Correlations without coincidence measurements: deciding between

J . Phys. Chem. 1990, 94, 8751-8762

8751

FEATURE ARTICLE

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Correlations without Coincidence Measurements: Deciding between Stepwise and Concerted Dissociation Mechanisms for ABC A B C

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C.E. M. S t r a m + and P . L . Houston* Deparrment of Chemistry, Cornell University. Ithaca, New York 15853- I301 (Received: May 14, 1990)

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A method is presented for deciding the degree to which a dissociation of the form ABC A + B + C is synchronous and/or concerted based on measurement of the dynamical properties of the fragments. The method considers the joint probability distribution, P, giving the coincidental occurrence for each possible set of the dynamical variables. Information theory is used to obtain the most probable joint probability distribution that is consistent with the experimental measurements available, cach of which is simply a projection of P. Once this "maximum-entropy" P is found, it can be used to predict the outcome of measurements that are less experimentally accessible; an important prediction is the angular distribution describing the fragmentation. A "concertedness" parameter is defined, which ranges from a value of zero for the stepwise limit of dissociation to a value of unity for the concerted limit. This parameter can be directly calculated from the angular distribution provided by P and can be used to estimate to what degree a reaction is concerted based on the experimental data used to calculate P. An exact prior distribution based on conservation of both energy and linear momentum is derived for a dissociation producing three atoms. Algorithms for calculating P are discussed.

1. Introduction: Correlations without Coincidence Measurements The concept that a chemical dissociation or rearrangement might be "Concerted" has intrigued chemists for decades; indeed, it was this fascination that led to the development of orbital symmetry rules for predicting when a rearrangement might take place in a concerted fashion.' Although there is still debate,* most chemists agree that some reactions, particularly pericyclic ones, take place not only in a concerted fashion (in a single kinetic step), but also synchronously; Le., with bond-making and bond~ recent attention breaking processes taking place in ~ n i s o n . More has focused on the dynamics of concerted reactions and in particular on the question of how to determine from experimental observables whether or not a reaction is concerted and/or synchronous. Particularly interesting examples involve photodissociative events producing three fragments. Can one discriminate from measurement of the energy disposal into the fragments betwecn an event in which the two bonds break at the same time in a concerted fashion and an event in which the bonds break in a sequential, or stepwise, fashion? Although several previous papers have addressed this issue for specific dissociation^,^-^ no general theory has yet emerged for answering such important questions. This paper attempts to provide a method for determining from observation of the products whether or not a reaction is synchronous and/or concerted. In order to avoid confusion, we define here what we mean by these two terms. Let A, B, and C be atomic or molecular fragments. We will call a reaction ABC A B + C concerted if all bonds holding the fragments together as a parent compound break within a "short" time, on the order of a few picoscconds. We will let the time scale be determined by the molecule itself. Figure I defines some relevant angles for the dissociation: Omor is defined as the B-A-C angle in the molecular frame, Ocom is angle between the recoil momentum vectors Pband b, in the center-of-mass frame of the three-particle system, and x is the angle between E, and the A-B bond. An extreme example of a concerted dissociation would be one that always produces,

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'To whom correspondence should be addressed. 'Current address: Los Alamos National Laboratory, Los Alamos, N M 87545.

0022-3654/90/2094-875 1 $02.50/0

out of all the possibilities satisfying conservation of energy and momentum, a particular triplet of momentum vectors pa,Pb,and fi,. For this particular combination, all of the angles defined in the figure, and cos x in particular, would be fixed; that is to say that the angular distribution of cos x would be a 6 function. Now consider an extreme example of a stepwise dissociation. If C were to dissociate first and if AB were able to rotate randomly about its center of mass (COM) before itself dissociating, then x will have a random value in the plane of dissociation. The rotational symmetry of dissociation planes about b, combined with a random value of the x in each plane leads to the conclusion that cos x should be uniformly distributed; Le., its distribution should be constant from cos x = -1 to cos x = I . Thus, a measure of how closely an actual cos x distribution approaches a 6 function or a constant will give a measure of whether the dissociation is characterized as concerted or stepwise.' We will use the rotational time scale of A-B as the dividing line between a concerted and a stepwise process. The line between synchronous and nonsynchronous is somewhat more restrictive than that between concerted and nonconcerted. Synchronism implies not only that the bonds break at the same time but also that in the average dissociation all bonds are broken (and formed) at the same rate. Consider a concerted dissociation of B-A-C, or for simplicity the symmetric version B-A-B. As an example of a synchronous reaction, imagine dissociation of the parent molecule along the symmetric stretching coordinate. While there might be a Gaussian distribution of trajectories about the symmetric one due to the zero-point vibrational motion in other coordinates, we would still expect that in the average dissociation (1) Woodward, R. B.; Hoffmann, R. The Conservation of Orbital Symmetry; Verlag Chemie, Academic Press: Weinheim, Germany, 1971. (2) Dewar, M. J . S.J. Am. Chem. SOC.1984, 106, 209-219. (3) Borden, W. T.; Loncharich, R. J.; Houk, K. N . Annu. Reo. Phys. Chem. 1988, 39, 213-236. (4) Kroger, P. M.; Riley, S.J. J . Chem. Phys. 1977, 67, 4483. (5) Venkataraman, 8. K.; Bandukwalla, G.;Zhang, Z.; Vernon, M. J. Chem. Phys. 1989, 90, 5510. (6) Zhao, X.;Miller, W.B.; Hintsa, R. J.; Lee, Y . T.J . Chem. Phys. 1989, 90, 5527. (7) Stepwise in this discussion refers to having the A-C bond break first and is indicated by a uniform distribution of cos x . Were the A-B bond to break first, it would be the distribution of cos x that would be uniform.

0 1990 American Chemical Society

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The Journal of Physical Chemistry, Vol. 94, No. 25, 19'90

Strauss and Houston content of equivalent fragments, can answer whether a particular dissociation is concerted or stepwise and whether it is synchronous or nonsynchronous. We begin by defining the joint probability for a dissociation producing three fragments at fixed total available energy Eav. Let P(E:,E:,E:,E~,EVb,E~,~,EC,,EF) denote the joint probability that the dissociation will produce fragment A with energy E: in rotation, energy E: in vibration, energy E: in translation, fragment B with energy E: in rotation, and so forth, subject to the conservation of energy condition E,, = E: + E: + E: + E: + Et + E! E; + We then note that any observable energy distribution is simply a projection of this joint probability distribution. For example, the vibrational energy distribution in fragment A is simply the integral of the joint probability distribution over all other variables consistent with conservation of energy:

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Figure 1. Dcfinition of dissociation angles. The point above B,, resents thc ccntcr of mass of the three-fragment system.

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both bonds would break at the same rate and that both B fragments should, on average, have the same energy. I n other words, the probability of finding both B fragments with the same energy should be higher than the probability of finding one with a large energy and the other with a small energy. Now consider a nonsynchronous (although still concerted) process in which the molecule dissociates via the antisymmetric stretch. I n this case, we imagine one bond stretching and breaking as the other one compresses, destined to break on the next vibrational phase. While both bonds still dissociate in concert, the rate at which the bonds break during an average dissociation is different. It is quite likely that the two B fragments will receive different amounts of energy in the average dissociation. Thus, a criterion for synchronism in the dissociation of a symmetric molecule is that the two equivalent fragments rcceivc on average the same energy; for them to receive different cnergics in the average dissociation would indicate nonsynchronism. The major difficulty in deciding whether a reaction is concerted or synchronous is that either determination requires us to know energy distributions in coincidence. Consider again the examples above. The angle x, whose distribution might tell us whether or not a reaction is concerted, can be determined by knowing the three fragment momenta or energies in coincidence, since the three vectors sum to zero only for a specific angular configuration. The angular distribution of cos x is thus related to the distribution of coincidental probabilities for finding triplets of fragment energies. Similarly, in order to determine the degree of synchronism, we need to know the probabilities that the two equivalent fragments will have a given pair of energies in Coincidence. Comparison of the probability that the two fragments will have a discrepant combination of energies with the probability that they will have the same energy can be used to measure synchronism. Thus, determining either the degree of "concertedness" or the dcgrec of synchronism requires knowledge of coincidence properties. Since direct experimental measurement of these coincidental probabilities, or so-called joint probability distributions, is experimentally difficult, we need to learn how to make the most of the experimental information that is more readily available. We present below a method for determining the joint probability distributions and for deciding from them whether a dissociation of the form ABC A + B + C is synchronous and/or concerted. The theory is based on three key ideas. The first is the recognition that all observable data for such a dissociation can be described as projections of a joint probability distribution. The second is that the joint probability distribution that gives the correct projections and is otherwise most probable can be determined by maximizing thc entropy using standard information theory techniqucs.8-" The third is that analysis of the joint probability distribution, particularly its predictions about the distribution of dissociation angles and its predictions about the coincident energy

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(8) Grandy, W. T. Maximum-Entropy and Bayesian Methods in Inverse Problems; Smith, C. R., Grandy, W. T., Eds.; Reidel: Dordrecht, Holland, 1985; p I . (9) Jaynes. E. T. Maximum-Entropy and Bayesian Methods in Inverse Problems; Smith, C. R.. Grandy. W . T., Eds.; Reidel: Dordrecht, Holland, 1985; p 21. (IO) Jaynes, E. T. Maximum Entropy Formalism; MIT: Cambridge, MA, 1978; p 15. ( I I ) Shannon. C. E. Bell System Tech. J . 1948. 27. 379. 623.

