Classical rate constants

The most important reactivity observable is the rate constant. The interest in calculating rate constants was already expressed in the development of the coUision theory and the TST. Such calculations require a reliable estimate of the reaction barrier, which remains a difficult task for ab initio calculations of polyatomic systems. The simplest way to make rate calculations is to use the energy barriers calculated by ISM in eq. (6.54) of the TST. However, in this empirical version of ISM, A V was obtained from the scahng to the activation energy of a reference system, and differs from the vibrationaUy adiabatic barrier AVai, used in the TST. An approximate relation between the activation energy and vibra-tionaUy adiabatic barrier can be obtained neglecting the tunnel effect and the minor difference between the thermal disfribution of the molecules in the transition state and in the 

We explicitly use AV rather than in the expression above, to emphasise that it was obtained nsing the ISM approximations for a gas-phase A+BC reaction with a Unear tri-atomic transition state, in molar concentration units. Under these approximations, and remembering that the ratio of the vibrational partition functions was assumed to be unity, we obtain a compact expression for the classical transition-state rate constants, eq. (6.33), 

The transition-state moment of inertia can only be calculated with the transition-state bond lengths. This is not a problem because the transition-state bond order, , obtained with eq. (6.68), can be employed in eq. (6.57) to yield the transition-state bond extensions. In summary, classical rate constants can be calculated with eqs. (6.57), (6.68), (6.76), (6.77) and (6.78), that only require bond dissociation energies, vibrational frequencies and bond lengths of the reactive bonds, ionization energies and electron affinities of the relevant radicals, and the masses of the atoms involved in the reaction coordinate. A few examples can make these calculations more clear. 

The ISM can also be applied to polyatomic systems, but now the partition function ratio of eq. (6.33) does not simplify into the pre-exponential factor of eq. (6.76). For linear transition states, the ratio of the partition functions for atom+diatomic and atom+polyatomic molecule has the same form 

However, some of the vibrations in polyatomic transition states are very loose vibrations and sometimes even regarded as hindered rotations. Thus, 0 should be rather close to the 

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In order to segregate the theoretical issues of condensed phase effects in chemical reaction dynamics, it is usefiil to rewrite the exact classical rate constant in (A3.8.2) as [5, 6, 7, 8, 9,10 and U]... 

Although Eqs. (4-1) and (4-2) have identical expressions as that of the classical rate constant, there is no variational upper bound in the QTST rate constant because the quantum transmission coefficient Yq may be either greater than or less than one. There is no practical procedure to compute the quantum transmission coefficient Yq- For a model reaction with a parabolic barrier along the reaction coordinate coupled to a bath of harmonic oscillators, the quantum transmission... 

Note that the conventional TST expression is simply the special case of VTST where evaluation is done exclusively for s = 0. As such, the VTST rate constant will always be less than or equal to the conventional TST rate constant (equal in the event that s = 0 minimizes Eq. (15.35)). Put differently, when very accurate potential energy surfaces are available, the conventional TST rate constant is typically an overestimate of the exact classical rate constant. (Note that it is possible, however, for a compensating or even offsetting error to arise from overestimation of the barrier height if the potential energy surface is not very accurate.)... 

Figure 7. Dissociation rates k as extracted from the quantum mechanical calculations (open circles). The statistical rates are represented by the step functions and the filled circles represent the classical rate constants as obtained from elaborate classical trajectory calculations. (Reprinted, with permission of the Royal Society of Chemistry, from Ref. 34.)...

For ds > 2, a random walker has a finite escape probability-microscopic behavior conducive to re-randomize the distribution of reactants around a trap and deplete the supply of reactive pairs, and thus a stable macroscopic reactivity as attested by the classical rate constant [296,297]. The scale of the self-organization is microscopic and independent of time, such that n (t) oc t (is linear) and k = n (t) is a constant, so the reaction kinetics are classical. 

The theoretical framework in the present discussion is transition state theory (TST), which yields the expression of the classical rate constant.9 For a unimolecular reaction, the forward rate constant is given below ... 

Lagana and co-workers utilized a wide variety of parallel supercomputers to calculate the rate constants and cross sections of various gas phase chemical reactions.2 79-284 They computed the quasi-classical rate constants and cross sections for the H + and D + reactions, using both shared-... 

If the classical dynamics is ergodic and intrinsically RRKM, one might expect that the classical rate constant approximates the average rate of the quantum mechanical state-specific rates. That is indeed the case for the dissociation of HO2 (Fig. 12 of Ref. 60) the classical rate is only slightly smaller than the average quantum mechanical rate. The same holds also... 

With the barrier results given in Table V it is possible to estimate the rate constant for ring rotation by using analytic models for the dynamics involved. The simplest assumption is to regard the rotation as a unimolecular process that can be treated by transition-state theory with a classical rate constant k equal to... 

Electron-transfer kinetics in solutions have often been analyzed and interpreted in the framework of the general adiabatic theory of Marcus (43). Although electron-transfer dynamics are not always characterized by a classical rate constant (44), a general formulation of the chemical reaction concerns the rate constant k, which can be expressed as ... 

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. 

Moreover, using a varia tional approach the "classical" expression (145. Ill) was obtained as an approximation to (141 111) which yields an upper boundary for the "classical" rate constant. 

The first part of the chapter focuses on the derivation of a mixed quantum-classical theory for rationalizing chemical reactions involving two electronic states. The central piece of this mixed quantum-classical rate constant is the appearance of a characteristic decoherence time. The second part of the chapter deals with numerical approaches at the atomic level that can be carried out to decipher the molecular mechanisms governing decoherence in real systems of biological interest. 

At our most fundamental level of description we consider molecular systems to be composed of atomic nuclei and electrons, all obeying quantum mechanical laws. The question of the kinetics of a physicochemical event is therefore related to the time evolution of such composite systems. In the first sub-section we recall the basic quantum mechanical equation-of-motions relevant in this context. We then consider approximations that can be operated to simplify the nuclear-electronic dynamics, leading to the derivation of the mixed quantum-classical rate constant expression. 

We then finally insert eqn (5.19) into eqn (5.2) to obtain a mixed quantum-classical rate constant expression ... 

Here k n and k m are the microscopic classical rate constants for conversion of state n to state m and vice versa. The subscript stochastic indicates that we are considering relaxations that depend on random fluctuations of the surroundings, not the oscillatory, quantum-mechanical phenomena described by Eq. (10.23). The ensemble will relax to a Boltzmann distribution of populations if the ratio k Jk m is given by tJ. —EnJk ). According to Eq. (10.27), relaxations of the diagonal elements toward thermal equilibrium do not depend on the off-diagonal elements of p, which is in accord with classical treatments of kinetic processes simply in terms of populations. 

Brunschwig et have reviewed existing nonadiabatic theories for comparison with data on bimolecular reactions. They define the semi-classical rate constant k c by equation (10) where ka is the classical ... 

In the current understanding of PCET reactions, both electron and proton are treated quantum-mechanically, and therefore the tunnelling probability must be accounted for both particles. In fact, concerted processes can be described as double tunnelling (proton and electron), with a single transition state. " For a description of the reaction coordinate, four adiabatic states (reactants, products and intermediates) described by paraboloids, are usually considered. The expression for the semi-classical rate constant in this case incorporates elements derived from electron and proton transfer theories... 

The consideration of statistical factors in the TST is a simple and necessary refinanent of this theory. When the statistical factor is included, the classical rate constant of TST is expressed as... 

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