Observation of a Condon Reflection Products State Distribution in the Collinear H + C1, Reaction BY M. S. CHILD -f AND K. B. WHALEY~ Theoretical Chemistry Department, University of Oxford, 1 South Parks Road, Oxford Received 5th December, 1978 A classical and semiclassical investigation of conditions required for the observation of a Condon reflection pattern in the products state distribution for the collinear H + C12 reaction is reported. Both vibrational and translation excitation of the reactants are considered. The use of semiclassical arguments in this context is justified by a high level of agreement with exact quantum mechanical results, although a significant threshold anomaly remains to be investigated. Competition with the inelastic channel above a certain threshold, which decreases with increasing reactant vibrational quantum number, is found to invalidate any simple Condon reflection prediction.The nature and importance of this competition is shown to be simply characterisable in terms of the properties of certain trapped trajectories, or nascent transition states, in the products valley of the potential surface. 1 . INTRODUCTION An important question for this General Discussion is what part quantum mechanics should play in the theory of reaction dynamics. Two obvious answers are in the treatment of the reaction threshold and of the resonances displayed by the collinear H + H2 reaction1y2 for example. Quantum mechanical interference can also yield a Condon reflection pattern in the product vibrational state distribution, whereby the distribution can in principle mirror the oscillatory form of the reactant wavefunction either as a function of product quantum number at given energy, or as a function of energy at given quantum n ~ m b e r .~ The former is clearly apparent in exact quantum mechanical results for the H + C1, rea~tion.~ It is also inevitably predicted by a variety of Franck-Condon models for the reaction me~hanism.~’ The experimental situation is less dear. A bimodal distribution from reactant state n = 1 has been reported for the H + C12 reaction,’ but this may be explicable on other than quantum mechanical grounds.10 While we recognise that the rotational degrees of freedom cannot be ignored in any direct comparison with experiment, we beIieve that the possi- ble observation of a Condon reflection products state distribution is sufficiently inter- esting to merit a detailed study of the conditions for its occurrence in a collinear model.The method of investigation is by classical and semiclassical mechanics, using the LEPS potential employed in previous classical l1 and quantum mechanical calcula- tions. Our purposes are (a) to examine the validity of available uniform semiclassical approximations 3~12-14 in a reactive context, (b) to identify the optimum conditions for observation of a Condon reflection distribution and (c) to discuss the branching between reactive and inelastic scattering, which is expected to invalidate any simple Condon reflect ion prediction. Emphasis is placed throughout on the disposition of relevant trajectory end points in reactant and product internal phase space.A particular “ barred ” representation l5 7 Visiting Fellow 1978-79. Institute for Advanced Studies, Hebrew University, Jerusalem, Israel. 0 Present address : Department of Chemistry, Harvard University, Cambridge, Mass., U.S.A.58 PRODUCT DISTRIBUTION FROM H + clz familiar to workers in the semiclassical field is employed in order to yield a picture which is independent of the choice of translational coordinate. This pictorial approach first clarifies the origin of any interference structure in the products state distribution; hence the conditions for its observation are readily understood. It also shows how the growth of the reaction zone in reactant space first grows to encompass successive quantum states as the energy increases, and then distorts and even begins to shrink as the advent of complex or snarled trajectories leads to increasing competition with the inelastic channel.Until the last few months the only approach to mapping this competition has been by laboriously noting the fate of many hundred traject~ries,l'-'~ but Pechukas and Pollak20 have recently shown how in principle knowledge of certain trapped trajec- tories on the potential surface may be used to generate such a map in a simple way. Families of such trapped trajectories have been identified for the H + H2 system2, and its isotopic variants,22 but the present paper contains the first report of their existence on a strongly exothermic surface. This is also the first practical implementation of the proposed mapping scheme.20 Our overall purpose is therefore to discuss the effects of both vibrational and trans- lation reactant excitation on the reaction dynamics. The results are specific to the H + Cl, system, but the general principles apply to any collinear atom-diatom reaction.2 . CONDON REFLECTION STRUCTURE Fig. 1 may be used to illustrate the principles behind the general Condon reflection prediction3 and to understand the requirements for its observation. It illustrates in product phase space the outcome of a set of classical trajectory calculations from a given reactant quantum state at a given energy [C12(n = 0) at E = 0.4 eV]. A special modified angle-action representation, outlined in the Appendix, was adopted to obtain this symmetrical picture, but it has no effect on the physical significance of the dia- gram.The key features are that the translate C; of the n,th reactant orbit necessarily encloses the same area [(n, + +)h] as the quantised orbit itself, and that the semi- 1 G1a.u. Cl' ,la.u. FIG. 1.