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P(E;) = JP(E;.E:.E:,E,, E,~,E~,E;,E&E;)x 6 ( Eav -Ea_Ea-Ea_Eb-Eb-Eb-EC-EC-EC) r v 1 I v t r v t dE; dE: dE! dE,b dEF dEC dEt dEC ( I ) In practice, any continuous distribution functions are replaced by discrete ones and the integrals replaced by sums. More complicated distributions are also easily obtained. For example, the E: E:, distribution of total energy in fragment a, Ea = E: is given by

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6 (Eav-E;- Et- E:- E!- E,b-Ep- E:- E:- E:) 6 (Ea-Ea-Ea-Ea) r v t dE; dEt dE: dE! dE: dEF dEC dEt dEC (2)

In principle, once the joint probability distribution is known, any observabledata that is a function of the variables in the distribution can be predicted. As mentioned above, an experimental determination of the joint probability distribution would require a very difficult coincidence measurement. For a single dissociative event, one would have to determine in coincidence the energy in each degree of freedom for each of the three fragments. Such experiments are not currently feasible. The data most often available is the ro-vibrational distribution or perhaps only the translational distribution for one or more of the fragments, but not obtained for different fragments in coincidence. Under these less-than-ideal conditions, it is reasonable to ask for the "most probable" joint probability distribution P that agrees with whatever data we have, even though the data set may be incomplete or fragmented. In other words, what is the most likely joint probability distribution that gives the observed projections? This problem is the same as that faced by a radiologist trying to determine the contents of a brain cross section from a small number of X-rays taken at different angles; the contents of the brain are analogous to the joint probability distribution, while the X-ray projections are analogous to our data. Mathematically it can be shown that an infinite number of X-rays is needed for a complete specification; hardly desirable if you are the patient! How well can the brain contents be determined from fewer X-rays? This "back-projection" problem occurs frequently in the fields of a ~ t r o n o m y , ' ~seismology,'O,'s ,'~ and medical tom ~ g r a p h y , 'and ~ ' ~it is fortunate both for the patient and for our dissociation problem that information theoretical techniques for ( I 2) This definition can easily be generalized to cover other degrees of freedom and a different number of total fragments. (13) Bracewell, R. N . ; Riddle, A . C. Asrrophys. J . 1967, 150, 427. (14) Frieden, B. R. Maximum-Entropy and Bayesian Methods in Inverse Problems; Smith, C . R., Grandy, W. T., Eds.; Reidel: Dordrecht, Holland, 1985; pp 133-169. ( I 5) Robinson, E. A. Maximum-Entropy and Bayesian Methods In Inuerse Problems; Smith, C. R.. Grandy, W. T., Eds.; Reidel: Dordrecht, Holland, 1985; pp 171, 211. (16) Shepp, L. A,; Kruskal, J. B. Am. Math. Mon. 1975, 85, 420. (17) Shepp, L. A.; Vardi, Y. IEEE Trans. Med. Imaging 1982, M I - / , 113-122. (18) Shepp, L. A . Proc. Symp. Appl. Math. 1982, 27, I . (19) Kaufman. L. IEEE Trans. Med. Imaging 1987, MI-6, 37

The Journal of Physical Chemistry, Vol. 94, No. 25, 1990 8753

Feature Article solving it are well developed.2*23 These techniques are briefly outlined here, while specific algorithms are discussed in Appendix

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I. Suppose that we have no data concerning the system. Then the most probable solution for the joint probability distribution will be the one of maximum entropy; i.e. M

S=

-E P, In ( P , / Q , )

= maximum

n= I

(3)

where P, is a shorthand notation for one of the elements of P(E;,E:,E:,E:,E:,EF,E;,EF,EF) and we have assumed that any continuous variables can be approximated by discrete ones. In (3), Q, is shorthand for one element of Q(E:,Etr,E:,E:,E~,EP,ES,EF,EC) and is defined as the number of ways it is possible to arrange E,, into the indicated partitions. The distribution Q is often called the "prior" distribution. We show in Appendix I I that Q = 4:937:4$7;9;, where the factors q are simply the degeneracies of the indicated rotational and vibrational levels. We now suppose that we have k experimental observations Pi, i = 1, ..., k , and that each one is related to the elements of P by some particular projection, so that

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Figure 2. The conservation of energy condition E,, = E: E! E f demands that the joint probability distribution P(Ef,EP,Ef) have nonzero values only on the surface of an equilateral triangle. E"=E,,

M

CPnf;.,= Fi, i n=I

= 1, ..., k

(4)

whereJn (usually a matrix of ones and zeros) simply provides the rule for making a particular projection. The distribution P, that satisfies (3) and matches the experimental observations is obtained by standard used of Lagrange multipliers:

subject to the normalization condition C P , = 1. In ( 5 ) , the pi are the Lagrange multipliers to be determined by demanding that the solution P,, conform to (4). Thus, the problem is changed from finding the M values P, to finding the k values pi, i = 1, ..., k , where k is the number of projections. In general, an iterative solution will be necessary; Le., a set of M~ is selected, (5) is used to calculate P,,, the projections are then calculated from (4) and compared to the data, and the comparison is used to select a corrected set of multipliers p,. Schemes for performing this "back-projection" iteratively are discussed in Appendix I. This procedure determines the "best" joint probability distribution by finding the distribution that has the maximum entropy while still agreeing with the experimental data. As new experimental observations are added, new multipliers pi can be included. Information theory assures us that the most probable distribution is physically relevant and that no other model that incorporates only the known data can make a better prediction. Once this best distribution is obtained, the most likely outcome of any experiment (even a gedanken experiment) can be predicted, based on all the experimental evidence in hand. This fact is significant, since it often occurs that the physical measurements that we can make experimentally are not intuitively related to the questions we would like to answer. In particular, questions of coincidence, of bimodality in energy distributions, and of angular distributions can be answered by making proper projections of the joint probability distribution even though they may be difficult to answer directly from the observed data. The remainder of this paper is organized as follows. We first consider the joint probability distribution for a simple thought experiment, the dissociation of a triatomic molecule into three atoms. This simple system serves to illustrate the connection between the potential energy surface controlling the dissociation and the joint probability distribution. We then apply the technique to three real dissociations using data reported in the literature. Finally, we discuss techniques for improving the calculations and (20) Levine. R. D. Annu. Rev. Phys. Chem. 1978, 29, 59-92. (21) Levine, R. D. Adu. Chem. Phys. 1981, 47, 239. (22) Levine. R . D. Maximum Entropy Formalism; MIT: Cambridge, MA, 1978; p 247. ( 2 3 ) Bernstein. R. B.; Levine. R. D. Ada. A i . Mol. Phys. 1975, / I , 2 1 5 .

