Ursell expansion

The prescription for determining the functions is that an m-decomposable model should be correct if the system consisted only of m molecules. Formulae such as Eqs. (6.2) and (6.3) involve the Mayer / function that, for a pair decomposable case, is / j) = exp [—/3m( )(0, y)] — 1. A natural accommodation of nonpair-decomposable interactions in this case takes the goal of insuring that successive terms in a virial expansion are ordered by the density. This is the historical approach (Ursell, 1927), and is called an Ursell expansion. In this language, fa (j) is an Ursell function (Stell, 1964 Munster, 1969). Again the idea is to require that the desired m-body Ursell function makes the product of Eq. (6.2) correct if just m molecules are involved. Thus for the case that only two molecules are involved... 

Write out the general term following from the Ursell expansion, and describe what to do if you have results for an n-alkane in water. 

Calculation of wave function components in Ursell-type expansion using... 

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). 

J. Cizek, On the correlation problem in atomic and molecular systems. Calculation of wavefunc-tion components in Ursell-type expansion using quantum-field theoretical methods. J. Chem. Phys. 45, 4256 (1969). 

Bartlett, R. J. 2000. Perspective on On the Correlation Problem in Atomic and Molecular Systems. Calculations of Wavefunction Components in Ursell-type Expansion Using Quantum-field Theoretical Methods Theor. Chem. Acc., 103, 273. 

The properties relating to decay of correlations have so far been discussed only in terms of Ursell functions, or functions generated from a cumulant expansion. This is obviously not a unique choice. Although there has been no systematic exploration of possible alternative families with, perhaps, better separation of rates of convergence than that expressed in Eqs. (12) and (13), or (17) and (18), it is of some interest to examine the Kirkwood12 approximation form... 

Systems. Calculation of Wave Function Components in Ursell-Type Expansion Using Quantum-Field Theoretical Methods. 

Figure 6.1 Mayer-Montroll expansion for the insertion probability p(0 X"). The notation here is fairly standard (see, for example, Hansen and McDonald, 1976 Andersen, 1977). The solid lines indicate factors of Mayer / functions introduced in Eq. (6.2) and are further discussed as Ursell functions beginning on p. 126. The inclusion-exclusion interpretation for hard-core cases is that the second term - assesses the m molecular volumes excluded to the...

The most compact graphical expansion we have is for the logarithm of S, so it is more convenient to consider a distribution function based upon a functional derivative of logH. This leads naturally to the Ursell cluster... 

The perturbation expansions that are used to study polymer solutions are generalizations of the Ursell-Mayer-Yvon expansions which apply to gas theory. This technique has been developed from 1972 onwards, especially in the United States and in Japan. The expansions start from a model. The most simple and common one is the famous continuous model which will be described in this chapter in great detail. 

The contributions of the general diagrams can be expressed in terms of those of the connected diagrams, and precise relations between these contributions will be given here in order to justify the results announced in the preceding section. For this purpose, it is convenient to use a very general simplified notation. Indeed, the results are not only true for the continuous model but also for the discrete-link model in continuous space (Ursell-Mayer-Yvon expansions) and for lattice models. 

The exponential cluster expansion has a long history in statistical physics, where it is known as the Ursell and Mayer linked-cluster expansion for the partition function [68]. A very general argument for the exponential form of the exact wave function can be found in a paper on the origins of the CC method by Kiimmel [69]. Nonetheless,... 

It is easy to verify that Eq. (84) is consistent with Eqs. (80) and (81), because each term in the Ursell function expansion of Wiv(l,..., N) is distinct from all of the others because the coordinates are all distinguishable. 

Equation (87) is not so trivial as it may first appear The terms in the Ursell function expansion of (Ws) are not all distinct from each other and this leads to the appearance of so-called combinatorial factors. They do not appear in Fig. 6 because they are included in the definitions of the graphs (Section 6.1). 

Jiri Cizek introduced the (diagrammatic) CC method into electron correlation theory in a paper On the correlation problem in atomic and molecular systems. Calculation of wavefunction components in Ursell-type expansion using quantum-field theoretical methods, pubhshed in the Journal of Chemical Physics, 45, 4256 (1966). The book Three Approaches to Electron Correlation in Atoms (Yale University Press, New Haven, CT, and London 1970), edited by Oktay Sinanoglu and Keith A. Brueckner, contains several reprints of the papers that cleared the path toward the CC method. 

J. Cizek, ]. Chem, Phys, 45, 4256 (1966). On the Correlation Problem in Atomic and Molecular Systems. Calculation of Wave Function Components in Ursell-Type Expansion Using Quantum-Field Theoretical Methods. J. Cizek, Adv, Chem. Phys., 14, 35 (1966). On the Use of the Cluster Expansion and the Technique of Diagrams in Calculations of Correlation Effects in Atoms and Molecules. 

These semiclassical approaches have their origin in the studies of the second and third virial coefficients of a gas of QHS [139-143]. Further developments involving cluster expansions of in terms of Ursell functions led to the conclu-... 

An alternative approach may be made without using a differential equation, but treating the problem completely statically. Such an approach was first made by Yamamoto and Teramoto as early as 1952 3). These authors used the Ursell-Mayer expansion method and evaluated the first order coefficient which agreed with Grimley s result. This method has been pursued by Yamakawa and Kurata and also by Fixman. 

We shall follow the same Ursell-Mayer expansion method. However, different from the previous theories, we shall not use the Gausdan chain approximation. In addition, we shall develop the theory for arbitrary inter-s mental intoactions. Althou a pearl-necklace model wiU be adopted for final results and especially for comparison with other results, our theory is rigorous and is applicable to chains of arbitrary lengths and interactions. Thus, even apart from applications the theory has its own merits. Hi tact, it is clear, as go higher orders, the Gansaan chain approximation becomes not applicable. 

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