Hessian matrix elements Hik=8E/as, Os of the energy surface at the Ts, one can express the variation of the potential energy along the 3N Cartesian coordinates s, of the molecule as follows: E(SK=E(0)+2x(aE/OSk)Sk +1/2 Ei kS;Hi kSk where E(O) is the energy at the Ts, and the is,i denote displacements away from the ts geometry. Of course, at the ts, the gradients all vanish because this geometry corresponds to a stationary point. As we discussed in the Background Material, the Hessian matrix Hk has 6 zero eigenvalues whose eigenvectors correspond to overall translation and rotation of the molecule. This matrix has 3N-7 positive eigenvalues whose eigenvectors correspond to the vibrations of the TS species, as well as one negative eigenvalue. The latter has an eigenvector whose components(s, along the 3N Cartesian coordinates describe the direction of the reaction path as it begins its journey from the ts backward to reactants(when followed in one direction) and onward to products( when followed in the opposite direction). Once one moves a small amount along the direction of negative curvature, the reaction path is subsequently followed by taking infinitesimal steps "downhill along the gradient vector g whose 3n components are(aE/as ) Note that once one has moved downhill away from the ts by taking the initial step along the negatively curved direction, the gradient no longer vanishes because one is no longer le stationary point Returning to the tsT rate calculation, one therefore is able to express the concentration [AB"of species at the ts in terms of the reactant concentrations and a4 Hessian matrix elements { Hj,k = ¶ 2E/¶sj¶sk}of the energy surface at the TS, one can express the variation of the potential energy along the 3N Cartesian coordinates {sk} of the molecule as follows: E (sk) = E(0) + Sk (¶E/¶sk) sk + 1/2 Sj,k sj Hj,k sk + … where E(0) is the energy at the TS, and the {sk} denote displacements away from the TS geometry. Of course, at the TS, the gradients all vanish because this geometry corresponds to a stationary point. As we discussed in the Background Material, the Hessian matrix Hj,k has 6 zero eigenvalues whose eigenvectors correspond to overall translation and rotation of the molecule. This matrix has 3N-7 positive eigenvalues whose eigenvectors correspond to the vibrations of the TS species, as well as one negative eigenvalue. The latter has an eigenvector whose components {sk} along the 3N Cartesian coordinates describe the direction of the reaction path as it begins its journey from the TS backward to reactants (when followed in one direction) and onward to products (when followed in the opposite direction). Once one moves a small amount along the direction of negative curvature, the reaction path is subsequently followed by taking infinitesimal “steps” downhill along the gradient vector g whose 3N components are (¶E/¶sk ). Note that once one has moved downhill away from the TS by taking the initial step along the negatively curved direction, the gradient no longer vanishes because one is no longer at the stationary point. Returning to the TST rate calculation, one therefore is able to express the concentration [AB*] of species at the TS in terms of the reactant concentrations and a