Figure 3. The triangle of energy conservation and the circle of momentum conservation. The significance of the points labeled 1-4 is discussed in the text.

summarize our conclusions. Appendices I-IV provide recipes for performing the calculations. 11. Dissociation of a Triatomic Molecule to Atomic

Fragments We consider the dissociation of a triatomic molecule ABC to atomic fragments A + B + C for several reasons. First, omitting electronic excitation, the joint probability distribution depends only on three variables, E:, E:, and E f . Since conservation of energy demands that E,, = E: + E: + E f , only two of these variables are independent, and the joint probability distribution P( E:,E:,EC) can, therefore, be represented as a density distribution on an easily visualized two-dimensional surface, the equilateral triangle shown in Figure 2. Second, the ABC system is simple enough so that features of the probability distribution on the triangle can be easily understood. Finally, trajectory calculations can be performed efficiently on assumed ABC potential energy surfaces in order to relate the features of the surface to the features of the joint probability distribution. Although we will explore only a few potential energy surfaces, the intent here is simply to develop an intuitive feeling for the results rather than to perform an exhaustive study. Figure 3 demonstrates some features of the joint probability distribution for the case when the atoms A, B, and C all have the same mass. Axes determining the values of the three variables are shown; for example the density at the indicated point is P(E; = x,EF = y,EC=z). Although any point on the triangle is permitted to have a nonzero probability density by conservation of energy, only those points within the indicated circle simultaneously satisfy conservation of linear momentum. At a point near the apex of the triangle, for example, fragment A has so much energy, and hence momentum, that the remaining energy of fragments B and C is such that their momenta cannot be large enough to satisfy

6,

+ j?b + j?,

=0

(6)

even if they are aligned together and opposite to the momentum of A. Each point within the circle corresponds both to a unique division of the energy among the three degrees of freedom and

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The Journal of Physical Chemistry, Vol. 94, No. 25, I990

-"Concerted"

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surfaces of Figure 4. (a) The probability distribution along the straight line joining points 1 and 3 in Figure 3. (b) The distribution function for the angle x defined in Figure I. Figure 5. Results of trajectories on the

Figure 4. Potcntial cncrgy surfaces used i n trajectory calculations.

to a single set of angles describing the recoil directions of the fragments. Finally, since all points within the circle satisfy both conservation of energy and conservation of linear momentum, in the absence of any other constraints the prior joint probability distribution will be given by a positive constant within the circle and zero elsewhere, as demonstrated in Appendix I!. Bond angle constraints on the dissociation would cause the probability distribution to deviate from the prior one. Some of the bond angles have been defined in Figure 1. Returning to Figure 3, the bottom circular arc joining points I , 2, and 3 represents the locus of nonzero probabilities under a constraint where Ocom = Om,, = 180'. For example, at point 2 the central atom A does not move in the center-of-mass (COM) frame, while atoms B and C move in opposite directions with equal momenta. The top circular arc joining points 1, 4, and 3 is the locus of nonzero probabilities if Ocom = Omol = 0'; for example, at point 4 atom A is moving in one direction, while atoms B and C have equal momentum to one another and are moving together in the opposite direction. The straight line connecting points 1 and 3 is the locus of nonzero probabilities for a fixed value of Omol = 90'. For a system constrained such that Omol = 90°, trajectory calculations were performed on the three potential energy surfaces shown in Figure 4. The surface of Figure 4b is simply the sum of two Morse oscillators, each of whose dissociation energy is a function of the distance from the other's origin: W,Y)= DeCv)[exp(-a(x-xo)) + 2 exp(-a(x-xo))l + De(x)[exp(-aCv-yo)) + 2 exp(-~Cv-.v~))I( 7 ) To this potential a hill (Figure 4a) or a depression (Figure 4c) is added at a point displaced from the Morse minima. Trajectories that were not localized near either coordinate axis in the asymptotic limit were taken as ones representing dissociation to three fragments. Figure 4a was chosen to represent a stepwise and necessarily nonsynchronous dissociation. A trajectory descending the repulsive wall on this surface will encounter a barrier in the symmetric stretching coordinate and will thus be deflected toward one of the axes. The bonds will break at a different rate. Figure 4c was choscn to rcprescnt ii concerted and synchronous dissociation. For this surfacc. the trajectories are focused toward the symmetric stretching coordinate, so that on average both bonds should break in synchrony. Of course, these surfaces do not begin

to explore all the possibilities, but the trajectories are instructive in developing intuition. The trajectory results are summarized in Figure 5 . Figure Sa plots the probability distribution along the line marked 1-3 in Figure 3. It demonstrates that the "concerted" (and synchronous) surface focuses probability toward the center of the triangle, indicating that the probability of finding the same energy in the two equivalent fragments is larger than that for finding them with discrepant energies. On the other hand, the "stepwise" surface dcfocuses probability toward the corners of the triangle, where the value of Ec/Eavis either large or small and Eb/EaVis coincidentally small or large. In the absence of any constraints other than conservation of energy, conservation of linear momentum, and Om,, = 90°,the probability distribution would be a constant between E C / E a ,= 0.33 and 0.67, a situation nearly approached by the dissociation on the surface of Figure 4b. Figure Sb demonstrates that the "concerted" surface also produces a much more sharply peaked cos x distribution than the "stepwise" one. We propose that these two features, the change in the probability distribution on the triangle and the cos x distribution, are useful in making conclusions about particular dissociations. The probability flow determines whether the fragment energy distributions will deviate from the statistical ones in the direction of bimodality. For example, in the dissociation of a symmetric molecule B-A-B, an increase in probability in the lower corners of the triangle relative to the statistical distribution would indicate that there were two types of B fragments, one with high energy and one with low. Such an uneven distribution of energy indicates that the dissociation is nonsynchronous. By contrast, a peak in the probability distribution along the center line of the triangle would be consistent with a synchronous dissociation. A convincing measure of "concertedness" can be obtained from the distribution of cos x, as discussed in the Introduction. A strongly peaked cos x distribution indicates a concerted process, whereas a uniform distribution from cos x = - I tp cos x = 1 indicates a stepwise process, one in which the fragment undergoing secondary dissociation has lived long enough to rotate before decomposing. A simple numerical measure of the shape of the cos x distribution is given by its "information content".24 For an angular distribution digitized into N bins, the quantity

(24) We are grateful to Professor R. D. Levine for suggesting this function rather than the more conventional x2 measure of the deviation.

rhe Journal of Physical Chemistry, Vol. 94, No. 25, I990 8755

Feature Article is a number that ranges from a value of zero when the distribution is a uniform to unity when it is a delta function. We propose C to be a “conccrtedness” measure with zero corresponding to the stepwise limit and unity corresponding to the concerted limit. The values of C for the distributions shown in Figure 5b are 0.157 for the “stepwise” surface, 0.1 50 for the “Morsen surface, and 0.291 for the “concerted” surface. Because the angle Ocom has been restricted to 90°, cos x > 0 and the values of C for the stepwise and Morse surfaces are somewhat larger than one might expect. Nonetheless. it is clear that C is even larger for the “concerted” surface. Before leaving this simple dissociation case, it is worthwhile to note that, as shown in Appendix 111, a statistical dissociation has an angular distribution for cos x which is identical with that for the limit of a stepwise dissociation; both show an uniform distribution in cos x. A consequence of this fact is that while a statistical dissociation always implies a uniform cos x distribution, the converse is not true. A uniform cos x distribution is consistent both with a statistical distribution and with a stepwise dissociation, the latter of which might be highly nonstatistical. In this sense, statistical dissociations can correctly be characterized as proceeding by a stepwise mechanism, whereas not all stepwise dissociations are statistical. A uniform cos x distribution is a good criterion for determining whether a dissociation is stepwise, whereas other criteria, such as whether fragment energy distributions agree with those predicted on the basis of degeneracies, are better for determining whether or not a dissociation mechanism is statistical.

111. Application to Real Molecular Systems A. The 157-nm Photodissociation of C 3 0 2 .As the first example of the application of the back-projection technique to an actual molecular system, we consider the 157-nm photodissociation of C302(O=C=C=C=O) to 2CO + C(’P), recently studied by Strauss et al.2s Briefly, the CO and C energy distributions were probed by V U V laser-induced fluorescence. C O internal levels were selected by tuning the probe laser to selected transitions of the CO A I l l X ‘Esystem; the internal energy distributions were determined by scanning the probe laser over vibrational and rotational transitions, while the translational energy distribution was determined for several ro-vibrational levels from the Doppler profile of an individual transition. The C(3P) translational energy was similarly determined from the profile of the 3D2 +- 3P2 transition. The internal energy data set spans roughly 60 CO rotational transitions in each of 4 vibrational levels, while the translational energy distributions were determined for roughly 18 representative C O internal states and for the C(jP). The joint probability distribution P ( E : , E ~ , E ~ , E P , E ~ , E(A F,E~) = carbon atom: B, C = C O molecule) spans seven dimensions and requires roughly 30 X 60 X 4 X 30 X 60 X 4 X 30 = 1.5 X IO9 elements to describe completely the data. In order ( I ) to be able to visualize the data in fewer dimensions, (2) to perform the calculation on a computer of reasonable capacity, and (3) to make a simple connection with the ABC dissociation discussed above, we have chosen to compress the data into far fewer elements. The simplest analogy to the ABC triatomic system recognizes that the equation for the triangle given as E,, = E: + E: + E:, can be replaced for more complex systems by the equation E,, = Ea + Eb + EC.where Em is the total energy (internal + translational) of fragment m. The joint probability distribution is then reduced to three dimensions, and the constraint of conservation of energy again dictates that the distribution can be plotted as a density distribution on an equilateral triangle. Note that the probability does not necessarily have to be zero near the apices of the triangle, since the translational energy of the high-energy fragment can be low enough to conserve linear momentum if its internal energy is high. Compression of the data into this total energy picture is reasonable. though not necessarily desirable, because the greatest distinction between stepwise and concerted is the order in which

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(25) Strauss. C. E.; Kable, S.H.; Chawla, G . K.: Houston, P. L.: Burak, I . J . Chem. Phvs.. submitted for publication.

