Binary collision isolated

For themial unimolecular reactions with bimolecular collisional activation steps and for bimolecular reactions, more specifically one takes the limit of tire time evolution operator for - co and t —> + co to describe isolated binary collision events. The corresponding matrix representation of f)is called the scattering matrix or S-matrix with matrix elements... 

At low solvent density, where isolated binary collisions prevail, the radial distribution fiinction g(r) is simply related to the pair potential u(r) via g ir) = exp[-n(r)//r7]. Correspondingly, at higher density one defines a fiinction w r) = -kT a[g r). It can be shown that the gradient of this fiinction is equivalent to the mean force between two particles obtamed by holding them at fixed distance r and averaging over the remaining N -2 particles of the system. Hence w r) is called the potential of mean force. Choosing the low-density system as a reference state one has the relation... 

Gelbart (1974) has reviewed a number of theories of the origins of the depolarized spectrum. One of the simplest models is the isolated binary collision (IBC) model of McTague and Bimbaum (1968). All effects due to the interaction of three or more particles are ignored, and the scattering is due only to diatomic collision processes. In the case that the interacting particles A and B are atoms or highly symmetrical molecules then there are only two unique components of the pair polarizability tensor, and attention focuses on the anisotropy and the incremental mean pair polarizability... 

This behavior can be attributed to the local CC>2-density around the solute and can be rationalized within the framework of the isolated binary collision model. The deactivation rate of the excited molecule in this case is proportional to the collision frequency Zip, T) which is correlated with the local density [23]. 

The isolated binary collision (IBC) theory is one of the earliest dephasing theories developed for liquids (77). Despite criticism of its assumptions (78), it remains popular for the interpretation of experimental data, because it relates the dephasing to easily measurable solvent properties. It assumes that the solvent dynamics are essentially gas-like, consisting of uncorrelated collisions of the vibrating molecule with solvent molecules. The solvent-solute interaction is taken as purely repulsive. The dependence of... 

Paige ME, Harris CB. A generic test of gas phase isolated binary collision theories for vibrational relaxation at liquid state densities based on the rescaling properties of collision frequencies. J Chem Phys 1990 93 3712-3713. 

Russell DJ, Harris CB. Vibrational relaxation in simple fluids a comparison of experimental results to the predictions of isolated binary collision theory. Chem Phys 1994 183 325-333. 

While a short-range isolated binary collision (IBC) picture, in its most extreme form — with the emphasis on collision — is on its face ludicrous for long-range force interactions, the corresponding isolated binary interaction (IBI) idea (5) could be examined. The IBI concept is that it is only the direct interaction with the solute of a single solvent molecule at a time that determines the VET this is of course also a key ingredient in the IBC picture, but in the IBI there is no a priori specification of exactly what the interaction force is. If the Fourier component of the fluctuating force tcf is decomposed as... 

The lifetime (Ti) of a vibrational mode in a polyatomic molecule dissolved in a polyatomic solvent is, at least in part, determined by the interactions of the internal degrees of freedom of the solute with the solvent. Therefore, the physical state of the solvent plays a large role in the mechanism and rate of VER. Relaxation in the gas phase, which tends to be slow and dominated by isolated binary collisions, has been studied extensively (11). More recently, with the advent of ultrafast lasers, vibrational lifetimes have been measured for liquid systems (1,4). In liquids, a solute molecule is constantly interacting with a large number of solvent molecules. Nonetheless, some systems have been adequately described by isolated binary collision models (5,12,13), while others deviate strongly from this type of behavior (14-18). The temperature dependence of VER of polyatomic molecules in liquid solvents can show complex behavior (16-18). It has been pointed out that a change in temperature of a liquid solute-solvent system also results in a change in the solvent s density. Therefore, it is difficult to separate the influences of density and temperature from an observed temperature dependence. 

The isolated binary collision (IBC) model for vibrational relaxation in liquids was developed by Herzfeld, Litovitz, and Madigosky. Its fundamental assumption is that the vibrational relaxation rate from level / to level y( //) given by the product of two factors... 

Since the introduction of the isolated binary collision model there has been considerable controversy over its applicability. A number of theoretical papers have challenged its basic assumptions and proposed corrections due to collective effects (usually within the weak coupling approximation of Section II.B), while others have supported and extended the model. In this section, we outline the development of the controversy over the binary collision approximation, which is not resolved even today. 

In the past several years systematic studies of the rates of vibrational energy relaxation in simple liquid mixtures have been carried out primarily by two research groups Ewing and co-workers and Brueck and Osgood. These authors have used their results to test the validity of isolated binary collision models. 

S. A. Adelman, R. Muralidhar, and R. H. Stote, /. Chem. Phys., 95, 2738 (1991). Time Correlation Function Approach to Vibrational Energy Relaxation in Liquids Revised Results for Monatomic Solvents and a Comparison with the Isolated Binary Collision Model. 

To determine if there is a correlation between the observed local density enhancements in the model system and the computed relaxation lifetimes, we use isolated binary collision theory [81-83], which, although approximate, has some empirically established validity [75,82,84], to provide a framework for our analysis [12]. Specifically, the isolated binary collision theory yields an expression for the vibrational relaxation rate. 

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