Cluster expansion formalism

Before describing the unifying theme, which will classify the various theoretical developments in the cluster expansion formalisms, it is pertinent to summarize first certain essential criteria that an... 

Although the closed shell CC theory has been sometimes used to describe some special open-shell situations/39,40/ (like single configuration triplets/39/) or dissociating species/40-42/, from the point of view of generality, and on physical grounds it is natural to look for open-shell generalizations of the cluster expansion formalisms as developed for the closed shells. 

D. Mukherjee and P. K. Mukherjee, Chem. Phys., 39, 325 (1979). A Response-Function Approach to the Direct Calculation of the Transition-Energy in a Multiple-Cluster Expansion Formalism. 

Having familiarized ourselves slightly with the cluster expansions let us now look in detail at a more difficult example involving long-range interactions where the quasi-chemical formalism appears less satisfactory. 

The discussion of the defect distribution functions and potentials of average force follows along rather similar fines to that for the activity coefficient. The formal cluster expansions, Eqs. (90)-(91), individual terms of which diverge, must be transformed into another series of closed terms. This can clearly be done by... 

The most recent effort in this direction is the work of Cohen,8 who established a systematic generalization of the Boltzmann equation. This author obtained the explicit forms of the two-, three-, and four-particle collision terms. His approach is formally very similar to the cluster expansion of Mayer in the equilibrium case. 

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. 

Our starting point is a density analogous to that used in [49] in treating the migration of excitons between randomly distributed sites. This expansion is generalization of the cluster expansion in equilibrium statistical mechanics to dynamical processes. It is formally exact even when the traps interact, but its utility depends on whether the coefficients are well behaved as V and t approach infinity. For the present problem, the survival probability of equation (5.2.19) admits the expansion... 

Obviously we may expect that the simple two- and three-particle collision approximation discussed in the previous sections is not appropriate, because a large number of particles always interact simultaneously. Formally this approximation leads to divergencies. In the previous sections we used in a systematic way cluster expansions for the two- and three-particle density operator in order to include two-particle bound states and their relevant interaction in three- and four-particle clusters. In the framework of that consideration we started with the elementary particles (e, p) and their interactions. The bound states turned out to be special states, and, especially, scattering states were dealt with in a consistent manner. 

In this section we shall discuss an approach which is neither variational nor perturba-tional. This approach also has its origin in nuclear physics and was introduced to quantum chemistry by Sinanoglu47, It is based on a cluster expansion of the wave function. A systematic method for the calculation of cluster expansion components of the exact wave function was developed by C ek48 The characteristic feature of this approach is the expansion of the wave function as a linear combination of Slater determinants. Formally, this expansion is similar to the ordinary Cl expansion. The cluster expansion, however, gives us not only the physical insight of the correlation energy but it also shows the connections between the variational approaches (Cl) and the perturbational approaches (e.g. MB-RSPT). 

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]

For the cluster expansion of the type (a2), we may thus conceive of two approaches. One is to invoke a single root strategy, and use either anonymous parentage or preferred parentage approximation. This approach by-passes the need for H, but by its very nature cannot generate a potentially exact formalism. The other is to use the multi—root strategy through the Bloch equation, and thereby produce a formally exact theory. We shall review these two types of schemes in Secs. 6.2 and 6.3 respectively. 

We see that the cluster expansion is formally the same, only instead of the c-set of expansion coefficients we have the d-coefficients (appearing in eqns. (4.25)-(4.27))4 Comparing coefficients standing before respective configurations gives us the following relations ... 

We shall now draw our attention to the practical use of the formalism of the cluster expansion. Our goal is to solve again the equation... 

In Volume 5 of this series, R. J. Bartlett and J. E Stanton authored a popular tutorial on applications of post-Hartree-Fock methods. Here in Chapter 2, Dr. T. Daniel Crawford and Professor Henry F. Schaefer III explore coupled cluster theory in great depth. Despite the depth, the treatment is brilliantly clear. Beginning with fundamental concepts of cluster expansion of the wavefunction, the authors provide the formal theory and the derivation of the coupled cluster equations. This is followed by thorough explanations of diagrammatic representations, the connection to many-bodied perturbation theory, and computer implementation of the method. Directions for future developments are laid out. 

The exponential character of the cluster expansion warrants the size-extensivity of the resulting formalism regardless of the truncation scheme employed, as implied by a comparison with the standard linear Cl expansion of (intermediately normalized) I T),... 

In 1966, Silverstone and Sinanoglu [77] and Kelly [80] published their extension of previous formal developments and applications of fhe cluster expansion of the wavefunction and MBPT for closed-(sub)shell ground states to analogous formalisms for open-(sub)shell sfafes. In facf, Kelly demonstrated his methods with an impressive calculation of parfs of elec-fron correlation in the oxygen atom, with emphasis on the correlation of pairs of electrons [80]. Among other things, he pointed to the inevitable appearance of terms that correspond to spin-orbital pair excitations such as 2p(-Fl ),2p(0+ 2p(-l+),4/(-F2+, which are called semi-internal by Silverstone and Sinanoglu [77], see below. 

Silverstone and Sinanoglu [77] wrote the cluster expansion of fhe nonrela-f ivisfic N-elecf ron eigenfunction in terms of a zero-order reference wavefunc-fion fhaf is multicmfigurational, in accordance wifh the earlier suggestion of Wafson [31] and the study of H-ND in Be [75, 76]. In their formalism, the one-, two-, three-, etc. correlation functions (i.e., the virtual electron-excitations in the language of Cl) are linked fo spin orbitals from an extended zero-order set of occupied and unoccupied spin orbitals. This set was named the Hartree-Fock sea (H-F sea). Optimally, the H-F sea spin orbitals are supposed to be computed self-consistently. 

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. 

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