Cluster expansion theory

SEMI-CLUSTER EXPANSION THEORIES FOR THE OPEN-SHELL STATES... 

FULL CLUSTER EXPANSION THEORIES IN HILBERT SPACE... 

FULL CLUSTER EXPANSION THEORIES IN FOCK SPACE 7.1 Preliminaries for a Fock Space Approach... 

The developments of the cluster expansion theories appear to have reached a stage where a clear perspective is beginning to emerge, although no comprehensive review of the various facets of the approach and a critical evaluation of the seemingly disparate formalisms put forward is available in the literature. There are, however, several reviews on closed-shell coupled cluster theories where the open-shell cluster expansion theories are also touched upon/18,19,21,22/. A few reviews on the open-shell MBPT describe in broad terms the cluster expansion techniques in so far as they relate to MBPT /20,23/. A concise survey of what we shall call full cluster expansion theories appears in a recent article by Lindgren and Mukherjee/94[Pg.293]

The next set of open-shell cluster expansion theories to appear on the scene emphasized the size-extensivity feature (al), and all of them were designed to compute energy differences with a fixed number of valence electrons. Several related theories may be described here - (i) the level-shift function approach in a time-dependent CC framework by Monkhorst/56/ and later generalizations by Dalgaard and Monkhorst/57/, also by Takahasi and Paldus/105/, (ii) the CC-based linear response theory by Mukherjee and Mukherjee/58/, and generalized later by Ghosh et a 1/59.60.107/,(iii)the closely related formulations by Nakatsuji/50,52/ and Emrich/62/ and (iv) variational theories by Paldus e t a I / 54/ and Saute et. al /55/ and by Nakatsuji/50/. 

As explained in Sec.2, the full cluster—expansion theories in Hilbert space are designed to compute wave-functions for the open-shell states that are explicitly size-extensive with respect to the total number of electrons N. The underlying cluster structure of all these developments is what was envisaged by Silverstone and Sinanoglu/48/s... 

In this section, we shall motivate towards the need for a Fock-space approach to generate core-valence extensive cluster expansion theories (i.e., of type (a3)), introduced in Sec.2. Since we have to maintain size-extensivity of the energy... 

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. 

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. 

In this section, we review some of the important formal results in the statistical mechanics of interaction site fluids. These results provide the basis for many of the approximate theories that will be described in Section III, and the calculation of correlation functions to describe the microscopic structure of fluids. We begin with a short review of the theory of the pair correlation function based upon cluster expansions. Although this material is featured in a number of other review articles, we have chosen to include a short account here so that the present article can be reasonably self-contained. Cluster expansion techniques have played an important part in the development of theories of interaction site fluids, and in order to fully grasp the significance of these developments, it is necessary to make contact with the results derived earlier for simple fluids. We will first describe the general cluster expansion theory for fluids, which is directly applicable to rigid nonspherical molecules by a simple addition of orientational coordinates. Next we will focus on the site-site correlation functions and describe the interaction site cluster expansion. After this, we review the calculation of thermodynamic properties from the correlation functions, and then we consider the calculation of the dielectric constant and the Kirkwood orientational correlation parameters. 

Aqueous solutions can be modeled by writing a virial equation such as (17.37) in which osmotic pressure replaces pressure. Friedman (1962) describes applications of cluster expansion theory, which include long-range Coulombic potentials as well as short-range square-well potentials that operate when unlike ions approach within the diameter of a water molecule. These models are mathematically quite cumbersome and are not easily used for routine calculations. They do predict the non-ideal behavior of simple electrolytes such as NaCl quite admirably at moderate concentrations however, they use the square-well potential as an adjustable parameter and so retain some of the properties of the D-H equation with an added adjustable term. For this reason these are not truly a priori models. 

Two other chapters in this volume, namely. Chapter 2 by Stell and Chapter 3 by Friedman and Dale also contain discussions of particular applications of cluster expansion theory. 

In this section we will define some of the important terms associated with the graphs in cluster expansion theory. 

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. 

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