Kramers-Restricted Coupled-Cluster Expansions

Another correlation method commonly used in nonrelativistic quantum chemistry is the coupled-cluster (CC) method. In this method the wave function is developed by applying an exponential wave operator to an A-particle reference function. 

Note that these excitation operators differ from those of section 9.1. They are given in terms of one-particle creation and annihilation operators as 

To simplify the appearance of the equations, we invoke the Einstein summation convention for this section, in which a summation over repeated indices is made. The expression for T reduces to 

The CCSD method (coupled cluster with single and double excitations) truncates the expansion of f after the double excitation term, T2. The t2 amplitudes obey the relations 

If the reference is a closed-shell determinant, it is symmetric under time reversal. We require that the coupled-cluster wave function is also symmetric under time reversal, because the wave function is nondegenerate and has an even number of electrons. Then 

---

Open-Shell Kramers-Restricted Coupled-Cluster Expansions... 

Here we have to use the operators E q and not the X q. The reason is that to ensure that the commutator expansions in coupled-cluster theory truncate, the wave operator must be expressed in terms of excitation operators between two disjoint one-particle spaces. This expression and the corresponding one for T2 form the basis for the Kramers-restricted CCSD (KRCCSD) method (Visscher et al. 1995). 

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). 

---