Cluster series expansion

Beyond the MSA Cluster Series Expansion and Generalized Mean... 

We proceed with cluster series which yield the integral equations. Evidently the correlation functions presented above can be defined by their diagrammatic expansions. In particular, the blocking correlation function is the subset of graphs of h rx2), such that all paths between... 

To derive working expressions for the dispersion coefficients Dabcd we need the power series expansion of the first-order and second-order responses of the cluster amplitudes and the Lagrangian multipliers in their frequency arguments. In Refs. [22,29] we have introduced the coupled cluster Cauchy vectors ... 

To find the power series expansion of Eq. (30) in ub, ojc, u>d we can thus replace the first-order responses of the cluster amplitudes and Lagrangian multipliers and the second-order responses of the cluster amplitudes by the expansions in Eqs. (37), (39) and (44) and express OJA as —ojb ojc — ojd- However, doing so starting from Eq. (30) leads to expressions which involve an unneccessary large number of second-order Cauchy vectors C m,n). To keep the number of second-order... 

Figure 3a shows the mean-field predictions for the polymer phase diagram for a range of values for Ep/Ec and B/Ec. The corresponding simulation results are shown in Fig. 3b. As can be seen from the figure, the mean-field theory captures the essential features of the polymer phase diagram and provides even fair quantitative agreement with the numerical results. A qualitative flaw of the mean-field model is that it fails to reproduce the crossing of the melting curves at 0 = 0.73. It is likely that this discrepancy is due to the neglect of the concentration dependence of XeS Improved estimates for Xeff at high densities can be obtained from series expansions based on the lattice-cluster theory [68,69].

One such systematic generalization was obtained by Cohen,8 whose method is now given the point of departure was the expansion in clusters of the non-equilibrium distribution functions. This procedure is formally analogous to the series expansion in the activity where the integrals of the Ursell cluster functions at equilibrium appear in the coefficients. Cohen then obtained two expressions in which the distribution functions of one and two particles are given in terms of the solution of the Liouville equation for one particle. The elimination of this quantity between these two expressions is a problem which presents a very full formal analogy with the elimination (at equilibrium) of the activity between the Mayer equation for the concentration and the series... 

These expressions are analogous to the series expansions of the equilibrium distribution functions in terms of the activity in which appear, in the coefficients, the integrals of the Ursell cluster functions Us (see, for example, ref. 30). 

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. 

In the mass spectrometric investigations of most hydrogen-bonded clusters, the protonated clusters form the dominant cluster ions as a result of a rapid proton transfer reaction (Mark and Castleman 1985). Castleman and coworkers have extensively investigated the neat expansions of methanol (Morgan and Castleman 1987, 1989 Morgan et al. 1989 Zhang et al. 1991) and observed the protonated cluster ions to form the most intense cluster series. In the case of methanol, the exoergic proton transfer reaction may be written as ... 

The exponential ansatz described above is essential to coupled cluster theory, but we do not yet have a recipe for determining the so-called cluster amplitudes (tf. If-- , etc.) that parameterize the power series expansion implicit in Eq. [31]. Naturally, the starting point for this analysis is the electronic Schrodinger equation,... 

Where H is the similarity-transformed Hamiltonian, eq (14), with respect to two independent cluster operators T and Z or, more precisely, with respect to the excitation operator T and the deexcitation operator Z The advantage of eq (36) over the expectation value of the Hamiltonian with the CC wave function, which can also improve the results for multiple bond breaking (28, 127), is the fact that EcC(z,j is a finite series in T and Z. Unfortunately, the power series expansions of (Z,7), eq (36), in terms of T and Z contain higher powers of... 

Limiting ourselves to the HF reference case for convenience, by combining a cluster-type expansion of ft given in Eq. (26) and its order-byorder series, the low-order contributions to the wave operator may be expressed as... 

In contrast, the force of interaction between two ions is long-range and at large distances is proportional to 1/r, where r is the distance between the ions. Thus the solution cannot be considered to be composed of noninteracting clusters, and power series expansions in concentration are not possible. Statistical mechanical treatments of this problem demonstrate that the coefficients of the power series expansions diverge for coulomb forces and that another representation for the properties of the solution must be found. The rigorous molecular considerations confirm the results of the Debye-Huckel treatment for dilute solutions and demonstrate that the assumptions of the Arrhenius hypothesis are incorrect. 

The smooth change in the polymer coil dimension passing through the 6-temperature is exemplified by the data of Pritchard and Caroline (1981) shown in Fig. 6.4. Here the hydrodynamic expansion factor is plotted as a fimction of the temperature. Contrary to the predictions of the blob theory, da /dr is clearly non-zero at the 0i -temperature, irrespective of the molecular weight. Its positive value is consonant with the predictions of the cluster series... 

The cluster expansions in this section are of the most primitive kind in the sense that they give A and the correlation functions relative to the corresponding functions for a reference system of noninteracting particles. Stell and Lebowitz have shown how to carry analogous procedures through for arbitrary reference systems, such as a suitable assembly of hard spheres. Other resummations and optimizations aimed at achieving better convergence of cluster series have been studied intensively by Andersen, Chandler, and Weeks. ... 

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