Vibrationally adiabatic theory

It is noteworthy that the threshold for the 0 + H2(v=l) reaction exceeds the conventional vibrationally adiabatic threshold. Presumably, the general vibrationally adiabatic theory and/or a P0DS analysis could account for this. 

J. M. Bowman, A. Kuppermann, J. T. Adams, and D. G. Truhlar, A direct test of the vibrationally adiabatic theory of chemical reactions, Chem. Phys. Lett. 20 229 (1973). 

Since chemical reactions usually show significant nonadiabaticity, there are naturally quantitative errors in the predictions of the vibrationally adiabatic model. Furthermore, there are ambiguities about how to apply the theory such as the optimal choice of coordinate system. Nevertheless, this simple picture seems to capture the essence of the resonance trapping mechanism for many systems. We also point out that the recent work of Truhlar and co-workers24,34 has demonstrated that the reaction dynamics is largely controlled by the quantized bottleneck states at the barrier maxima in a much more quantitative manner than expected. 

As noted in Section III.C.2, the adiabatic method allows one to separate "slow" rotational motion from "fast" vibrational motion. The evaluation of vibrational distributions that has been described is based on this feature of adiabatic theory. In many cases one can also similarly ignore "slow" bending motion. However, advances in experimental methods have led to measurements of rotational distributions of photofragments (see Okabe and Jackson, this volume) and thus the evaluation of these distributions has become a timely and interesting problem. 

The non-adiabatic theory indicates that electron - vibration (phonon) interaction, at stabilization (minimization) of the fermionic ground state... 

Abdolsalami, M. and Morrison, M.A. (1987). Calculating vibrational-excitation cross sections off the energy shell A first-order adiabatic theory, Phys. Rev. A 36, 5474-5477. 

Extending the theory to interpret or predict the rovibrational state distribution of the products of the unimolecular dissociation, requires some postulate about the nature of the motion after the unimolecularly dissociating system leaves the TS on its way to form products. For systems with no potential energy maximum in the exit channel, the higher frequency vibrations will tend to remain in the same vibrational quantum state after leaving the TS. That is, the reaction is expected to be vibrationally adiabatic for those coordinates in the exit channel (we return to vibrational adiabaticity in Section 1.2.9). The hindered rotations and the translation along the reaction coordinate were assumed to be in statistical equilibrium in the exit channel after leaving the TS until an outer TS, the PST TS , is reached. With these assumptions, the products quantum state distribution was calculated. (After the system leaves the PST TS, there can be no further dynamical interactions, by definition.)... 

If the PCET reaction involves electronically adiabatic ET and PT and is vibrationally adiabatic, the system moves on the lowest two-dimensional energy surface equivalent to the lowest eigenvalue of the matrix H. In this case, the rate could be calculated with the multidimensional generalization of the Grote-Hynes theory [29-31]. 

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. 

To clarify these questions we have studied the quantal and semiclassical theory of reactive resonances. In section II the Diagonal corrected Vibrational Adiabatic Hyperspherical (DIVAH) model 21, ) is reviewed. This theory was the first to provide quantitative predictions (with a typical accuracy of 0.1 kcal/mole) of collinear quantal resonance energies. Furthermore the DIVAH model led us to the new and very surprising phenomenon of vibrationally bonded molecules (29-41). The connection and interrelation with semiclassical RPO theory (25.30.42-47) which predicted and interpreted these results, is presented in section III. Having... 

Adiabatic, or as it has been termed vibrationally adiabatic,15 transition state theory has its origin in a paragraph in an article by Hirschfelder and Wigner.16 The treatment was developed further by a number of authors.17 In this type of transition state theory the eigenvalues of the system at each R, which are the vibrationally adiabatic eigenvalues, are plotted versus R. The N j in Eq. (2.1) then becomes the number of such states whose maximum energy on this plot does not exceed E, that is, N%j now denotes the sum of all open adiabatic reaction channels. 

A more accurate treatment of the reaction uses variational TST, in which the dividing surface is allowed to move off the saddle point, or equivalently, uses the adiabatic theory as described in Section 27.2. The vibrationally adiabatic potential... 

The results of the present calculations that the zero-point vibrational energy of the reactants can pass smoothly into that of the intermediate complexes is entirely consistent with the basic postulate of Eyring s theory that activated complexes are created from the reactants in equilibrium states. It is easy to show that the vibrationally adiabatic model, coupled with the assumption that collision cross sections are the same for all vibrational levels, leads to the conclusion that there is a Boltzmann distribution between the vibrational levels in the activated state. Thus, consider the situation represented by the energy diagram shown in Fig. 6 two levels are shown for the initial and activated states— the ground level and the nth vibrationally excited... 

The simplest way of taking account of vibrational effects is to assume vibrational adiabaticity during the motion up to the critical dividing surface [27]. As mentioned already in the Introduction, much of the earlier work on vibrational adiabaticity was concerned with its relationship to transition-state theory, especially as applied to the prediction of thermal rate constants [24-26]. It is pointed out in [27] that the validity of the vibrationally adiabatic assumption is supported by the results of both quasiclassical and quantum scattering calculations. The effective thresholds indicated by the latter for the D + H2(v =1) and O + H2(v =1) reactions [37,38] are similar to those found from vibrationally adiabatic transition-state theory, which is a strong evidence for the correctness of the hypothesis of vibrational adiabaticity. Similar corroboration is provided by the combined transition-state and quasiclassical trajectory calculations [39-44]. For virtrrally all the A + BC systems studied [39-44], both collinearly and in three... 

With all of the v, and v2 assignments made, some interesting trends in the fitted parameters kt and W7 appear. From Table 2, we see that the significant deviations of kt from unity are generally found for the highly bend excited transition state levels. Therefore, most of the breakdown of transition state theory appears to be associated with only those few levels. In addition, we see in Table 2 that W7 becomes larger as v2 is increased from 0 to 2 for a given value of v,. This is consistent with the discussion below Eq. (13) since the vibrationally adiabatic potentials become narrower as v2 increases (8,16). This same trend in W7 is even more apparent in the results discussed below for H + H2, 7=1, where both even and odd values of v2 are allowed. 

The Cl + HC1 quantized transition states have also been studied by Cohen et al. (159), using semiclassical transition state theory based on second-order perturbation theory for cubic force constants and first-order perturbation theory for quartic ones. Their treatment yielded 0), = 339 cm-1 and to2 = 508 cm"1. The former is considerably lower than the values extracted from finite-resolution quantal densities of reactive states and from vibrationally adiabatic analysis, 2010 and 1920 cm 1 respectively (11), but the bend frequency to2 is in good agreement with the previous (11) values, 497 and 691 cm-1 from quantum scattering and vibrationally adiabatic analyses respectively. The discrepancy in the stretching frequency is a consequence of Cohen et al. using second-order perturbation theory in the vicinity of the saddle point rather than the variational transition state. As discussed elsewhere (88), second-order perturbation theory is inadequate to capture large deviations in position of the variational transition state from the saddle point. 

B. C. Garrett and D. G. Truhlar, WKB approximation for the reaction-path Hamiltonian Application to variational transition state theory, vibrationally adiabatic excited-state barrier heights, and resonance calculations,/. Chem. Phys. 81 309 (1984). 

---