-Reactive trajectory end points in the products phase space at E = 0.4 eV (0.0147 a.u.). Cz is the product phase space orbit for n2 = 5, C; the translate of the reactant orbit for nl = 0. The shaded area determines the semiclassical phase difference between the two nl = 0 to nz = 5 root trajectories. The axes B and are defined in the Appendix.M. S. CHILD AND K. B . WHALEY 59 classical phase responsible for the interference between the two root trajectories contributing to a particular transition is given by the relevant shaded area in the diagram (applicable to the n, = 0, n2 = 5 transition in this case).The root trajec- tories themselves appear as intersections between C', and C,. The observation of Condon reflection structure as a function of product quantum state requires first that successive quantised orbits should cut off areas lying between 0 and (n, + 4)h. This is impossible if C; encloses the origin for example, in which case the reflection pattern would remain undeveloped, just as in diatomic spectroscopy a fully developed pattern within the discrete spectrum requires an adequate separation between the potential wells. A second requirement is that C; should suffer only two intersections with any given C2. The existence of more than two root trajectories would lead to higher interference structure.Finally the contour Ci should be com- plete, implying no significant branching between reactive and inelastic scattering. Similar arguments apply to the variation of a given n, -+ n2 transition probability as a function of energy, because C; will typically move outwards as the energy in- creases, again cutting off an increasing area between 0 and (n, + +)h. Only the second two requirements then apply, but they are much less easily satisfied, particularly for excited reactant states, due to increasing competition with the inelastic channel at higher energies. The nature of this competition for the H + C1, system is discussed in detail in section 4. 3. SEMICLASSICAL TRANSITION PROBABILITIES An extensive comparison between semiclassical and available exact quantum mechanical r e s ~ l t s , ~ was performed in order to test the above semiclassical con- clusions, and to assess the reliability of specific uniform approximations, namely Airy,12 Bessel,13 Laguerre l4 and h a r m ~ n i c .~ The semiclassical calculations were performed in the conventional way 23 with the semiclassical phases deduced from the action integrals along the trajectories. Our only computational contribution is to use the constraints imposed by the phase space picture in fig. 1 to solve a troublesome phase discontinuity problem.2k27 This discontinuity occurs during a transformation of phase between Cartesian and angle action representations, which is applied in the initial and final asymptotic region^.^**^^ It arises because the necessary classical generator involves inverse tri- gonometrical functions, which are returned by any computer routine as the principal value.Such discontinuities may cause errors of &(2n, + 1)n & (212, + 1)n in the phase difference between the contributing trajectories. Fig. 1 implies however that the maximum phase difference is (n, + +)n, because the total area of Ci is (n, + +)h or (2n, + l)nh, and the Airy12 and Bessel13 approximations require that the area of the smaller of the two divisions of C; by C2 should be adopted; the Laguerre14 and har- monic* approximations give the same result for either of the two areas. Hence the correct phase difference is obtained simply by adding or subtracting terms (2n, + 1)n (2n + 1)n and (2n2 + 1)n to the raw value returned by the program until the answer lies between 0 and (n, + 4)~.Given this procedure the results obtained by the different approximations are in agreement with 5 x. The Laguerre result is given below. Fig. 2 gives a comparison between the exact quantum mechanical and semi- classical results for all classically allowed transitions from the n, = 0 and n1 = 1 reactant states at energies for which the trajectories are 100 % reactive. Three features may be noted. First the semiclassical results are in good qualitative agreement with the exact Similar Bessel results are available in the l i t e r a t ~ r e . ~ ~60 PRODUCT DISTRIBUTION FROM H -/- clz results for all transitions. The quantitative agreement is typically within 0.05 probability units for transitions from the n, = 0 state, and within 0.1 probability units for the n, = 1 state.The right hand sections of the diagram show that the agreement is certainly sufficient to confirm the Condon reflection behaviour observed in the quantum mechanical results4 which also extend to n1 = 2. PO n 4 n f/eV A 0.3 0.2 0.1 * 0.1 t / e V n :::I 0.5 P,, 0.4 ::;h 0.1 1 2 3 4 5 6 7 n ( b ) FIG. 2.