GENERAL DISCUSSION Dr. P. G. Burton (Uniuersity of Wollongong) said: My comment relates to the H2 dimer intermolecular potential ; specificially at this stage the isotropic component of this potential. A comment is made concerning the leading anisotropy of this potential following Reuss's paper. We have been intrigued from the theoretical point of view by the odd characteristics of the small but important discrepancy between the recent ab initio potential of Schae- fer and Meyer and several semi-empirical H2-H2 isotropic potentials. Of the latter, the apparently good agreement of the Silvera and Goldman " solid H, " pair potential and the independently derived " total differential scattering " pair potential of Buck et ~ 1 . ~ combined with the multiproperty-fit potential of Mc- C ~ n v i l l e , ~ suggests that the low part of the repulsive wall of the Schaefer-Meyer potentials is too " hard " or repulsive, and that for example their ab initio R, value (zero crossing of the isotropic component of the potential) of 3.09 A is too great by 0.07-0.10 A.Reference to Buck's fig. 6 illustrates how the Schaefer-Meyer potential would have to be adjusted inwards to agree with these semi-empirical potentials, and we were struck by the apparently rapid onset at ca. 3.5 A of such an adjustment to the ab initiu potential. It was difficult to see from the theoretical point of view what might have been missing from the ab initiu calculations that could account theoretically for such a difference. One possibility of course was a deficiency in the theoretical models upon which experimental data fits were based; that the choice o f " damping " terms in the HFD models to interpolate between the long-range Van der Waals coefficient expansion of the potential and the short-range Hartree-Fock data typically underestimated the distance of separation beyond which it is appropriate to replace the interaction potential by a purely asymptotic multipole form.The use of unzform-fields to compute atomic or molecular polarizability is ultimately inappropriate to the real case of a localised, nearby perturbation (characterised by radiating lines of force), and accord- ing to Koide s significant modification of the conventional symptotic expansion should extend well beyond an intermolecular separation of 8 ~ 0 , where in any case overlap and other effects (which must be determined through supermolecule computations) are already significant.However, since the Schaefer-Meyer potentials are more repulsive in the low part of the repulsive region of the isotropic component than all the recent semi-empirical potentials, a second possibility which we have addressed in our own H2-H, calcu- lations was that the basis sets used in the Schaefer-Meyer electronic structure com- putations of the H,-H, are unable to account for more than ca. 92% of the electron correlation energy of H,, and the additional electron correlation beyond that accessible in their calculations might have softened the molecular interaction. I n contrast to the approach of Schaefer and Meyer, where different basis sets are used in the computation of different features of the overall interaction, cf.ref. (7), we have consistently used the same basis for each H, throughout our supermolecule calculations, and this basis set accounts for 96.3% of the electron correlation energy of an isolated H2 molecule.* With such a basis of 39 functions per H, we compute the R, value for the isotropic component of the interaction to lie between 2.96 and 3.02 A,276 GENERAL DISCUSSION depending on whether CEPA2-PNO or the PNOCI method was used to allow for correlation effects in the supermolecule caIculations (see comment on the paper by Stone and Hayes concerning these methods). We can see then in our results for H,-H, that there is a justification from the theoretical point of view for adjusting the Schaefer-Meyer potential to be more attractive in the low part of the repulsive wall. What is certainly not apparent from our calculations is why the significantly increased correlation within each H, and between the H2 molecules that we find in our calculations (compared with the Schacfer-Meyer results) shouId not extend right out to the range of R = 11.0 a,, where the asymptotic mdtipole form can take over.In our calculations, the softening of the low repulsive part of Vooo is intimately associated with a deepening of the computed well depth. This effect extends out beyond 8ao, making our rigid-rotor H,-H, potentials significantly deeper than all the recent semi- empirical determinations of the isotropic component of the potential [E,(PNOCI) 2 27.5 cm-I, cm(CEPA2-PNO) = 33.6 cm-’1.Whether more than six different angular conformations need to be explicitly treated to define the spherical average potential * precisely enough in the region of the Van der Waals minimum, whether the rigid-rotor approximation itself is too severe due to the large vibrational amplitudes of H2 in the H,-H, complex and “ intramolecular ” vibrational energies (including the zero points) are modified by the interaction, or whether the experiments are especially sensitive to a preferred geometry or geometries of interaction, for which the effective potential is very different from our isotropic potential (which is determined by equal weighting of all angles in the spherical averaging), remains to be seen, J. Schaefer and W, Mcyer, J.Chent. Phys., 1979, 70, 344; J. A. Schaefer, personal communic- ation of 1980 results which onIy slightly modify the isotropic potential. 1. F. Silvera and V.V. Goldman, J. Chem, Plzys., 1978, 69, 4209. (0) M. G. Dondi, U. VaIbusa and G. Scoles, Chrtir. Phys. Letf., 1972, 17, 137; ( / I ) 3. M. Farrar and Y.T. Lee, J . Chem. Phys., 1972, 57, 5492; (c) A. M. Rulis and G. Scoles, Chem. Phys., 1977, 25, 183 ; ( d ) A. M. Rulis, K. M. Smith and G. Scoles, Cmi. J . Phys., 1973,56, 753. G . T. McConville, J . Chem. I’hys., 1981, 74, 2201. A. Koide, J. Phys. B, 1976, 9, 3173. P. G . Burton and U. E. Senff, J. Chent. Phys., 1982,76, 6073. P. G. Burton, F. D. Gray and U. E. Senff, Mol. Pliys., submitted for publication. ’ W. Meyer, P. C . Hariharan and W. Kutzelnigg, J .Chem. Pliys,, 1980, 73, 1880. Dr. J. M. Wutson (Universify of Waferloo) said : I fully agree with Prof. Klemperer that anisotropic iiitermolecular forces at small distances are not well represented by 1-1 low-order Legendre expansion; we have obtained indirect evidence for this when fitting potential surfaces for the rare-gas-HC1 systems simultaneously to molecular-beam spectra and rotational line-broadening cross-sec1ions.I * z However, I do not think that the isotopic substitution experiment proposed by Prof. Klemperer would provide a probe of the high-order Legendre terms. Odd A,j transitions in systems such as He + I4Nl5N and Ar + HD arise from odd-order Legendre terms in the intermoIecular potentials, and these odd-order terms themselves arise because the geometrical centre of the diatomic molecule does not coincide with the centre of mass of the isotopically substituted species. The dominant contribution to odd Aj transitions in these systems arises from theP,(cos 0) term in the potential due to the centre-of-mass transf~rmation,~ and the most important contribution to this comes from the isotropic term in the potential for the unsubstituted species.* We followed Schaefer and Meyer, using thc same geometries and the same weighting co- efficient s ,GENERAL DISCUSSION 277 Nevertheless, there is a need for information on the high-order anisotropic terms in intermolecular potentials, and there are a number of experiments which would probe these directly. (1) Direct observation of high A j transitions in inelastic scatter- ing experiments.The j = 0-4 transition in an atom-diatom system, for example, is mainly sensitive to the P,(cos 0) anisotropy. (2) High-resolution spectra of Van der Waals complexes. McKellar’s recent spectra of rare-gas-H, complexes may be at a sufficiently high resolution to determine a P,(cos S) contribution to the potential anisotropy. (3) Observation of product state distributions for vibrational pre- dissociation of Van der Waals molecules. The fraction of rotationally hut fragments produced will depend strongly on the higher-order terms in the potential. ’ J. M. Hutson and 3. J. Howard, Moi. Phys., 1981,43, 433. J. M. Hutson and B. J. Howard, Mol. Phyi;., 1982, 46, 769. H. Kreek and R. J. Le Roy, J. Chem. Phys., 1975, 63, 338. Prof. U.Buck (Max- PlaPrck-Institut fur Stromungsforschung, Gotringen) said : I do not think that an isotopic substitution will heIp in probing higher-order terms in the Legendre expansion and I agree completely with the comment of Dr. Hutson on this subject. We extensively studied the systems HD-Ne and D,-Ne, both experimentalIy and theoretically, and found that 80% of the anisotropy for HD, the P,(cos 0) term, results from the slope of the isotropic potentiaI term of the homonuclear system 1 ~ 2 due to the shift of the centre of mass from the centre of symmetry. U. Buck, F. Huisken, J. SchIeusener and J. Schaefer, J. Chem. PJzys., 1980, 72, 1512. J. Andres, U. Buck, F. Huisken, J. Schleusener and F. Torello, J . Cliern. f h y s . , 3980,73, 5620. Prof. J. J. M. Beenakker (University ofleiden) said : T want to draw attention to the fact that measurements of transitions out of low j states might have another window on the potential surface than the transitions that are of importance in studying bulk properties.Prof. U. Buck (Max-Plunck-lnstitut fiir Stromuizgsforschung, Gottingen) said : In reply to Prof. Beenakker’s remarks concerning the use of only low rotational states and low energies in beam experiments I would like to make two comments: First, the use of the lowest rotational state j = 0 as initial state is the best possible choice. The well known factorization formula shows that any other transition starting from different j can be directly related to the one starting from j = 0 with some angular coupling coefficients. Although this relation is based on the infinite-order sudden approximation it seems to be of much rnure general validity as shown in many calcu- lations.In my contribution collision energies between 30 and 90 meV have been used, which correspond roughly to the range from liquid-nitrogen to room temperature, a range which is quite normal for macroscopic properties. ‘ See D. Kouri, in Atom-Molecule Collision Theory, ed. R. B. Bernstein (Plenum Press, New York, Secondly, the final j-state depends obviously on the collision energy. 