Oscillators reaction probabilities

Based on this physical view of the reaction dynamics, a very broad class of models can be constructed that yield qualitatively similar oscillations of the reaction probabilities. As shown in Fig. 40(b), a model based on Eckart barriers and constant non-adiabatic coupling to mimic H + D2, yields out-of-phase oscillations in Pr(0,0 — 0,j E) analogous to those observed in the full quantum scattering calculation. Note, however, that if the recoupling in the exit-channel is omitted (as shown in Fig. 40(b) with dashed lines) then oscillations disappear and Pr exhibits simple steps at the QBS energies. As the occurrence of the oscillation is quite insensitive to the details of the model, the interference of pathways through the network of QBS seems to provide a robust mechanism for the oscillating reaction probabilities. 

Aside from merely calculational difficulties, the existence of a low-temperature rate-constant limit poses a conceptual problem. In fact, one may question the actual meaning of the rate constant at r = 0, when the TST conditions listed above are not fulfilled. If the potential has a double-well shape, then quantum mechanics predicts coherent oscillations of probability between the wells, rather than the exponential decay towards equilibrium. These oscillations are associated with tunneling splitting measured spectroscopically, not with a chemical conversion. Therefore, a simple one-dimensional system has no rate constant at T = 0, unless it is a metastable potential without a bound final state. In practice, however, there are exchange chemical reactions, characterized by symmetric, or nearly symmetric double-well potentials, in which the rate constant is measured. To account for this, one has to admit the existence of some external mechanism whose role is to destroy the phase coherence. It is here that the need to introduce a heat bath arises. 

In the following, we will discuss DMC simulations on the CO oxidation on the Pt(lOO) surface, that were done in our laboratories. The simulations show oscillations in the CO2 production rate as well as several types of spatio-temporal pattern formation. In essence, it is an extension of the ZGB model with desorption and diffusion of A, finite reaction rates and surface reconstruction. We will discuss it to illustrate the complexity of the models with which DMC simulations can be done nowadays. For clarity, we will stick to the A and B2 notation employed in the previous section. Species A corresponds to CO and B2 corresponds to 02- Furthermore, we will speak in terms of reaction rates instead of relative reaction probabilities. This terminology is entirely justified in the DMC approach that we used. 

Fully coupled (luantum results for the total reaction probabilities P( for t = 1, 2, 3, 4 are plotted in Figure 4 as a function of energy. Figure 4 shows that the fully coupled results are in-between the adiabatie and diabatie results close to threshold, with the effective thresholds being at 0.009 for Pi. tmd 0.012 for P2 mid P3. The similar behavior of P2 and P3 is certainly closer to the adiabatie results, but a key difference between the uncoupled and coupled results is that P2 tmd P3 show oscillations in the coupled results, and neither probability rises above 0.9 until the energy is above the 0( D) threshold. 

With respect to sharp oscillations in reaction probability that occur at higher energy due to reflection from the excited singlet, we expect that this is a real effect that should be captured by TSH calculations. Indeed, somewhat analogous... 

We end this section by showing reaction probabilities calculated using accurate boundary conditions as described and by aj)j)lying approximate boundary conditions. The approximate boundcU y conditions are applied directly in the hyper-spherical coordinates[46, 47]. Fig. 3 shows how the state-to-state reaction probabilities oscillate as a function of the hyperradius when using the approximate boundary conditions. The illustration here is for the Cl-l-CH/i — HCI-I-CH3 reaction. The straight line is an average value. Here the REA model[9] was used and CH4 had initially two quanta in a C-H stretch mode while HCl formed in the first excited vibrational state. 

Darakjian et al. (164) computed accurate densities of reactive states for He + H2 - HeH+ + H with 7 = 0. The raw CRP exhibited many rapid oscillations, indicative of narrow trapped-state resonances. By averaging the cumulative reaction probability for... 

Figure 5.H The top panel shows the cumulative reaction probabilities A/"exact( ) (black oscillatory curve) and A/"weyi( ) (red smooth curve) for the Eckart-Morse-Morse reactive system with the Hamiltonian given by Eq. (66) with e = 0. It also shows the quantum numbers (rii, rii) of the Morse oscillators that contribute to the quantization steps. The bottom panel shows the resonances in the complex energy plane marked by circles for the uncoupled case e = 0 and by crosses for the strongly coupled case e = 0.3. The parameters for the Eckart potential are o = 1, A = 0.5, and 6 = 5. The parameters for the Morse potential are = 1, Dj 3 = 1.5, and Om = = 1. Also, h ff = 0.1.

At low total energies, less than the saddle-point of the potential surface, the calculations of TRUHLAR et al /75/ yield non-zero values for the transition probabilities, which is evidence of nuclear tunneling. Another important result is the observation of oscillations of the reaction probability at high energies just above the threshold for vibration excitation (Pig.14). 

The oscillator model for proton transfer was first developed by DOGONADZE and KUSNETSO /147/. The above treatment proposed by CHRISTOV /37e/ is based on the general "theory of reaction rates applied to the two-frequency oscillator model. It reproduces the essential results of earlier work concerning electronically and protonically non-adiabatic reactions and yields, moreover, simple, explicit expressions for adiabatic reactions never derived before. This shows the utility of certain new methods in calculating reaction probabilities, developed in Chapter II, which allow an application of the most suitable formulations of the rate theory. 

Extremely accurate results for reaction probabilities, e.g. were obtained in a few cases (e.g. I+HI and isotopic variants, [22,52] for H+MuH [22], see Figure 6), using purely semiclassical techniques for scattering phaseshifts. Formula (15) allows to predict oscillations due to interference between even and odd propagation in the energy dependence of probabilities. These oscillations have been found in numerical work [7]. As shown in Table I (from [22]) excellent agreement with exact calculation was obtained for resonances in H+MuH (see also Figure 5). 

Here we look at some details of the cumulative reaction probability of the reaction of H + H2 and discuss what picture can be obtained from the analytical expressions of the current theory. Figure 7.7 shows the cumulative reaction probability N E) as a function of the energy E both in linear and log scales. The result of the harmonic approximation (dotted curve) and that of the NF theory (solid curve) are compared. The calculation uses the potential energy surface of Mielke et al. The total angular momentum is fixed to zero for simplicity. This allows us to take the zeroth order approximation as in eqn (7.11), i.e. a collection of harmonic oscillators and a parabolic barrier. Readers interested in the extension of the theory to include the rotational motions should refer to the literature. ... 

I had a similar experience in 1954 when I spoke, probably for the first time, about the possibility of oscillating reactions. At that time, I had published a short paper with Radu Balescu on the possibility that far from equilibrium we could have chemical oscillations, in contrast with what happens near equilibrium. This work was connected with involvment in the so-called "universal evolution criterion", derived with Paul Glansdorff. My lecture of 1954 had no more success than the one of 1946. The chemists were very skeptical about the possibility of chemical oscillations and in addition, said an outstanding chemist, even if it would be possible, what should be the interest The interest of chemical kinetics was at that time the discovery of well-defined mechanisms, and specially of potential energy surfaces, which one could then connect with quantum mechanical calculations. The appearance of chemical oscillations or other exotic phenomena seemed to him to be of no interest in the direction in which chemical kinetics was traditionally engaged. All this has changed, but to some extent the situation of chemistry in respect to physics remains under the shadow of this distrust of time. 

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