Faraday Discuss. Chem. Soc., 1982, 73, 187-203 Rotationally Inelastic Scattering of Hydrogen Molecules and the Nan-spherical Interaction BY UDO BUCK Max-Planck-Institut fur Stromungsforschung, 3400 Gottingen, West Germany Received 30th November, 198 1 Molecular-beam scattering experiments on state-to-state differential cross-sections for rotational transitions of hydrogen moIecules are a sensitive probe for both the isotropic and the anisotropic interaction potential with other atoms and molecules. New experimental results on the weakly coupled system D2 + Ar allow reliable determination of the repulsive part of the complete potential. It is compared with the well known D2-Ne potential and recently proposed model potentials which predict the general behaviour but not the details.For the important H2-H2 interaction differential cross-sections for 0+2 transitions in HD + D2 and Dz + H2 collisions are presented. The evaluation of these data leads to repulsive anisotropic potential terms which show the same shift of 0.1 8, to smaller R-values with respect to the best ab inirio calculation found previously for the isotropic interaction. Scattering experiments provide a direct tool for the determination of the interaction forces of Van der Waals molecules. Elastic scattering experiments essentially probe the isotropic potential whereas rotationally inelastic processes are due to the angular dependent non-central forces. For the determination of isotropic interactions as well as the attractive part of the complete potential surface many other methods are available.' However, the large-angle differential cross-sections for single rotational transitions are one of the few sources for information on the repulsive anisotropy of the system under study.For the interactions of neutral species cross-sections of this type are only available for few systems, namely for H, and its isotopes scattered from He,, Ne3-5 and Ar,6*7 for Na, scattered from He,8 Ne '3" and Ar,','' for N,, 0, and CO scattered from He l2 and for CH, scattered from He,', Ne l 3 and Ar.I3 The only molecular system studied in this way is H, + H, in its different isotope configurations HD + HD,14 HD + D2 and D, + H2.18 Among these, the hydrogen-molecule- rare-gas interactions and the hydrogen-molecule dimers are the best studied systems, involving as they do the analysis of many experimental properties.In particular, high- precision diffusion coefficients l9 and other transport phenomena,,' the spectros- copy of the Van der Waals c ~ r n p l e x e s , ~ ~ - ~ ~ bulk relaxation data,,, and molecular beam experiments on total total differential 26 and integral cross-sections with oriented molecules 27 have been used. In addition, these small systems are well suited to perform highly reliable calculations of the ab initio type with configuration inter- action included 28-30 or of the model type where the dispersion attraction is added to the SCF repulsion with suitable corrections for the overlap r e g i ~ n . ~ ' - ~ ~ The theoretical analysis of rotationally inelastic transitions of HD and D, molecules scattered from rare-gas atoms in the energy range from 30 to 90 meV gave results as follows. (i) The inelastic cross-sections are peaked in the backward direction, that is, the coupling essentially occurs in the repulsive part of the anisotropy. (ii) If we expand the interaction potential in Legendre polynomials V(R9.J) = VO(R) + ~,(R)~z(c0s.J)188 INELASTIC SCATTERING OF HYDROGEN MOLECULES where R is the distance between the atom and the centre of mass of the molecule and y is the angle between the molecular axis and R, two terms are sufficient to describe the scattering process for the homonuclear species.(iii) The 0 + 2 transitions of D2 are directly related to the repulsive V,(R) term. Since the coupling is weak we were able to construct an inversion procedure based on the exponential distorted wave approxim- ation which directly gives V2(R) once CJ (0 -+ 2; 3) is k n ~ w n .~ ~ ~ ~ ~ The magnitude of the coupling is determined by V, where 0 depends quadratically on V,. However, the range is given by the elastic wavefunctions which can be approximated by the classical turning point Ro of Yo. Therefore, Vo also has to be known for the analysis. If V,, and thus R,, decreases, generally a larger value of V, is probed and the inelastic cross- sections are approximately determined by the ratio V2/V0. (iv) The 0 + 1 transitions of HD are produced by the U,(R) term of the potential which is mainly sensitive to the Yo(&) term of the homonuclear interaction. Only a small dependence (ca. 10%) on the V2 term is left.Combining the rotationally inelastic transitions with the total differential cross- sections, we were able to derive a very reliable complete interaction potential for H,-Ne. Similar considerations hold for the scattering of hydrogen dimers. The analysis of the total differential cross-section and the 0 --+ 1 transitions of HD in HD + D, collisions leads to a very reliable isotropic potential for this system. The comparison with several ab initio potentials indicated that the surface calculated by