MOLECULAR BEAM SCATTERING * BY J. C. POLANYI Department of Chemistry, University of Toronto, Toronto M5S 1Al Receit-ed 13th June, 1973 Surprisingly, this has been the first general discussion that our Division has held on the topic of molecular beam scattering. In fact, of course, the past days have not been restricted to a discussion of that technique. Molecular beams, when they are not an end to one's means, are a means to an end. Rather than make the molecular beam method our topic, we have centred our discussion on the phenomena that this remarkable technique has made accessible. As a consequence, this meeting has been the intellectual heir to the General Discussion on the " Molecular Dynamics of Chemical Reactions of Gases " held five years ago in Toronto. That Discussion, in its turn, was the successor to the General Discussion on the " Inelastic Collisions of Atoms and Simple Molecules " that took place five years prior to that, in Cambridge.[I have traced our indebted- ness to still earlier Discussions previously and shall have occasion to refer to it again]. The present Discussion has encompassed the topics of these two earlier ones-and more. If one could have said it in a single breath, it would have been entitled a General Discussion on the Elastic, Inelastic and Reactive Scattering of Atoms and Molecules in Gases. Following tradition we began with a session devoted to theory. This has come to take the place of an opening prayer ; a prayer for understanding. We then passed on, rather briskly, from the sublime to the experimental. The experiments were ordered into a hierarchy of ascending complexity. First, elastic scattering in which only translation suffered change, then inelastic scattering in which one form of motion was transposed into another, and finally reactive scattering in which there was transfer not only of motion but also of mass.ELASTIC, INELASTIC AND REACTIVE SCATTERING Let me start by summarizing, very briefly indeed, how things seem to stand in each of the sub-areas into which our discussion has been divided : elastic, inelastic and reactive scattering (see tables 1, 2 and 3, respectively). The fundamental objective in each area is the same, namely to improve our understanding of the interatomic forces. In the case of elastic scattering the inter- atomic potential U(r) (whose slope gives the force) has a form which is now fairly well understood. The constants which govern the potential in its various formulations include E and r,; respectively the depth and location of the long-range attractive potential-well arising from the London dispersion forces.It is encouraging that, at least for pairs of atoms, the magnitudes of these physical quantities are quite weakly " model-dependent ".2s 39 It is only in the realm of elastic scattering that we are able to make this sort of claim. * Summarizing Remarks. 389390 MOLECULAR BEAM SCATTERING Elastic scattering is the most highly-developed of the three fields in which beam scattering studies are made. This is, of course, because the large cross-sections for elastic scattering make the detection of scattered particles easier than for most inelastic or reactive processes. To say that it is easier, is not to say that it is easy.For example it has only recently become possible to obtain sophisticated elastic scattering data for labile collision partners (atomic hydrogen and electronically-excited atomic TABLE 1 .-ELASTIC SCA~ERING Considilrable data. Wide range of collision energies ; 10-2-10 53 eV. Form of U(r) (rot. av’d) known. r, and E only weakly dependent on form of U(r). Know rm to 0.01-0.1 A. Know E to 0.01-0.1 eV. Need studies of anisotropy of A+ BC. Need studies for labile partners. TABLE z.-INELASTIC SCATTERING Few scattering studies ; field opming up. Current approx. data include : T-R, T t , V, T-+Dissocn., Collision-energies 0.1-100 eV.Anisotropy of U(r, 0) important. Aligned studies needed. Configuration-space somewhat more limited than for reaction, hence theoretical problems sometimes less severe. T+R-,V, V-+E,T-+E. TABLE 3.-REACTIVE SCATTERING Numerous studies. Much scope for extension and sophistication. Considerable final-state selection (energies and angles). Increasing initial-state selection (energies and angles). Collision-energies (neutrals) 0.1-10 eV. Less-sophisticated experiments restricted to 02 1 A’. Moresophisticated experiments restricted to 02 10 A’. Qualitative features of hypersurfaces emerging : barrier-locations, regions of max. energy-rise or -fall, potential-wells, surface-crossings. Quantitative features derived, but with questionable uniqueness.mercury 4, ; there is scope for more development of elastic scattering studies in this direction (Otto Stern would have been surprised ; his disciples were doing magnetic moment studies on beams of atomic H and 0 in the late 1920’s. Things get harder once it becomes necessary to scatter one beam off another, in order to define collision energies and scattering angles). A development that is (so far as I know) still awaited, though it is within the reach of present day technology, is elastic scattering in which one of the collision partners is an oriented molecule, for example a dipolar molecule partially aligned in an electric field (cf. ref. ( 5 ) and (6)). The case of molecular hydrogen which we have heardSUMMARIZING REMARKS 39 1 discussed at this meeting is, as the authors note, likely to be exceptional in presenting a very isotropic interaction potential for elastic scattering.' The phenomena encompassed by elastic scattering occur at energies up to and beyond the threshold energy at which inelastic scattering sets in.The study of in- elastic collisions by means of molecular beams is in one sense a well-established tech- nique. It has what in this field is a venerable history, extending back 8 years.' An impressive list of energy-transfer phenomena have been observed (in table 1, T, R, V and E symbolize translational, rotational, vibrational and electronic energies). Nonetheless it seems fair to say that we know rather little in detail about any of these phenomena. Few people have, until now, chosen to work in a field in which even the simpler problems tend to raise experimental difficulties on the level of a rather sophisti- cated reactive-scattering experiment.Inelastic scattering experiments require energy-selection of the collision partners (i.e., some knowledge of T, R, V or E ) as well as energy-selection of the scattered " products ", since the objective is to measure 6T, 6R, 6 V or 6E. To be of real value 6T, 6R, 6 V or 6E should be measured over a range of T, R, Vor E. These difficult experimental problems are now beginning to find solutions. New developments reported in the course of this Discussion include reports of limited but encouraging successes with the measurement of rotational excitation in ion-molecule collisions,* vibrational deactivation in triple-beam experiment^,^^ and also rotational and vibrational excitation in energetic neutral-neutral collisions (the first two results were based on time-of-flight analysis, the last on infra-red emission).The inversion of inelastic scattering data to obtain the form of the potential in the relevant region of configuration-space is a problem that is only beginning to be tack- led.' The degree of success will no doubt be high so long as the data is fragmentary or approximate, i.e., the demands being made on the potential-function are modest. What we want, however, are unique solutions. These we may hope to get when the detail in the experimental data becomes somewhat richer than it is today, or the theoretical input from ab initio calculation becomes a good deal greater.(From the latter standpoint, the case of Li-!- + H2 discussed at this meeting looks particularly promi sing). 