Pair clusters

Calculations for nniforin systems showed that the perturbative treatment usually overestimates the fluctuative contribution . Thus more refined, cluster inetods have been developed, such as the ( VM and CFM". They can be extended to nonuniform systems. In particular, the pair cluster (PCA) expression for F c, can be written out analytically" ... 

Different bismuth clusters are apparently formed in reactions between bismuth and bismuth trichloride-aluminum trichloride mixtures, from which the salts (Big) " (AICl4 )3, and (Bi8) + (AICl4 )2 have been isolated 43). A trigonal-bipyramidal structure was predicted for Big (a closo 6 skeletal bond pair system cf. C2B3Hg), and a square anti-prismatic structure for Bi8 , as appropriate for an arachno 8-atom, 11 skeletal bond pair cluster. Similar polyhedral shapes appear likely for other clusters not only of bismuth but also of other heavy main group metals (the anion Pbg ", for example, is isoelectronic with Bi9 " ). 

We have carried out several applications showing the promise of this procedure [63,64], as well as addressed the question of the size-consistency and size-extensivity [65-67], to which we wish to turn our attention again in this paper. Finally, we have also extended the idea of externally corrected (ec) SR CCSD methods [68-70] (see also Refs. [21,24]) to the MR case, introducing the (N, M)-CCSB method [71], which exploits an Preference (A R) CISD wave functions as a source of higher-than-pair clusters in an M-reference SU CCSD method. Both the CMS and (N, M)-CCSD allow us to avoid the undesirable intruder states, while providing very encouraging results. 

Let us recall, finally, that ec CCSD approaches exploit the complementarity of the Cl and CC methods in their handling of the dynamic and nondynamic correlations. While we use the Cl as an external source of higher than pair clusters, Meissner et al. [10,72-74] exploit the CC method to correct the Cl results (thus designing the CC-based Davidson-type corrections). This aspect will also be addressed below. 

Another way to exploit the complementarity of Cl and CC approaches was explored earlier by Meissner et al. [10]. Instead of using Cl as a source of higher-than-pair clusters and correcting CCSD, it exploits the CC theory to correct the MR CISD results. In the spirit of an earlier work on Davidson-type corrections for SR CISD [10], Meissner et al. formulated a CCSD-based corrections for both SR [72] and MR [74] CISD. The latter was later extended to higher lying excited states [73]. 

The three- and four-body clusters play an even more important role in MR CC theories. In contrast to the SR formalism, where the energy is fully determined by one- and two-body clusters, the higher-than-pair clusters enter already the effective Hamiltonian. Consequently, even with the exact one- and two-body amplitudes, we can no longer recover the exact energies [71]. Here we must also keep in mind that the excitation order of various configurations from is not uniquely defined, since a given configuration... 

The above presented data clearly demonstrate the usefulness of the ec CC approaches at both the SR and MR levels. While in the SR case the energy is fully determined by the one- and two-body clusters, and the truncation of the CC chain of equations at the CCSD level can be made exact by accounting for the three- and four-body clusters, the MR case is much more demanding, since the higher-than-pair clusters appear already in the effective Hamiltonian. An introduction of the external corrections is thus... 

Numerous earlier studies of the ec GG methods clearly indicate that the modest size MR GISD wave functions represent the most suitable and easily available source of higher-than-pair clusters for this purpose (see Ref. [21] for an overview). Indeed, these wave functions can be easily transformed to a SR form, whose cluster analysis is straightforward. Moreover, the resulting three- and four-body amplitudes represent only a very small subset of all such amplitudes, namely those which are most important, and which at the same time implicitly account for all higher-order cluster components that are present in the MR GISD wave functions. 

Defect thermodynamics, as outlined in this chapter, is to a large extent thermodynamics of dilute solutions. In this situation, the theoretical calculation of individual defect energies and defect entropies can be helpful. Numerical methods for their calculation are available, see [A. R. Allnatt, A. B. Lidiard (1993)]. If point defects interact, idealized models are necessary in order to find the relations between defect concentrations and thermodynamic variables, in particular the component potentials. We have briefly discussed the ideal pair (cluster) approach and its phenomenological extension by a series expansion formalism, which corresponds to the virial coefficient expansion for gases. 

Table VII. Use Pair Clusters in Merck Index Date Base, Derived by Alphanumeric Clustering Method III.

Examples of thermal skeletal isomerization are as follows. The 11-electron pair cluster c/oi o-Cp2CoNiCB7Hg (Scheme 6) has a structure based upon a bicapped square antiprism (Figure 1). This polyhedron has two distinct types of vertex (i) two apical sites each of which has a connectivity of four within the cage and (ii) eight equatorial or square antiprism sites each of which has a connectivity of five within... 

At very low dopant concentration (less than 1%) oxygen migration is adequately modeled by taking into account the formation equilibrium of simple pair clusters (equations (8.8) or (8.9) with n = 1). so that the concentration of charge carriers (oxygen vacancies) is regulated by the following equations (in the case of a trivalent dopant) ... 

FIGURE 16 A premise-conclusion pair (cluster center) of H NMR spectral knowledge. 

FIGURE 17 A premise-conclusion pair (cluster center) of IR spectral knowledge. Data partially adopted from IR Mentor (BIO-RAD Laboratories, Sadtler Division, 1994). To identify the conclusion substructure, seven IR peaks should be observed in the listed seven bands simultaneously. Intensity is indicated as s, strong, m, medium, and v, variable. 

FIGURE 18 A premise-conclusion pair (cluster center) of mass spectrometry knowledge. Data adopted from Silverstein, Bassler, and Morrill. The terpene substructure has only one free bond, but it may appear anywhere in the substructure. The OH and CH2 groups may be connected to any position on the benzene ring. These Markush structures can be more complicated. 

Our recently developed reduced multireference (RMR) CCSD method [16, 21, 22, 23, 24, 25] represents such a combined approach. In essence, this is a version of the so-called externally corrected CCSD method [26, 27, 28, 29, 30, 31, 32, 33, 34] that uses a low dimensional MR CISD as an external source. Thus, rather than neglecting higher-than-pair cluster amplitudes, as is done in standard CCSD, it uses approximate values for triply and quadruply excited cluster amplitudes that are extracted by the cluster analysis from the MR CISD wave function. The latter is based on a small active space, yet large enough to allow proper dissociation, and thus a proper account of dynamic correlation. It is the objective of this paper to review this approach in more detail and to illustrate its performance on a few examples. 

Although the existing applications of RMR CCSD, and of other versions of the ecCCSD method, have shown considerable promise, much work remains to be done in order to establish optimal sources of higher than pair clusters that would be both reliable and computationally affordable, as well as to determine the limits of applicability of this type of approaches. Here we shall only present a few typical examples that illustrate the potential of this technique, drawing on both the existing applications and recently generated new results. 

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. 

Obviously the pair-cluster function S2 is invariant with respect to a two-particle transformation, i.e. alternatively to (34) we can express S2 as... 

Forty years later, measurements of masses of galaxies from rotation curves, binary pairs, cluster velocity dispersions, and other indicators had accumulated to the point where two brief 1974 reviews by an Estonian trio (3) and an American trio (4) tipped the consensus of the community in favor of... 

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