Vibrationally adiabatic barrier

Such calculations have been performed by Takayanagi et al. [1987] and Hancock et al. [1989]. The minimum energy of the linear H3 complex is only 0.055 kcal/mol lower than that of the isolated H and H2. The intermolecular vibration frequency is smaller than 50cm L The height of the vibrational-adiabatic barrier is 9.4 kcal/mol, the H-H distance 0.82 A. The barrier was approximated by an Eckart potential with width 1.5-1.8 A. The rate constant has been calculated from eq. (2.1), using the barrier height as an adjustable parameter. This led to a value of Vq similar to that of the gas-phase reaction H -I- H2. 

As stated by inequality (2.81) (see also section 4.2 and fig. 30), when the tunneling mass grows, the tunneling regime tends to be adiabatic, and the extremal trajectory approaches the MEP. The transition can be thought of as a one-dimensional tunneling in the vibrationally adiabatic barrier (1.10), and an estimate of and can be obtained on substitution of the parameters of this barrier in the one-dimensional formulae (2.6) and (2.7). The rate constant falls into the interval available for measurements if, as the mass m is increased, the barrier parameters are decreased so that the quantity d(Vom/mn) remains approximately invariant. 

Since the transverse vibration frequency at the barrier top is usually lower than co+, the vibrationally adiabatic barrier is lower than the bare one. 

In this case, again only a one-dimensional instanton exists. The transverse frequency increases when moving along the MEP from the well to the transition state. So we have an effect of dynamically induced barrier formation in which the height of the vibrationally adiabatic barrier exceeds V. The analysis of this squeezed potential by Auerbach and Kivelson [1985] shows that the vibrationally adiabatic approximation is valid when... 

The MEP for inversion corresponds to 6 = 0 and is characterized by the barrier height VWhen C/2V, > 1, apart from this MEP, there is a path that includes two segments described by Eq. (8.42) and a second-order saddle point. The barrier along this path is greater than V, and equal to U,(l + 2V0/C). The transverse frequency along the straight-line MEP for inversion has a minimum at the saddle point q = 0, 0 = 0 consequently, the vibrationally adiabatic barrier is lower than the static one. 

Figure 6. The dependence of the harrier height on the Jacobi angle y for the potential energy surfaces used in the present cross section calculations for O + H2 and O + HCl (DCl) reactions (—) pure electronic barrier (—) vibrationally adiabatic barrier (v = 0) [21].

The idea that the vibrational enhancement of the rate is due to the attraetive potential for excited vibrational states of the reactant is closely related to the observation made long ago based on transition state theoiy [25,26]. Poliak [25] found that for vibrationally highly excited reactants the repulsive pods (periodic orbit dividing surface) is way out in die reactant valley, and the corresponding adiabatic barrier is shallow. Based on this theory one can explain why dynamical thresholds are observed in reactions with vibrationally excited reactants. The simplicity of the theory and its success for mostly collinear reactions has a real appeal. However, to reconcile the existence of a vibrationally adiabatic barrier with the capture-type behavior - which seems to be supported by the agreement of the calculated and experimental rate coefficients [23] -needs further study. 

On the other hand, the RP method was recently used by Poliak to locate a quasiperlodlc orbit near the entrance channel v=l vibrational adiabatic barrier In the 3D H+H2 reaction (13). Using this method,... 

Method Classical barrier Zero-point energy Vibrational adiabatic barrier... 

More subtle than the lack of ZPE in bound modes after the collision is the problem of ZPE during the collision. For instance, as a trajectory passes over a saddle point in a reactive collision, all but one of the vibrational (e.g., normal) modes are bound. Each of these bound modes is subject to quantization and should contain ZPE. In classical mechanics, however, there is no such restriction. This has been most clearly shown in model studies of reactive collisions (28,35), in which it could be seen that the classical threshold for reaction occurred at a lower energy than the quantum threshold, since the classical trajectories could pass under the quantum mechanical vibrationally adiabatic barrier to reaction. However, this problem is conspicuous only near threshold, and may even compensate somewhat for the lack of tunneling exhibited by quantum mechanics. One approach in which ZPE for local modes was added to the potential energy (44) has had some success in improving reaction threshold calculations. 

Vibrational zero-point energy is an inherent contribution to the total energy, and it is therefore also the vibrational adiabatic barrier which is relevant for the extent of tunneling. The increased barrier height for the lighter isotope thus reduces significantly the tunneling probabihty in comparison with that expected based on a one-dimensional approach. 

Table 3. Vibrational adiabatic barrier heights (in eV) for 0( P)+H2> D2> HD and DH forward and reverse reactions.

All of these applications suffer from one major defect. In the Born-Oppenheimer approximation, the difference in mass of the nuclei and electrons is large enough to admit a natural set of adiabatic coordinates. This is not the case in the collision of an atom with a diatom. For any curvilinear coordinate system (u,v) one obtains different adiabatic surfaces. To do away with these ambiguities we will use the pods to define the (u,v) coordinate system. We will then find that a pods is necessarily a vibrationally adiabatic barrier or well and that at the pods the adiabatic approximation is exact. 

As we have already seen in the previous section, finding adiabatic barriers and wells of the n-th quantal vibrational adiabatic potential surface for the y dependent Hamiltonian h(y) is equivalent semiclassica-lly to finding periodic orbits of h(y) with quantised action - (n+l/2)h if the periodic orbit is over a simple well potential. The time dependent coordinates and momenta of the (y dependent) periodic orbit are denoted r (t y), R (t y), Pr(t y), and PR(t y), and the period of the orbit is T (y). We thus find for each value of y a vibrationally adiabatic barrier or well at energy E (y), a stability frequency o)n(y) and effective mass M (uq) (cf. Eq. 27) for motion perpendicular to the... 

Figure 4 shows that the effect of reagent vibrational excitation on the cross sections is small, but with several well-defined propensities. For (000) the reactive threshold is at 14.4 kcal/mol, which is close to the vibrationally adiabatic barrier for 1,3 H-migration. The thresholds for the other states are 12.9 kcal/mol for (100), 14.4 kcal/mol for (01 0) and 11.7 kcal/mol for (001). At energies well above threshold, the cross sections all rise to a value of about 3 a. The excited state cross sections rise somewhat more quickly than the ground state with the largest effect being associated with the (100) state. 

In this equation, a allows for the number of equivalent reaction paths, for example, cr = 4 for H + CH4, ( /F), is the per volume partition function for the specified species (z), denotes the transition state species, and is the difference in energy between the zeroth levels in the transition state and in the reactants, sometimes referred to as the vibrationally adiabatic barrier. It should be noted that, because of zero-point effects, the energy of the classical barrier, V, is related to but not the same as AE and that, because the preexponential term on the right-hand-side of eqn (1.14) depends on temperature, ds.E is also not identical with Eaa as defined in eqn (1.8). 

The vibrationally-adiabatic barrier can be obtained adding the zero-point energy to the classical energy at each point along the reaction path, taken as the bond order of the product HB under the assumption that the bond order is conserved along the reaction coordinate, " = hb= 1 ha. 

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