RRKM theory vibrational frequencies

Variational RRKM theory is particularly important for imimolecular dissociation reactions, in which vibrational modes of the reactant molecule become translations and rotations in the products [22]. For CH —> CHg+H dissociation there are tlnee vibrational modes of this type, i.e. the C—H stretch which is the reaction coordinate and the two degenerate H—CH bends, which first transfomi from high-frequency to low-frequency vibrations and then hindered rotors as the H—C bond ruptures. These latter two degrees of freedom are called transitional modes [24,25]. C2Hg 2CH3 dissociation has five transitional modes, i.e. two pairs of degenerate CH rocking/rotational motions and the CH torsion. 

Vibrational frequencies for various normal modes must be estimated and active as well as inactive energies should be decided. Numerical methods may be used to calculate rate constant k at various concentrations obtained by RRKM theory. The rate constant has been found to be same as given by conventional transition state theory, i.e. 

Results of a PEPICO study of the dissociation dynamics of 2-bromobutane ions have been analysed with tunnelling-corrected RRKM statistical theory using vibrational frequencies obtained from ab initio MO calculations. It has been concluded that the slow rate of loss of HBr, to form the but-2-ene ion, occurs via a concerted mechanism in which tunnelling is a feature of the proton transfer. 

The procedure adopted here is to make use again of RRKM theory to calculate k2/k l as a function of the relative barrier height. In this case, the transition state for the, reaction is taken as the loose ion-molecule complex at the Langevin capture distance. The transition state for the reaction k2 is taken as the tetrahedral intermediate RCOYX ". By a suitable choice of the vibrational frequencies and moments of inertia, this type of calculation shows that E 0-E0 for Cl- + CH3COCl should be around — 7 kcal mol 1 in order to reproduce the experimental efficiency. This amounts to an E 0 of 4 kcal mol-1. 

Because QRRK theory was developed long before computing became readily available, it had to employ significant physical approximations to obtain a tractable result. The most significant assumption was that the molecule is composed of s vibrational modes with identical frequency i and that other molecular degrees of freedom are completely ignored. RRKM theory relies on neither approximation and thus has a much sounder physical basis. In the limit of infinite pressure, RRKM theory matches the transition state theory discussed in Section 10.3. 

RRKM theory uses the actual vibrational frequencies of the molecule. The density of molecular states (i.e., the number of quantum states per unit energy range) is obtained using direct counting techniques. Modem high-speed computing and efficient algorithms make this aspect of the theory quite accurate [33,375,430]. 

The dimension of the factor IIf=1z/j/IIlz v is that of a frequency. If the frequencies of the reactant and the activated complex are not too different, this frequency is roughly a typical vibrational frequency vr (typically in the range 1013 to 1014 s 4). Since the energy-dependent factor is less than one, we have that the microcanonical rate constant k(E) < i/r, i.e., it is less than a typical vibrational frequency. The energy dependence as a function of the number of vibrational degrees of freedom was illustrated in Fig. 7.3.2, and as shown previously in Eq. (7.38) it can be interpreted as the probability that the energy in one out of s vibrational modes exceeds the energy threshold Eq for the reaction. Note that if we make the identification vr n =1z/i/n 11i/ , we have recovered RRK theory, Eq. (7.39), from RRKM theory. 

A more general discussion of the dependence of the decomposition rate on internal energy was developed by Marcus and Rice [4] and further refined and applied by Marcus [5] (RRKM). Their method is to obtain the reaction rate by summing over each of the accessible quantum states of the transition complex. The first-order rate coefficient for decomposition of an energised molecule is shown to be proportional to the ratio of the total internal quantum states of the transition complex divided by the density of states (states per unit energy) of the excited molecule. It is a great advance over previous theory because it can be applied to real molecules, counting the states from the known vibrational frequencies. 

These problems arise because of the use of the classical density of states rather than the proper vibrational energy levels, and, of course an alternative would be to use the more difficult QRRK theory. If this is done, acceptable fits may be obtained to experimental data using the full number of oscillators. However, QRRK theory is not easily applicable with a realistic spectrum of vibrational frequencies, and it is preferable to use an alternative theory such as the RRKM theory instead. 

Whereas in the old RRK theory the v was simply an adjustable parameter (Rice and Ramsperger, 1927, 1928), it can here be calculated from the vibrational frequencies of the TS and the molecule. The classical rate constant in Eq. (6.77) cannot be compared to experimentally measured rate constants because the vibrational density of states is dominated by quantum effects. On the other hand, classical RRKM rate theory is highly useful for comparing with rate constants obtained from classical trajectory calculations. 

Although the entropy is evaluated in terms of the vibrational frequencies of the reactant and the transition state, it is a single parameter. Thus, in spite of the large number of frequencies, the RRKM equation, is in first order a low-parameter theory. 

The effect on the slopes of the k(E) curves for the dissociation of the bro-mobenzene ion with various assumed entropies, based on the frequencies in table 7.2, and energies of activation are shown in figure 7.3 (Baer et al., 1991). While the bromobenzene ion has no barrier in the dissociation channel, it is here treated with a vibrator TS. It is evident that two parameters can generate a whole family of k E) curves. Thus, if neither the activation energy nor the transition state structure is known, any set of data can be fit with RRKM theory. The lower the TS frequencies, the steeper the slope. However, if either the activation energy is known from other information, or if the frequencies are known from calculations, then the RRKM equation reduces to a one parameter model in which either the magnitude of the rate or the slope can be adjusted, but not both. 

The variational version of RRKM theory (VTST) can be used to locate the transition state on the basis of the minimum sum of states. However, if this level of effort does not appear appropriate for the particular reaction, it is perfectly possible to fit a given data set with the vibrator model of the RRKM theory simply by adjusting the transition-state vibrational frequencies until a fit is obtained (as was done in the calculations of figures 7.3 and 7.4). In fact, such a fitting procedure is one means for determining whether the reaction is characterized by a loose or a tight transition state. 

Application of the RRKM theory (used for to reactions in the gas phase) to surface reactions required detailed analysis of contributions from vibrations that belong to the matrix to the statistical sums of the prereaction state and transition state [29,68]. Inclusion of the low-frequency phonons of the solid (<200 cm ) leads to appreciable dependence of the preexponential of the rate constant (feo) on the temperature. It should be noted that the ko values calculated on the basis of the experimental therma-grams (TPD-MS) at various temperature sections on the assumption of the nondependence of on T can differ by three orders of magnitude (Table 37.7). 

Poor agreement is observed between the experimental and theoretical values [calculated by a combined method, that is, quantum-chemical calculation of the activation energy (Table 37.7 and Table 37.8) and the frequencies of the vibrations of the bonds in the prereaction complex and in the transition state and calculation of the rate constants on the basis of RRKM theory] for certain processes where the effects of electron correlation and the contributions of the excited electronic configurations are not predominant [68-73]. 

With the development of computers, accurate calculations using theoretical models better able to represent the behavior of real molecules has become widespread. A very important extension of the original theory, due to Marcus, is known as RRKM theory. Here, the real vibrational frequencies are used to calculate the density of vibrational states of the activated molecule, N E). The number of ways that the total energy can be distributed in the activated complex at the transition state is denoted W E ). Note that the geometry of the transition state need not be known, but the vibrational frequencies must be estimated in order to calculate W E ). n calculating the total number of available levels of the transition state, explicit consideration of the role of angular momentum is included. The RRKM reaction rate constant is given by ... 

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