The Reaction Path Hamiltonian and Variational Transition State Theory

THE REACTION PATH HAMILTONIAN AND VARIATIONAL TRANSITION STATE THEORY 

In 1980, Miller, Handy, and Adams further entrenched the reaction path idea with the derivation of the classical Hamiltonian for a simple potential based on the MEP. The motivations for the development of the reaction path Hamiltonian included the emerging ability to calculate derivatives of the Born—Oppenheimer potential energy (i.e., forces and force constants) directly via electronic structure methods and the desire to develop practical methods for 

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. [8] as previously discussed 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. 

The curvature of the reaction path at a nonstationary point can be calculated from only the gradient and the force constants via Eq. [14]. Then from Eq. [20], the curvature coupling elements are given by 

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Chapter 2, Michael L. McKee and Michael Page address an important issue for bench chemists how to go from reactant to product. They describe how to compute reaction pathways. The chapter begins with an introduction of how to locate stationary points on a potential energy surface. Then they describe methods of computing minimum energy reactions pathways and explain the reaction path Hamiltonian and variational transition state theory. 

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