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The hydrogen exchange reaction: Discrepancies between experimental state-resolved differential cross sections and 3-D quantun dynamics
Volume 168, number 6
CHEMICAL PHYSICS LETTERS
18 May 1990
THE HYDROGEN EXCHANGE REACTION: DISCREPANCIES BETWEEN EXPERIMENTAL STATE-RESOLVED DIFFERENTIAL CROSS SECTIONS AND 3-D QUANTUM DYNAMICS
Steven A. BUNTIN I, Clayton F. GIESE and W. Ronald GENTRY Chemical Dynamics Laboratory, University ojMinnesota, 207 Pleasant St. SE, Minneapolis, MN 55455, USA
Received 2 1 March 1990
Measured time-of-flight (TGF) distributions, corresponding to speed- and angle-resolved differential cross sections for the reaction D+H*+HD+H, are compared with recently published 3-D quantum calculations. The comparison is made in the manner least susceptible to distortion: the energy-, angle- and state-resolved theoretical results in center-of-mass (cm. ) coordinates are projected onto the laboratory angle-time axis, and averaged over all experimental resolution parameters to predict the measured TOF distributions. Features directly visible in the experimental data are: ( 1) relative populations of the v’= 0 and u’ = 1 product vibrational states, (2) the extent of HD pmduct rotational excitation, and (3) the ratio of differential cross sections near 180” cm. to those near 70” cm. Significant discrepancies with theory are found with respect to features (2) and (3).
1. Introduction
The hydrogen exchange reaction has for decades provided the most promising meeting ground for a rigorous comparison between experimental and theoretical chemical dynamics [ 11. Only in the last \few years, however, have experimental data become available which are sufficiently detailed to offer a really definitive challenge to the best theoretical descriptions. These recent experiments have been carried out for several isotopic combinations, and include measurements of both integral and differential cross sections. Zare et al. [ 21 and Valentini et al. [ 31 have used different but complementary laser spectroscopic methods to measure the kinetic-energy-dependent integral reaction cross sections for fully resolved product rotational and vibrational states. Our own efforts have been directed toward crossed-beam measurements of angle-resolved differential cross sections for the reaction D + H2+HD+ H, in which product kinetic and internal energy distributions are determined by a time-of-flight (TOF) method [ 41. Some similar experiments have now been carried out ’ Present address: National Institute of Standardsand Technology, Chemistry B248, Gaithersburg, MD 20899, USA.
by Continetti, Balko and Lee [ 51. The crossed-beam experiments avoid the averaging over scattering angle which OCCUIS in the bulb experiments, and also achieve better resolution in initial kinetic energy, but at the sacrifice of complete product rotational state resolution. Ultimately, of course, it is expected that a theoretical description will have to reproduce all of these complementary experimental results simultaneously in order to be judged successful. The purpose of this Letter is to report the present status of the comparison between the experimental differential cross sections for the reaction D+H2 -tHD+H and the best available theoretical results. There are three components of the theory to be tested: the ab initio calculation of points on the potential energy hypersurface, the fitting of those calculated points with an analytic function which permits interpolation and extrapolation, and the calculation of quantum dynamics on the analytic hypersurface. Fortunately, we now have available two independent 3-D quantum dynamics calculations for this reaction in the experimental energy range. Zhang and Miller [ 61 have obtained converged quantum dynamics for the Truhlar-Horowitz representation [ 7 ] of potential points calculated by Liu and Siegbahn [ 81 (the LSTH surface) and for the initial reactant rotational
0009-2614/90/$ 03.50 0 Elsevier Science Publishers B.V. ( North-Holland )
513
Volume 168, number 6
CHEMICAL PHYSICS LETTERS
state j=O. Zhao, Truhlar, Schwenke and Kouri [ 91 now have used an independent method to obtain converged quantum dynamics for a different fit to the points of Liu and Siegbahn, augmented by later calculations of Blomberg and Liu [ lo] and Varandas et al. [ 111 (the DMBE surface [ II] ), and for both j= 0 and j= 1 initial states. Both of these groups were kind enough to supply us with extensive tables of their results, at more closely spaced intervals of collision energy and scattering angle than were available in the published versions.
