Furuduy Discuss. Chem. Soc., 1991, 91, 17-30 Resonances in Heavy + Light-Heavy Atom Reactions: Influence on Differential and Integral Cross-sections and on Transition-state Photodetachment Spectra George C. Schatz" Department of Chemistry, Northwestern University, Evanston, IL 60208-31 13, USA D. Sokolovski and J. N. L. Connor Department of Chemistry, University of Manchester, Ilfanchester M13 9PL, UK The effect of resonances on the observable properties of chemical reactions is studied theoretically, in particular for I + H I - + I H + I and CI+HCI-+CIH+Cl. All of our calculations use hyperspherical coordinates and accurate coupled- channel solutions to the Schrodinger equation for reactive scattering in three dimensions. For I + HI, we investigate the effect of potential surface variation on transition-state IHI- photodetachment spectra for total angular momen- tum quantum number equal to zero.Four different I + HI London-Eyring- Polanyi-Sat0 (LEPS) surfaces are used, with classical barrier heights varying from 0.048 to 0.243eV. The low-energy portion of the photodetachment spectra are examined in detail; it is found that peaks in the spectra arising from resonances and from direct scattering move up and broaden in energy as the barrier height is increased. An approximate match of the theoretical peak widths and spacings with the experimental ones is obtained for one of the surfaces. However, the peak intensities differ, suggesting that LEPS surfaces may not be adequate to characterize fully the observed spectra. For C1+ HCI, we perform a centrifugal-sudden hyperspherical calculation in order to examine the scattering properties of the single isolated resonance with transition-state quantum numbers ( v l , v2, v3) = (0, 0,2) (where v l = symmetric stretching, v2 = bending, v3 = asymmetric stretching quantum numbers).This relatively narrow (width 4 meV) resonance produces sub- stantial peaks in certair. state resolved reaction probabilities, which leads to a smooth step-like behaviour in the integral cross-sections, and Breit- Wigner (or Lorentzian) peaks in the differential cross-sections. Simple dynamical models are developed which explain these results 1. Introduction During the past two years, the study of resonances associated with the transition-state region of bimolecular reactions has made exciting progress.These resonances were originally seen in reduced dimensionality calculations',* for reactions like H + H2 -+ H2 + H, and then subsequently in accurate three-dimensional (3 D) quantum scattering compu- tatiow3 The experimental observation of resonances has proven difficult, however, as illustrated by the continuing uncertainty over the interpretation of the Lee group experiments on F+ H2 -+ HF+ H.4 In addition, experimental5 and theoretical6 results for H + H2 have now refuted earlier experimental data' which reported observable resonance effects in the state-to-state integral cross-sections. 1718 Resonances in H + LH Atom Reactions EIeV Fig. 1 Comparison of experimental2' ( a ) and theoretical16 ( b ) IHI- photodetachment spectra us.E. The J = 0 theoretical spectrum, which has only been calculated for E < 0.5 eV, has been shifted up in energy by 0.1 eV More fruitful progress has taken place in studies of heavy + light-heavy (H + LH) atom reactions, such as C1+ HCl + ClH + C1 and I + HI - IH + I. Resonances in col- linear (1D) models of these reactions have been known for some and simple adiabatic models8710911 are able to determine resonance energies with great precision. Often the 1D resonances are narrow (width <10-4eV), reflecting the weak coupling between the fast H-atom motion, where energy is trapped in the resonant state, and the slow halogen atom motions that are responsible for the decay of the resonance. The 3D analogue of these resonances has recently been found in accurate quantum reactive scattering calculations for both C1+ HCl,12-14 and I + HI.15716 Many properties of these resonances, such as the resonance energies and spacings, are what one would