Open-Shell Kramers-Restricted Coupled-Cluster Expansions

Open-Shell Kramers-Restricted Coupled-Cluster Expansions 

Open-shell coupled-cluster theory can be formulated in terms of the one-particle spinor or spin-orbital basis. However, spin-restricted or Kramers-restricted open-shell theories are complicated by the ambiguous role of the open-shell orbital or Kramers pair. We develop here the basic outline of open-shell Kramers-restricted coupled-cluster theory. 

We consider first the case of a single electron outside a closed-shell core, for which the reference iV-particle functions form a Kramers pair of states 

Because the open-shell spinor is different for the two partners, the Kramers-unrestricted wave operator cannot be the same. At the least it must differ in the terms that involw the open-shell Kramers pair. Hence we must have two excitation operators, T and t, such that 

This relation gives us a connection between the t amplitudes and the i amplitudes, but it does not provide any relation between t amplitudes for the Kramers partners such as we get for the closed-shell case. This stands to reason, because the amplitudes represent the configuration mixing due to correlation, and we cannot expect the correlation to be the same for a and spin in an open-shell doublet. The incorporation of spin-orbit interaction makes no change to this picture, in which the Pauli repulsion between spin-orbitals of the same spin is transferred to spinors of the same irrep row. 

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The only way to get a wave operator that is symmetric under time-reversal symmetry is to impose the restriction from the beginning. While this fixes the relations between the amplitudes, it also forces the occupied and the unoccupied Kramers components of the open shell to be treated equivalently. This equivalence is what introduces the ambiguity in the treatment of the open shell the open-shell Kramers pair must behave as both a particle and a hole, and the result is that the truncated commutator expansions in the coupled-cluster equations are much longer than in closed-shell theory. The alternative is to use an unrestricted wave operator with the Kramers-restricted spinors. The use of the latter provides some reduction in the work due to the relations between the integrals, but not a full reduction (Visscher et al. 1996). 

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