2. Experimental The experimental geometry is illustrated schematically in fig. 1. The D atom beam is produced by photolysis under nearly collisionless conditions of a pulsed D$ beam with the 193 nm, 250 mJ output of a pulsed ArF* excimer laser, and collimated with a pair of defining slits. The resulting 2.2 eV, 0.9 us D-atom pulse then crosses a well-collimated pulsed Hz beam at a point 29.8 cm from the electron-impact ionizer of a mass spectrometer detector, which records time-of-flight (TOF) distributions for the scattered HD products. The kinetic energy in center-of-
18 May 1990
mass (c.m.) coordinates was varied by changing the intersection angle of the two beams. For the comparison given below, it is essential to know accurately all of the experimental parameters which affect the TOF peak shapes or intensities. All of these were either measured directly or calculated from the apparatus geometry. By far the largest factor determining the experimental resolution function was the speed distribution of the D-atom beam, which was measured directly by TOF. It displayed an intense fast peak with Av/v= 11% full-width at halfmaximum (fwhm), followed by a slow tail having a shape which reflects mostly the internal state distribution of the DS photofragments. The angular distribution of D atoms was estimated from the slit geometry, the measured angular distribution of the D2S pulse, and the measured laser beam profile. The spatial profile and the velocity vector distributions of the Hz beam were measured directly with a pair of fast ionization gauge detectors [ 121. The effective geometry at the beam crossing point is determined by the overlap of the two beams with the volume viewed by the detector. This volume, which for simplicitly is illustrated as a cylinder in fig. 1, is actually determined by three apertures in the detector which define the element of solid angle subtended as a
H2 BEAM SOURCE
D ATOM
Fig. I. Schematic illustration of the experimental geometry.
514
GENERATION
Volume168,number6
18May 1990
CHEMICAL PHYSICSLETTERS
function of the angle with respect to the detector symmetry axis. This function was calculated numerically for the measured sizes and positions of the three apertures. The effective length of the ionizer was estimated to be 5 mm from the geometry of the frlaments and electron beam defining slits. Actually, however, this parameter contributes negligibly to the overall TOF resolution function. 0.6 3. Results and discussion In the course of this work, experimental TOF distributions were measured for nominal initial kinetic energies of 0.85, 0.90, 0.95, 1.00, 1.05 and 1.20 eV, with a spread of kinetic energies of 0.22 * 0.02 eV fwhm [ 131. In each case, the laboratory scattering angle was selected so that the slowest HD products observed corresponded to c.m. scattering angles near 180”, relative to the initial cm. velocity vector of the D atom. Fig. 2 shows the TOF distribution measured at a kinetic energy of 0.85 eV, together with a velocity vector diagram for this case. The two peaks at longer times correspond to U’= 0 and v’ = 1 backscattered products, while the first peak corresponds to products scattered near 70” c.m. which are not state-resolved. The inner and outer circles in the velocity vector diagram of fig. 2 correspond to HD products in the j’ =O state of U’= 1 and U’=O, respectively. Any rotational’excitation of the products results in a smaller final cm. speed and a shift of the TOF toward the middle of the distribution [ 41. It can easily be seen in fig. 2 that the backscattered peaks are shifted and broadened by rotational excitation. Because we know the instrumental resolution function very well, it is possible to extract information on the rotational state distributions from the locations and shapes of the vibrationally resolved peaks, as we did previously [4]. However, it must be recognized that the rotational distributions derived in this way correspond to the constant laboratoryscattering angle of the experiment, not a constant c-m. scattering angle. Each product rotational state is therefore detected with a distribution of c.m. scattering angles in which both the most-probable value and the width vary with the state. If the experimental resolution in collision energy and scattering angle were perfect, we could
50 Flight
100
150
Time (psec)
Fig. 2. Measured HD product time-of-flight distribution for 0.85 eV initial kinetic energy of collision. The velocity vector diagram shown corresponds to the most-probable reactant velocities. The dotted lines indicate the expected most-probable times of flight for products in the indicated vibrational states and the ground rotational state. The solid line is the simulation of this experiment using the theoretical results of Zhao et al. [ 91.