1979), p, 324. Dr. M. Faubel (Max-Planck-Institut fur Srromungsforschung, Giittingen) said : I share Prof. Buck’s view in his repIy to Prof. Beenakker’s comment. The factorization formula derived for energy sudden rotational collisions of an atom with a diatomic molecule is definitely telling you that you should start from the rotational ground state in order to obtain the maximum possible information on the potential.Tt would therefore be very interesting to see if a corresponding factorization can be derived for say the collision of two diatomic molecules,’ which might be considered as mure typical in studying bulk properties. However, near-resonant transitions between rotational2 78 GENERAL DISCUSSION o r rotational and vibrational states (for example), will, presumably, require measure- ments (and accompanying theoretical investigations), with excited initial states. See L. H. Beard et al., J . Chem. Phys., 1982, 76, 3623. Dr. G. C. Maitland, Dr. V. Vesovic and Dr. W. A. Wakeham (Imperial College, London) said: Prof.Buck refers to confirming the accuracy of the isotropic part of potentials by evaluation of diffusion coefficients and comparison with experimental data. For many years it has been the widely held view that gas transport coefficients, particularly viscosity and diffusion, are insensitive to the anisotropic part of the inter- molecular potential and so only give information about the spherically symmetric part of the interaction. However, recent work in our laboratory has shown that this is not necessarily the case. The paper by Smith and Tindell, and the subsequent dis- cussion remarks, will show clearly that transport coefficients for anisotropic systems are not determined solely by the isotropic potential. The point we wish to emphasise here is that even when the full anisotropy of the potential is taken into account, extreme care must be taken in the method of calculating transport cofficients.The calculation of dilute gas transport properties, which is relatively straight- forward for monatomic systems,' is complicated in the case of polyatomic molecules by (i) the non-spherical nature of the intermolecular pair potential; (ii) the occurrence of inelastic collisions. Although a formal semi-classical kinetic theory for dilute polyatomic gases was developed in the 1960s by Wang Chang, Uhlenbeck and De Boer (W.C.U.B.),2 it has not been possible to carry the full calculation scheme through for any realistic potential model. This is because the evaluation of the state-to-state differential scattering cross- sections, I(j'+j), for binary collisions which occur in the expressions for the transport coefficients must in general be performed by a solution of the full Schrodinger wave equation for the process.The computation time and costs involved are prohibitive, certainly for routine calculations. This led to the development of approximate methods, the most famous of which is the Mason-Monchick approximation (M.M.),3 which makes use of physically reasonable assumptions to simplify both the W.C. U.B. expressions and the dynamics of the collision process. Recently various sudden approximations have been developed which involve simplifications of the full close- coupling equations so as to reduce the computational effort required to obtain the scattering cross-sections I(j'+-j).The W.C.U.B. expressions may then be evaluated exactly, subject only to the sudden approximation for I . We have evaluated the transport properties of a series of model atom-diatomic molecule systems in this way, treating the collision dynamics within the infinite-order sudden approximation (I.0.S.A.).4 These calculations show the influence of the anisotropy of the inter- molecular pair potential on different transport coefficients and enable the accuracy of the widely used M.M. approximation to be assessed. They have been carried out over the temperature range 200-1000 K for a series of hypothetical atom (A)-rigid- rotor (BC) systems of varying rotor rotational temperature Or (= h2/2I,,k) interacting through a series of model potential-energy surfaces differing markedly in their degree of anisotropy.A comparison to first-order in the W.C.U.B.-I.O.S. and M.M. approximations is given in table 1 for two hypothetical molecular systems at 500 K. This shows that use of the M.M. approximation can lead to considerable errors, especially where large rotational energy-level spacings and/or highly anisotropic potentials are involved. The major conclusions of the complete study may be sum- marked as follows: (i) The M.M. approximation is good only for evaluating viscosity coefficients for small potential anisotropies and low Or. (ii) The deviations of M.M. from W.C.U.B.4.O.S. are in all cases greater for diffusion than viscosity. (iii) TheGENERAL DISCUSSION 279 deviations of M.M. from W.C.U.B.-I.O.S.increase with the degree of potential anisotropy and with rotational energy-level spacing but are relatively insensitive to the value of the thermodynamic temperature. (iv) The diffusion coefficient for rotational energy (Drat) can be significantly different from that for mass (D) in a dilute gas. The ratio D,,,/D can be both greater or less than unity, unlike in the M.M. TABLE 1 .-COMPARISON OF W.C.U.B./I.O.S.A. AND M.M CALCULATIONS OF TRANSPORT COEFFICIENTS FOR HYPOTHETICAL ATOM-RIGID ROTOR GASES AT 500 K potential surface A anisotropy parameter a 0.06 B 0.28 C 0.54 molecular system ' He-N2 ' ' He-C02 ' ' He-N2 ' ' He-C02 ' ' He-N2 ' ' He-C02' WK 2.87 0.56 0.995 1 .oo 0.74 0.97 0.41 0.8 1 ,,pi.M./21 W.C.U.B. 12 d 1.01 1 .oo 0.88 1.01 0.53 0.94 A a8M.M. 1A"W.C.U.B.e 1-01 1 .oo 1.18 1.05 1.29 1.15 By2. M :/Dy2. C. U . B. C rot l D Y * u. B. 1.01 1.01 1.05 1.08 0.74 1.11 DW .C.U .B. a Based on rate of change of classical turning point with relative orientation. Hypothetical; Self-diffusion coefficients. Collision integral ratio A* Z 5npDI2/3ql2 where n = number density and D,,, is the diffusion coefficient for rotational energy; ratio is unity in M.M. names used merely indicate molecular masses of species involved. p = reduced mass. approximation. Interaction viscosities. approximation where it is equal to 1.0. (v) Evaluation of thermal conductivity from viscosity using the M.M. approximation is in general unsatisfactory, in agreement with recent studies for real polyatomic gases.5 In the light of these conclusions, we would like to ask Prof.Buck and Dr. Faubel which method of calculation was used in the evaluations of transport coefficients reported in their papers and to comment on the extent to which these calculations provide a true test of the proposed potentials. G. C. Maitland, M. Rigby, E. B. Smith and W. A. Wakeham, Intermolecular Forces: Their Origin and Determination (Clarendon Press, Oxford, 1981). C. S. Wang Chang, G. E. Uhlenbeck and J. De Boer, Studies in Statistical Mechanics (North Holland, Amsterdam, 1964), vol. 11, part C. E. A. Mason and L. Monchick, J. Chem. Phys., 1961,35, 1676; 1962, 36, 1622. G. A. Parker and R. T. Pack, J. Chem. Phys., 1978, 68, 1585. G. C. Maitland, M. Mustafa and W. A. Wakeham, J. Chem. SOC., Favaday Trans. I , 1983,79, in press (2/752).Prof. U. Buck (Max-Planck-Institut f u r StrBrnungsforschug, Gottingen) said : In response to the question of Dr. Maitland I would like to say that the calculations of the diffusion coefficient for H,-Ne and H,-Ar mentioned in the paper have been performed only with the spherical component of the interaction potential. We assume that inelastic collisions and the anisotropic component make a negligible contribution (< 1%) to the collision integrals. This behaviour is confirmed by the calculation of viscosity cross-sections for the H, system by Kohler and Schaefer using close-coupling methods on the complete ab initio potential surface.' W. E. Kohler and J. Schaefer, J. Chem. Phys., 1982, in press.280 GENERAL DISCUSSION Prof F. R. W. McCourt (University qf Waterloo, Canada) said : In partial response to the query in the comment by Drs.Maitland, Vesovic and Wakeham on Prof. Buck's paper and in support of the H2-Ne potential obtained by Prof. Buck, I would like to report on one result of calculations that I have been making using the potential surface of Rodwell and Scoles.' Prof. Buck has shown us the excellent agreement between his experimental H,-Ne surface and the semi-empiricai surface of Rodwell and ScoIes, and he has indicated the rather good agreement between the diffusion coefficient DI2 measured by Trengove and Dunlop at 300 K and that calculated using the centrifugal sudden approximation and the isotropic part of his H,-Ne surface. For the hydrogen- isotope-rare-gas interactions, the anisotropies are very weak and so the value of D,, will indeed be largely determined by Vo.of D1, = I , 171 cm2 s-' at 300 K is only 0.4% lower than the value of Trengove and Dunlop of Ol2 = 1.176 cm2 s-l. I have performed a full close-coupled calculation of Ol2 and the interaction viscosity pAB(0) for the H,-Ne system using the RodwelI-ScoIes potential surface and obtained values of D,, = 1.176 cm2 s-I at 300 K and pAB(0) = 13.28 pPa s at 298 K, to be compared with the Trengove and Dunlop value for D,, and the value pAB(0) = 13.05 pPa s obtained by Clifford et aZ.4 from their experimental measurements. The calculated value of pAB(0) Iies within the estimated uncertainty of &2% estimated by Clifford et al.' Thus it appears that the Rodwell-Scoles-Buck surface for H,-Ne is quite reliable. Buck's value W.R. Rodwell and G. Scoles, J. Phys. Chem., 1982,86, 1053. 1981, preprint. A. A. Clifford, J. Kestin and W. A. Wakeham, Ber. Bunsenges. Phys. Chem., 1981,85,385. ' R. D. Trengove and P. J. Dunlop, 8th International Symposium on Thermophysical Properties, ' U. Buck, personal communication. Prof. R. J. Le Roy (Unicersity qf Waterloo) said: One of the more intriguing results in Prof. Buck's paper ' is the observation that at a given collision energy, D2(j = 0+2) inelastic transitions occur much more readily in collisions with Ne than with Ar. Since internal-rotational predissociation of a Van der Waals molecule may be thought of as an inelastic transition occurring below threshold, his result is in excellent accord with our prediction that internal-rotational predissociation level widths of H,-inert-gas complexes increase from H,-Xe to H,-Kr to H,-Ar.These results appear to present a paradox, since the splitting of energy levels of the H,-inert-gas complexes caused by the potential anisotropy, and hence the effective anisotropy strength determined from the discrete spectra of these molecules, increases with the size of the inert-gas partner. This apparent discrepancy is explained by the facts that these two types of pheno- mena are sensitive to different parts of the anisotropy strength function V,(R), and that the functions associated with the heavier inert-gas partners are displaced to smaller distances relative to the position of the zero of the isotropic potential. In particular, the level energies observed in the discrete spectra depend mainly on the anisotropy in the classicalIy allowed region at distances greater than the zero of the isotropic potential.In this region V,(R) is mainly negative, preferring a collinear relative orientation, and (as does the potential as a whole) its strength increases with the size of the inert-gas partner. On the other hand, the matrix elements which govern the rotational inelasticity and predissociation processes depend mainly on V,(R) at shorter distances, where it is positive (preferring a T-shaped configuration). and grow rapidly with decreasing R. The trends observed for these properties there- fore simply reflect the fact that the change in sign of V2(R) occurs at relatively smaller distances for the heavier inert-gas p a r f n e r ~ .~ , ~ Note that application of this argument to the diatom bond-length-dependent partGENERAL DISCUSSION 28 1 of the potential leads to the prediction that the ease of vibrational excitation of hydro- gen by inert-gas collision partners increases from Xe to Kr to Ar. U. Buck, Furuday Discuss. Chem. SOC., 1982, 73, 187. R. J. LeRoy, G. C. Corey and J. M. Hutson, Furuday Discuss. Chem. SOC., 1982, 73, 339. R. J. LeRoy and J. Van Kranendonk, J. Chem. Phys., 1974,61,4750. (a) R. J. LeRoy, J. S. Carley and J. E. Grabenstetter, Furuday Discuss. Chem. SOC., 1977, 62, 169; (6) J. S. Carley, Furuday Discuss. Chem. SOC., 1977, 62, 303. Prof. U. Buck and Mr. H. Meyer (Max- Planck-Institut fur Strornungsforschung, Gottingen) (communicated) : The question of the conformality of angular-dependent potential surfaces can now be answered on the basis of very precise experimental re- sults for the hydrogen-molecule-rare-gas systems.The reduced potentials for H,-Ne and H,-Ar, which are essentially derived from state-selective differential cross-sections (O+O and 0+2 rotational transitions) and the spectroscopy of the bound dimer, are 8 6 w 2 4 a= x . 4 3 2 c-( ._ c.’ a 0 -1 I I I 0.8 1.0 1.2 distance, R/R, FIG. 1.