expect from the 1D results, with the simple addition of transition-state bending motions in going from 1D to 3D.A number of simple models which accurately predict the 3D resonance energies have also been given.I7-l9 The resonance widths obtained in the accurate 3D calculations12-16 are, however, significantly broader (by factors of ca. lo2) than in either the 1D calculation^^^^ or in 3D models that include bending motions adiabatically. l 9 The accurate 3D calculations show that the resonances decay primarily into excited rotations of the product diatomic molecule,16320 suggesting that the larger widths in 3D are the result of the decay of the resonance via bending rather than stretching motions.Probably the most exciting feature of resonances in H+LH reactions is that they have been observed experimentally via photodetachment spectroscopy by Neumark and his g r ~ u p . ~ ' - ~ ' (This type of experiment was suggested in Ref. 17.) The most conclusive results to date are for I+HI,21 where the stable molecule IHI- has been photodetached to produce IHI close to its transition state. Fig. l ( a ) shows the measured spectrum, plotted as a function of the total energy E of IHI, measured relative to the minimum of the HI potential well. The spectrum shows three groups of peaks separated by ca. 0.19 eV. This separation is in good agreement with theoretical predictions16720 for twice the asymmetric stretch spacing of the IHI transition state (only even states are Franck- Condon allowed).The three groups of peaks correspond to an asymmetric stretchingG. C. Schatz, D. Sokolovski and J. N. L. Connor 19 quantum number v3 of 0 (at E = 0.25 eV), 2 (0.44 eV) and 4 (0.63 eV). Fig. l ( 6 ) shows the predicted spectrum from Ref. 16. It is clear there is a reasonable correspondence between theory and experiment for both the coarse and fine structure in the spectrum. Note particularly that the v3 = 2 group of peaks in both the theoretical and experimental spectra consists of three main peaks separated by ca. 0.01 eV. From the theoretical work, it is known that these correspond to individual resonances having a symmetric stretching quantum number vl equal to 0, 1 and 2, and a bending quantum number v2=0.Since rotational broadening is negligible, the peak widths directly give the resonance lifetimes, ca. 0 . 0 5 ~ s in the experimental results and 0.3ps in the theory. Although the agreement is far from perfect, it is good enough to provide confidence in the assignment of these peaks as resonances. The v3=0 group of peaks is more complex, with theory predicting two narrow resonances at low E and then broader ‘rotational threshold’ peaks16 at higher E. The measured spectrum agrees better with the rotational threshold peaks, and indeed the observed peak locations are consistent with HI rotor levels ranging from j = 11 to j = 16. Thus it appears that the v3=0 peaks are not due to resonances.From the point of view of theory, although the results in Fig. 1 are encouraging, there are still several significant issues that need to be resolved (and which are therefore the subjects of this paper). One is that the comparison between theory and experiment for IHI- photodetachment is still far from perfect. The v3 = 0 peaks in Fig. 1 are not in a one-to-one correspondence, probably as a result of inaccuracies in the I+HI potential surface used in the calculations. In addition, there is a 0.1 eV shift in energy between the theoretical and experimental spectra that was subtracted when plotting Fig. 1. Although this shift may be due to experimental energy uncertainties, it is more likely to arise from errors in the potential surface.To study these problems, we have calculated IHI- photodetachment spectra for four different potential surfaces. The comparison with experiment will comprise section 2 of this paper. Another significant issue concerns the influence of resonances on differential and integral cross-sections. All experiments that have attempted to observe resonances so far (except for the photodetachment