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Figure 6 . Data and calculated results for the 157-nm photodissociation of C 3 0 2 . The solid lines in the top two panels represent the measured data, convoluted into distributions of total fragment energy (internal + translational). The dotted lines in these panels represent the prediction of the prior distribution. The triangles in these panels show the total fragment energy distributions projected following calculation of the best joint probability distribution; they agree adequately with the data. The distribution function for cos x is given in the bottom panel. An even distribution with the same binning is shown as the horizontal line.

the bonds break, and this order determines the total energy available to the fragments. The data are reduced as follows. For the carbon atom we can safely neglect population of excited spin-orbit levels so that the E” distribution is simply equal to the translational energy distribution of C(3P0). For the C O molecule, the measured vibrational rotational, and translational energy distributions were convoluted to obtain the Eb = ECdistributions. For a symmetric potential energy surface, such as that for C 3 0 2(or acetone or symmetric tetrazine discussed below), there is no experimental distinction between the two identical fragments; the joint probability for Eb = y and EC= z is equal to that for E b = z and EC = y . This fact has two consequences: the distribution of probability at locations on the triangle will be symmetric about the Ea axis, and the single, measured distribution of the repeated fragment is equal to both the Eb and Ec distributions. The total fragment energy distributions for the carbon and C O fragments arc given by the solid lines in Figure 6. Calculation of the prior distribution, Q(Ea,Eb,Ec), is described in Appendix IV. By definition, in the absence of any data the prior distribution would be the best possible guess at the joint probability distribution, and its projections would give the Ea, Eb, and ECdistributions. As shown by the dotted lines in the top two panels of Figure 6, these distributions agree closely with the experimental ones. It is thus expected that the best value of P will not be very different from Q. Equations 5 and 4 were solved iteratively by use of techniques described in Appendix I . As described in the Introduction, the resulting distribution P is the joint probability distribution which both agrees with the data and has the maximum entropy. To show

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The Journal of Physical Chemistry, Vol. 94. No. 25, 1990

Strauss and Houston

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Figure 7. Plot of the difference between the calculated joint probability

distribution P and the prior one Q for the 157-nm photodissociation of C,O,. The contours are lines of even difference, with negative values

shown as dashed lines. The percentage differences between P and Q at the lowest and highest contours are approximately -49% and 144%, respectively. that the distribution actually does reproduce the data, we have plotted as triangles in the top two panels of Figure 6 the projections of P corresponding to the C O and carbon fragments. As can be seen from this figure, the agreement with the data is excellent. Figure 7 displays a contour diagram of the different between the calculated best joint probability distribution, P, and the prior distribution, Q. The percentage differences between P and Q a t the lowest and highest contours are approximately -49% and 1447c, rcspcctively. There is a decided flow of probability away from the center and toward the lower corners. Projection of this distribution onto either CO axis shows that, with respect to the prior distribution, the actual distribution deviates slightly in the direction of a bimodal energy distribution in the CO; there is a tendency for a more energetic CO to be produced in coincidence with a less energetic one. Such a deviation might be expected for a dissociation mechanism that diverges from a statistical one in the direction of the stepwise limit. It definitely indicates that the dissociation is nonsynchronous. Thc best joint probability distribution, P, can also be used to predict unmeasured quantities, such as the angular distributions. The bottom panel of Figure 6 displays the distribution of cos x calculated from the best joint probability distribution, P , using the method described in Appendix IVA. The distribution is very similar to a uniform one and has a "C parameter" (8) of 0.0015. Thus, the dissociation can adequately be described as stepwise. Comparison of the measured and statistical energy distributions in the top panels of Figure 6 suggests that the dissociation is also very nearly statistical, despite small deviations toward a bimodal energy distribution in the C O fragment. We conclude that the dissociation of C 3 0 zis described very adequately as nonsynchronous, stepwise, and nearly statistical, in agreement with the conclusions of the authors,2s which they reached by less direct means. B. The 193-nm Photodissociation of Acetone. As a second example of an application of these techniques to a real molecular system, we consider the 193-nm photodissociation of acetone, which yields two methyl fragments and one CO. Extensive data on the internal and translational energy distributions of these photofragments have been reported by Trentelman et Briefly, the CO internal and translational distributions were determined by vacuum-UV laser-induced fluorescence as in the case of C 3 0 2 . The methyl internal distribution was determined by multiphoton ionization, while its translational distribution was measured by use of a one-dimensional imaging technique.*' As in the case of C30z.a very large number of elements would be needed to describe the joint probability distribution in such a way as to completely represent the data. Consequently, the data was reduced ~~

( 2 6 ) Trcntclmdn. K A , Kdblc, S H , Moss, D B , Houston. P L J Chem Phys 1989, 91, 7498-7513 (27) Loo, R Ogorzalek. Hall, G E , Haerri. H - P , Houston. P L J Phys Chem 1988. 92 5

h^ n

e

a 0.05 0.00 00

0.5

1.5

10

20

t-

i

0.020

0.000 ' -1.0

'

I

'

I " -0.5

' '

1 ' 0.0

' '

I

'

0.5

I

'

"

1

1 .o

cosx Figure 8. Data and calculated results for the 193-nm photodissociation of acetone. The solid lines in the top two panels represent the measured

data, convoluted into distributions of total fragment energy (internal + translational). The dotted lines in these panels represent the prediction of the prior distribution. The triangles in these panels show the total fragment energy distributions projected following calculated of the best joint probability distribution; they agree adequately with the data. The distribution function for cos x is given in the bottom panel. An even distribution with the same binning is shown as the horizontal line.

by convoluting for each fragment the internal and translational energy distributions into three total energy distributions: Ea, representing the CO, and Eb = E', representing the two methyl radicals. Conservation of energy again restricts the probability distribution to a triangular surface. The distributions of total fragment energy are given by the solid lines in Figure 8. The prior distribution Q(Ea,Eb,EC)was calculated as described in Appendix IVB. The dotted lines in the top two panels of Figure 8 give the fragment total energy distributions based on projections of the statistical prediction. As can be seen, these differ substantially from the actual distributions. Using the techniques described in Appendix I, (5) and (4) were again solved iteratively to obtain P,the joint probability distribution which both agrees with the data and has the maximum entropy. The triangles in the top two panels of Figure 8 give the fragment total energy distributions predicted by this best joint probability distribution; they agree with the data, as they should. Figure 9 displays a contour diagram of the difference between the calculated best joint probability distribution, P, and the prior distribution, Q. The percentage differences between P and Q a t :he lowest and highest contours are approximately -93% and +4353%, respectively. There is a decided flow of probability toward the apex of the triangle, Le., toward high CO energy. Thus, the dissociation is decidedly nonstatistical. On the basis of only the contour diagram, it is not clear whether the dissociation is stepwise or concerted, since the concentration of probability flow toward the symmetry axis could indicate a synchronous process in which both methyls receive the same energy. A more definitive answer is obtained from the angular distribution. The cos x distribution obtained from the best joint probability distribution is shown in the bottom panel of Figure 8 . The distribution dcviaics significantly from a uniform one. and

The Journal of Physical Chemistry, Vol. 94, No. 25, 1990 8757

Feature Article

co

A

~10'

Fragment Translational Energy (cm-')