8 * 41 The anisotropy of U(r) is unquestionably going to be of importance in inelastic scattering. It will determine the torque that produces rotational excitation. It will also figure in descriptions of vibrational excitation, for the same reasons that it is of importance in determining the cross-section for reactive encounters. (A collision that excites internal motion in a molecule is an abortive reactive encounter. In the exceptional case that the collision partners are incapable of forming a new bond, the abortive reaction can only be coilisional dissociation, A + BC-A + B + C. The importance of angle-of-approach in collisional dissociation has been remarked on at this meeting lo).Clearly it would be most valuable to have inelastic scattering data for aligned collision partners. We can expect such experiments before long. Reactive scattering has become an extremely active field in recent years, justifying the optimistic predictions of its devotees. Though the Millenium is not yet upon us, it is in sight. To a limited extent, and for a limited range of reactions, final-state selection of angles (including polarization of the molecular angular momentum with respect to the velocity of the attacking atom ' I ) and energies (translation, vibration and rotation) has been achieved. The laser-induced fluorescence method described at this meeting l2 is a recent and powerful addition to the tools available for measuring product energy-distribution.It has been used for vibrational distribution measure- ments, and should ultimately also be of value in measuring rotational distributions. Following the previous Discussion in this series,' I had the pleasure of announcing the392 MOLECULAR BEAM SCATTERING betrothal of beam scattering to spectroscopy. I can now read the second banns of marriage. If final-state selection is possible then so also, in principle, is initial-state selection. The difficulty is that at this extreme of exclusivity the flux of products is likely to be too small for measurement. For this reason it is exceptional for initial-state and final- state selection to be combined in reactive scattering experiments. We have heard, nonetheless, how initial-state orientation 5* has been selected, and (for the first time) how this information can be incorporated into a classical trajectory study,5* 6* l3 ie., into the search for an acceptable potential-function.We have also heard of attempts to determine the effect on reaction cross-section and product energy distribution of variation in reagent translation 5 * l4 vibration l4 and r0tati0n.l~ As an example of the extreme difficulty of obtaining good data at this high level of detail, it may be noted that for the well-studied reaction K + CH31, which has a reactive cross-section - 35 A2, the differential cross-section and product translational energy-distribution have so far been obtained over a range of reagent collision-energies from 0.077 to 0.16 eV (a range of slightly less than 2 kcal m~l-~).' The total cross-section, by contrast, has been studied over 10 x the energy range ; 0.1-1.0 eV (with most intriguing results).POTENTIAL FUNCTION The potential-energy function for chemical reaction is a hypersurface. Inversion of the experimental data regarding reaction dynamics to obtain the potential-energy hypersurface is a hazardous procedure. However, it seems fair to say that over the past decade there has, for example in the case of the K + CH31 reaction just referred to, " been a steady growth in the reliability of the potential-energy hypersurface that can be extracted from the experimental data. . . ,'.I3 This has been achieved in the main through a series of attempts to isolate the principal features of the hypersurfaces that govern the dynamics, thus greatly reducing the effective number of variables.A most helpful conceptional aid in grappling with the potential-energy surfaces governing chemical reactions was described by M. G. Evans and M. Polanyi at the General Discussion of the Faraday Society on " Reaction Kinetics ", in 1937. (Both Michael Polanyi, and Sir Eric Rideal, whose far-sighted interest in what were then called " Molecular Rays " led to the first textbook on this are in the audience today). The O.E.P. approach (the 0 was R.A. Ogg, who initiated the method in earlier work with M. Polanyi) represents the collinear potential-energy surface by means of a sequence of 2D cuts (for reaction A+BC these would be U(rAB) and u(rBC)). On the collinear potential-energy surface these cuts would be perpendicular to one another.In the O.E.P. representation they are connected in sequence to form a single plot of U(r), for which the abscissa Y changes sequentially from rAB to rBc, then back to rAB, and finally back to rBc. The corresponding route across the collinear energy-surface is shown at the lower left of fig. 1. Reagents A + BC exist at large rAB (symbolized A-B in the figure) ; products are at large rBC. The O.E.P. route across the surface follows the curve of the minimum- energy path only in a rough qualitative way. Its merit is that it decomposes the potential-energy changes along the minimum-energy path into two-body interaction terms. Inspections of the potential-energy " profile " for the reactive case (bottom of fig. 1, central column) shows that the initial potential rise is ascribed to reagent repulsion R, and the residue of the potential-rise en route to the barrier crest is attri- buted to energy expended in stretching the bond under attack, S.In the descent from the barrier-crest the tension in the new bond is first relaxed with energy release S', and subsequently the products separate releasing the repulsive energy R'. The barrier-SUMMARIZING REMARKS 393 crest is characterized by R + S = R' + S'- Q (where Q is the energy released by the reaction). Ogg, Evans and Polanyi used this approach as a means to the crude calculation of barrier-heights or (more commonly) changes in barrier-height. Our present concerns in the field of reaction dynamics go beyond barrier-height to such questions as barrier- location, and the location of the downward-slope representing the release of the P-E SURFACE P-E PROFILES T V I m A-B Ez s" s, A - B j I A-B+ I 1 1 +A-B B-C+ 7 w I a [I N E LAST IC] cm 4 1 I \ I I 1 I I I ' ; I I 1 I .. . t-A-B$ B-C--+ B-c+ &-B IREACTIVE] ,+ L- B-C+ +A-B T w I a 1 W I CL I , . I +A-B B-C- I 1 I +A-B B-C+ FIG. 1 .-Column 1 compares, on a collinear potential-energy surface, the regions of configuration- space explored by elastic, inelastic (T+ V ) and reactive scattering. Column 2 does the same using O.E.P. energy-profiles. Column 3, second row, exemplifies the essential features for inelastic scattering of the type T-tE. Column 3, bottom row [taken from M. H. Mok and J. C. Polanyi, J. Chem. Phys., 1969, 51, 14511 suggests a simplified representation appropriate to a " typical " reaction (exothermic from left to right, endothermic from right to left).reaction energy. We can use the O.E.P. pictures to advantage in these new contexts. Where O.E.P. attempted to reduce their energy-profiles to still simpler representations we can usefully do the same-though in the light of our present understanding of the " typical " energy-surface we should, I think, choose a different reduced-representation from theirs. I shall have occasion to refer to these points in a little more detail, later.394 MOLECULAR BEAM SCATTERING Fig. 1 makes a comparison between the regions of configuration-space which ,we explore in elastic scattering experiments, inelastic scattering and reactive scattering.The comparison is made both in terms of the collinear potential-energy surface for reaction A+BC-+AB+C (column l), and in terms of the corresponding O.E.P. potential-energy profiles (column 2). The diagrams assume that elastic or inelastic scattering, as the case may be, have been studied both in scattering experiments A + BC and also in experiments on AB + C. The figure is intended to underscore the point already made by the organising committee for this Discussion when they drew up the programme, namely that elastic, inelastic and reactive scattering studies are related-not only in experimental approach, but also at a fundamental level. We can look forward to the time when we have studies of reactive, inelastic and elastic scattering for the same system, in sufficient detail that it becomes possible to splice together the portions of the interaction-potential revealed by the individual