Coupled cluster theory particle excitation operator

First, we note that the determination of the exact many-particle operator U is equivalent to solving for the full interacting wavefunction ik. Consequently, some approximation must be made. The ansatz of Eq. (2) recalls perturbation theory, since (as contrasted with the most general variational approach) the target state is parameterized in terms of a reference iko- A perturbative construction of U is used in the effective valence shell Hamiltonian theory of Freed and the generalized Van Vleck theory of Kirtman. However, a more general way forward, which is not restricted to low order, is to determine U (and the associated amplitudes in A) directly. In our CT theory, we adopt the projection technique as used in coupled-cluster theory [17]. By projecting onto excited determinants, we obtain a set of nonlinear amplitude equations, namely,... 

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). 

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