studies) have involved scattering of some t ~ p e , ~ , ~ . ’ but none of them has studied H+LH reactions. Since the observation of resonances for other types of reaction has proved to be difficult, it is important to find out whether resonances cause measurable changes in scattering properties for H + LH reactions. We have chosen to consider for this part of our paper the ( v l , u2, v3) = (0, 0,2) resonance in the C1+ HCI reaction.This resonance has similar properties to those for the v3 = 2 group of I+HI shown in Fig. 1. However, unlike I+HI, it is an isolated resonance (with no nearby v,>O counterparts), and therefore it seems a better candidate for observation in scattering experiments. In addition, the potential surface for C1+ HC1 is known better than for I + HI,24 and the non-resonant scattering properties of C1+ HC1 have been studied more t h o r o ~ g h l y . ~ ~ - ~ ~ Our calculations on C1+ HCl will be described in section 3. The two studies presented in this paper have in common that they use the same method to solve the time independent Schrodinger equation for a reactive collision. In this method,33 hyperspherical coordinates are employed to set up coupled-channel equations, and the equations propagated by standard numerical techniques to obtain the scattering wavefunction. Details concerning the application of this method to I + HI and C1+ HCI have been described p r e v i ~ u s l y ’ ~ ~ ’ ~ ~ ’ ~ and have not changed in the present work.In the I+HI calculations, only the partial wave with total angular momentum quantum number J = O has been considered, as this is sufficient to determine the photodetachment spectrum. For C1+ HCl, we have calculated all the partial waves (J,,, = 160) necessary to converge the reactive differential and integral cross-sections. For this purpose a centrifugal sudden (CS) approximation has been used. The accuracy of this approximation (denoted CSH) has been discussed previously.14333 In Ref.14 we20 Resonances in H+LH Atom Reactions Table 1 Properties of the four LEPS IHI potential surfaces quantity A B C D S 0.1900 0.1608 0.1490 0.1200 VS/eV 0.048 0.128 0.161 0.243 r'l Qo 3.376 3.395 3.403 3.424 o / cm- ' 146 144 143 141 w,/cm-' 362 3 64 365 366 o,/cm-' 447i 761i 855i 1051i V' + zero-point energy/eV 0.102 0.182 0.215 0.297 have presented CSH integral cross-sections that complement the new results reported in this paper. 2. Photodetachment Spectra for IHI- --+ I + HI + e- The potential surface used to generate the results presented in Fig. l was surface A of Manz and Romeltg (here denoted simply as surface A). This surface is based on the extended LEPS function,34 using a single Sat0 parameter, S = 0.190.The barrier height for this surface is 0.048 eV. Prior to the appearance of the photodetachment spectrum, there were no experimental data available to help calibrate the accuracy of the surface. Since the photodetachment spectrum indicates that the surface is inaccurate, with a barrier that is probably lower than it should be, we have used three new LEPS surfaces, which we denote B, C and D, to study the variation of the photodetachment spectrum with surface. Because the calculation of the photodetachment spectrum for each surface requires substantial computational effort, we will only consider the u3 = 0 region of each spectrum. In order to keep our study simple, the new surfaces we use still only have a single Sat0 parameter. By analogy with C1HC1,24727,35 it is likely that the correct IHI surface has a non-collinear saddle point (whilst all our LEPS saddle points are collinear).However, there is no simple way to sample systematically surfaces with non-collinear saddle points without introducing many unknown parameters, so we have chosen to ignore this problem in the present study. Table 1 presents the saddle-point properties (barrier height V*, H-I distance r*, harmonic frequencies w I , w 2 , w 3 , and V*+ zero-point energy) for the four surfaces A-D. Surface B was constructed to have a barrier height 0.08 eV higher than that of surface A, an energy difference suggested by an early comparison between theoretical and experimental photodetachment ~pectra.'