-Reduced potential surfaces for the isotropic (Vo) and the anisotropic ( V2) potential term: ( a - .) H2-Ne; (-) H2-Ar. displayed in fig. 1. In spite of the large differences in the potential well depths co = 2.84 meV and E , = 0.27 meV for H,-Ne and c0 = 6.31 meV and E, = 0.71 meV for H,-Ar, the curves are rather similar.However, they are only conformal with respect to the rare-gas atom and not with respect to the isotropic potential Vo or the aniso- tropic potential V,. This result is in contrast to the result of the Tang-Toennies model, where a complete conformality is found. Note that the potential models used in evaluating our data are different for the two systems.2 82 GENERAL DISCUSSION Prof. F. A. Gianturco (Uniuersity of Rome) said: In relation to the paper by Faubel et al. on the determination of the He-N, interaction from experimental data and from theoretical models, I think it of interest also to report a similar study that we have carried out for another system, the 0, molecule interacting with He atoms, where we have carefully fitted experimental data via theoretical calculations of infinite-order sudden (10s) partial and total differential cross-sections.These very accurate measurements (carried out, in fact, at the same laboratory in Gottingen) observed the total differential cross-section over a wide angular range at a collision energy of ca. 26.8 meV, together with the partial, rotationally inelastic differential cross-section under the same conditions and for the (1+3) excitation process. Since we had already obtained an effective, averaged potential for this very system,, an attempt was made at generating its anisotropic component by analogy with the be- haviour of the 0,-Ar system for which detailed studies had also been made by our g r o ~ p . ~ , ~ Thus, various anisotropic coefficients were introduced in an Exp-Spline- Morse-Spline Van der Waals potential from the interaction, by writing each of its required parameters P in the following form : P = B [I + a,~,(cos R - r)] (1) and by choosing the values of B required from the averaged 0,-He interaction and the ap from their values for the 0,-Ar full interaction.Calculations were then carried out within the 10s approximation of the dynamics, and the individual parameters were thus adjusted to fit the energy-averaged experi- mental 'data transformed in the centre-of-mass system used to perform the calcu- lations. Fig. 2 reports the actual fit obtained, where one sees that a very high accuracy was achieved down to rather small scattering angles. It is worth noting, in fact, that the relative values of the total cross-section (TDCS, curve a) with respect to the partial cross-section (IDCS, curve b) are not adjusted at all but are those directly produced by experiments as well as theory, once the latter was fixed at the arbitrary units of the former when determining the TDCS.One can also observe the following: (i) The P values in eqn (1) finally used are all very close to the initial choice from the average 0,-He interaction. (ii) The 0,-He anisotropy for the well depth and position turns out to be markedly larger than the one derived from the 0,-Ar interaction, thus producing rather different values of ap in eqn (1). (iii) The anisotropic characteristics of the wall region and of the dispersion tail appear instead to be similar in both cases, at least within the sensitivity of the present procedure.(iv) Ab initio calculations for the C,, and C,, geometries of the 0,-He were carried out by Jaquet and Staemmler at various internuclear distances and exhibited shallower minima than those suggested by our present calculations. M. Faubel, K. H. Kohl and J. P. Toennies, unpublished results. F. Battaglia, F. A. Gianturco, P. Casavecchia, F. Pirani and F. Vecchiocattivi, Faraday Discuss. Chem. SOC., 1982,73,257. F. Pirani and F. Vecchiocattivi, Chem. Phys., 1981, 59, 387. F. Battaglia and F. A. Gianturco, Chem. Phys., 1981, 59, 397. M. Faubel, K. H. Kohl, F. A. Gianturco and J. P. Toennies, unpublished results. R. Jaquet and V. Staemmler, personal communication. Dr. M. Faubel (Max-Planck-Institiit fur Stromungsforschung, Gottingen) said : The infinite-order sudden approximation, because of its numerical speed, is ideally suited to the purpose of fitting a potential to experimental cross-sections.We therefore checked the accuracy of this approximation in the present range of experimentalGENERAL DISCUSSION 283 0 0 D U 0 lo-$- 1 1 1 1 1 , ~ , ~ 1 , , , , 1 , , , , 1 , 1 1 1 I 1 1 , , 0.0 10.0 20.0 30.0 4 0.0 50 0 @cm FIG. 2.-Total differential cross-sections (a) and partial, inelastic differential cross-sections (b) for the O,-He system. The continuous lines refer to calculations within the 10s approximation, while the open circles are from experiments. The collision energy was 268 meV and the inelastic process refers to the ( j , =: l+jf = 3) rotational excitation of 02.energies and scattering angles by comparison of 10s cross-sections * with the close- coupling He-N, results given in this paper. With the “ HTT ” potential and at E,, = 27.3 meV the general agreement between the “ respective ” 10s and CC cross-sections was in the order of some 10% in the scattering angle range from 10 to 50”. At scattering angles below lo”, where the influence of the potential attractive well is important, the TOSA can be expected to frtil, and indeed does. For example the first diffraction minimum in the j = O-+j’ = 2 cross-section is too small by an order of magnitude. Keeping this limitation on the angular range of validity in mind the IOSA is suitable to obtain realistic potential fits from the experimental He-N, as well as from the quite similar He-02 cross-section data.Prof. B. Schramm (Uniuersity of Heidelberg) said : We have calculated He-N, second interaction virial coefficients with the Habitz-Tang-Toennies potential using the classical formula B(T) = dVA/mrn ( I - exp[- V(R,y)/kT]}RZsiny dy dR. 0 0 * I thank Russel Pack for permission to use his 10s program.284 GENERAL DISCUSSION A comparison with experimental data taken from the literature 1 , 2 is shown in fig. 3. Most of the data from different sources agree very well. The second virial coefficient at 90 K was calculated from the measured excess quantity E = B(He-N,) - +[B(He) + B(N2)] using B(N,) = - 180 cm3 mol - '. Recent measurements 3*4 indicate that this value should be -204 cm3 mol-'. The virial coefficient at 90 K therefore has to be lowered from 12.4 to 0.4 cm3 mo1-'.The corrected value is shown in fig. 3. If quantum corrections As can be seen, the calculations do not fit the data. i 4 I 4 2 1 . 4 -1 X X X TIK FIG. 3.-Second interaction virial coefficients of He-N,. Experimental data from different sources are shown with different symbols. Dashed line : calculated with Habitz-Tang-Toennies (HTT) potential. Full line: calculated with Faubel's modification of the HTT potential. are taken into account the disagreement becomes even worse and is therefore not shown here. The modification of the Habitz-Tang-Toennies potential proposed by Dr. Faubel does not give better results, as can also be seen from fig. 3. In order to lower the calculated second virial coefficients the attractive bowl of the potential has to be enlarged, probably by making it wider.* J. H. Dymond and E. B. Smith, The Virial Coefficients of Pure Gases and Mixtures (Oxford ' T. N. Bell and P. J. Dunlop, Chem. Phys. Lett., 1981, 84, 99. University Press, 1980). G. Pocock and C . J. Wormald, J. Chem. SOC., Faraday Truns. I , 1975, 71, 705. B. Schramm and R. Gehrmann, J. Chem. SOC., Faraduy Trans. I , 1979, 75, 479. Dr. M. Faubel (Max-Planck-Institut fur Str6mungsforschung, Gottingen) said : Both Prof. Schramm and Prof. McCourt noted a serious disagreement between experimental second virial coefficients and the values calculated from the HTT model potential for He-N,. In applying our estimated correction for the potential well depth Prof. Schramm found that the discrepancy became worse.As is obvious from fig. 9(6) of our paper the oversimplistic procedure for deepening the potential well by multiplyingGENERAL DISCUSSION 285 the vo part of the potential by a constant factor of ca. 1.2 is unintentionally increasing the repulsive barrier part of the potential at the same time. This latter effect is counteracting the decrease of the virial coefficients obtained by only deepening the well but leaving the repulsive barrier unchanged. We expect that a potential fit to the scattering data will finally give a potential well 20% deeper than the model and with slightly inward-shifted uo and u2 repulsive terms with an essentially unchanged repulsive barrier " deformation " of potential. Mr. R. R. Fuchs (University of Waterloo) said: In this paper the measurements of total and state-to-state differential collision cross-sections are reported for He-N,, and these are used to test the validity of a corresponding semi-classical semi-empirical anisotropic intermolecular potential derived from the Tang and Toennies mode1.l This potential is hereafter referred to as the HTT potential.Our group has determined two semi-empirical anisotropic potentials (soon to be published) for this system of the Hartree-Fock-plus-dispersion type. These will be referred to as HFDl and HFD2, respectively. Other anisotropic intermolecular potentials for He-N, referred to are those of Keil et aL4 (KSK), and a modified version of KSK labelled KKM3 in ref. (5). For the above potentials the following physical quantities were evaluated: (i) the second virial coefficient B,,(T), (ii) the viscosity qmix(0), (iii) the binary diffusion co- efficient o,,, (iv) the total differential collision cross-sections (TDCCS), and (v) the state-to-state differential collision cross-sections (SSDCCS) ;* these were then com- pared with experiment.All the scattering calculations were made in the infinite-order sudden approximation (IOSA). Second virial coefficients were calculated [including the first quantum-mechanical correction to B12(T)] in the temperature range 90-748 K. These calculations agree very well with those of Schramm lo for the HTT potential. Some representative values of B12(T) are listed in table 2. The HFDl potential is found to be in very good TABLE 2.-sECOND VIRIAL COEFFICIENT BIZ( T ) IN Cm3 n l O l - T K 11 3.26 148.27 321 .78 523.2 598.2 exptl 10.43 15.32 21.7 22.41 5 0.19 21.73 3 0.42 HFDl 10.8 15.4 21.6 22.3 22.3 HFD2 14.8 18.6 23.6 23.8 23.7 HTT 20.4 23.8 27.8 27.8 27.5 KSK - 8.2 17.3 18.8 18.8 agreement with experiment throughout the temperature range studied, whereas the HTT and HFD2 potentials are ca.40% and 20% too large, respectively. KKM3 and KSK can be up to ca. 30% and 45% too low, respectively, in the temperature range studied. It is found that the HTT potential is in excellent agreement with experiment, whereas HFDl and HFD2 are not in such good agreement.? * We thank Dr. M. Faubel for his calculations of these latter two quantities using our potentials. t At 373 K preliminary results show HFDl to agree much better than HTT with experiment.Although morej-states must be included in this calculation than in the 298 K one, we doubt that the relative agreement will change. Thus it seems probable that although one potential is good at a given temperature, it may not be as good as others at another temperature. The mixture viscosity qmix(0) results at 298.15 K are listed in table 3.286 GENERAL DISCUSSION The binary diffusion coefficients Ol2 calculated are listed in table 4. These show the HTT and KKM3 potentials to be in better agreement with experiment than the HFDl and HFD2 potentials. Calculations of the TDCCSs show that both the HTT and HFD2 potentials agree well with experiment while HFDI is shifted towards higher angles. On the other TABLE 3.-VISCOSITY q.lmix(O) IN PUP mole fraction X,, T/K exptl l 1 HFDl HFD2 HTT 0.2 298 183.2 186.2 185.5 183.2 0.4 298 188.6 194.7 193.2 188.6 0.6 298 194.4 202.8 200.8 194.3 0.8 298 199.2 207.7 205.7 199.2 hand, calculations of SSDCCSs have the HTT and HFDl potentials agreeing well in both magnitude and phase, while HFD2 is not in as good agreement on the phase.In the paper in question Faubel et al. mention that the Yo part of the He-N, potential should be ca. 20% deeper than the Vo part of the HTT potential. In fact, the Vo parts of HFDl and HFD2 are 34% and 16% deeper than V, (HTT), respectively. The TABLE 4.-BINARY DIFFUSION COEFFICIENT D12 IN Cm2 S-’ AT T = 300 K exptl l2 HFDl HFD2 HTT KSK KKM3 0 1 2 0.713 0.824 0.807 0.745 0.859 0.801 reason however that HFD2 does not agree better than HTT with the scattering is that the V2 and V, parts of the potential are different for both these potentials.From the above results it is quite clear that a definitive anisotropic intermolecular potential for He-N, is not as yet available. For errata see J. Chem. Phys., K. T. Tang and J. P. Toennies, J. Chem. Phys., 1977, 66, 1496. 