GENERAL DISCUSSION Dr. P. G. Burton (University of Wollongong) said: In supporting his error esti- mate of 5 1 kcal mol-' in the values he quoted for the strength of the Van der Waals interactions between the first-row hydrides, Prof. Pople only referred to the approach to convergence, through several orders of perturbation theory, of the estimate of the electron correlation contribution to these interaction strengths. However, an independent source of uncertainty in these quantities is the systematic errors stemming from the limited basis sets employed. Even with the largest 6-31G** bases used, errors in the individual molecule multipole moments at the (SCF and) CI level will surely affect the computed interaction strengths to a greater degree than A1 kcal mol-'. In addition, since such small basis sets are incapable of accounting for a large fraction (ca.30%) of even the valence correlation energy of the molecules studied, no matter what order of perturbation theory is used, there is a systematic error in the computed total energies of the molecules which is two orders of magnitude greater than the claimed precision. Prof. Pople has presented no evidence that this large systematic error is unimportant in the comparison of the Van der Waals complex and separated molecule energies, nor any evidence as to the magnitude of the basis set superposition artefact which is involved in these comparisons. Prof. J. A. Pople (Carnegie Mellon University, Pittsburgh) said : An error estimate of & 1 kcal mol-' would be appropriate only for the weaker interactions between first-row hydrides.The larger interaction energies between systems such as BH3 may indeed be subject to greater error for reasons of basis-set limitation, as you indicate. Prof. J. Jortner (Tel-Aviv University) said : The charge-transfer contribution to intermolecular Van der Waals bonding may be amenable to experimental detection by the observation of the enhancement of the infrared intensity of one of the components. Intermolecular charge flow is expected to exhibit a vibronic-type contribution to the infrared intensity.' This mechanism is well documented for " charge-transfer " complexes in condensed phases. Such a charge-transfer correlation effect implies a relation between the intensity enhancement and an energetic shift of the fundamental frequency towards lower energies.R. S. Mulliken and R. Pearson, Molecular Complexes (Wiley-Interscience, New York, 1969). Prof. V. Magnasco (University of Genoa) said: 1 fully agree with the philosophy inspiring Dr. Stone's paper. To make progress in the understanding of intermolecular interactions (possibly including chemical reactions) we ought to resort to some kind of perturbation approach, where knowledge of the properties of the isolated molecules would be of great value, as Prof. Buckingham said in the course of this meeting. The proper treatment of the small overlap existing between atoms or molecules at relatively large internuclear separations is essential to give a correct description of the intermolecular interaction and its breakdown into components of different orders.On similar premises, we have recently developed a perturbation treatment which is second order in the intermolecular potential and infinite order in the intermolecular overlap. The method, which makes use of the whole set of bonding and antibonding orbitals of the separate systems, allows for a full account of the non-orthogonality existing in second order between dzflerent interacting partners, giving results which110 GENERAL DISCUSSION are easily analysable into components with a direct physical meaning. In this context, our previous work on first-order interactions has been improved by allowing for polarisation and delocalisation (charge transfer) described in terms of single excitations, and applied both to the study of the orientation dependence of the intermolecular inter- actions in the short to medium range ' and to the problem of rotational barrier in ethane.3 Further work is in progress mostly on intramolecular interactions.The overall results are close to those obtained from conventional MO-SCF calculations. Electron correlation might be included as well in terms of double excitations. Secondly, turning to Dr. Stone's paper, I would like to make a few comments. which I believe can be of rather general interest. (i) In decomposing its first-order energy into components, Dr. Stone does not mention explicitly the A-correction term to the interaction energy, which is zero if v0 is an eigenfunction of the unperturbed Hamiltonian Ho. Now, although it is well known that this term is small for Hartree-Fock wavefunctions of the separate mole- cules, giving a correction which vanishes at least as the fourth power in the intermole- cular ~ v e r l a p , ~ * ~ its importance has been seen to increase considerably for the heavier rare-gas atoms and at larger di~tances.~ A possible answer to that might be in the fact that Dr.Stone defines the exchange repulsion energy [eqn (6)] implicitzy as the difference between the total (electronic) energy to first order [eqn (4)] and the electro- static (or Coulombic) energy. The A-correction might then well be absorbed into the exchange term. However, we would expect a sensible loss in accuracy in obtaining such differences when the quantities involved are almost of the same order of magnitude, and a lack of physical transparency in the definition of the exchange component.Starting from the same zeroth-order wavefunction in the form of a single deter- minant of non-orthogonal spin-orbitals {vi"), we have given an expIicit expression for the first-order exchange energy among several overlapping charge distributions in terms of the one-electron density matrices occ pR(0011 ;l') = 2 V/i"(l)V/i"*(l'), I fdxlpA(OO1 1 ;1) = NA (1) A all i j pf(0011;l') = - 2 2 vio(l)Aijvjo*(l'), Jdx1pf(00/1;1) = 0 ( 2 ) where eqn (2) is the overlap density matrix and Aij = (SM-l)i,j with M = 1 + S the matrix of the unperturbed spin-orbital basis. If we denote by the induction potential due to nuclei and unperturbed Hartree-Fock wavefunction of the separate molecules electron density on B, for the first-order exchange is The first term in eqn (4) is the true two-electron exchange, which survives in the case of no overlap, and coincides with eqn ( 5 ) of Dr.Stone's paper in the case of twoGENERAL DISCUSSION 111 molecules. The second term gives the interaction of the overlap density (2) in the Coulomb-exchange field provided by nuclei and unperturbed electron density of all other groups. The last term gives the Coulomb-exchange interaction of the overlap densities among themselves. Since eqn (4) describes the penetration of charge clouds due to their overlap, the exchange component of the first-order interaction energy has also been called by us first-order penetration. We can notice that all terms in eqn (4) do arise from either charge overlap or non-local exchange potentials, and that all non-additiuity of intermolecular forces in first order stems from the last two terms in (ii) The use of basis sets of insufficient size (i.e.below Hartree-Fock) in yo is expected to introduce errors into the Coulombic and exchange overlap components of the first-order interaction. Such errors are known to be rather large for the heavier rare gases, even using two-term SCF wavefunctions based upon Slater orbital^.