This means that the momenta conjugate to each(mass-weighted) coordinate xi, obtained in the usual way as p, a[T-va(dx/dt)=dx/dt, all have identical (unit)mass factors associated with them To obtain the working expression for the reaction path Hamiltonian(RPh),one must transform the above equation for the kinetic energy t by replacing the 3N Cartesian mass-weighted coordinates (x)by the 3N-7 eigenmode displacement coordinates sX i1. the reaction path displacement coordinate 8S, and iii. 3 translation and 3 rotational coordinates The 3 translational coordinates can be separated and ignored(because center-of-mass energy is conserved) in further consideration. The 3 rotational coordinates do not enter into the potential e, but they do appear in T. However, it is most common to ignore their effects on the dynamics that occurs in the internal-coordinates; this amounts to ignoring the effects of overall centrifugal forces on the reaction dynamics. We will proceed with this approximation in mind Although it is tedious to perform the coordinate transformation of T outlined above, it has been done and results in the following form for the rPh H=2K-13N-712[PK +OK(S)]+E(S)+ 1/2 Ips-2K K'=13N-7 PK 8XK. Bkx /(1+F) where13 This means that the momenta conjugate to each (mass-weighted) coordinate xj , obtained in the usual way as pj = ¶[T-V]/¶(dxj /dt) = dxj /dt, all have identical (unit) mass factors associated with them. To obtain the working expression for the reaction path Hamiltonian (RPH), one must transform the above equation for the kinetic energy T by replacing the 3N Cartesian mass-weighted coordinates {xj} by i. the 3N-7 eigenmode displacement coordinates dXj , ii. the reaction path displacement coordinate dS, and iii. 3 translation and 3 rotational coordinates. The 3 translational coordinates can be separated and ignored (because center-of-mass energy is conserved) in further consideration. The 3 rotational coordinates do not enter into the potential E, but they do appear in T. However, it is most common to ignore their effects on the dynamics that occurs in the internal-coordinates; this amounts to ignoring the effects of overall centrifugal forces on the reaction dynamics. We will proceed with this approximation in mind. Although it is tedious to perform the coordinate transformation of T outlined above, it has been done and results in the following form for the RPH: H = SK=1,3N-7 1/2[pK 2 + wK 2 (S)] + E(S) + 1/2 [pS - SK,K’=1,3N-7 pK dXK’ BK,K’] 2 /(1+F) where