GENERAL DISCUSSION Prof. E. Zaremba (Queen’s University, Kingston, Ontario) said: I would like to comment on the relationship between the repulsive part of the helium surface potential and the electronic charge density, specifically for metallic surfaces. It is frequently stated that Esbjerg and Norskov have ‘established’ the proportionality of the potential to the charge density, an appealing result from the point of view of surface structure determinations. That there is a correspondence between the potential and charge density is obviously true, but a strict proportionality is not. The original Esbjerg-Norskov suggestion was based on calculations of the embed- ding energy of helium in a uniform electron gas for which a linear dependence on the mean electron density is indeed found.Even if such a result were to apply to the surface potential problem, the coefficient of proportionality would be modified by the fact that the surface electron density is inhomogeneous. However, the true situation is more complex in that the repulsive potential is actually determined by an energy average of I-dependent scattering amplitudes weighted by an appropriately defined local density of states.’ Only in the asymptotic region beyond that of physical interest is a proportionality to density valid (as first pointed out by Zaremba and Kohn), but the constant of proportionality is substrate-dependent rather than a universal constant. Although the Esbjerg-Norskov suggestion can provide a useful guide, it cannot be assumed to be quantitatively reliable. E.Zaremba and W. Kohn, Phys. Rev. B, 1977,15, 1769; J. Harris and L. Liebsch, J. Phys. C, 1982, 15, 2275. Dr J. E. Black (Brock University, St. Catharine’s, Ontario) said: Dr Cardillo remarked informally that: ‘only altering force constants, when the surfaces also relax and reconstruct, may not be appropriate’, to which I would comment: The models used in the present lattice dynamical studies are admittedly simple, but they are the best available. While it is true that the gold(ll1) surface reconstructs, evidence is that others, such as silver( 11 1) undergo no reconstruction or relaxation. Prof. M. W. Cole (Pennsylvania State University, U.S.A.) said: I would like to make several comments concerning the interaction V ( r ) between a He atom and the system consisting of a Kr layer adsorbed on graphite.Before doing so, let me make two remarks. One is that there is, overall, rather gratifying consistency between the alternative potentials being discussed and hence with the experimental data of Larese et aL;’ the remaining discussion pertains to 10% effects. The second comment is that we have not performed close coupling calculations, such as those of J6nsson and Weare (henceforth denoted JW), which are necessary ultimately for resolving differences between ‘competing’ potential models. Nevertheless, we believe that our conclusion’ that three-body effects are present may well be correct, in disagreement with that of JW. When we began our study, we enquired about the role of three-body effects. John Barker suggested that we adopt the Axilrod-Teller-Muto (ATM) triple-dipole potential; this term alone is found to be successful in predicting in detail the behaviour of bulk noble-gas system^.^ A recent review by Meath and Aziz4 found use of the ATM term to be ‘the method of choice for representing non-additive 5758 GENERAL DISCUSSION effects, with due caution'.In the surface domain, the ATM term has been found to be appropriate in calculating both the surface tension of liquid Ar5 (J. Miyazaki, J. A. Barker and G. M. Pound, J. Chem. Phys., 1976,64, 3364); and in treating the statistical mechanics of adsorbed gases,6 where the three-body interaction is the ca. 15% 'McLachlan correction' due to the substrate. Based on this experience, we included the ATM term in our He potential.The net difference between our potential (CHC) and that of JW has several contributions, two of which are nearly negligible; this leaves A V V&, - VJW = VAT,( r ) - ( C:Hc - Ciw)( z + b)-3 where the choice b = 1.9 A is derived by Chung et al. as an accurate approximation to the sum over graphite planes. We refer here to the potential model 1 of JW for specificity. Here is the important point: we chose CFHC = 180 meV - A3, the theoretical value of Vidali et al.,' while JW used C:" = 110 meV - A3 semiempirically. One can discuss at length the merits of either choice. Instead, I prefer to discuss the relevant issue of uniqueness. To be specific, A V is -0.22 It 0.02 meV in the region 3 < z / A < 4 above the Kr layer. This is a very small magnitude (<20% of the total graphite contribution and 5% of the total potential); it is so small because the larger attraction of the CHC potential is partially cancelled by the repulsive ATM term. Thus the presence (or absence) of the latter is detectable only if the C3 disagreement is resolved.This requires, in part, a more precise assessment of the shallowest bound states for graphite. If the previous data' are confirmed, interesting questions will be raised about the theoretical value of C,. In