^ (iii) Use of orbital energy differences in the calculation of excitation energies [eqn (7)], being tantamount to assuming an uncoupled procedure where the details of the many-electron nature of the wavefunction are lost, is expected to underestimate second- order eqn (4) * V. Magnasco and G.F. Musso, Chem. Phys. Lett., 1981, 84, 575. V. Magnasco, Atti Accad. Ligure Sci. Lett., 1981, 38, 3 and references therein. G. F. Musso and V. Magnasco, J . Chem. SOC., Faraday Trans. 2, 1982,78, 1609. J. N. Murrell and A. J. C . Varandas, Mol. Phys., 1975, 30, 223. A. Conway and J. N. Murrell, Mol. Phys., 1972, 23, 1143; 1974, 27, 873. V. Magnasco, Mol. Phys., 1979,37, 73. ' V. Magnasco and G. Roncallo, Chem. Phys. Lett., 1981, 79, 125. Dr. A. J. Stone (Cambridge University) said: The calculations reported in the paper do not use big enough basis sets to give useful values for the dispersion and polarization energies. We have since done calculations on Ar * * - HCI using an [8~4p2d/3~2p/7~5p2d] basis, and find that the minimum appears at about the distance found by Hutson and Howard, but that the well is ca.10% deeper. The largest attractive contribution by far is the dispersion term, but there is also a substantial contribution from the " charge-transfer correlation " term, which arises from double excitations of the form AOCCBOCC-+AVirtAVirt. The single-excitation charge-transfer term is smaller, and the polarization term larger, than for the smaller basis. Similar results were obtained for Ar - We have done some calculations using Epstein-Nesbet energy denominators instead of the Moller-Plesset denominators described in the paper. The com- putational labour is much increased, and it is not clear whether the results are better. It must be remembered that in this calculation a more negative energy (which the Epstein-Nesbet formalism does give) is not necessarily a better result, because there is no variational principle for the energy diflerence that we seek.The A-correction mentioned by Dr. Magnasco arises when the first-order energy is partitioned in a certain way. Our calculation uses a different partitioning scheme, merely separating out the electrostatic and exchange terms. The total first-order energy is the same in both schemes. We do not believe that our scheme leads to any significant loss of numerical accuracy. HF using an [8~4p2d/3~2p/5~4p2d] basis. Dr. P. G. Burton (University of Wollungong) said: My first comment relates to the relative importance of higher than double excitations, compared to single and double excitations, in contributing to the Van der Waals interactions. While I agree that the higher substitutions are less important than the single and double excitations, and so the intermolecular perturbation theory should concentrate first on the latter, our own112 GENERAL DISCUSSION results on very small systems (e.g.He-He and H2-H,) indicate that these higher excitations are directly responsible for a significant fraction of the total Van der Waals interaci ion. Recently we have directly compared the Van der Waals well depths, computed with and without these higher excitations, for the He-He and H2-H, systems using large scale CI supermolecule calculations and moderately large orbital basis sets (ca. 40 functions per electron pair). While we do not yet have full CI for these four-electron systems with these basis sets, we have variational PNO-CI (limited to double excit- ations) and the non-variational but size-consistent CEPA2-PNO result^.^ The pair natural orbitals (PNO) used in these calculations prediagonalize the C1 expansions for these systems and lead to very compact CI wavefunctions, with minimal loss of accuracy (ca.0.7% of E,,,, when a selection threshold of The CEPA2-PNO calculations allow, in an approximate but systematic way, for the important independent pairs of double excitations which are required to simultan- eously correlate both electron pairs of these systems, i.e. to provide for statistically independent simultaneous correlation of both subsystems. In the He-He system we find that the more limited PNO-CI accounts for only 90% of the Van der Waals well depth, while in H2-H2 the higher-order excitations are even more important and account for 18% of the computed well depth (PNO-CI = 82% of CEPA2-PNO well depth). In the light of these results it seems inadvisable to exclude higher-order excitations from consideration as important contributors to the Van der Waals inter- action of systems with a great many more electrons.My second comment relates to the treatment of the basis set superposition correc- tions in the intermolecular perturbation theory for Van der Waals molecules. Al- though Boys and Bernardi introduced the " function counterpoise " correction in the context of correlated wavefunctions for intermolecular force calculation^,^ it has been most widely applied at the SCF level of wavefunction computation.The magnitude of the basis-set extension or superposition error is quite different (and typically smaller) at the SCF than at the (limited or) full CI level, simply because the less constrained correlation Ansatz greatly increases the opportunity for an arbi- trary basis function to contribute to the description of a molecular system. When a molecule is represented using a given basis set, the maximum extent of the basis set extension error that is possible in an intermolecular force supermolecule calculation is given at the SCF level by the difference between the computed SCF energy and the corresponding Hartree-Fock