compare the extracted state distribution with theory by performing the appropriate laboratory-to-c.m. transformation of probabilities and comparing with theoretical values corresponding to a different value of angle for each j’. Such a comparison is problematic, however, when the experimental resolution function is not narrow, but is instead a complicated function of many variables. Of particular concern is the possibility that dynamical resonances predicted by theory might be quite narrow in energy compared to the experimental distribution. We have therefore chosen a different method of comparing theory and experiment, which we believe is the most rigorous possible at present. The principle of the method is to take the theoretical state-tostate differential cross sections as functions of both collision energy and scattering angle, and to use these as inputs to a Monte Carlo program which samples the probability distribution functions of all experi515
Volume 168, number 6
CHEMICAL
PHYSICS LETTERS
mental variables, thus simulating the experiment by averaging over all experimental parameters. The details of this simulation method will be presented elsewhere [ 131. Briefly, random samples are taken from the D and H, primary beam velocity vector and pulse profile distributions, and combined with random choices of spatial coordinates for the D+H2 collision and for the detector ionization of the HD product. This defines a unique velocity diagram and TOF bin for each final (u’ ,j’ ) state and thus a unique Jacobian for the c.m.-to-laboratory transformation. Each sample, simulating a collision between an individual pair of reactants, is then weighted with the product of the probability distribution functions for each experimental parameter and with the theoretical state-to-state differential cross section from an interpolation of the tabulated results in both collision energy and scattering angle. We had available from both Zhang and Miller [ 61 and Zhao, Truhlar, Schwenke and Kouri [ 91 their converged quantum results at five values of collision energy from 0.5 1 to 1.08 eV, and c.m, angle intervals of 5”. Both theoretical groups calculated state-to-state differential cross sections for the initial H2 reactant state j=O. These results are in good agreement 191, indicating that the dynamics are fully converged in both cases, and also that the differences due to the different functional forms in the LSTH and DMBE analytic potential energy surfaces are small. As expected, inserting the two independent sets of results into our simulation program produced negligible differences in the predicted TOF distributions. The experiments were done with a pulsed supersonic beam of normal-H2 from a room-temperature source, having a spread in speed of about 7% fwhm and a translational temperature of a few kelvin. The rotational degrees of freedom of H2 are, however, not very well cooled in supersonic expansion. From careful energy-balance measurements [ 141, the effective rotational temperature under these experimental conditions was estimated to be about 170 K, although significant deviations from a Boltzmann distribution may occur. The corresponding rotational populations are 20% j= 0, 74% j= 1, 5% j=2, 1% j=3, with negligible contributions from higher states. At 0 K, the populations would be 25% j=O and 75% j= 1. Since Zhao et al. [9] carried out calculations for 516
18 May 1990
