Semi-classical Rate Constants

Hydrogen-atom or proton-transfer transfers between heavy atoms often lead to vibrationally adiabatic paths with two maxima. Formally, the resolution of this problem should be made in the fi amework of the canonically unified theory [5], In practice, given the approximate nature of our treatment, it suffices to use TST with AFj equal to the value for the maximum point. For these cases, the tunnelling correction of the particles wifli energies between that of the highest of the two maxima and the minimum between tiiese is calculated for the highest barrier only. 

The semi-classical expression for the reaction rate constant given by transition-state theory 

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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 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. 

In the light of the path-integral representation, the density matrix p Q-,Q-,p) may be semi-classically represented as oc exp[ —Si(Q )], where Si(Q ) is the Eucledian action on the -periodic trajectory that starts and ends at the point Q and visits the potential minimum Q = 0 for r = 0. The one-dimensional tunneling rate, in turn, is proportional to exp[ —S2(Q-)], where S2 is the action in the barrier for the closed straight trajectory which goes along the line with constant Q. The integral in (4.32) may be evaluated by the method of steepest descents, which leads to an optimum value of Q- = Q. This amounts to minimization of the total action Si -i- S2 over the positions of the bend point Q. ... 

W. H. Miller, Semi-classical theory for non-separable systems construction of good action-angle variables for reaction rate constants, Faraday Disc. Chem. Soc. 62, 40 (1977). 

The rate of hydrogen transfer can be calculated using the direct dynamics approach of Truhlar and co-workers which combines canonical variational transition state theory (CVT) [82, 83] with semi-classical multidimensional tunnelling corrections [84], The rate constant is calculated using [83] ... 

In this semi-classical treatment, effects of isotopic substitution on equilibria and rates in chemical reactions reflect the changes in zero-point energy differences between (i) reactants (R) and products (P) - equilibrium isotope effects (EIE), or (ii) between reactants (R) and transition structures (TS) - kinetic isotope effects (KIE). In both cases, these changes reflect changes of force constants at the isotopically substituted site. When bonds are broken and made at the isotopic atom, the effects are described as primary, otherwise they are described as secondary. 

CVT approach is particularly attractive due to the limited amount of potential energy and Hessian information that is required to perform the calculations. Direct dynamics with CVT thus offers an efficient and cost-effective methodology. Furthermore, several theoretical reviews60,61 have indicated that CVT plus multidimensional semi-classical tunneling approximations yield accurate rate constants not only for gas-phase reactions but also for chemisorption and diffusion on metals. Computationally, it is expensive if these Hessians are to be calculated at an accurate level of ab initio molecular orbital theory. Several approaches have been proposed to reduce this computational demand. One approach is to estimate rate constants and tunneling contributions by using Interpolated CVT when the available accurate ab initio electronic structure information is very limited.62 Another way is to carry out CVT calculations with multidimensional semi-classical tunneling approximations. 

Thus, the semi-classical Marcus theory of non-adiabatic ET expresses the ET rate constant in terms of three important quantities, namely Vel, A, and AG°. It therefore follows that an understanding of ET reactions entails an understanding of how these three variables are dependent on factors such as the electronic properties of the donor and acceptor chromophores, the nature of the intervening medium and the inter-chromophore separation and orientation. 

Classical, semi-classical, and quantum mechanical procedures have been developed to rationalize and predict the rates of electron transfer. In summary, the observed rate of a self-exchange reaction can be calculated as a function of interatomic distances, force constants, electronic coupling matrix element, and solvent parameters. These model parameters are either calculated, estimated, or determined by experiment, in each case with a corresponding standard deviation. Error propagation immediately demonstrates that calculated rates have error ranges of roughly two orders of magnitude, independent of the level of sophistication in the numerical procedures. 

Most electron-transfer systems belong to classes I and IIA. In terms of the semi-classical model the rate constant for electron transfer in these systems is given by ... 

A number of modern and quite sophisticated treatments of H-transfer in the condensed phase are not that dissimilar from the approach taken by Bell, in that they formulate the rate constant for H-transfer as a semi-classical term multiplied by a tunneling correction factor, e.g. Eqs. (10.16a) and (10.16b). 

Joseph, T.R., Steckler, R. and Truhlar, D.G. (1987) A new potential energy surface for the CH3 + H2 CH + H reaction Calibration and calculation of rate constants and kinetic isotope effects by variational transition state theory and semi-classical tunneling calculations, J. Chem. Phys. 87, 7036-7049. 

The semi-classical temperature dependence of electron transfer rate constants has been inferred from Equations (l)-(3). However, it is frequently observed that these equations are more successful at describing the overall behavior of the activation free energy, AG a than that of the component activation enthalpy, and entropy, A5 da, terms. Newton has suggested... 

The content of the chapter is arranged as follows. Section 12.2 describes the semi-classical approximation of thermal rate constants for ET in a fast solvent relaxation limit. Section 12.3 discusses solvent-controlled ET theories. Section 12.4 presents the TDWPD approach and section 12.5 shows several applications. Concluding remarks are given in section 12.6. 

If we can successfully predict the influence of basic molecular properties such as size or mass on the rate constant k of electron-transfer reactions, we can in principle calculate the value of k in terms of those properties, using some model (classical, semi-classical or quantum-mechanical). The comparison of such theoretical values with experimental ones... 

Figure 9.11 Correlation of rate constants for electron-exchange reactions with molecular dimensions (3). The plot is of-[ln(/ obs// afnf ) + Gjjj // 7 ], against f(Ado) /RT (representing AG j/ffr). The semi-classical...

Experimental determinations of rate constants for intramolecular electron-transfer, for a series of related donor-bridge-acceptor (DBA) molecules in which the donor, the acceptor, or the bridge has been varied, usually show a fairly smooth relation between the ergonic-ity (—AC ) of the reaction and ket or AG (. Often the plot of log ket against -AG is linear over the observable range (usually limited by that of AG ) sometimes it is appreciably curved. Such plots can be used to examine the classical and semi-classical Marcus equations. 

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