Theory, cluster integral expansion

One of the newer theoretical treatments, based on the pioneering statistical thermodynamic work of McMillan and Mayer (6) y as mathematically formulated by Friedman W, does appear to hold significant promise as a theory of sufficient generality that it may eventually embody other working theories as demonstrated special cases. This theory, known as the cluster integral expansion theory (j ) or simply as cluster theory (9)y has been developed to the point where applications have been made to calculating... 

This was accomplished by expanding an existing model based on the cluster integral expansion theory of electrolyte solution structure into a comprehensive thermodynamic model describing the major and minor components of brines, including metals present in trace amounts. 

Note added in -proof. The application of the usual integral equation theories of the liquid state 2> to water has not been successful.1) A recent study by H. C. Andersen [J. Chem. Phys. 61, 4985 (1974)] promises to change this situation. Briefly, Andersen reformulates the well known Mayer cluster expansion of the distribution function 2> by consistently taking into account the saturation of interaction characteristic of hydrogen bonding. Approximations are selected which satisfy this saturation condition at each step of the analysis. Preliminary calculations (H. C. Andersen, private communication) indicate that even low order approximations that preserve the saturation condition lead to qualitative be-... 

This feature of incorrectly giving the behavior of h(r) for r < d is common to many types of approximation schemes, for example, the mode expansion. This is a formal theory of the equilibrium properties of fluids that is not based on cluster expansion methods but whose results can be expressed in terms of cluster integrals. In the mode expansion, a useful strategy was to redefine or optimize the perturbation for rself-consistent F-ordered result. Moreover, it can also be usefully applied even to approximations that do not suffer from the difficulty of giving physically unreasonable values of h for roptimization method will be discussed more fully in the next section. 

If we apply Mayer s theory of condensation to AHS systems, we find that in the cluster expansions (7.13a) for the molar volume v and (7.13b) for the pressure p. The coefficients (cluster integrals) bi are constmcted by the special form... 

As already mentioned, one of the merits of the virial equation is that it has a firm foundation in statistical thermodynamics and molecular theory. The theoretical derivation of the series has been described in numerous texts and will not be discussed in detail here. The most complete derivation for a mixture containing an arbitrary number of components is made by means of an expansion of the grand partition function. This leads to expressions for the virial coelficients in terms of cluster integrals involving two molecules for B, three molecules for C etc. These expressions are completely general and involve no restrictive assumptions about the nature of molecular interactions. Nevertheless, to simplify the expressions for the virial coefficients, a number of assumptions are often made as follows ... 

The first contribution to h(r) is the direct correlation function c(r) that represents the correlation between a particle of a pair with its closest neighbor separated by a distance r. The second contribution is the indirect correlation function y(r), which represents the correlation between the selected particle of the pair with the rest of the fluid constituents. The total and direct correlation functions are amenable to an analysis in terms of configurational integrals clusters of particles, known as diagrammatic expansions. Providing a brief resume of the diagrammatic approach of the liquid state theory is beyond the scope of this chapter. The reader is invited to refer to appropriate textbooks on this approach [7, 9, 18, 26]. 

Modern theory of associative fluids is based on the combination of the activity and density expansions for the description of the equilibrium properties. The activity expansions are used to describe the clusterization effects caused by the strongly attractive part of the interparticle interactions. The density expansions are used to treat the contributions of the conventional nonassociative part of interactions. The diagram analysis of these expansions for pair distribution functions leads to the so-called multidensity integral equation approach in the theory of associative fluids. The AMSA theory represents the two-density version of the traditional MSA theory [4, 5] and will be used here for the treatment of ion association in the ionic fluids. 

It is also of interest to develop theories for systems where the particles are nonspherical in shape. Recently, Lupkowski and Monson have shown how the ideas described above may be extended to systems of nonspherical particles made up of interaction sites. They have formulated the interaction site cluster expansion of the site-site pair connectedness function, h y r), which measures the probability that two sites on different particles are separated by a distance r and are members of the same cluster. The extensions of the SSOZ and CSL integral equations to the site-site connectivity problem are readily obtained. For example, the connectivity SSOZ equation is... 

We then carry out the integration term by term. This is the method referred to as cluster expansion, which was developed in the theory of condensation of interacting gases [15]. For a polymer chain, the condition of linear connectivity is added. 

The first two of these results (4.141) and (4.142) are of little use unless we know or can approximate the function c(ri. t2 ap) for all values of a. They were obtained first by Stillinger and Buff by a cluster expansion and by Lebowitz and Percus by using functional integration, but we owe to Saam and Ebner the comment that since F[p] is a unique functional of p(i) then the values of F and 0 calculated in this way are independent of the path in p-space (4.136). The third result (4.146), although restricted to pair potentials, is a useful starting point for the development of perturbation theories of both bulk liquids and of the gas-liquid surface. 

The only way to get a wave operator that is symmetric under time-reversal symmetry is to impose the restriction from the beginning. While this fixes the relations between the amplitudes, it also forces the occupied and the unoccupied Kramers components of the open shell to be treated equivalently. This equivalence is what introduces the ambiguity in the treatment of the open shell the open-shell Kramers pair must behave as both a particle and a hole, and the result is that the truncated commutator expansions in the coupled-cluster equations are much longer than in closed-shell theory. The alternative is to use an unrestricted wave operator with the Kramers-restricted spinors. The use of the latter provides some reduction in the work due to the relations between the integrals, but not a full reduction (Visscher et al. 1996). 

The most simple and instructive way to study the main elements of F12 theory is to start with second-order Moller-Plesset perturbation theory, MP2. We will show in the following how the conventional, orbital-expansion based theory is supplemented by additional geminal functions and how the working equations change. We will then discuss how expensive many-electron integrals are avoided by reducing them to products of two-electron integrals, and finally, how F12 theory is transferred to second quantization, which in particular is required for the formulation of coupled-cluster theory with F12 terms. 

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