-Comparison between exact (solid line) and semiclassical (dashed line) transition probabilities for (a) nl = 0, (b) nl = 1. The right hand section of each diagram gives the distribution as a function of nz at E = 0.32 eV (0.117 a.u.). Points are the quantum mechanical and circles the semiclassical Secondly the major semiclassical anomaly occurs at threshold for each state al- though it is much weaker than that observed for the H + H2 reaction.31 This is tentatively attributed to neglect of possible complex trajectories (arising from complex reactant angle variables) in the construction of the uniform approximation.The available uniform approximations all require only two root trajectories but the non- sinusoidal shape of the quantum number n2 against initial angle q1 curve in fig. 3 while admitting only two real roots, suggests the possibility of nearby complex roots. A similar anomaly observed in spectroscopic applications of the uniform Airy approxi- mation3’ has recently been removed by use of a four transition point appr~ximation.~’ Finally, the energy range amenable to the semiclassical analysis, and hence to a firm prediction of Condon reflection behaviour, is sharply reduced in going from n, = 0 to n1 = 2.This reduction occurs at low energies due to the increasing threshold, and at higher energies due to the more rapid onset of competition with the inelastic channel for higher reactant quantum numbers. The factors underlying this competition are discussed in the following section in relation to the number and loca- tion of possible trapped trajectories on the potential surface. values.M. S . CHILD AND K . B . WHALEY 6 - 4- 2 - 61 1 - - I I I "2 FIG. 3.-Variation of the product quantum number n2, with modified reactant angle g1 (see Appendix for nl = 1 at E = 0.22 eV (0.0081 a.u.). 4. EXISTENCE AND SIGNIFICANCE OF TRAPPED TRAJECTORIES P e ~ h u k a s ~ ~ and, more recently, Pechukas and Pol1ack2O have underlined the im- portance of the number and location of certain periodic trajectories trapped between equipotentials of the surface.We therefore first demonstrate the existence of such trajectories for the H + C1, system and then discuss their significance for the present investigation. This is the first such study for a strongly exothermic reaction, previous investigations having been limited to the H + H2 system21 and its isotopic variants.22 Fig. 4 shows one such trajectory at E = 0.20 eV and the four most important of a possibly infinite family at E = 0.35 and E = 0.50 eV. Also shown in the lower part of each diagram is the map in reactant phase space of trajectory end points obtained by applying a small perturbation in the reactants direction at successive points along the single trapped trajectory in fig.4(a) and the two outermost trajec- tories in fig. 4(b) and (c). The single trapped trajectory, passing close to but not necessarily through the saddle point, constitutes the strict transition at this energy, and the translate of the trajectory into the reactant phase space divides this space into reactive and inelastic scattering regions. Transition state theory is exact under these conditions, with the microcanonical reaction probability given by the ratio of the reactive area to the total area within the available energy shell. Finally the percentage classical reactivity coefficient for a given quantum state is simply the fraction of the relevant orbit lying within the reaction zone.Thus the threshold for 100 % classical reactivity occurs at an energy such that the reaction zone just encloses the orbit in question. The appearance of further trapped trajectories at higher energies complicates the picture, [fig. 4(b) and (c)]. The most significant of these additional trajectories for the present discussion is the one closest to the products region, because it acts as the ultimate point of no return. Some trajectories therefore passing through the first " transition state " may fail to reach it. Hence it provides a division between the " directly reactive " and the " complex " or " snarled " trajectories. As expected in view of the complex nature of these trajectories the effect of a smaII perturbation of this product side trajectory in the reactant direction is not always to lead to reactants, but our calculations show as in fig.4(b) and 4(c) that the resulting translate forms a closed, or almost closed contour in reactant phase space, lying necessarily inside the The significance of this diagram will now be discussed. The low energy case depicted in fig. 4(a) is relatively simple.62 PRODUCT DISTRIBUTION FROM H + c12 - - - - - - - - - - - I I I I I I I 5 10 x1a.u. x/a.u. FIG. 4.