limit energy for that molecule. With a given limited or full CI Ansatz, the corresponding upper limit to the superposition error is given by the difference between the computed CI energy and the exact non-relativistic energy for the molecule. We can illustrate these ideas numerically by considering a basis set developed for our recent H,-H, study, where we have the advantage of the precise knowledge of the exact energies of H2 at either the SCF or CI level.The partially optimized basis designed for H, consists of 39 independent basis functions, with characteristics defined by the quantities EsCF = - 1.133 40 E h (- 1.133 63 E J , E,, = - 1.172 45 Eh (- 1.174 47 Eh) and Q2" = 0.4516 (0.4574). The values in brackets are those for the Kolos and Wolniewicz wavefunctions for H2 at the same bond distance (we make these comparisons at re, although ro was used for the intermolecular force compu- tations).The error at the SCF level by restricting the H, basis to these 39 functions is seen to 0.23 x Eh (441 cm-I), an order of magnitude larger. E h (882 cm-I). In practice the actual superposition corrections we found with this basis Eh is used). Eh (50 cm-'), while the error at the CI level is 2.02 x Thus the upper limit to the CI superposition error for H,-H, is 4.04 xGENERAL DISCUSSION 113 for each H2 in the H2-H2 calculations \\*ere greatest for shortest intermolecular dis- tances (midbond to midbond) considered (R = 3.0 au) and even here in the worst case geometry (the T configuration) the CI superposition correction was less than 0.097 x low3 E h (21.2 compared to the total interaction energy for this same geometry of 43 230 X E h (9487.9 cm-I). The corresponding correction for this geometry, at R = 6.5 au near the Van der Waals minimum for the isotropic potential, was 0.009 X E h (1.95 cm-I).Although the actual superposition corrections are seen to be orders of magnitude less than the theoretical upper limit for these corrections, these corrections nevertheless enter at the SCF level and CI level in a similar ratio to that suggested by the respective limiting values, over the whole range of the potential. The reason that the actual superposition corrections are such a small fraction of the theoretical maximum value of the corrections is that at intermolecular distances near the Van der Waals minimum, the overlap of the basis functions of one molecule with the electron distribution of the other molecule is so small.How large the theoretical maximal SCF and CI superposition errors are in a particular supermolecule calculation depends on (a) how close to a full C1 the chosen CI Ansatz represents and (b) the basis set employed. The very small values for the superposition corrections that we have quoted above arise only because the H2 basis set employed accounts for 96.3% of the correlation energy of this molecule; typically nothing like this approach to full correlation is achieved with conventional basis sets (2-5 basis functions per electron), certainly not for systems as large as ArHF. Given the expectation that the SCF superposition error in a given basis might be expected to be an order of magnitude less than the corresponding error at the CI level, the fact that Stone and Hayes have quoted SCF superposition corrections in their ArHF and ArHC1 studies which are, even at the SCF level, comparable to the well depth in the vicinity of the Van der Waals minimum, implies that the corresponding CI superposition corrections should be an order of magnitude greater than the well depth with the basis sets employed.P. G. Burton, J. Chem. Phys., 1979,70,3112; Chem. Phys. Lett., 1981,82,335; and unpublished results. R. Ahlrichs, H. Lischka, V. Staemmler and W. Kutzelnigg, J, Chem. Phys., 1975, 62, 1975; (6) R. Ahlrichs, Cumput. Phys. Commun., 1979, 17, 31 ; (c) S. Koch and W. Kutzelnigg, Theor. Chim. Acta, 1981, 59, 387. S. F. Boys and F. Bernardi, Mol. Phys., 1970, 19, 553. N. S. Ostlund and D. L. Merrifield, Chem. Phys. Lett., 1976, 39, 612.P. G. Burton, P. D. Gray and U. E. Senff, Mol. Phys., to be published, * P. G. Burton and U, E. Senff, J. Chem. Phys., 1982, 76, 6073. Dr. A. J. Stone (Cambridge University) said: The basis-extension error is certainly larger for the double-excitation terms, which correspond to the electron-correlation part of the interaction energy, than for the single-excitation terms. The magnitude of the error in the latter case is of the order of 0.5 x E h for Ar - - HF in the region of the minimum, as can be seen from fig. 1 of our paper. The double-excitation terms, however, separate very conveniently, some of them being free of basis extension error entirely, and some being wholly basis extension error, at least to zeroth order in overlap. The former include the dispersion energy, arising from excitations of the form AoccBocc-+AvirtBvirt, and the “ charge-transfer correlation” AoccBocc+AvirtAvirt- These contribute cn.-1.0 x El, and -0.7 x loa3 Eh, respectively, in the region of the minimum of Ar - - - HF. The “ extension-correlation ” AOCCAOCC+ AvirtBvirt is an example of a term which is entirely attributable to basis extension error, to zeroth order in overlap, and its magnitude at the minimum for Ar - * - HF is -20.5 x Eh. Naturally it is not included in the total interaction energy. It is evident that, as Dr. Burton suggests, the basis-extension effect is very much larger for114 GENERAL DISCUSSION the correlation terms than for the single excitations; but it seems to be a relatively straightforward matter to isolate it and correct for it within the perturbation form- alism.Dr. J. M. Hutson (University of Waterloo) said : Douketis and Scoles have recently attempted to model intermolecular forces in the Ar - * * HF system using a semi- empirical Hartree-Fock + dispersion (HFD) model. In calculations of this type it is not clear what point to use as the origin of the dispersion forces, but the results depend critically on the point chosen; in particular, there is no reason to suppose that the H F centre of mass is the appropriate origin, although this assumption has fre- quently been made in the past. Drs. Stone and Hayes have found that the electrostatic and exchange/repulsion energy curves for Ar - - - HF and Ar - * - FH are similar if the intermolecular distance is measured to a point 0.4 au along the F-H bond.Is there any evidence for similar behaviour in the induction and dispersion energies ? C. Douketis and G. Scoles, 1982, unpublished