phenomena.The reactions of alkali metal atoms with alkali dimers, that we have heard discussed, l 6 could be a candidate for such a three-pronged attack. The reaction may be governed by a potential in which the London dispersion-force attraction passes smoothly into the long-range portion of a shallow potential-well that owes its existence to “ chemical ” forces (i.e., to the delocalisation of charge). With one exception, the potential-energy surfaces and profiles in fig. 1 assume that the process can be understood in terms of nuclear motions in the field of a single electronic state. This is a tempting simplification (though we cannot yet be certain how often it is warranted) since the problem of inversion of the scattering data is already so difficult.The undoubted exception (fig. 1, column 3, row 2) is inelastic scattering in which translational motion of the nuclei is converted to electronic excitation; T-+E. If the quantisation of vibronic states were included, this figure would exhibit a large array of curve crossings. Overlap between adjacent crossing regions (due to the operation of the uncertainty principle) could then complicate the interpretation. Fortunately a single-curve-crossing provides an adequate represent- ation for short interaction times. Multiple crossing of a different sort, due to the involvement of three electronic states, is shown at the right of the T+E schematic drawing (fig.1) : R’-R’*++R’**. This sort of process was postulated over 30 years by K. J. Laidler as an explanation for the quenching of atomic sodium (“P-,~S) in collisions with molecular hydrogen. The intermediate state, R’*, in his work, was an ionic state arising out of the transfer of an electron from the alkali metal, M, to the collision partner, BC. The concept has been developed by Nikitin and co-workers, and shows promise of providing a frame- work within which the new details of T+E transfer in M+AB collisions, presented at this meeting,18 can be understood. The remainder of these remarks will be devoted in large part to comments on the topic of reactive scattering in relation to the present Discussion. The phenomenon of reactive scattering, as fig. 1 indicates, embodies the subject matter of inelastic and elastic scattering. Most often the concepts that are familiar to us in the inelastic and elastic scattering regimes are obscured by the dynamical processes peculiar to reaction. However, this is not always the case.In view of the scope of the present Discussion it is particularly interesting to look for cases in which the simpler phenomena are evident within the more complex one. The O.E.P. representation of the energy-changes in chemical reaction as being R-+S-+S’+R’ (central picture, bottom row of fig. 1) is less familiar than is reduction of this sequence to the rudimentary picture of S-+ R’, i.e., a rise in potential due to the stretching of the bond under attack followed by a fall due to repulsion (not shown in 4 *SUMMARIZING REMARKS 395 fig.1 ; for an example see ref. (1)). An alternative reduction, which seems likely to have more general applicability, is indicated in the lower right-hand comer of fig. 1 : R+S’+R’. In the exothermic direction, reagent repulsion is held to be principally responsible for raising the potential-energy to the barrier-crest value. Thereafter, a portion of the product energy, S’, is released as the new bond relaxes to its normal equilibrium separation, and the remainder, R’, is released as the products separate. The quantities S’ and R’ can be approximately identified with the “ attractive ” and the “ repulsive ” components of the energy release ; sQI and 91. 9 @ Jzr ,-4 7 0 Jd *=4 = 4 O/O = 0 O/O = 96 rABlA FIG. 2.-Three exothermic LEPS potential-energy surfaces (reagents at lower right, proLxts at upper left) for collinear rcaction A + BC-tAB + C.The surfaces are arranged vertically in ascending order of “ attractiveness ”. At the left percent attractive energy-release is given according to the “ per- pendicular ” energy-drop measured on the surface. At the right is recorded the percentage attractive, mixed and repulsive energy-release for the L+ HH mass-combination, obtained by the “ trajectory ” method (a single collinear trajectory). Contour energies are in kcal mol-I ; all three reactions have the same exothermicity. (The symbol t denotes a mass of 1 a.m.u., H E 80 a.m.u.). The barrier crest, designated by X, shifts to “ earlier ” locations along ~ A B as the surfaces become more attractive, according to the general expectation for a chemically-related (“ homologous ”) series of reactions.[Surfaces taken from P. J. Kuntz, E. M. Nemeth, J. C. Polanyi, S. D. Rosner and C. E. Young, J. Chem. Pliys., 1966, 44, 11681. Separation of exothermic energy-release into attractive and repulsive components was suggested by the speculations of M. G. Evans and M. Polanyi regarding the dynamics of the sodium-flame reactions M + X2 and X + M1, presented at the General Discussion of the Faraday Society on Luminescence, in 1938. By the time of the 1967 Discussion on the Dynamics of Chemical Reactions, classical trajectory studies (which began to appear in 1962) had revealed some of the strengths and weaknesses of this method of categorization. There is no doubt that the “ attractive ”-“ repulsive ” antithesis is fundamental and revealing.It even seems worthwhile to employ a quantitative index of “ percent attraction ” and “ percent repulsion ”, %d1 and %91, based on a perpendicular measurement on the collinear energy-surface.396 MOLECULAR BEAM SCATTERING Provided a typical trajectory for the system in question does follow a more-or-less perpendicular path across the energy-surface, %dl will correlate with %(I?$), the mean percentage of the total energy-release that appears as product vibration. Fig. 2 shows three collinear potential-energy surfaces, increasing in %dl from surface @ to @ to @. Fig. 3 shows the correlation between %dl and %(I?$) for these three surfaces, and also for a variety of other surfaces.(The computations which yielded the data of fig. 3 were not restricted to collinear reaction; they were performed in 2D. The values of %(E+) would differ by an insignificant amount for 3D). In every case recorded in fig. 3 the mass-combination was L + HH (i.e., atom A was light, and atoms B and C were heavy) and the energy-surface was of the LEPS variety (London, Eyring, Polanyi, Sat0 surfaces, favouring collinear approach). For L + HH the characteristic path across the potential-energy surface is sufficiently rectilinear to result in a strong correlation between %dI and %(Ek) (the droop in the the curve at high %dl, will be discussed later). I R /’ 20 I I I i 20 40 60 80 100 dl, % FIG. 3.-Mean vibrational excitation in the products of L+HH reaction on 8 potential-energy hyper- surfaces, plotted against the (perpendicular) attractive energy-release.(Fall-off in <E+) at high d~ is due to indirect reaction). [P. J. Kuntz, E. M. Nemeth, J. C. Polanyi, S. D. Rosner and C. E. Young, J. Chern. Phys., 1966,44, 11681. Several L + HH reactions have figured in our discussion at this meeting ; H + C12+ HCl+ C1,I4 CH3 +I,+CH31 + 1,’’ F+ 12+F1 +I,43* 44 and Na+ Cs,-+NaCs + Cs.16 The fact that the second of these reactions can be approximated as L + HH (analogous to H + 12+HI +I) has been noted.20 In going from H + C12 to H +I2 or CH3 +I2, there is good reason to suppose that the preferred angle of approach alters from collinear for H+Cl, to bent for H+12 20* 21 and for CH3+12.19* 2o If this is the case then the definition of %dI, and its utility as a measure of %(&) (or more likely, %(E++El;), where EL is product rotational excitation) needs to be investigated for surfaces involving varied direction of approach. This has not yet been done.The case of the L + HH reaction Na + Cs, also involves special features. In the first place,SUMMARIZING REMARKS 397 the exothermicity is minimal, and secondly the dynamics may be dominated by the location of a potential-well.16 It should be noted that the correlation between %.dl and %(E$) illustrated for L+HH in fig. 2 applies in a qualitatively similar fashion