GENERAL DISCUSSIONProf. M. Karplus (Harvard University) said: In the original formulation of thequasiclassical trajectory method for calculating cross-sections for exchange reactionsof the type A+BC+AB+C, the impact parameter b was selected from a randomdistribution between 0 and b,,,, the latter value being chosen so that the reactionprobability P, is zero for bzb,,,. On the other hand, the initial internal stateof the molecule BC was selected from the quantum-mechanically allowed valuesof the angular momentum and the rotation-vibration energy. We have now followedthe suggestion of Marcus and explored the effect of selecting b in accordance withthe quantum-mechanically allowed values of the orbital angular momentum L,given the initial relative velocity VR.When b is selected at random for a fixed V,, the calculated values of P, for thereaction H+H2+H2+H can be fitted to a good approximation by the simpletwo-parameter functionTreating b as a continuous variable and using eqn. (1) for P,(E), one may obtainthe reaction cross-section for the given value of VR by the simple integration(" ' ) a bi.bP,(b)db = 4 -Alternatively, restriction of impact parameter to the quantum-mechanical valuesof L [i.e., bl = (h/pVR){2(2+l))l/2, or b, = (h/pVR)(2+l/2), where 2 = 0, 1, 2, .. .]TABLE NUMERICAL COMPARISON OF EQN. (2) and (3).S4eqn. (311a VRa J 'rn 'ma, 'Aeqn. (2)1 L = [r(l+ l)l+h L = [i++)h1.2 0 0.39 1.85 11 1-940 a.u. 1.95 a.u. 1.945 a.u.0.95 0 0*13& 0.95 4 0.171 a.u. 0.178 a.u. 0.175 a.u.0.87 0 0.025 0.375 2 0.0051 a.u.0.0068 a.u. 0.0057 a.u.Q, velocity in units of 0.979 x lo6 cmlsec.b, this value is a correction to table 3 of ref. (1).and use of the quantum-mechanical degeneracy factor (22+ 1) together with eqn.(1) for P,(bi), gives S, as the sumNumerical comparisons of the reaction cross-section obtained from eqn. (2) and(3) at three values of V' are givenin table 1. Restriction of the impact parameterto quantum-mechanically allowed values does not produce a significant change inS, even for small b, values and velocities near threshold. Fig. 1 presents a graphicalrepresentation of eqn. (2) and (3) for values for the parameters a and b,, whichcorrespond to V' = 0.95 in units of 0.979 x lo6 cm sec-l. The initial molecularangular momentum is zero in each case.The area under the solid curve in theM. Karplus, R. N. Porter and R. D. Sharma, J. Chem. Physics, 1965, 43, 3259.7GENERAL DISCUSSION 77figure is S, as given by eqn. (2); the area under the step function is S, as given byeqn. (3).The results reported here show that quantization of the impact parameter,particularly according to the semi-classical relationship L = pVRb = (1+1/2)h,0.3 c0 0.2 0.4 0.6 0-8 1.0b(a.u.)FIG. 1.-Probability of reaction Pr against impact parameter b. The curve corresponds to thecalculated probability and the bar graph to its approximation in eqn. (3).has a negligible effect on the reaction cross section for the H+H, system. Asto the quantization of the spatial orientation, which was also suggested by Marcus,we have made no quantitative tests ; however, qualitative considerations suggestthat the effect would be very small.Dr.D. A. Micha (University of Calgornia, Saiz Diego) said: One of the presentdifficulties in the quantum mechanical description of direct molecular reactionsis to aaccunt for the effect of inelastic processes (rotational and vibrational excitations)on the reaction cross-sections. A possible way of dealing with these effects wouldbe to use optical potentials. A formal exact theory of rearrangement collisionsin terms of optical potentials has been known for some time and we have thoughtit of interest to discuss its application to molecular reactions of the type A+BC+AB+C. The scattering wavefunction Yi+) for a collision where BC is in the stateva is given bywhere E(+) = E+k, E+O, CD, describe the system in the absence of the initial channelinteraction 0, and K , is given by 2 = IC,+O,.An optical (complex) potential u i + )for the initial channel is defined bywith = (y, I Y$+))q,. The real part of this potential describes the effect ofdistortion of the relative motion and of the internal motion of BC on the elasticscattering, and its imaginary part accounts for the effect of inelastic and reactivetransitions.Y:') = @,+(E'+'- tiJ-'o,Yi+),y:;) = c"D +(E'+'-li,)-'u~+'y~,+',It may be written explicitly asd+) = (7, I vnE+) I I?,>,M. H. Mittleman, Physic. Rev., 1961, 122, 1930; 1962, 126, 373.M. L. Goldberger and K. M. Watson, ColZision Tiwory (John Wley & Sons, Inc., New York,1964), chap.1178 GENERAL DISCUSSIONwhereSimilarly, a potential u r ) describes the elastic scattering of C colliding with ABin state yrs. Using these potentials the reaction scattering matrix takes the formwhere the symbol (-) indicates incoming waves and = U , - V $ + ) . This exactexpression is a useful starting point for the introduction of approximations. In areaction which proceeds through a direct mechanism it would be enough to keepthe first term in TB,, since the second term corresponds to reactions which proceedthrough multiple inelastic and rearrangement collisions. For direct reactionsIf, furthermore, inelastic and reactive collisions do not affect the initial and finalchannels, the optical potentials may be approximated by their lowest order in theinteractions,and similarly II~-)cz$, in which case Y:;) and Y($ reduce to the distorted wave-functions xi+) and xb-), and Tpa is given by the distorted wave Born approximationNevertheless, for reactions where the nature of the interaction or the range of relativekinetic energies studied lead to large probabilities of inelastic transitions it will bebetter to use T&), which takes into account those inelastic processes.Prof. M.Karplus (Harvard University) said: To provide a quantitative test oftransition state theory and its underlying assumptions, K. Morokuma and I havebeen examining the initial, transition, and final-state energy, and the phase-spacedistributions of appropriately selected classical trajectories for