Dr. Aniruddha Chakraborty

In 1930s, Eyring and Polanyi studied a chemical reaction, providing the first use of the potential energy surface. This surface contains one minimum associated with the reactant and another minimum for product. They are separated by a barrier that has to be crossed for the chemical reaction to occur. Eyring and Polanyi defined the surface’s transition state as the path of steepest ascent from the barriers saddle point in coordinate space. Once crossed, this transition state can never be re-crossed. The notion of a transition state as a surface of no return defined in coordinate space was soon recognized as fundamentally flawed, because re-crossing is in general possible. So transition state theory provides upper bound of the exact classical reaction rate. The exact classical reaction rate in general will depend on the choice of dividing surface. This is the key idea behind the variational transition state theory, in which one searches the space between reactants and products for a dividing surface that minimizes the reaction rate.

E. Wigner was probably the first person to recognize that in order to develop a rigorous theory of reaction rates, one must extend the notions above from configuration space to phase space. In the late 1970’s P. Pechukas shows that, for systems with two degrees of freedom, at energies sufficiently close to the energy at the saddle point, the classical trajectory perpendicular to the reaction path that passes through the saddle point is an unstable periodic orbit. If the orbit of this periodic trajectory is plotted in phase space, it defines a surface known as a periodic orbit dividing surface or PODS - this corresponds to the surface of minimum flux. The stable and unstable manifolds of this orbit provide an invariant partition of systems’s energy shell into reactive and nonreactive dynamics. The defining periodic orbit also bounds a surface in the energy shell (at which the Hamiltonian is constant) partitioning it into reactant and product regions.

This is also called repulsive PODS, since any trajectory that crosses it never comes back and so PODS define a surface of no return and yields an unambiguous measure of the flux between the reactants and products. It can be shown that as long as there is a single PODS, transition state theory is exact, while the energies at which there are multiple PODS, transition state theory is no longer exact.

In systems with three or more degrees of freedom, periodic orbits and their associated stable and unstable manifolds do not partition the energy shells because their dimensionality is insufficient. So one needs to search for higher dimensional analogue of PODS  i.e. NHIM (Normally Hyperbolic Invariant Manifold) . Quite recently, the general method for extracting no return transition states for many degrees of freedom system has been established by Wiggins.