Restricted Open Shell Hartree-Fock

The Roothaan-Hall equations are unsuitable for an open-shell system, and require some modification. One approach is to allow some orbitals to contain only an electron of $\alpha$ spin [22,23]. Under such a scheme the energy expression becomes
 $$\begin{align} \begin{split} E = & \sum_{i}2(\psi_{i}\vert\hat{h}\vert\psi_{i}) +... ...psi_{i}\psi_{i})-(\psi_{i}\psi_{s}\vert\psi_{i}\psi_{s})] \end{split}\end{align}$$
where i,j denote doubly occupied orbitals and s,t denote singly occupied orbitals.

Proceeding as before, we consider the variations
$$\begin{align}\begin{split} \psi_{i} \rightarrow & \psi_{i} + \lambda \psi_{a} \\... ...} \\ \psi_{s} \rightarrow & \psi_{s} - \lambda \psi_{i}. \end{split}\end{align}$$
Substituting these into equation (1.40) and minimizing the energy gives the self consistent conditions
 $$\begin{align} \begin{split} F_{ai} = & \; 0 \\ F_{sa} - \frac{1}{2}K_{sa}^{O} = & \; 0 \\ F_{si} + \frac{1}{2}K_{si}^{O} = & \; 0 \\ \end{split}\end{align}$$
where
$$\begin{align}\mathbf{F} = \mathbf{H} + 2\mathbf{J}^{C}-\mathbf{K}^{C}+\mathbf{J}^{O}-\frac{1}{2}\mathbf{K}^{O} \end{align}$$
with the superscripts C and O denoting summation over closed shell and open shell orbitals respectively. From this we can define Fock matrices for the $\alpha$ and $\beta$ electrons

$$\mathbf{F}^{\alpha} = \mathbf{F} - \frac{1}{2}\mathbf{K}^{O}$$ (5.3)

$$\mathbf{F}^{\beta} = \mathbf{F} + \frac{1}{2}\mathbf{K}^{O}.$$ (5.4)

Orbitals can then be found that satisfy the conditions (1.42) by diagonalisation of the block matrix
$$\begin{align}\left( \begin{array}{ccc} (\mathbf{F}-\mathbf{K}^{O}) & \mathbf{F}... ...thbf{F}^{\alpha} & (\mathbf{F}-\mathbf{K}^{O}) \end{array} \right) \end{align}$$
where the three blocks refer to doubly occupied, singly occupied and virtual orbitals. The diagonal blocks do not affect the stationary conditions, so can be defined in any desired way. Roothaan's original definition has been used above.

The ROHF scheme places an unphysical constraint on the wavefunction. $\alpha$ and $\beta$ electrons in an open shell molecule may feel different potentials, yet their spatial orbitals are constrained to be the same. This has the effect of raising the variational energy. The ROHF wavefunction must also be of high spin.