to reactions of other mass- combination, provided that the mass-combination does not alter significantly as %.dl is changed. This has figured in the discussion that we have heard of the family of reactions Ba + HX(H + HL), to which more-detailed reference will be made shortly.A correlation that has come to the fore since the Discussion of 1967 is one that can be seen by inspection of the selection of collinear energy-surfaces shown in fig. 2. In chemically-related series of reactions a lower energy-barrier correlates strongly with an “ earlier ” energy-barrier, which in turn correlates with increased percentage attractive energy-release (the series of surfaces (iJ-)@-)@ shows all three effects). If one makes the plausible assumption that there is (even over a restricted region of rAB) parallelism between the energy-profiles for a related series of reactions, then these correlations can be made graphic. In fig. 4 this has been done for an energy-profile that can be regarded as a smoothed version of the R-+S’-)R’ pattern shown at the lower-right of fig.1. If the parallelism between the profiles for reactions 0 and Q, fig. 4, extends into the exit valley of the energy-surface (i.e., along r,,, to the right of the vertical “ equal bond-extension ” line) then there will in addition be the well-known O.E.P. correlation between increased reaction-energy and diminished barrier-height ; - AE = a- AQ.’ However, this is a weaker correlation, hence it is shown as a likely but not necessary condition in fig. 4. Fig. 4(b) makes the point that cross-section and percent attractive 8 ( b ) ONSET a EARLIER 0 7 GREATER&(%) FIG. +Schematic energy-profiles along the path of minimum energy for chemically-related pairs of reactions.Exothermic reaction proceeds from left to right. Along the first half of the abscissa r- is decreasing (reagents approach), along the second half ~ B C is increasing (products separate). The vertical line marks the point along the reaction coordinate at which new and old bonds are equally extended ; ~ A B - T ~ B = r g c - r i C . In (a) the hypothetical reaction (21 has a lower energy barrier, hence an “ earlier ’’ energy barrier and greater percentage attractive energy release d, than reaction CD. In (b) the pair of reactions have negligible energy barriers, but CD has an earlier onset of attrac- tion and hence greater %.d. [M. H. Mok and J. C. Polanyi, J. Gem. Phys., 1969, 51, 1451).398 MOLECULAR BEAM SCATTERING energy-release can correlate even in the absence of any activation barrier, if the variation in cross-section is simply due to increased range of interaction ; earlier onset of interaction (0 in the fig.4(b)) takes the place of an earlier barrier. This could be a better representation in comparing members of the homologous series M +XR (M = alkali metal atom, X is a halogen atom and R is an alkyl group; m a homologous series either M is varied, or X, or R). These simple amsiderations are of no value in comparing widely differing chemical reactians, such as M+Xz compared with M+XR. Both have large cross-sections but whereas the former has a large %dl) the latter has a relatively small %dl.'* l3 In such cases, as well as in the matter of preferred intermediate configurations, empiricism must be linked to an appreciation of the changing nature of the chemical binding.The application of simple molecular orbital theory constitutes a significant step along that road ; (see ref. (19), (37) and earlier references cited there). The correlation exemplified in fig. 4(a) has been applied in this Discussion to the family of reactions Ba+HX (X = F, C1, Br, I).', As X is varied there is an order- of-magnitude change in total reactive cross-section, 0,. Increased cross-section could be due to decreased barrier-height. This in turn would (in general) imply " earlier " barriers and increased %&I. It is, in fact, found that the %(E;) increases with increasing 0,. * The fact that (E;) is practically invariant with change in the isotopic mass of the central atom conforms to previous experimental and theoretical evidence concerning CI+HI-+ClH+I compared with Cl+DI-+ClD+I, H+Cl,+HCl+Cl compared with D + C12 + DCl + C1, and F + H, + FH + H compared with F + D2+ FD + D.The total reactive cross-section may change somewhat as a consequence of isotopic substitution, but this modest kinematic effect has insignificant consequences for (E;). MASS-COMBINATION Large kinematic effects due to substantial changes in mass-combination are important. A change from the mass-combination L + HH l4 to H + LH(F + HC1 14) or H+HL(Ba+HX 12) results in markedly different reaction dynamics on a given energy-surface. In the latter cases, product vibrational excitation will be greater, and the angular scattering of the molecular product will be more-forward than for L +HH (see below).This is because the repulsive part of the energy-release tends to occur while the heavy attacking atom is still some distance from the atom under attack. This is called " mixed energy release ", corresponding to trajectories that cut the corner of the potential-energy surface. The extreme mass-combinations that appear to favour mixed energy-release are mA rz rn, $ m,, and mA z mB % m, ; i.e., the attacking atom much heavier than either B or C in the molecule under attack. In terms of the skewed and scaled potential-energy surface, the former condition (H + LH 14) corresponds to a strongly skewed surface, and the latter (H+HL 12) to one that has been scaled so that the exit valley is wide compared with the entry valley.The case in which A is heavier than both B and C also favours mixed energy release. In the examples under discussion 12* l4 the repulsion in HL-H or HHoL (the dot indicates the site of the repulsion) tends to force the central atom against the heavy attacking atom, to produce internal excitation in the newly formed bond. The fact that %(E;) is in the surprisingly low range of approximately 10-40 % for the family of reactions Ba+ HX is, therefore, indicative of a highly repulsive potential-energy surface and/or an exceptionally narrow reaction-channel which forces the trajectories into a perpendicular reaction-path (diminishing the " corner-cutting " referred to above). In the reactive system, the separation of C1H.X gives some insight into V-TS U M M A R I Z I N G R E M A R K S 399 mixing for non-reactive scattering HCl +X, under conditions that are intriguing since they correspond to a partially-aligned inelastic scattering event (the products of Cl+HX are most likely to separate as ClHOX, whereas the products of H+ClX l4 separate as HCLX).As always, however, the results from reactive scattering cannot be transposed uncritically to inelastic scattering ; the different initial states for the two phenomena imply that the system is exploring different intermediate configurations. As studies of the individual phenomena reveal more detail, these differences will be- come of central importance. L+ H+ SbRFACE I 8 4 I 3 4 LL LH HL HH BACK FOR&. DECD. REPN. INCD. ATTN. FIG. 5.