H, H2 reactivecollisions.Since this analysis is pertinent to the question of vibrational adiabaticity, we reportsome results for the simplest case of linear collisions.There are two degrees offreedom, which can be associated with the molecular vibration and relative atom-molecule co-ordinates in the initial state and in the final state and with the symmetricand asymmetric stretching vibration in the transition state. In fig. 1, (a) and (b),we plot the energy in the symmetric vibration at the saddle-point against the initiallypresent in molecular vibration. Each dot represents one reactive trajectory selectedfrom a set with fixed initial translational energy and a Boltzmann distribution ofvibrational energies (T = 900°K).The line corresponds to the energy partitioningpredicted by the adiabatic hypothesis. Significant deviations from adiabaticityoccur, and these are in the expected direction : if the initial translational energy isbelow the classical barrier (0.396 eV) the reactive trajectories have less than the adiabaticenergy in the symmetric mode; if the initial translational energy is higher than thebarrier, the symmetric mode energy is above the adiabatic value. An alternativecomparison can be made for a system with fixed initial vibrational energy (6.2 kcal,the zero-point energy of H2) and a Boltzmann distribution of translational energies(T = 900°K). Here the average symmetric mode energy of reactive trajectoriesis 2.3 kcal as compared with the adiabatic value of 3.1 kcal.Thus, more energyis available for crossing the barrier (i.e., in the asymmetric stretching mode) than issuggested by the adiabatic model. However, the final state molecular vibrationaGENERAL DISCUSSION 79energy in all cases is close to the initial state energy. Thus, the reaction is vibration-ally adiabatic in terms of initial and final-state variables. This implies that for theH, potential, which is symmetric about the saddle-point, the redistribution of energythat occurs on the incoming part of the trajectory is approximately counterbalancedon the outgoing portion.ER = 0.48 eVTvib = 900°Krc.04-rFIG. 1.-Relation between vibrational energy in the symmetric mode at the saddle-point and theinitial molecular vibrational energy (a) initial translational energy = 0.48 eV, (b) initial translationalenergy = 0.33 eV.The initial vibrational energy is selected from a Boltzmann distribution at 900°K.The solid line corresponds to the adiabatic result.Prof. R. A. Marcus (University of I h o i s ) said: I should like to indicate somespecific examples of the kinetic energy expression, based on natural collision co-ordinates, to illustrate the method of solution. For reaction in a plane one find80 GENERAL DISCUSSIONapart from several terms which are small at both small and large I s I. (Here,pR2 is I in my paper and f is cos(c- ~) there.) Evidently, p4 is a constant of themotion. Introduction of an adiabatic approximation for the r and y motions isrelatively straightforward.Non-adiabatic corrections can be calculated using amethod similar to that used in ref. (1) of the paper.For a reaction in space the “low mass” approximation is readily introduced asfollows. (For brevity of presentation the classical orbital plane, i.e., the 6, $-plane,which is constant in this zeroth order approximation, is taken to occur at 6 = rc/2).Apart from terms which are small at both small and large I s I one findsAgain, p4 is a constant and an adiabatic approximation is readily introduced foradiabatic-separation of the r- and the (y,X)-motions. The solution of the y,x-equation needs further analysis. It may involve methods similar to those used torelate, i.e., the asymmetric top to the symmetric top behaviour. A detailed dis-cussion of these equations and of approximate solutions will be given in a futurepublicat ion.Prof.R. Wolfgang (University o j Colorado) said: This comment regards theapplicability and usefulness of statistical or “ phase-space ” theory in treating directreactions, i.e., those occurring on the time scale of one molecular rotation or less.Perhaps the best single experimental criterion as to whether a reaction is direct, isthe angular distribution of products. If this is symmetric about the centre of mass,a long-lived intermediate is indicated. If it is asymmetric, the mechanism mustbe partly or wholly direct in that the products have retained a memory of where theycame from. This in turn means that the strong coupling assumption, which requiresthat decomposition of the intermediate complex be uncorrelated with its formation(except through the conservation laws) does not hold.Light acknowledges that statistical theory will hold best for reactions wherethere is a complex It is not clear to me that it can be useful at all for reactions,such as KfHBr, which have been shown to be direct by their angular distribution.It may be argued that even in these cases, fluctuation in product properties withsmall changes in reaction conditions (such as energy and impact parameter) willprovide some averaging over possible states in phase space.This may well be true.But it is also true that in almost any conceivable direct reaction certain large volumesof phase space are excluded or disproportionately populated.What these volumeswill be one cannot say a priori unless the mechanism of the reaction is alreadycompletely understood. Thus while it may be possible to fit the theory to a directreaction, it would seem that the parameters so fitted would have little physical signifi-cance. If the statistical theory is indeed applicable only to reactions in which along-lived complex decomposes independently of its mode of formation, it mightthen be appropriate to regard it as a theory of unimolecular decay.Prof. J . C. Light (University of Chicago) said: I agree with the basic premise ofWolfgang’s comment. If a particular reaction is known from experiment to yielda very asymmetric distribution of products in the centre of mass system, then nostrong coupling complex in the sense of the statistical theory is formed.In thiscase agreement between averaged quantities calculated from the statistical theoryand those found from experiment must be regarded as largely fortuitous. It caGENERAL DISCUSSION 81happen only if the numbers of states actually populated in each channel are pro-portional, on the average, to the total numbers of states (available) in each channelas calculated in the statistical theory. The reaction K+HBr may, however, beone in which a " direct " reaction yields a symmetric product distribution Inthis case the averaging over unobserved initial and final states may yield good agree-ment between quantities observed and calculated from the statistical theory (see alsocalculations for this reaction in ref.