Mayer and Schaefer 30 was the best one, although the agreement is not perfect. In the present contribution we summarize the state of the H,-Ne surface and present a similar analysis for D,-Ar. Based on the angular dependence of the 0 --f 2 transitions of D, + Ar collisions and a reliable Vo from the total differential cross- section, a complete surface is constructed.The results are compared with predictions from model potentials and also with the most recent version of the D,Ne surface. For the hydrogen dimer we summarize the present state of the attempt to derive a unique interaction potential for this important molecule-molecule system which correctly predicts the numerous data in the literature. We will, in particular, discuss the influence of the recently measured differential cross-sections for 0 -+ 2 transitions in HD + D, [ref. (17)] and D2 + H2 [ref. (IS)] collisions on the determination of the anisotropic part of the surface. It turns out that a particular combination of initially rotating molecules not only allows one to determine the usual anisotropy which corres- ponds to V , in the atom-diatom case but also the term which asymptotically corres- ponds to the quadrupole-quadrupole interaction.This term plays an important role in phase transitions of the molecular hydrogen solid at low temperatures and high pressures.38$ 39 EXPERIMENTAL The experiments have been carried out in a crossed-molecular-beam apparatus which has been described el~ewhere.~,~ The two colliding beams, which cross at 90°, are produced as nozzle beams from two differentially pumped chambers. The angular dependence is measured by rotating the source unit with respect to the fixed detector assembly. The scattered particles are detected by a doubly differentially pumped mass spectrometer operat- ing at pressures <lO-'O mbar.* Elastic and inelastic events are separated by time-of-flight analysis of the scattered particles using the pseudorandom chopping technique.The flight path has been extended to 589 mm: that is to say it is larger than in the previous experimental arrangement. In this way the velocity distributions of the scattered particles were measured with a resolution of cu. 2%, where effects from the finite ionization volume, the shutter * 1 bar = lo5 Pa.U . BUCK 189 function and the channel width have been included. Collection and processing of the data were executed by a minicomputer. The pumping facilities for the two beam sources permit the use of pressures up to 150 bar with 18 pm diameter nozzles. Therefore, the two intersecting beams not only have velocity distributions with a full width at half maximum of better than 5% but also expand in such a manner that nearly all molecules are in their lowest available rotational states.In addition, the accumulating time for measuring time-of-flight spectra could be reduced by a factor of 25 compared with expansions at 2 bar stagnation pressure. The actual beam conditions for three scattering experiments are given in table 1 . For the TABLE BEAM DATA parameter 0-Dz Ar HD n-D, o-D, n-H2 ~~ nozzle diameter/pm 10 source pressure/bar 192 source temperature/K 304 peak velocity/ms- I 2 042 speed ratio, S 29 rotational temperature/K 70 fraction in j = 0 0.89 j = 1 . . . . j = 2 0.1 1 100 3 304 560 21 ... ... . . . ... 50 20 10 80 164 306 1930 2040 31 30 < 10 68 1.00 0.60 .. . . 0.33 .... 0.07 18 150 303 2 131 37 56 0.95 0.05 .... 20 70 305 2 874 39 115 0.24 0.75 0.01 Ar secondary beam the pressure has been kept at low values in order to avoid condensation. Similar considerations hold for the cooled HD beam. The homonuclear hydrogen molecule beams at room temperature are not limited by condensation. Since there are two nuclear spin modifications of normal (n)-hydrogen molecules these beams consist of 5 o-D2(i p-H2) and + p-D2($ o-H,) for the even and odd rotational states, respectively. In order to increase the contribution in j = 0, the gas lines could be fed to converter cryostats to produce pure o-D, or p-H, beams. The degree of rotational population of the different beams has been obtained by a careful energy balance, which also accounts for real gas e f f e ~ t s , ~ ~ ' ~ and a suitable extrapolation of Raman studies for these nozzle beams at lower stagnation pressures.4o The time-of-flight spectra (see for instance fig.6 and 8) obtained at different laboratory angles are normalized to their maximum intensity after subtracting a large background contribution (signal-to-background ratio ca. 1 %). The accumulating time varies between 2 and 4 h. The peaks which appear at flight-times corresponding to inelastic transitions are clearly resolved. The procedure to derive cross-section ratios from these data has been discussed in detail in ref. (5). Briefly, as a first step the distribution functions in the final velocity space are calculated by a Monte-Carlo procedure for the elastic and possible inelastic transitions using the angular and velocity spread of the two intersecting beams and the trans- mission function of the time-of-flight analyser.As for the second step the measured spectra are fitted to the calculated distribution