1977, 67, 375 and 1978,68, 786. P. Habitz, K. T. Tang and J. P. Toennies, Chem. Phys. Lett., 1982, 85, 461. C. Douketis, G. Scoles, S. Marchetti, M. Zen and A. J. Thakkar, J. Chem. Phys., 1982,76,3057. M. Keil, J. T. Slankas and A. Kuppermann, J . Chem. Phys., 1979, 70, 541. W. K. Liu, F. R. McCourt, D. E. Fitz and D. J. Kouri, J. Chem. Phys., 1981,75,1496. K. R. Hall and F. B. Canfield, Physica, 1969, 47, 219. J. Brewer and G. W. Vaughn, J.Chem. Phys., 1969,550,2960. T. N. Bell and P. J. Dunlop, Chem. Phys. L e f f . , 1981, 84, 99. R. J. Witonsky and J. G. Miller, J . Am. Chem. SOC., 1963,12,282. J. Kestin, S. T. Ro and W. A. Wakeham, J. Chem. Phys., 1972,56, 4036. lo B. Schramm, this Discussion. l2 P. S. Arora and P. J. Dunlop, J. Chem. Phys., 1979, 71, 2430. Mr. C. Douketis, Dr. M. Keil and Prof. G. ScoIes (University of Waterloo) said: The experiments of Faubel et al. forcefully demonstrate the usefulness of state-to-state scattering data when investigating anisotropic intermolecular potentia1s.l Since such detailed studies are very difficult to perform, however, it would be useful to know the information content of less detailed measurements. In particular, we would like to know whether differential cross-sections (DCS), without resolving elastic from rotation-GENERAL DISCUSSION 287 ally inelastic transitions, contain information on the anisotropy of the Van der Waals potential.As discussed earlier,2 little anisotropic information is obtained from these experi- ments unless the " total " DCS show pronounced damping of the oscillatory structure normally present for atom-atom scattering. The paper by Faubel et al. makes this point quantitatively. On the other hand, more strongly anisotropic potentials do result in significant damping of the DCS oscillations, and consequently have been investigated by total DCS measurement^.^ We have used a model potential, based on the HFD procedure and extended to anisotropic systems, to compare to experimentally obtained potentials for He + C02.3 This is done in the same spirit as the comparison of Faubel et al.with the Tang- Toennies model.' In fig. 4 we compare the experimentally determined anisotropic I I I I 3 4 5 6 rlA FIG. 4.-Intermolecular potentials for He + COz for orientation angles of y = 0 and 90". The experi- mental potential (--) is the EGMSV potential obtained from the total DCS measurements of ref. (3); the HFD potential (- - - -) is obtained by extending the model of ref. (4). potential with the HFD potential. These potentials are seen to be in qualitative agreement, even though the model HFD potential has not been optimized. These results show that total DCS data exhibiting damping of the diffraction oscil- lations reasonably may be expected to contain information on the potential anisotropy.Secondly, both the results of Faubel et al. and those just discussed indicate that the respective anisotropic model potentials would serve as good starting points for analyses of experiments sensitive to the anisotropic potential.288 GENERAL DISCUSSION 0.30 0.20- 0.10- M. Faubel, K. H. Kohl, J. P. Toennies, K. T. Tang and Y. Y. Yung, Faraday Discuss. Chem. SOC., 1982, 73, 205. M. Keil, J. T. Slankas and A. Kuppermann, J. Chem. Phys., 1979, 70, 541. M. Keil, G. A. Parker and A. Kuppermann, Chem. Phys. Lett., 1978,59,443; G. A. Parker, M. Keil and A. Kuppermann, J. Chem. Phys., in press. C. Douketis, G. Scoles, S. Marchetti, M. Zen and A. J. Thakkar, J. Chem. Phys., 1982,76,3057. - Prof. Ph. BrCchignac (Uniuersity of Paris, Orsay) said: I understand from Dr.Faubel's paper that it is a dangerous procedure to derive an " effective " potential for anisotropic systems from total differential cross-sections only, while inelastic differen- tial cross-sections seem to be more reliable. However, such detailed data still need a lot of hard experimental work. 1 wish to point out that a significant amount of information is already contained in integral inelastic cross-sections, which, in spite of some averaging, have the big advantage that they can be obtained from bulk experi- ments (LIF or various kinds of double resonance). Fig. 5 shows the state-to-state I 0 I P 4 I I I I I 0 T I 4 6 8 10 FIG. 5.-Relative values of state-to-state integral inelastic crosa-sections for CO ( j = 7) - H2 as a function of finalj.The open circles correspond to theory, the full circles correspond to experiment. (a) T = 77 K; (b) T = 293 K. integral inelastic cross-sections obtained from an IRDR experiment for the Co-H, system.' The Aj = 1 and Aj = 2 cross-sections are very similar, and the higher Aj smaller. This fact appears to be the direct reflection of the shape of the intermolecular potential. Indeed the ab initio surface computed by Kochanski et al., is characterized by extremely similar R-dependence of the polynomial expansion coefficients u,,,(R) and v,,,(R) which are, respectively, responsible for Aj = 1 and Aj = 2 quantum jumps. Fig. 6 shows another set of integral inelastic cross-sections obtained from a steady- state IRDR experiment for the NO-NO ~ystern.~ The quantum jumps A j = 1, 2, 3GENERAL DISCUSSION 289 were all found to contribute significantly.This indicates that the anisotropy is not dominated by multipolar interaction. The dynamical calculations of the cross-sec- tions necessary to test specific surfaces have not yet been done. However, due to the lL-doubling of the 2rc electronic ground state of NO a more complete study of parity 1 -3 T -1 0 * 1 I T + 3 i CHANGE b i YES NO YES YES NO YES PARITY CHANGE FIG. 6.--Relative values of rate constants for NO('R~,~, j % 18.5) + NO as a function of angular momentum and parity changes. T = 300 K. changes accompanying the changes of angular momentum would be necessary, which would give a better insight into the corresponding intermolecdar potential.I Ph. Brkhignac, A. Picard-BerselIini, R. Charneau and J . M. Launay, Chem. Phys., 1980,53,165. D. R. Flower, J. M. Launay, E. Kochanski and J. Prissette, Chem. Phys., 1979, 37, 355. Ph. Brechignac, in preparation. Dr. R. Candori, Dr. F. Pirani and Dr. F. Vecchiocattivi (Uniuersity qf Perugia) and Prof. F. A. Gianturco (Univevbity of Rome) said : In the paper by Faubel et al.' experi- mental scattering cross-sections have been used to test the validity of a theoretical potential-energy surface (p.e.s.) for the He-N, system. We made a similar test using other scattering data from our Iaboratory and we found an uncertainty larger than that estimated by Faubel et al. We measured the absolute integral cross-sec- tions, for the He-N, system, as a function of the colIision ve10city.~ These results 5ave been obtained with a molecular-beam apparatus which has been described previously.4 The cross-sections are reported in fig.7 and are compared with the calcdation performed using the V,(R) component of the p.e.s. by Habitz et aL2 These data are expected to be almost completely sensitive only to the spherical average of the ~ . e . s . ~ As it is evident from the figure, both the value and the dope of the cal- culated cross-sections are in disagreement with the experimental results. Moreover, a disagreement is found also with the relative cross-sections by Butz et aL6 and the second virial coefficients calculated with this potentid show large deviations from the experimental These inconsistencies of the p.e.s. in the Vo(R) component could also be reflected in the angular dependence of the interaction. A much better p.e.s.can be obtained by a combined analysis of several experimental data, including inelastic differential and totaI cross-sections by Faubel et ul.,' as has been done for He-U,.8 M. Faubel, K. H. Kohl, J. P. Toennies, K. T. Tang and Y . Y . Yung, Farada~ Discuss. Chem. Soc., 1982, 73, 205. P, Habitz, K. T. Tang and J. P. Toennies, Chem. Phys. Lett., 1982, 85, 461. R. Candori, F. Pirani and F. Vecchiocattivi, unpublished results. V. Aquilanti, G. Liuti, F. Pirani, F, Vecchiocattivi and G. G. Volpi, J. Chem. Phys., 1976, 65, 4751. Such work is in progress.290 GENERAL DISCUSSION l Z O t ’ ti I I \ ‘\ \ ++++ \ \ \ - 1 I I 1 1.0 1.5 2.0 velocity/km s-l FIG. 7.-Effective absolute integral cross-sections for He-N, collision as a function of the Iaboratory velocity.The dashed line is the calculation performed with a theoretical potential (see text). The calculated cross-sections are convoluted to the experimental conditions. F. Pirani and F. Vecchiocattivi, Chem. Phys., 1981, 59, 387. H. P. Butz, R. Feltgen, H. Pauly and H. Vehmeyer, Z . Phys., 1971, 247, 70. ’ J. Brewer and G. W. Vaughn, J. Chem. Phys., 1969,50,2960. F. Battaglia, F. A. Gianturco, P. Casavecchia, F. Pirani and F. Vecchiocattivi, Furaduy Discuss. Chem. SOC., 1982,73,257. Dr. G. C. Maitland, Dr. V. Vesovic and Dr. W. A. Wakeham (Imperial College, London), said: We have been examining a similar question to that considered by Smith and Tindell : how an effective spherical potential obtained by inversion of transport coefficients for atom-linear-molecule systems is related to the full anisotropic potential- energy surface.The system chosen for study was “Ar-C02” interacting with a hypothetical potential-energy function based on electron-gas calculations.’ Simulated diffusion and interaction viscosity coefficients were evaluated for this system over the temperature range 100-2400 K, which represents the maximum range over which experimental data might reasonably be expected to exist for any real system. The Mason-Monchick approximation,2 which other calculations 394 suggest should be reasonably accurate for this particular system, was used to evaluate the transport co- efficients. These data were then inverted by the standard procedures developed for monatomic systems to generate a spherical inverted potential V,(r) ; this is illustrated in fig.8. This confirms the conclusion of Smith and Tindell that within the M.M. approximation diffusion and viscosity coefficients lead to the same inverted potential. This effective spherical potential V,(r) is significantly different from the unweighted spherical average of the full potential from which the transport coefficient data were calculated, in line with the findings of Smith and Tindell for diatom-diatom systems.GENERAL DISCUSSION 29 1 120( 80( 4 O( & z n ( . . K -lo( - 20( - 30( -40( 0.30 0.34 0.38 0.42 0.46 0.52 r/nm FIG. 8.-Comparison of potential V,(r) obtained by inversion of model ' Ar-C02 ' transport co- efficients (0, viscosity; o, diffusion) with the unweighted spherical average ( S ) and two fixed orienta- tion components of the full anisotropic potential used to generate the data.It is also very different from the potentials corresponding to head-on ( y = 0') or side- ways-on ( y = 90') collisions (see fig. s), nor does it correspond to any simple fixed orientation potential. Various other weighted averages of the full potential, such as Boltzmann and free-energy averages, were investigated but none of these coincided with the inverted potential over a wide range of separations. However, when the data are plotted in reduced form (kV/e against r/a where E is the well depth and a the separation where V = 0, see fig. 9), the inverted potential is remarkably close to the spherical average of the full potential over the entire range.Hence it appears that the shape of V,(r) for atom-linear molecular interactions may be inferred from the full anisotropic potential V(r,y) but that there is no simple relation between the values of its scaling parameters, E and a, and those of V(r,y). We would like to ask Smith and Tindell whether they have observed a similar con- formality between V,(r) and the spherical average of the true potential in their work on diatom-diatom systems. Should such a conformality prove to be universal, it may have some connection with the observations reported in the earlier papers by Buck and by Faubel et al. of the near conformality of V,(r) and V,(r) for some sys- tems. Although this may prove to be simply a coincidence for the particular cases studied, it suggests that an investigation of how the E and 0 parameters for V,,(r) and292 GENERAL DISCUSSION w n L .z II n L W 2 I I 1 I 1 .o 1.25 1.5 1.75 r* = ria FIG. 9.