Nested commutators

The Kronecker delta functions, 5 and 6,p, resulting from Eq. [21], cannot be simplified to 1 or 0 because the indices p and q may refer to either occupied or virtual orbitals. The important point here, however, is that the commutator has reduced the number of general-index second-quantized operators by one. Therefore, each nested commutator from the Hausdorff expansion of H and T serves to eliminate one of the electronic Hamiltonian s general-index annihilation or creation operators in favor of a simple delta function. Since f contains at most four such operators (in its two-electron component), all creation or annihilation operators arising from f will be eliminated beginning with the quadruply nested commutator in the Hausdorff expansion. All higher order terms will contain commutators of only the cluster operators, T, and are therefore zero. Hence, Eq. [52] truncates itself naturally after the first five terms on the right-hand side. ° This convenient property results entirely from the two-electron property of the Hamiltonian and from the fact that the cluster opera-... 

Now consider the first doubly nested commutator from Eq. [106]. The term involving the Eock operator expands to give... 

A similar analysis for the remaining two terms of the doubly nested commutator gives... 

This shows, first that a single commutator of this form results in a single creation operator, and secondly, because of the nested commutator form of Eq. (82), that all such terms in this expression will reduce to a single creation... 

We have used moment expansions in terms of nested commutators in the determination of propagators, and these concepts are useful also in this context. The notation... 

Since f is not anti-Hermitian, it gives rise to a nonunitary transformation and the similarity-transformed Hamiltonian operator is therefore non-Hermitian. Naively, we would expect the BCH expansion (3.1.7) of the similarity-transformed Hamiltonian to yield an infinite sequence of nested commutators. Nevertheless, we shall see that the expansion terminates after five terms. 

Let us now consider the nested commutator of a general number-conserving m-particle operator 6 ... 

The use of a similarity-transformed Hamiltonian in linked coupled-cluster theory means that the energy and amplitude equations contain terms that consist either of the Hamiltonian itself or of nested commutators of the Hamiltonian with cluster operators. For a system containing two noninteracting subsystems A and B, these nested commutators separate additively into nested commutators, each involving a single subsystem, for example. 

The higher-order, nested commutators vanish as discussed in Section 13.2.8. Expanding fi in (13.7.21) and evaluating the resulting commutators using (2.3.36), we may express the transformed creation operator as a linear combination of untransformed operators ... 

It was suggested in Section 14.2.4 that it is advantageous to express the Mpller-Plesset energy corrections in terms of nested commutators. We shall in this subsection see that such commutator expressions are convenient since they make the energy corrections termwise size-extensive, just like the similarity-transformed formulation of coupled-cluster theory makes the coupled-cluster equations termwise size-extensive, as shown in Section 13.3.2. We first consider the perturbed wave function for two noninteracting systems and then go on to consider the separability of the Mpller-Plesset energies. 

The reader may verify that the intersystem terms are similarly absent from the third-order energy correction (14.2.49). Thus, separating the fluctuation potential and the amplitudes into terms pertaining to each subsystem and expanding the nested commutators, we obtain... 

This follows from commutation of the excitation operators amongst each other and the fact that every new nested commutator eliminates one of the maximally four general indices of the electronic Hamiltonian. 

Let us finally consider nested commutators. Nested commutators may be simplified by the same techniques as the single commutators, thus giving rise to rank reductions greater than 1. For example, the following double commutator is easily evaluated using (1.8.14) ... 

The particle rank is reduced by 2. In manipulating and simplifying nested commutators, the Jacobi identity is often useful ... 

In general, therefore, the commutators in the BCH expansions of (3E.4.3)-(3E.4.6) are very simple. Separating the expansion of (3E.4.3) into terms involving even and odd numbers of nested commutators, we obtain... 

Comparing this with the expression for Sk in (3E.5.8), we conclude that the general expression for the nested commutators in (3E.5.3) is given by... 

At K = 0, the zero-order GBT vector is ielectronic gradient E< > (10.1.29), whereas the Jacobian (i.e. the first-derivative matrix) W< > diftes from the electronic Hessian E<2> (10.1.30) in that the nested commutator is not symmetrized. From Section 10.2.1, we conclude that and E are identical at stationary points at nonstationary points, they differ in terms that are proportional to the GBT vector. 

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