-Twenty-one computed product angular distributions (obtained from batches of 3D traject- ories) for AB formed in reactions A+ BC-tAB + C on the three LEPS potentialenergy hypersurfaces whose collinear sections are shown in fig. 2.The collision energy was 1 kcal mol-I above the barrier height, corresponding to approximately room-temperature reaction. In each distribution backward scattering is at the left of the graph, forward scattering at the right. For the matrix at the left of the figure, the attacking atom A was light (L I 1 a.m.u.). For the matrix at the right, A was heavy (H = 80 a.m.u.). Only seven mass-combinations are shown ; the eighth, H+ HH, would be identical to the first, L+ LL. [J. C. Polanyi and J. L. Schreiber, in Physical Chemistry-An Advanced Treatise. eds. H. Eyring, W. Jost and D.Henderson (Academic Press, New York, 1973), chap. 9, to be pub- lis hed] . The utility of the simple separation of energy-release into attractive and repulsive contributions extends beyond the correlation with product energy-distribulion docu- mented in fig. 3 to the correlation with product angular distribution shown in fig. 5. The three potential-energy hypersurfaces used to obtain these data were the same as the three shown in collinear section in fig. 2, and designated by arrows in fig. 3. They become increasingly attractive from @-+@+@. It is evident that there is a shift to more-forward scattering of the molecular product as the surfaces become more attractive (less repulsive). As before, the qualitative effect is the same for any particular mass-combination, though on a given surface there is evidence of important kinematic effects as a consequence of changing mass-combination. Mass-combinations that favour mixed energy-release tend to give more-forward scattering.This is because the repulsion that otherwise would propel the new molecule AB back along the direction from which the attacking atom A came, becomes, instead, internal motion of AB. In view of the highly-repulsive surface implied by the low ( E ; ) for Ba+HX, the400 MOLECULAR B E A M SCATTERING scattering in this case is likely to be backward-peaked, despite the fact that the mass- combination is somewhat favourable for forward scattering. The mass-combination for K + ICH, 5 * and for K + ICF3,6 namely L + HL in the three-particle approxima- tion, favours backward-scattering.In these cases the breadth of the backward- peaked distribution provides a useful indicator of the extent to which the energy-release is repulsive. Fig. 5 , on which these comments are based, refers to potential-energy surfaces that favour reaction as a result of roughly collinear approach from the reacting end of the molecule under attack (i.e., the B end of BC). The aligned experiments on K+IR, with R = CH, or CF3 and K approaching from the I end of IR, do show back- ward-peaked scattering. For K+ICH,, approach from the R end to yield KI is disfavoured; this is likely to be the normal case. For K+ICF, rear-approach (from the CF3 end) to form KI represents a reaction path of comparable probability to frontal-approach. (Could this be due to initial formation of K++CFJI-as if the thermoneutral reaction to form KF were about to take place-followed by negative- charge migration to the slowly-moving I ?).Whatever the reason for the facility of reaction from the CF3 end of CFJ, the outcome is of considerable interest. The KI formed by this route is scattered forward. This is simply explained if one supposes that, perhaps due to the aligning field, the CF,-I axis does not rotate to a major extent during the time required for reaction. The initial direction of motion of the attacking atom, K, defines the forward direction. The recoil of I away from CF, (that produced the backward scattering in the reaction from the I end) is, as before, SIDE SCATT. ( FORW. ) BACK SCATT. SCATT. L 4 FIG. 6.-Correlation between direction of approach of atomic species A and angular scattering of the molecular product, AB.The labelling indicates the expected direction of scattering of AB with respect to the direction of approach of A. (Approach of A from the C end of BC to give product AB will be disfavoured in many systems, consequently it is indicated parenthetically). The correlation will be most evident for the masscombination L+ HH-+LH+ H, on a repulsive surface. directed along the CF3-I axis. However, with K approaching from the rear (CF,) end of CF,-I, the recoil velocity of I is directed (approximately) along the continua- tion of the direction of approach of K. For normal " frontal " approach of A to the B end of BC, the reIease of the B-C repulsion gives rise to a momentum in B directed backwards, and sufficiently large to carry B backward. If the encounter is successful in producing AB, then AB must follow B into the backward hemisphere.By contrast the (less common) approach of A from the rear (C) end of BC to form the same product, AB, will, by a strictly parallel argument, give rise to forward scattering. Fig. 6 was drawn for the mass-combination L+HH, since for this mass-combina- tion the molecule under attack has been shown (in trajectory calculations) in general to rotate by only a small amount during the time of a reactive collision. This suggested Forward scattering of KI results. Fig. 6 illustrates this.SUMMARIZING REMARKS 401 to D. R. Herschbach and his associates that, for this particular mass-combination, one can infer the preferred direction of approach of A to BC directly from the observed angular scattering of AB.If this is so, then the (reasonably well-established) poten- tial-energy surface for H + C12+ HCI + C1 that favours collinear approach should be replaced in the case of H+12-,HI+I 20* 21 or CH3+12-+CHsI+I 2o by a surface that favours more-nearly sideways approach (since the latter reactions, as noted above, scatter the product approximately sideways '). Though this interpretation is very likely to be correct, it is worth noting that even in this case, which by virtue of the " light-atom anomaly " is so favourable for the inversion of reactive scattering data to the interaction potential, ((E;) is unusually sensitive to %dJ, the postulate of a non-linear direction-of-approach does not yet offer a unique interpretation of the experimentally-observed sideways-peaked scatter- ing.21 Nonetheless a " unique '' potential (i.e., one having qualitative features that appear uniquely able to explain the observed scattering) should be within reach, in view of the detailed information that we have, or are currently obtaining for the 4 3 5 9 2 I 0' I I ' 0 I 1' ' x emol/deg FIG.7.-Correlation between increased impact parameter b and increased forward-scattering of product AB, in a system having the mass combination H+LH+HL+H. The points refer to trajectories computed in 3D for the reaction C1+ HI+ClH+ I, with reagents Monte-Carlo selected from a 303 K distribution. Reactive encounters with a collision energy T< 1.24 kcal mol-I are symbolized by x 's, T> 1.24 kcal mol-' by 0's.(Stratified sampling increased the number of points below the lower broken line by 5 x , and decreased the number above the upper broken line by 0.5 x ). The curve indicates the dependence of Omol on b for hard-sphere elastic scattering, with a distance of closest approach equal to the sum of the normal equilibrium separations, r&-i+rh, = 2.9 A. [C. A. Parr, J. C. Polanyi and W. H. Wong, J. Chem. Phys., 1973,58, 5).402 MOLECULAR BEAM SCATTERING H + X2 and CH3 + X2 families of reaction. The reaction M + ICH3 is an example of a case where an extensive search for an acceptable potential-energy surface produced a solution which did not appear to admit of further wide variation in the parameters.For certain mass-combinations the kinematics are such that the particles ride, so to speak, rough-shod over the energy surface, only registering a crude impression of its features. The case of H+LH(X+HY where X and Y are halogen^'^) provides an extreme example. Fig. 7 shows the computed product angular scattering as a function of the reagent impact parameter for such a case. Even though the collision energy is only - 1-2 kcal mol-', the momentum of the heavy attacking atom, A, relative to the heavy atom in the molecule under attack, C , dominates the outcome to such an extent that the motion of the central atom (responding to the potential field embodied in the energy-surface) has little effect on the outcome. The product angular distribution is adequately described simply by the elastic scattering of A off C (solid line in fig.7). SIMPLE MODELS Of course, the cases in which one can usefully describe the reactive scattering event in terms of elastic or inelastic (non-reactive) scattering, will be few. A much more PRODUCT- FORCE MODELS A + BC+A--BXC+AB+C (a) I M P @ & J (d DIPR @ M FIG. 8.-Five types of simple " product-force models ". In all five models the outcome of reaction is calculated from the relaxation of A--B*C in its retreat from the activated state. In each case a force (pictured here as repulsive) is assumed to be located between the separating particles, B-C. Var- ious assumptions are made regarding the new bond A--B. Model (a) is the Impulsive model, (6) is the Constant Force model, (c) the Simple Harmonic Force model, (d) the DIPR model (Direct Interaction with Product Repulsion), and (e) the FOTO model (Forced Oscillation in a Tightening Oscillat~r).