work. Dr. A. J. Stone (Cambridge University) said: We have not examined the induc- tion and dispersion terms yet for Ar - - FH in a large basis. Nor have we investi- gated the interaction potential for non-linear geometries. Consequently the evidence for an effective interaction centre 0.4 bohr along the bond is at present very slight. Our present belief is that attempts to model intermolecular forces in terms of a single interaction site for each molecule are misguided. A distributed multipole analysis of HF shows that the electrostatic forces are very much better described in terms of a two-site or three-site model, and a suitable two-site model gives a good fit to the data of Hutson and Howard for Ar - * - HC1.2 A distributed multipole model also accounts well for the dielectric second virial coefficient of HCl, which a one-centre multipole expansion completely fails to A.J. Stone, Chem. Phys. Lett., 1982, 83, 233. C. G. J o s h and A. J. Stone, 1981, unpublished work. ' A. J. Stone and S. L. Price, 1982, unpublished work. Prof. J. Jortner (Tel-Aviv University) said : An interesting relation may exist between the thermal excitation of intermolecular vibrational modes in (N2)n clusters and phase transitions in solid N2. Thermal excitation of intermolecular vibrations in the cluster will result in orientational disorder, which is reminiscent of the K- (ordered)-+P(disordered) phase transition in the solid.A cardinal question is: What is the smallest size of the (N2),* cluster expected to exhibit a solid-state-like phase transition? Obviously, the dimer (n = 2) is too small, and solid-state-like behaviour is expected to set in for larger values of n. In this context, the calculation of the thermodynamic properties of these clusters will be of considerable interest. In alluding to the intriguing question concerning the onset of " solid-state proper- ties '' in finite clusters it is imperative to distinguish between local properties of clusters, e.g. vibrational excitations of intramolecular vibrations, electronic excitations of a guest molecule and global (collective) properties, such as phase transitions.These local and global solid-state features will set in at different coordination numbers (size) of the clwter. Prof. A. van der Avoird (University of Nijmegen) said : Concerning the question at which cluster size Van der Waals molecules begin to acquire solid-state properties, I would like to quote a recent paper by Etters et a2.l who have calculated thermodynamicGENERAL DISCUSSION 115 properties of (CO,), clusters for 2 2 n 2 13 using a classical Monte Carlo scheme. It is demonstrated in this paper that surprisingly small clusters, even dimers, already show rather well defined orientational order-disorder " phase " transitions and melt- ing. E. D. Etters, K. Flurchick, R. P. Pan and V. Chandrasekharan, J. Chem. Phys., 1981, 75, 929. Prof. W. Klemperer (Haruard University) said: We have attempted to study the structure of (N,), by molecular-beam methods.We found, however, that the electric dipole moment was below the limit of detectability by deflection methods.' We should like to call attention to the observation of several rotational resonances for the perhaps closely related isoelectric system (CO),. This study is at present incomplete. We believe it is possible to extend this work, and thus we suggest that it would be valuable to have detailed studies of the inter- molecular potential for (CO), as have been discussed for the N2 dimer. S. E. Novick, P. B. Davies, T. R. Dyke and W. Klemperer, J . Am. Chem. Soc., 1973,95, 8547. P. A. Vanden Bout, J. M. Steed, L. S. Bernstein and W. Klemperer, Astuophys.J, 1978,234,503. Prof. Ph. Brechignac (University of Paris, Orsay) said : Coming back to the N2 dimer and to the similarity between the two molecules N, and CO, I wish to point out that such similarity arguments should be used with much caution. Indeed, we measured a few years ago by infrared double resonance the propensity rules for the rotational-energy transfer in gaseous CO, which are governed by the anisotropy of the CO-CO potential. The results were found to be very different from what would be inferred from a N2-N, surface. In the latter case odd-AJ collisional transitions are symmetry-forbidden, and because of the quadrupole moment (1.5 D A) of the N2 molecule the rotational relaxation is dominated by A J = 2 changes. The CO molecule has a quadrupole moment very close (2.2 D A) to that of N, and a fairly small dipole moment (0.1 D) so that similar behaviour would be predicted by a first Born-approximation kind of calculation from the long-range multipolar anisotropic interaction.However, the experimental findings are, very unexpectedly, that 2/3 of the rotationally inelastic transitions proceed by A J = 1 , while 1/3 proceed by A J = 2 and < 10% by A J >, 3. The only way in which we can understand these results is if the short-range anisotropy is essential for the outcome of inelastic collisions. The CO-CO potential should have a rather large both P,- and P,-shaped short-range anisotropy to be responsible for such selection rules. The P,-like part is necessary to account for the efficiency of A J = 1 transitions, while the P,-like part has to allow for some cancellation of the A J = 2 Q-Q contribution by destructive inter- ference during the dynamics.In conclusion, it seems impossible to predict from simple arguments the rotational-energy-transfer behaviour of such weakly polar molecules. Of course any structural information on the CO dimer would be very interesting in order to see whether it confirms the double-resonance results. Prof. A. van der Avoird (University of Nijmegen) said: I do believe that CO molecules behave rather differently from N,, mainly due to the importance of P, terms in the short-range anisotropic interaction with other collision partners. Support for the importance of such terms can be found in a recent calculation of the CO-H, interaction potential, followed by a spherical expansion [cf.eqn (3) of our paper]. It appears that, among the anisotropic vLA,LB,L(R) terms in the potential (where LA refers to CO and LB to H,), vl,o,l is smaller than v , , ~ , ~ but considerably larger than u , , ~ , ~ , at least in the short range. At the same time it should be remembered that the116 GENERAL DISCUSSION short-range contribution to u2,2,4 is substantial. (This term does not just contain the qua drupole-quadrupole interaction .) M. C. van Hemert, Thesis (University of Leiden, 1981). Dr. R. Altman, Dr. M. Marshall and Prof. W. Klernperer (Haruard University) said: It is well known that the isoelectronic species N2 and CO are quite similar. This similarity is useful in illustrating the likely complexity in charge distributions of weakly bound complexes.The geometric structure of CO and N2 complexes is extremely similar. The dipole polarizability of CO and N, are also very similar. The dipole moments of complexes involving these species are however quite different. We illustrate this point below. co 1.95 2.60 2.886 0.592 0.482 2.41 0 1.5178 0.388 NZ COIN2 1.76 1.1 1 2.38 1.09 2.875 1 .oo 0.35 0.35 1.38 2.41 3 1 .oo 1.2497 0.2525 1.53 We note that the ratios of induced moments do not scale simply with polarizability nor are they constant from one complex to another. In this sense the electrical properties appear to reflect the complexity of bonding in these Van der Waals acceptor- donor complexes. Prof. A. D. Buckingharn (Cambridge University) said: Prof. Klemperer has given us interesting results for the electric dipole moments of BF3C0, BF3N2, HClCO and HClN,.He concluded, from a comparison of the dipole moments of the correspond- ing complexes of CO and N2, that the induced dipoles are not proportional to the polarizabilities. However, the electric field at the CO and N2 molecules is far from uniform. The absence of a centre of symmetry in CO means it has a dipole-quad- rupole polarizability A that will produce a dipole in a field gradient. The induced dipole linear in the field is Apa = aa&p + +Aa@&y + . where AaE, is the gradient of the electric field. The tensor Aaay is responsible for rotational Raman scattering by compressed CH4, for in a tetrahedron aaa is isotropic but AXy2 # 0. In the non-uniform field near an atom or molecule in an optical field, the rotating CH4 molecule has a fluctuat- ing induced dipole proportional to A,,, which couples its rotation to the radiation field.2*3 The tensor Aapy may be considered to describe the asymmetric distribution of the polarizability within the molecule ; it vanishes in centrosymmetric molecules.The dipole induced in CO by a field gradient can be evaluated since its A-tensor has been determined by ab initio cornp~tation.~ For a linear structure of the type +CO, the ratio of the dipoles proportional to A and to a isGENERAL DISCUSSION 117 where n = 3 if a dipole and 4 if a quadrupole, produces the field. For CO, Azzz/azz = -0.474 A where the z axis is from C to 0.4 Thus if R = 3 8, ApA/Apa = 0.237 for a dipole and 0.31 6 for a quadrupole.Thus the simple model Ap = a Eis not adequate. However, the general long-range form Ap = a * E + +A*AE + * . * + $p: E2 +Y&Y i E3 + . . . may be appropriate. Non-linear contributions to Ap, determined by the hyper- polarizabilities p,y etc. probably contribute only a few percent of Ap. A. D. Buckingham, Ado. Chem. Phys., 1967, 12, 107. A. D. Buckingham and G. C. Tabisz, Mol. Phys., 1978, 36, 583. D. P. Shelton and G. C. Tabisz, Mol. Phys., 1980,40, 299. R. D. Amos, Chem. Phys. Lett., 1980,70,613. Miss S. H. Ling, Mr. R. R. Miledi and Dr. M. Rigby (Queen Elizabeth College, London) (communicated). Dr. Barker has pointed out the value of empiricism in the de- velopment of intermoleqular potentials. We have attempted to establish an accurate potential for nitrogen using procedures which have proved reliable when applied to the inert gases.Our approach has been to use results from quantum-mechanical calculations in the regions for which they are believed to be reliable, at short and long range. We believe that the behaviour in the well region cannot yet be obtained with sufficient accuracy in this way, and have used comparisons with thermophysical properties to establish the shape of the potential well. The form of the short-range potential was studied by carrying out ab initio cal- culations for the dimer (N2)2 in several relative orientations and at a number of separ- ations. The results were in good general agreement with those recently published by other worker~,l-~ and have been fitted to a simple analytic function based on a diatomic site-site interaction model.The long-range function was based on the results of Mulder et aZ.,4 which includes several electrostatic and dispersion contributions. Electron-cloud overlap effects on the quadrupole-quadrupole energy were taken into account using the procedure of Ng et aL5 The total potential function was represented as a sum of these two interactions, with the long-range functions modified in the well region by a variety of cutoff functions similar to those employed in the HFD potentials for the rare gases.6 Various damping functions have been studied, and the most suitable form determined on the basis of second virial coefficients and 0 K lattice properties. Allowances were made for the contribution to the sublimation energy due to non-additive triple-dipole contribution^,^ and quantum corrections to the virial coefficients were estimated on the basis of approximate potential models.Calculations of dilute-gas shear viscosity were then carried out, using the Mon- chick-Mason scheme, for some of the better potentials. No single potential was found which was capable of completely reconciling the lattice properties, virial coefficients and viscosities. This may reflect the inadequacy of the Monchick-Mason procedure. For our best potential functions, the maximum attractive energy for the N2 dimer was found for the crossed structure, in agreement with the findings of Berns and van der Avoird.8 However, our potentials differed in several respects from those of these workers, and the unweighted angle averaged potential from our models had deeper wells and less steep repulsion than their potentials.F. H. Ree and N. W. Winter, J. Chem. Phys., 1980,73,322. B. Jonsson, G. Karlstrom and S. Romano, J. Chem. Phys., 1981, 74, 2896. F. Mulder, G. van Dijk and A. van der Avoird, Mol. Phys., 1980,39,407. K-C. Ng, W. Meath and A. R. Allnatt, Mol. Phys., 1979, 38, 449. * D. G. Bounds, A. Hinchcliffe and C. J. Spicer, Mol, Phys., 1981,42,73.118 GENERAL DISCUSSION R. Ahlrichs, R. Penco and G. Scoles, Chem. Phys., 1976,19,119. R. M. Berns and A. van der Avoird, J. Chem. Phys., 1980,72, 6107. ' P. Monson and M. Rigby, Mol. Phys., 1980, 39, 1163; 1981,42, 249. Dr. J. Tennyson and Prof. A. van der Avoird (University qf Nijmegen) said: In the last few months we have performed fully variational ro-vibrational