Disconnected cluster amplitudes

In principle, the coupled-cluster ansatz for the wave function is exact if the excitation operator in Eq. (8.234) is not truncated. But this defines an FQ approach, which is unfeasible in actual calculations on general many-electron systems. A truncation of the CC expansion at a predefined order in the excitation operator T is necessary from the point of view of computational practice. Truncation after the single and double excitations, for instance, defines the CCSD scheme. However, in contrast with the linear Cl ansatz, a truncated CC wave function is still size consistent, because all disconnected cluster amplitudes which can be constructed from a truncated set of connected ones are kept [407]. The maximum excitation in T determines the maximum connected... 

With respect to the determinant /xv), the amplitude is referred to as a connected cluster amplitude and tf t, as a disconnected cluster amplitude. In general, high-order excitations can be reached by a large number of processes or mechanisms, each contributing to the total amplitude with a weight equal to the product of the amplitudes of the individual excitations. 

In agreement with the MPl expression (14.2.21), the first-order wave function (14.3.28) contains contributions only from the connected doubles. To second order (14.3.29), there are contributions from the disconnected quadruples as well as from the connected singles, doubles and triples - in agreement with the MP2 expression (14.2.40). However, whereas the MPPT expression was obtained after extensive algebraic manipulations, (14.3.29) was obtained in a simple manner from the genera] expressions of CCPT. To high orders, a large number of disconnected cluster amplitudes appear in the wave-function corrections - see for example (14.3.30). 

Indeed, when Mr < Mt, the disconnected component of the left-hand side of Eq. (97), i.e. the expression P Ck,open Hn,oPen )i vanishes, since cluster amplitudes defining T, Eq. (41), satisfy equations (78) with n = 1,..., Mr-Equation (99) represents a generalization of the exact Eq. (88) to truncated EOMXCC schemes. Again, the only significant difference between the EOMXCC equations (98) and (99) and their EOMCC analogs (48) and (47), respectively, is the similarity transformed Hamiltonian used by both theories. As in the EOMCC theory, Eqs. (98) and (99) have the same general form (in particular, they rely on the same similarity transformed Hamiltonian) for all the sectors of Fock space. 

Clearly, the resulting wave function has contributions from all Slater determinants, whose expansion coefficients are determined by the cluster amplitudes. The doubly excited determinants, for example, have contributions both from pure double excitations Xfj and from products of two independent single excitations X Xj. The former excitations are known as connected, the latter excitations are known as disconnected. In this manner, the amplitudes of different excitation processes contribute to the same expansion coefficients of the FCI wave function in Eq. (14). 

Exploring the cluster analysis of a finite set of FCI wave functions based on the SU CC Ansatz [224], we realized that by introducing the so-called C-conditions ( C implying either constraint or connectivity , as will be seen shortly), we can achieve a unique representation of a chosen finite subset of the exact FCI wave functions, while preserving the intermediate normalization. (In fact, any set of MR Cl wave functions can be so represented and thus reproduced via an MR CC formalism.) These C-conditions simply require that the internal amplitudes (i.e. those associated with the excitations within the chosen GMS) be set equal to the product of aU lower-order cluster amplitudes, as implied by the relationship between the Cl and CC amplitudes [223], rather than by setting them equal to zero, as was done in earlier IMS-based approaches [205,206] (see also Ref. [225]). Remarkably, these conditions also warrant that all disconnected contributions, in both the elfective Hamiltonian and the coupling coefficients, cancel out, leaving only connected terms [202,223]. 

In the next step, we analyze the structure of the various terms generated after the application of the WT to the matrix element in our working equations and establish that we can systematically eliminate the disconnected portion of M, if we keep track of which components of the composites containing F and G are connected. This particular analysis requires the concept of cumulant decomposition [75, 80, 88, 89] of the density matrix elements of Fjt for various ranks k. Since the final working equations are connected after the elimination of the disconnected terms, the cluster amplitudes of F are connected and are compatible with the connectivity of G. ... 

Here we have to note that eq. (4.48) only reduces to eq. (4.49) in the case when the cluster amplitudes are fully converged. We see that the cancellation of disconnected contributions in the bwccd theory is achieved iteratively and, furthermore, exact cancellation is achieved by the full convergence of cluster amplitudes. It can therefore be concluded that bwccd theory is fully equivalent to the standard ccd approximation, which exploits the linked cluster theorem. Similar arguments have been used to demonstrate the equivalency of bwccsd and ccsd theories [9]. The extension of these arguments to other coupled cluster approximations, such as bwccsdt and ccsDT theories, is straightforward. 

At this stage, it is interesting to compare the Mpller-Plesset corrections (14.2.21) and (14.2.40) with the coupled-cluster wave function as analysed in Section 13.2.2 - see, in particular (13.2.12)-(13.2.15). The MPl correction contains only the connected first-order doubles - no disconnected terms appear at this level. To second order, the MP2 correction contains connected contributions from the second-order singles, doubles and triples amplitudes as well as a disconnected contribution from the first-order doubles - there are no eontributions from the connected quadruples to the MP2 wave function. 

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