(25)).Finally there are two basic differences between the statistical theory and thenormal (RRKM) theory of unimolecular reactions. In the statistical theory aneffort is made to avoid parameterizing the theory in terms of unobservable molecularproperties of initial and final states. Secondly, the statistical theory includes theeffects of the total angular momentum of the complex. This often has a largeeffect on the mode of decomposition as well as the internal energy distributions ofthe products. Thus, although the statistical theory would be applicable to unimole-cular decomposition (particularly if alternative channels are available) it is differentfrom the RRKM theory.Prof. R. D. Levine (University of Wisconsin) said : The formalism of the statisticalapproximation is usually described in terms of a model where the colliding moleculesform a complex which then breaks down in a statistical fashion.It is important,however, to realize that this model is not necessary in order to derive the formalism,but is just a possible physical interpretation of the formal expressions obtained.that a possible parametrization of the average inelastic cross-section from channela to channel p , isUsing the concept of an average S matrix which we denote (S) one can showwherepa = 1- I ( S a a ) I 2*This proposed paramerization is independent of any specific model, and reducesto the Hauser-Feshbach formula for nuclear collisions and to the identical eqn.(1 1) of Light's paper when we adopt a collision complex model.In our derivationeqn. (1) does not hold for p = a, but<n-'k%a,a> = I 1 - (saor> I +PaPa/Cpp. (2)BTo evaluate Pa one needs to know (S). Bernstein, Dalgarno, Massey and Perciva14have shown that for strongly coupled channels it is reasonable to assume thatwhere R is an arbitrary orthogonal matrix. Taking R = -I it follows that(Saa> = 0 and Pa = 1. We shall define the statistical approximation by eqn. (3).With this definition eqn. (1) is equivalent to Light's theory. In this case = N,where N is the number of strongly coupled channels. (Only channels with thesame values of all good quantum numbers can be strongly coupled.)(S) = <RS) (3)Bsee C. Riley, K. T. Gillen and R.B. Bernstein, WIS-TCZ-256X, (24 July 1967).R. D. Levine, Quantum Mechanics of Molecular Rate Processes (Oxford University Press,Oxford, 1968), 9 3.51.W. Hauser and H. Feshbach, Physic. Rev., 1952, 87, 366.R. B. Bernstein, A. Dalgarno, H. S. W. Massey and I. C. Percival, Proc. Roy. Sue. A , 1963,274, 42782 GENERAL DISCUSSIONThere are two immediate experimental consequences of this statistical approxi-random-phase mation. (i) For N = 1, eqn. (2) reduces to the Massey-Mohrapproximationwhich predicts correctly the average elastic cross-section. (ii) Irrespective of thevalue of N, the total average cross-section obtained by summing eqn. (1) over p $: aand adding eqn. (2) satisfiesThus the total cross-section (for a given initial channel) has the random phase value,irrespective of the occurrence of inelastic collisions.2 The collision complex theoriesseem to omit the first contribution to eqn.(2) and thus obtain the erroneous valueof unity for eqn. (4) and (5).Eqn. (3) also implies( ~ i - kzc,,,) = 2, (4)(rc-'k:o,> = 2. ( 5 )< I sba i 2> = 1 / ~ * (6)Recent exact (close coupling) computations of I Spa J2 for strong coupling seemto confirm eqn. (6), in that the diagonal element can not be distinguished from otherelements of the same column.The point is that neither eqn. (1) nor eqn. (3) depend on the physical conceptof a collision complex, although they can be interpreted in terms of such a physicalmodel. Moreover, eqn. (I) is independent of eqn. (3). Eqn.(3) simply providesa computational prescription for the Pa. Further discussion will be publishedelse~here.~ It is a pleasure to acknowledge the benefit of many discussions withProf. R. B. Bernsteirz and Dr. W. A. Lester, Jr.Prof. J. C. Light (University of Chicago) said: Although the average S matrixapproach is pleasing because of its brevity and generality, one must still have aprescription for choosing the sub-set of strongly coupled channels from the entireset of channels allowed by the conservation laws. The value, if any, of a physicalprescription of a strong coupling complex lies in the ease with which physical intuitioncan be used to define the sub-set of strongly coupled channels. The alternativeof choosing the set of strongly coupled channels a priori and letting this choicedefine the " complex " seenis more difficult physically.Dr.J. Troe and Prof. H. Gg. Wagner (University of Giittingen) said: The statementof Hoare and Thiele, that the detailed balance condition prevents any simpleMarkovian theory from becoming very different from a simple Kassel theory forthe specific rate constants k(E), seems to be very important, because it is not easy tocheck this assumption experimentally. This is due to at least two reasons. (a) Theaccuracy of measured rate constants in the '' fall-off " region of unimolecular reactionsseems to be not yet sufficient to check deviations of the specific rate constants k(E)from those given by Kassel's expression, because different models of k(E) can resultin similar shapes of " fall-off " curves.However, from independent measurementsof k(E) it could be shown