both j=O and j= 1, we performed simulations using both sets of results and combined the two predicted TOF distributions with the weighting 251 j=O and 75% j= 1 to make the best comparison with experiment which is possible with the available calculations. The result is shown in fig. 2 as the solid line, which has been scaled to the experimental data at the largest peak. The discrepancies are easily seen. The calculations get the vibrational state ratio for backscattered products about right, but there is a large discrepancy in the ratio of forward-scattered to backward-scattered intensities. In addition, the experiments show more rotational excitation in the backscattered products, as exhibited by the shifting and broadening of the peaks in the experimental TOF spectrum relative to those predicted theoretically. This discrepancy becomes more pronounced at higher collision energies [ 13 1. In comparing the TOF spectra predicted theoretically for the 25%j=O, 75Wj= 1 mixture with that predicted for pure j=O, including j= 1 causes a slight shift toward higher product rotational excitation. However, the shift is small compared to the difference between theory and experiment, making it unlikely that the discrepancy between theory and experiment could be removed by including small populations of higher H, rotational states in the calculations. Since the two independent dynamics calculations carried out with two different analytic representations of the potential energy hypersurface are in good agreement with each other for the j=O initial state, it seems most likely that the source of the discrepancy between theory and experiment lies in the original calculated potential energy points [ 8 1. Although these points are probably among the most accurately calculated for any potential energy surface, it is not known just what level of accuracy is actually required to reproduce the experimentally measured features. It is also possible that the number of pointed calculated, particularly for bent geometries, is not sufficient to permit a sufficiently accurate global representation. Kliner, Rinnen and Zare [ 15 ] have recently reported rotationally resolved integral cross sections for the reaction D+ Hz+ HD (u’ = 1,j’ ) f H at a collision energy of 1.05 eV, and have also compared their results with the calculations of Zhang and Miller [ 61 and of Zhao et al. [ 91. In this case, the agreement between theory and experiment was very good, de-
Volume 168, number 6
CHEMICAL PHYSICS LETTERS
spite the fact
that higher populations of states+ 1 were present in the room-temperature H2 reactants than in our supersonic Hz beam. It appears, as one might expect 141, that the differential cross sections for this reaction are considerably more sensitive to the details of the potential energy surface and/or the dynamics than are the integral cross sections. In particular, the effects of dynamical resonances, which can easily be washed out in
the integral cross sections
bital
angular
momentum
by averaging states
of
the
over
or-
reactants
[ 6,9,16- 19J, are expected to persist to a much greater degree in the product angular distributions. Whatever the cause, the discrepancy reported here between theory and experiment must be resolved before it can truly be claimed, as stated in a recent news report, that “Sixty years after its initial development, quantum mechanics has finally succeeded in completely describing the simplest possible chemical reactions.” [ 201.
Acknowledgement
This research was supported by the National Science Foundation, Grant No. CHE-8705611. The authors are grateful to Dr. Jue Wang and Mr. Mark Nachbor for performing some supplementary experiments which contributed to the interpretation of the data presented here. We are also grateful to Dr. Zhang and Dr. Miller and to Dr. Zhao, Dr. Truhlar, Dr. Schwenke and Dr. Kouri for providing us with the extensive tables of their theoretical results needed for the simulation of the experiment. We also appreciate the comments of Dr. Truhlar on the manuscript.
18 May 1990
References