functions. If the half-width of the distribution func- tion is smaller than the corresponding energy loss due to the inelastic transition, the problem can be solved in a simple manner. The fitted amplitude Aif is directly related to the cross- section oif(g,g) by where K is a constant with respect to velocity variables and only depends on the laboratory deflection angle 6, p i is the relative population of the initial state, Jif the Jacobian for the transformation from the centre of mass (CM) to the laboratory system, g the relative velocity and 8 the CM scattering angle.The bars indicate average values within the small distribution function. It may happen that several individual cross-sections contribute to one energy-loss peak. If we normalize these values to the elastic contribution we can derive cross-section ratios. To obtain absolute inelastic cross-sections the total differential signal is used to- gether with a suitable calibration based on a calculation. Depending on the information190 INELASTIC SCATTERING OF HYDROGEN MOLECULES available for a system we will present the data as cross-section ratios or absolute inelastic cross-sections in the further course of this work. RESULTS HYDROGEN-MOLECULE-RARE-GAS SCATTERING RESULTS FOR H,-Ne For this system the complete potential surface has been accurately determined from molecular-beam scattering data.5 The high-resolution total differential cross- section for D2 + Ne and the rotationally inelastic cross-section for the 0 + 1 tran- sition of HD + Ne, together with the second virial coefficients as constraints for the long-range attractive part, were used to determine the isotropic part of the potential.For the anisotropy, the angular dependence of the 0 -+ 2 transitions of D, + Ne has been used together with the integral cross-section for different orientations 27 as constraints for the attractive part. The main input data are essentially sensitive to the position, slope and magnitude of the repulsive part of the potential as shown in a direct inversion pr0cedu1-e.~~ The accuracy of the isotropic potential was nicely confirmed by a comparison of the high-precision diffusion data l9 with predictions of several potential models. Among all H,-rare-gas systems this potential was the only one which predicted the measured data correctly.For the anisotropy small dis- crepancies existed in the attractive part. Thus a new measurement of the hyperfine dimer spectrum for H,-Ne by the magnetic-beam resonance technique was used to slightly modify the V, term in the attractive part.23 Fig. 1 shows a comparison of a calculation of the differential cross-section for the 0 + 0 and 0 + 2 transitions of D2 + Ne based on this modified V, value 23 and the V, value of ref. ( 5 ) with the measured 10’ 1 oo * I LI 10- ’ 0 50 100 150 9.1 a FIG.1.-Measured differential cross-sections for O+O and 0+2 rotational transitions of Dz + Ne and calculations based on the potential surface of ref. (5) with slightly modified attractive anisotropy [ref. (23)] (see fig. 4). E = 84.9 meV.U. BUCK 191 data. The agreement is excellent, so we can consider this potential (see also fig. 4) as being well-understood in both the attractive and repulsive regions. It can be used as a good test case for the accuracy of ab initio calculations. RESULTS FOR H2-Ar In contrast to the case of H,-Ne, the H2-Ar potential is only well known in the attrac- tive part. A careful determination of this potential by Le Roy and Carley from spec- troscopic data of the Van der Waals complexes ,' was confirmed by total differential cross-sections 26d and low-energy integral c r o s ~ - ~ e c t i o n ~ .~ ~ ~ Therefore, it is very interesting to perform a similar analysis as for D2-Ne where essentially data are used 10 n C c) ." c1 Y Y 1.0 0 . l l I I I 1 I I I I I 1 10 20 30 40 50 laboratory deflection angle,O/" FIG. 2.-Measured total differential cross-sections for Dz + Ar in the laboratory system. The calculations (solid line) are based on the potential of this work (see table 3 and fig. 3). E = 83.2 meV. which are sensitive to the repulsive part of the interaction. Fig. 2 shows a measure- ment of the total differential cross-sections at the energy of E = 83.2 meV.7 The diffraction oscillations are clearly resolved and serve as a precise measure for the zero point of the isotropic potential and the form of the repulsive wall. The large-angle part, which has been obtained up to 8 = 70°, gives the slope of the repulsion.In- elastic 0 + 2 transitions have been measured at E = 85.0 meV and are given in table 2. Because of large background problems there are only three points available. First, we compare our data with the best potential of Le Roy and Carley,,I called BG, where the collapsed diatom limit has been incorporated in the interaction potential. It is of the general Buckingham-Corner form Vi (R)= Aiexp(-BiR) - (C6JF + C8iR-8)f(R)192 INELASTIC SCATTERING OF HYDROGEN MOLECULES with There are four free parameters, Ai, pi, C,i and Csi, for each potential i = 0, 2. For convenience two of these parameters, A i and Csi, are replaced by the parameters of the potential minimum, Rmi and ci, the distance and the well depth, respectively.The calculated total differential cross-sections based on this potential show very small deviations in the oscillatory regime and are slightly too large in the monotonic large- angle part. Such behaviour is to be expected, since this potential is not able to TABLE 2.