that Kassels description is essentially correct. (6) TheH. S . W. Massey and C . B. 0. Mohr, Proc. Roy. SOC. A , 1934,144, 183.R. B. Bernstein, A. Dalgarno, H. S . W. Massey and I. C. Percival, Proc. Roy. SOC. A , 1963,274, 427.W. A. Lester, Jr., and R. B. Bernstein, private communication, July 1967; see also Ckrern.Physics Letters, 1967, 1, 207.R. D. Levine, Quantum Mechanics of Molecular Rate Processes (Oxford University Press,Oxford, 1968), 3 3.51GENERAL DISCUSSION 83“ fall-off ” curves are effected by lifetime distributions (e.g., assumption (ii)).However, they are also influenced by intermolecular energy exchange processes(e.g., assumption (iii)).When interpreting experimental results on low pressurelimiting rates particularly of small molecules, it became evident,l that a much betterfitting of the data could be obtained, if assumption (iii) was replaced by a “ ladder-climbing ” model. Because of this additional complication the separation of theinfluences of lifetime distributions and intermolecular energy exchange processesin the “ fall-off ” region becomes very complicated. This would still hold, if onecould obtain very accurate experimental data in the transition region between low-and high-pressure limiting rates.Dr. E . R. Buckle (Imperial College, Londorz) said: I have been trying to devise akinetic theory of clustering in monatomic gases which is based on Kassel’s treatmentof unimolecular reactions.The ultimate aim is to provide an alternative to Volmer’squasi-thermodynamic treatment of condensation. The models I use for unimoleculardecay and bimolecular growth of clusters are necessarily crude, as they are intendedto apply at various stages of growth from the monomeric state. For example,cohesion is attributed to pairwise additive dispersion forces. The equilibrium distri-butions in argon are in rough qualitative agreement with the shape of Bentley’smass spectrometric curve for clusters up to the 22-mer in C02.2 The dimer molefraction at p = 0.44 atm and T = u/k deg is about 1 x in fair agreement withthe more exact calculations of Stogryn and Hir~chfelder,~ but while in both casesthe same value for u, the pair energy, is used their concentration falls less rapidlyat higher T.My model of the dimerization reaction closely resembles the three-atom radical-molecule mechanism described by P ~ r t e r . ~One of the simplifications of the cluster theory is to ignore, at all sizes, possibleeffects due to the distribution of angular momentum. In view of assumption (b)in Light’s paper it would be interesting to know in what way the results of ratecalculations using a theory which omits this assumption, but which otherwise agreeswith his definition (a) of a strong coupling theory, would be expected to differfrom those using a theory which includes assumption (b).With regard to the uniform distribution Q(EJ) found by Hoare and Thiele forthe energy of the critical oscillator after collision with another oscillator, are theyconfidient that detailed balancing can be applied without reservation to their hypo-thetical mechanism? They offer no physical description of a “collision ”, and Ifind it hard to think of reversibility in terms of normal vibrations.Dr.M . R. Hoare (Bedford College, London) said: In reply to Buckle, even if onehas doubts about the ultimate validity of the detailed balance condition, one cannotlegitimately raise them in the context of our theory. We are constructing a stochasticmodel which is supposed to mirror as closely as possible more detailed mechanicalprocesses for which microscopic reversibility is reasonably well established ; its useor not is hardly at our disposal.And in constructing valid transition kernels thecondition is virtually unavoidable. Although detailed balance (eqn. (6)) andnormalization (eqn. (10)) are sufficient rather than necessary conditions for passageto equilibrium (eqn. (11)) it is difficult to find kernels which lead to equilibrium,J. Troe and H. Gg. Wagner in Receirt Advances in Aerotherinochernistry (AGARD, Oslo1966) and Bc7r. Bunsenges. Physik. Chenz., 1967, in press.P. G. Bentley, Nature, 1961, 190, 432.D. E. Stogryn and J. 0. Hirschfklder, J. CJzeni. Physics, 1959, 31, 1531. ‘ G. Porter, Disc. Faraduy SOC., 1962, 33, 19884 GENERAL DISCUSSIONi.e., have an eigenvalue unity and the appropriate distribution as its eigenfunctionwithout this being due to the “ built-in ” property, (6).Dr.M. R. Hoare (Bedford College, London) said: The prospect of applying noisetheory to unimolecular reactions must have tempted many people and it is a pitythat the original work of Kramers has never been followed up, I admit to havingspent time searching for buried treasure in the electrical and mechanical engineeringliterature, without much success. Unfortunately mechanical engineers are interestedmainly in dissipative systems, usually with distributed mass, while electrical engineerslive in a linear universe admitting only the whitest of white noise.1 agree that the study of the driven harmonic oscillator will surely throw lighton the unirnolecular problem, but Kac would admit that the situation here is differentfrom the idealized one that he is treating.For example, in the molecule the drivingspectrum would not be white-there would be a high-frequency cut-off, peaks aroundthe normal-mode frequencies, plus side-bands, sub-harmonics etc. Even a Debyespectrum would be better than white noise. There is little evidence from computerexperiments that energy transfer occurs by a Brownian-motion-type mechanism.Thus, one loses the simplicity of the Kramers equation. Furthermore, one shouldreally be driving an anharmonic oscillator. Finally, there is the question of energyconservation, which would normally be violated in a driven system? To extractan energy-dependent rate constant the molecule would have to be large enoughfor heat-bath-like behaviour to occur. In this respect our model seems to be astep in the right direction. Because of the rather artificial energy-description weare able to “ drive ” a single oscillator with s- 1 other oscillators at constant totalenergy E.Actually, for s> -10, we can let the heat-bath become infinite, use acanonical ensemble formulation with D = s/E and obtain virtually indentical results.Altogether, in view of the heat-bath idea it is surprising how little attention is paid inconventional unimolecular theories to special results valid for large s.As to doubts about treating lifetimes in terms of a well-defined g(E,E, . . . etc.)we tend to agree. This is roughly what we had in mind in our concluding remarksabout the importance of “ hidden variables ” behind the energy-description.Prof.W. C. Gardiner (University of Texas) said : Kupperinan, Stevenson andO’Keefe find that their relative energy distribution function is determined by theelastic scattering cross-section and is essentially independent of the assumed reactivecross-section. Since the inelastic cross-sections corresponding to rotational andvibrational excitation of the H2 heat-bath molecules by the hot D atoms shouldbe comparable to the reactive cross-sections, it would seem that the lack of dependenceupon reactive cross-section implies a lack of dependence upon the inelastic non-reactive cross-section. Is this correct ?Prof. R. N. Porter (University of Arkansas) said: We have treated the generalproblem of reactions of hot atoms in thermal media by a method which makes useof the concept of integral reaction probabi1ity.l This is defined as the probabilityA,@) that an atom whose initial laboratory energy is E will be removed from thesystem by reaction i upon some future collision with component c.The integralequation satisfied by A&) has the simple forGENERAL DISCUSSION 85where Fc(E) is the collision fraction for the cth component, pi,(E) is the probabilityof reaction i per collision with component c, and the kernel K(E,E') is determinedby the total and differential cross-sections for the various possible collision processes.For a single reaction with a single reactive component, eqn. (1) takes the formA(E) = p(E) + [ 1 - p(E)]/ co P(E,E')A(E')dE', (2)0where P(E,E') is the normalized scattering kernel for a single non-reactive collision.It has been shown that eqn.(2) is mathematically equivalent to the integral equationfor the collision density n(E,E') viz.,n(E,E') = 6(E - E') + [l - p(E")~n(E,E'f)P(E'',Ef)dE", (3) sa combined with the Miller-Dodson equation,A(E) = /En(E,Ef)p(E')dE'. 0 (4)Recently, we have explored the compatibility of eqn. (3) and (4) with the steady-stateformalism which Kuppermann has employed and a similar formalism used by Kostinet al.3 Our formalism in terms of probabilities is equivalent to the steady-stateBoltzmann formalism, provided the probability for a given scattering process isdefined as the steady-state rate of the process divided by the total steady-state collisionrate.TABLE 1.A(E)eqn. (2) eqn. (3) and (4) E(eW0.6 0*01131 0.01 1290.9 0.05600 0.055961.0 0 07481 0.07477By way of numerical illustration, we compare in table 1 the results for A(E)calculated from eqn.(2) with those calculated from the steady-state Boltzmann for-malism [i.e., eqn. (3) and (4)]. The reactive and non-reactive cross-sections whichwere assumed for the calculations both have realistic energy dependence for thesystem D+H2. The small disagreement in the results is within the round-off andtruncation error for the trapezoidal integration routine. The computation timerequired for solving eqn. (2) to give A(E) at 0.001 eV energy intervals over the range0-6<E< 1.0 eV was slightly less than that required to obtain A(E) at one energyvalue by eqn. (3) and (4).Besides the ease of computation of A(E), an advantage offered by the formalismof eqn.(2) is that the equation is readily written in the formA(@-/ P(E,E')A(E')dE'1 - 1 E , P(E,E')A(E')dE'E' m =R. N. Porter, J. Chem. Physics, 1966, 45, 2284.J. M. Miller and R. W. Dodson, J . Chem. Physics, 1950, 18, 865.M. D. Kostin, J . Appl. Physics, 1965,36,850 ; R. M. Felder and M. D. Kostin, J. Chem. Physics,1965, 43, 3082 ; D. M. Chapin and M. D. Kostin, J . Chem. Physics, 1967, 46,250686 GENERAL DISCUSSIONThus, when experimental energy-dependent yields and differential non-react ivecross-sections are known, a simple integration gives the reaction probability percollision, a quantity closely related to the reaction cross-section.l A more completeand rigorous discussion will be published elsewhere.The stimulation of several discussions with Professor A.Kuppermann and thefinancial assistance of Public Health Service Research Grant no. GM 13253 aregratefully acknowledged.Prof. R. Wolfgang (University of Colorado) said : Theories of hot atom thermaliza-tion, such as that given by Kuppermann and that of Porter,2 and Kostin and Felder,3can potentially be useful in extracting information on the energy dependence ofreaction. However, although their formalism is complete, their present applicabilityseems limited by our ignorance of energy loss processes in non-reactive collisions.Such data are critical for any theory of hot atoms moderation. At low velocitiesand for simple systems (such as T+H,), energy loss appears to occur largely inelastic collision and can therefore be reasonably well estimated.But at higherenergies ( - 5 eV) hot-atom studies have shown that, with methane and more complexmolecules, energy degradation occurs largely in highly inelastic collision^.