~ Surfaces C and D were constructed to have still higher barriers, after it was discovered that the shift in the spectrum on going from A to B was substantially smaller than the difference in barrier heights.Surface D's barrier height of 0.243 eV is probably an upper bound to the correct result, since the IHI barrier height should be lower than ClHCl, which has a barrier height of ca. 0.32 eV in a6 initio calculation^.^^ Note that most of the saddle-point properties, other than barrier height and w 3 , are approximately the same for the four surfaces. We have used the four surfaces just described to calculate J = 0 reaction probabilities and photodetachment Franck-Condon factors. The IHI- wavefunction was identical to that used p r e v i o ~ s l y . ' ~ ~ ' ~ ~ ~ ~ The results are presented as a function of E in Fig. 2 and 3.Fig. 2 shows the J = 0 cumulative reaction probability PJ"(E), which is the sum over all open initial and final rovibrational states-of the state-to-state reaction probability ~l;2 v 3 j ( ~ ) , i.e.G. C. Schatz, D. Sokolovski and J. N. L. Connor 1.0 0.0 1.0 0.0 21 - A - - - ' Fig. 2 h 4 4 v 0 Y I1 2*o Cumulative reaction probability PJeO( E) for I + HI us. E for the four LEPS surfaces A- D EIeV Fig. 3 Calculated IHI- photodetachment Franck-Condon factors S"'(E) us. E for the LEPS surfaces A-D. Also shown is the v3 = 0 part of the experimental spectrum from Fig. 1, which has been shifted down in energy by 0.04eV Fig. 3 presents the total Franck-Condon factor SJ'"(E), which is a sum over all asymptotic HI states of the square of the overlap between the IHI- ground-state vibrational wavefunction and the IHI scattering wavefunction.The SJYo( E ) values were all normalized to a peak value of unity over the energy range considered. Fig. 2 and 3 show that as the height of the barrier increases, the two resonances seen at low E move up in energy and broaden considerably, disappearing completely on surface D. The features associated with direct scattering in Fig. 2 and 3 also move up in energy with increasing barrier height. It is interesting to note that this shift is greater22 Resonances in H+LH Atom Reactions for PJ”(E) than for SJ’O(E). To understand this effect, we first note that the reaction probability threshold occurs at an energy that corresporids approximately to the height of the adiabatic barrier to reaction.On surface A, this barrier is well removed from the saddle p ~ i n t ~ . ~ ~ because the barrier is low and the change in zero-point energy between I +HI and IHI is large. Thus in Fig. 2 the surface A threshold occurs at 0.165 eV, which is much higher than 0.102 eV, the V*+zero point energy value (see Table 1 ) . As the barrier height increases, the adiabatic barrier also becomes higher and moves closer to the saddle point. As a result, the threshold for surface D, ca. 0.270 eV, is now below V* + zero-point energy. Consider next the spectra in Fig. 3 . For surface A, the intensity is large over a wide range of E near the reaction threshold. This means that the adiabatic barrier must be in the Franck-Condon region. The latter occurs at an 1-1 distance of 7.332a0.I5 As the barrier height increases, the spectral features broaden considerably (reflecting a steeper potential), and the peak intensities occur at energies that are below the height of the adiabatic barrier.This means that the Franck-Condon region (still at 7.332a0) is now located out towards the reagents or products region, away from the barrier top. This is consistent with the fact that the adiabatic barrier has moved in towards the saddle point. The saddle point 1-1 distance is 6.8ao for all four surfaces. In Fig. 3 we have also superimposed the v3 = 0 portion of the experimental data on the surface C spectrum to show that the width and spacing of the measured peaks approximately matches the calculated spectrum. However, the match is still far from satisfactory, as the intensity drops off more quickly in the calculated spectrum.Also, to improve the agreement between theory and experiment, the measured spectrum has been shifted down in energy by 0.04 eV. A shift of this size is well within the experimental uncertainty, but the discrepancy in the intensity is still serious. Finally, we note that the rotational quantum numbers associated with the surface C peaks ( j = 6 - 1 1 ) are lower than those reported by Waller et aL2’ ( j = 11-16) based on a match to the asymptotic HI states. 