Perturbation theory integral equation methods

Integral equation methods provide another approach, but their use is limited to potential models that are usually too simple for engineering use and are moreover numerically difficult to solve. They are useful in providing equations of state for certain simple reference fluids (e.g., hard spheres, dipolar hard spheres, charged hard spheres) that can then be used in the perturbation theories or density functional theories. 

In this chapter we shall present a necessarily partial review of the main theoretical approaches so far developed to treat liquid systems in terms of physical frmctions. We shall restrict ourselves to two basic theories, integral equation and perturbation theories, to keep the chapter within reasonable bounds. In addition, only the basic theoretical principles imderly-ing the original methods will be discussed, because the progress has been less rapid for theory than for numerical applications. The latter are in fact developing so fast that it is an impossible task to try to give an exhaustive view in a few pages. 

The classical models of adsorption processes like Langmuir, BET, DR or Kelvin treatments and their numerous variations and extensions, contain several uncontrolled approximations. However, the classical theories are convenient and their usage is very widespread. On the other hand, the aforementioned classical theories do not start from a well - defined molecular model, and the result is that the link between the molecular behaviour and the macroscopic properties of the systems studied are blurred. The more developed and notable descriptions of the condensed systems include lattice models [408] which are solved by means of the mean - field or other non-classical techniques [409]. The virial formalism of low -pressure adsorption discussed above, integral equation method and perturbation theory are also useful approaches. However, the state of the art technique is the density functional theory (DFT) introduced by Evans [410] and Tarazona [411]. The DFT method enables calculating the equilibrium density profile, p (r), of the fluid which is in contact with the solid phase. The main idea of the DFT approach is that the free energy of inhomogeneous fluid which is a function of p (r), can be... 

Computer simulations are likely to be useful in two distinct cases the first in which numerical data of a specified accuracy are required, possibly for some utilitarian purpose the second, perhaps more fundamentally, in providing guidance to the theoretician s intuition, for example, by comparing numerical results with those from approximate analytical theories (such as perturbation theory, mean-field approximation, integral equation methods). Naturally, the models used should reflect those generic properties of polymers that are the result of the chain-like stmcture of macromolecules. 

An analogy may be drawn between the phase behavior of weakly attractive monodisperse dispersions and that of conventional molecular systems provided coalescence and Ostwald ripening do not occur. The similarity arises from the common form of the pair potential, whose dominant feature in both cases is the presence of a shallow minimum. The equilibrium statistical mechanics of such systems have been extensively explored. As previously explained, the primary difficulty in predicting equilibrium phase behavior lies in the many-body interactions intrinsic to any condensed phase. Fortunately, the synthesis of several methods (integral equation approaches, perturbation theories, virial expansions, and computer simulations) now provides accurate predictions of thermodynamic properties and phase behavior of dense molecular fluids or colloidal fluids [1]. 

The idea of Kirkwood (25) is combined with the Rouse model by Pyun and Fixman (14). The theory allows a uniform expansion of the bond length by a factor a such as introduced by Flory. The nondiagonal term of the Oseen tensor is considered but only to the first order by a perturbation method. Otherwise, their theory is identical to Zimm s theory in Hearst s version in the treatment of the integral equation (14). 

Computer simulations, Monte Carlo or molecular dynamics, in fact appear to be the actual most effective way of introducing statistical averages (if one decides not to pass to continuous distributions), in spite of their computational cost. Some concepts, such as the quasi-structure model introduced by Yomosa (1978), have not evolved into algorithms of practical use. The numerous versions of methods based on virial expansion, on integral equation description of correlation functions, on the application of perturbation theory to simple reference systems (the basic aspects of these... 

Initial results obtained for TPA and for photoelectron spectra of small systems, show that anharmonicity must be included in the calculation of EC factors to reproduce experiment [54, 77, 104]. However, it is difficult to treat larger anharmonic systems by means of perturbation theory. Such systems can be handled by applying the variation/perturbation methods of electronic structure theory that have been, and continue to be, extended to the vibrational Schrodinger equation as discussed earlier. The EC integrals tliat appear in the equations for resonant (hyper)polarizabilities may be calculated employing approaches like VSCF, VMP2, VCI and VCC, That will allow us to Include anharmonic contributions to all orders and thereby remove the intrinsic limitations of the perturbation expansion in terms of normal coordinates. 

Simple open shell cases may also be treated via this kind of perturbation theory. The high spin case with one electron outside a closed shell is of course easy when an unrestricted formalism is used. Dyall also worked out equations for the restricted HE formalism and the more complicated case of two electrons in two Kramers pairs outside a closed shell [32]. Also in this method the crucial step remains the efficient formation of two-electron integrals in the molecular spinor basis. 

The collision probability is one of several possible formulations of integral transport theory. Three other formulations are the integral equations for the neutron flux, neutron birth-rate density, and fission neutron density. Oosterkamp (26) derived perturbation expressions for reactivity in the birth rate density formulation. The fission density formulation provides the basis for Monte Carlo methods for perturbation calculations (52, 55). 

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