Excitation operators commutators

Second, the order of the operators in (13.1.7) is unimportant since the excitation operators commute ... 

In Boxes 13.1 and 13.2, we have listed only those commutators that contain the single-excitation operators. Commutators involving double and higher exeitations are easily expressed in terms of the single excitations, for instance ... 

Since the active-active excitation operators commute with the core string and annihilate the vacuum state, we may write... 

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. 

The commutator between two elementary excitation operators is in next section shown to be... 

The commutator relation (3 8) for the excitation operators together with the adjoint relation (3 9) leads to the following symmetry properties for the coupling coefficients (for real-valued wave functions) ... 

Express the spin dgeantbS in terms of annihilation and creation operators. Then show that the excitation operators Ey commute with these spin operators. 

We start with the equation for the orbital gradient (4 9). We then have to compute the commutator between the Hamiltonian (3 24) and an excitation operator E. To do that we use the commutator relation for the excitation operators as given in equation (3 8). The result of the computation is ... 

Let us finally take a closer look at the orbital Hessian matrix, H(00). The calculation now involves the evaluation of commutators between the Hamiltonian and products of excitation operators according to equation (4 14). In spite of the rather tiring algebra, the result takes a surprisingly simple form ... 

Consider the excitation operator k (k corresponding to the elementary transition relating k (root state) to a target state k) F,f acting from the right to commutator [H, k)(k ] yields ... 

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... 

Linear terms are absent because of the Brillouin theorem. The coefficients Ap p. and Bap p, can be calculated by equating the nonzero matrix elements of the RPA Hamiltonian [Eq. (122)], in the basis of Eq. (121), to the corresponding matrix elements of the exact Hamiltonian [Eq. (23)] in the same basis. From the translational symmetry of the mean field states it follows that the A and B coefficients do not depend on the complete labels P = n, i, K and P = n, /, K1, but only on the sublattice labels /, AT and /, K. The second ingredient of the RPA formalism is that we assume boson commutation relations for the excitation and de-excitation operators (Raich and Etters, 1968 Dunmore, 1972). 

We note that the excitation operator Dt defined by (2.41) is expressed in terms of the linearly independent operators in the sets BT and TBr T, which form a new basis we note that the set BT as before contains exactly p linearly independent elements. It is evident that the coefficients in this basis are no longer necessarily optimal, and we may hence again start out with the double-commutator relations (1.49) and (1.50), etc.,... 

Still the purpose of this article is to advocate that the time is now ripe to attack the Liouvillian eigenvalue problem LC = vC directly in terms of single-commutator methods and secular equations of the type (2.16). This approach should further be combined with ket-bra methods of the type developed in Section II in order to decompose the eigenelements C associated with degenerate eigenvalues v into components having the form of excitation operators of the type C = TfXTj. ... 

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. 

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). 

Since the excitation operators (13.1.2) commute among one another, there are no problems with the order of the operators in this expression. The resulting wave function (13.1.5) corresponds to a particularly simple realization of the coupled-cluster model, in which only double excitations are allowed the coupled-cluster doubles (CCD) wave function. For a complete specification of this model, we must also describe the method by which the amplitudes are determined. We shall... 

Commutation occurs since the excitation operators always excite from the set of occupied Hartree-Fock spin orbitals to the virtual ones - see (13.1.2) for the double-excitation operators. The creation and annihilation operators of the excitation operators therefore anticommute. 

Each excitation operator in (13.2.5) excites at least one electron from an occupied to a virtual spin orbital. Since there are N electrons in the system, the expansion (13.2.5) terminates after T/v. We further note that the excitation operators given by (13.2.6) and (13.2.7) satisfy the commutation relation (13.1.9) and that the following relationship holds... 

The projected coupled-cluster Schrodinger equation (13.2.32) therefore yields at most quartic equations in the cluster amplitudes - even for the full cluster expansion. The BCH expansion terminates because of the special structure of the cluster operators, which are linear combinations of commuting excitation operators of the form (13.2.6) and (13.2.7). 

As noted in Section 13.6.1, we would like the EOM-CC excited states (13.6.2) to be orthonormal. Orthonormality in the usual sense presents problems of the sort discussed in Section 13.1.4. Indeed, even the calculation of the norm of the ground state (CC CC> is cumbersome since the excitation operators in T do not commute with the de-exdtation operators in 7. We solve this problem by resorting to biorthogonality, expanding the bra states in a set of configurations that, toother wddi the ket states (13.6.2), constitute a biorthonormal set Adc )ting the notation... 

In Box 13.1, we have listed all nonzero conunutators between the Hamiltonian operator and the single-excitation operators. Usually, we are not so much interested in the commutators themselves as in their application to the Hartree-Fock state see Box 13.2. In these boxes [22], the operators l carry out permutations of the pair indices in the following manner ... 

This expression follows since, upon expansion of the commutators, all terms that contain excitation operators to the left of the Hamiltonian vanish. We may therefore write the CCSD eneigy (13.7.12) in the form... 

Box 13.2 The Hamiltonian and its nonzero commutators with the single-excitation operators applied to the Hartree-Fock state. The commutators are listed in Box 13.1... 

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 excitation operators, which were introduced in Section 14.2.1, commute among themselves but not with their adjoints. In the spin-orbital basis, the determinants generated by the excitation operators (14.3.3) are orthonormal. 

Before we study the CCPT expansion in any detail, it is convenient to rewrite the electronic energy and the Schrodinger equation in terms of the Fock operator and the fluctuation potential. From the commutator (14.2.14) between the Fock operator and the excitation operators, we obtain the following commutators between the Fock and cluster operators... 

In setting up these expressions, we have used the fact that 7" vanishes (14.3.18) and that the cluster operators commute. Expanding the terms in excitation levels according to Table 14.2, we... 

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