--a(0-+2)/0(0-+0) FOR D2 + Ar AT E = 85 meV IN PERCENT O(lab)/" I ~ ~ ~ ( C M ) / " 8,,,,(CM)/" experiment this work LC a TT RS 60 72.7 72.2 6.0 f 1.2 6.1 5.6 3.4 5.2 70 85.5 85.5 8.9 & 1.0 8.5 7.8 4.6 7.1 80 98.3 98.8 10.3 f 1.2 10.8 10.0 5.9 8.8 a Ref. (21), BC3 experiment; ref. (31), model; ref. (33), model. predict the diffusion coefficients of Dunlop and Tengro~e,'~ which also probe the repulsive part.The calculated inelastic cross-sections are lower than the measured values (see table 2). Since, on the other hand, the deviations are rather small we decided to use the same potential form and to determine mainly the parameters p1 by the large-angle measurements, keeping the other three parameters as close as possible to the values determined by spectroscopy. The result of such an analysis is a potential which differs in RmO = 3.60 0.03 A by 0.8% and in Po = 3.4 & 0.2 A-' by 5.8%. Then, following the procedure described in detail in ref. ( 9 , the complete potential surface is obtained from the inelastic 0 --f 2 differential cross-sections. To solve the coupled equations, the coupled-states approximation has been used. With a basis of j = 0, 2, 4 convergence was reached.The final result for the two potentials is dis- played in fig. 3 and table 3. The calculations based on this result are shown in fig. 2 and table 2 in comparison with the experiment. As expected, the new isotropic potential Vo is a little softer than Le Roy's potential, whereas V, is nearly the same. Since, as discussed in the introduction, the inelastic transitions depend on both V, and Yo, the lower value predicted by the BC, potential of Le Roy is easily understandable.* A calculation of the diffusion coefficient gives a value of 0.813 cm2 s-l in reasonable agreement with the experimental value of 0.824 cm2 s-l. Therefore, we think that this potential is also very reliable concerning both the isotropic and the anisotropic part. What still has to be done is the incorporation of the spectroscopic data of the infrared absorption and the hyperfine structure of the Van der Waals molecules in the evaluation of the data.This will be done in a forthcoming paper and ought not to change the results very much, since the attractive part has been kept nearly fixed in the present procedure. DISCUSSION We now have two reliable potential surfaces of the H2-rare-gas interaction, which makes it interesting to compare the results. This is done in fig. 4, where we plot the * This conclusion also holds for the cross-section ratio, since the elastic cross-section introduces only a much weaker additional dependence on Vo.U. BUCK 193 40 30 20 2 .- 4 4 3 g 10 u 0 -10 I I I 2.0 3.0 3.5 distance/W 0 FIG. 3.-Potential curves for D,(H,) + Ar: (-) this work; (- .-) ref. (21); ( x ) ref. (31); (0) ref. (33). result for H2-Ne on the same scale as the H,-Ar result of fig. 3 and the ratio V,/Vo for the two potentials. The two potential surfaces are very similar. The potential well for the isotropic part is roughly a factor of 10 deeper than for the anisotropic part. Both potentials cross just above the zero point and Y2 is generally much smaller than V,. A comparison of the absolute potentials shows the expected behaviour. The TABLE 3.-POTENTIAL PARAMETERS FOR D2 f Ar Vo 6.307 3.595 3.175 1686.18 3.40 16.676 191.71 Vz 0.711 3.725 3.335 719.55 3.69 1.6738 3 1.746 value of the position of the potential minimum is shifted by ca. 0.3 A and the minimum itself is larger by a factor of two for the H,-Ar system, obviously due to the larger polarizabilities.The same shift is also observed in the repulsive region. An import- ant quantity for the interpretation of rotationally inelastic data is the value of V, at the classical turning point of Yo, which can be expressed by the ratio V,/VO and which is roughly responsible for the strength of the transition. The comparison shows the surprising result that this ratio is larger for H,-Ne than for H2-Ar. Therefore, 0 -+ 2 inelastic transitions are larger for D2-Ne than for D2-Ar at the same collision energy, which can be seen directly by looking at the measured time-of-flight spectra. Since no ab initio potentials are available in the literature. we compare our results194 INELASTIC SCATTERING OF HYDROGEN MOLECULES with predictions of the so-called Hartree-Fock dispersion m ~ d e l , ~ ’ .~ ~ where the cal- culated repulsion is added to the calculated dispersion with some suitable corrections for the vanishing contribution of the dispersion in the overlap region. In the Tang- Toennies (TT) model 31 this is done by the Drude model, whereas Ahlrichs et aL3, use an empirical damping function for all coefficients. In a refined version Rodwell and Scoles (RS) 33 introduce individual damping functions for each coefficient. First we LO 30 20 % E 8 . - .d C w 10 0 -10 I I I 0.21 D z + A r / X I 1 1 2.5 3.0 3.5 i distance/%i FIG. 4.