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 gradient
11 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
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 massweighted, 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
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) where
13 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
1+F)=[1+∑ δXgB In the absence of the so-called dynamical coupling factors BK.. and Bks, this expression for h describes (1)3N-7 harmonic-oscillator Hamiltonia 1/2[px +ok (S)] each of which has a locally defined frequency ok(S)that varies along the reaction path(i.e, is S-dependent) : (2)a Hamiltonian 1/2 p+ e(S) for motion along the reaction coordinate S with E(S) serving as the potential In this limit (i.e, with the B factors turned off ), the reaction dynamics can be simulated in what is termed a"vibrationally adiabatic" manner by placing each transverse oscillator into a quantum level vx that characterizes the reactants population of this mode i1. assigning an initial momentum ps(o) to the reaction coordinate that is characteristic of the collision to be simulated(e.g, ps(O)could be sampled from a Maxwell-Boltzmann distribution if a thermal reaction is of interest, or p(O) could be chosen equal to the mean collision energy of a beam-collision experiment); ii1. time evolving the S and ps, coordinate and momentum using the above Hamiltonian, assuming that each transverse mode remains in the quantum state vx that it had when the reaction began The assumption that Vx remains fixed, which is why this model is called vibrationally adiabatic, does not mean that the energy content of the Kth mode remains fixed because the frequencies ok(S)vary as one moves along the reaction path. As a result, the kinetic
14 (1+F) = [1 + SK=1,3N-7 dXK BK,S] 2 . In the absence of the so-called dynamical coupling factors BK,K’ and BK,S, this expression for H describes (1) 3N-7 harmonic-oscillator Hamiltonia 1/2[pK 2 + wK 2 (S)] each of which has a locally defined frequency wK(S) that varies along the reaction path (i.e., is S-dependent); (2) a Hamiltonian 1/2 pS 2 + E(S) for motion along the reaction coordinate S with E(S) serving as the potential. In this limit (i.e., with the B factors “turned off”), the reaction dynamics can be simulated in what is termed a “vibrationally adiabatic” manner by i. placing each transverse oscillator into a quantum level vK that characterizes the reactant’s population of this mode; ii. assigning an initial momentum pS (0) to the reaction coordinate that is characteristic of the collision to be simulated (e.g., pS (0) could be sampled from a Maxwell-Boltzmann distribution if a thermal reaction is of interest, or pS (0) could be chosen equal to the mean collision energy of a beam-collision experiment); iii. time evolving the S and pS , coordinate and momentum using the above Hamiltonian, assuming that each transverse mode remains in the quantum state vK that it had when the reaction began. The assumption that vK remains fixed, which is why this model is called vibrationally adiabatic, does not mean that the energy content of the Kth mode remains fixed because the frequencies wK(S) vary as one moves along the reaction path. As a result, the kinetic
energy along the reaction coordinate 1/2 ps will change both because E(S)varies along S and because Xx=L3N-7 ho (S)vx+ 1/2] varies along S Lets return now to the rPh theory in which the dynamical couplings among motion along the reaction path and the modes transverse to it are included. In the full RPH, the terms Bkk. ( S)couple modes K and K,, while bxs(s)couples the reaction path to mode k. These couplings are how energy can flow among these various degrees of freedom. Explicit forms for the Bkk. and Bx.s factors are given in terms of the eigenvectors vk,vs) of the mass-weighted Hessian matrix as follows BkK.=<dv/ds vx, Bxs =<dv/sIvs> where the derivatives of the eigenvectors (dvds, are usually computed by taking the eigenvectors at two neighboring points S and s along the reaction path dv/dS=v(S)-V(S)/(S In summary, once a reaction path has been mapped out, one can compute along this path, the mass-weighted Hessian matrix and the potential e(S). Given these quantities, all terms in the rPh H= EK=l3N-7 1/2[PK +OK(S)]+E(S)+ 1/2 [.=13N-7pk 8XK BkK/(
15 energy along the reaction coordinate 1/2 pS 2 will change both because E(S) varies along S and because SK=1,3N-7 hwK 2 (S) [vK + 1/2] varies along S. Let’s return now to the RPH theory in which the dynamical couplings among motion along the reaction path and the modes transverse to it are included. In the full RPH, the terms BK,K’(S) couple modes K and K’, while BK,S(S) couples the reaction path to mode K. These couplings are how energy can flow among these various degrees of freedom. Explicit forms for the BK,K’ and BK,S factors are given in terms of the eigenvectors {vK, vS} of the mass-weighted Hessian matrix as follows: BK,K’ = <dvK/dS| vK’>; BK,S = <dvK/dS | vS> where the derivatives of the eigenvectors {dvK/dS} are usually computed by taking the eigenvectors at two neighboring points S and S’ along the reaction path: dvK/dS = {vK(S’) – vK(S))/(S’-S). In summary, once a reaction path has been mapped out, one can compute, along this path, the mass-weighted Hessian matrix and the potential E(S). Given these quantities, all terms in 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) are in hand