Cluster commutation

Although the similarity-transformed Hamiltonian is quartic in the cluster amplitudes, the equations for the cluster amplitudes (13.2.32) need not contain all the amplitudes to this order. In Section 13.2.8, we use the cluster-commutation condition (13.2.36) to show that, for a general operator O of particle rank mo, the state... 

To summarize the theory dynamic correlations are described by the unitary operator exp A acting on a suitable reference funchon, where A consists of excitation operators of the form (4). We employ a cumulant decomposition to evaluate all expressions in the energy and amphtude equations. Since we are applying the cumulant decomposition after the first commutator (the term linear in the amplimdes), we call this theory linearized canonical transformation theory, by analogy with the coupled-cluster usage of the term. The key features of the hnearized CT theory are summarized and compared with other theories in Table II. 

In addition to the encouraging numerical results, the canonical transformation theory has a number of appealing formal features. It is based on a unitary exponential and is therefore a Hermitian theory it is size-consistent and it has a cost comparable to that of single-reference coupled-cluster theory. Cumulants are used in two places in the theory to close the commutator expansion of the unitary exponential, and to decouple the complexity of the multireference wave-function from the treatment of dynamic correlation. 

It is seen from (60)-(61) that there are two alternative ways to calculate the density variation i) through the transition density and matrix elements of Qfc-operator and ii) through the ground state density. The second way is the most simple. It becomes possible because, in atomic clusters, Vres has no T-odd Ffc-operators and thus the commutator of Qk with the full Hamiltonian is reduced to the commutator with the kinetic energy term only ... 

As we have taken the groupings A,B etc., to refer to true linked clusters TA, Tg etc., the operators T"A, Tg must appear as physically connected entities in the occupation number representations. Eft, Eg etc., will also then appear as connected entities - as a consequence of the multi-commutator expansion generated by eqs. (3.8). Since the groupings are... 

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. 

In this expression, hp = pW q) represeiits a matrix element of the one-electron component of the Hamiltonian, h, while (pqWrs) s ( lcontains general annihilation and creation operators (e.g., or ) that may act on orbitals in either occupied or virtual subspaces. The cluster operators, T , on the other hand, contain operators that are restricted to act in only one of these spaces (e.g., al, which may act only on the virtual orbitals). As pointed out earlier, the cluster operators therefore commute with one another, but not with the Hamiltonian, f . For example, consider the commutator of the pair of general second-quantized operators from the one-electron component of the Hamiltonian in Eq. [53] with the single-excitation pair found in the cluster operator, Tj ... 

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-... 

Using the truncated Hausdorff expansion, we may obtain analytic expressions for the commutators in Eq. [52] and insert these into the coupled cluster energy and amplitude equations (Eqs. [50] and [51], respectively). However, this is only the first step in obtaining expressions that may be efficiently implemented on the computer. We must next choose a truncation of T and then derive expressions containing only one- and two-electron integrals and cluster amplitudes. This is a formidable task to which we will return in later sections. 

Since the two cluster operators act on the reference determinant to produce a total excitation level of +2, we require the same Hamiltonian -2 diagram fragment used in Eq. [164]. Also, because the cluster operators act before the Hamiltonian operator in the matrix element, they are placed at the bottom of the diagram. Furthermore, because the operators commute, their vertical... 

Using the fact that the (0,0) sector cluster operator commutes with the others (16), we can decouple the (0,0) sector calculation from the rest of the FSCC calculation. This requires redefining the wave operator,... 

If we demand that K commutes with the cluster operator 7j we find that... 

If the cluster operator is connected, one can easily show that the dressed Hamiltonian and the matrix elements are also connected via multi-commutator expansion. Hence, the proof of the connectedness of the first term of Eq. (7) is quite... 

Following Sekino and Bartlett[26], the EOM-CCl approach to optical rotation involves elimination of (a) the quadratic terms in Eq. (25) (b) the commutator between the perturbation operator and the perturbed cluster operators in the linear term and (c) the disconnected contributions to the A equations consistent with Eq. (23), resulting in the simpler expression. 

The first commutator has been neglected in in both Eqs. (131) and (132), whereas the remaining two commutators were neglected only in the T2 equation. The removal of entire commutators assures the eize-extensivity of the CCSD(F12) energy [5, 41]. The Eqs. (130)-(132) are of a general form that is not yet suitable for the implementation. In the present work very often the expressions vector function and residual are used. They always refer to the many-index quantity, defined by the right hand sites of these equations. The working expressions of the coupled-cluster Ti, T2 and T2/ residuals are discussed in next subsections. 

Hence, only pairs of creation and annihilation operators survive. Moreover, when the full second-quantized Hamiltonian of Eq. (8.140) enters Eq. (8.244) four such elimination steps leave only the excitation/de-excitation operators of the cluster ansatz for the wave function, such as a a which commute with T. 

---