-Potential curves for D2(H2) + Ne: (-) ref. (5) and (23); ( X ) ref. (31); (0) ref. (33). Insert: Ratio of VJV0 for the best experimental potentials of fig. 3 and 4 as a function of the classical turning point Ro( Vo) expressed by the values Vo of the isotropic potential.compare the predictions of these models for the measured ratio of the 0 -+ 2 transition to the 0 -+ 0 transition of D, + Ar. Both predictions are too low, ca. 40% for the TT model and 15% for the RS model, indicating that the ratio V2/ V, is too low. An inspection of the individual potentials in comparison with our best-fit results in fig. 3 shows that the deviations for the TT model are essentially due to V, being too large, while the RS model gives too small V, values. The comparison of the D,-Ne potentials shows deviations for the TT model from the experimental best-fit potential which partly compensate each other for the prediction of 0 -+ 2 transitions and a good agreement for the RS model.Thus the predictive character of the two models is satisfactory if the requirement for accuracy is not too strong. The high-precision measurements presented here can deviate appreciably, as shown for D, + Ar and the TT model.U. BUCK 195 HYDROGEN-MOLECULE DIMER SCATTERING THE INTERMOLECULAR POTENTlAL The interaction potential for a diatom-diatom system can, as in ref. (30), be expanded in complete orthonormal sets of sperical harmonics Wl,YZ,R) = 2 A I , I , L ( W 2 , R ) 2 (4m1~2m2l~J2W YAlrnI(4) rnz,z(+2) Y*Irn(R) . u 2 L mlrnlni where Y, and Y, are the internuclear vectors of the two molecules and R is the separation of the centres of mass. (a b c dl * - - .) is the usual Clebsch-Gordon vector coupling coefficient.Fig. 5 gives the leading potential terms for H2-H, as obtained from the 10 a 6 4 2 0 - 2 f ’ I I I I 2.2 2.6 3 .O 3.4 3.0 RIA FIG. 5.-Potential terms for H2 + H2 in spaced fixed coordinates of the ab initiu calculations of ref. (41) and (42). The dashed line is the isotropic potential determined from ref. (16). The arrow marks the position of the classical turning point of Aooo at E = 89.1 meV. most elaborate ab initio calculation by Meyer and S ~ h a e f e r . ~ ~ It is an improved version of the results published in ref. (30) which is based on a six-term rigid-rotor expansion at vibrationally averaged distances ri = 1.449 a.u. of the ground state u = 0, j = 0. These results have been corrected first for the different vibrationally averaged distances for differentj-states (vib-rotor) and second for a 19-term expansion at small distances.42 The first term, Aooo/(4n)3’2, is exactly the isotropic term.The leading anisotropic terms are The first corresponds to the Vz term of the = A022 and A224.196 INELASTIC SCATTERING OF HYDROGEN MOLECULES diatom-atom potential and is responsible for the 0 +- 2 transitions of one molecule. The second is asymptotically due to the long-range quadrupole-quadrupole interac- tion of the two molecules and couples only two rotating molecules. There are numerous experimental investigations of this system, ranging from equilibrium properties of gases and solids 43 and transport and relaxation phenom- ena 20*24 to scattering experiments 25-27 and the spectroscopy of the dimer.22v44 The isotropic interaction potentials, which are in most cases derived, differ appreciably from each other and suffer from the fact that they are mostly determined by one measured quantity. However, there are some general trends which converge to the same solution. In a comparative study Rulis and Scoles 26c found that the potentials derived by Silvera and Goldman 43 from solid-state equilibrium data also fitted their total differential cross-sections.Independently we ended up with nearly the same potential from the measurement of the total differential cross-section for D2 + H2 and the differential cross-sections for the 0 -+ 0 and 0 -+ 1 transitions of HD scattered from D2.* This potential is also plotted in fig. 5. All these measurements are mainly sensitive to the near-repulsive part of the potential from the minimum upwards.As is clearly seen there is a small but significant difference from the ab initio potential of Meyer and Schaefer, being essentially a shift of the zero point by 0.08 A to lower values. A very recent multiproperty analysis of high- and low-temperature virial coefficients, bound- state energies and low-energy integral and differential cross-sections confirms this conclusion, giving a potential zero of Ro = 3.018 A [ref. (45)], in close agreement with our value of A, = 3.00 0.03 A. All experiments which essentially probe the attractive part of the potential, such as low-energy integral cross-sections 25a-c or transport data,20 are predicted within 510% by the ab initio potential.No other calculated potential is precise enough to reproduce all these data, which are sensitive to the isotropic potential. A very similar conclusion holds for the experimental test of the anisotropic terms. A comparative study of rotational relaxation showed that none of the published potentials were able to fit