calculations on the nitrogen dimer, (N2)2, utilizing a secular equation method based on the close-coupling approach.We have formulated the problem in body-fixed coordinates using the Hamiltonian given by eqn (7) of our paper. As in eqn (8) the basis set is constructed from R - 'xn(R>g42 d e A , vA,& ,PB)D~, M A a A O ) (1) (2) with y$k".jB = 2 yjA,m(oA,qA) y j B , k - m(eB,qB)(ikljA,m ; j B , k - m). m Following recent work on atom-diatom systems the radial functions were written in terms of associated Laguerre polynomials (3) (4) Xn(R) == p+ NnaY(a+1)/2e-Y/2La "01) Y = A exp[--P(R -Re)] where the parameters A , p and Re (and hence a) are optimized variationally. paper, the integrals over the angular coordinates can be performed analytically With this basis and the spherically fitted dimer potential, eqn (3) and (4) of our <gj';$' 1 vAB(R,'A,oB,vA - qB)Iq$kzJB) @A'pA'@B'IB A B 2 uL L(R)(-l).i--k+jA+jB z= bkkl A B, LA, LB, L x [j'(j' + 1)jA(jA + 1)jtXjE; + 1)L(L + l ) L A ( L A + 1)LB(& + l)j(j + l)jA ( j A + 1 ) j d j B + I)]+ For J (the total angular momentum) = 0 use of all basis functions with n 5 4 and jA,jB I 9 leads to a secular problem of dimension 3350.This can be greatly simplified, however, by use of the permutation and inversion symmetry of the system. Schematically, functions of the form (6) cDpi(j),p(jA) = 2 - y w j k , j B a & g j i : j A ) ; j A + j B even and two-vectors describe a basis for the irreducible representations when J = 0, where p ( i ) is the parity of i.This allows the problem to be solved in ten blocks, and means that for J = 0 no secular matrix larger than 475 x 475 needs to be diagonalised. It is possible to use the local symmetry of basis functions (6) and (7) about the equilibrium structure (6, = o B = 90", v, = qA - qB = 90" or 270") to make a corres- pondence between the present results and the fundamental vibrational levels given by the harmonic-oscillator rigid-rotor model. Table 1 makes such a comparison. TheGENERAL DISCUSSION 119 TABLE 1 .-" FUNDAMENTAL " FREQUENCIES OBTAINED WITH THE FULL CALCULATIONS AND THE HARMONIC OSCILLATOR MODEL, USING THE SPHERICAL POTENTIAL OF BERNS AND VAN DER AVOIRD coordinate symmetry frequenciedcm - DZd SJ 0 Ci harmonic full R A1 A: 39.2 33.2 8,,8B E E+ 22.1 14.2 v B1 B: 13.9 8.I zero-point energy 48.7 47.1 anharmonicity in the potential causes the harmonic-oscillator model to overestimate the fundamental frequencies by up to 40%. The zero-point energy, however, is well represented, suggesting that the ground state is fairly harmonic in character. Besides these vibrationally excited levels there is one other low lying level (sym- metry B y ) which lies only 2.4 cm-' above the ground state. This is due to the splitting of the ground state by tunnelling through the low barrier, 25.5 cm-l, in the 9 co- ordinate which separates the two equivalent equilibrium structures at 9 = 90" and 270". Fig. 1 shows cuts through various wavefunctions for UA = OB = 90". The 360 q / o 180 0 5.0 360 180 0 7.0 9.0 5.0 7.0 360 180 d" 0 5 .O 7.0 9.0 RlG 9 .o FIG. 1.-Amplitude of the wavefunctions of the lowest states of A:, B; and B t symmetry with J = 0.The cuts are with OA = 8, = 90". Solid and dashed contours represent regions of positive and negative amplitude, respectively.120 GENERAL DISCUSSION A,+ and B; states lie well below the barrier; the Bf state is the highest state " local- ised " in the q coordinate. This is in good qualitative agreement with the analysis by Long et aL2 of their infrared spectrum. Higher excitations should become increasingly free-rotor like. This trend is difficult to detect due to coupling with vibrations in other coordinates and in particular the effect of the slightly larger barrier in the 8 directions. Many of the observed free internal rotor levels seem to be resonances in the continuum which would be difficult to reproduce with a secular equation method.Fig. 2 shows the ground state and those states that we identify with the R and 180 180 O,/" 90 - 0 I I 0 5.0 7.0 9.0 5.0 7.0 9 .o 180 90 @I O 0 5.0 7.0 9.0 Ria0 FIG, 2.-Amplitude of the ground-state, stretch fundamental and (@A,&) fundamental wavefunctions with J = 0. The cuts are for p = OB = 90". Solid and dashed contours represent regions of positive and negative amplitude, respectively. (8,,8,) fundamentals. In terms of the harmonic-oscillator model the stretching fundamental is strongly mixed with a state deriving from an overtone in the (OA,O,) bending coordinate. Mixing such as this would be difficult to simulate within the BOARS approximation used by Barton and Howard for (HF),.With the basis outlined above, we have obtained all the bound states with J = 0 and 1. The J = 1 states were calculated neglecting Coriolis interactions, making k a good quantum number. This approximation has been tested by performing full calcu- lations for certain symmetry blocks. The effect of the Coriolis interactions is found to be small.GENERAL DISCUSSION 121 The nitrogen dimer can be regarded as a collision complex made up of two N2 monomers each having either ortho or para nuclear spin symmetry. This symmetry is fully reflected by our symmetry adapted basis functions (6) and (7). Conservation of this symmetry means that for ortho-para complexes the J = I , k = 1 state is lower than the J = 0 state.The full set of results is currently being prepared for publi- cation. J. Tennyson and B. T. Sutcliffe, J. Chem. Phys., 1982, Oct. 15th. C. A. Long, G. Henderson and G E. Ewing, Chem. Phys., 1973,2,485. Prof. R. J. Le Roy (University of Waterloo) said: It is very striking to note that the calculations described by Barton and Howard and by Tennyson and van der Avoird effectively involve the accurate solution of systems of up to several hundred coupled differential equations using, respectively, the corrected Born-Oppenheimer method of Hutson and Howard or the secular equation method of Grabenstetter and LeRoy [see ref. (4) for example]. I wonder if these authors might have any comments on the relative merits of these two schemes for such very large scale applications? A.E. Barton and B. J . Howard, Faraday Discuss. Chem. SOC., 1982, 73, 45. J. Tennyson and A. van der Avoird, Favaduy Discuss. Chem. SOC., 1982, 73, 118. J. M. Hutson and B. J. Howard, Mol. Phys., 1980,41, 1123. R. J. LeRoy and J. S . Carley, Ado. Chem. Phys., 1980,42, 353. Dr. J. Tennyson (University of Nijmegen) said: I would like to make three points in reply to Prof. LeRoy. First the BOARS