Mayer cluster theory

In these remarks, one can see a pioneering suggestion of a cluster mixture theory of liquids with short-range (exchange-like) forces, along the lines of Mayer cluster theory (Sidebar 13.5) or quantum cluster equilibrium theory (Section 13.3.4). 

Note that (6.23) differs from the expression given in Section 1.8. The latter is obtained from the former if the total potential energy C/3(Xi, X2, X3) is pairwise additive. Equations (6.22) and (6.23) are special cases of a more general scheme which provides relations between virial coefficients and integrals involving interactions among a set of a small number of particles. This is known as the Mayer cluster theory [see, for example, Mayer and Mayer (1940), Hill (1956), and Munster (1969)]. 

Friedman (1962) has used the cluster theory of Mayer (1950) to derive equations which give the thermodynamic properties of electrolyte solutions as the sum of convergent series. The first term in these series is identical to and thus confirms the Debye-Huckel limiting law. The second term is an I2.nl term whose coefficient is, like the coefficient in the Debye-Huckel limiting law equation, a function of the charge type of the salt and the properties of the solvent. From this theory, as well as from others referred to above, a higher order limiting law can be written as... 

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-... 

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... 

Hill, Isihara, and Kaufman and Watson. Hill s method parallels Mayer s treatment of the compressibility factor of an imperfect gas. Kaufman and Watson and Isihara utilize cluster theory in an attempt to obtain a solution for -cP in closed form. We will follow the treatment of Buckingham and Pople, primarily for reasons of simplicity. 

Our general approach is a proper adaptation and generalization of the gas-type theories of McMillan and Mayer and of Kirkwood and Buff. These were originally developed for simple (monomer) solutions. We use the cluster development of McMillan and Mayer, which itself is an adaptation of the original (Ursell)-Mayer cluster development. We... 

Somewhat simplified version of the Mayer cluster expansion theory for a single component gas has been presented by E. E. Salpeter in Annals of Physics 5, 183 (1958). There some of the combinatorial algebra is replaced by topological considerations. 

The thermodynamics of a l-d Fermi system can be perfectly mapped onto the thermodynamics of a two-component classical real gas on the surface of a cylinder. The relationship between these two infrared problems (cf. Zittartz s contribution) is exploited as follows. We treat the classical plasma by a modified Mayer cluster expansion method (the lowest order term corresponding to the Debye Hiickel theory), and obtain an exponentially activated behavior of the specific heat (cf. Luther s contribution) of the original quantum gas by simply reinterpreting the meaning of thermodynamic variables. 

A bond function is defined as some function of the sets of variables Xi and %2 for two particles. The fundamental bond function in the cluster theory of classical fluids is the Mayer / function... 

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... 

The products in the integrand correspond to the ordered sequence of the bond vectors. The evaluation of the right hand side integral may be made following Mayer s cluster theory of imperfect gases. We arrive at a cluster series... 

The remainder of Section I is devoted to a rather brief review of earlier work in the field in order to gain a little perspective. In Sections II to IV the basic results of the cluster method are derived. In Section V a very brief account of the application of the formal equations to some systems with short-range forces is given. Section VI is devoted to a review of the application to systems with Coulomb forces between defects, where the cluster formalism is particularly advantageous for bringing the discussion to the level of modern ionic-solution theory.86 Finally, in Section VII a brief account is given of Mayer s formalism for lattice defects69 since it is in certain respects complementary to that principally discussed here. We would like to emphasize that the material in Sections V and VI is illustrative of the method. This is not meant to be an exhaustive review of results obtainable. 

The virial coefficients B(T), C(T), D(T),... are functions of temperature only. Although these coefficients might be treated simply as empirical fitting parameters, they are actually deeply connected to the theory of intermolecular clustering, as developed by J. E. Mayer (Sidebar 13.5). More specifically, the second virial coefficient B(T) is related to the intermolecular potential for pairs of molecules, the third virial coefficient C(T) to that for triples of molecules, and so forth. For example, knowledge of the intermolecular pair potential V(R) (see Sidebar 2.8) allows B T) to be explicitly evaluated by statistical mechanical methods as... 