-The data of fig. 8 plotted in reduced units [O, VF(r*)]. V,(r) are linked, if at all, may give further insight into the nature of the effective poten- tials obtained by inversion of bulk properties. G. A. Parker, R. L. Snow and R. T. Pack, J . Chem. Phys., 1976,64, 1668. G. C. Maitland, M. Rigby, E. B. Smith and W. A. Wakeham, Intermolecular Forces: Their Origin and Determination (Clarendon Press, Oxford, 1981). G. C. Maitland, V. Vesovic and W. A. Wakeham, to be published. V. Vesovic, PhD Thesis (University of London, 1982). Mr. C. D. Eley, Dr. E. B. Smith and Dr.A. R. Tindell (Uniuersity of Oxford) (communicated) : Following the findings of Maitland, Vesovic and Wakeham that the reduced unweighted spherical average of the anisotropic potential function ( Vo) was conformal with the potential obtained on inversion of transport properties calculated for an atom-diatom potential, we were interested to see if this was also applicable to our system of two interacting diatomic molecules. We calculated the second virial coefficients and transport properties from the di- Lennard-Jones potential for a wide range of anisotropy using the Mason-Monchick approximation.l These properties were then inverted to give an effective spherical potential. In fig. 10 we show, for a relatively high degree of anisotropy (comparable to that proposed for bromine), that the reduced effective spherical potential is confor- mal with the reduced unweighted spherical average of the anisotropic function.This phenomenon of conformality was investigated by a more quantitative pro-GENERAL DISCUSSION 293 ?x (8: QX -1.0 Ox xsdp 0 9 10 11 12 13 1.4 15 16 FIG. 10.-Comparison of the reduced unweighted spherical potential ( x j with the reduced effective spherical potential (0) for an anisotropy of 0.6584 (as defined in the paper by Smith and Tindcll). cedure in the following way. The unweighted spherical potential was rescaled by choosing suitable parameters for the well depth and collision diameter in order to obtain a best fit to the second virial coefficients calculated from the anisotropic potential function.These " best fit " parameters were then compared with those of the effective spherical potential obtained from inversion. The agreement, as shown in table 5, is extremely good and was found to hold for the wide range of anisotropy investigated. TABLE 5.-POTENTIAL PARAMETERS effective spherical rescaled V,, deviation of (e/k)/K lOa/nm /cm3 (&/k)/K 1 Ou/nm an isotropy potential (&/kj/K 1 Oo/nm calculation, B vo 0.3292 100 1 3 3.60 102.7 3.575 0.08 87.0 3.70 0.4938 90 4: 5 3.77 90.5 3.784 0.23 59.0 4.12 0.6584 80 1: 6 3.98 81 .o 3.963 0.29 40.0 4.59 10.02 10.02 10.03294 GENERAL DISCUSSION These results, then, support the observation of Maitland et al. that conformality exists between V, and the effective spherical potential produced on inversion.This, in view of the large differences in the two potentials (see fig. 6 of the paper by Smith and Tindell) is a rather surprising result. It cannot be explained for the atom-diatom system by the conformality of Yo and Yz which, as illustrated in fig. 11, are quite different in shape for r > Y,. L. Monchick and E. A. Mason, J. Chem. Phys., 1961 , 35, 1676. U* 0.5C 0.2: 0. c -0.25 -0.50 -0.75 -1.00 1 I I I 1.0 1.5 2.0 R* FIG. 1 1 .-Comparison of the reduced unweighted spherical potential V,, (full line) and the reduced second Legendre term, V2, (broken line) for an atom-diatom interaction for anisotropy of 0.3292 (as defined in the paper by Smith and Tindell). Prof. B. Schramm (University of Heidelberg) said: Some years ago we tried simul- taneously to describe second virial coefficients and viscosity data with a spherical potential function U(r).' We used modified Lennard-Jones potentials that were buiIt by connecting two (different) Lennard-Jones ( 4 6 ) potential curves at their minima.We chose one of the potentials and determined the potential parameters E and rmin with the help of the Boyle temperature [where B(TB) = 01 and of the second virial coefficient at the temperature 0.5TB. Then we calculated the viscosities and looked how far they were shifted from the data calculated in the same way with the Lennard- Jones (12,6) potential. Calculations of the same kind using Stockmayer potentials with different dipole parameters showed much larger shifts from the ca!culations with the Lennard-Jones (12,6) potential than all the spherical potentials.This indicates that second virialGENERAL DISCUSSION 29 5 coefficients and viscosities calculated with a Stockmayer potential cannot be re- produced with an effective spherical potential. It would be interesting to answer the following questions: Is the shape of the inverted effective potential significantly different from a modified Lennard-Jones potential? Can an effective potential be found from virial coefficients and viscosities that were originally calculated with a Stockmayer potential? Can an effective potential be found that describes second virial coefficients and viscosities of nitrogen within the experimental error bounds? ' B. Schramm, R. Wiesler and T. Merz, Ber. Bunsenges. Phys. Chem., 1975,79,1240. Dr. A. R. Tindell (University of Oxford) (communicated) : Schramm reports that it is not possible to fit second virial coefficients calculated from the Stockmayer potential to a modified form of the Lennard-Jones (n,6) potential, in which each branch of the potential has a different exponent n.He asks if it is possible to produce an effective spherical potential, using the inversion techniques advocated in our paper,3 which will reproduce the second virial coefficients calculated from a Stockmayer potential. We calculated virial coefficients from the Stockmayer potential using parameters ( p = 1.83 I), Elk = 400 K, o = 0.35 nm and t* = 0.5) which are in accord with those for an organic compound with high dipole moment (e.g. CH,Cl). We found it was possible to invert these virial coefficients and produce an effective potential capable of reproducing the second virial coefficients to within 1 cm3 over a wide temperature range 360-4000 K. The failure of the approach of Schramm probably results from the double (n,6) functional form of his potential.When the functional form is unconstrained as in the inversion method, there appears to be no difficulty in obtaining an effective spherical potential which can reproduce the virial coefficients, B. Schramm, R. Wiesler and T. Merz, Ber, Bunsenges, Phys. Cheiti., 1975, 79, 1240. W. H. Stockmayer, J. Chem. Phys., 1941, 9, 398, E. B. Smith and A. R. Tindel!, Faraday Discuss. Chem. SOC., 1982,13, 221. Pro€. J. J. M. Beenakker (University oJ'Leiden) said: I want to point to a dificulty that occurs in analysing gas-transport properties of polyatomic molecules.While for noble gases the expansion in '' Sonine polynomials " converges rapidly, the situation in this respect for polyatornics is more complicated. Direct expcrirnents on the non- equilibrium distribution function (Douma et a[.)' suggest that the scalar dependence on the molecular velocity will remain similar to the situation in noble gases. There are, however, strong indications that the situation for the dependence on the rotational state is more complicated. show for example that higher polynomials in the internal energy contribute loo/, and more to the volume viscosity. Experiments that are more directly measuring the influence of molecular orient- ation on transport propertics such as field effects and flow birefringence point in thc same direction.These difficulties will be absent in low-temperature properties of systems involving hydrogenic molecules as long as only one rotational state is excited, Recent realistic model calculations for N,-N, by Turfa For a survey of this last aspect see ref. (3). ' 13. S. Douma, H. F. P. Knaap and J . J . M. Bcenakker, C'hern. fhys. Left., 1980,74, 421. A. F. Turfa, H. F. P. Knaap, H. J . Thijsse and J. J. M. Beenakker,Physic.a, 1982, in press. E. Mazur, TI1esi.T (Leiden, 3981 ; Physicl-r, to be published.) Dr. E. B. Smith (Oxford University) said: A number of authors have raised the296 GENERAL DISCUSSION possibility that certain properties including second virial coefficients, gaseous viscosity coefficients etc.can be regarded as depending almost entirely on the " isotropic part " of the potential surface (the unweighted spherically averaged potential, Vo). 1 do not believe this to be the case and would endorse the comments of Dr. McCourt on bulk gas transport data. No thermophysical properties we have investigated appear to be determined, to any useful accuracy, by the isotropic part of the potential alone. The anisotropy plays an important role, although as the paper by Smith and Tindell shows this can be taken into account by employing effective potentials that are strikingly different from the isotropic contribution. Dr. S. L. Price (Cambridge Uniuersity) said: The problem which this paper has left unsolved, that is the relationship between the effective spherical potential and the anisotropic potential, is perhaps more than an interesting intellectual problem.At the moment, the accurate calculation of transport properties for even diatomic molecules is so computationally expensive that experimental transport coefficients can only be used to test proposed diatom-diatom intermolecular potentials, and cannot be used in an iterative fashion to help determine the potential parameters as part of a multiproperty analysis. I would like Dr. Smith's opinion as to whether a knowledge of the relationship between the effective spherical potential and the actual anisotropic potential would enable us to use some information from experimental transport in determining the potential. Also, might we hope that such work could lead to a cheap method of calculating transport properties, or do the authors already know that the errors would be too large for any such method to be useful? Dr.E. B. Smith (Oxford University) said: If we could understand the relationship between our effective spherical potentials and the anisotropic potential then this could indeed lead not only to a deeper understanding of the role of anisotropy but to an economical method of evaluating the transport properties of polyatomic molecules. (It is interesting to note in this context that the remote origins of the modern inversion methods for spherically symmetric potentials lay in a method devised to allow the transport properties to be calculated more economically.)' However, our calculations of transport properties were not exact and I believe that the use of transport properties in the elucidation of the forces between polyatomic molecules will be of limited value until we can assess the accuracy of the various approximate models employed.Extremely laborious calculations must be under- taken (both close coupled, and using the classical formulation of Taxman, as attempted by Evans).2 Unfortunately an accuracy of better than 1% for the bulk gas transport properties will be necessary if useful information about intermolecular forces is to be deduced. This accuracy presents a formidable challenge particularly in view of the fundamental difficulties that Prof. Beenakker has pointed out. J. 0. Hirschfelder and M. A. Ellison, Ann. N. Y. Acad. Sci., 1957, 67, 451. D. J. Evans, Mol.Phys., 1977, 34, 103. Dr. M. La1 (Unileuer Research, Port Sunlight) (communicated) : The procedure adopted by the authors for computing the second virial coefficient, B, is equivalent to the use of the equation B = -2nN {exp[-(U12(r))/kt] - l)r2dr I: where (U,,(r)) is the potential of mean force defined asGENERAL DISCUSSION 297 the angular brackets denoting the average taken over the configurational ensemble corresponding to a pair of molecules at fixed r, the energy of a state in the ensemble being equal to the sum of the internal energies of the two molecules (i.e. U' = U: + Ui for a state i). True inversion of B, therefore, should reproduce (UI2(r)). The in- version approach adopted by the authors is able to yield only the temperature- independent potential functions, whereas, as the authors rightly point out, for a high degree of anisotropy one would expect an appreciable dependence of the potential of mean force on temperature.This imposes a serious limitation on the applicability of the method to systems with large I*. Would the authors indicate the upper limit of 1" beyond which the inversion procedure will break down? Another class of systems possessing temperature-dependent potentials of mean force is that of chain molecules. In such systems the temperature dependence is due not only to the orientational effects but also to the variability of the internal energy of the molecules with their configurational states.'v2 M. La1 and D. Spencer, J. Chem. Soc., Faraday Trans. 