I

HCN

CH3

Figure 9. Plot of the difference between the calculated joint probability distribution P and the prior one Q for the 193-nm photodissociation of acetone. The contours are lines of even difference, with negative values shown as dashed lines. The percentage differences between P and Qat the lowot and highest contours are approximately -93% and 4353%,

respectively. the system demonstrates a mild angular preference. There is a tendency to favor the cos x = -1 end of the distribution, i.e., toward having the methyls recoil roughly in the same direction. A value of C = 0.014 is calculated for this distribution. We conclude that the dissociation of acetone at 193 nm cannot be characterized by a stepwise mechanism; there is a significant deviation in the direction of a concerted reaction. However, the angular distribution is still quite far from one approaching a 6 function, so that acetone should be viewed as an intermediate case. C . The So Photodissociation of Symmetric Tefrazine. A third application of the technique was made to the So dissociation of symmetric tetrazine, using data recently reported by Zhao et aL6 These authors measured the velocity distributions of the N, and HCN fragments using time-of-flight analysis with mass spectrometric detection, but their technique did not allow them to determine the internal energy distributions. Although the actual translational energy distributions for the N 2 and HCN were not reported, these can be derived by applying an approximate laboratory-to-COM transformation to the published arrival time distributions and then transforming from velocity to energy coordinates. The data set then consists of P( E;) (A = N,) and P(Ep) = P ( E f ) (B, C = H C N ) . Even though there is no information available from the experiment on the total energies in each of the fragments as there was for the cases of C 3 0 2and acetone, the technique developed here for determining the best joint probability distribution works no less rigorously. Since less data is available and because the measurement does not provide the distribution of total energy in each fragment, we chose to use the complete set for the calculation rather than to reduce the data. Thus, we seek the most probable whose projections distribution P(E:~E:.E;,E~,E~,EP,E~,EF,E~) agree with the measured P(E:) and P(EP) = P(EC), shown by the solid lines in the top two panels of Figure IO. In order to make the calculation more compact, we note that since no experimental information is available on the internal energy distributions, the maximum entropy procedure dictates that the best guess at these distributions is the statistical one. We thus integrate over the statistical internal distributions in both P and Q to define P(E;,EP,Er) and Q(Ey,EP,Er), respectively: P(E;.EP,EC) P(Et,E:,E:,EP,E,b,E,,Er,E~,E:) dE; dE," dE," dE,b dEC dE:

1

(9)

Q(Ea Eb E:)

=

S'Q~E:,E:.E:,E~.EI,EB.E:.E~,E:) dE; dE: dE: dE,b dEC dE: (10)

Because each of the variables Et, E:, and E: can rangc from 0 to E,,, subject only to E: + E: E: IE,,, the probabilities P and Q are distributions on a three-dimensional pyramid rather than on a two-dimcnsional triangle.

+

0.10

,

Fragment Translational Energy (em-') , , , I 1 , I , , , I I I , I I

,

,

A

0.08

."o a"

.lo'

0.04 0.02

o.oo-' -1.0

-0.5

' J 4 - I

0.0

' '

I U

0.5

'

-4 1.0

cosx

Figure 10. Data and calculated results for the So photodissociation of symmetric tetrazine. The solid lines in the top two panels represent the measured data. the dotted lines in these panels represent the prediction of the prior distribution. The triangles in these panels show the total fragment energy distributions projected following calculation of the best joint probability distribution; they agree adequately with the data. The distribution function for cos x is given in the bottom panel. An even distribution with the same binning is shown as the horizontal line.

The prior distribution Q(E:,E:.EF) was calculated as described in Appendix IVC. The dotted lines in the top two panels of Figure I O gives the fragment translational energy distributions based on the statistical prediction. As can be seen, these differ substantially from the actual distributions. Using the techniques described in Appendix I , (5) and (4) were again solved iteratively to obtain P. The triangles in the top two panels of Figure 10 give the fragment translational energy distributions predicted by this best joint probability distribution; they again agree with the data, as they should. The distribution function P contains all the information we need to predict the outcome of any distribution based on the variables, even distributions not directly measured. In order to see how the actual probability distribution P deviates from the statistical one Q, we have reconstructed the joint probability distribution P(Ea,,Eb,EC),a function of fragment total energies, based on our knowledge of the statistical internal energy distribution and the calculation described above for P(E:,EP,Er), a function of fragment translational energies. The difference between P(Ea,Eb,F) and Q(Ed,Eb,Ec)is displayed in Figure I 1 as a contour diagram. The percentage differences between P and Q at the lowest and highest contours are approximately -76% and 476%,respectively. The plot shows that there is a decided flow of probability toward a very restricted region near the center of the triangle. The fact that the new probability is centered on the symmetry axis suggests a synchronous process. That the density is in only a restricted region indicates that the dissociation is decidedly nonstatistical and suggests a concerted process. This interpretation is confirmed by examination of the cos x distribution, shown in the bottom panel of Figure I O . The distribution shows a strong peak in a very restricted angular range; a value of C = 0.347 is calculated. Figure 12 displays the distribution of 8,,, computed from the best joint probability dis-

8758

N2

\

\

I -S-Tetrazine

:i\ I

I

f

Strauss and Houston

The Journal of Physical Chemistry, Vol. 94, No. 25, 1990

I

0.3

a 0.2

t

%m

Figure 12. Angular distributions for Ocom for the dissociations of sym-

metric tetrazine, C302. and acetone. tribution for the symmetric tetrazine dissociation as well as for the dissociations of acetone and C302.The symmetric tetrazine Ocom distribution is strongly peaked at 1 2 5 O , in agreement with the analysis of Zhao et a1.6 We thus conclude that the dissociation of symmetric tetrazine through So is a synchronous and concerted process.

IV. Discussion A . Achievements, Limitations, arid Future Prospects of the Back-Prujrcrion Technique. The principal conclusions from the data analyzed thus far are ( I ) that the information theory technique provides an cxccllcnt method for back-projection from the data to thc joint probability distribution and ( 2 ) that analysis of the joint probability distribution can help to answer to what degree a dissociation is conccrtcd or stepwise. It is clear from the experimental examples that the projections actually contain much more information about the joint probability distribution than we are accustomed to thinking about. Even more is actually available. since in the examples of C 3 0 2and acetone the data set was actually reduced significantly in order to picture it in two dimensions and in order to perform the calculations on a readily available computer. What are the prospects for dealing with the entire data set? The main problem in treating all of the data is that the number of elements in P , and Q, can become very large. Assuming three diatomic fragments, each with four vibrational levels in the available energy window, and assuming that we want a resolution of, say. 30 bins in cach of the rotational and translational distributions, then thcrc would be ca. (3600)3= 4.6 X 10l0elements in each of P and Q. Computer storage will certainly become a problcm. However, each element can be calculated rather simply, I n the above example. only 192 Lagrange parameters are needed

to calculate any particular P,, and the calculation of Q, is also relatively simple. These could be calculated each time they are needed in the iteration scheme. A trial run with our algorithm showed that calculating P, each time it was needed did not noticeably degrade the execution time. Thus, the prospect of solving the complete problem, even giving an extensive data set of projections, is not beyond modern computer technology. The problem is made somewhat easier because normal data on vibrational, rotational, and translational distributions often include coincidence measurements. For example, from spectroscopic measurements it is almost always the case that E: (m = a, b, c) is obtained in coincidence with E:, and if the Doppler technique is used to measure E:, this distribution will also be obtained for a particular vibrational and rotational level. What is not usually measured, however, is a coincidence between different fragments. although such measurements may eventually be possible. T o the extent that the data include even partial coincidence measurements, the calculation becomes somewhat easier since the data are then a less complicated projection of the joint probability distribution. Because of the pervasive use of back-projection techniques, particularly in the field of m e d i ~ i n e , ' ~it- 'is~ likely that improvements will be made on the simple iterative schemes used in this study. Indeed, new schemes are already in use, even ones employing vectorized machines,28 though it is not yet clear how robust they will be when applied to problems such as that considered here. The algorithms used for the back-projection can easily be modified to take account of noisy data, although we have not included this modification in the current study. The simplest method for including noise would be to weight each element of the difference between the predicted and observed projection inversely in proportion to the uncertainty in the observed data. This technique would work for both the simulated annealing and the vector methods described in Appendix I . It should be noted that the final values of the Lagrange parameters are most certain when they correspond to projections of high probability and least certain when they correspond to projections of low probability. B. The Physical Constraints. Although the p, of (5) provide a set of constraints from which to calculate P, from Qn, they do not provide a very simple or easily interpreted set. It is natural to ask what is the simplest possible set of constraints on Ea, Eb, and EC that would give precisely the same projections. The mathematical answer to this question brings us as close as we can come to answering the basic physical question "what caused the projections'?'' For the cases described above, where there are three axes of projection, it is possible to reproduce the projections using only three constraints:

where c" (m = a, b, c) are constants, are the projections along the axes Em,respectively, and p," are the corresponding Lagrange multipliers, which can be treated for each fragment as a function Lm(Em).lsIn general, there will be such a function for each axis of projection. The functional Lm(Em)can be interpreted in a manner similar to ordinary ~ u r p r i s a l s . For ~ ~ example, ~~~ if each Lm(Em)were a straight line, e.g. La(Ed)= cvEa + @, then the constraints of ( 1 I ) would become ( u E m @ ) = c" or ( E m )= constant. We could then state that a set of constraints which would produce the same set of projections is that ( E " ) ,(Eb),and ( E ' ) are equal to constants. Although these may not be physically relevant constraints,

+

( 2 8 ) Kaufman, L. Ann. Oper. Res. 1990, 22, 325-353. (29) Actually. a simpler constraint exists, but i t lacks the natural interpretation of the axial constraints: a ( L " ) h ( L b ) + c ( L c ) = CP + cb + CT (30) Procaccia. 1.: Levine. R . D. Chem. Phj,s. L e r f . 1975, 33. 5 .