summary, quite similar potentials have been proposed by CHC and JW. The difference between them is smaller than or comparable to both the triple-dipole term and the uncertainty in the graphite potential. Thus I believe that it is premature to draw any conclusion about the importance of the various terms contributing to these potentials.Note added in proof: New measurements by Chung et aL9 revise the situation for He reacting with bare graphite. The data are consistent with predictions of Toigo and Cole" based on the theoretical value of C3; this confirms, in my opinion, the analysis of Chung et aL2 of the adlayer problem. ' J. Z. Laresse, W. Y. Leung, D. R. Frankl, N. Holter, S. Chung and M. W. Cole, Phys. Rev. Lett., ' S. Chung, N. Holter and M. W. Cole, Phys. Rev. B, 1985, 31, 6660. 1985, 54, 2533. J. A. Barker, in Rare-gas Solids, ed. M. L. Klein and J. A. Venables (Academic Press, New York, 1976), vol. 1. W. J. Meath and R. A. Aziz, Mol. Phys., 1984, 52, 225. J. Miyazaki, J. A. Barker and G. M. Pound, J. Chem. Phys., 1976, 64, 3364; J.S. Rowlinson and B. Widom, Molecular Theory of Capillarity (Oxford University Press, Oxford, 1982), p. 184. L. W. Bruch, Surf: Sci., 1983, 125, 194. G. Derry, D. Wesner, W. Carlos and D. R. Frankl, Surf: Sci., 1979, 87, 629. S. Chung, A. Kara and D. R. Frankl, SurJ Sci., in press. l o M. Toigo and M. W. Cole, Phys. Rev. B, 1985, 32, 6989. ' G. Vidali, M . W. Cole and C. Schwartz, Surf: Sci., 1979,87, L273. Mr H. Jonsson and Prof. J. H. Weare ( University of California, San Diego, U.S.A.) (communicated): The potential used by Chung, Holter and Cole' is sufficiently different from our best-fit potential to give intensities in clear disagreement with the experimental data. It is furthermore qualitatively different from our theoretical potential in the approach to the He-graphite contribution.Fig. 7-9 in our paperGENERAL DISCUSSION 59 show the results of close-coupling calculations using a potential (potential 2) very similar to that of Cole and coworkers. The difference in the bound-state energy levels is only *0.02 meV, except for the n = 2 level where the potential used in the calculation is 0.05 meV closer to the best-fit value. There is very clear disagreement between this calculation and the experimental data, whereas our best-fit potential gives extremely close agreement with experimental data (fig. 10 in our paper). The selective absorption spectrum of highly corrugated surfaces is a remarkably sensitive probe of the bound-state energy levels of the atom-surface potential. It is precisely this high sensitivity which makes it possible to see the many-body correction. In our view there is no reason to assume the C3 coefficient calculated using a dielectric function fitted to reflectivity data, is highly accurate (not to mention C,, C5 etc.).Our approach is to use experimental data on the He-graphite interaction itself to construct a complete surface potential. Unless the experimental data are in error, this approach determines the He-graphite potential in the relevant region quite well. In ref. (2) Cole and coworkers presented these scattering data in support of the empirical rule of only including long-range three-body corrections. We agree with Prof. Cole’s current position that it is premature to draw such conclusions. ’ S. Chung, N. Holter and M. W. Cole, Phys.Rev. B, 1985, 31, 6660. J. Z. Larese, W. Y. Leung, D. R. Frankl, N. Holter, S. Chungand M. W. Cole, Phys. Rev. Let?., 1985,54, 2533. Dr J. A. Barker (IBM, Sun Jose, U.S.A.) said: I want to comment on the question raised in the subtitle of the paper by Jonsson and Weare,’ ‘What are the Non-additive Many-body Corrections,’ and in particular on the comment ccncerning earlier work by myself and my colleagues on rare gases, that ‘the good agreement (with experi- ment) obtained when only the Axilrod-Teller-Muto term was included appears to be fortuitous’. This latter statement is based on and echoes the conclusion of a paper by Meath and Aziz.’ That paper exhibited excellent agreement between ‘experimental’ many-body energies of rare-gas crystals at zero temperature and pressure and theoretical values calculated using only long-range many-body interac- tions (see table l ) , but concluded that this must be regarded as fortuitous since the long-range many-body interactions appeared to be substantially cancelled by first- order exchange three-body interactions.The agreement shown in table 1 confirms extensive earlier work, reviewed for example in ref. (3). Given that there is no adjustable parameter in the many-body calculation it is unlikely to be fortuitous. If this is coincidence it could more aptly Table 1. Total binding energies and long-range non-additive many-body energies of rare-gas crystals, from Meath and Aziz’ (units are J mol-’) Ne Ar Kr Xe total binding -1927*6 -7675*52 -11 190*84 -15902*89 total binding -1933*8 -7722I-t 11 -11 148* 13 -15 784*35 energy, calculated energy, experimental energy, calculated energy, experimental long-range many-body 81 * 6 663 f 52 1042*84 1397*89 total many-body 75*8 616* 11 1084* 13 1515*3523 22 21 20 19 18 17 16 15 vo~ume/cm~ mol-' Fig.1. Pressure-volume relation for argon at 0 K [from ref. (3), chap. 41. The solid curve includes pair potential and long-range many-body interactions, the dash-dotted curve includes also the nearest-neighbour first-order three-body exchange interaction; the dashed curve uses pair interactions alone. 