[ I] D.G. Truhlar and R.E. Wyatt, Ann. Rev. Phys. Chem. 27 (1976) 1. [2]D.A.V. Kliner and R.N. Zare, J. Chem. Phys. 92 (1990) 2107, and references therein. [3] J.-C. Nieh and J.J. Valentini, J. Chem. Phys. 92 ( 1990) 1083, and references therein. [4] S.A. Buntin, CF. Giese and W.R. Gentry, J. Chem. Phys. 87 (1987) 1443. [S] R.E. Continetti, B.A. BaIko and Y.T. Lee, Proceedings of the International Symposium on Near-Future Chemistry in Nuclear Energy Field, Ibaraki-Ken, Japan ( 1989). [6] J.Z.H. Zhang and W.H. Miller, J. Chem. Phys. 91 ( 1989) 1528. [ 71 D.G. Truhlar and C.J. Horowitz, J. Chem. Phys. 68 ( 1978) 2466;71 (1978) 1514. [ 81 B. Liu, J. Chem. Phys. 80 ( 1984) 58 1; P. Siegbahn and B. Liu, J. Chem. Phys. 68 (1978) 2457. [9] M. Zhao, D.G. Truhlar, D.W. Schwenke and D.J. Kouri, J. Phys. Chem., in press. [lo] M.R.A. Blomberg and B. Liu, J. Chem. Phys. 82 ( 1985) 1050. [ II] A.J.C. Varandas, F.B. Brown, CA. Mead, D.G. Truhlarand N.C. Blais, J. Chem. Phys. 86 (1987) 6258. [ 121 W.R. Gentry and C.F. Giese, Rev. Sci. Instr. 49 (1978) 595. [ 131 S.A. Buntin, J. Wang, M. Nachbor, C.F. Giese and W.R. Gentry, to be published. [ 141 G. Hall, K. Liu, M.J. McAuliffe, CF. Giese and W.R. Gentry, J. Chem. Phys. 84 ( 1986) 1402. [ 15] D.A.V. Kliner, K.-D. Rinnen and R.N. Zare, Chem. Phys. Letters 166 (1990) 107. [ 161 M. Mladenovic, M. Zhao. D.G. Truhlar. D.W. Schwenke. Y. Sun and D.J. Kouri, Chem. Phys. Letters 146 (1988) 358. [ 171 J.Z.H. Zhang and W.H. Miller, Chem. Phys. Letters 153 (1988) 465. [18]M. Zhao, D.G. Truhlar, D.J. Kouri, Y. Sun and D.W. Schwenke, Chem. Phys. Letters 156 ( 1989) 281. [ 191D.E. Manolopoulos and R.E. Wyatt, Chem. Phys. Letters 159 (1989) 123. [20] R, Pool, Science247 (1989) 413.
517
CHEMICAL PHYSICS LETTERS
18 May 1990
THE HYDROGEN EXCHANGE REACTION: DISCREPANCIES BETWEEN EXPERIMENTAL STATE-RESOLVED DIFFERENTIAL CROSS SECTIONS AND 3-D QUANTUM DYNAMICS
Steven A. BUNTIN I, Clayton F. GIESE and W. Ronald GENTRY Chemical Dynamics Laboratory, University ojMinnesota, 207 Pleasant St. SE, Minneapolis, MN 55455, USA
Received 2 1 March 1990
Measured time-of-flight (TGF) distributions, corresponding to speed- and angle-resolved differential cross sections for the reaction D+H*+HD+H, are compared with recently published 3-D quantum calculations. The comparison is made in the manner least susceptible to distortion: the energy-, angle- and state-resolved theoretical results in center-of-mass (cm. ) coordinates are projected onto the laboratory angle-time axis, and averaged over all experimental resolution parameters to predict the measured TOF distributions. Features directly visible in the experimental data are: ( 1) relative populations of the v’= 0 and u’ = 1 product vibrational states, (2) the extent of HD pmduct rotational excitation, and (3) the ratio of differential cross sections near 180” cm. to those near 70” cm. Significant discrepancies with theory are found with respect to features (2) and (3).
1. Introduction
The hydrogen exchange reaction has for decades provided the most promising meeting ground for a rigorous comparison between experimental and theoretical chemical dynamics [ 11. Only in the last \few years, however, have experimental data become available which are sufficiently detailed to offer a really definitive challenge to the best theoretical descriptions. These recent experiments have been carried out for several isotopic combinations, and include measurements of both integral and differential cross sections. Zare et al. [ 21 and Valentini et al. [ 31 have used different but complementary laser spectroscopic methods to measure the kinetic-energy-dependent integral reaction cross sections for fully resolved product rotational and vibrational states. Our own efforts have been directed toward crossed-beam measurements of angle-resolved differential cross sections for the reaction D + H2+HD+ H, in which product kinetic and internal energy distributions are determined by a time-of-flight (TOF) method [ 41. Some similar experiments have now been carried out ’ Present address: National Institute of Standardsand Technology, Chemistry B248, Gaithersburg, MD 20899, USA.