+

The Journal of Physical Chemistry, Vol. 94, No. 25, 1990 8759

Feature Article they are mathematically correct. Any other set of feasible constraints will be simple coordinate changes from these. C . Anisotropic Spatial Distributions of the Recoil Fragments. The development so far has relied only on scalar properties of the dissociation dynamics. Although the examples in section Ill have made it clear that scalar properties alone are sufficient to provide important information about the joint probability distribution, it is clear that vector properties can augment the description. For example, the anisotropy of fragment recoil, or its absence, can indicate the time scale of the di~sociation.~’We note here without elaboration that the formalism developed for back-projection can incorporate recoil anisotropy with modest effort. The method of incorporation is most easily seen in the case when the anisotropy does not depend on the energy, Le., when the angular coordinates are separable for the speed coordinates. The basic equation is (A11.6). For excitation of the parent molecule with polarized light, we must include in the integrand a term expressing the distribution of transition dipoles in the excited molecules: p(fl,) = 3 / [ ( 4 r )cos2e,,], where e d is the angle between the z axis (taken to be the polarization axis of the dissociating light) and the transition dipole moment of the parent molecule.32 In order to perform the integration in (AII.6) over d cos 0, d$,, sin 6, sin ( cos we need to recognize that cos ed = cos 6, cos $, where 0, and 4, are the polar angles between the z axis and the vector pa, and is the angle between the transition dipole moment and pa. Integration over d cos 6, d$, reveals that the density is proportional to [ 1 2P2(cos ()P2(cos e,)], where P2(x) is the second Legendre p ~ l y n o m i a l .Thus, ~ ~ for axial recoil, i.e., when the dissociation is rapid compared to parent rotation, the angular distribution for B, will vary as [ 1 pP,(cos e,)], where p = 2P2(cos 5 ) . Similar equations can be derived for the angular distributions of pb and pc. Since the speed distribution does not depend on angle, the density in (ATT.7) is unchanged. If the angular and speed coordinates cannot be separated, then pa will depend on ed. so that this dependence must be included in (A11.6) before the integration. As a result, the density in (A11.7) will also depend on e d . Although the calculation of the prior distribution is then much more complicated, the overall strategy for back-projection is still the same. D. Incorporation of Other Vector Correlations. It is less clear how to incorporate higher vector correlations, such as those between fragment velocity and angular momentum vector^.^^,^^ In the development so far, we have treated degenerate mJ states as being equally populated. Inclusion of these states as separate entities in Q,and P, would provide a method for determining the directional properties of fragment rotation and velocity. However, the prior distribution of Appendix 11, while conserving energy and linear momentum, is not constructed specifically to conserve angular momentum. It is unlikely that the current construction affects the results for any of the molecules discussed above, since angular momentum conservation would not substantially limit the number of states for such heavy particles. Nonetheless, more work is needed to incorporate angular momentum into the back-projection technique in order to be able to describe u-J correlations. For this further development, it should be possible to draw on the substantial progress already been made by Case, McClelland, and Hers~hbach.~~ E . Concerted Dissociations Producing Only Two Fragments. I n a dissociation producing three fragments, at least two bonds must break. The success of the back-projection method in interpreting whether such dissociations are concerted or stepwise leads one to ask whether dissociations breaking two bonds but producing only two fragments can be similarly interpreted. The

+

+

+

~~

~~

(31) Jonah, C. J . Chem. Phys. 1971, 55, 1915-1922. (32) Bersohn. R.; Lin, S. H. Ado. Chem. Phys. 1969. 16. 67. (33) Hall. G . E.: Sivakumar, N.; Houston. P. L.; Burak, I . Phys. Reu. Leu. 1986.56, 1671-1674. Houston, P. L. J . Phys. Chem. 1987, 91, 5388-5397. Hall, G . E.; Sivakumar. N.; Chawla, D.; Houston, P. L.; Burak, I . J . Chem. Phys. 1988, 88, 3682-3691. (34) Dixon, R . N . J . Chem. Phys. 1986, 85, 1866-1879. (35) Case, D. A.: Herschbach, D. R.J . Chem. Phys. 1978,69, 150. Case, D. A.: McClelland. G . M.: Herschbach, D. R . Mol. Phys. 1978, 35, 541-573. McClelland. G . M.; Herschbach, D. R . J . Phys. Chem. 1979. 83, 1445.

“difficulty“ with having only two fragments is that the angular distribution, so helpful for three-fragment dissociations, is trivially simple, and thus less informative: the two fragments always separate in opposite directions on a line. Even without a more informative angular distribution, one can imagine properties of the joint probability distribution that might lead to informative interpretations. An essential feature, we believe, is to let the molecule itself define the time scale on which the degree of “concertedness” is to be measured. In the three-fragment case, the time scale was determined by the rotational motion of the fragment undergoing secondary dissociation. In the two-fragment case, a reasonable molecular time scale might be the redistribution time for vibrational energy. If both bonds breaqk more rapidly than the time for intramolecular relaxation, then the energy in the fragments will be distributed nonstatistically, and the entropy of the joint probability distribution, defined as S = - x P , In ( P J Q , ) ,will be low. It is conceivable that the entropy itself could act as a “concertedness” parameter. However, it should be noted that some stepwise reactions can also take place in a restricted volume of momentum space, though less restricted than that for a concerted process. In addition, non-RRKM behavior might also produce low entropy, so that the intramolecular vibrational redistribution time is not a perfect time scale. Nonetheless, whatever the time scale definition and the “concertedness” parameter, it is clear that the joint probability distribution should contain the information on which to base a decision. The difficulty will be in determining a universal measure. V. Conclusions A general method has been developed for deciding the degree to which a dissociation of the form ABC A B C is concerted. The method determines the best joint probability distribution P of the dynamical variables by using information theoretical methods to determine the distribution P that agrees with the measured projections and that also has the highest entropy. The joint probability distribution is then used to determine the angular distributions between the three fragments. Restricted angular ranges are associated with concerted processes. The method is illustrated by considering experimental data from three molecular systems. The dissociation of C3O2 at 157 nm to produce C + 2CO is found to be nonsynchronous,nearly statistical, and stepwise. The dissociation of acetone at 193 nm to produce C O + 2CH3, was found to be an intermediate case. Although the range of angles is broad, the dissociation cannot be described as stepwise. Dissociation of symmetric tetrazine to produce N 2 + 2HCN proceeds by a concerted mechanism. The joint probability distribution is consistent with a synchronous process, and the angular distributions are strongly peaked. Only a very small region of momentum space is populated. An exact prior distribution for a dissociation producing three fragments is calculated (Appendix 11) and is shown to have a surprisingly simple form. The constraints include conservation of energy and linear momentum, but conservation of angular momentum has not yet been incorporated. It should be noted that the conclusions of the information theoretic approach have been found empirically to be rather insensitive to the exact form of the prior distribution, so that approximations in evaluating densities of states36 can readily be employed for more complex systems. The joint probability distribution can be determined even when the data set is incomplete, and since the method for its determination is simple and rapid, the joint probability distribution should probably be used routinely as a tool for data analysis. It is useful for a n y reaction or dissociation yielding one or more fragments with two or more independent degrees of freedom. Two principal directions for further research are indicated. A more complete description of the joint probability distribution should include higher order vector correlations. It is likely that these can be incorporated by treating the mJ variables explicitly. Applications of the back-projection technique should be made to molecules dissociating via processes in which two bonds break but

-

(36) Forst. W . Chem. Reu. 1971, 71, 339.

+ +

8760

Strauss and Houston

The Journal of Physical Chemistry, Vol. 94, No. 25, 1990

Figure 13. Definition of vectors and angles used in Appendix 111.

only two fragments are produced. The degree to which such a reaction is concerted is likely to be determined from features of the joint probability distribution.

Acknowledgment. This work was supported by the National Science Foundation under Grant CHE-8920404. We gratefully acknowledge Professor R. D. Levine for stimulating conversations concerning this work; X . Zhao, S. H. Kable, K. A. Trentelman, D. B. Moss, and I. Burak for discussions of the experimental data; and Professors B. K . Carpenter and R. Hoffmann for critical readings of the manuscript.