20.0 22.0 21.0 volume/cm3 rnol-' Fig. 2. Pressure-volume relation for argon on the melting line in fluid (upper curve) and solid (lower curve) phases. From data in ref. (1 1). The curves are Monte Carlo results using pair interactions and long-range many-body interactions; the circles are experimental data.GENERAL DISCUSSION 61 3000 L 2000 z t' E 1000 0 I I I I I I I 1 100 200 300 400 T / K 10 Fig.3. Third virial coefficients for argon [from ref. (3), chap, 41. The dashed curve uses pair interactions alone, the dotted curve includes triple-dipole interaction, the dashed curve also includes third-order dipole quadrupole interactions and the solid curve also includes fourth- order dipole interactions. The circles are experimental data. be called conspiracy on the part of nature. Furthermore, the agreement is not confined to zero temperature and pressure or to the solid phase, as shown by comparison of calculated and experimental pressures for solid and fluid argon in fig. 1 and 2.Similar results are given in ref. (3) for neon, krypton and xenon. Independent evidence is provided by third virial coefficients as shown in fig. 3; values calculated with pair interactions alone are much too small, while those calculated with long-range three-body interactions agree well with experiment. Also shown in fig. 1 are pressures for solid argon calculated by including the first-order three-body exchange energy for nearest-neighbour triangles as calculated by Bulski and Chala~inski.~ These values are in disagreement with experiment. If this interaction is to be cancelled by some dther many-body interaction so as to give agreement with experiment, then this hypothetical interaction must have essen- tially the same density dependence (and opposite sign). This seems unlikely; it is almost equally likely that the calculated values represent an overestimate of the first-order three-body exchange energy.It is noteworthy that Wells and Wilson' find, for the Hartree-Fock contribution to the three-body energy of three helium atoms in an equilateral triangle of side 5.6a, a value which is smaller by a factor of 4.7 than the value of the first-order three-body exchange energy found by Novaro and Beltran-Lopez,6 who used methods roughly comparable with those of ref. (4). It is clear that overlap-dependent many-body interactions are much more difficult to calculate ab initio than the long-range interactions which can be related to oscillator strengths, and I suggest that the problems with many-body interactions raised by Meath and Aziz,* by Jonsson and Weare,' and in a later paper in this Discussion by Bruch and Ni7 are really problems of ab initio theory and computation; the experimental evidence that the long-range many-body interactions give the dominant many-body contribution is extremely strong.I would also like to suggest that the difficulty experienced by Jonsson and Weare in fitting the data for He scattering from Kr overlayers may be due at least in part to the He-Kr pair interaction, whi'ch I believe has larger uncertainty than they62 GENERAL DISCUSS I 0 N Table 2. He-Kr pair potentials r.m.s. deviation of scattering data r.m.s. deviation of depth, room second virial potential ( € / k ) / K liquid N2 temperature coefficients ESMSV~ U ( 0, K)'O HFD-19 HFD-29 24.7 0.6 1.6 29.1 0.6 1.7 33.7 1.2 1.5 30.8 0.4 2.2 0.7 suggest, especially with respect to the attractive well to which the selective-adsorption data are most sensitive.This is illustrated by the data in table 2, taken from ref. (9) and (lo), which suggest that potentials with depths as low as 24.7 K and as high as 33.7 K can fit the differential scattering cross-section data and second virial coefficients, which are the data most sensitive to the attractive well. The fourth potential in table 2 is that adopted by Jonsson and Weare.' The ESMSV potential does not fit gas-transport data, but these are not sensitive to the attractive well. Clearly these are extreme variations, but it does not seem unreasonable to suppose that the depth of the attractive well is uncertain by ca.5% or perhaps even 10%. The spectroscopic bound-state data which determine the attractive well very closely for like-atom pairs are not available for unlike pairs. ' H. Jonsson and J. H. Weare, Furuduy Discuss. Chem. Soc., 1985, 80, 29. W. J. Meath and R. A. Aziz, Mol. Phys., 1984, 52, 225. Rare-gas Solids, ed. M. L. Klein and J. A. Venables (Academic Press, New York, 1976), vol. 1. M. Bulski and G. Chalasinki, Chem. Phys. Lett., 1982, 89, 450. V. A. Novaro and V. Beltran-Lopez, J. Chern. Phys., 1972, 56, 815. L. W. Bruch and X-Z. Ni, Furuduy Discuss. Chem. SOC., 1985, 85, 217. C. H. Chen, P. E. Siska and Y . T. Lee, J. Chem. Phys., 1973, 59, 601. K. M. Smith, A. M. Rulis, G. Scoles, R. A. Aziz and