Commutation relations cluster operators

Note that commutation of cluster operators holds only when the occupied and virtual orbital spaces are disjoint, as is the case in spin-orbital or spin-restricted closed-shell theories. For spin-restricted open-shell approaches, where singly occupied orbitals contribute terms to both the occupied and virtual orbital subspaces, the commutation relations of cluster operators are significantly more complicated. See Ref. 36 for a discussion of this issue. 

Abstract The purpose of this paper is to introduce a second-order perturbation theory derived from the mathematical framework of the quasiparticle-based multi-reference coupled-cluster approach (Rolik and Kallay in J Chem Phys 141 134112, 2014). The quasiparticles are introduced via a unitary transformation which allows us to represent a complete active space reference function and other elements of an orthonormal multi-reference basis in a determinant-like form. The quasiparticle creation and annihilation operators satisfy the fermion anti-commutation relations. As the consequence of the many-particle nature of the applied unitary transformation these quasiparticles are also many-particle objects, and the Hamilton operator in the quasiparticle basis contains higher than two-body terms. The definition of the new theory strictly follows the form of the single-reference many-body perturbation theory and retains several of its beneficial properties like the extensivity. The efficient implementation of the method is briefly discussed, and test results are also presented. 

Along this line, in a recent paper [37] we introduced the so-called quasiparticle-based MR CC method (QMRCC). The mathmatical structure of QMRCC is more or less the same as that of the well-known SR CC theory, i.e., the reference function is a determinant, commuting cluster operators are applied, normal-ordering and diagram techniques can be used, the method is extensive, etc. The point where the MR description appears is the application of quasiparticle slates instead of the ordinary molecular orbitals. These quasiparticles are second-quantized many-particle objects introduced by a unitary transformation which allows us to represent the reference CAS function in a determinant-like form. As it is shown in the cited paper, on one hand the QMRCC method has some advantages with respect to the closely related SR-based MR CC theory [22, 31, 34] (more... 

First, the cluster operator may be chosen to include only those excitation operators that conserve the spin projection of the RHF state so as to satisfy the commutation relation... 

The CCPT energies derived in this subsection are all expressed in terms of commutators. To clarify the relation to the RSPT expression (14.3.32), we may expand the commutators in (14.3.34), retaining only those terms where all cluster operators appear to the right of 4>. The remaining terms may be discarded since vanishes for all k. The resulting expression agrees with the... 

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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