Connected cluster amplitudes

As emphasized above the coefficients can be represented in terms of a cluster expansion involving connected cluster amplitudes, using any acting as the pivotal function ... 

As already pointed out in Ref. 13, the externally corrected CCSD is equivalent to (truncated) CCSDTQ with zero-iteration on and T4 amplitudes that are in turn obtained from some external sources. Depending on the source of these amplitudes, we usually deal with only a proper subset of all possible T3 and T4 amplitudes. This subset is fixed in the externally corrected CCSD calculations. The RMR CCSD is then a special case of the general externally corrected CCSD in which the MR CISD wave function is used as the external source. The RMR CCSD method represents in fact a multireference approach in the sense that it is uniquely defined by the choice of the reference space and the fact that the RMR CCSD wave function involves the same number of connected cluster amplitudes as the corresponding genuine MR CCSD, such as the state-universal CCSD employing the same reference space. 

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. 

Without truncation, the FCI and full coupled-cluster functions contain the same number of parameters since there is then one connected cluster amplitude for each determinant. In this special case, the Cl and coupled-cluster models provide linear and nonlinear parametrizations of the same state and there is then no obvious advantage in employing the more complicated exponential parametrization. The advantages of the cluster parametrization become apparent only upon tmncation and are related to the fact that, even at the truncated level, the coupled-cluster state contains contributions from all determinants in the FCI wave function, with weights obtained from the different excitation processes leading to the determinants. 

T introduces true n-particle correlation, and products like T T etc., arising out of the expansion eq.(3.2 ), generate simultaneous presence of k-particle amd m-particle correlations in a (k+m)-fold excited determinants etc. The truncation of T == T then corresponds to the pair—correlation model of Sinanoglu, while incorporating higher excited states with several disjoint pair excitations induced through the powers T - The amplitudes for T may be called linked or connected clusters for n electrons. The difficulty of a linear variation method such as Cl lies in its inability to realize the cluster expansion structure eq.(3.2),in a simple and practicable manner. 

Mukherjee/69/, use of the sufficiency conditions (7.3.9) amounts in effect to assuming that ft is a valence-universal wave-operator. In fact Haque has explicitly demonstrated/123/ that the use of a valence-universal ft in the Fock-space Bloch equation leads automatically to eqn (7.3.9) with the ad-hoc sufficiency requirement. We give the sketch of a general proof here, since it shows that the extra information content of a Fock-space ft, as opposed to a Hilbert space, can be used to advantage for ensuring the connectivity of the cluster amplitudes of S/93/. For a valence-universal ft, the Fock-space Bloch equation (6.1.15) leads to... 

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). 

Eq.(65) are the set of working equations for determining the cluster amplitudes. Since each term of (V o K-// exp(r) o V o) consists of a connected cumulant joining K, H and exp(T) o, the cluster amplitudes obtained from eq.(65) will be connected. Since E from eqs.(64) and (63) is generated also from the connected cumulants of (V olfif exp(T) o V )), it is also connected. This proves the size-extensivity of the formalism. Also, since / o separates correctly to the fragments - being a CAS-type function, it also follows that the formalism is size-consistent. 

In the first paradigm, we deliberately retain the linearly dependent ( redundant ) cluster-amplitudes in the wave-operator, but invoke suitable sufficiency conditions which not only provides the extra equations needed for their determination but also ensures that the cluster amplitudes are manifestly connected. In the second paradigm, we use a singular-value decomposition technique to extract only the linearly independent cluster-amplitudes in as unbiased a manner as is possible - without any pivotal implied in the generation of the linearly independent amplitudes. 

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]. 

The basic idea of the externally corrected CCSD methods relies on the fact that the electronic Hamiltonian, defining standard ah initio models, involves at most two body terms, so that the correlation energy is fully determined by one (Ti) and two (T2) body cluster amplitudes, while the subset of CC equations determining these amplitudes involves at most three (T3) and four (T4) body connected clusters. In order to decouple this subset of singly and doubly projected CC equations from the rest of the CC chain, one simply neglects all higher than pair cluster amplitudes by setting... 

The split-amplitude strategy represents the total amplitudes as the sum of an a priori known approximate value, obtained from some external source, and an unknown correcting term. Assuming, further, that the known amplitudes represent a good approximation to the true ones, the unknown corrections can be obtained to a high degree of accuracy from a set of linear equations. The results of this article show that when a proper reference space is used, the connected clusters obtained from the MR CISD wave function represent indeed a very good approximation, and the almost linear versions of the RMR CCSD method performs very well. 

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. ... 

We now prove that the cluster amplitudes of 7) are not only connected but also extensive. To begin with, we of course assume that there are three sets of T Taa, Tbb and Tab, and all appear in a coupled manner in aU the three Eqs. (66), (68), and (69). The extensivity of T follows by... 

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... 

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). 

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