V. Nain, J. Chem. Phys. 1977, 67, 152.' B. H. Wells and S. Wilson, Mol. Phys., 1985, 55, 199. lo K. Watanabe, A. R. Allnatt and W. J. Meath, Chem. Phys., 1982, 68, 423. " J. A. Barker and M. L. Klein, Phys. Rev. B, 1973, 7 , 4707. Prof. W. J. Meath (University of Western Ontario, London, Ontario) said: In reply to Dr Barker's comments, I would like to begin by reviewing, briefly, the state of affairs regarding various non-additive interactidn energies that can have the same magnitude as the triple-dipole energy for relevant interatomic or intermolecular configurations. Of particular interest are configurations of importance in the evalu- ation of the binding energy of rare-gas crystals. For details, including many refer- ences, see Meath and Aziz,' ref. (39) of Mr J6nsson and Prof. Weare's paper.Using Ne, Ar, Kr and Xe as examples, it is clear that the non-additive contribu- tion, AE(O), to the crystal binding energy E ( 0 ) of the rare-gas solids at OK is substantial, i.e. much larger than the experimental error in E ( 0 ) ; AE(0) is ca. 10% of E ( 0 ) for Ar, Kr and Xe and is evaluated' by subtracting the additive contribution to E(O), as calculated using reliable two-body potentials, from the experimental results for E ( 0 ) . Often the non-additive part of the many-body potential is repre- sented by the triple-dipole dispersion energy; this is the three-body analogue of the dipole-dipole dispersion energy occurring in the interaction between two atoms or molecules. The results so obtained for AE(0) are in good agreement with the known values of the non-additive contribution to the crystal binding energy.ForGENERAL DISCUSSION 63 what follows it is important to realize that 60% of AE(0) is from the first-shell (nearest-neighbour) lattice sum. This contribution arises from equilateral triangular configurations of the rare-gas atoms of side equal to the nearest-neighbour distance Ro in the crystal; Ro == R,, the position of the van der Waals minimum in the two-body potential. There are other terms in the non-additive part of the many-body potential: the question is, are there any having the same magnitude as the triple-dipole dispersion energy for important atomic configurations? Relatively speaking much less is known about these non-additive energies than is known about the energies in the two-body potential that have the same magnitude as the (analogous) two-body dipole-dipole dispersion energy.However, what is known suggests that there are non-additive energies that can compete with the triple-dipole energy. The best understood ‘competitor’ is the three-body first-order exchange energy Ex(’) (NA) for which results are available for (He),, (Ne),, (Ar), and the quartet spin state of H3. The quartet state of H3 can be regarded as a model for rare-gas trimers that is analogous to using H2(3Z3 to model rare-gas dimers.* Exact results for E,“’(NA) for H3 have been available for some time. Reliable results for the rare-gas trimers, for relevant interatomic distances, have been available only recently; they have been computed’ in the Heitler-London method using quality SCF wavefunctions to represent the isolated atoms and in the SCF trimer scheme with corrections for superposition errors (both techniques agree well with each other).These are difficult and expensive calculations, and data are sparse, but fortunately results for the first-order three-body exchange energy are available for the atomic configurations of most importance to the crystal binding energy calculations. In all cases Ex(’)( NA) is negative for equilateral triangular configurations of side R , and cancels part, all, or more than all, of the triple-dipole contribution to AE(0) leading to a disagreement with experiment if the non-additive part of the many-body potential is represented by the sum of the triple-dipole and the first-order three-body exchange energies. There is evidence to support the view that the SCF atom is smaller than the real atom and therefore the magnitude of E,‘”(NA) is underesti- mated, not overestimated, by using SCF wavefunctions to represent the isolated atoms; for example dipole polarizabilities evaluated at the SCF level of calculation are smaller than the exact polarizabilities for the type of species of interest to this discu~sion.~ Further the recent calculations of Wells and Wilson’ support the results of Jeziorski et aL3 for the first-order three-body exchange energies of (He), in equilateral triangular configurations and not those of Novaro and Beltran-Lopez.6 This can be seen by plotting In lEx(l)(NA)l, as a function of the side of the equilaterial triangle, from the three sources of the calculations (fig.4). The calculations in ref. (6) take no account of superposition effects and their reliability decreases rapidly as the side of the equilaterial triangle increases past 5 bohr. I would like to emphasize that Dr Aziz and I in no way suggested using the sum of the triple-dipole and first-order three-body exchange energies to represent the non-additive part of the interaction energy. Rather we used this sum of two energies to give an example