method used by Barton and Howard for (HF), represents only an approximation to the secular equation method with the same angular basis functions (channels) and a saturated radial basis set. The methods thus do not yield the same level of accuracy, and it is clear from our results on the nitrogen dimer * that certain couplings (Fermi resonances) which we find significant are neglected in the BOARS approximation.Secondly, the bottleneck in the secular equation method is diagonalisation, even for the diatom-diatom problem where the potential matrix elements involve sums over 9 - j symbols. The current generation of vector processors can greatly speed up this step, meaning the method is suitable for more taxing problems. Finally, I note that both Hutson and Howard and ourselves used methods formu- lated in body-fixed coordinates, unlike the method of Grabenstetter and LeRoy. Besides being computationally simpler than the equivalent space-fixed formulation, body-fixed coordinates allow the simplifying approximation of neglecting the Coriolis interactions. We have found this approximation to be surprisingly A.E. Barton and B. J . Howard, Faraday Discuss. Chem. Soc., 1982,73,45. J. Tennyson and A. van der Avoird, J. Chem. Phys., 1982, Dec. 1st. R. J. LeRoy and J. S. Carley, Adu. Chem. Phys., 1980,42,353. J. Tennyson and B. T. Sutcliffe, J . Chem. Phys., 1982, Oct. 15th. Dr. B. J. Howard (Oxford University) said: Although the BOARS method presented by us uses a Born-Oppenheimer separation of angular and radial variables as a first approximation, it is not true that the final result is any less accurate than the close coupling and secular equation methods. All the non-adiabatic couplings between channels (including Coriolis interactions) are included by perturbation theory using the correction Born-Oppen heimer (CBO) method of Hutson and Howard.' The advantage of the CBO method is that it leads to a considerable saving in computing time over the other methods.It is also capable of dealing with near degeneracies and Fermi resonance if Brillouin-Wigner rather than Rayleigh-Schrodinger perturbation theory is used.2122 GENERAL DISCUSSION J. M. Hutson and B, J. Howard, Mol. Phys., 1980,41, 1113 and 1123. J. G. Frey and B. J. Howard, Chem. Phys., submitted for publication. , Prof. R. J. Le Roy (University of Waterloo) said: The absence of short-range 9- dependent terms in the potential energy surface of Barton and Howard reflects the lack of sensitivity of the available spectroscopic data to such terms. However, our studies of the H,-inert-gas systems suggest that level broadening due to internal- rotational predissociation is relatively much more sensitive to the short-range potential anisotropy than are level spacings and expectation values of the type analysed by Barton and Howard.Measurements of predissociation lifetimes should therefore be a sensitive probe of the short-range potential anisotropies. A. E. Barton and B. J. Howard, Faraday Discuss. Chem. SOC., 1982, 73,45. Prof. J. M. Lisy (University of Illinois) said: The gas-phase vibrational spectra of hydrogen-bonded systems can now be measured using vibrational predissociation Such information will be quite useful in testing new intermolecular potentials. The vibrational predissociation spectrum of (HF)2 contains a combin- ation band involving intramolecular and intermolecular modes. This would be con- sistent with a hydrogen-bond vibrational energy of ca.90 cm-’. It would be useful to compare this value with the harmonic vibrational frequencies of the (HF)2 potential surface of Barton and H ~ w a r d . ~ J. M. Lisy, A, Tramer, M. F. Vernon and Y. T. Lee, J. Chem. Phys., 1981,75,4733. Lee, Faraday Discuss. Chem. SOC., 1982,73, 387. A. E. Barton and B. J. Howard, Faraday Discuss. Chem. SOC., 1982,73,45. ’ M. F. Vernon, J. M. Lisy, D. J. Krajinovich, A. Tramer, H. S. Kwok, Y. R. Shen and Y. T. Dr. B. J. Howard (Oxford University) said: The potential-energy surface for (HF)2 determined in the present paper leads to vibrational frequencies considerably larger than those estimated from an analysis of the predissociation s p e c t r ~ m . ~ ’ ~ Within the harmonic approximation, the frequency for the stretching of the hydrogen bond is calculated to be 178 cm-’ while the symmetric and antisymmetric bends are deter- mined to 337 and 520 cm-l, respectively.The remaining torsional vibrational frequency is of a similar magnitude but is poorly determined. Because of large an- harmonicity effects, especially in the bending motions, the above vibrational frequen- cies are too large. Far better estimates of the vibrational intervals are 148, 160 and 304 cm-l, respectively ; these values are obtained directly from the first-order Born- Oppenheimer separation calculations outlined in the paper.l A. E. Barton and B. J. Howard, Faraday Discuss. Chem. SOC., 1982,73,45. M. F. Vernon, J. M. Lisy, D. J. Krajinovich, A. Tramer, H. S. Kwok, Y. R. Shen and Y.T. Lee Faraday Discuss. Chem. SOC., 1982, 73, 387. ’ J. M. Lisy, A. Tramer, M. F. Vernon and Y. T. Lee, J. Chem. Phys., 1981,75,4733. Prof. G. E. Ewing (Indiana University) said: I feel a need to emphasize that molecu- lar beam experiments on Van der Waals molecules have offered a rather restrictive pic- ture of the intermolecular potential surface. The complexes, produced by supersonic nozzles, reside in energy levels near the minimum of this surface, and radiofrequency and microwave transitions do not interrogate energy levels far from this minimum. While these experiments do provide valuable information on the equilibrium geo- metries of Van der Waals molecules, details of the intermolecular surface essential to understanding other properties are missing. Consider the case of (HF), for example.Here the elegant analysis by Barton and Howard of the pioneering molecular beam experiments of Dyke et al., reveal theGENERAL DISCUSSION 123 semi-rigid geometry of (HF)2 locked into its bent configuration by the strong aniso- tropic hydrogen bond. It is the details of the intermolecular surface minimum which dictate this structure. However, there is another view of this surface which is revealed by quite different experiments. Imagine two HF molecules in high rotational states which are the products of predissociation of vibrationally excited (HF)2.3 Or in another type of experiment, vibrationally excited H F molecules collide and transfer their energy into rotational motion^.