by Continetti, Balko and Lee [ 51. The crossed-beam experiments avoid the averaging over scattering angle which OCCUIS in the bulb experiments, and also achieve better resolution in initial kinetic energy, but at the sacrifice of complete product rotational state resolution. Ultimately, of course, it is expected that a theoretical description will have to reproduce all of these complementary experimental results simultaneously in order to be judged successful. The purpose of this Letter is to report the present status of the comparison between the experimental differential cross sections for the reaction D+H2 -tHD+H and the best available theoretical results. There are three components of the theory to be tested: the ab initio calculation of points on the potential energy hypersurface, the fitting of those calculated points with an analytic function which permits interpolation and extrapolation, and the calculation of quantum dynamics on the analytic hypersurface. Fortunately, we now have available two independent 3-D quantum dynamics calculations for this reaction in the experimental energy range. Zhang and Miller [ 61 have obtained converged quantum dynamics for the Truhlar-Horowitz representation [ 7 ] of potential points calculated by Liu and Siegbahn [ 81 (the LSTH surface) and for the initial reactant rotational
0009-2614/90/$ 03.50 0 Elsevier Science Publishers B.V. ( North-Holland )
513
Volume 168, number 6
CHEMICAL PHYSICS LETTERS
state j=O. Zhao, Truhlar, Schwenke and Kouri [ 91 now have used an independent method to obtain converged quantum dynamics for a different fit to the points of Liu and Siegbahn, augmented by later calculations of Blomberg and Liu [ lo] and Varandas et al. [ 111 (the DMBE surface [ II] ), and for both j= 0 and j= 1 initial states. Both of these groups were kind enough to supply us with extensive tables of their results, at more closely spaced intervals of collision energy and scattering angle than were available in the published versions.
2. Experimental The experimental geometry is illustrated schematically in fig. 1. The D atom beam is produced by photolysis under nearly collisionless conditions of a pulsed D$ beam with the 193 nm, 250 mJ output of a pulsed ArF* excimer laser, and collimated with a pair of defining slits. The resulting 2.2 eV, 0.9 us D-atom pulse then crosses a well-collimated pulsed Hz beam at a point 29.8 cm from the electron-impact ionizer of a mass spectrometer detector, which records time-of-flight (TOF) distributions for the scattered HD products. The kinetic energy in center-of-
18 May 1990
mass (c.m.) coordinates was varied by changing the intersection angle of the two beams. For the comparison given below, it is essential to know accurately all of the experimental parameters which affect the TOF peak shapes or intensities. All of these were either measured directly or calculated from the apparatus geometry. By far the largest factor determining the experimental resolution function was the speed distribution of the D-atom beam, which was measured directly by TOF. It displayed an intense fast peak with Av/v= 11% full-width at halfmaximum (fwhm), followed by a slow tail having a shape which reflects mostly the internal state distribution of the DS photofragments. The angular distribution of D atoms was estimated from the slit geometry, the measured angular distribution of the D2S pulse, and the measured laser beam profile. The spatial profile and the velocity vector distributions of the Hz beam were measured directly with a pair of fast ionization gauge detectors [ 121. The effective geometry at the beam crossing point is determined by the overlap of the two beams with the volume viewed by the detector. This volume, which for simplicitly is illustrated as a cylinder in fig. 1, is actually determined by three apertures in the detector which define the element of solid angle subtended as a
H2 BEAM SOURCE
D ATOM
Fig. I. Schematic illustration of the experimental geometry.
