Classical RRKM

As a result of possible recrossings of the transition state, the classical RRKM lc(E) is an upper bound to the correct classical microcanonical rate constant. The transition state should serve as a bottleneck between reactants and products, and in variational RRKM theory [22] the position of the transition state along q is varied to minimize k E). This minimum k E) is expected to be the closest to the truth. The quantity actually minimized is N (E - E ) in equation (A3.12.15). so the operational equation in variational RRKM theory is... 

The analysis is performed for the calculations with rrot=0 K for the CH3C1 reactant, so that the angular momentum distribution for the complex P(j) is the distribution of orbital angular momentum for complex formation P(i). This latter distribution is given in ref. 37. Jm , the quantum number for j, varies from 282 for Enl = 0.5 kcal/mol to 357 for rel = 3.0 kcal/mol. The term k iEJ) in equation 24 is written as k (.EJ)=k Ejyf E), where k EJ) is the classical RRKM rate constant with the CH3C1 intramolecular modes inactive and / ( ) is treated as a fitting factor. 

As in the development of RRK theory, the analysis starts with the assumption that the dynamics of the molecule are classical. The essential approximations of the classical RRKM treatment are that... 

Although the use of the correct energy levels for calculating the density of states is strictly a quantum correction of the classical RRKM theory, there are two other effects that are much more fundamental to the theory quantum mechanical tunnelling and fluctuations. The first of these is dealt with in Chapter 2, and the second is the main subject of Chapter 3. 

The classical RRKM expression provides rate coefficients that are generally in error by orders of magnitude. The simple replacement of the classical expressions for the transition state number of states and the reactant density of states with their quantum mechanical counterparts... 

The quantized results of Eqs. (2.8) and (2.10) negate, at least in principle, the variational property of classical RRKM theory, since both the one-dimensional tuimeling probabilities and the step function may exceed the true reactivities. In practice, however, variational minimizations still prove useful. 

Figure 17 Calculated state-specific rates k — V/h for HCO and HNO vs. excess energy (small dots). The symbols indicate the fast (triangles), the slow (squares) and the average (diamonds) classical rates (see Sect. 8). The solid and dashed lines are the SACM and the classical RRKM rates, respectively. Fleprinted, with permission of the Bunsengesellschaft fiir Physikalische Chemie, from Ref. 39.

Gray, S.K., Rice, S.A. and Davis, M.J. (1986) Bottlenecks to unimolecular reactions and alternative form for classical RRKM Theory. J.Phys.Chem. 90. 3470-3482. 

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. 

It is also interesting to consider the classical/quantal correspondence in the number of energized molecules versus time N(/, E), Eq. (8.22), for a microcanonical ensemble of chaotic trajectories. Because of the above zero-point energy effect and the improper treatment of resonances by chaotic classical trajectories, the classical and quantal I l( , t) are not expected to agree. For example, if the classical motion is sufficiently chaotic so that a microcanonical ensemble is maintained during the decomposition process, the classical N(/, E) will be exponential with a rate constant equal to the classical (not quantal) RRKM value. However, the quantal decay is expected to be statistical state specific, where the random 4i s give rise to statistical fluctuations in the k and a nonexponential N(r, E). This distinction between classical and quantum mechanics for Hamiltonians, with classical f (/, E) which agree with classical RRKM theory, is expected to be evident for numerous systems. 

A number of MD studies on various unimolecular reactions over the years have shown that there can sometimes be large discrepancies (an order of magnitude or more) between reaction rates obtained from molecular dynamics simulations and those predicted by classical RRKM theory. RRKM theory contains certain assumptions about the nature of prereactive and postreactive molecular dynamics it assumes that all prereactive motion is statistical, that all trajectories will eventually react, and that no trajectory will ever recross the transition state to reform reactants. These assumptions are apparently not always valid otherwise, why would there be discrepancies between trajectory studies and RRKM theory Understanding the reasons for the discrepancies may therefore help us learn something new and interesting about reaction dynamics. 

S. K. Gray, S. A. Rice, and M. J. Davis, /. Phys. Chem., 90, 3470 (1986). Botdenecks to Unimolecular Reactions and an Alternative Form for Classical RRKM Theory. M. J. Davis and S. K. Gray, /. Chem. Phys., 84,5389 (1986). Unimolecular Reactions and Hiase Space Bottlenecks. 

The classical RRKM rate constant may be written as an average flux through the transition state. " To illustrate this, the number of quantum states for the reactant molecule at energy E may be expressed as ... 

The classical RRKM rate constant written this way is seen to be an average flux through the transition state. 

RRKM theory has been used widely to interpret measurements of unimolecular rate constants. However, harmonic state counting procedures are usually used in the RRKM calculations. This is not because enharmonic effects are thought to be unimportant, but because they are difficult to account for. The only comprehensive attempt to include the effect of anharmonicity has involved treating the vibrational degrees of freedom as separable Morse oscillators. However, since this correction is an obvious oversimplification it has not been widely used. The importance of anharmonicity is illustrated by comparing the trajectory unimolecular rate constant for C2H5 H + C2Hi dissociation at 100 kcal/mol (Fig. 4b), which is about 4.7 X 10 with that predicted by harmonic classical RRKM... 

---