3. Resonant Differential and Integral Cross-sections for C1+ HCI + CIH + C1 Our calculations use the LEPS surface of Bondi et aL8 (which is denoted BCMR). The non-resonant dynamics of Cl+HCl on this surface have been extensively studied in In particular, thermal rate coefficients computed by the cen- trifugal-sudden distorted-wave (CSDW) method match experiment a c ~ u r a t e l y .~ ~ Although the position of the saddle point on the BCMR surface (which is collinear) does not match the best ab i n i t i ~ ~ ~ predictions (non-collinear), this difference has a relatively small effect on the reaction dynamics.24727 In an earlier study of the (0, 0 , 2 ) resonance for C1+ HCl,14 we examined the state-to-state J = 0 reaction probabilities in detail, and found that only certain of them are strongly perturbed by the resonance. Specifically, we found that HCl rovibrational states (v,j) having v = 0 and j = 14-16 in either reagents or products are perturbed, as well as transitions having v = 1 and j = 0-8.We have selected three transitions from this set of quantum numbers for study. They are: past work. 8-1 2- 14,19,24-33 ( a ) v = O , j = 1 5 -+ v’=O,j’=15 ( b ) v = l , j = 5 ---* u’=O,j’=15 ( c ) v = l , J = 5 + v ’ = l , j ‘ = 5 These transitions were chosen because each has the largest J = 0 reaction probability for the indicated v and u‘ combination at a total energy of E = 0.642 eV, which is the (0, 0 , 2 ) resonance energy.G. C. Schatz, D. Sokolovski and J. N. L. Connor h 4 v a - 0.62 0.64 0.66 0.68 0.70 23 E/eV Fig. 4 J = 0 reaction probability us. E for C1+ HCl for transitions ( a ) v = 0, j = 15 + u’ = 0, j ’ = 15 (solid curve), (b) u = l , j = 5 + v‘=O,j’=15 (dashed curve) and ( c ) v = l , j = 5 + u ’ = l , j = 5 (dotted curve) In the following we will use the results from our CSH computations to study reaction probabilities, S-matrix phases, and integral and differential cross-sections for transitions ( a ) - ( c) for energies close to the (0, 0,2) resonance ( E = 0.620-0.700 eV).For J = 0, we calculated scattering matrices at 0.001 eV intervals over the indicated range (81 energies). For 0 < J S 160, calculations were done for every partial wave at 0.620, 0.642, 0.660, 0.680 and 0.700 eV. In addition, at 0.640,0.646 and 0.650 eV calculations were performed for every J with 0 J S 30, and then for every fifth J thereafter up to J,,, = 160. Reaction Probabilities Fig. 4 shows the energy dependence of the J = 0 reaction probabilities Pi;. v 3 g ( E ) for the three transitions (a)-( c). Near E = 0.642 eV, each probability possesses a typical resonance shape, with transitions (6) and (c) having nearly Breit-Wigner (or Lorentzian) behaviour, i.e. where Ere, = 0.642 eV, r = 0.004 eV, A is P;;: ,j.,( Ere,) and P”,zc,( E ) is a small, slowly varying, direct scattering contribution. [Subscript, uj -+ v y ’ , and superscript, J = 0, have been omitted from most of the terms in eqn. ( 1 ) for notational convenience.] Transition ( a ) has a larger background contribution, and shows significant interference between the direct and resonant scattering. S-matrix Phases Fig. 5 shows the phases 8;;: .,,(E) = arg S;;?, ”),( E ) of the S-matrix elements for the three transitions as a function of E. All three phases decrease with E, as would be expected for direct reactive scattering on a repulsive potential.” For E =r E,,,, the phases show small wiggles, although their slopes are negative in all cases (implying negative24 Resonances in H + LH Atom Reactions E/eV Fig.5 J = 0 phase of