^ Suchcollisions may be expected to be velocity dependent, and are difficult to evaluatetheoretically. The consequent limitation on the usefulness of theories of hot-atommoderation emphasizes the need for detailed experimental information on inelasticinteractions in the range of a few electron volts and higher.Prof. I. Amdur (Massachusetts Iiwtitute of Technology) (communicated) : Thequestion has been raised as to the magnitude of non-reactive, inelastic scatteringin comparison with elastic scattering for systems such as those treated by Kuppermann.A.L. Smith and M. C. Fowler 5 * have recently completed experiments in ourlaboratory which bear directly on this question. They have measured total cross-sections for scattering of fast He atoms through laboratory angles greater than 0.1deg. by H2, D2, and HD at room temperature. Although the fast atoms had kineticenergies in the range 300-1900eV, it is the maximum energy lost by them inexperiencing their small-angle deflections, not their initial relative kinetic energy,which is available for possible transfer to internal energy of the H2 isotopes. Thisenergy loss which, for elastic scattering, is the potential energy of the system at thedistance of closest approach, is in the range 0.25-1-35 eV. This range is almost thesame as the range of initial laboratory energies of D atoms considered by Kuppermannfor the reaction D+H2 = DH+H.The results of our scattering experiments may be summarized as follows.(i) Theexperimental cross-sections for He + H2 and He + D, are identical over the entireenergy range, as are the average potential energies of interaction calculated on theassumption that the scattering is completely elastic. (ii). The measured He + H2(and He + D2) cross-sections are in excellent agreement with orientation-averagedcross-sections calculated from a theoretical angular dependent He + H2 potentialR. N. Porter, J. Chem. Physics,'1966, 45, 2284.R. N. Porter, J. Chem. Physics, 1966,45, 2284.M.D. Kostin, J . Appl. Physics, 1965,36,850 ; R. M. Felder and M. D. Kostin, J . Chem. Physics,1965,43, 3082 ; D. M. Chapin and M. D. Kostin, J. Chem. Physics, 1967,46,2506.J. W. Root and F. S. Rowland, J. Chem. Physics, 1963,38,2030; 1967,46,4299 ; R. Wolfgang,J. Chem. Physics, 1963, 39, 2983 ; A. H. Rosenberg and R. Wolfgang, J. Chem. Physics, 1964,41, 2166 ; D. Seewald and R. Wolfgang, J. Chem. Physics, 1967, 47, 151.A. L. Smith, Ph.D. Thesis, (M.I.T., 1965).M. C. Fqwler, Ph.D. Thesis, (M.I.T., 1967)GENERAL DISCUSSION 87obtained by Mies and Krauss. In calculating the cross-sections from the HefH,potential it was assumed that no inelastic scattering occurred. (iii) Several differentmethods for calculating cross-sections for rotational excitation for the He + M22nd He+D2 systems shown that these inelastic cross-sections are less than 0.5 %of the corresponding elastic cross-sections.(iv) The measured cross-sections forHe + HD are consistently higher than the measured cross-sections for He + H,or HefD,. The difference ranges from about 1 % at the lowest beam energy toabout 3 % at the highest. Because of the displacement of the centre of mass in HD,the He+HD has a larger anisotropic component than the He+H, or He+D2potentials. However, orientation-averaged cross-sections for He + HD calculatedon the assumption that all scattering is elastic are indistinguishable from the corres-ponding He+H2 or He+D2 cross-sections. On the other hand, inclusion in thecalculation of the effects of rotational excitation and de-excitation of HD by Heaccounts for most of the observed enhancement in cross-sections.Estimates of theeffects of vibrational excitation suggest that these are much smaller than the rotationaleffects.In summary, scattering of He by hydrogen isotopes indicates that for H2 or D2inelastic effects are negligible, and for HD, the detectable scattering resulting fromrotational excitation is of small magnitude, about 2 %, on the average. Kuppermann’sneglect of non-reactive inelastic scattering appears to be well justified.Prof. R. A. Marcus (University of Illinois) said : The vibrational energy resultsof Karplus (p. 78) are extremely interesting and can be used to test the near-adiabaticexpressions for the instantaneous vibrational energy of a reacting system, eqn.(33and 35, in ref. (1). The latter contain adiabatic, statistical adiabatic and non-adiabaticterms and, under certain conditions, display just the behaviour observed in fig. laas well as that in the “ annealing process” (an approximate return of the finalvibrational energy to the adiabatic value after undergoing statistical adiabaticand non-adiabatic effects).For example, at low initial vibrational energy E: the envelope of points in fig.la is seen to be a parabola about an axis parallel to but displaced upwards from adiagonally-drawn adiabatic line. This vertical shift, according to (35), is the non-adiabatic term and the scatter about the shifted-adiabatic line reflects the statistical-adiabatic term.When only the sine Fourier transform in (35) and (33) need beconsidered one finds that (i) the vibrational energy at the end of reaction shouldequal the adiabatic value ; (ii) the vibrational energy at s+ should equal (El+I)-2(E3* sin 6, where E% is the adiabatic value at s#, I is the square of the sine transform,and 6 is the initial vibrational phase angle. Both conclusions are modified somewhatwhen the cosine transform cannot be neglected.Item (i) agrees with the findings of Karplus. From item (ii) the shifted adiabaticline is seen to be E, = E,”+I and the envelope to be displaced 2(E;1)4 from it, inreasonable agreement with fig. la.3 A more detailed analysis of the computerR. A . Marcus, J. Clzem. Physics, 1966, 45, 4500. Altnough (35) was derived for the case ofi = O at s = s#, one can show that it remains applicable for # 0, provided w is replaced by w2.Also, as a first approximation Ireplaced x in the last term of (32) by xo.Note that the function in the transform is, like a sine term, small atE.g., the value of I at EI: = 0 is the intercept of the shifted-adiabatic line on the &-axis, andis seen from fig.l a to be about 0.04 eV. The value of 2(E3)* calculated from it at Et = 0,0.05, and 0-10 eV, is 0, 0.06 and 0.09 eV, respectively, while the observed vertical distancefrom the envelope to shifted adiabatic line in fig. la is 0. 0.055, and 0.085 eV, respectively.(Strictly speaking, the I varies with Ei and one should use the formula in item (ii), computingeach I.)= s#88 GENERAL DISCUSSIONresults, together with an Q priori calculation of the cosine and sine transforms usingthe method in ref.