of a cancellation effect between two non-additive energies of the same magnitude that occurs for relevant interatomic separations in a well defined application. We then went on to discuss other possible non-additive interaction energies, or other effects, that could be significant in helping to supply the ‘missing’ contribution to AE (0).Dr Aziz and I tried very hard to provide an unbiased presentation of the difficulties associated with the representation of many-body interaction energies.64 GENERAL DISCUSSION 3 4 5 6 Fig. 4. Plot of In lEx(’)(NA)/ against the side of the equilateral triangle for (He), from the three calculations discussed in the text: x, Polish group; A, Wells and Wilson; 0, Novaro and Beltram-Lopez. side We have apparently succeeded in this since, for example, at this Discussion three papers used our work to support three different points of view, i.e. (1) that the triple-dipole energy is not adequate, (2) that it is adequate and (3) that the representa- tion of many-body potentials is not well understood. Perhaps the best way to end this comment is to provide a few quotations’ from ref.(39) of Prof. Weare’s paper. ‘What is needed are reliable calculations of the many-body potential that are consistent through all terms of the same order of magnitude.. .’; ‘The authors hope this paper will serve as a stimulus for further a6 initio calculations of many-body interaction energies that are designed to help resolve this difficult problem.. .’; ‘An important point must be made clear. There is little doubt that the total many-body interaction often seems to be effectively represented by the triple-dipole interaction energy especially when coupled to a reliable two-body potential. The evidence for this can be found in papers by Barker and coworkers, and others, that are concerned with the evaluation of third virial coefficients as well as liquid and crystal properties of the rare gases; the references in ref.(1) provide details. Since, at present, very little is known about the other relevant non-additive forces needed to counter-balance the cancellation between the non-additive dispersion and the three-body first-order exchange energies, this approach is still probably the method of choice for representing non-additive effects,GENERAL DISCUSSION 65 with due caution, in the immediate future. However, there are excellent reasons for concern and a recent example augments the earlier discussions on this point’. (This is discussed briefly, with references, in my next comment.) ’ W. J. Meath and R. A. Aziz, Mol. Phys., 1984, 52, 225. These model dimer potentials have been quite successful; see references in ref.(1) and, for example, the detailed review by R. A. Aziz, Springer Ser. Chem. Phys., vol. 34, ed. M. L. Klein (Springer, Berlin, 1984), pp. 5-86. Of particular importance in this context are the relatively recent calculations of G. Chalasinski, M. Bulski, B. Jeziorski and L. Piela, Chem. Phys. Lett. 1982, 89, 450; 1981, 78, 361; Theor. Chim. Am, 1980, 56, 199; Znt. J. Quantum Chem., 1976, 10, 281. For example see H. J. Werner and W. Meyer, Phys. Rev. A, 1976, 13 and Mol. Phys., 1976,31,855. B. H. Wells and S. Wilson, MoZ. Phys., 1985, 55, 199. U. A. Novaro and V. Beltran-Lopez, J. Chem. Phys., 1972, 56, 815. Prof. S. J. Sibener ( University of Chicago, U.S.A.) said: I would like to mention that the well regions of the pair potentials between helium and heavy rare-gas pair potentials (Ne, Ar, Kr, Xe) may still be a little uncertain. The last experimental differential scattering cross-sections were taken at collision energies that were, I believe, larger than the well depths of these systems.It would be really nice if data were available from experiments which used cryogenically cooled He expansions operating at $10 K. Such measurements would be expected to confirm the accuracy of the current potentials, or lead to further (slight) refinements. It is obviously of critical importance that the most accurate pair potentials be used as input to gas-surface scattering calculations which seek quantitatively to assess higher-order contributions to the interaction.Prof. D. R. Frank1 (Pennsylvania State University, U.S.A.) said: Much of the discussion of this paper has hinged on the accuracy of the experimental energy levels. This is very difficult to assess. In addition to the statistical error limits, there may be systematic errors due to various misalignments and to lack of perfect planarity of the crystal surface. These problems are particularly severe for the higher-lying levels, which are the most important for the determination of C,. In addition to the measurement errors, there are the further problems of identifying the resonance signatures and of relating their energies to the actual bound-state energies. For these reasons, the kinematically determined energies should be regarded as, at best, a starting point for a complete scattering calculation.Agreement of the calculated beam intensity pattern is, as Prof. Weare and others have emphasized, the critical test. Mr H. J6nsson and Prof. J. H. Weare ( University of