these data.24 Up to now only the potential surface of ref, (41) and (42) gave good agreement for the rotational relaxation 42 and tensor cross- sections in the field dependence of transport p h e n ~ m e n a , ~ ~ as well as for low-energy transport data 2o and the integral cross-sections with oriented molecules.44 It should be noted that for those quantities 42,46 which are essentially sensitive to the repulsive part of the interaction the correction due to the nineteen-term expansion proved to be significant in reaching agreement with the measured quantities.We summarize: the corrected potential surface of Meyer and Schaefer is the best description of all the data available, with the exception of the repulsive part of the isotropic potential near the zero point, which should be a little softer. MEASUREMENTS AND RESULTS To test the anisotropy of the H2-H2 potential by microscopic properties we have performed measurements of the differential cross-section for 0 -+ 2 transitions for the HD + D2 system at 70.3 meV l7 and the D, + H2 system at 89.1 meV.18 We will discuss both systems separately since the analysis showed that they are sensitive to different parts of the potential surface. * A reevaluation of the total differential cross-section for D2+H2 presented in ref.(16) gave the parameters for the potential model of ref. (16) with the damping function exp { - [2.29(C8/C6)1'2/R - 112} : A = 101.4 eV; p == 2.779 A-1, y = 0.08 A-2; C, = 7.264 eV A6; c8 = 36.008 A8 eV; C,, = 225.56 eV; e = 2.92 meV and R, = 3.436 A.U. BUCK 197 HD + D2 The measured time-of-flight spectra are shown in fig. 6. The possible transitions are listed in table 4 and the arrows in fig. 6 mark these transitions in the time-of-flight spectra. The spectra are characterized as follows: (i) At small angles the elastic transition (0) dominates. (ii) The contribution of the 0 -+ 1 transition of HD(1) increases with increasing scattering angle and exceeds that of the elastic transitions at 1 .o 0.8 -5 0.6 h !3 -5 0.4 8 75! 0.2 G o u a .- G I 30" 5 0" Loo I 012 3 I 01 2 3 60" 0 1 2 3 - loops flight time FIG.6.-Measured time-of-flight spectra for HD + D2 at different laboratory scattering angles The numbered arrows mark the transitions listed in table 4. large angles. (iii) The 0 -+ 2 transitions of D2 give almost no detectable contribution over the measured angular range. (iv) Transition (3), which consists of the 0 -+ 2 transition of HD and the simultaneous transition 0 -+ 1 of HD and 0 --f 2 of Dz, increases with increasing scattering angle. Now we have to explain why transition (3) is much larger than transition (2) although it needs 50% more energy to be excited. In addition, we would like to know to which part of the potential surface transition (3) is sensitive. To answer these questions, we first have to transform the H2-H2 potential to the HD + D2 coordinate system.The details of this procedure have been described in ref. (16). TABLE 4.---TRANSITIONS FOR HD + D2 energy loss/ no. meV transition (HD, D2) 0 0.00 oo+oo, 01-+01,02+02 1 11.06 00-+10,01+11,02-+12 10.91 02-20 2 22.20 004 20 3 33.1 1 00+20,01-+21,02-+22 33.26 OO+ 12198 INELASTIC SCATTERING OF HYDROGEN MOLECULES Then we have to perform close-coupling calculations based on this transformed potential. The calculations were performed as described in ref. (16) and (30). Only open channels have been retained in the close-coupling basis expansion, which means that the relatively large cross-sections, which have been measured in the experiment, are considered converged within a few percent.The calculations were performed for the latest version of the ab initio potential of Schaefer and Meyer 41942 and a potential surface where only the Aooo term was replaced by the experimentally determined rigid- rotor potential derived from ref. (16) * keeping all the other terms of the ab initio potential and performing the necessary transformations. The results are given in table 5 for two characteristic angles. In addition we artificially changed the AZo2 TABLE CALCULATED ROTATIONAL TRANSITIONS FOR HD + D2 AT E = 67.1 meV a potential $(CM)/" 00 10 20 12 02 remark ref. (42) and (43) 60 1.875 0.545 0.035 0.013 0.011 - 180 0.321 0.524 0.117 0.033 0.015 ref. (42) and (43) 60 1.710 0.505 0.042 0.020 0.022 reduced Aooo: ref.(16) * 180 0.398 0.444 0.122 0.044 0.022 isotropy ref. (30) 60 1.854 0.547 0.036 0.017 0.015 - 180 0.303 0.560 0.102 0.042 0.023 ref. (30) 60 1.846 0.556 0.026 0.010 0.004 reduced 0.67 4202 180 0.358 0.588 0.094 0.027 0.007 anisotropy a The initial state is always 00; the values are in A2 sr-l. term of the H2-H, interaction by 33% in order to study the influence of a pure anisotropic change of the potential. The potential surface used is a previous version of the ab initio potential of Meyer and Schaefer 30 where the anisotropy is somewhat larger. The calculations confirm the experimental result. The fact that the 00 -+ 02 transition of D2 is excited with much less probability than the transitions 00 --+ 20 and 00 --f 12, where essentially HD is excited, is obviously due to the different coupling elements.The former transition is