Unrestricted Hartree-Fock

A simpler way to expand RHF to open shell systems is to introduce separate spatial orbitals for electrons of $\alpha$ and $\beta$ spin:
$$\begin{align}\begin{split} \chi_{2i-1}({\bf x}) = \psi_{i}^{\alpha}({\bf r}) \al... ... \chi_{2i}({\bf x}) = \psi_{i}^{\beta}({\bf r}) \beta(s). \end{split}\end{align}$$
This allows Hartree-Fock theory to give excess $\beta$ electron density at points in the molecule, something which has been seen in experiment and is only possible in RHF if the wavefunction is expanded beyond a single determinant. By the variational principle the UHF energy will be lower than (or equal to) the RHF energy.

These orbitals lead to two density matrices,
$$\begin{align}\begin{split} P_{\mu \nu}^{\alpha} = \sum_{i}^{occ} C_{\mu i}^{\alp... ... = \sum_{i}^{occ} C_{\mu i}^{\beta} C_{\nu i}^{\beta} \\ \end{split}\end{align}$$
and two Fock operators,
$$\begin{align}\begin{split} \hat{F}^{\alpha}({\bf r}_{1}) = \hat{h}({\bf r}_{1}) ... ...at{J}^{\beta}({\bf r}_{1}) - \hat{K}^{\beta}({\bf r}_{1}) \end{split}\end{align}$$
which are combined to form the Pople-Nesbet [24] equations,
$$\begin{align}\begin{split} \sum_{\nu}(\phi_{\mu}\vert\hat{F}^{\alpha}-\varepsilo... ...psilon_{i}^{\beta}\vert\phi_{\nu})C_{\nu i}^{\beta} = 0, \end{split}\end{align}$$
the solution of which gives the molecular orbitals.