Appendix I . Computational Algorithms for Determining Maximum Probability Although (3)-(5) provide a framework for calculating the probability distribution that best matches the prior distribution and simultaneously agrees with the projections, the solution to these equations is, in fact, transcendental, and the equations themselves do not specify an iterative scheme for correcting the multipliers pi based on the deviation between the predicted projections and the data. We consider here various approaches to this complicated problem. Common techniques for nonlinear optimization such as steepest descent, conjugate gradient, or iterative optimization of variable subsets fail in the face of high-dimensional, rough surfaces. The common tendency of all such methods is to find the nearest local m i n i m ~ m . ~Instead, ’ we employed a hybrid of simulated annealing and a vector potential method. Simulated annealing3’ searches for the global minimum in a way phenomenologically similar to a cooling crystal. Instead of generating a scquence of trial solutions in which the next improves on the last by greedily looking for a steepest descent, the simulated annealing method includes at each iteration a finite chance of accepting a trial solution which is a poorer fit than the previous one. Over many iterations this chance is “cooled” toward zero. Thus the algorithm samples more of the surface, and at “high temperaturc” the variable parameters will be “jostled” out of local minima in favor of a deeper well. Since gradient information is no longer used, each trial solution (a vector of Lagrange parameters) was selected in our implementation by a random perturbation of the last. The algorithm is inherently stable and will converge to virtually the same result regardless of the initial trial solution. The cost for this prescient behavior is prolonged execution time. I n our implementation, the “goodness-of-fit” x function was the entropy itself. An additional feature was that the magnitude of the perturbations was reduced whenever the rate of acceptable new trial solutions fell below 30%. I n this way, the jostling of the parameters cooled in concert with the thermal fluctuation probability. (37) Press, W. H . ; Flannery, B. P.; Teukolski, S. A.; Vetterling, W . T. Numerics/ Recipes: The Arr ofScienrific Compuring Cambridge University Press: Cambridse. UK. 1987.

An alternative scheme used a uector goodness-of-fit formed with elements of the unsquared difference between the observed constraint and the corresponding trial function p r o j e ~ t i o n . ~The ~.~~ length of this vector-the sum of its squared elements-is the ubiquitous “x2”. Reference 38 has pointed out that the special mathematical form of the maximum entropy solution will allow this vector to be written in closed form as the gradient of a concave function. A concave function has but a single local/global minimum, at which the gradient length is necessarily zero and the solution therefore optimized. I n our implementation we find the sole minimum of the constructed concave surface by the Fletcher-Powell method (a greedy minimizing proced~re).~’ The solution is very fast (a 465-element matrix rarely exceeded 2 min of run time on an IBM PS/2 Model 80). Each iteration consisted of calculating a new probability matrix from a line minimization along a conjugate gradient computed functionally from the previous triangle. Problems using this technique occur when the gradient length does not approach zero near the minimum, when the projections are linearly dependent, and when the data are inconsistent. These are discussed in detail elsewhere.40 The two algorithms, the simulated anneal and the concave potential, can be used in concert. The concave potential method accelerates the minimum search by speeding past most of the local minima until the onset of instability caused by nonideal data. Then the simulated anneal takes over. In addition to providing stability, this combination gives us the pragmatic advantage of still using the redundant linearly dependent observations.

Appendix 11. The Prior Distribution for Three Fragments Conserving Momentum The following calculation determines the density in phase space for arranging three fragment atoms with translational energies ea, t b , and tCand zero net momentum. This calculation is algebraically involved but straightforward. The result, however, is somewhat surprising: every combination of ea, tb, and e, permitted by momentum conservation is equally likely-independent of the magnitudes of the energies. We set up the phase-space integral as follows: (a) the spatial coordinates are irrelevant and (b) the momentum space integral is over all nine dimensions of the three particles, (c) the constraints imposed by the conservation of momentum and energy are handled explicitly with Dirac 6 functions rather than implicitly in the limits of integration, and therefore (d) the limits of integration are unrestricted. For three particles with energies ea, tb, and t, with zero net momentum the volume in phase space is proportional to .

. ..

dc#Jadc#Jb dc#Jc6(Ea-ta) 6(Eb-tb) 6(EC-t,) 6@,+,6,+fiC)

(AII.1)

where the energies are defined as Ea pa2/2ma,Eb E Pb2/2mb, and EC pc2/2m,. We define fibc to be the vector sum of fib and fi, and arbitrarily choose to measure o b with respect to the fi, axis (in which case 8, becomes the angle between fib and fi,). From the law of cosines: cos

(e,) = - @bc2

- Pb2 - P c 2 )

(AI 1.2)

2Pbpc

The vector 6 function can be written in polar form: (38) Agmon,

N.;Alhassid, Y.; Levine, R. D.

Maximum Enfropy For-

molism; MIT: Cambridge, MA, 1978: p 207.

(39) Agmon, N.;Alhassid, Y . ;Levine, R . D. J . Comput. Phys. 1979,30, 250. (40) Strauss, C. E. M . Ph.D. Thesis, Cornell University, 1990. (41) Alhassid. Y.; Agmon, N.;Levine, R. D. Chem. Phys. Lerr. 1978, 53. 22. (42) Alhassid, Y.; Agmon, N . ; Levine. R. D. J . Chem. Phys. 1977, 67, 4321. (43) This assumes that angular momentum conservation is not a strong constraint in determining the prior distribution. Although this assumption will often be valid, it must be reconsidered when prior distributions are calculated for nonstatistical m, level distributions.

The Journal of Physical Chemistry, Vol. 94, No. 25, 1990 8761

Feature Article

principle result of this section is that the statistical density of states is uniform in cos x. We begin with the expression for the density of states D(fa,tb.CC) dt, dtb de, where a(@) is understood to be taken mod 2a. Substituting for pa and transforming the coordinates inside the 6 function,44the first term becomes

(AI I .4)

{zero of &function argument) =

cos (6,)

Reassembling these into the original integral and canceling terms reduces the expression to

(AIII.1)

-

+

pa2= ps2 + pra2- 2p,gs cos (x) = 2maEa

Pb2 + P c 2 - Pa2

2Pbpc

I de, deb dec

(subject to the triangle inequality) and change coordinates to a system where momentum conservation is assured: (Ca,tb&c) (pS,pc,x),where the vectors and angle x are defined in Figure 13. Vector pc is the momentum of fragment C in the center-of-mass (COM) system of the three atoms; p s is the momentum of the B (or A) fragment in the B-A COM frame; x is the angle between Ps and P c : pb2 = p: Pr: 2Pr$, COS (X) = 2mbEb

+

where

0:

where m b + ma m b + ma Prb = -Pr. mb ma

(AI 11.2)

= Pc

The Jacobian for the transformation is

Next we transform from pap& dp, dpb dp, to ma dEa mb dEb m, dE,: mambmcl

11

[6(Ea-ta) a(Eb-tb)

a(Ec-~c)

dEa dEb dEC1 x

deal [6(d'a-(d'bc-a)) dd'al [b(cos (eb)-cos (ob?) d COS (e,)] d COS(8,) d& ddc (A11.6)

[s(ea-(r-hc))

Here we have suggestively arranged the integrand into collapsing groups. So far no integral has been evaluated so the range on each of the integrated coordinates remains independent of the others. Since 0, and are, by construction, actual angles, the two 6 functions including those angles will always integrate to unity. (This is a statement that the recoil direction of the b-c center of mass is oppositely directed to that of fragment a.) On the other hand e,', is not a physically constructed angle: the angle will be imaginary wherever the triangle inequality for the momenta is not satisfied. Conversely, it will be real, and the 6 function will integrate to 1 wherever the triangle inequality for momentum holds. The term d cos (0,) d$b dd', provides some constant factors upon integration. Finally, the remainder of the equation is the kernel of an expression for D(ta,tb,t,), the desired expression for the density of states in the energy coordinates:

D ( ~ a , ~ b , ~ o: c)

I

1

if the magnitudes of the momenta implied by the energies satisfy the triangle inequality

0

otherwise (A11.7)

The proportionality factor is determined by the overall degeneracy of the system and normalization. For fragments which are not atoms, the full prior distribution Q ( E : . E : ~ E : , E : , E ~ , E P , E ~is, ~proportional ,~) to the density in phasc spacc for cach triplet of translational energies times the density of internal energy levels. The former is a constant for any triplet conserving momentum, while the latter is simp1 the product of the degeneracies of the internal levels: q:q:q:&fq:.

Appendix 111. The Recoil Angle Distribution Here we proceed backward from the result of Appendix I I to a "starting point" in a more intuitive coordinate system. The (44) Messiah, A . Quanrum Mechanics. Vol. I; Wiley: New York. 1966; Appendix A l l , Properties of the &Function, p 468 ff.