The origins of percolation theory are usually attributed to Flory and Stock-mayer [5-8], who published the first studies of polymerization of multifunctional units (monomers). The polymerization process of the multifunctional monomers leads to a continuous formation of bonds between the monomers, and the final ensemble of the branched polymer is a network of chemical bonds. The polymerization reaction is usually considered in terms of a lattice, where each site (square) represents a monomer and the branched intermediate polymers represent clusters (neighboring occupied sites), Figure 1.4 A. When the entire network of the polymer, i.e., the cluster, spans two opposite sides of the lattice, it is called a percolating cluster, Figure 1.4 B. 

In the next section we shall present a simplified expansion theorem of osmotic pressure which was first obtained by McMillan and Mayer. This cluster expansion theory will be further extended in Section 3 to distribution functions, and medn results of Kirkwood and Buff will be recovered. A new and simple derivation of the cluster expansion of the pair distribution function is also given. Section 4 presents a new expression for the chemical potential of solvents in dilute solutions. Section 5 shows how the general solution theory may be applied to compact macromolecules. Finally, Section 6 deals with the second osmotic virial coefficient of flexible macromolecules and is followaJ in Sa tion 7 by concluding remarks. 

Rosch, N., Kriiger, S., Mayer, M. andNasluzov.V. A. (1996) The Douglas-Kroll-Hess approach to relativistic density functional theory Methodological aspects and applications to metal complexes and clusters. In Recent Developments and Applications of Modem Density Functional Theory (ed. J. M. Seminario), pp. 497-566. Elsevier. 

As noted in Section 2, the fundamental bond function in cluster expansion theory is the Mayer / function defined in Eq. (8). Both si and g can be expressed very simply in terms of graphical series containing / bonds. 

Special emphasis is placed upon the McMillan-Mayer theory (Sections 4 and 5) and on cluster expansions (Section 6), as these represent aspects that are both difficult and strongly established, but that are seldom given detailed exposition. Other developments that are easily accessible in the literature are treated more lightly, as are the many aspects of theory of fluids that are not yet completely developed for application to ionic solution problems. 

Note that these look just like the corresponding expansion coefficients in gas theory except for one important difference the potential of mean force takes the place of the intermolecular potential. Since the potential of mean force is not, in general, pairwise additive, the familiar technology of Mayer /functions and cluster diagrams are not available to the solution theorist. It is interesting to note that the emphasis on osmotic pressure in McMiUan-Mayer theory seems to bring one back to the ideas of van t Hoff. 

In this theory the single-contact term, the double-contact term, and so forth are taken into consideration. The cluster diagrams developed by Ursell (1927) and Mayer et al. (1940, 1945) are employed to calculate the mean-square end-to-end distance. Here we give the results obtained by Yamakawa (1971) ... 

F. Coester, Nucl. Phys, 1958, 7, 421-424 F. Coester and H. KUmmel, Nucl. Phys., 1960, 17, 477-485 It should be noted that cluster-type expansions have been first used in statistical mechanics (see, for example, J. E. Mayer and M. G, Mayer, Statistical Mechanics Wiley, New York, 1940, Chap. 13). A brief account of the history of CC theory has been given by J. Click and J. Paldus Phys. Scripta, 1980, 21, 251-254). 

Formula (7.8.13) expresses the configurational partition function (7.8.6) as the sum of contributions from all possible subdivisions of the iV-ceUs into cell clusters. A similar expression is well known in the theory of imperfect gases where Q is expressed as a sum of contributions from all possible molecular clusters (cf. for example Mayer and Mayer [1940], p. 277). 

In a general theory of solutions, McMillan and Mayer demonstrated the formal equivalence between the pressure of a gas and the osmotic pressure of a solution. Hence the ratio of the osmotic pressure O of a dilute solution to the concentration (number density) p of the solute can be expanded in a power series in p and the coefficients of the series can be expressed, as in the theory of a real gas, in terms of cluster integrals determined by intermolecular potential energy functions. The only difference is, as already mentioned, that in the solution these potentials are effective potentials of average force, which include implicitly the effects of the solvent, modelled as a continuum. 

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