2, 1973, 69, 1502. M. La1 and D. Spencer, J.Chem. SOC., Faraday Trans. 2, 1974, 70, 910. Dr. E. B. Smith and Dr. A. R. Tindell (University of Oxford) (communicated): As noted by Dr. La1 the expression for the second virial coefficient at a particular temper- ature T (exp[-(U(r))/kT] - l)r2dr leads to a definition of ( U ( r ) ) , Thus second-virial-coefficient data can be used to define a series of effective isotropic potential functions which depend on temperature. Our inversions are entirely consistent with this observation. However, the important and unexpected conclusion of our paper is that there exists one isotropic temperature-independent potential function which will reproduce thermo- physical properties over a wide temperature range. Furthermore as shown in our fig. 6 this conclusion is equally valid at high degrees of anisotropy.Just how significant the anisotropy can be in the systems we investigated can readily be seen in fig. 6 and is reflected in the enormous difference between the isotropic potential derived from inversion of the properties and the unweighted spherical average of the di-Lennard Jones function. Prof. A. W. Castleman Jr, Dr. B. D. Kay, Dr. F. J. Schelling and Dr. R. Sievert ( University of Colorado) (communicated) : At several points in the discussion following the presentation of Barker the question was raised as to how the results presented for small Van der Waals molecules relate to the nucleation processes by which individual molecules aggregate to form a condensed phase. Alternatively, one may ask at what size do molecular clusters begin to evidence properties of the bulk phase.Several aspects of our research on both neutral and charged clusters address questions of this nature and have yielded new insight. Research involving neutral clusters formed in a supersonic jet has provided inform- ation on the mechanisms of cluster growth and on their structure. A study of the co- clustered nitric-acid-water system is particularly intriguing since the results have shown that properties normally associated with the condensed phase are displayed for very small numbers of molecules. High-pressure mass spectrometry experiments on charged species have also contributed significantly to our understanding of nucle-2 98 GENERAL DISCUSSION ation phenomena, and allow a quantitative interpretation of molecular interactions in Van der Waals clusters of electrolyte-containing systems. An analysis, in terms of classical nucleation theory, of the thermodynamic data obtained as a function of cluster size highlights the structural changes required on forming the liquid phase.Investigation of successive gas clustering has been directed at determining the nature of ion solvation and of ion-ligand orientational effects; these are experimentally difficult problems to study in bulk solution. A study of clusters produced by co-expanding nitric acid and water vapour has provided data particularly relevant to the above questi0ns.l Detailed experiments were performed on mixed clusters of these molecules and their deuterated analogue. Neutral clusters produced in a supersonic expansion are detected by employing a phase-sensitive electron-impact quadrupole mass spectrometer.During the course of this investigation, numerous mass spectra were taken at a variety of stagnation and ionization conditions using argon, helium and carbon dioxide as carrier gases at stagnation pressures ranging from 200 to 800 Torr. Nitric acid concentrations employed in the study range from 2.4 to 11.2 mol dm-3 and ionization voltages range from 15 to 100 eV. The general shape of the envelope of the intensity distributions shown in fig. 12 is FIG. 12.-Plot of the equivalent mole fraction of modes per cluster as a function of the number of nitric acid molecules in the cluster.GENERAL DISCUSSION 299 found to depend on the conditions in a complex manner, but in each case the observed mass spectra exhibit certain reproducible characteristic features which are independent of all experimental variables.An extensive study suggests that the proton associated with the clusters following ionization is contributed from a water molecule. In the case of the clusters containing one nitric acid molecule, it i s noted that there is a dis- tinct minimum at the cluster with five waters (fig. 12). For clusters with more than one nitric acid molecule and between one and six waters, clusters having less than a particular number of waters are not observed. Furthermore, the pure nitric acid clustcrs (dimer, trimer, etc.) are not detected, even through the dimer may be observed in experiments involving anhydrous nitric acid.The fact that thc position of the m inirna and onset compositions are independent of the stagnation temperature and pressure, carrier gas, acid concentration and ionizing voltage, strongiy suggests that they arise due to the intrinsic chemical nature of the system. Isotopic effects were ruled out, as identical results were obtained for the deuterated species. An attractive explanation for the position of the minima and onsets observed, and the attendant fact that clusters with greater than a certain nitric acid concentration are unstable, is that they are indicative of some change in the properties of the system. Such a situation could occur if a complex became sufficiently hydrated that it began to display properties normally associated with concentrated nitric acid solutions.These are known to be both thermally and photochemically unstable, giving rise to the evolution of NO, and oxygen. Plotting the equivalent mole fraction of water per cluster (associated with the unstable cluster sizes found in the experiments) as a func- tion of the number of nitric acid molecules in the cluster (insert, fig. 12), shows an approach to a specific limiting value. This value represents the critical composition range for which liquid solutions of nitric acid become unstable. This result indicates that neutral clusters begin to display liquid phase properties at very small sizes. Complementing ow research on the properties of neutral species are a number of studies of the thermodynamics of ion-molecule clusters, performed using the technique of high pressure mass spe~txometry.~*~ Such clusters bear a close relationship to neutral Van der Wads complexes, as a recent review paper discusse~.~ This work is also closely allied to the study of nucleation, in that the early steps in the addition of Iigand molecules to an ion may be approximately followed.The results of these studies, for example, made it possible to evaluate the limitations of the ciassical liquid drop model of ion-induced nucleation. By sampling the equilibrium cluster distri- bution as a function of temperature, the enthaIpy and entropy changes for clustering reactions may be extracted. Experimental hydration and ammoniation enthalpies determined in this manner were found to be in excellent agreement with the values predicted by the classical Thornson equation for clusters containing as few as four to six rnole~ules.~ The comparison with theory for the clustering of ammonia about several ions is given in fig.13(a).5 Predictions of entropy changes determined from the appropriate derivative of the free energy, as expressed by the Thomson equation, differed significantly from the experimental values, as fig. 13(6) shows, again for the clustering of NHJF5 More negative values are found experimentally than are predicted by theories based on bulk liquid properties. This finding reflects the more highly-ordered structure of smdl systems relative to the disordered nature of liquids. Such structure is useful in qualita- tively explaining the observation that while the ‘Thornson equation agrees with experiment for some systems, large discrepancies are found for others.Of related interest is the observation that for a variety of atomic and molecular ions of both positive and negative charge, the buIk heat of solvation is rapidly approached as the number of ligands In addition, stability breaks, reflected in values3 00 GENERAL DISCUSSION of the enthalpy changes as a function of cluster size, are closely related to known solution properties.2 In agreement with condensed phase studies, stabilization in- duced by the filling of solvation shells and by the formation of coordination com- plexes are obser~ed.~ These results indicate that the properties of small clusters are intimately related to those of the condensed phase, and that the study of their proper- ties provides a unique bridge between those of Van der Waals molecules and that of bulk solution.B. D. Kay, V. Hermann and A. W, Castleman Jr, Chem. Phys. Lett., 1981,80,469. A. W. Castelman Jr, P. M. Holland, D. M. Lindsay and K. I. Peterson, J. Ant. Chem. Soc., 1978, 100, 6039. I I I 5 10 100 I( 01 1 n I 00 FIG. 13.-(u) Gas-phase ammoniation enthalpie~:~ a, theory; V, Na+ ; A, K+ ; 0, Rb" ; El, Bi' ; 0, NHZ, (b) Gas-phase arnmoniation entropies [symbols as in (a)].GENERAL DISCUSSION 301 R. G. Keesee, N. Lee and A. W. Castleman Jr, J . Am. Chem. SOC., 1979,101,2599. P. Hobza and R. Zahradnik, “Van der Waals Systems”, in Topics in Current Chemistry, 1980,93, A. W. Castleman Jr, P. M. Holland and R. G. Keesee, J. Chem. Phys., 1978,68, 1760. N. Lee, R.G . Keessee and A. W. Castleman Jr, J. Colloid Interface Sci., 1980,75, 555. 539-90. ’ P. M. Holland and A. W. Castleman Jr, J. Chem. Phys., 1982,76,4195. Dr. G. C. Maitland (Imperial College, London) (communicated): The Parker and Pack IOSA calculation of transport coefficients referred to in the paper of McCourt and Liu makes the further assumptions that the energy exchange on collision is small (E = E’) and that the summation over the differential scattering cross-sections I(j‘+ jlO), for the rotational transitionj’tj, in the expressions for the transport cross-sections can be carried out t o j ’ = co. It is hence entirely equivalent to the Mason-Monchick (MM) approximation. However, the maximum value ofj’ is strictly restricted by the energy conservation expression Ej.max = E j + E where E~ is the energy of the rotor in statej and E is the initial relative kinetic energy of the atom and the rotor.Our recent calculations using the IOS approximation taking this into account have shown that making the additional MM assumptions can lead to considerable errors in the transport cross-sections, especially for high potential anisotropies and large rotational energy level spacings. Are the N,-Ar calculations of McCourt and Liu carried out using the Parker-Pack procedure or have they evaluated the full unapproximated 10s expressions? Use of the latter procedure would be expected to lead to higher values for both mixture viscosity and binary diffiusion coefficients. The papers by McCourt and Liu and by Smith and Tindell adopt different views on the ability of bulk thermophysical properties to give detailed information on potential function anisotropy.Smith and Tindell are pessimistic about their usefulness in this respect; however, all their calculations are based on the MM approximation. McCourt and Liu do see a role for bulk properties in this area although they do not demonstrate the extent to which they may be used to define the angular dependence as opposed to the average magnitude of the anisotropy. I share their optimism but for a different reason. Our full 10s calculations show that the deviations from MM-type calculations are very different for diffusion and viscosity coefficients and extremely sensitive to the anisotropic part of the potential. This suggests that dzferences between bulk transport properties could provide useful information on non-spherical interactions in the future.However, exploitation of this sensitivity is dependent on progress being made in two areas: (i) All present calculations are based on the Wang C hang-Uhlenbeck first-order solutions of the Boltzmann e q ~ a t i o n . ~ There are indi- cations that the uncertainties in these expressions could be significantly greater than those in the experimental data which are now available for polyatomic systems. There is a need, therefore, to extend the rigorous kinetic theory expressions for polyatomic gases to at least second-order. (ii) Although there are sound reasons