~~ [Cf.J. C. Polanyi and J. L. Schreiber, in Physical Chemistry-An Advanced Treatise, eds. H. Eyring, W. Jost and D. Henderson (Academic Press, New York, 1973), Chap. 9, to be published].S U M M A R I Z I N G REMARKS 403 important meeting ground between these categories of phenomena will be found in the detailed treatments of reactive scattering (classical, semi-classical or quantum mech- anical 22* 38) that concurrently yield data on elastic, inelastic and reactive events. A still more intimate meeting ground is to be found in the realm of simple models. These tend to be models for elastic or inelastic scattering which have been slightly elaborated so that the mechanics to which they give rise can reasonably be regarded as embodying the central features of at least some types of reactive encounters. The five simple models indicated pictorially in fig.8 are all of this type. Despite their crudity, simple models are important since their " moving parts " are open to in- spection-in contrast to the oftentimes mysterious workings of potential energy hypersurfaces. INCD. COLLN ENERGY AND/OR 0.3 c. 1 3. I - INCD. INTERNAL ENERGY FIG. 9.-Solid lines show three-dimensional DIPR model predictions for product angular distribu- tions (top row) and energy distributions (bottom row ; Eiot is product rotational excitation, Efnt is product vibrational plus rotational excitation).The total product repulsion used in the DIPR model calculations was decreased in four stages, from left to right in the figure. The model therefore simulates the effect of decreased %B?, implying increased %d. In the DIPR formulation this is mathematically equivalent to the effect of increased collision energy (hence the arrow at the top of the figure). The qualitative consequences of these changes are summarized at the foot of the figure. The dots (open or closed) record the corresponding distributions from 3D trajectories on comparable potential-energy hypersurfaces. [Based on P. J. Kuntz, M. H. Mok and J. C. Polanyi, J. Cheni. Phys., 1969, SO, 46231. The models indicated in fig. 8 are all based on the notion that the products are retreating from the barrier-crest ; i.e., the salient features of the energy-profile (cf.fig. 1, lower right-hand corner) are S' and R'. The models are more sophisticated than the O.E.P. representation in fig. 1, however, since they allow for the coizcicrrent release of S' (extension in the new bond, A--B) with R' (repulsion in B C ) . Model (a), the impulsive model, is the most rudimentary. It assumes an instantaneous release of force in B C , and derives the outcome from momentum conservation. Model (6) provides for a constant B C force of finite duration, forcing a simple harmonic oscillator (S.H.O.) that can be initially under tension. Model (c) resembles404 MOLECULAR BEAM SCATTERING (b), but with a linearly decaying (i.e., S.H.) force in BC. Model (d), the DIPR model (an acronym for Direct Interaction with Product Repulsion), assumes that a generalised force produces a known total impulse between atoms B and C.(In the case illustrated, this is an exponentially decaying force). The product translational energy, T', is obtained from the recoil of C, rotational excitation, R', from conserva- tion of angular momentum, and Y' from conservation of energy. The simple DIPR model is able to explain some fundamental correlations, des- cribed above. Foremost among these are the correlations between increased attract- ive energy-release (decreased product repulsion, in the model) and increased product forward scattering, with concurrent increase in the internal excitation, Y' + R'. The predictions of the DIPR model, in both these regards, are recorded in fig.9. It is instructive that so crude a model as the DIPR model is able to simulate these correlations. They are not, it would seem, particularly subtle effects. If we want to explain the effect of mass-combination on the outcome of chemical reaction, or to account for still more subtle phenomena such as the effect of a redistribution of the reagent energy between Tand V, the DIPR model will be inadequate. Models (b) and (c) (the Constant Force, and the S.H. Force models) come closer to reality since they couple A to the repulsive interaction of B and C, from the outset. They are deficient, however, in ignoring the remarkable fact that in the second half of a chemical reaction the B *C repulsion is forcing oscillation into an oscillator that is increasing its char- acteristic frequency and decreasing its equilibrium separation all the while.i.e. it is a bond in the process of being formed. Model (e), the FOTO tries to make good this omission by describing the reactive event as Forced Oscillation in a Tighten- ing Oscillator. The model (at present restricted to the collinear case) accounts quite well for the effect of changes in mass-combination on reaction dynamics, in a manner which lends itself to interpretation in terms of changing attractive, mixed and repulsive energy-release. REAGENT ENERGY Up to this point, these comments have dealt with the characteristics of product energy and angular distributions, their dependence on the nature of the interaction and their dependence on the masses of the reacting species. However, this Discussion has been distinguished by reports of direct (though sometimes crude and limited) measure- ments of the effect on reaction dynamics of changes in reagent energy.These have appeared under various headings. We have heard of the unexpectedly sharp maxi- mum in the total cross-section for the reaction K+ICH,-,KI+CH, at 0.18 eV (4.1 kcal mol-') collision en erg^.^ This feature poses a challenge to the0~y.l~' 35* 36 There has been discussion (both theoretical and experimental) of the intriguing question of the relative effectiveness of reagent translation and vibration, T and V, in promoting exothermic reacti~n,'~ and their relative effectiveness in promoting endo- thermic r e a ~ t i o n . ~ ~ ' ~ ~ Theory and (approximate but direct) experimental evidence seem to agree that reagent translation is likely to be more effective than vibration in promoting exothermic reaction.I4 Theory 2 5 and experiment 2 5 * 26 agree that the converse applies in the case of endothermic reaction ; reagent vibration tends to be more effective than translation.The simplified O.E.P. energy-profile shown in the lower-right-hand corner of fig. 1 (which has already served us well) can be used to render these latter findings plausible. It postulates that A-B repulsion, R, is the major requirement for barrier- crossing in the exothermic direction. It is to be expected that relative motion of A with respect to B (reagent translation) will most readily lead to A.B. In the reverse, endothermic, direction there is, in addition to a requirement for repulsion R' in B*C, aSUMMARIZING REMARKS 405 W I n very significant energy-rise labelled S’.This corresponds to stretching of the bond being broken in the endothermic reaction, A--B. The potential-rise S’ can be achieved most readily if that portion of the reagent energy for endothermic reaction is placed directly into AB, i.e., if a substantial part of the reagent energy is present as vibration in the bond under attack. (This rationale implies that the characteristic energy- profile for endothermic reaction resembles that in the exothermic direction, at least qualitatively. This is a good approximation for endothermic reagent energies not greatly in excess of the endothermic barrier height 25). TA I 1’ W I a. L I I 0--0--0 B-C- t - A - 8 FIG.10.