The Reaction Hamiltonian

In a pivotal development. Miller, Handy and Adams [12] derived the classical Hamiltonian for a simple potential based on the MEP. The idea of the reaction path Hamiltonian is, conceptually, to consider the potential as a trough or as a stream bed along with 3N-7 harmonic walls that are free to close in or widen out as one proceeds along the trough. The potential energy surface is approximated as the potential energy of the MEP Vo(s) plus a quadratic approximation to the energy in directions perpendicular to the MEP, 

Here the Q are the generalized normal coordinates and the to are the associated harmonic frequencies. They are obtained at each point on the path by diagonalizing the force constant matrix for which the reaction path direction as well as directions corresponding to rotations and translations have been projected out. The projected force constant matrix has seven zero eigenvalues corresponding to overall rotations, translations, and the reaction path direction. It also has 3N-7 nonzero eigenvalues corresponding to vibrations transverse to the path. 

To obtain the reaction path potential via electronic structure calculations, one must begin at the saddle point and numerically integrate Eq. (2) as discussed previously to obtain the MEP. The force constant matrix is then needed at several points along the path in order to perform the normal mode analyses to obtain the generalized normal modes. 

To obtain the Hamiltonian function for this reaction path potential, it is necessary to express the kinetic energy in terms of the momenta congugate to the reaction path coordinates. The result, for the J=0 case, is [12] 

Several approaches to obtaining both qualitative and quantitative information on polyatomic reaction dynamics via reaction path potentials have been discussed by Miller [13], Truhlar and Gordon [14], Kraka and Dunning [1], Collins [15] and Hammes-SchifFer [16]. 

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