Vibrationally adiabatic zero-curvature

We first consider the case where the reaction probabilities are computed for the adiabatic model with the reaction-path curvature neglected, the so-called vibrationally adiabatic zero-curvature approximation [36]. We approximate the quantum mechanical ground-state probabilities P (E) for the one-dimensional scattering problem by a uniform semiclassical expression [48], which for E < is given by... 

FIGURE 3. Reaction probability for collinear H+H2- H2+H on the Porter-Karplus potential energy surface. EQ denotes the exact quantum mechanical values, VAZC the results of the vibrationally adiabatic zero curvature approximation, and the points the results of the present SCP-IOS reaction path model. 

A direct test of the vibrationally adiabatic approximation for H + H2 has also been made (Bowman et al.. 1973). This test was done by projecting accurate wavefunctions on the vibrationally adiabatic functions for zero curvature, and measuring deviations of the resulting probability weight from unity. The symmetric stretch motion was found to be adiabatic to within 10 % for total energies between 0-51 and 0-72 eV, but adiabaticity was lost at lower and higher energies. 

Hancock et al. [1989] used a version of the small curvature semiclassical adiabatic approach introduced by Truhlar et al. [1982] to calculate the temperature dependence of the rate constant, as shown in Figure 6.29. Variations in k(T) below the crossover point (25-30 K) are due to changes in the prefactor due to zero-point vibrations of the H atom in the crystal. Obviously, the gas-phase model does not take these into account. The absolute values of the rate constant differ by 1-2 orders of magnitude from the experimental ones for the same reason. 

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]. 

The LS state is characterised by the adiabatic potential surface i LS(r) having a minimum ifo.Ls at a certain geometry (metal-ligand distances rLS) with some definite curvature (the second derivative representing a force constant /LS). Within this nearly parabolic function a set of vibration levels e, LS occurs, the lowest one corresponding to the zero-point vibration, e0LS. The HS state differs in these characteristics in such a way that the relationships... 

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