for believing that the 10s approximation is in most cases a good one for transport cross-sections, confidence in its use would be increased by direct tests of its accuracy against either close-coupling calculations in the quantum-mechanical limit or trajectory calculations in the classical limit.We are currently pursuing both these targets. G . A. Parker and R. T. Pack, J. Chem. Phys., 1978,68, 1585. V. Vesovic, Ph.D. Thesis (University of London, 1982). C. S. Wang Chang, G. E. Uhlenbeck and J. De Boer, Studies in Statistical Mechanics (North Holland, Amsterdam, 1964), vol. 11, part C .302 GENERAL DISCUSSION Prof. F. R. W. McCourt and Dr. W-K. Liu (Uniuersity of Waterloo, Canada) (communicated): Dr. Maitland raises a number of interesting points, two of which we wish to respond to here. (1) Both the N,-He and the N,-Ar calculations reported in our discussion paper were evaluated using exact expressions relating the transport-relaxation cross-sections to the S-matrix elements of scattering theory; the S-matrix elements were calculated using the infinite-order sudden approximation.Thus, no approximations of the type introduced by Monchick and Mason have been employed in our work. (2) Apart from the pioneering effort of Shafer and G ~ r d o n , ~ we are aware only of our own crude attempt to extract information about the anisotropic parts of inter- molecular potential surfaces from bulk transport and relaxation phenomena. In both cases, use was made of phenomena which depend crucially upon potential anisotropies : in the case of Shafer and Gordon use was made of nuclear magnetic relaxation data, sound absorption data and spectral line-shape data, while in our case use was made of the magnetoviscosity Senftleben-Beenakker effect.To illustrate the type of con- clusion reached in both these efforts, it is instructive to consider a Lennard-Jones potential representation suggested some years ago by Pack.' It can be demonstrated that the relaxation cross-section determining the value of the ratio of magnetic field strength to gas pressure, H/p, at which the longitudinal shear viscosity coefficients attain half their saturation values is rather sensitive to the angular variation of as reflected by the value of the anisotropy parameter uM, and much less sensitive to the angular variation of as reflected by the value of the parameter aD. Thus, if R,, and co have been accurately determined from some other experimental results, magnetoviscosity measurements can be utilized to obtain a meaningful value of aM, for example.In the case of H,-He studied by Shafer and G ~ r d o n , ~ R,, and E~ are accurately determined from molecular- beam scattering data (although it is to be remembered that they did not employ a Lennard-Jones potential form!). In the case of N,-He we4 had assumed that R,, and E~ were accurately determined from molecular-beam total differential scattering data,6 but following the work of Faubel et u I . ~ it seems clear that this is not the case, so that an experimental " best " potential will clearly only be obtained if data from a large number of phenomena are analysed simultaneously. Insufficient sensitivity studies have been carried out to this date to allow us to know just how useful data from various phenomena will be for this purpose.&(e) = &()[I $- aDP,(COS e)] Much has yet to be done. W-K. Liu, F. R. McCourt, D. E. Fitz and D. J. Kouri, J. Chem. Phys., 1979, 71, 415. ' E. A. Mason and L. Monchick, J. Chem. Phys., 1961,35, 1676; 1962, 36, 1622. R. Shafer and R. G. Gordon, J. Chem. Phys., 1973,58, 5422. W-K. Liu, F. R. McCourt, D. E. Fitz and D. J. Kouri, J. Chem. Phys., 1981,75, 1496. R. T. Pack, Chem. Phys. Lett., 1978, 55, 197. M. Keil, J. T. Slankas and A. Kuppermann, J. Chem. Phys., 1979, 70, 541. Soc., 1982, 73, 205. ' M. Faubel, K. H. Kohl, J. P. Toennies, K. T. Tang and Y . Y . Yung, Faraduy Discuss. Chem. Prof. F. R. W. McCourt (University of Waterloo, Canada) said: Dr.Maitland has mentioned in his comment problems related to the relative importance of uncertainties in experimental measurements and those inherent in the lowest-order Chapman- Enskog solutions of the Wang Chang-Uhlenbeck-de Boer (WCUB) equation. I am basically in agreement with what he says regarding the need for careful and detailed consideration of higher-order solutions. There are, however, other aspects of theGENERAL DISCUSSION 303 theoretical treatment at this lcvel which, although mentioned in the monograph by Maitland et al.' and familiar to the transport aficionado, will not be familiar to many of those attending this symposium. I speak of the type of treatment needed in order to deal with rotating molecules, with their consequent degenerate states.One aspect is that the WCUB equation applies strictly only to molecules possessing non-degenerate internal states (e.g. most vibrational states) and, in particular, does not allow for phenomena which may depend upon the orientation of the rotational angular momentum of the colliding molecules. Thus all tensorial phenomena are excluded. This does not, however, render the situation entirely grave. It has been shown many years ago that, if tensorial polarization corrections are neglected, it is appropriate to use the WCU B formulation provided that the collision cross-sections appearing in the WCU B results are interpreted as degeneracy-averaged collision cross- sections. An estimate of the error involved in doing this can be had by examining the saturation values of the appropriate Senftleben (for paramagnetic gases) or Senft- leben-Beenakker (for diamagnetic gases) effects: these are typically 0.5% or less contributions.Unfortunately, at the 0.1% level of reproducibility even polarization contributions cannot be neglected. Another aspect related to higher-order corrections and mentioned at this sym- posium by Prof. Beenakker is that the dependence of transport phenomena on internal energy or, for linear diamagnetic molecules, equivalently on J 2 may be much more sig- nificant for polyatomic species than our experience with the Sonine polynomial expansion in monatomic gases would indicate. indicates that for pure N, higher polynomials in J 2 contribute of the order of lox to rotational relaxation and virtually explain all discrepancies between the lowest-order calculations and experiment. I t is entirely possible that a similar behaviour will be found also in the thermal conductivity and a number of the field effects.The question of scalar expansion terms has to be thoroughly reconsidered. A recent paper from the Leiden group G. C. Maitland, M. Rigby, E. B. Smith and W, A. Wakeham, Itztermolecular Forces: Their Origin and Determination (Clarendon Press, Oxford, 1981). F. R. McCourt and R. F. Snider, J, Chem. Phys., 1964, 41, 3185. A. F. Turfa, H. F. P. Knaap and J . J . M. Beenakker, Physica, 1982, 112A, 18. Prof. Ph. Brkchignac (University o j Paris, Orsay) said : As mentioned in Prof. McCourt's paper one of the properties related to the anisotropic interactions is the pressure broadening (PB) of spectroscopic lines, I would like to comment about the sensitivity of PB measurements to determine the anisotropy, particularly in systems like N,-He where the IOSA works well.The first thing that a potential should be able to reproduce is the magnitude of the PB cross-sections. However, it is well known that it is possible to find several potential surfrtces fitting the experimental value with completely different shapes. So that this magnitude is not by itself an observable vury sensitive to the interaction. But more can be learned from the variation of the PB cross-sections with temperature or rotational quantum number, J. Concerning the latter the variation is explicitly given within the IOSA by the factorization relationships due to Goldflam, Green and Kouri.' The PB cross-section of an i.r.line for a diatomic molecule is expressed as:304 GENERAL DISCUSSION where the set of primary cross-sections are the individual inelastic cross-sections out of levelj =- 0. The algebraic coefficients decrease sharply with x and all the odd coefficients are zero. Then the PB cross-section is nearly constant with,j, except for the very first transitionsj 7 0+1 a n d i = 1 -+2, and the amount of variation is related to the magnitude of This behaviour can be illustrated by the CO- He system, interesting because both theoretical calculatioiis and experimental data are available. Two ah initio potentials exist: the electron-gas surface reported by Green and Thaddeus (GT) and the recent surface by Thomas, Kraemer and Dicrcksen (TKD) computed with large SCF and CJ calculations with extended basis sets.Fig. 14 shows how different these surfaces I I I I I I I r Rlau 8 10 I I I I I I i 6 8 10 Rlau FK. I4,-Plot of the first three Coefficients of the Legentlre polynomial expansion for thc potential energy surfaces of ref. (3) (solid line) and ref. (4) (dashed line), are, for both of the isotropic and anisotropic parts. Note that the magnitudes of the PB cross-sections obtained by close-coupling calculations from the two surfaces are almost the same. However, the difference is reflected in the individual inelastic cross- sections, as apparent in table 6. The larger value of (70-r2 obtained from the GT surface leads to a bigger variation withj of the PB cross-section, although it is still not very strong.Thus PB data able to discriminate hctween the two surfaces have to beGENERAL. DISCUSSION 305 TABLE 6.-INDIVIDUAL INELASTIC CROSS-SECTIONS potential TKD GT %+1/A2 13.7 10.0 c0--t21A2 7.2 18.0 d l ) ( O , l ;O, 1)/d1)(4,5;4,5) 1.07 1.17 very accurate. Owing to the high-resolution capabilities of diode-laser spectroscopy the measurements achieved by Picard-Bersellini et are quite reliable. The value d l ) ( O , l ; 0,1yd1y4,5; 43) = 1.08 + 0.03 obtained in this work seems to favour the TKD surface. In conclusion, the rather poor sensitivity of PB data to the anisotropic interaction in similar systems has to be compensated by high-accuracy measurements. ' R. Goldffam, S . Green and D. J. Kouri, J. Chem. Phys., 1977, 67, 4149.S. Green and P. Thaddeus, Astrophys. J . , 1976, 205, 760. L. D. Thomas, W. P. Kraemer and G. H. F. Djercksen, Chern. Phys., 1980,51, 131. A. Picard-Bersellini, R. Charneau and Ph. Brechignac, J. Mu/. Srruct., in press. Mr. A. J. B. Cruickshank (Ui7icersit.y of Bristol) (commiriicated) : Battaglia and Gianturco refer to specifications for the effective, spherically symmetric, potentials for the oxygen-argon and the oxygen-oxygen systems. Although their paper is con- cerned Iargely with vibrational relaxation processes, the potentials cited [eqn (2) and (31, with table I ] may be relevant to solving a quite different problem, namely that the liquid-phase mixture argon 2 oxygen exhibits an unexpectedly large, positive voIurne of mixing. Smith and Tindell, in their paper, argued that effective spherical potentials for diatomic molecules are obtained from experimental data in a notably consistent way, by the use of their inversion procedure.Barker, in the next paper, asserts that con- densed-phase properties of the rare gases may be accurately predicted from the inter- action potentiah deduced from the low density gas-phase properties, provided only that account is taken of the triple-dipole three-body' interaction. It is therefore tempting to wonder if effective spherical potentiah for diatomics might not be as useful in relation to the condensed phases as Smith and Tindell have shown them to be in relation to gas-phase properties. We have already the Barker, Fisher and Watts potential fur Ar-Ar, and now Battaglia and Gianturco present effective spherical potentials for Ar-0, and 02-02.I would therefore ask Dr. Barker for his view on whether it might be worthwhile to attempt a new approach to the liquid-phase system Ar + O2 using the potentials listed above. Dr. E. B. Smith and Dr. A. R. Tindell (Oxfurd Uniuersity) (communicated): Mr. Cruickshank raises the question of how far our effective spherical potentials might, after appropriate non-pairwise contributions have been incorporated, account for the properties of condensed phases. We would be surprised if they were useful for the solid-state but have made no calculations in this area. However, we have embarked