Correlation functions cluster expansion

SAC-CI can explicitly treat multiply excited electrons and holes allowing for the excited states to be quantified as single or MEs. For example, the SAC-CI used below explicitly includes electron correlation through cluster expansion of the ground state wave function (WF) ... 

Theoretical investigations of quenched-annealed systems have been initiated with success by Madden and Glandt [15,16] these authors have presented exact Mayer cluster expansions of correlation functions for the case when the matrix subsystem is generated by quenching from an equihbrium distribution, as well as for the case of arbitrary distribution of obstacles. However, their integral equations for the correlation functions... 

We proceed with cluster series which yield the integral equations. Evidently the correlation functions presented above can be defined by their diagrammatic expansions. In particular, the blocking correlation function is the subset of graphs of h rx2), such that all paths between... 

The direct correlation function c is the sum of all graphs in h with no nodal points. The cluster expansions for the correlation functions were first obtained and analyzed in detail by Madden and Glandt [15,16]. However, the exact equations for the correlation functions, which have been called the replica Ornstein-Zernike (ROZ) equations, have been derived by Given and Stell [17-19]. These equations, for a one-component fluid in a one-component matrix, have the following form... 

The equlibrium between the bulk fluid and fluid adsorbed in disordered porous media must be discussed at fixed chemical potential. Evaluation of the chemical potential for adsorbed fluid is a key issue for the adsorption isotherms, in studying the phase diagram of adsorbed fluid, and for performing comparisons of the structure of a fluid in media of different microporosity. At present, one of the popular tools to obtain the chemical potentials is an approach proposed by Ford and Glandt [23]. From the detailed analysis of the cluster expansions, these authors have concluded that the derivative of the excess chemical potential with respect to the fluid density equals the connected part of the fluid-fluid direct correlation function (dcf). Then, it follows that the chemical potential of a fluid adsorbed in a disordered matrix, p ), is... 

The cluster expansions of the correlation functions and potentials of mean force can be found by studying the semi-invariant expansion of logg( n ) and by the use of the linked cluster theorem. The method is a straightforward extension of those given in the preceding section and details can be found in the paper by Allnatt and Cohen.3 The linked cluster method is simpler and... 

Density expansion. The method of cluster expansions has been used to obtain the time-dependent correlation functions for a mixture of atomic gases. The particle dynamics was treated quantum mechanically. Expressions up to third order in density were given explicitly [331]. We have discussed similar work in the previous Section and simply state that one may talk about binary, ternary, etc., dipole autocorrelation functions. 

In addition to the distribution functions / a second sequence of symmetric functions gm, called correlation functions, will be useful. They are defined in terms of the / by the following cluster expansion ... 

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 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]. 

That means, as for g(r), the direct correlation function c(r) can be, in principle, derived from experiment. As can be seen, the OZ relation deals with a priori two unknown functions h(r) and c(r). The cluster expansion provides an important exact feature for c(r) with respect to the potential interaction at large distances... 

An analysis of clusters expansion to higher order (as compared to PY equation) leads to the hypernetted-chain (HNC) approximation [44—46]. In other words, directly solving the OZ relation in conjunction with Eq. (28) is possible, under a drastic assumption on B(r). The total correlation function is given simply by... 

In the mean spherical approximation (MSA) treatment of the ion association in aqueous solutions, the linearity of the relative permittivity and of the hydrated cation diameters with the electrolyte concentration was taken into account and a good fit of the experimental activity and osmotic coefficient was obtained [72-75]. The MSA model was elaborated on the basis of cluster expansion considerations involving the direct correlation function the treatment can deal with the many-body interaction term and with a screening parameter and proved expedient for the interpretation of experimental results concerning inorganic electrolyte solutions [67,75-77]. 

Besides the mentioned aperiodicity problem the treatment of correlation in the ground state of a polymer presents the most formidable problem. If one has a polymer with completely filled valence and conduction bands, one can Fourier transform the delocalized Bloch orbitals into localized Wannier functions and use these (instead of the MO-s of the polymer units) for a quantum chemical treatment of the short range correlation in a subunit taking only excitations in the subunit or between the reference unit and a few neighbouring units. With the aid of the Wannier functions then one can perform a Moeller-Plesset perturbation theory (PX), or for instance, a coupled electron pair approximation (CEPA) (1 ), or a coupled cluster expansion (19) calculation. The long range correlation then can be approximated with the help of the already mentioned electronic polaron model (11). 

Sinanoglu was the first who suggested a practical method for calculating the correlation energy based on the cluster expansion of the wave function. By the approximate treatment of the problem (4,42) he arrived for the function... 

A first contribution to J-p is caused by the bonding between two atoms which constitute one molecule. This direct intra-molecular interaction between the atoms is taken into considerations as a f -bond. In addition there are intra-molecular interactions of indirect nature between the both atoms of a molecule. These atoms affect each other indirectly by n point interactions with all remaining atoms and combinations of atoms. The so-called intra-molecular pair cavity function y (ryw) expresses the ensemble of all indirect interactions which appear between the atoms of a molecule in f-bonds [13] and establishes the searched correlation function for all indirect interactions between the atoms inside a molecule. TTie molecular DFT approach evaluates the cluster expansion to calculate y (rjvr) using TPT. This approximation takes into account only presentations with vertices n <— 2, for what reason it is called the single chain approximation (TPT1)[12]. [7,8]... 

Coupled cluster response calculaAons are usually based on the HF-SCF wave-function of the unperturbed system as reference state, i.e. they correspond to so-called orbital-unrelaxed derivatives. In the static limit this becomes equivalent to finite field calculations where Aie perturbation is added to the Hamiltonian after the HF-SCF step, while in the orbital-relaxed approach the perturbation is included already in the HF-SCF calculation. For frequency-dependent properties the orbital-relaxed approach leads to artificial poles in the correlated results whenever one of the involved frequencies becomes equal to an HF-SCF excitation energy. However, in Aie static limit both unrelaxed and relaxed coupled cluster calculations can be used and for boAi approaches the hierarchy CCS (HF-SCF), CC2, CCSD, CC3,... converges in the limit of a complete cluster expansion to the Full CI result. Thus, the question arises, whether for second hyperpolarizabilities one... 

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. 

They refer to Eq. (3.4.3) as the proper integral equation in view of the fact that the direct correlation function so defined does correspond to the sum of the nodeless diagrams in the interaction site cluster expansion of the total correlation function. Following Lupkowski and Monson, we shall refer to Eq. (3.4.3) as the Chandler-Silbey-Ladanyi (CSL) equation. Interestingly, the component functions have a simple physical interpretation. The elements of Ho correspond to the total correlation functions for pairs of sites at infinite dilution in the molecular solvent. The elements of the sum of Hq and //, (or H ) matrices are the solute-solvent site-site total correlation functions for sites at infinite dilution in the molecular solvent. 

An alternative to the calculation of higher-order terms in the k expansion is provided by analysis of the interaction site cluster expansions of the correlation functions and Helmholtz free energy into contributions arising... 

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