only caused by the coupling element Ah22, whereas AiO2 (which is larger than Ah,,) and A;21 are reponsible for the latter ones.7 In addition, successive transitions of Aj = 1 steps might also be possible for these transitions, which are caused by the much larger Aiol coupling term. The results of the calculations for the 0 -+ 2 transitions based on different potential variations are summarized as follows: (i) Decreasing the isotropic part Aooo by ca. 30% leads to larger cross-sections for 0 -+ 2 transitions by approximately the same amount. Only the 00 --f 20 transition (HD) in the backward direction is less affected. As discussed in detail for atom-molecule scattering, this general behaviour is caused by probing the anisotropy at smaller R values where it is larger.(ii) Decreasing the anisotropic part A202 of the H2-H, interaction by 33% leads to a decrease of all 0 -+ 2 transitions, again with the exception that the backward scattering of the 00 --+ 20 transition is less affected. We conclude that 0 -+ 2 transitions in HD + D2 collision are determined by equal amounts of the isotropic and the anisotropic potentials. However, in the extreme backward direction the 00 --+ 20 transitions of HD are more influenced by the A'lOl t Note that the prime (') is used for the transformed potential in HD + D2 coordinates. For * See previous footnote. details and a figure, see ref. (16).U . BUCK 199 term, which causes double transitions and which mainly depends on the isotropic potential.As described in detail in ref. (16), this behaviour leads only to small changes in the transition probability when varying the isotropic potential, since its influence on AAoo and Aiol almost compensate each other. Fig. 7 shows the comparison of the measured data with calculations based on the B(lab)/" I I I I I I I 0 50 90 130 170 deflection angle, SF,(CM)/O FIG. 7.-Comparison of measured differential cross-section ratios for HD + D2 with calculations based on the ab initio potential of ref. (41) and (42). For the transitions involved see table 4 and text. E = 70.3 meV. ab initio potential of ref. (41) and (42). We have plotted the expression a3/(ao + a,) with a3 = po [O(OO -+ 20) + a(O0 -+ 12)] + p1a(Ol -+ 21) + p2402 --+ 22) 0 0 = po a(O0 -+ 00) + p,a(Ol -+ 01) + p,a(02 + 02) o1 = po ~ ( 0 0 -+ 10) + plo(Ol + 11) + p2 [a(02 -+ 12) + a(02 --+ 20)].This quantity is easily derived from the time-of-flight distribution. For the calculations we have to sum all the different components. The comparison clearly shows deviation at large angles but agreement at intermediate angles. The agreement at intermediate angles indicates that the ratio of the anisotropic potential to the isotropic one is correct. However, the disagreement at large angles (probably being determined by successive 0 -+ 1 transitions) indicates that the isotropic potential is not correct. Using the experimental fit potential for Aooo would give larger values for the theory. To get agreement with experiment the anisotropy has also to be lowered.200 INELASTIC SCATTERING OF HYDROGEN MOLECULES Dz 3- H2 To check the conclusion of the last section we have tried to measure the very weak 0 -+ 2 transitions of the homonuclear system.We started with the system 0-D, + n-H, at E = 89.1 meV. The beam data are given in table 1. The result of a time-of- flight spectrum at the laboratory angle of 6' = 60" which corresponds to 9. = 160" in the centre-of-mass system is shown in fig. 8(b). An inelastic peak, which is due to the I 00-0020 02 1 01-01 21 I t I ~ ~ I ~ I ~ I L ~ ~ ~ ~ I ~ I ~ ~ 8 , a l l i l l l , l . , i l i , , , , , . . o r 20 0 300 200 3 00 flight time/,us flight timeips FIG. &-Measured time-of-flight spectra for D2 + Hz at E = 89.1 meV and &lab) = 60". The inelastic transitions are also shown in 5-fold enlargement.The solid lines are the results of a Monte- Czrlo simulation of the spectra with the cross-section ratios given by the ab initio potential of ref. (41) and (42). (a) o-Dz +p-H2. (b) o-Dz + n-H2. 0 -+ 2 transition of D,, is clearly resolved. To demonstrate this fact this portion of the spectrum is enlzrged by a factor of 5 in the figure. Since the scattering partner is n-H,, 75% of these molecules are i n j = 1 before and after the collision. A careful inspection of the coupling matrix elements 30 shows that such a transition 01 -+ 21 is due to both the AZo2 and the A,,, tzrm. The quadrupole-quadrupole term enters because H2 rotates before and after the collision. To separate this term from the usual anisotropy we performed a similar experiment for 0-D, + p-H, where the H2 molecules do not rotate before and after the collision.The result is presented in fig. 8(a). Now two very weak inelastic peaks appear which are due to the excitation of D,(OO -+ 20) and H,(OO -+ 02). The striking feature is that the 00 -+ 20 transition is much less in intensity than the same transition 01 -+ 21 with a rotating collision part- ner. This is obviously caused by the different matrix elements which are responsible for this transition. Thus it is possible to extract both anisotropic terms from these two sets of data. The calculation