Surprisal plots

The H + X2 reactions give non-linear surprisal plots for vibrational and translational energy disposal and are approximately quadratic in form. By including an additional minimal-momentum transfer or Frank—Condon-like constraint [250, 251], the translational distributions are reproduced by a surprisal that is Gaussian in momentum. A corresponding vibrational form can be derived [241]. 

Many semi-classical and quantum mechanical calculations have been performed on the F + H2 reaction, mainly being restricted to one dimension [520, 521, 602]. The prediction of features due to quantum-mechanical interferences (resonances) dominates many of the calculations. In one semi-classical study [522], it was predicted that the rate coefficient for the reaction F (2P1/2) + H2 is about an order of magnitude smaller than that for F(2P3/2) 4- H2, which lends support to the conclusion [508] that the experimental studies relate solely to the reaction of ground state fluorine atoms. Information theory has been applied to many aspects of the reaction including the rotational energy disposal and branching ratios for F + HD [523, 524] and has been used for transformation of one-dimensional quantum results to three dimensions [150]. Linear surprisal plots occur for F 4- H2(i> = 0), as noted before, but non-linear surprisal plots are noted in calculations for F + H2 (v < 2) [524],... 

Initial analyses of the product vibrational energy distributions for X + HX suggested that they could be described by a linear surprisal plot. However, more recent work [545] suggests that there is a deviation... 

Figure 3 Surprisal plots (18) for the HF vibrational state distribution from the exoergic H atom abstraction reaction F + (CH,)4C - (CH,),CCH2 + HF(v). (Bottom panel) The observed (by D. J. Bogan and D. W. Setser, J. Chem. Phys. 64 586 (1976)) distribution, P(v), open dots connected by a line, and the (so called, prior) distribution, P (v) full symbols, vs. the HF vibrational energy. The prior distribution is the one expected when all products final states are equally probable (18). The observed distribution is qualitatively different from the prior one and their deviance, the surprisal, —In(P(v)/P"(v)) is plotted vs. E/Ev, where Ev is the HF vibrational energy and E is the total energy, in the upper panel. One can interpret the linear dependence of the surprisal on the HF vibrational energy as reflecting the presence of a quantity which is conserved by the dynamics. (See, for example, ref. (108)). In this sense, surprisal analysis is analogous to the search for quantum numbers that are not destroyed by the intramolecular couplings.

Figure 21 Surprisal plot for the vibrationally hot HF product from the four-center elimination reaction CH,CF, - CH2 = CF + HF. The energy rich, long living, CH,CF, is produced via two routes as shown. The HF vibrational distribution is rather nonstatistical, but is almost the same for both routes. (Adapted from E. Zamir and R. D. Levine, Chem. Phys. 52 253 (1980).) For recent experimental studies of elimination reactions see E. Arunan, S. J. Wategaonker, and D. W. Setser, J. Phys. Chem. 95 1539 (1991) T. R. Fletcher and R. Leone, J. Chem. Phys. 88 4720 (1988).)...

Surprisal plots of I(v ) against v or v have now been produced for several reactions. With very few exceptions they are linear or nearly so. That shown in Figure 2 for the reaction... 

Reaction (21) is omitted from Table 1 because the product state vibrational distribution does not yield a linear surprisal plot. It appears that this is another manifestation of the light atom anomaly referred to earlier. In particular it should... 

Surprisal plots are shown in Figure 5 for all of the HF dimer ro-vibrational bands studied to date. In all cases the striking feature in these plots is the fact that the most important channels are those which correspond to one HF fragment being highly rotationally excited while the other is not. On the other hand, the channels for which Jj J2 are consistently smaller. This preference can be explained if we assume that the dimer dissociates impulsively from a geometry which is essentially that of the equilibrium structure, as shown in Figure 6. 

For direct reactions, the calculation of Fim(T) is a complicated dynamical task. However, it has often been found that the results of both dynamic calculations and of experiments can be described by the distribution Fim(T) which is in turn relatively simply expressed via Ff (T) (the so-called surprisal plot) [39, 270]... 

The v=0 populations, given for some entries, were obtained from surprisal plots (62) or from rule-of-thumb estimate, as described in the original publications. 

These values were obtained by extrapolation of the model III surprisal plots. If model II or I were used, the relative population of v=0 would be higher by factors of v>2 and 3 respectively. Recently acquired results (80) using an improved reaction vessel and interferometric observation of the emission suggest the v /v ratio is somewhat higher than given in the table. 

The surprisal plots tend to be linear for all three prior distributions unless dynamical restrictions are present. Then surprisal plots for I and II become nonlinear. Extrapolation of the surprisal plots to = 0 facilitates assignment of the relative v=0 population. The vibrational energy disposal for the CHjX(X = F, Cl, Br, I) series are all characterized by the same X . 

The population of the highest accessible level, v, is very sensitive to slight variations in i.e., a small decrease in because of an Increase in Dj(R-H) markedly lowers P (v ). Surprisal plots (with all models) confirm this sensitivity, in that reduction of the energy causes the I(f )point to rise above the line formed by the I(fv) points, vtoluene reaction, using the full thermochemical energy, showed a very serious deviation of this sort. Since the benzyl radical is resonance stabilized, we reasoned that F + toluene might obey the three-body model if the available energy... 

Figure 4. Vibrational surprisal plots for models I, 11, and III for F + CHsX reactions, taken from (31. Recent work suggests that the v = 3/v = 2 and v = 2/v = 1 ratios may be somewhat higher than those used in making this plot. The general conclusions (31) based upon the surprisal analyses would be unchanged but the —A.V values may be increased slightly.

Because I(fjj fy) is conditional upon v and E, the proper reduced variable for rotational surprisal plots is g = f /(1-f ). By convention, values of P°(f fy) often are expressed without the (2J+1) factor. However, sxnce experiments do not resolve the Mj states of given J, for practical applications the prior must include (2J+1)(62,72,73). For many reactions the surprisal values are near zero and there is little rotational disequilibrium. The surprisals generally are not linear, but it is impossible to decide whether this is a problem with the estimated "initial" distributions or whether the plots are truly non-linear. There-foFe, analysis in terms of the most probable rotational level,... 

So far we have concentrated on the analysis of detailed rate data, as distinct from their synthesis. It has been implied that vibrational surprisal plots are frequently linear because of some, qualitatively common, dynamic constraint, but this has not been identified. Bernstein and Levine have elegantly reviewed the statistical mechanical basis of the relationship between real distributions and the constraints which lead to them. The general principle is that a system will adopt the distribution with maximum entropy, which is also consistent with all the constraints. Consequently, a complete distribution could be synthesized if the constraints could be independently determined. Alternatively, it should be possible, in principle, to deduce the constraints by observing the distribution. 

Recently, Pollak has demonstrated that the linearity of vibrational surprisal plots is consistent with an exponential gap law for the detailed rate constants, k v v T), of the form ... 

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