California, San Diego, U.S.A.) (communicated): The evidence, Dr Barker points out, for the validity of only including long-range many-body corrections, is certainly very impressive. There remains, however, the theoretical problem of explaining why this rule works. Theoretical calculations of three-body corrections to the exchange repulsion do not support it. The scattering experiment is in many ways different from the condensed matter experiments discussed by Barker (quasibound state us. bound, heteronuclear us. homonuclear) and applying this empirical rule to the scattering experiment is an extrapolation.Our tentative conclusion is that a significant discrepancy exists between the experimental bound states of He-Kr/graphite and the theoretical potential constructed from best estimates of two-body potentials and Barker’s rule for many-body corrections. However, as we point out in our paper, this experiment can not66 GENERAL [>ISCUSSION yet be considered conclusive. If, as Prof. Frank1 suggests, there is a systematic error in the experimental bound states reported for He-graphite, or if, as Dr Barker suggests, the well depth of the He-Kr pair potential is off by 10°/O, the discrepancy could be removed. By presenting our numbers at this discussion with quantitative estimates of error bars based on some literature values, we hoped to stimulate critical reevaluation of these factors.Prof. Scoles and coworkers are currently remeasuring the He-graphite scattering with very high resolution, and Prof. Aziz er al. have adjusted the He-Kr pair potential simultaneously to all available experimental (two-body) data (seven different properties).’ This new He- Kr pair potential gives larger discrepancy between the measured and calculated bound state of He-Kr/graphite (0.44 meV instead of 0.35 meV). Possibly, additional high-resolution experimental data which particularly probe the well depth of the pair potential are needed, as Prof. Sibener points out. Dr Barker points out a discrepancy between the a6 initio calculations performed by Wells and Wilson’and those performed by Novaro and Beltran-Lopez3 for He, at R = 5 .6 ~ . We have recently been doing first-order calculations following the formalism of Bulski and c o ~ o r k e r s . ~ We find quite good agreement with the calculations of Wells and Wilson, with difference of only 8%. For smaller distances, the restricted Hartree-Fock calculations of Novaro et al. give values that are close to the first-order calculation. At this large distance (5.6u), however, there is a discrepancy, but we note that Novaro et al. only report one significant figure in that case. We calculated the He- Kr-Kr three-body correction to the first-order exchange repulsion. When the Kr atoms are placed one lattice spacing apart (4.26 A) and the He atom placed at the van der Waals distance away from each Kr atom (3.7 A) to form an isosceles triangle, this correction is larger in magnitude by 10% and opposite in sign to the ATM correction.However, the scattering experiment is sensitive to the bound states of the surface potential averaged over the unit cell. Because of the large corrugation and rapid drop in the three-body correction to the exchange repulsion, the shift in the lowest bound state energy level of the average potential is very small, only 10% of the triple-dipole correction. Theoretically, Barker’s rule is therefore better justified for the scattering experiment than the condensed phase experiments. We are grateful to Prof. Meath for pointing out typographical errors in the values given for the ATM coefficients in the preprint of our paper. ’ R. A. Aziz and M.J. Slaman, 9th S>.nip. on 7hermopli,t,qiccrl Propcwies, June 23-27, 1985, University ’ B. H . Wells and S. Wilson, Mol. PI1y.s.. 1965, 55, 199. of Colorado. Prof. G . Scoles (Uniueryity oj’ Waterloo, Ontario) said: ( i ) With respect to the magnitude of the error in the two-body tcrms of the He-Kr-graphite potential the figures quoted by Dr Barker are definitely too large. Indeed for the case of He interacting with Ar it has been established beyond any doubt’ that the well-depth values obtained in Berkeley, quoted by Barker as possible numbers, were much too low, while the results obtained pleviousi): in Waterloo, with well depths around 30 K, were correct ( i i ) While it is true thdt high-resolution differential collision cross-sections similar to those csed to seti’n the He-Ar case, which were obtained in the laboratory of Dr LJ.Ruck in Gottingeil, have not yet been measured for the He-Kr case, it is also true that there is absolutely no reason to believe that the latter system would behave any dittetently from the former one. (iii) Moreover. the recent work of Aziz and coworkers has substantially increased the level of precision with which two-body intermolecular forces are being studied. Since many properties which depend on the two-body interaction only are being fitted at the same time, the resulting potentials have been proved to be as accurate, if not better, than thoseGENERAL DISCUSSION 67 obtained via the highest-resolution scattering and spectroscopic