the S-matrix element us. E for the same transitions as in Fig. 4 for C1+ HCl time delays).” In earlier it was found that the phase of the resonant part of an S-matrix element often increases rapidly on passing through a resonance. However, this behaviour can be masked by a rapid energy dependence in the phase of the direct scattering. Fig. 5 shows that the S-matrix phases do not increase by 27r on passing through the resonance. Evidently, the E dependence of the phases is not simple because of interference between the resonant and direct scattering.There is a slight second wiggle in the phase of transition (6) near 0.678 eV, which coincides with a dip in the corresponding reaction probability in Fig. 4. This could be due to a weakly excited symmetric stretching resonance labelled by the quantum numbers ( 1 , 0, Z), as the energy spacing between the (1, 0,2) and (0, 0,2) resonances (0.036 eV) is consistent with that deduced from the harmonic symmetric stretching frequency at the saddle point (namely 0.043 eV). This resonance is extremely weak, however, so we will ignore it in our discussion. Probably the most important conclusion to be drawn from Fig. 4 is that the phase of the direct scattering contribution has a dependence on E that is almost as rapid as the resonant contribution. This makes it difficult to separate out the resonant part of the S matrix (for the purpose of determining resonant time delays), and it also makes an analysis based on Argand diagram^'^ not very useful.This problem with the direct contribution is probably more important for H + LH reactions than most others, because the phase is proportional to the asymptotic wavenumber which is large when the translational reduced mass is large. Integral Cross-sections The crosses in Fig. 6 show the E dependence of the degeneracy-averaged integral cross-sections Q , - j , ( E ) obtained from our CSH calculations. For transitions (6) and ( c ) , there is a sudden rise near Ere, = 0.642 eV with a width equal to the width of the peaks in Fig. 4. Transition ( a ) , by contrast, shows only a small wiggle at 0.642 eV, presumably reflecting the larger contribution from direct scattering.It is possible to explain the results in Fig. 6 using the following simple m0de1.I~ The cross-section Q , - L , , , j ( E ) is related to the reaction probability P i , + o,,p( E ) via the usualG. C. Schatz, D. Sokolovski and J. N. L. Connor 25 N O U \ h 4 v a t 0 Y 3- O - r 0.1 0.0 I I I 0.62 0.64 0.66 0.68 0.70 EIeV Fig. 6 Degeneracy-averaged CSH integral cross-sections us E for C1+ HCl for the same transitions as in Fig. 4. The smooth curves are results from the J = 0 approximate model [eqn. (4)] while the crosses are the full partial wave summed CSH results formula (2) I T w Q v j 4 .?,(E) = (2j+ 1)-' 7 C ( 2 J + l ) P i j , v 3 p ( E ) kuj J = O where k, is the translational wavenumber for the initial state.Since a large number of J s contribute to the cross-section (ca. 160) and P i j - .?,(E) varies smoothly with J, the sum in eqn. (2) can be approximately replaced by an integral. We also invoke a J-shifting a p p r o x i m a t i ~ n ' ~ ' ~ ~ ' ~ ~ ~ ~ ~ in order to relate the J and E dependence of Pi,- ".,(E) via Pij - L,,,,( E ) Pi:: .?,[ E - B*J( J + l ) ] (3) where B* is the rigid rotator constant for ClHCl at the transition state. The validity of eqn. (3) has been established in Ref. 14 and 30. Combining eqn. (2) and (3), we find This formula allows the cross-sections to be calculated from the J = 0 reaction prob- abilities in Fig. 4. The continuous curves in Fig. 6 were obtained by numerical evaluation of eqn.