My question stems from Table 1 of this paper:
"Generalized Hartree–Fock Description of Molecular Dissociation, by Carlos A. Jiménez-Hoyos, Thomas M. Henderson, and Gustavo E. Scuseria" https://pubs.acs.org/doi/10.1021/ct200345a
In particular, I am confused about the notation for the subscript of the elements in the orbital coefficient matrix. For a real RHF ansatz for the two-electron/two-orbital system, for example, we should have
\begin{gather} |\psi\rangle = C_{11}^{\sigma\sigma}|\phi_1\alpha,\, \phi_1\beta\rangle + C_{22}^{\sigma\sigma}|\phi_2\alpha,\, \phi_2\beta\rangle, \end{gather} (where I have changed notation slightly from the original paper by putting the sigma's in the superscript) while for a real UHF, we have \begin{gather} |\psi\rangle = C_{11}^{\sigma\sigma}|\phi_1\alpha,\, \phi_1\beta\rangle + C_{22}^{\sigma'\sigma'}|\phi_2\alpha,\, \phi_2\beta\rangle, \end{gather} where the $\phi_1$ and $\phi_2$ are spatial orbitals and $\alpha$ and $\beta$ are the spin orbitals.
Does the superscript $\sigma\sigma$ vs $\sigma'\sigma'$ signify the variational coefficients? For example, does $C_{11}^{\sigma\sigma}$ correspond to a variation of the alpha and beta spin coefficients with the same variational parameter, while a term say $C_{12}^{\sigma\sigma'}$ corresponds to varying alpha and beta with different parameters? I don't believe this notation is explained in the paper.
I also find similar notation here (unfortunately, no online pdf). Now, RHF, UHF, GHF, etc. are defined by the matrix-structure of the orbital coefficient matrix. It looks like the block-diagonal terms correspond to how we vary the spatial component in the HF ansatz with respect to the different spin sectors (correct?) Do off-diagonal terms in the orbital coefficient matrix correspond to variation of the spin sectors themselves? Is there a more accessible reference that discusses this reasoning in more detail?