Normal mode coordinates reaction path

Qk is the vibrational (normal mode) coordinates for the vibrational motion perpendicular to the reaction path, pk are the corresponding momenta, VQ(s) is the potential along the path and... 

A chemical reaction may be described by the reaction path Hamiltonian in order to focus on the intrinsic reaction coordinate (IRC) measuring the motion along the drain-pipe bottom (reaction path) and the normal mode coordinates orthogonal to the IRC. 

Miller, Handy, and Adams have recently shown how one can construct a classical Hamiltonian for a general molecular system based on the reaction path and a harmonic approximation to the potential surface about it. The coordinates of this model are the reaction coordinate and the normal mode coordinates for vibrations transverse to the reaction path these are essentially a polyatomic version of the natural collision coordinates introduced by Marcus and by Hofacker for A + BC AB 4- C reactions. One of the important practical aspects of this model is that all of the quantities necessary to define it are obtainable from a relatively modest number of db initio quantum chemistry calculations, essentially independent of the number of atoms in the system. This thus makes possible an ab initio theoretical description of the dynamics of reactions more complicated than atom-diatom reactions. 

Molecular mechanics calculations are an attempt to understand the physical properties of molecular systems based upon an assumed knowledge of the way in which the energy of such systems varies as a function of the coordinates of the component atoms. While this term is most closely associated with the conformational energy analyses of small organic molecules pioneered by Allinger (1), in their more general applications molecular mechanics calculations include energy minimization studies, normal mode calculations, molecular dynamics (MD) and Monte Carlo simulations, reaction path analysis, and a number of related techniques (2). Molecular mechanics... 

However, a reaction coordinate can never be parallel to a degenerate normal mode in a transition state as this would imply that there would be more than one direction of negative curvature in the transition state. In such a situation one can always find a lower potential energy path that goes around the hill (61). The structure... 

A normal-mode representation of the Hamiltonian for the reduced system involves the diagonalization of the projected force constant matrix, which in turn generates a reduced-dimension potential-energy surface in terms of the mass-weighted coordinates of the reaction path [64] ... 

The in-1 vibrational frequencies, C0 (s), are obtained from normal-mode analyses at points along the reaction path via diagonalization of a projected force constant matrix that removes the translational, rotational, and reaction coordinate motions. The B coefficients are defined in terms of the normal mode coefficients, with those in the denominator of the last term determining the reaction path curvature, while those in the numerator are related to the non-adiabatic coupling of different vibrational states. A generalization to non-zero total angular momentum is available [59]. 

Flexible RRKM theory and the reaction path Hamiltonian approach take two quite different perspectives in their evaluation of the transition state partition functions. In flexible RRKM theory the reaction coordinate is implicitly assumed to be that which is appropriate at infinite separation and one effectively considers perturbations from the energies of the separated fragments. In contrast, the reaction path Hamiltonian approach considers a perspective that is appropriate for the molecular complex. Furthermore, the reaction path Hamiltonian approach with normal mode vibrations emphasizes the local area of the potential along the minimum energy path, whereas flexible RRKM theory requires a global potential for the transitional modes. One might well imagine that each of these perspectives is more or less appropriate under various conditions. 

Other qualitative rules for the study of reaction paths have been derived independently. For unimolecular reactions, it has been found that conditions favorable to a given path exist if there is a low-energy excited state of the same symmetry as the normal mode corresponding to the reaction coordinate, the transition density is localized in the region of nuclear motion and the excitation energy decreases along the coordinate 32>. 

The one-dimensional potential along the tunneling coordinate, represented by t/c(x) in Eq. (29.20), is a crude-adiabatic potential evaluated with the heavy atoms fixed in the equilibrium configuration, i.e. with y = Ay ,y5 = Ay it is equivalent to the potential along the linear reaction path. This symmetric doubleminimum potential has a maximum U (0) = Ug at x = 0, minima U( ( Ax) = 0 at X = +Ax, and a curvature in the minima given by the effective frequency Qq which accounts for the contribution of the normal modes of the minima to the reaction coordinate [27]. Eor the shape of the potential in the intermediate points we use an interpolation formula based on the calculated energies and curvatures near the stationary points. We have found that in many cases the simple quartic potential of the form... 

Reaction path curvature is actually a vector with 3N — 7 components [80]. Each component is associated with a particular generalized normal mode, and it measures the extent to which the system curves into a particular direction as it progresses along the MEP. Corner-cutting tunneling involves a coupled motion involving the reaction coordinate and all the generalized normal modes that are associated with nonzero curvature components [81]. 

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