on computer simulation studies to test how far these potentials can reproduce liquid- state properties.Our first studies have compared the properties of di-Lennard-Jones Auids with those calculated using our effective potential and have found that the internal energies agree to within 4%. Dr. J. A. Barker (ISM, San Jose) (conrmtmicated): The suggested study of the Ar -f- U2 liquid-phase system would undoubtedly be of considerable interest. The306 GENERAL DISCUSSION unexpectedly large positive volume of mixing could be due to differences in shape of the three isotropic pair potentials. On the other hand it could also depend on the anisotropy of the potentials involving 0, or on an exotic many-body interaction in 02. Certainly one will never known until one tries, and it would make sense to examine the simplest possibility first. The calculation could be done by perturbation theory, by the Monte Carlo method, or by both methods. This is a nice example of the kind of question concerning condensed phases to which potential studies can contribute.Conversely this kind of experimental fact can also contribute to our knowledge of potentials. Prof, G . E. Ewing (Indiana Uniwrsify) said: I find the calculated vibrational relax- ation times for O,-O, puzzling. Why should they be two orders of magnitude faster than for 0,-Ar? Infrared spectra of the Van der Waals molecules 0, - - * 0, and O2 * * Ar suggest that they have comparable barriers to internal r0tation.l That is to say, there is nothing dramatically different in the intermolecular potential surfaces of 02-02 and 0,-Ar. Might the answer have something to do with the reduced moment of inertia (for internal rotation) of 02-02 which will be half that of 0,-Ar? An efficient vibration-rotation relaxation channel may therefore be open for O,-O, but closed for 0,-Ar.It will be interesting to see how more detailed calculations can resolve this puzzle. G. Henderson and G. Ewing, J. Chem. Phys., 1973, 59, 2230. Mr. B. H. Wells (Univeusily of Oxfurd) said: Tt is desirable that intermolecular pair potentials be based on the widest possible range of avaiIable data as even multi- property fits can sometimes prove surprisingly inadequate. This can be illustrated by reference to the spherically averaged potential of Battaglia et al. for oxygen. We have caIcuIated viscosity coefficients for oxygen using the potential they employ. The r.m.s. deviation of the calculated from the experimentaI viscosities was 8% i.e.the calculated viscosities are less than the experimental by an amount considerably greater than experimental error (ca. 1 %>. If, however, we make use of both transport data and second virial coefficients, we may employ an alternative method of obtaining an effective spherically averaged potential for oxygen. Experimental viscosities and second virial coefficients were fitted to a di-Lennard-Jones model The resulting best-fit parameters are E,B = 44.5 K, a,p = 0,309 nm, d* = d/omb = 0.29 where d is the distance between the two interaction sites on the same molecule. Jt is known that the unweighted Spherical average of such a potential gives a poor representation of gas-phase pro perties. However, the paper of Smith and Tindell suggests an alternative route to an effective spherical potential: second virial co- efficients were calculated using the di-Lennard-Jones potential and inverted as ex- plained by Smith and Tindell to obtain a spherically symmetric potential with para- meters c = 145 K, 0 = 0.333 nm (those of Battaglia e t d .are e = 132.3 K, 0 = 0.350 nm). The effective spherical potentiaI we obtain fits both the experimental second virial coefficients and viscosities, whereas, as stated above, that of Battaglia ef a[. does not fit the latter property. Total integral cross-sections for the scattering of oxygen were calculated using theGENERAL DISCUSSION 30 7 JWKB approximation. Both the potential of Battaglia et al. and the one we propose adequately reproduce the data of Brunetti et The effective spherical potential we propose has the virtue of not being constrained to any preconceived functional form.Such constraints may be partly responsible for the inadequacy of some multiproperty potentials. The method we employ for ob- taining an effective spherical potential could easily be applied to other diatom-diatom and atom-diatom systems. G. P. Matthews, C . M. S . R. Thomas, A. N. Dufty and E. B. Smith, J . Chern. Soc., Faraday Trans. I , 1976, 72, 238. ’ J. H. Dymond and E. B. Smith, The Virial Coeficients of Pure Gases and Mixtures (Clarendon Press, Oxford, 1980). B. Brunetti, G. Liutti, E. Luzzatti, F. Pirani and F. Vecchiocattivi, J. Chern. Phys., 1981, 74, 6734. Dr. P. Casavecchia, Dr. F.Pirani and Dr. A. Vecchiocattivi (University of Perugia, Italy) and Dr. F. Battaglia and Dr. F. A. Gianturco (University qf Rome, Italy) said : Concerning the remarks made by Prof. Ewing, it is worth noting that the differences between the vibrational relaxation times of 0,-0, and 0,-Ar are indeed related to their reduced moments of inertia, as we also state in the text of our paper. Because of the real potential-energy surface for 0,-0, being more anisotropic than for 0,-Ar, we expect that there should be a more efficient presence of (V,R)-T relaxation channels in the former rather than in the latter. Thus, the use of effective potentials to compute (V,R) relaxation times is likely to hold more realistically either with systems exhibiting weak anisotropy in their inter- actions or for those temperature ranges where vibrational inelasticity dominates, hence making the angular coupling a less effective process for relaxation whenever close (small impact parameter) encounters dominate the collisions.This also seems to be the result of our present calculations for the above systems. Going back now to the comments on our data by Mr. Wells, one should make clear at the outstart that the correct orthornormal expansion of a general function uniquely defines its spherical part, V A = ~ ( ~ ) . This is the correct spherical potential yielded by an unweighted spherical average of the converged expansion (1). Any truncated version of eqn (1) necessarily produces coefficients which depend on the level of truncation: l m a x V(r,Q) = 1=0 Therefore the efficiency of the summation on the right-hand side of Lqn (2) in represent- ing V(r,e) may conceivably be different in different regions of space.This means that one can define an efective spherical potential which depends on the chosen I.,,, and that may fit only some of the chosen molecular properties, especially if the experi- ments involved sample different regions of the whole interaction. One may also decide to use the prescription suggested by Smith and Tindell in their paper; then one obtains another way of defining an effective, spherical potential which may not agree with the previous results when different properties are fitted in order to test their respective qualities. As an example, our 0,-0, interaction is an effective spherical potential that represents the correct spherical coefficient of eqn (1) rather closely in the regions around the well and at the onset of the attractive tail, while is probably less realistic for the repulsive-wall representation of it.308 GENERAL DISCUSSION The one proposed by Wells, on the other hand, is fine-tuned for virial coefficients and viscosities, while it does not fit the spectroscopic data, as we found from WKB calculations carried out with their effective potential.Their unweighted spherical average, on the other hand, now fits those spectra but no longer reproduces the scatter- ing data. In conclusion, we feel that here we have both generated eflectiue spherical potentials that were each fine-tuned to slightly non-overlapping sets of properties and therefore could not completely agree with each other.Thus, any form of truncated expansion (2) is likely to produce different effective coefficients, which become stable against changes only after the level of convergence for expansion (1) is reached. Dr. J. M. Hutson (Uniuersity of Waterloo) and Dr. B. J. Howard (Oxford Uni- uersity) (communicated) : Obtaining anisotropic potential-energy surfaces by multi- property analysis is clearly advisable, since most experiments probe only part of the surface. When performing analyses of this kind it is particularly important to know which parts of the potential are well determined by the experimental data and which are not. by simultaneous least-squares fitting of potential parameters to moleculear-beam rotational spectra of the Ar-HC1 complex, pressure-broadening of HC1 rotational lines by Ar and second virial coefficients of Ar + HCl mixtures. The potential surfaces obtained in this way were well determined near the absolute minimum at the linear Ar-H-C1 geometry and on the repulsive wall, but the behaviour of the potential near the linear Ar-Cl-H geometry (0 = 180") was uncertain. the parameterisation chosen did not allow a secondary minimum at 0 = 180", but such a minimum was not excluded by the experimental data. More recently, information has emerged which suggests that there is actually a secondary minimum at 6 = 180". Most importantly, the MBER spectra of Ne-HCl demonstrate that a secondary minimum is present there, and it seems unlikely that Ar-HCl and Ne-HCl will be dissimilar in this respect. We have therefore obtained a new potential for Ar-HC1 (M5 potential,2 fig. 15), in which there are local minima at both linear geometries; the absolute minimum is still at linear Ar-H-Cl. The M5 potential reproduces the spectroscopic and virial data as accurately as the M3 poten- tial, but is to be preferred because of its closer resemblance to the Ne-HCl potential around 8 = 180". We recently obtained potential-energy surfaces for Ar-HCl In our M3 potential J. M. Hutson and B. J. Howard, Mol. Phys., 1981, 43, 493. J. M. Hutson and B. J. Howard, Mol. Phys., 1982, 45, 769. Prof. U. Buck (Max-Planck-Institut fur Stromungsforschung, Gottingen) (communi- cated) : The rainbow scattering of the total differential scattering cross-sections is usually used to fix the well depth of the isotropic potential. In order to check this re- lation for the potential surface derived by Hutson and Howard for Ar-HCl we per- formed a 9-channel coupled-states calculation on their M3 potential and compared the result with our measured cross-sections.l The M3 potential accurately predicts the experimental position of the rainbow maximum and the large-angle scattering, but not the amplitude of the rainbow minimum (see fig. 16). The new potential M5 with a secondary minimum at 8 = 180" should improve the fit to the scattering data since additional damping is introduced for the potential well depth. We have performed the same type of calculation for the M5 potential and the result is shown in fig. 16. The M5 potential not only gives a slightly better fit to the rainbow minimum but also predicts the general form of the large-angle scattering much better than M3. This isGENERAL DISCUSSION 309 3.0 1 I i I I I I 0 30 60 90 120 150 180 6 ; ' FIG. 15.-Contour plot of the fully optimised M5 potential for Ar-HC1. The absolute minimum is at the linear Ar-H-Cl geometry, and the absolute well depth is 180.5 cm-'. Contours are at 10 cm-' intervals relative to the absolute minimum. t L laboratory deflection angle, 8'" FIG. 16.-Experimentai and calculated angular distributions in the laboratory system for the poten- tial surfaces of Hutgon and Howard: t - ) M5,(---) M3. E - 88.1 meV.310 GENERAL DISCUSSION obviously due to a more realistic description of the potential anisotropy which is re- sponsible for the cross-sections at this angular range. This is a clear experimental indication that the subsidiary minimum at 0 = 180" is also correct for Ar-HCI. The fit could still be improved, but considering the experimental and possible theoretical errors (use of the CS approximation) present, the fit is satisfactory. An improvement could only be obtained by directly measuring the inelastic contributions. Our analysis gave differences in the cross-sections of more than a factor of two for 0+2 rotational transitions calculated using the M3 and M5 potentials. U. Buck and J. Schleusener, J. Chern. Phys., 1981,75, 2470.