Disconnected clusters

There are a number of useful points to be drawn from this analysis of the water cluster distribution. First, it appears that the cumulative water cluster distribution with a properly chosen cutoff distance is a metric that allows one to see clearly differences in connectivity of the aqueous domain as a function of both water content and polymer architecture. Clearly, Fig. 9(b) shows the connectivity of the aqueous domain moving from many small disconnected clusters to a single sample-spanning cluster as a func-... 

Cl approach, the above noted cancellation would mean numerical (a posteriori) justification of Davidson s correction. The last entry listed in the family of "various quantities" in Table 4,10, mimics the effect of augmenting the Cl wave function with the quadruply excited configurations. Agreement between the fourth-order MB-RSPT and Cl is remarkable on this point. It should be realized, however, that in MB-RSPT a part of disconnected clusters (those from higher orders) is mis-... 

Simple computer experiments (which employ 6-8 million water molecules) in which various fractions of H-bonds in ordinary ice are allowed to break are presented (6.1-6.2). The results of our calculations show that the small fraction of broken H-bonds (13-20%), which is usually considered enough for melting, is not sufficient to break up the network of H-bonds into separate clusters. Consequently, liquid water can be considered to be a deformed network with some ruptured H-bonds. The cooperative effect, first suggested by Frank and Wen, was examined by combining an ab initio quantum mechanical method with a combinatorial one (6.2). In agreement with the results obtained in (6.1), it is shown that 62-63% of H bonds must be broken in order to disintegrate a piece of ice (containing 8 million water molecules) into disconnected clusters. 

A rapid increase in the importance of higher-than-pair clusters is clearly illustrated by the sequence of SR Cl results. Even the SR CISDTQ NPEs amount to 12.6 and 38 mhartree. This clearly indicates the role played by higher-than-4-body (both connected and disconnected) clusters as R —> oo. 

Unfortunately, investigations of disconnected clusters are not going to inform us as to what happens on a face of AgBr>I. 

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... 

Unlike the situation in the many body methods, the connected and disconnected cluster contributions to each excited CSF are inextricably combined in the ci formalism. 

The purpose of the present section is to introduce the coupled-cluster model. First, in Section 13.1.1, we consider the description of virtual excitation processes and correlated electronic states by means of pair clusters. Next, in Section 13.1.2, we introduce the coupled-cluster model as a generalization of the concept of pair clusters. After a discussion of connected and disconnected clusters in Section 13.1.3, we consider the conditions for the optimized coupled-cluster state in Section 13.1.4. 

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. 

The most common approximation in coupled-cluster theory is to truncate the cluster operator at the doubles level, yielding the coupled-cluster singles-and-doubles (CCSD) model [5]. In this model, the T2 operator describes the important electron-pair interactions and T carries out the orbital relaxations induced by the field set up by the pair interactions. The CCSD wave function contains contributions from all determinants of the FCl wave function, although the highly excited determinants, generated by disconnected clusters, are in general less accurately described than those that also contain connected contributions. However, the disconnected contributions may... 

In the coupled-cluster calculations, the error decreases by about the same factor at each excitation level. Also, at a given truncation level, the error is significantly smaller than that of the Cl energy, which converges in a less systematic and satisfactory manner because of the neglect of higher-order disconnected clusters. As an example, the CCSD error falls between the CISDT and CISDTQ errors... 

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). 

Except for the explicit expression of D anti-red in d > 2, the main lines of reference [9] are correct. The equality of the disconnecting and reconnecting bonds leads to an equivalent number of clusters containing connected and disconnected clusters of size s Ncp Njjp N N being the number of boxes of side Rg mean radius of s-clusters covering the rimeter. Applied to the Bethe lattice this gives,... 

Chapter 13 discusses coupled-cluster theory. Important concepts such as connected and disconnected clusters, the exponential ansatz, and size-extensivity are discussed the Unked and unlinked equations of coupled-clustCT theory are compared and the optimization of the wave function is described. Brueckner theory and orbital-optimized coupled-cluster theory are also discussed, as are the coupled-cluster variational Lagrangian and the equation-of-motion coupled-cluster model. A large section is devoted to the coupled-cluster singles-and-doubles (CCSD) model, whose working equations are derived in detail. A discussion of a spin-restricted open-shell formalism concludes the chapter. 

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