Connected and 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. 

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. 

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. 

Notice that unlike in the right eigenvalue problem, Eq. (38), where only connected part of Hat,open Rk,open is needed, both connected and disconnected components of the operator product LkHn,open enter Eq. (51). This is a consequence of the fact that the SRCC cluster operator T, which satisfies Eq. (35), cannot simultaneously satisfy a system of equations (. Hr7V,openG l1 " n ) = 0. As a result, the disconnected part of ( LjcJ 7v,open, i-e. ( ff/v,oPeni/f> does not vanish (cf., e.g., Refs. 59, 60, 64). 

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-... 

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. ... 

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. 

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 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). 

Fig. 1.2. Portion of a random bond percolating cluster backbone, connecting the points A and B. Here, the thick black lines represent the singly connected bonds or red bonds which, if cut, will disconnect the connection between A and B. The bonds in the blob portions are indicated by dotted lines. The dangling bonds are indicated by thin black lines (cf. StauflPer and Aharony 1992).

Percolation describes the geometrical transition between disconnected and connected phases as the concentration of bonds in a lattice increases. It is the foundation for the physical properties of many disordered systems and has been applied to gelation phenomena (de Gennes, 1979 Stauffer et al., 1982). At just above gelation threshold, denoting the fraction of reacted bonds as p and p=Pc + A/ , pc the critical concentration (infinite cluster), the scaling laws (critical exponents) for gel fraction (5oo) and modulus E) are ... 

Again the terms to be included were based on a perturbation order argument.6,16 The contribution of connected triples was shown numerically to be not inconsequential in applications of perturbation theory.19,20 More recently, Purvis and Bartlett16 reported the equations and initial implementation of a full coupled-cluster singles and doubles model (CCSD) this theory includes all terms in the first five parentheses, C0-C4, of Eq. (6) except for Ti, r4, and T Ti. The inclusion of disconnected terms is known to enhance the numerical stability of the coupled equations.21,22... 

Another way of introducing cluster operators is to define the operator 7) to sum only connected /-fold excitation diagrams in P mbpt, and by virtue of defining Cl = exp(T) the disconnected but linked mbpt diagrams are summed as the quadratic and higher terms in the exp(T) expansion. This is the essential relationship of MBPT to coupled-cluster theory. 

Fig. 5.19. A serine octamer cluster built from zwitterionic L-serine molecules. Left Space filling representation Right favorable interactions (electrostatic attraction, hydrogen bonds) holding the cluster together. Bottom View showing a cyclic array of hydrogen bonds connecting the serines OH groups with carboxylates from adjacent serines. Changing the stereochemistry of only one serine will disconnect this array and thus destabilize heterochiral forms.

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). 

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