Kassel model

At present there exist [72, 73] different models to describe monomolecular reactions. We use the quantum version of the Kassel model [72], which combines the obviousness and simplicity of the formulae with the high extent of rigour. Within the framework of this model the molecule is considered as a set of equivalent oscillators having the frequency co. Energy exchange between the oscillators is assumed to be fast. The function f(n) of the distribution of the molecules over the energy range is derived from the equation... 

This rate L(E) will be different for each detailed assignment of the energies E, to the oscillators and is a very complex function of the values E. For sufficiently large molecules with values of E /kT not too small, Slater has estimated that this detailed model behaves like a Kassel model [Eq. (X.5.1)] in which the number of Kassel oscillators n = (s + l)/2. 

This would seem to imply that the Kassel model for energy transfer is not too bad for this complex molecule. If we correct the probable values of Table XI. 2 to the high frequency factor and the lower gas density at 450 C, we obtain a value of about 13 oscillators, also in agreement. 

Kassell (Takahashi et al., 1993) has described the activity of a novel tropolone U88999E in a rabbit model of cerebral vasosjrasm. U88999E inhibits lipid peroxidation and acts as a calcium antagonist. Kassell showed that the compound relaxed preconstricted arterial rings in vitro (potency slightly less than flunarizine or dil-tiazem) and that it reduced vasospasm of basilar arteries after rabbit subarachnoid haemorrhage (Takahashi etal., 1993). 

A stimulus-filter model of nicotine reinforcement has been proposed that suggests that it helps screen irrelevant stimuli from awareness (Kassel 1997). An alternative explanation asserts that nicotine induces attentional narrowing and facilitates perceptual processing. Human nonsmokers given an acute dose of nicotine (administered in gum) showed dose-related trends toward decreased accuracy and increased response time (Heishman et al. 1993). 

Kariv, I., Rourick, R.A., Kassel, D.B., and Chung, T.D.Y. Improvement of hit-to-lead optimization by integration of in vitro HTS experimental models for early determination of pharmacokinetic properties. Comb. Chem. High Throughput Screen. 2002, 5, 459-472. 

Rice, Ramsperger, and Kassel [206,333,334] developed further refinements in the theory of unimolecular reactions in what is known as RRK theory. Kassel extended the model to account for quantum effects [207] this treatment is known as QRRK theory. 

An even more complex model that of Kassel... 

The RRK (after Rice, Ramsperger, and Kassel) theory is, like the Slater theory, a model for a unimolecular reaction rather than a faithful representation. The molecule is again represented by a collection of s uncoupled harmonic oscillators, which is an exact representation close to a stationary point on the potential energy surface. One of these... 

In more detail, our approach can be briefly summarized as follows gas-phase reactions, surface structures, and gas-surface reactions are treated at an ab initio level, using either cluster or periodic (plane-wave) calculations for surface structures, when appropriate. The results of these calculations are used to calculate reaction rate constants within the transition state (TS) or Rice-Ramsperger-Kassel-Marcus (RRKM) theory for bimolecular gas-phase reactions or unimolecular and surface reactions, respectively. The structure and energy characteristics of various surface groups can also be extracted from the results of ab initio calculations. Based on these results, a chemical mechanism can be constructed for both gas-phase reactions and surface growth. The film growth process is modeled within the kinetic Monte Carlo (KMC) approach, which provides an effective separation of fast and slow processes on an atomistic scale. The results of Monte Carlo (MC) simulations can be used in kinetic modeling based on formal chemical kinetics. 

Thermal unimolecular reactions usually exhibit first-order kinetics at high pressures. As pointed out originally by Lindemann [1], such behaviour is found because collisionally energised molecules require a finite time for decomposition at high pressures, collisional excitation and de-excitation are sufficiently rapid to maintain an equilibrium distribution of excited molecules. Rice and Ramsperger [2] and, independently, Kassel [3] (RRK), realised that a detailed theory must take account of the variation of decomposition rate of an excited molecule with its degree of internal excitation. Kassel s theory is still widely used and is valid for the chosen model of a set of coupled, classical, harmonic oscillators. 

The simple model outlined in the previous section would require that be a linear function of [M]". In fact, such plots of experimental data show marked curvature. The simple scheme fails because the mean time for decomposition of X decreases with its energy. In Kassel s theory [3], the Lindemann scheme is taken to be valid for a small energy range and ft, and fe3 are evaluated as a function of energy. 

To consolidate the discussion, it is instructive at this stage to reduce the general equation, eqn. (8), to the particular case of a set of coupled, classical, harmonic oscillators, i.e. Kassel s model [3]. 

The first usable results were obtained by Rice and Ramsperger" and Kassel,who were able to deduce from a simplified model of a molecule, consisting of a set of harmonic oscillators, that... 

The Kassel theory is of course quantized and in essential agreement with the classical model. N. B. Slater, Proc. Roy. Soc. Edinburgh 64, 161 (1955), has given a quantum version with results somewhat different from his classical model. 

While the Slater model does not lend itself to a simple solution in terms of the quantum theory, the fact that it agrees in form with the simple quantum model of Rice-Ramsperger and Kassel suggests that we can write the following expression for the mean rate of decomposition of a critically energized molecule of energy E E ... 

Bossel H., Metzler W. and Schafer H., Dynamik des Waldsterbens. Mathematisches Modell und Computer-Simulation. Bericbte der Arbeitsgruppe Matbematisierung. Sonderiieft 2, Gesamthochschule Kassel, 267 p. (1984). 

The classical RRK theory was proposed very soon after the quantum theory of Schrodinger, and so it is scarcely surprising that it uses a classical model of the vibrations. However, very soon afterwards Kassel proposed an alternative version of the RRK theory in which the oscillators were quantum harmonic oscillators [13]. In the simplest version of the theory all the oscillators have the same frequency v, although Kassel did also present a version in which the oscillators are divided into two classes with different frequencies. 

Another advantage of the quantum calculations is that they provide a rigorous test of approximate methods for calculating dissociation rates, namely classical trajectories and statistical models. Two commonly used statistical theories are the Rice-Ramsperger-Kassel-Marcus (RRKM) theory and the statistical adiabatic channel model (SACM). The first one is thoroughly discussed in Chapter 2, while the second one is briefly reviewed in the Introduction. Moreover, the quantum mechanical approach is indispensable in analyzing the reaction mechanisms. A resonance state is characterized not only by its position, width and the distribution of product states, but also by an individual wave function. Analysis of the nodal structure of resonance wave functions gives direct access to the mechanisms of state- and mode-selectivity. 

---