Prior distributions Surprisal

There are two ways to implement this program. One is to directly discuss the distribution of final states. This is known as surprisal analysis . In this simpler procedure one does not ask how this distribution came about Instead, one seeks the coarsest or most statistical" (= of maximal entropy) distribution of final states, subject to constraints. The last proviso is, of course, essential. If no constraints axe imposed, one will obtain as an answer the prior distribution. It is the constraints that generate a distribution which contains just that minimal dynamical detail as is necessary to generate the answers of interest. Few simple and physically obvious constraints are often sufficient [1, 3, 23] to account for even extreme deviations from the prior distribution. 

Information theory was first applied [177] to chemical reactions in an attempt to compact and classify the energy distributions of reaction products. This is achieved by surprisal analysis, where the observed product energy distribution, say for vibration, P(v ), is compared with a non-specific prior distribution P°(v ). Then, the surprisal, I(v ), is given by... 

In calculating the prior distribution, P°(v ), it is usual to assume that all product quantum states have an equal probability of being populated at a given total energy. Many reactions show linearity of the surprisal... 

The discovery, the linear surprisal, due to Kinsey, Bernstein, and Levine is about a rule on microcanonical rate constants ( /( /)) or the associated product distribution (p/(s/)) experimentally observed in a chemical reaction, in which a final state, for instance, in a vibrational level of a given energy Ej is specified. A statistically estimated product distribution pj (s ) corresponding to Pj(Ej) is called the prior distribution, which is usually evaluated in terms of the volume of a relevant classical phase space and is frequently represented in terms of energy parameters. Their remarkable finding [2-5] is an exponential form... 

However, paying no attention to physics behind actual events, the MEP simply maximizes information content under the constraints stated above. It is not physically clear how the prior distributions should be specified. In fact, several different methods have been proposed to determine a suitable candidate of Pjis-j) [46]. It is also not clear why the product distributions themselves do not directly obey the exponential distributions as in Eq. (76) in chemical reactions. It should also be noted that the appearance of exponential distributions is not always the actual case [43,44]. (We are not discussing the notion of surprisal synthesis due to Levine and Bernstein [2-5]). 

Here, we show, in terms of a variational principle of statistical mechanics, that a temperature proportional to Xj-1 can be naturally defined to characterize the first-order feature (fluctuation) around the peak position of a distribution in a state space that is projected onto an appropriate coordinate. It is also shown that the necessity of the so-called prior distribution can naturally result along with its clear physical meaning. Several new features of the linear surprisal theory are uncovered through the analysis. 

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.

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 maximum entropy method and the associated surprisal theory are an outgrowth of information theory. They involve a comparison between the actual shape of the KERD and the hypothetical, most statistical, so-called prior distribution . Two precious pieces of information can be derived from this comparison (i) an identification of the constraint that operates on the dynamics and prevents it from being statistical and (ii) the magnitude of the entropy deficiency which can be related to the fraction of phase space effectively sampled by the transition state. Values of 75-80% have been obtained in the case of the halogenobenzene ions. 

The prior distribution does not seek to match experimental results. Rather, it provides a reference against which to compare. The (logarithmic) deviation of the actual (observed or computed) distribution Irom the prior one is known as the surprisal. Then, armed with detailed balance, we will also seek to characterize the energy requirements of chemical reactions. Consistent with the arguments already presented, tiie measnre of specificity of energy disposal in the forward reaction turns out to be equal to the measure of selectivity of energy requirement in the reverse reaction. 

This raises the question of why should the surprisal be a simple function. One answer is the following. For a statistical distribution of final states we take it that all final quantiun states are equally probable. The practical version of this expectation is the prior distribution. Because of the dynamics the final quantum states may not be equally probable. What one then needs is a quantitative way to impose the inequality of states as enforced the dynamics. How to do it ... 

The surprisal is the same for all the states in the sum. So the sum is the number of such states. The prior distribution is just this number, divided by a normalization (the total number of states) to render the number into a probabihty. If there is no dynamic constraint the numerical value of ky is zero and the distribution is the prior one. We reiterate that the prior distribution is not the same as a uniform distribution. Rather, it depends on how many products quantum states fall into the group of states of interest. For example, in Figure 6.13 the prior vibrational distribution falls rapidly with increasing vibrational excitation and so looks thermal-hke. Problem H shows that in the hmit where the products have many atoms so that the fraction of energy in any particular vibrational mode is hkely to be small, the prior distribution is exactly thermal. But in an A + BC reaction there is only one vibrational mode in the products and so the correct form of the prior distribution is required. 

J. Compute the mean energy in the vibration for a distribution with a linear surprisal, Eq. (6.35), and plot it vs. A-v, both positive and negative. To do so you will need, as an intermediate step, to compute A.o as a function of A,v. By using a prior distribution from Eq. (6.31) all of this can be done analytically. 

Section 2.6.1) of the one-dimensional statistical distribution,J rather than use a surprisal analysis based upon the information theory prior distribution. We will follow this convention and, unless otherwise specified, the predicted statistical result for Et will refer to a one-dimensional translational distribution. 

Prior to 1965, all we had in our armoury were the a and it Hiickel theories, and a very small number of rigorous calculations designated ab initio (to be discussed later). The aims of quantum chemistry in those days were to give total energies and charge distributions for real molecules, and the seventh decimal place in the calculated properties of LiH. Practical chemists wanted things like reliable enthalpy changes for reactions, reaction paths, and so on. It should come as no surprise to learn that the practical chemists therefore treated theoreticians with scepticism. 

The failure of a simple statistical model is not too surprising, since most evidence indicates alkyl iodide dissociation is a rapid, direct process. The recoil angular distributions which we have measured show that photodissociation occurs prior to significant molecular rotation, and the lack of any vibrational structure in the ultraviolet absorption spectra is also consistent with dissociation before significant vibrational motion. 

The reactions and product distributions thus far reported have been exclusively concerned with hexene. It was of interest to see whether the high specificity of positional substitution could be maintained with the other hexene isomers. By positional substitution specificity is meant ester attachment on ether of the carbons involved in the original carbon-carbon double bond. Table VII shows the results of these studies. The internal olefins reacted more slowly than the a-olefin, and with both palladium chloride-cupric chloride and 7r-hexenylpalladium chloride-cupric chloride systems high substitutional specificity (> 95% ) was also maintained with 2-hexene (Table VII). However, with 3-hexene the specificity is considerably lower (80%). Whether this is caused by 3-hexene isomerization prior to vinylation or by allylic ester isomerization is not known. A surprisingly high ratio of 2-substitution to 3-substitution is found ( 7 1) in the products from 2-hexene. An effect this large... 

---