Kramers-Restricted Open-Shell Methods

The answer to this question is a qualified yes , and depends on the nature of the wave function. For a single electron outside a closed shell, the Kramers-restricted wave function is a single determinant, and it is relatively easy to define the Kramers-restricted Fock matrices for the closed-shell and open-shell electrons. Using the expressions for the Fock matrix in the molecular basis, (11.34), with indices i and j for doubly occupied Kramers pairs, t for the singly occupied Kramers pair, and a for the empty or virtual Kramers pairs, we can define the three nonredundant sections of the Fock matrix as follows  

These expressions are actually derived from the average energy of the two degenerate states. If we were to use only one of the states we would arrive at different expressions for the barred and unbarred spinors. The same is true in nonrelativistic theory. The expressions have also been divided by the relevant occupation number, so that 

Technically, it is possible to define methods in which other restrictions than those on spin are lifted, but the most common unrestricted method is the spin-unrestricted method, and we follow the common usage of UHF for the spin-unrestricted Hartree-Fock method. 

Nevertheless, let us analyze the Fock matrices for the wave functions based on the linear combinations. The closed-empty, closed-open and open-empty Fock matrices are 

In these expressions we have added canceling terms such as it tt) — it tt) and (iu iiu) to make it clear that the Fock matrices for the open-shell electrons are equivalent. The sum over v covers the two values t and u, and w can be either t or u. We must also consider the open-open Fock matrices. The off-diagonal matrix element has no contribution from interaction between the open-shell electrons, 

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Simple open shell cases may also be treated via this kind of perturbation theory. The high spin case with one electron outside a closed shell is of course easy when an unrestricted formalism is used. Dyall also worked out equations for the restricted HE formalism and the more complicated case of two electrons in two Kramers pairs outside a closed shell [32]. Also in this method the crucial step remains the efficient formation of two-electron integrals in the molecular spinor basis. 

However, if the product of the fermion irreps for the open shells belongs to a singly degenerate boson irrep, the wave function is symmetric under time reversal, and hence there ought to be a way of devising a Kramers-restricted method. We write the reference as... 

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