based on the ab initio potential of Schaefer and Meyer 4 1 3 4 2 is shown in fig. 9. The measured behaviour is confirmed. All inelastic transitions are smaller by more than one order of magnitude compared with the elastic transitions.The 01 --+ 21 transitions are approximately a factor of two larger than the 00 --f 20 tran-U. BUCK 20 1 sitions. The peaking in the forward direction clearly stresses that this difference is essentially due to the A,,,-term which is the largest anisotropic contribution at long- range distances and which is responsible for the small-angle scattering. To compare these calculations with the measured points we have carefully evalu- ated the measured time-of-flight spectra as described above. Since the amplitudes are 0 30 60 90 120 150 180 angle/" FIG. 9.-Calculated differential cross-section for rotational transitions of il(Dz)j,(H,)-ti:(Dz)i;(H,) at E = 89.1 meV based on the ab initio potential of Meyer and Schaefer [ref.(41) and (42)]. The experimental points are normalized to the elastic cross-sections and have been corrected for the measured level population: 0,01+21; A, 00-+20; 0,00-+02. rather small it proved very important to simulate the experimental distribution func- tions. The solid lines in fig. 8 are based on such a simulation with the cross-section ratio from the ab initio potential of ref. (41) and (42). The final results normalized to the calculated elastic cross-sections are displayed in fig. 9. In contrast to the large- angle result for HD + D2, the points are reproduced by the ab initio potential within their experimental error. This result is in agreement with the investigations on the rotational relaxation 42 and tensor cross-sections in transport properties 46 for H2 + H2, which were all predicted correctly by the ab initio potential.According to the discussion of atom-molecule scattering we know that 0 -+ 2 transitions of homo-202 INELASTIC SCATTERING OF HYDROGEN MOLECULES nuclear molecules are determined by the ratio of the anisotropy to the isotropic potential accounting for the fact that the coupling range is changed by the classical turning point. Since we know from several studies that the Aooo term has to be lowered 16*43945 compared with the ab initio calculation we conclude also that the anisotropic terms and A224 have to be shifted by the same amount. Indeed first calculation with a complete potential surface shifted by 0.1 8, gave the same results for the inelastic transitions." As for the accuracy of the final result it should be noted that the cross-sections depend quadratically on the potential, which means that even with experimental errors of 20% the potential is known with an accuracy of 10%.This result also has interesting consequences for the phase transitions in solid H2,38*39 which were not predicted correctly by the theoretical quadrupole interaction. The experiment required a smaller interaction in agreement with our conclusion. Almost the same result was obtained by the analysis of the hyperfine spectrum of the hydrogen- molecule dimer, which mainly probes the attractive interaction. The measured quantities could only be explained by shifting the potentials, both the isotropic and the anisotropic ones, to smaller values.22 SUMMARY The measured angular dependence of 0 --f 2 rotational transitions of D2 in D2 + H2 collisions proved to be a sensitive probe for the repulsive anisotropic part of the H2-H, interaction potential.Depending on the rotation of the scattering partner H2 both the normal anisotropy A202(R) and the quadrupole-quadrupole term A,,,@) are determined. As for the isotropic part AOo0, a shift of 0.1 8, to smaller R values is found compared with the most elaborate ab initio potential computed by Meyer and Schaefer. Note that the inelastic cross-sections are correctly predicted by this potential. However, since the cross-sections are sensitive to the ratio of anisotropic to isotropic forces, the necessary correction for the isotropic part found in earlier investigations has also to be applied to the anisotropic forces.I thank Drs F. Huisken, H. Meyer, A. Kohlhase and D. Otten for their contri- butions to this paper. I also thank Dr J. Schaefer for performing the close-coupling calculations for HD + D, and D2 + H2. I am grateful to Prof. R. B. Le Roy for helpful correspondence on his H,-Ar potential and to Prof. R. T. Pack for providing me with a copy of his program for calculating diffusion coefficients. Faraday Discuss. Cheni. SOC., 1982,73. W. R. Gentry and C. F. Giese, J. Chem. Phys., 1977, 67,5389. U. Buck, F. Huisken, J. Schleusener and H. Pauly, Phys. Reu. Lett., 1977,38, 680. U. Buck, F. Huisken and J. Schleusener, J. Chem. Phys., 1980,72, 1512. ' J. Andres, U. Buck, F. Huisken, J. Schleusener and F. Torello, J.Chem. Phys., 1980, 73, 5620. J. Andres, U. Buck, F. Huisken, J. Schleusener and F. Torello, in Electronic and Atomic Collisions, ed. N. Oda and K. Takayanagi (North-Holland, Amsterdam, 1980), p. 531. ' U. Buck, H. Meyer and R. B. Le Roy, to be published. * K. Bergmann, U. Hefter and J. Witt, J. Chem. 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