How do restricted open shell calculations mathematically proceed in the context of the self consistent field iterations?

Question

I'm specifically concerned with HF theory. I understand restricted closed shell calculations use the Roothaan equations. If I understand correctly these equations assume each filled molecule orbital is doubly occupied. Each of the electrons in a pair is described with the same spatial wave function.

I also think I understand the Pople-Nesbet equations which allow the wavefunction for each electron to be described with their own spatial wavefunctions furthermore their spins are not constrained hence spin contamination can result.

I however, have not been able to figure out how restricted open shell calculations work. All I understand is that the spin is constrained. I do not understand if the wavefunctions of the paired electrons are described with the same wave function (similar to the Roothaan restricted closed shell approach) while the unpaired electrons are simply allowed to have their own spatial wave function. Coming back to the title question is there a bunch of equations with a famous person’s name (Roothaan, Pople-Nesbet) that are used to describe the scf procedure for restricted open shell calculations or does it use some combination of Roothaan and Pople-Nesbet?

Answer 1

The paper by Tsuchimochi and Scuseria describes it quite well.

Briefly, Roothan-Hall equation is the same but the ROHF matrix is constructed from the $\alpha$ and $\beta$ Fock matrices from UHF.

$$\mathbf{F}_\mathrm{ROHF} = \begin{bmatrix} \mathbf{R}_{cc} & \mathbf{F}_{co}^\beta & \mathbf{F}_{cv}^{cs} \\ \mathbf{F}_{oc}^\beta & \mathbf{R}_{oo} & \mathbf{F}_{ov}^\alpha \\ \mathbf{F}_{vc}^{cs} & \mathbf{F}_{vo}^\alpha & \mathbf{R}_{cc} \\ \end{bmatrix}$$

where $c$ is doubly occupied, $o$ is singly occupied, $v$ is virtual and

$$\begin{align} \mathbf{R}_{cc} &= A_{cc}\mathbf{F}_{cc}^\alpha + B_{cc}\mathbf{F}_{cc}^\beta\\ \mathbf{R}_{oo} &= A_{oo}\mathbf{F}_{oo}^\alpha + B_{oo}\mathbf{F}_{oo}^\beta\\ \mathbf{R}_{vv} &= A_{vv}\mathbf{F}_{vv}^\alpha + B_{vv}\mathbf{F}_{vv}^\beta \end{align}$$

Different values for these parameters have been suggested in the literature. Although they do not affect the ROHF wave function and energy, they affect orbital energies whose physical meaning is obscured because of this dependence.

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1. Takashi Tsuchimochi and Gustavo E. Scuseria, J. Chem. Phys. 2010, 133, 141102. DOI: 10.1063/1.3503173; available as ePrint: arXiv:1008.1607 [cond-mat.str-el].