-Trapped trajectories and division of the reactant phase space at (a) E = 0.2 eV (0.0074 a.u.), (b) E = 0.35 eV (0.0129 a.u.) and (c) E = 0.5 eV (0.0184 a.u.). Potential contours are given at intervals of 0.01 a.u. from -0.06 to 0.04 a.u. The axes are given by x = rHCl -I [mCl/mHCI]rCICl y =.[(mH + 2mc1)/4 mH]+rclcl. The boundary of each phase space picture is determined by the available energy. A' and B' denote the reactant contours asymptotic to the outer trapped trajectories A and B, respectively. Fig. 4(a) is divided into an inner, direct reactive, and an outer, direct inelastic, scattering region. The intermediate shaded region in fig. 4(b) and (c) belongs to the complex trajectories.M. S. CHILD AND K. B . WHALEY j2a.u. I 63 T1a.u. I ( b ) j1a.u.64 PRODUCT DISTRIBUTION FROM H -/- c12 translate of the reactant trapped trajectory. This provides a division of the available space into three parts, direct reactive, complex and direct inelastic, a division which would otherwise require a laborious examination of the nature and fate of several hundred traject~ries.'~-'~ The deeply incursive part of the inner curve at E = 0.35 eV [fig.4(b)] was in fact first charted in this way. The curve obtained is indistinguishable by eye from that shown in fig. 4(b). The significance of these results in relation to the previous discussion, is that the increasing incursion of the " complex " area into the reaction zone is directly respon- sible for the increasing preponderance of inelastic scattering at higher energy, particu- larly since, for the present system, most complex trajectories appear to be ultimately non-reactive. The effect is most marked for the higher reactant quantum numbers because the shape of the incursion cuts more deeply into this part of the reactant phase space. It therefore appears that the dominant features of the reaction may be understood without reference to the central, dashed periodic trajectories in fig.4(b) and 4(c). These differ in nature from the outer trajectories in that trajectories starting from neighbouring points on the equipotential appear to run towards and then cross them, whereas those starting close to the outer ones always diverge away from them. It is conjectured that the former are closely associated with the observation of reactivity bands 17-19,25,35 and also possibly with the existence of quantum mechanical reso- nances.'P2 Many questions related to the existence of stable classical motions in regions far from the saddle point remain, however, to be investigated. 5. CONCLUSIONS Existing semiclassical theory based on real classical trajectories from states which are 100 % reactive has been tested, and shown to be typically accurate to 5-10 %.The most serious anomaly occurs at threshold. The energy range over which such calculations can be performed, and hence over which a firm prediction of Condon reflection behaviour can be made, decreases sharply with increasing reactant vibrational state, due to significant branching between reactive and inelastic scattering. The ranges found in the present study are 0.15-0.5 eV at n, = 0 and 0.2-20.32 eV at n, = 1. It has been shown that the present H + CI2 surface can support not merely a single " transition state trapped trajectory " passing close to the saddle, but that at higher energies whole families of such trajectories exist.These provide insight into the occurrence and nature of complex or snarled trajectories. They also offer a simple device for mapping the reactant phase space into direct reactive, direct inelastic, and complex trajectory end points. The authors are grateful for stimulating discussions with Dr. P. M. Hunt and One of them (M.S.C.) wishes to acknowlege the hospitality of the Dr. E. Pollack. Hebrew University, Jerusalem where the final part of this work was completed.65 M . S . CHILD AND K . B . WHALEY APPENDIX The conventional modified angle variable 1 5 9 1 6 q is defined in terms of the true angle q,36 the local vibrational frequency o, and the translational coordinate R, momentum P and reduced mass m Q = q - mcoR/P. An associated modified coordinate r‘ and momentump may be defined by substituting for q in the appropriate formulae for the type of oscillator in que~tion,~’ which is a Morse oscillator in the asymptotic parts of a LEPS surface.Any semiclassical results are, however, invariant to a further modification of the form because the Jacobian [(t, n)/(q, n)] is unity. The definition 4 = Q + y ( 4 y ( n 2 ) = - 3 [ q a ( n 2 ) + qb(n2)1 (3) where qa(n,) and q b ( n 2 ) are the modified angles derived from the two root trajectories from the given n, to any n2, has been used to obtain the symmetric representation in fig. 1. S. F. Wu and R. D. Levine, Mol. Phys., 1971, 22, 881. D. J. Diestler, J. Chem. Phys., 1972, 56, 2092. M. S. Child, Mol. Phys., 1978, 35, 759. M. Baer, J. Chem. Phys., 1973, 60, 1057.M. J. Berry, Chem. Phys. Letters, 1974, 27, 73. U. Halavee and M. Shapiro, J. Chem. Phys., 1976,64,2826. B. C. Eu, Mol. Phys., 1976, 31, 1261. G. C. Schatz and J. Ross, J. Chem. 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