1

p:p:

sin

(x)

(A111.3)

mambmc

where we have used the identity pr, + pr, = p,. Applying this to (Alll.1) yields ~@,,Ps,X)dP, dPs d cos (XI

a

Pc2Ps2dPc dPs d cos (XI (AI I I .4)

We point out that this coordinate system is the natural basis for a stepwise event (in which fragment C splits from fragment B-A first): pc2 is proportional to the energy available from the first bond cleavage while ps2 is proportional to the energy from the second bond cleavage. Nonetheless, it is still an entirely valid, if unsymmetrical, coordinate system to describe any general case of three fragments conserving momentum. The conceptual interpretation of (AIII.4) is that the density is the direct product of the independent degrees of freedom. The quantity pc2 is proportional to the number of ways of arranging the energy between m, and the center of mass of mb ma so that m, receives e,, while p: is proportional to the number of ways of disposing the remaining energy (less that reserved for recoil from m,) between the mb and ma fragments. The cosine dependence arises from the out-of-plane identical rotations of the system. The statistical distribution of angles at fixed energy, in general, will be given by the energy convolution of this density of states with the density of states for other degrees of freedom (internal energy in the case when A, B, and C are not atoms). Since the energy is independent of x, we may abstractly summarize this by multiplying D@,,ps,x) by a probability density P(pc,ps) and integrating over the momenta. However, because expression AlII.4 for the density of states is a separable function in x, the angular diffusion is indifferent to these operations; that is, x is not coupled to (pc,ps)in the dissociation. By inspection this leaves

+

D(x)d cos (x)

0:

1 d cos (x) = sin (x) d x (A111.5)

Appendix IV. Calculation of Prior Distributions A. C 3 0 2 . The prior distribution Q(Ea,Eb,Ec)for C 3 0 2 was calculated by performing a direct count over all pairs of CO internal energy levels in the rigid rotor, harmonic oscillator approximation. I f momentum conservation could be obeyed, then for each pair of internal levels ( E : + E:, + g ) ,subject to E: + Eb I E b and E: + E: I E,, a degenerac corresponding to 4:4&7E was added to Q(Ea,Eb,Ec),where c$ = 2Jb + 1 and qf = 25, + 1 are the degeneracies of the rotational levels and qt = 4: = I are the degeneracies of the vibrational levels. The translational energies of the three fragments E: = Ea, E: = Eb - E: - E:, and E: = EE - Ef were used not only to check that

e

J . Phys. Chem. 1990, 94, 8762-8766

8762

momentum could be conserved for each pair of internal levels (E! + Et, ET + g )but also to calculate cos x using (AIII.2). Thus, at each triplet of Ea, Eb,E,, the calculation provided not only a number proportional to the density of states, but also a distribution of cos x. The continuous range of E,, Eb, and E, between the energies of 0 and E,, = 15 386 cm-l was discretized into 30 bins. After all pairs conserving energy were counted, the distribution Q(Ea,Eb,EC)was normalized. The angular distribution in the bottom panel of Figure 6 was calculated by summing each of the above determined cos x dis), Ptributions weighted by P ( E a , E b , E c ) / Q ( E a , E b , E cwhere (Ea,Eb.EC)is the distribution whose projections agree with the data and which is otherwise of maximum entropy. B . Acetone. The calculation of Q(Ea,Eb,Ec)for acetone proceeded in a manner similar to that for C , 0 2 . The internal-state densities for CO and CH3 were calculated separately up to a total energy of 20 000 cm-' and stored in arrays of dimension 200. For each triplet of internal levels (E: Et, E: Et, E; E t ) , subject to E: E: IEa,E: Eb < Eb, and E; Et IE,, a degeneracy corresponding to q:q:q!~&qt was added to Q(Ea,Eb,EC),where were used as and 9; = (25, q; = 2J, 1, 9: = ( 2 J b the degeneracies of the rotational levels and 4:. qt, and qF are the

+

+

+

+

+

+

+

+

+

degeneracies of the vibrational levels. The square of 25 + 1 was used as the methyl rotational degeneracy to account implicitly for K values. The available energy was taken as E,, = 18 540 cm-I, and the final results were again discretized into 30 bins. C. Symmetric Tetrazine. The calculation of Q(E:,EF,G) for symmetric tetrazine proceeded somewhat differently since the distribution function ranges over a pyramid rather than a triangle. With the use of the rigid rotor, harmonic oscillator approximation, the internal-state densities for N2 and HCN were calculated separately up to a total energy of 40000 cm-I and stored in arrays of dimension 1000. For each triad (E: + Et, E: + Et, E; E;) the product of the three densities was added to each element of Q(E:,EF,EF) that satisfied

+

E: IE,, - E; - E:,

E! 5 E,, - E,b - E:, E: IE,, - E; - E;, and E: 4- E! + Et = E,, - E: - Et - E! - E t - E: - E:

based on a grid of translational energies of dimension 30 between E , = 0 and E, = E,, = 39033 cm-I. Q was then normalized. Registry No. C , 0 2 , 504-64-3; (CH3)*CO,67-64-1 ; tetrazine, 29096-0.

ARTICLES Conformation In 2,3-Difluorobutanes G. Angelhi,*-+E. Gavuzzo,f A. L. Segre,f and M. Speranzas Istituto di Chimica Nucleare del C N R , Area della Ricerca di Roma, C.P. I O , 00016 Monterotondo Stazione, Roma, Italy; Istituto di Strutturistica Chimica del C N R , Area della Ricerca di Roma, C.P. 10, 00016 Monterontodo Stazione, Roma, Italy; and Universitd degli Studi della Tuscia, Viterbo, Italy (Received: January 22, 1990; In Final Form: May 30, 1990)

A conformational analysis for m e m and d,l-2,3-difluorobutanes has been carried out, employing 'Hand "F N M R and theoretical calculations. The real configurations of the two isomers were assigned by specific optical rotation measurements of the products coming from an optically active precursor. Gauche conformations were predominant for meso (E) isomer, while the d.1 (T) isomer showed all possible staggered rotamers almost equally populated. Finally remarks for NMR peak assignment of homoand copolymers partially fluorinated by using gauche additive effects are given.

Introduction

Conformational analysis of haloalkanes was the subject of considerable interest in the past decades.Is2 Most of the studies were carried out by ' H NMR spectroscopy,3-iialthough other techniques, such as IR,I2 microwaves,i2 and electron diffraction s p e c t r o ~ c o p yoften . ~ ~ supported by conformational analysis,I4 and a b initio calculation^^^ were employed. N o difficulties were met with the rationalization of experimental N M R data concerning haloalkanes containing CI, Br, and 1, whereas with those containing fluorine atoms some uncertainty is still present. This is partly due to the spin of I9F, a nucleus that complicates N M R spectra of F alkanes. A further complication arises from the so-called "gauche effect",16 which seems to play an important rolc only in F alkanes and not in the CI, Br, and I homologues.

' lstituto di Chimica Nucleare.

' lstituto di Strutturistica Chimica

9 Universitj.

degli Studi Tuscia

0022-3654/90/2094-8762$02.50/0

Semiempirical potential energy calculations have often been employed on model compounds to predict the configurational ( I ) Mizushima, S. The Structure of Molecules and Internal Rotation; Academic Press: New York, 1954. (2) Lister, D. G.; MacDonald, J . N.; Owen, N.L. Internal Rotation and Incersion; Academic Press: London, 1978. ( 3 ) Emsley, J. W.; Feeney, J.; Sutcliffe, L. H. High Resolution N.M.R. Specfroscopy; Pergamon Press: Oxford, 1967; Vol. 1. ( 4 ) Abraham, R. J.; Pachler, K. G. R. Mol. Phys. 1963, 7 , 165. (5) Kingsbury, C. A.; Best, D. C. J . Org. Chem. 1966, 32, 6. (6) Abraham, R . J.; Cavalli, L.; Pachler, K. G.R. Mol. Phys. 1966, / I , 471. (7) Abraham, R. J. J . Chem. Phys. 1969, 73, 1192. (8) Abraham, R. J.; Gatti, G. J . Chem. Soc., Perkin Trans. 2 1970, 961. (9) Abraham, R. J.; Kemp, R. H. J . Chem. Soc., Perkin Trans. 2 1971, 1240. ( I O ) Phillips, L.; Wray, V. J . Chem. Soc., Perkin Trans. 2 1972, 536. ( I I ) Anderson, J. E.; Doecke, C. W.; Pearson, H. J . Chem. Soc., Perkin Trans. 2 1976, 336. (12) Wilson, E. 8. Chem. SOC.Rec. 1972, I , 293. ( I 3) Clark, A. H.Electron Diffraction Studies and Rotational Isomerism. I n Internal Rotation in Molecules: Orville-Thomas, W. J., Ed.; Wiley: Chichester. 1974: Chapter IO.

0 1990 American Chemical Society