$$\begin{align}\gamma_{1}({\bf x}_{1}',{\bf x}_{1}) &= N \int\cdots\int\Psi_{N}({\... ...1}{\bf x}_{2}\cdots{\bf x}_{N}) \, d{\bf x}_{3}\ldots d{\bf x}_{N}. \end{align}$$
Note that the first-order density matrix integrates to the number of electrons, and the second-order density matrix integrates to the number of electron pairs. Obviously, $\gamma_{1}$ can be obtained from $\gamma_{2}$ by integration,
 $$\begin{align} \gamma_{1}({\bf x}_{1}',{\bf x}_{1}) = \frac{2}{N-1}\int \gamma_{2}({\bf x}_{1}'{\bf x}_{2},{\bf x}_{1}{\bf x}_{2})\, d{\bf x}_{2}. \end{align}$$

Most operators of interest do not involve the spin coordinates, so it is common to integrate over spin, forming the spinless density matrices [75],
$$\begin{align}\rho_{1}({\bf r}_{1}',{\bf r}_{1}) = \int \gamma_{1}({\bf r}_{1}'s_... ...1}{\bf r}_{2}'s_{2},{\bf r}_{1}s_{1}{\bf r}_{2}s_{2})\,ds_{1}ds_{2}. \end{align}$$
The diagonal element of $\rho({\bf r}_{1}',{\bf r}_{1})$ is simply the electron density, $\rho({\bf r}_{1})$. There is a shorthand for the diagonal element of $\rho_{2}$,
$$\begin{align}\rho_{2}({\bf r}_{1},{\bf r}_{2}) = \rho_{2}({\bf r}_{1}{\bf r}_{2},{\bf r}_{1}{\bf r}_{2}). \end{align}$$

Using this new notation the expectation value of the electronic Hamiltonian can be written as
 $$\begin{align} E = \int [-\frac{1}{2}\nabla_{{\bf r}}^{2}\rho_{1}({\bf r}',{\bf r... ...{r_{12}}\rho_{2}({\bf r}_{1},{\bf r}_{2})\,d{\bf r}_{1}d{\bf r}_{2}. \end{align}$$
For restricted HF the last term simplifies to
$$\begin{align}J[\rho]-K[\rho_{1}] = \frac{1}{2}\iint\frac{1}{r_{12}}\rho({\bf r}_... ...\rho_{1}({\bf r}_{1},{\bf r}_{2})\vert^{2}\,d{\bf r}_{1}d{\bf r}_{2} \end{align}$$
where the first-order reduced density matrix, the Fock-Dirac density matrix, is defined in terms of the HF orbitals
$$\begin{align}\rho_{1}({\bf r}_{1},{\bf r}_{2}) = 2 \sum_{i}^{N/2} \psi({\bf r}_{1})\psi^{*}({\bf r}_{2}). \end{align}$$