(4) using these J=O probabilities. The agreement between the (partial wave summed) CSH and J = 0 model cross-sections is generally good. A simple analytical expression for the resonant contribution to the cross-section can be obtained by substituting eqn. (1) into eqn. (4). This results in where Q$YtU3,(E) is the cross-section arising from the direct probability in eqn. ( 1 ) . The resonant part of eqn. (5) has a smooth step-like dependence on E that is consistent with the accurate CSH results. The step arises from the sum over resonances for many26 Resonances in H + LH Atom Reactions =l,j=5--v‘=l,j’=5 (x5.0) 0.700 eV f o.680 ev 0.660 eV (b)v=l,j=5-v‘=O,j’=15 (x4.0) 0.5 Fig. 7 Differential cross-section aOj- u‘,r( 8,) us. OR for C l f HCl for transition (a) u = 0, j = 15- u ’ = O , j ’ = 1 5 , ( b ) u = l , j = S - * u ’ = O , j ’ = 1 5 and ( c ) u = l , j = 5 + u ’ = l , j ’ = 5 .The cross- sections plotted are from CSH partial wave sum calculations at E = 0.620 eV (solid curve), 0.642 eV (dots), 0.660 eV (short dash), 0.680 eV (long dash), 0.700 eV (dash-dot) Js, each being shifted by its rotational energy B*J(J+ 1) from the J = 0 resonance, but otherwise having approximately the same widths and amplitudes. This energy depen- dence should be typical for resonances in H+LH systems, as it only requires that the resonance widths be larger than the transition-state rotational constant (B* = 1.25 x eV for CIHCI) and that J-shifting be accurate, both of which are accurately satisfied in calculations done so far.’2-16930 Differential Cross-sections Fig.7 presents the differential cross-sections uuj+ u3,( 6,) as a function of the scattering angle t?R (Ref. 24) for the three transitions ( a ) - ( c) at the five energies where all partial waves have been calculated. For transitions (6) and (c), the cross-section at 0.620 eV is very small; it therefore appears as a horizontal line on the scale of the plot. Fig. 7 shows that, for all three transitions, there is a shift from backwards to sideways peaking as E increases from 0.620 to 0.700 eV. Since transition ( a ) is dominated by direct scattering while (6) and (c) have stronger resonance contributions, this shift in the angular distributions does not appear to be a unique property of either mechanism. Evidently the identification of resonance effects in angular distributions needs to be investigated more carefully.In the present case, this can be achieved by generalizing a semiclassical optical model that we have previously shown to work well in describing CSDW angular distributions at lower E for C1+ HCl,24*26 and C1+ DCl,24 (see also Ref. 29). In this model, we use the classical expression for the reactive differential cross-section where P , --. b( OR)] is the reaction probability as a function of the impact parameterG. C. Schatz, D. Sokolovski and J. N. L. Connor 27 Fig. 8 Differential cross-section uu, + n r j , ( 6,) us. 6, for C1+ HCl for the same transitions as in Fig. 7 but using the J-shifted semiclassical optical model expression eqn. (8). For transitions ( b ) and ( c ) , the value of Ork calculated from eqn.( 1 1 ) for each energy (>Oh42 eV) is indicated by an arrow on the abscissa 6( OR). If 6( 6,) and db( 6,)/deR are evaluated using the hard-sphere expression 6( 6,) = 2r cos( 6R/2), we find (7) u u j + u,jt(6R) = r2P"j -. u T j p [ b( OR)] where r is the hard-sphere radius. Next we equate P, -. ,>,( 6) with the quantum reaction probability Pij + ">.( E), making the approximation 6 = A J / p , where p is the initial (state-dependent) relative momentum. If J-shifting [ i.e. eqn. (3)] is also assumed, then eqn. (7) becomes 'Tuj - t,'j'( 6,) = rZPi,=2 o'j'{ E - B*J( 6 R ) [ J ( 6,) + 11) = r 2 P i , 2 .,JE - (4~'r'p'/ A') cos2 ( 6 , / 2 ) ] (8) where in eqn. (8) we have assumed J(J + 1) = J 2 , and have approximated J ( 6,) by the hard-sphere formula.Eqn. (8) provides a simple relation between the J = 0 reaction probability and the differential cross-section, which is similar in form to previously derived expression^.^^-^' Fig. 8 shows the angular distributions corresponding to Fig. 7, that have been calculated using eqn. (8). The radial parameter r was adjusted to optimize agreement of the peak positions with Fig. 7; this resulted in r = 2 . 4 4 ~ ~ for transition (a), r = 2 . 4 0 ~ ~ for (b), and r = 2 . 