514
GENERATION
Volume168,number6
18May 1990
CHEMICAL PHYSICSLETTERS
function of the angle with respect to the detector symmetry axis. This function was calculated numerically for the measured sizes and positions of the three apertures. The effective length of the ionizer was estimated to be 5 mm from the geometry of the frlaments and electron beam defining slits. Actually, however, this parameter contributes negligibly to the overall TOF resolution function. 0.6 3. Results and discussion In the course of this work, experimental TOF distributions were measured for nominal initial kinetic energies of 0.85, 0.90, 0.95, 1.00, 1.05 and 1.20 eV, with a spread of kinetic energies of 0.22 * 0.02 eV fwhm [ 131. In each case, the laboratory scattering angle was selected so that the slowest HD products observed corresponded to c.m. scattering angles near 180”, relative to the initial cm. velocity vector of the D atom. Fig. 2 shows the TOF distribution measured at a kinetic energy of 0.85 eV, together with a velocity vector diagram for this case. The two peaks at longer times correspond to U’= 0 and v’ = 1 backscattered products, while the first peak corresponds to products scattered near 70” c.m. which are not state-resolved. The inner and outer circles in the velocity vector diagram of fig. 2 correspond to HD products in the j’ =O state of U’= 1 and U’=O, respectively. Any rotational’excitation of the products results in a smaller final cm. speed and a shift of the TOF toward the middle of the distribution [ 41. It can easily be seen in fig. 2 that the backscattered peaks are shifted and broadened by rotational excitation. Because we know the instrumental resolution function very well, it is possible to extract information on the rotational state distributions from the locations and shapes of the vibrationally resolved peaks, as we did previously [4]. However, it must be recognized that the rotational distributions derived in this way correspond to the constant laboratoryscattering angle of the experiment, not a constant c-m. scattering angle. Each product rotational state is therefore detected with a distribution of c.m. scattering angles in which both the most-probable value and the width vary with the state. If the experimental resolution in collision energy and scattering angle were perfect, we could
50 Flight
100
150
Time (psec)
Fig. 2. Measured HD product time-of-flight distribution for 0.85 eV initial kinetic energy of collision. The velocity vector diagram shown corresponds to the most-probable reactant velocities. The dotted lines indicate the expected most-probable times of flight for products in the indicated vibrational states and the ground rotational state. The solid line is the simulation of this experiment using the theoretical results of Zhao et al. [ 91.
compare the extracted state distribution with theory by performing the appropriate laboratory-to-c.m. transformation of probabilities and comparing with theoretical values corresponding to a different value of angle for each j’. Such a comparison is problematic, however, when the experimental resolution function is not narrow, but is instead a complicated function of many variables. Of particular concern is the possibility that dynamical resonances predicted by theory might be quite narrow in energy compared to the experimental distribution. We have therefore chosen a different method of comparing theory and experiment, which we believe is the most rigorous possible at present. The principle of the method is to take the theoretical state-tostate differential cross sections as functions of both collision energy and scattering angle, and to use these as inputs to a Monte Carlo program which samples the probability distribution functions of all experi515
Volume 168, number 6
CHEMICAL
PHYSICS LETTERS
mental variables, thus simulating the experiment by averaging over all experimental parameters. The details of this simulation method will be presented elsewhere [ 131. Briefly, random samples are taken from the D and H, primary beam velocity vector and pulse profile distributions, and combined with random choices of spatial coordinates for the D+H2 collision and for the detector ionization of the HD product. This defines a unique velocity diagram and TOF bin for each final (u’ ,j’ ) state and thus a unique Jacobian for the c.m.-to-laboratory transformation. Each sample, simulating a collision between an individual pair of reactants, is then weighted with the product of the probability distribution functions for each experimental parameter and with the theoretical state-to-state differential cross section from an interpolation of the tabulated results in both collision energy and scattering angle. We had available from both Zhang and Miller [ 61 and Zhao, Truhlar, Schwenke and Kouri [ 91 their converged quantum results at five values of collision energy from 0.5 1 to 1.08 eV, and c.m, angle intervals of 5”. Both theoretical groups calculated state-to-state differential cross sections for the initial H2 reactant state j=O. These results are in good agreement 191, indicating that the dynamics are fully converged in both cases, and also