In terms of the 3N-6 mass-weighted Hessian s eigen-mode directions(vx and Vs), the potential energy surface can be approximated, in the neighborhood of each such point on the reaction path S, by expanding it in powers of displacements away from this point. If these displacements are expressed as components 1. 8X along the 3N-7 eigenvectors vx and os along the gradient direction vs, one can write the Born-Oppenheimer potential energy surface locally as E=E(S)+vs8S+1/2o32 1/2o-2δX Within this local quadratic approximation, E describes a sum of harmonic potentials along each of the 3N-7 modes transverse to the reaction path direction. Along the reaction path, E appears with a non-zero gradient and a curvature that may be positive negative, or zero The eigenmodes of the local (i.e, in the neighborhood of any point S along the reaction path )mass-weighted Hessian decompose the 3N-6 internal coordinates into 3N-7 along which E is harmonic and one(s)along which the reaction evolves. In terms of these same coordinates, the kinetic energy t can also be written and thus the classical Hamiltonian h=t+ v can be constructed because the coordinates we use are mass- weighted, in Cartesian form the kinetic energy T contains no explicit mass factors 2,, (ds dt)=1/2 2,(dx,/dt)12 In terms of the 3N-6 mass-weighted Hessian’s eigen-mode directions ({vK} and vS ), the potential energy surface can be approximated, in the neighborhood of each such point on the reaction path S, by expanding it in powers of displacements away from this point. If these displacements are expressed as components i. dXk along the 3N-7 eigenvectors vK and ii. dS along the gradient direction vS , one can write the Born-Oppenheimer potential energy surface locally as: E = E(S) + vS dS + 1/2 wS 2 dS 2 + SK=1,3N-7 1/2 wK 2 dXK 2 . Within this local quadratic approximation, E describes a sum of harmonic potentials along each of the 3N-7 modes transverse to the reaction path direction. Along the reaction path, E appears with a non-zero gradient and a curvature that may be positive, negative, or zero. The eigenmodes of the local (i.e., in the neighborhood of any point S along the reaction path) mass-weighted Hessian decompose the 3N-6 internal coordinates into 3N-7 along which E is harmonic and one (S) along which the reaction evolves. In terms of these same coordinates, the kinetic energy T can also be written and thus the classical Hamiltonian H = T + V can be constructed. Because the coordinates we use are mass￾weighted, in Cartesian form the kinetic energy T contains no explicit mass factors: T = 1/2 Sj mj (dsj /dt)2 = 1/2 Sj (dxj /dt)2