3 0 ~ ~ for (c). All these values are close to those derived in a CSDW ~ t ~ d y ~ ~ , at lower energies for non-resonant scattering. The agreement between Fig. 7 and 8 is generally good, with the peaks in Fig. 7 being somewhat broader at most energies. Note also that at 0.642 eV, the absolute magnitudes of the differential cross-sections given by the J-shifted optical model for 6 , = 180" are in good agreement (ca.20%) with the CSH results. For transitions (6) and (c), the model shows that the resonance produces the dominant backward peak, which rapidly moves to smaller 6R as E increases. The direct scattering also contributes to aVj+ ">/( 6,) at large OR for E 3 0.660 eV, but its contribution can be clearly distinguished at most energies. For transition (a), the dominant peak at OR = 130" for E = 0.660 eV is due to direct scattering. In the J-shifted optical model28 Resonances in H+LH Atom Reactions calculations, the resonance produces a wiggle near the top of this peak, but even this is largely washed out in the CSH results of Fig.7. We can obtain a more detailed description of the resonant contribution to the differential cross-section by substituting eqn. ( 1 ) into eqn. (8). This yields which shows that the resonant part has a Breit-Wigner (or Lorentzian) dependence on cos2 (&/2). If E s E,,,, the peak in uuj- u'i7( 8,) occurs at e y k = 180". If we denote by (&)1/2, the value of OR where the cross-section equals half its value at epReak = 180°, then for E = Ere, we find (&)1/2 is determined by The factor (r/2B')1'2 in this formula is approximately the maximum partial wave that contributes to the resonant part of the cross-section when E = Ere, (which will be denoted JE:x),14 while 2 r p / h is approximately Jk:x, the maximum value of J that contributes to the hard-sphere cross-section. The angle (&)1/2 is thus determined by the ratio of these two quantities, i.e.cos [ (8,),,,/2] = J',"ix/ J k z x . One way eqn. (10) might help in analysing experimental data is to use the measured value of (&)1/2 together with estimates of the geometrical parameters BS and r, to determine the width r. It is also possible to use the resonant part of eqn. (9) to study the shift of the peak in uuj - u'jn( 6,) with energy for the case E > E,,,. For example, the peak in uuj - uj-.( 0,) moves to OpReak=Oo when E - Ere, 3 4B*r2p2/ A2 = B*(Jk:x)2 Forward scattering thus occurs when E - E r e , matches the rotational energy for an angular momentum that corresponds to scattering at an impact parameter equal to the hard-sphere diameter (i.e.6k:x = 2 r ) . B ' ( J ~ : ~ ) ~ is given by The value of for which uvj - .>,( 0,) is a maximum for 0 s E - E,,. To see how well this expression works, note from the arrows in Fig. 8 that OKak for transition ( c ) matches that in Fig. 7 accurately at all the energies plotted, with r fixed at r = 2 . 3 0 ~ ~ . For transition (6) the correspondence between Orak and Fig. 8 is less good, as the direct and resonant contributions to the cross-section in Fig. 7 are badly overlapped for E > 0.660 eV. A practical application of eqn. ( 1 1 ) to the interpretation of data would be to extract the transition-state rotational constant B* from plots of cos ( 6rak/2) vs. ( E - 4. Conclusions We have examined the influence of transition-state resonances on a number of measurable quantities for chemical reactions.Photodetachment intensities show great sensitivity to resonances, although rotational thresholds can also contribute. Even these thresholds are sensitive to the form of the potential surface, as is evident from our results in Fig. 3 for IHI-. Our attempt at improving agreement between theory and experiment by varying the Sat0 parameter in the LEPS function has only been moderately successful, which suggests that further improvement will require a non-LEPS surface. For CI + HCI, we have studied the energy dependence of both integral and differential cross-sections. The integral cross-sections show a smooth step-like increase at the resonance energy which can be used to find the J = 0 resonance energy and width. ThisG.C. Schatz, D. Sokolovski and J. N. L. 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