-These energy profiles record only the diference between the energy changes for thermal reaction and energy changes appropriate to the case of enhanced reagent translational energy (AT, in excess of the minimum energy required to cross the barrier, Ec) or enhanced reagent vibrational energy (A V> Ec). The activated state is indicated by i. For ATit is compressed, for A Vit is most often extended (the dynamics depend on the initial phase of vibration). In either case += is at a potential energy substantially in excess of Ec. There has also been discussion for exothermic reactions of the effect of reagent- energy enhancement, AT or AY in excess of the barrier-height, on the product energy di~tributi0n.l~ Experiment and theory are in general accord that AT-+AT’+ AR’ (enhanced product translation and rotation), and A V+A Y’ (enhanced product vibrational excitation). This “ adiabaticity ” with respect to additional energy in406 MOLECULAR BEAM SCATTERING excess of the barrier-height has been traced to the fact that AT gives rise to a more compressed intermediate configuration than would otherwise be obtained, whereas AY gives rise (most often) to a more-extended intermediate configuration.The enhanced product translation in the former case and enhanced product vibration in the latter case can, accordingly, be understood as being due to a measure of " induced repulsive energy-release " arising from AT, and " induced attractive energy-release " arising from AV. Fig. 10 makes use of the O.E.P. type of representatiOn to clarify this.All that is recorded in the figure is the dipereme in energycprofile in the presence of the enhanced reagent energy, as compared with that in the absence of this additional energy. For AT this difference takes the form of enhanced R and R' ; for A V it takes the form of enhanced S and S'. (Fig. 10 of ref. (14) shows the corresponding changes as they would appear on a collinear energy-surface). A well-known effect of enhanced reagent translational energy is to increase the percentage of forward scattering of the molecular product. On a previous occasion I suggested that one might usefully define a " stripping threshold energy ", T,, which could be the collision-energy above which >90 % of the molecular product is scat- tered into the forward hemisphere.In part, as we have seen, the value of T, will depend on the reactant mass-combination; in part it will depend on the extent of interaction between atom B and C.' If we can make an allowance for the former effect, then we can use T, to obtain a measure of the interaction between B and C. In the common case that B and C repel, we would be obtaining a measure of the repulsive energy release, 9. Of course there are other clues to the magnitude of 9 than T,. The suggestion that T' be used as the index of 9 was based on the notion that it would eventually be experimentally feasible to vary T for many reactions, and then to codify these reactions uniformly in terms of their T,. The time when we can do this experimentally has not yet arrived.We can, however, exemplify the value of such a procedure from a theoretical standpoint, out of the present Discussion. In an ambitious 6-atom trajectory study of Rb+ICH, l 3 the collision energy was increased from 0.025 to 4 eV. The result was that the scattering of the RbI product shifted from backward to sideways to forward. The stripping threshold energy has a value T,>5 eV. This is surely connected with the fact that the bCH3 repulsion is substantial. For the DIPR model the effect of increased T is to decrease the time during which the B-C force operates, i.e., to decrease the repulsive impulse. As a result, increased col- lision energy is mathematically equivalent to decreased repulsion, as indicated in the arrow over fig. 9. (The model also predicts the gross changes in product energy distribution as the spectator-stripping limit is approached.It fails, however, to take any account of such subtle effects as " induced repulsive energy-release ', since there is no mechanism for momentum of A to become enhanced B-C repulsion). If some simple form is chosen for the repulsive force as a function of time, F(t), then the value of T, can be used to obtain the parameters governing F(t), and hence the repulsive impulse at any other collision energy. This would be a still more useful exercise in terms of models such as (b), (c) and (e) of fig. 8, which make some allowance for the role played by the relative masses of A, B and C, as distinct from the form of the potential-energy function. Simple models can make the relationship between T, and 9 explicit." I N D I RE C T " R E A C T I 0 N The account given here of various types of interaction, and their consequences for reaction dynamics, has up to this point been incomplete in one important regard ; no mention has been made of the possibility that reaction products instead of separatingSUMMARIZING REMARKS 407 cleanly might suffer secondary, tertiary, and further, encounters. I have spoken of reaction, throughout, as if it were a “ direct ” process. This is evident in the five models pictured in fig. 8. The trajectory calculations on potential-energy hyper- surfaces do, however, allow for the possibility of direct or indirect encounters.’ Looking back at the evidence for a positive correlation between attractive energy release and product vibrational excitation obtained from the trajectory calculations (recorded in fig.3) it is clear that at high %d1 the mean vibrational excitation %(E+) falls. Inspection of the trajectories shows that this is due to the effect of indirect reaction. There is too little product-repulsion to separate the products before oscillation in the new bond can give rise to secondary encounters. Incipient vibration in the new bond is, therefore, dissipated, in the course of secondary en- counters, into product translation and rotation ; (&) starts to diminish toward its statistical value. This Discussion has seen a considerable flowering of interest and information regarding the difficult domain of indirect (complex) enc~unters.~~ The results are generally characterized by an outcome that corresponds to the statistical expectation in some regards, and to deterministic behaviour in others.The extensive studies of unimolecular decomposition in crossed molecular beams exemplify this nicely.’’ The ratios of cross-sections for alternative pairs of products (e.g., +C4H8+ C4H8F,*tt-+C3HSF + CH3- or +C4H7F + He) is statistical, whereas the translational- energy distribution in the products is not. The simplest interpretation is that the choice of reaction path is determined in a statistical region of strong mixing (e.g., in C4H8F*++), but that the product energy-distribution is determined by specific forces operative in the exit channels. These simple explanations (along with those advanced throughout the present paper) should be treated with reserve, in view of the fact that there are often contrary, no less convincing, explanations to account for precisely the opposite type of behav- iour. Thus, in the course of a wide-ranging study of four-centre exchange reactions 2 8 it has been shown that the reaction CsCl+KI+CsI+KCI proceeds through a long- lived collision complex (many vibrations, - 1 rotation).The angular and velocity distributions of the scattered molecules are consistent with a simple statistical model, resembling the RRKM (Rice, Ramsperger, Kassel, Marcus) theory for unimolecular decay. In this case, however, the property most directly obtained from KRKM theory, the ratio of nonreactive to reactive decay, was found to be 2-3 times larger than the statistical prediction.There may, of course, be no conflict at all between this example and the previous one. There is reason to suppose that the 4-atom intermediates MXM’X’ have preferred atomic arrangements. 28 Trajectory computa- tions (exceptionally difficult to perform for these long-lived species) confirm that with the involvement of two forms of intermediate dimer, the non-statistical yield of products is e~plicable.~~ The (largely attractive) forces operating between the products may turn out to be consistent with statistical angular and energy distri- butions ; this remains to be established. NEW THEORETICAL APPROACHES Theoretical discussion at this meeting began with a consideration of the funda- mental practical question ; the choice of optimal theoretical approaches from among the considerable range of scattering treatments available.22 Discussion then focused on the semi-classical method, which has great attractions in the treatment of phen- omena for which the purely classical method is definitely unsati~factory.