(l), will permit a more detailed testing of (33) and (35).At very low initial vibrational energies (<O.l25 eV) the data in fig. lb are belowthe adiabatic threshold. At these and at the higher Ei as well, the upper envelopeof points in fig. 16 corresponds to the maximum allowable by energy conservationunder these conditions.' Below this maximum, but at the higher Ei one couldagain test the equation in item (ii) above.Dr. D. A. Micha (University of California, San Diego) said: I would like tocomment briefly on the range of energies investigated in the contribution by Karplusand Tang. The distorted wave approximation for rearrangement collisions doesnot account for the coupling between inelastic and reactive scattering channels.For H and H2 colliding with relative kinetic energies above about 0.5 eV, vibrationalexcitations from ZI = 0 to ZI = 1, 2, .. . become energetically possible and bothrotational excitation and vibrational excitation cross-sections are comparablein magnitude to the reaction cross-section. This indicates that it would be important,to improve the quantum mechanical results at energies above 0.5 eV and to comparethem with classical calculations, to go beyond the distorted wave approximation byincluding the effects of vibrational and rotational excitations at those high energies.Prof. R. A. Marcus (University of ZZZinois) said: It is of interest to compare theresults of Karplus and Tang with those of a statistical-dynamical theory of reactioncross- section^.^" (The classical version of the latter and the classical mechanicalcomputer results agreed well in the energy region of interest, without introductionof adjustable parameter^.^') When the reaction co-ordinate is treated classically,but the vibrations of the activated complex are treated in a quantum manner, oneobtains a step-like cross-section against energy curve, instead of the smooth classicalone.The classical curve passes about mid-way through the ~tep-risers.~" Thestep-like nature of the new curve arises from the increasing accessibility of quantizedbending states of the activated complex with increasing energy and is closely relatedto the cusps mentioned by Child in his paper.When the reaction co-ordinateis treated quantum mechanically, there is a rounding-off at the foot of each stepbecause of tunnelling and a rounding-off at the top of each step because of theusual quantum mechanical reflection for motion just above the top of a barrier.4cAny non-adiabaticity of the bending modes also has a rounding-off effect.According to these arguments the quantum curve should be fairly close to theclassical one, except at threshold. Comparison with fig. 7 of Karplus and Tangshows that in both cases the threshold energy of the quantum case exceeds that ofthe classical but that their curve is appreciably below ours at the energies of interestin thermal reaction, a factor of 5 below, perhaps.At high energies, fig. 7 displaysthe usual " blow-up " of the distorted wave approximation.With the usual co-ordinates, the distorted wave approach at low energies givesa chemical reaction cross-section which is too low perhaps because of the difficultyE.g., the barrier in Karplus and Tang's paper is 0.396 eV, and so the initial energy defect is0.396-0-333 or 0.063 eV. The upper envelope in fig. lb is seen, in fact, to correspond ratherclosely to (E:-0*063) eV. The corresponding limit in fig. l a is (Ei+048-0*396) eV, whichis above the observed E,f at low Ei, but which may be a limiting factor at higher Et.A. C . Allison and A. Dalgarno, Proc. Physic. SOC., 1967, 90, 609.D. Rapp and T. E. Sharp, J . Chem. Physics, 1963, 38,2641(a) R.A. Marcus, J. Chem. Physics, 1966, 43, 2630 ; (b) J. Chem. Physics, 1967, 46, 959 ;(c) eqn. (19), (26), (29) and (31) in ref. (la) and eqn. ( 5 ) of ref. (lb) are used.M. Karplus, R. N. Porter and R. D. Sharma, J. Chem. Physics, 1965,43, 3259GENERAL DISCUSSION 89of choosing a wave function giving good overlap of the reactants and the productscontributions. Perhaps a calculation based on the natural collision co-ordinateswill offer an easier opportunity far getting the correct wavefunction.Dr. J. L. J. Rosenfeld (“ Shell ” Research Ltd. Chester) said: Hurle’s data onhydrogen atom recombination at temperatures between 2800 and 7000°K havebeen re-analyzed in detail, Fig. 1 shows the new results for the rate constant ofthe three-atom recombination reactionH + H + H -+HZ + H.The mean line through Hurle’s data and the extreme upper and lower envelopes(indicating approximately the scatter in the experimental results) are shown, togetherwith the results of Sutton,2 Patch,3 Rink,4 and Jacobs et aZ.j Patch and Rinkassumed a temperature-dependence of T-l in their analysis, and their results cantherefore only be used as confirmation of the magnitude of the rate constant atthese temperatures..\BENNETT and’-. -~B>ACKMORE15.0 .\. -. ‘.\ -0 1000 2000 3OOO 4000 5000 6000 7000temperature T, O KFIG. 1 .-Temperature-dependence of the rate constant of the three-atom recombination reaction,Both Sutton’s and Hurle’s results indicate a steep (P4 to P6) temperature-dependence at high temperatures, with a gradual reduction in the slope between4000 and 3000°K.There are indications of a maximum at about 3000°K. Thesefeatures gain significance in the light of the estimate at 300°K (also shown in fig. 1)by Bennett and Blackmore of an upper limit for the constant much lower thanthe shock-tube values at 3000°K. Other things being equal, one might expecthydrogen atoms to be about as efficient as Ar or H2 molecules in acting as energyH+ H+ H +H2 + H.’ I. R. Hurle, 11 th Int. Symp. Combustion (The Combustion Institute, Pittsburgh, 1964), p. 827.E. A. Sutton, J. Chern. Physics, 1962, 36, 2923.R. W. Patch, J. Chern. Physics., 1962, 36, 1919.J. P. Rink, J. Chern. Physics, 1962, 36, 262.T. A. Jacobs, R. R. Giedt and N. Cohen, 1.Chern. Physics, 1967,47, 54.J. E. Bennett and D. R. Blackmore, private communication (to be published)90 GENERAL DISCUSSIONsinks in the recombination. The rate constant kHZ for recombination with H2as third body is also shown in fig. 1 for comparison. These results taken togethersuggest the existence of a " resonance " in the three-atom rate constant due to somemechanism other than straightforward collisional stabilization. Stabilization byexchange is a likely possibility. If the vibrational mode is assumed to be adiabaticduring exchange then one may expect the activation energy E, for the exchangereaction to be independent of the degree of vibrational excitation. The detailedmechanism is then as follows.We imagine two hydrogen atoms, H(l) and H(2), to collide to form an orbitingpair.Nearly all the bond energy goes into the vibrational mode, the remainder,together with the relative kinetic energy Ety goes into the rotational mode. Whena third hydrogen atom, H(3), approaches, H(2) leaves with an excess kinetic energyequal to the activation energy, leaving the H2(l, 3) molecule with that much lessrotational energy. Thus,H(1) + H(2)+H;JH";(l,2)+H(3)-+H';J;(1,3)+H(2).The rotational energies areEj = E,,+D,; EjP = Ej-E,,where D, is the dissociation energy from the vth vibrational level.The products are stabilized only if EJ.cD,, and hence only collisions betweenH(l) and H(2) with E,,<E, must be counted. This leads to an effective three-bodyrecombination rate constant k,, wherecm6 sec-'.(I)Here [Hg*] and [HI are the concentrations of the orbiting pair and free atomsrespectively, and A exp (- E,/RT) equals the rate constant for the exchange reaction.When E, = 7-5 kcal mole-' (the experimental value when v = 0), the temperature-dependent part of k, goes through a maximum at 3000"K, in agreement with theobservations. Although the decrease of this function alone at higher temperaturesis much too slow, one might expect the ratio [H;*]/[Hl2 to diminish rapidly withincreasing temperature, as the effective potential gets shallower at higher energies.Thus it might still be possible to reconcile eqn. (1) with the experiments.To reproduce the observed magnitude of the rate constant at 3000"K, a valueof about lo3 c1n3 mole-' must be assumed for [H;*]/[Hl2.Under the experimentalconditions (typically 50 % Ar, 50 % H2, 10 dissociation at 3000°K and 1 atm)CH1-3 x mole ~ m - ~ , so that [H;*] N lo-* mole ~ m - ~ or one three-hundredthof the free atom concentration. This is physically acceptable in view of the approxi-mate nature of the calculations. It would seem therefore that exchange stabilizationwith vibrational adiabaticity may explain the observed " resonance " in three-atomrecombination rate constant.Prof. R. A. Marcus (University of Illinois) said : The results of Child are interesting,and some comparison with our related study is appropriate. The orbital-rotationalpart of the kinetic energy based on the natural collision co-ordinate system describedJ.C. Polanyi, Atomic and Molecular Processes, ed. D. R. Bates, (Academic Press Inc., NewYork, 1692), p. 807GENERAL DISCUSSION 91in my paper is similar to Child’s outside of the saddle-point region. Near thatregion, however, I was able to choose the co-ordinates so that the orbital-bendingkinetic energy cross-term became small, as it is for a typical molecule. (My co-ordinates pass snioothly from those in fig. l a of Child’s paper to those in hisfig. l b as s varies from - 03 to + m.) On this basis the details of Child’s calculationat the saddle-point are open to question, but this question does not effect theseat fairly negative or fairly positive s. Publication of a more detailed comparisonis planned.Child’s calculation of a rate constant which agrees with activated complex theoryis an interesting concrete example of a more general, formal result in the literature,viz., a vibrationally-adiabatic derivation of activated complex theory.Prof. M . Karplus (Harvard University) said : Child has presented an interestingtwo-dimensional calculation of the H + H2 exchange reaction which predicts thatthe product H2 molecule is formed in the same rotational state as the reactant molecule.This result is surprising, particularly in view of the rate of ortho-H, to para-H,conversion, which must involve a rotational state change if it proceeds via the exchangereaction. Moreover, molecular rotation, in contrast to vibration, is predicted tobe significantly non-adiabatic on the basis of an approximate quantum mechanicaltreatment and exact quasi-classical calculations on the potential surface used byChild. Would Child comment on this point, discuss the nature of the approximationswhich lead to rotational adiabaticity in his model, and indicate possible refinementswhich might alter the conclusions ?Dr. M. S . Child (Oxford University) said: Karplus has raised an importantquestion. The prediction he has questioned is based on the extreme adiabaticapproximation which rests on the neglect of certain coupling terms in deriving eqn.(1). The most important of these are determined by the rate of change of the internalrotational part of wavefunction and the velocity along the reaction co-ordinate asthe system enters the bending vibrational region; their effect also depends on theinternal rotational energy spacing in this region. The system H -t- H,, therefore,appears most favourable for the approximation because rotational energy spacingsare large, and the velocity along the reaction co-ordinate is reduced by the presenceof the activation barrier. The point can be settled however, only by further calcula-tion. Comparison with the exact quasi-classical calculations of Wyatt and Karpluswould then be of the greatest interest.In various degrees of generality (curvilinearity of reaction co-ordinate, etc.) this derivationmay be found in J. 0. Hirschfelder and E. Wigner, J. Chem. Physics, 1939, 7, 616; M. A.Eliason and J. 0. Hirschfelder, J. Chenz. Physics, 1959,30,1426 ; L. Hofacker, 2. Naturforsch. A,1963, 18, 607 acd R. A. Marcus, J. Chem. Physics, 1965,43, 1598 ; 1967, 46,959.M. Karplus and K. T. Tang, this Discussion ; see particularly fig. 6 and the related text.R. Wyatt and hf. Karplus, to be published