Modes orthogonal

The situation simplifies when V Q) is a parabola, since the mean position of the particle now behaves as a classical coordinate. For the parabolic barrier (1.5) the total system consisting of particle and bath is represented by a multidimensional harmonic potential, and all one should do is diagonalize it. On doing so, one finds a single unstable mode with imaginary frequency iA and a spectrum of normal modes orthogonal to this coordinate. The quantity A is the renormalized parabolic barrier frequency which replaces in a. multidimensional theory. In order to calculate... 

Fig. 4.5 Schematic projection of the energetics of a reaction. The diagram shows the Born-Oppenheimer energy surface mapped onto the reaction coordinate. The barrier height AE has its zero at the bottom of the reactant well. One of the 3n — 6 vibrational modes orthogonal to the reaction coordinate is shown in the transition state. H and D zero point vibrational levels are shown schematically in the reactant, product, and transition states. The reaction as diagrammed is slightly endothermic, AE > 0. The semiclassical reaction path follows the dash-dot arrows. Alternatively part of the reaction may proceed by tunneling through the barrier from reactants to products with a certain probability as shown with the gray arrow...

A standard reaction path Hamiltonian is derived from a quadratic expansion of the potential in the modes orthogonal to the reaction coordinate [55-59]. For zero total angular momentum this Hamiltonian is expressed as... 

Equations (27.1) and (27.2) are hybrid quantized expressions in which the bound modes orthogonal to the reaction coordinate are treated quantum mechanically, that is, the partition functions (T, s) and 0 T) are computed quantum mechanically... 

Treating bound modes quantum mechanically, the adiabatic separation between s and u is equivalent to assuming that quantum states in bound modes orthogonal to s do not change throughout the reaction (as s progresses from reactants to products). The reaction dynamics is then described by motion on a one-mathematical-dimensional vibrationally and rotationally adiabatic potential... 

Figure 27.9 (a) Potential energy and ground-state adiabatic potential curves (solid curves) and SCTeffective mass (dashed curve) as a function of reaction coordinate for the intramolecular H-tranferin 1,3-pentadiene. (b) Harmonic frequencies for modes orthogonal to the reaction coordinate. 

Recently, Carter and Handy have addressed this very effectively by incorporating the approach taken in MULTIMODE into the Reaction Path Hamiltonian (RPH) [60]. In this approach one special, large amplitude mode is singled out and the u-mode coupling idea is applied to the normal modes orthogonal to this mode. The kinetic energy operator is somewhat complex and is given elsewhere [60]. This version of MULTIMODE is denoted MULTIMODE-RPH or abbreviated as MM-RPH. 

Each of the 10 energy levels of the H3 quantized transition state can be associated with a set of linear-triatomic quantum numbers (66) [vjv ] where v, and v2 are the stretch and bend quantum numbers respectively for modes orthogonal to the reaction coordinate... 

As one moves along a reaction path connecting a saddle point and potential minimum, harmonic frequencies can be found for the 3A-7 vibrational modes orthogonal to the reaction path motion (Miller et al., 1980 Kato and Morokuma, 1980). However, since all derivatives of the potential dV/dq, are not zero as one moves along the path, these vibrational modes cannot be found by diagonalizing the mass-weighted... 

The Bks are called the cutvature coupling constants, because they would equal zero if the IRC were a straight line. They couple the translational motion along the reaction coordinate with the vibrational modes orthogonal to it. All the above eoupling constants B depend on s. 

Curvature coupling constant Bks links the motion along the reaction valley with the normal modes orthogonal to the IRC (Fig. 14.7) ... 

Figure 1. Harmonic frequencies for bound Figure 2. Potential energy along the minimum modes orthogonal to the reaction path as a energy path and relative ground-state adiabatic function of the reaction coordinate for the potential energy [eq. (2)] as a function of...

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