experiments. Since the recent He-Kr potential of Watanabe, Allnatt and Meath, used by Jonsson and Weare, has also passed through the same filter and is in substantial agreement with the Waterloo collision cross-section data, we conclude that the He-Kr interaction is presently known with a very good degree of confidence. Minor adjustments (< 1 '/o on the R axis and <4% on the energy axis) may be still possible but larger ones are quite unlikely.In summary, while Jonsson and Weare, quite correctly, call our attention to all possible sources of error that could invalidate their conclusions, their tentative conclusion about the inadequacy of the triple-dipole correction to account for all many-body effects is not likely to require drastic changes owing to future increases in the precision with which the two-body interaction is known.On the other hand, and for completeness, it should be noted here that our present knowledge of the He-graphite and Kr-graphite interactions, while rather good, is far from perfect and that more work in this area would be very useful. ' R. A. Aziz, P. W. Riley, U. Buck, G. Maneka, J. Schleusenes, G. Scoles and U. Valbusa, J. Chem. Phys., 1979, 71, 2637. Prof. W. J. Meath (University of Western Ontario, London, Ontario) said: The evaluation of the bound states of He interacting with a Kr overlayer on graphite involves the use of He-Kr and He-graphite two-body potentials. The He-Kr potential' is apparently a very good two-body potential which reproduces2 most of the experimental gas-phase properties for HeKr to within experimental error.However, it is based on the simplest form of the XC (exchange-coulomb) potential model, which involves only a single overall damping-corrector function to convert the long-range dispersion energy series into a representation of the second- and higher-order coulomb energies.lv3 The parameter in the representation of the exchange energy was adjusted to fit experimental data for the second virial coefficient, as were the dispersion-energy coefficients within their estimated uncertainty. This simple model is adequate for two-body rare-gas potentials, since the uncertainties in the higher-order dispersion energy coefficients are relatively large. An individually damped dispersion energy version4 of the model is necessary for other types of interactions and will also probably be needed for rare-gas interactions when the dispersion energy coefficients are known more accurately.It is to be emphasized that the adjustments in the He-Kr potential were made to best-fit mixed second virial coefficients for He-Kr. The fact that the potential reproduces most of the other gas-phase properties for the He-Kr system as well gives credence to the model, but it is important to realize that no attempt was made to improve the fit to all the available experimental data. Given the errors inherent in the gas-phase data used to construct and test the XC potential, which are generally larger for mixtures than for pure gases, and the possibility that gas-phase and gas-surface properties may well be sensitive to different ranges of interatomic separations, there is a distinct possibility of small but significant errors being introduced into the calculation of the He-surface binding energies through the use of the XC ;potential.More impor- tantly it seems that the He-graphite potential may well be less well understood than the He-Kr potential. The binding energy co evaluated using the two-body potentials is 2.5% higher than the estimated true energy, which has an error of *2.1% associated with it. On the other hand E~ evaluated using the two-body potentials plus the triple-dipole energy is 6% higher than the estimated true energy, with the contribution due to the triple-dipole energy being positive and only 3.4% of the magnitude of the true68 G E N E RAL r) I sc us s I o N value for e0. The authors are correct in stating that the shift due to the addition of the triple-dipole term is in the wrong direction to yield an overall calculated result value for eo.The authors are correct in stating that the shift due to the addition of the triple-dipole term is in the wrong direction to yield an overall calculated result, in agreement with the estimated correct value of E ~ . This is suggestive of a problem in using the triple-dipole energy to represent the many-body effects in this calculation. However, as much as I would like it to be so, 1 do not think this is a definitive example showing the inadequacy of the triple-dipole energy as a representation of non-additive interaction energies. For example, taking the most unfavourable inter- pretation of the error in the correct result for E~ leaves a discrepamy of 3.9% between calculation and the estimated correct value of E ~ , a difference that could be accounted for by the combined errors in the two-body potentials.The authors do suggest the possibility of their error bars being too optimistic and also discuss another contribution to eO, namely that due to thermal vibrations. There is a more definitive example in the literature that does strongly suggest that