Notation for orbital coefficient matrix in RHF vs UHF

Question

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?

Answer 1

They are spin-blocking the canonical MO coefficients $\mathbf{C}$, in other words treating the spin orbitals as two-component spin vectors. I think this spin-structure is easier to see with one-body operators like the Hamiltonian (Fock operator) or density matrix, but we can still apply the spin-blocking in this case.

Say we have an atomic spin orbital \begin{gather} |\Phi\rangle = |\phi\rangle|\sigma\rangle = \pmatrix{\phi_\alpha\phi_\beta}, \end{gather} where $|\phi\rangle$ is the spatial component and $|\sigma\rangle \in \{|\alpha\rangle, |\beta\rangle\}$ (spin-up and spin-down, respectively).

Then the general canonical spin orbital $|\Psi\rangle$ can be obtained from the coefficient matrix $\mathbf{C}$ as \begin{gather} |\Psi\rangle = \mathbf{C} |\Phi\rangle = \pmatrix{ C_{\alpha\alpha} & C_{\alpha\beta} \\ C_{\beta\alpha} & C_{\beta\beta} } \pmatrix{ \phi_\alpha \\ \phi_\beta }. \end{gather}

In the RHF case, $C_{\alpha\alpha} = C_{\beta\beta}$, and so \begin{gather} |\Psi\rangle = \mathbf{C} |\Phi\rangle = \pmatrix{ C_{\alpha\alpha} & 0 \\ 0 & C_{\alpha\alpha} } \pmatrix{ \phi_\alpha \\ \phi_\beta } = C_{\alpha\alpha} \pmatrix{ \phi_\alpha \\ \phi_\beta }, \end{gather} which describes two spin-paired electrons in a single spatial orbital.

In the UHF case, $C_{\alpha\alpha} \neq C_{\beta\beta}$, and so \begin{gather} |\Psi\rangle = \mathbf{C} |\Phi\rangle = \pmatrix{ C_{\alpha\alpha} & 0 \\ 0 & C_{\beta\beta} } \pmatrix{ \phi_\alpha \\ \phi_\beta } = \pmatrix{ C_{\alpha\alpha} \phi_\alpha \\ C_{\beta\beta}\phi_\beta }, \end{gather} which describes two opposite spin electrons in a separate spatial orbitals.

In the GHF case, where we relax the requirement for electrons to be eigenfunctions of $\hat{S}_z$, we can write the general case as \begin{gather} |\Psi\rangle = \mathbf{C} |\Phi\rangle = \pmatrix{ C_{\alpha\alpha} & C_{\alpha\beta} \\ C_{\beta\alpha} & C_{\beta\beta} } \pmatrix{ \phi_\alpha \\ \phi_\beta } = \pmatrix{ C_{\alpha\alpha} \phi_\alpha + C_{\alpha\beta} \phi_\beta\\ C_{\beta\alpha} \phi_\alpha + C_{\beta\beta}\phi_\beta }. \end{gather} All the $C_{\sigma\sigma'}$ coefficients are independent variational parameters, subject to orthonormality constraints. Moreover, coefficients may be real or complex-valued.