Reaction Path methods

Except for reactions with low barriers (i.e. 10 kcal/mol at 7 = 300 K), or at high temperatures, the quantity /k T is large, and the last series can be neglected. The tunnel correction is then given completely in terms of the magnitude of the imaginary frequency. For small values of the first term may be Taylor expanded to give 

The first order term is known as the Wigner correction.  

It is possible to derive tunnel corrections for functional forms of the energy barrier other than an inverted parabola, but these cannot be expressed in analytical form. Since any barrier can be approximated by a parabola near the TS, and since tunnelling is most important for energies just below the top, they tend to give results in qualitative agreement with the Bell foimula. 

Inclusion of dynamical effects allows calculation of corrections to simple Transition State Theory, often described by a transmission coefficient k to be multiplied with the TST rate constant (Section 12.1), or used in connection with variational TST (Section 12.3). Classical dynamics allow corrections due to recrossing to be calculated, while a quantum treatment is necessary for including tunnelling effects. Owing to the stringent 

The tunnel effect is sometimes approximated by inclusion of a semi-classical 

In the spirit of the Car-Parrinello approach, the whole set of variables (nuclei and wave function parameters) may also be allowed to evolve simultaneously by solving the time-dependent Schrodinger equation. Orhn and coworkers have developed an 

Electron-Nuclear Dynamics (END) method, where both the orbitals describing the electronic wave function and the nuclear degrees of freedom are described by expansion into a Gaussian basis set, which moves along with the nuclei. Such an approach in principle allows a complete solution of the combined nuclear-electron Schrodinger equation without having to invoke approximations beyond those imposed by the basis set. Inclusion of the electronic parameters in the dynamics, however, means that the fundamental time step is short, and this results in a high computational cost for even quite short simulations and simple wave functions. 

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The reaction path methods discussed in Section 5.9.3 may be helpful in determining these pathways. 

THE INTERFACE BETWEEN ELECTRONIC STRUCTURE THEORY AND REACTION DYNAMICS BY REACTION PATH METHODS... 

At present, reaction path methods represent the best approach for utilizing ab initio electronic structure theory directly in chemical reaction dynamics. To study reaction dynamics we need to evaluate accurately the Born-Oppenheimer molecular potential energy surface. Our experience suggests that chemical reaction may take place within in a restricted range of molecular configurations (i.e., there is a defined mechanism for the reaction). Hence we may not need to know the PES everywhere. Reaction path methods provide a means of evaluating the PES for the most relevant molecular geometries and in a form that we can use directly in dynamical calculations. 

Reaction path methods have great promise for future progress because they can be made direct or automated. Direct ab initio dynamics is taken to mean the ab initio evaluation of the molecular energy whenever it is required in a dynamical calculation. At present this is prohibitively expensive. A more traditional approach uses electronic structure methods to provide information about a PES which is then fitted to a functional form. The dynamics employed thereafter is restricted only by the limitations of current dynamical theories. However, the process of fitting functional forms to a molecular PES is difficult, unsystematic, and extremely time consuming. The reaction path approaches that use interpolation of the reaction path data are affordable methods aimed toward direct dynamics, as they avoid the process of fitting a functional form for the PES. Such methods have been automated (programmed) and are therefore readily usable. Thus reaction path methods have been applied with various levels of ab initio theory to statistical theories of the reaction rate, to approximate quantum dynamics, and to classical trajectory studies of reactions. 

Another area where technical advances are in progress is the treatment of the vibrations perpendicular to the reaction path. A central assumption of all reaction-path methods, including variational transition state theory and small-curvature tunneling methods, is the separation of the coordinates into three sets external coordinates describing the overall translation and rotation, a reaction coordinate describing the motion of the system along some direct route from reactants to products, and the remaining coordinates, which will be called the bound vibrational coordinates. 

All the trajectories are reactive as in any reaction path method. This is in contrast to initial value normal and MTS MD methods, in which many trajectories do not end at the desired state. This also enhances the efficient use of computational resources. 

The SDEL formulation is general. It is not limited to processes with large energy barriers, single barriers, or those with exponential kinetics. This makes SDEL more versatile than other reaction paths methods. 

The length formulation makes it difficult to estimate the time scale of the process. SDEL can provide information about the relative sequence of events but not absolute times. This is a limitation shared by all reaction path methods. 

Summary. The reaction path method is a selfcontained approximate theory which allows an analysis of complex chemical reactions and the calculation of state resolved reaction rate-constants using information on gradients and hessians of the nuclear potential energy surface. 

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