The gradient and Hessian matrices along these new coordinates (x can be aluated in terms of the original cartesian counterparts (S)=g(S)(m) Hik=Hik(m,my) The eigenvalues and eigenvectors (v) of the mass-weighted Hessian Hcan then be determined. Upon doing so, one find 6 zero eigenvalues whose eigenvectors describe overall rotation and translation of 3N-7 positive eigenvalues (OK) and eigenvectors vx along which the gradient g has zero(or nearly so) components ii1. and one eigenvalue os( that may be positive, zero, or negative)along whose eigenvector vs the gradient g has its largest component. The one unique direction along vs gives the direction of evolution of the reaction path (in these mass-weighted coordinates). All other directions (i.e, within the space spanned by the 3N-7 other vectors vx)) possess zero gradient component and positive curvature This means that at any point s on the reaction path being discussed one is at a local minimum along all 3N-7 directions vx) that are transverse to the reaction path direction(i.e, the gradient direction) one can move to a neighboring point on the reaction path by moving a small (infinitesimal)amount along the gradient11 The gradient and Hessian matrices along these new coordinates {xj} can be evaluated in terms of the original Cartesian counterparts: gk ’(S) = gk (S) (mk ) -1/2 Hj,k’ = Hj,k (mjmk ) -1/2 . The eigenvalues {wk 2 } and eigenvectors {vk} of the mass-weighted Hessian H’ can then be determined. Upon doing so, one finds i. 6 zero eigenvalues whose eigenvectors describe overall rotation and translation of the molecule; ii. 3N-7 positive eigenvalues {wK 2 } and eigenvectors vK along which the gradient g has zero (or nearly so) components; iii. and one eigenvalue wS 2 (that may be positive, zero, or negative) along whose eigenvector vS the gradient g has its largest component. The one unique direction along vS gives the direction of evolution of the reaction path (in these mass-weighted coordinates). All other directions (i.e., within the space spanned by the 3N-7 other vectors {vK}) possess zero gradient component and positive curvature. This means that at any point S on the reaction path being discussed i. one is at a local minimum along all 3N-7 directions {vK} that are transverse to the reaction path direction (i.e., the gradient direction); ii. one can move to a neighboring point on the reaction path by moving a small (infinitesimal) amount along the gradient