C Reaction Path Hamiltonian Theory Let us review what the reaction path is as defined above. It is a path that 1. begins at a transition state (ts)and evolves along the direction of negative curvature on the potential energy surface(as found by identify ing the eigenvector of the Hessian matrix H k =aE/as, as, that belongs to the negative eigenvalue); i1. moves further downhill along the gradient vector g whose components are g =aE/as iii terminates at the geometry of either the reactants or products( depending on whether one began moving away from the ts forward or backward along the direction of negative curvature) The individual"steps" along the reaction coordinate can be labeled So, S, S,,... Spas they evolve from the ts to the products(labeled Sp)and SR,SR+I,. So as they evolve from reactants(S-g)to the Ts. If these steps are taken in very small (infinitesimal) lengths, they form a continuous path and a continuous coordinate that we label s At any point S along a reaction path, the Born-Oppenheimer potential energy surface E(S), its gradient components g (S)=(aE(S)as)and its Hessian components HK (S)=(OE(SyOS, Os ) can be evaluated in terms of derivatives of E with respect to the 3N Cartesian coordinates of the molecule. However, when one carries out reaction path dynamics, one uses a different set of coordinates for reasons that are similar to those that arise in the treatment of normal modes of vibration as given in the back ground material In particular, one introduces 3N mass-weighted coordinates x;=sj(mj)/2 that are related to the 3N Cartesian coordinates s, in the same way as we saw in the background material10 C. Reaction Path Hamiltonian Theory Let us review what the reaction path is as defined above. It is a path that i. begins at a transition state (TS) and evolves along the direction of negative curvature on the potential energy surface (as found by identifying the eigenvector of the Hessian matrix Hj,k = ¶ 2E/¶sk¶sj that belongs to the negative eigenvalue); ii. moves further downhill along the gradient vector g whose components are gk = ¶E/¶sk ’ iii. terminates at the geometry of either the reactants or products (depending on whether one began moving away from the TS forward or backward along the direction of negative curvature). The individual “steps” along the reaction coordinate can be labeled S0 , S1 , S2 , … SP as they evolve from the TS to the products (labeled SP ) and S-R, S-R+1, …S0 as they evolve from reactants (S-R) to the TS. If these steps are taken in very small (infinitesimal) lengths, they form a continuous path and a continuous coordinate that we label S. At any point S along a reaction path, the Born-Oppenheimer potential energy surface E(S), its gradient components gk (S) = (¶E(S)/¶sk ) and its Hessian components Hk,j(S) = (¶ 2E(S)/¶sk¶sj ) can be evaluated in terms of derivatives of E with respect to the 3N Cartesian coordinates of the molecule. However, when one carries out reaction path dynamics, one uses a different set of coordinates for reasons that are similar to those that arise in the treatment of normal modes of vibration as given in the Background Material. In particular, one introduces 3N mass-weighted coordinates xj = sj (mj ) 1/2 that are related to the 3N Cartesian coordinates sj in the same way as we saw in the Background Material