~~~ 47 Such408 MOLECULAR BEAM SCATTERING phenomena are in the minority, but this may not always be the case.For example, with increasing energy-resolution it will become possible to explore partial and total reactive cross-sections in their threshold-energy regions. The form of these functions in the threshold region can be markedly affected by quantum restrictions on the products, and by tunneling processes (whose probabilities are too low to be of import- ance well above threshold). The semi-classical m e t h ~ d , ~ * ~ ~ * 48 which uses data from exact classical trajectory calculations to evaluate the quantities required for a quantum mechanical description, has been shown in a number of cases to describe quantum effects accurately.Since its point of departure is a classical trajectory calculation, there is no restriction on the potential-energy hypersurface that can be treated by this means. In the present Discussion we have heard how this method can be applied selectively to only those degrees of freedom that are strongly quantised, treating the remainder classically,30 and we have also seen how the method can be applied to the type of reactive encounters that span the gap between " direct " and statistical reaction. Looking into the future at the close of our previous Discussion I said that I would be surprised if ab initio calculations of potential-energy surfaces did not figure sig- nificantly at our next meeting.I can now report that ab initio surfaces have not figured significantly, and that I am surprised. The brightest hope that they offered us in the course of this Discussion was in connection with the inelastic scattering experiments on Li++H2.8 For this case information is coming available on the extent to which the ab initio surface is able to account for the experimental finding^.^' The difficulty of making ab initio calculations to a chemical accuracy over a sufficient range of configurations to be of value for prediction, hardly needs to be restated. Though it has not been to the fore in the present Discussion, very sub- stantial progress has in fact been made in the field of ab initio computation in recent years.The importance of this line of endeavour perhaps does need to be stressed. It will provide an extremely valuable standard against which our future empirical and semi-empirical efforts can be measured. The information-theoretic approach to reaction dynamics represents an intriguing new departure.24 The outcome of a dynamical process is considered in relation to the statistical outcome. The more the former deviates from the latter the greater the " surprisal ". If, for example, a reaction deposits a fraction f$ of its energy into vibration, with a probability P(f;), and if the corresponding statistical expectation for a closed system (a microcanonical ensemble) were Po(f+), then the surprisal would be defined as Z(f;) = -log[P(f;)/P*(f;)].The surprisal is therefore a local measure (for a given V, in the example cited) of the deviation of the observed probability from the statistical expectation. In some cases, the surprisal is found to vary linearly with the energy-mode to which it refers. In these cases an extreme " compaction " of data is achievable ; a matrix of - 100 detailed rate constants k(u', J') (rates into specified quantum states of vibration and rotation obtained from an infra-red chemi- luminescence study of a single reaction) can be represented in terms of two numbers ; 1". and &, respectively the slopes dI(f$)/df+ and dZ(fd)/df{. At present no reason is known why the surprisal should vary linearly with energy (corresponding to a monotonous logarithmic change in P/P0).339 34 However, there are already enough cases in which the variation of the surprisal with energy is simpler than the variation of the reaction outcome with energy (the raw experimental data), to make this new vantage point a noteworthy one.It would seem to be particularly interesting to apply the approach to reactions which involve indirect (complex) encounters. For these reactions, the statistical quantity Po is the natural datum against which to measure the observed outcome, P.SUMMARIZING REMARKS 409 A great deal has now been begun in the field of molecular dynamics. A little has been concluded. On the days that we incline to modesty we may say of our work, as Francis Bacon did of his, that it " seemeth .. . not much better than that noise or sound which musicians make while they are tuning their instruments; which is nothing pleasant to hear, but yet is a cause why the music is sweeter afterwards . . .". In truth that is quite a brave assessment, since it implies a faith that some considerable orchestration will follow. In the light of the discussion we have heard at this meeting (dissonant at times, but we are dealing with modern music) this appears to be a very reasonable expectation. ' J. C. Polanyi, Disc. Furuday SOC., 1967, 44, 293. D. D. Fitts and M. L. Law, this Discussion, p. 179. R. W. Bickes, B. Lantzsch, J. P. Toennies and K. Walaschewski, this Discussion, p. 167. T. A. Davidson, M. A. D. Fluendy and K. P. Lawley, this Discussion, p. 158. R. B. Bernstein and A. M. Rulis, this Discussion, p. 126. P. R. Brooks, this Discussion, p. 299. A. Kuppermann, R. J. Gordon and M. J. Coggiola, this Discussion, p. 145. H. E. van den Bergh, M. Faubel and J. P. Toennies, this Discussion, p. 203. A. M. G. Ding and J. C. Polanyi, this Discussion, p. 225. lo Y. T. Lee, this Discussion, based on work of F. P. Tully, H. Haberland and Y. T. Lee. l 1 D. S. Y. Hsu and D. R. Herschbach, this Discussion, p. 116. l 2 H. W. Cruse, P. J. Dagdigian and R. N. Zare, this Discussion, p. 277. l 3 D. L. Bunker and E. A. Goring-Simpson, this Discussion, p. 93. l4 A. M. G. Ding, L. J. Kirsch, D. S. Perry, J. C. Polanyi and J. L. Schreiber, this Discussion, p. 252. ' R. G. J. Fraser, Molecular Rays, ed. E. K. Rideal (Cambridge University Press, London, 193 1). l6 J. C. Whitehead and R. Grice, this Discussion, p. 320. M. S. Child, this Discussion, p. 30. E. Gersing, H. Pauly, E. Schadlich and M. Vonderschen, this Discussion, p. 211. l 9 C. F. Carter, M. R. Levy and R. Grice, this Discussion, p. 357. 2o J. C. Polanyi and J. L. Schreiber, this Discussion, p. 372. '' J. D. McDonald, this Discussion, p. 372. 22 R. G. Gordon, this Discussion, p. 22. 23 M. D. Pattengill and J. C. Polanyi, this Discussion, p. 63. 24 R. D. Levine and R. B. Bernstein, this Discussion, p. 100. 2 5 D. S. Perry, J. C. Polanyi and C. Woodrow Wilson Jr., this Discussion, p. 127. 26 D. J. Douglas, J. C. Polanyi and J. J. Sloan, this Discussion, p. 310. 27 J. M. Parson, K. Shobatake, Y. T. Lee and S. A. Rice, this Discussion, p. 344. 28 D. L. King and D. R. Herschbach, this Discussion, p. 331. 29 P. Brumer and M. Karplus, this Discussion, p. 80. 30 W. H. Miller and A. W. Raczkowski, this Discussion, p. 45. 31 R. A. Marcus, this Discussion, p. 34. 32 J. N. L. Connor, this Discussion, p. 51. 33 J. C. Polanyi, J. L. Schreiber and J. J. Sloan, this Discussion, p. 124. 34 R. D. Levine, this Discussion, p. 125. 35 R. B. Bernstein, R. A. LaBudde, P. J. Kuntz and R. D. Levine, this Discussion, p. 120. 36 R. M. Harris and D. R. Herschbach, this Discussion, p. 121. 37 D. A. Dixon, D. L. King and D. R. Herschbach, this Discussion, p. 375. 38 G. G. Balint-Kurti and B. R. Johnson, this Discussion, p. 59. 39 U. Buck, H. 0. Hoppe, F. Huisken and H. Pauly, this Discussion, p. 185. 40 M. S. Chou, F. F. Crim and G. A. Fisk, work described by D. L. King, H. J. Loesch and D. R. Herschbach, this Discussion, p. 222. 41 G. M. Kendall and J. P. Toennies, this Discussion, p. 227. 42 A. Ben-Shaul, this Discussion, p. 307. 43 C. F. Carter, M. R. Levy, K. B. Woodall and R. Grice, this Discussion, p. 381, 385. 44 Y. C. Wong and Y. T. Lee, this Discussion, p. 383. 45 Comments in this Discussion, p. 113-1 16, p. 377-381. 46 W. H. Miller, this Discussion, p, 119. 47 G. D. Barg, H. Fremery and J. P. Toennies, this Discussion, p. 59. 48 Comments in this Discussion, p. 68-79, p. 119-120.