Restricted Open-shell Hartree-Fock

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Restricted Open-shell Hartree-Fock (ROHF) is a variant of Hartree-Fock theory for open shell molecules. It uses doubly occupied molecular orbitals as far as possible and then singly occupied orbitals for the unpaired electrons. This is the simple picture for open shell molecules but it is difficult to implement.

As with restricted Hartree-Fock theory for closed shell molecules, it leads to a Roothaan equations written in the form of a generalized eigenvalue problem

$\mathbf{F} \mathbf{C} = \mathbf{S} \mathbf{C} \mathbf{\epsilon}$

Where F is the so-called Fock matrix, C is a matrix of coefficients, S is the overlap matrix of the basis functions, and $ε$ is the (diagonal, by convention) matrix of orbital energies. Unlike restricted Hartree-Fock theory for closed shell molecules, the form of the Fock matrix is not unique. Different so-called canonicalisations can be used leading to different orbitals and different orbital energies, but the same total wavefunction, total energy, and other observables.

In contrast to Unrestricted Hartree-Fock (UHF), the ROHF wave function is a satisfactory eigenfunction of the total spin operator - $\mathbf{S}^2$ .

Developing post Hartree-Fock methods from the ROHF wave function has proved to be more difficult than from the UHF wave function, but a restricted open shell version of Møller-Plesset perturbation theory to second order (ROMP2) has been developed.