that the differences due to the different functional forms in the LSTH and DMBE analytic potential energy surfaces are small. As expected, inserting the two independent sets of results into our simulation program produced negligible differences in the predicted TOF distributions. The experiments were done with a pulsed supersonic beam of normal-H2 from a room-temperature source, having a spread in speed of about 7% fwhm and a translational temperature of a few kelvin. The rotational degrees of freedom of H2 are, however, not very well cooled in supersonic expansion. From careful energy-balance measurements [ 141, the effective rotational temperature under these experimental conditions was estimated to be about 170 K, although significant deviations from a Boltzmann distribution may occur. The corresponding rotational populations are 20% j= 0, 74% j= 1, 5% j=2, 1% j=3, with negligible contributions from higher states. At 0 K, the populations would be 25% j=O and 75% j= 1. Since Zhao et al. [9] carried out calculations for 516
18 May 1990
both j=O and j= 1, we performed simulations using both sets of results and combined the two predicted TOF distributions with the weighting 251 j=O and 75% j= 1 to make the best comparison with experiment which is possible with the available calculations. The result is shown in fig. 2 as the solid line, which has been scaled to the experimental data at the largest peak. The discrepancies are easily seen. The calculations get the vibrational state ratio for backscattered products about right, but there is a large discrepancy in the ratio of forward-scattered to backward-scattered intensities. In addition, the experiments show more rotational excitation in the backscattered products, as exhibited by the shifting and broadening of the peaks in the experimental TOF spectrum relative to those predicted theoretically. This discrepancy becomes more pronounced at higher collision energies [ 13 1. In comparing the TOF spectra predicted theoretically for the 25%j=O, 75Wj= 1 mixture with that predicted for pure j=O, including j= 1 causes a slight shift toward higher product rotational excitation. However, the shift is small compared to the difference between theory and experiment, making it unlikely that the discrepancy between theory and experiment could be removed by including small populations of higher H, rotational states in the calculations. Since the two independent dynamics calculations carried out with two different analytic representations of the potential energy hypersurface are in good agreement with each other for the j=O initial state, it seems most likely that the source of the discrepancy between theory and experiment lies in the original calculated potential energy points [ 8 1. Although these points are probably among the most accurately calculated for any potential energy surface, it is not known just what level of accuracy is actually required to reproduce the experimentally measured features. It is also possible that the number of pointed calculated, particularly for bent geometries, is not sufficient to permit a sufficiently accurate global representation. Kliner, Rinnen and Zare [ 15 ] have recently reported rotationally resolved integral cross sections for the reaction D+ Hz+ HD (u’ = 1,j’ ) f H at a collision energy of 1.05 eV, and have also compared their results with the calculations of Zhang and Miller [ 61 and of Zhao et al. [ 91. In this case, the agreement between theory and experiment was very good, de-
Volume 168, number 6
CHEMICAL PHYSICS LETTERS
spite the fact
that higher populations of states+ 1 were present in the room-temperature H2 reactants than in our supersonic Hz beam. It appears, as one might expect 141, that the differential cross sections for this reaction are considerably more sensitive to the details of the potential energy surface and/or the dynamics than are the integral cross sections. In particular, the effects of dynamical resonances, which can easily be washed out in
the integral cross sections
bital
angular
momentum
by averaging states
of
the
over
or-
reactants
[ 6,9,16- 19J, are expected to persist to a much greater degree in the product angular distributions. Whatever the cause, the discrepancy reported here between theory and experiment must be resolved before it can truly be claimed, as stated in a recent news report, that “Sixty years after its initial development, quantum mechanics has finally succeeded in completely describing the simplest possible chemical reactions.” [ 201.
Acknowledgement
This research was supported by the National Science Foundation, Grant No. CHE-8705611. The authors are grateful to Dr. Jue Wang and Mr. Mark Nachbor for performing some supplementary experiments which contributed to the interpretation of the data presented here. We are also grateful to Dr. Zhang and Dr. Miller and to Dr. Zhao, Dr. Truhlar, Dr. Schwenke and Dr. Kouri for providing us with the extensive tables of their theoretical results needed for the simulation of the experiment. We also appreciate the comments of Dr. Truhlar on the manuscript.
18 May 1990
References
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