Absolute semi-classical rates

The empiricism of the ISM is eliminated using ab initio data, rather than experimental data, to obtain the value of d. The most exact ab initio calculations on reactive systems are those of the H -f system. Varandas and co-workers [41] used such ab initio calculations to build a DMBE PES, which has the properties presented in Table 1. The ab initio sum of bond extensions at the transition state of this surface is 0.3746 A, and eq. (6.60) with n = 0.5 leads to a = 0.182. The ISM is now scaled to structural data and does not involve kinetic information. On the other hand, the model now gives classical (electronic) potential energy barriers, free of ZPE or tunnelling corrections, rather than activation energies. They are directly comparable with the classical barriers of ab initio calculations, but require a method to calculate ZPE corrections along all the reaction coordinates before they can be employed in the TST to calculate tunnelling corrections and semi-classical rate constants. 

The need for an analytical description of the complete reaction coordinate can be satisfied with an interpolation between the energies of the reactants and products, using the continuity of the reaction coordinate n, following earlier work by Agmon and Levine [42], 

Equating the first derivative of the classical reaction path, above, to zero gives the location of the classical transition state 

Using the parameters of the H2 Morse curve presented in Appendix III, and knowing that a symmetrical reaction has n = 0.5, we calculate A = 42.3 kJ mol for the H + Hj reaction. The classical reaction barrier for this reaction is 40.4 kJ moU according to the DMBE PES, Table 6.1, and 41.5 kJ mol according to the most exact ab initio calculations presently available. Table 6.4 presents the classical potential energy barriers calculated with the semi-classical ISM (scISM) and compares them with the classical barriers obtained by the ab initio methods. The agreement is quite satisfactory for such simple calculations without adjustable parameters. 

The similarity of scISM and DMBE reaction paths suggests that their tunnelling corrections should be similar. The simplest realistic tunnelling correction is that of the Eckart barrier. This barrier can be fitted to the scISM reaction path using its asymptotic limits, the barrier height AV j and the curvature of this path at the classical transition state. This latter 

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The tunnel correction is not now a fundamentally defined number rather it is defined by the equation Q = kobJk, where kobs is the observed rate constant for a chemical reaction and k is that calculated on the basis of some model which is as good as possible except that it does not allow tunnelling. In this chapter the definition used for k is that calculated by absolute reaction rate theory [3], i.e., k = KRT/Nh)K where X is the equilibrium constant for the formation of the transition state. The factor k, the transmission coefficient, is also a quantum correction on the barrier passage process, but it is in the other direction, that is k < 1. We shall here follow the customary view (though it is not solidly based) that k is temperature-independent and not markedly less than unity. The term k is used following Bell [1] the s stands for semi-classical, that is quantum mechanics is applied to vibrations and rotations, but translation along the reaction coordinate is treated classically. 

The availability of three isotopes of hydrogen allows the comparison between the behaviour of any two of these experimentally and the comparison to that calculated by absolute reaction rate theory. Since the isotope effect cancels out the major uncertainty in absolute reaction rate theory in calculations of rate, namely the barrier height, the theory puts some sharp restrictions on the magnitude of the isotope effect and its temperature dependence [10]. However, the semi-classically calculated isotope effect k"//cP is multiplied by the factor a number greater than unity which increases as the... 

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