there are problems in using the triple-dipole energy to represent non-additive energies in the calculation of bulk properties.556 It involves the analysis, by Teitsma and Egelstaff,6 of the virial expansion of the direct correlation function for Kr gas using measurements of the structure factor of the gas determined by neutron- diffraction techniques.Their analysis of the data gave results for a generalization C(q, T ) of the usual third virial coefficient C(0, T ) , which differs significantly, for q # 0, from results obtained by using the triple-dipole energy as a representation of the many-body forces. C(q, T ) is quite sensitive to many-body effects. K. Wsanabe, A. R. Allnatt and W. J. Meath, Chem. Phys., 1982, 68, 423. ’ R. A. Aziz, Springer Ser. Chem. Phys., vol. 34, ed. M. L. Klein (Springer, Berlin, 1984), pp. 5-86. See, for example, A. Koide, W. J. Meath and A. R. Allnatt, Mol. Phys., 1980, 39, 895; R. A. Aziz, W. J. Meath and A. R. Allnatt, Chem. Phys., 1983, 78, 295 (erratum, 1984, 85, 491). W. J. Meath, D. J. Margoliash, B. L. Jhanwar, A. Koide and G. D. Zeiss, in Intermolecular Forces, ed B.Pullman (Reidel, Dordrecht, 1981), pp. 101-115. W. J. Meath and R. A. Aziz, Mol. Phys., 1984, 52, 225. A. Teitsma and P. A. Egelstaff, Phys. Rev. A, 1980, 21, 367; P. A. Egelstaff, Adv. Chem. Phys., 1983, 53, 1. Prof. G. Scoles (University of Waterloo, Ontario) said: The very high sensitivity of the atom-surface scattering data to errors in the two-body term of the interaction, which is a prerequisite for safe conclusions about the contributions due to three-body forces should not, in my opinion, be a deterrent to seeking progress in this direction. This is because of three reasons. First, as I mentioned in the previous remark, with little extra work the two-body terms may indeed be obtained with sufficient accuracy. Second, there is, in my opinion, sufficient information in the gas-surface scattering data to extract from them, with a careful analysis, both a better two-body interaction and the three-body contributions.Third, because of the broken symmetry at the surface, three-body forces at the gas-solid interface are bound to be different from their counterparts in thF oulk and to teach us a lot about their importance in liquid cavities, solid defects and similar situations. This makes the problem much more interesting and should represent an extra incentive for working toward its solution. Prof. M. W. Cole (Pennsylvania State University, U.S.A.) said: Could Dr Hutson describe to us the procedure he mentioned for evaluating a two-body’ dispersion interaction for the case of an ionic crystal?GENERAL DISCUSSION 69 Dr J.M. Hutson and Dr P. W. Fowler (Cambridge University) replied: We construct the atom-surface potential using a procedure similar in spirit to that of Celli et aZ.' in that we start with a pairwise additive sum over He-F- and He-Li+ potentials, and represent the pair potentials using an SCF repulsive part and a damped dispersion series for the attractive part. However, there are a number of points of difference. ( 1 ) We calculate SCF repulsive energies involving anions in the crystalline environment, rather than free anions. The crystalline environment is simulated by performing calculations on a cluster made up of a central F- ion at the surface of the crystal and its five nearest-neighbour cations. The entire cluster is embedded in a point charge lattice, so that the effects of the Madelung field are correctly represented. The resulting SCF repulsion energies are considerably smaller than those used by Celli et al. This reflects two effects. First, that an in-crystal anion is considerably smaller than a free anion; and secondly, that first-order SCF calculations significantly overestimate interaction energies involving anions. (2) Instead of fitting an empirical He-F- c6 coefficient to the measured bound states,' we compute c6 and C, coefficients for He-F- and He-Li' interactions using coupled Hartree-Fock calculations, with the in-crystal ions modelled as described above. The polarizabilities and dispersion coefficients of surface F- ions are found to be ca. 10% larger than those of ions in the bulk, but very much smaller than those of free F- ions. The detailed manner in which we include non-additive terms and construct the full He-LiF interaction potential is too complicated to give in full here, but will be described in a forthcoming paper.2 With no adjustable parameters, we obtain a He-LiF potential which has a well depth of 8.11 meV; this may be compared with the value of ca. 8.7 meV required to reproduce the measured resonance energies. It should be possible to bring the well depth into agreement with experiment with fairly small modifications of the a6 initio parameters. We hope that the resulting potential will give as good an account of the experimental data as that of Celli et aZ., but will have much more realistic asymptotic behaviour at both long and short range. ' V. Celli, D. Eichenauer, A. Kaufhold and J. P. Toennies, J. Chem. Phys., 1985,83, 2504. P. W. Fowler and J. M. Hutson, Phys. Rev. B, in press.