^ In the limit of exceedingly high rotational states the anisotropic intermolecular forces will be averaged out for the product molecules of these experiments.The two HF molecules will appear to each other as, essentially, isoelectronic Ne atoms. The reality of this effective surface is revealed by the isotropic interaction term for HF + H F (the result of ab initio calculations) which appears in fig. 4 or 5 of the paper by Barton and Howard.' The effective isotropic hard-sphere diameter [i.e. where Vmm(R) = 01 for HF + HF is 2.9 A, remarkably close to that for Ne + Ne with CT = 2.8 A.5 In order to understand vibrational-energy- transfer experiments we therefore need the details of the intermolecular potential surface at two extremes: near the minimum and at high energies above the dissoci- ational limit of the Van der Waals bond.The effective shape of the potential surface for the vibrational relaxation experiments is discussed Of course there is only one intermolecular potential surface for a Van der Waals molecule, and many types of experiments are required to map it completely. For- tunately we are provided with a great variety of these experiments at this meeting. A. E. Barton and B. J. Howard, Faraday Discuss. Chem. Suc., 1982,73,45. T. R. Dyke, B. J. Howard and W. Klemperer, J. Chem. Phys., 1972,67, 2442. M. F. Vernon, J. M. Lisy, D. J. Krajnovich, A. Tramer, H-S. Kwok, Y . Ron Shen and Y. T. Lee, Faraday Discuss. Chem. Suc., 1982, 73, 387. 0. D. Krough and G. C. Pimentel, J. Chem. Phys., 1977,67, 2993. J. 0. Hirschfelder, C. F. Curtiss and R. B. Bird, The Molecular Theory of Gases and Liquids (Wiley, New York, 1954).L. L. Poulson, G . D. Billing and J. I. Steinfeld, J. Chem. Phys., 1978,68, 5121. ' G . Ewing, Chem. Phys., 1981, 64,411. Dr. A. Tramer (University of Paris, Orsay) said: Infrared intensities for X-H stretching vibrations yield important information about charge distribution in hydro- gen bonded complexes, but it seems of interest to extend this study to overtone spectra: in a number of hydrogen-bonded systems in solution the enhancement of the X-H fundamental intensity is accompanied by a drastic decrease in the oscillator strength for the Av = 2 transition. This peculiar behaviour has not been explained: it may be due to the effects of " electric anharmonicity " [higher terms in the p =f(Q) development] cancelling those of " mechanic " anharmonicity.The estimation of ap/aQ and of a2p/i3Q2 (where Q is the normal coordinate of the X-H stretching mode of the hydrogen-bonded complex) from an ab initio treatment would be of great interest. Exact experimental data for infrared intensities of funda- mentals and overtones in isolated Van der Waals molecules are also needed. Mr. J. T. Brobjer (University of Sussex) said : In their paper Barton and Howard have used a multipole expansion to represent the attractive (essentially the electrostatic) part of their potential, It must, however, be questioned whether the multipole expansion is accurate at these relatively short distances, and how many terms in the expansion are needed for a certain level of accuracy. By using a Morokuma-type partitioning technique it is possible to calculate the exa:t electrostatic energy between two monomer wavefunctions, and this can be compared with the multipole expansion using the multipole moments of the monomer wavefunctions.An alternative method124 GENERAL DISCUSSION of calculating the electrostatic energy is to use a point-charge representation of the monomer charge distribution. Prof. Murrell and I have constructed a set of point- charge models which are constructed so as to reproduce the monomer multipole moments. The simplest model for H F is of two charges positioned on the nuclei whose magnitude is determined to give the dipole moment (either the experimental moment or that of the above-mentioned wavefunction depending on the mode’s inten- ded use).The most accurate model we have constructed for HF has three point charges and it reproduces all multipole moments up to and including the hexadecapole moment. We have shown that these multipole-fitted point-charge models give a more accurate electrostatic energy than an equivalent multipole expansion. The multipole expansion for the HF dimer with dipole-dipole, diple-quadrupole and quadrupole-quadrupole moments is in error by ca. 50% at R = 2.67 A. The point- charge model which fits dipole and quadrupole moments is in error by ca. 15% at the same distance. At the experimentally determined intermolecular distance the best point-charge model gives a dimer structure in agreement with experimental findings, A more detailed account of our results for (HF)2 is avai1able.l ’ J.T. Brobjer and J. N. Murrell, J. Chem. SOC., Faraday Trans. 2, 1982,78, 1853, Prof. J. S. Muenter (Uniuersity of Rochester) said: In reply to Stephen Berry’s informally-posed question of whether the isotropy of the Ar-SO, potential might arise from the size of the sulphur atom, we present fig. 3. Indeed, the van der Waals radius FIG. 3.-Schematic representation of Ar*S02. of sulphur dominates the sulphur dioxide shape. However, this view of SO2 does not address the large dipole and quadrupole moments of SO2 which contribute to the anisotropy of the complex. Ian Mills pointed out, privately, that the Ar-SO, force constant data in table 3 of our paper were inconsistent, with the off-diagonal element being too large. This problem arose from the extreme correlation in the data analysis. This is another indication of the isotropic potential of Ar-SO,.The absence of this problem in Ar-SO, is consistent with its semi-rigid nature. Prof. A. W. Castleman Jr, Dr. B. D. Kay, Dr. F. J. Schelling and Dr. R. Sievert (Uniuersity of Colorado) (communicated). For a number of years we have been actively engaged in a laboratory investigation of the properties, energetics of form- ation, stabilities and binding energies, and structures of both Van der Waals and ion clusters. Some of the results have potential significance to problems in the atmo-GENERAL DISCUSSION 125 spheric sciences. As part of the work on neutral Van der Waals species we have examined the formation dynamics and have carried out electrostatic deflection measurements for a variety of substances, including multiple clusters of SO2 and H20, SO3 and H20, and clusters of sulphuric acid, Other relevant studies have been made on clusters of up to forty water molecules, mixed clusters of nitric acid and water, and the nitric-acid/water/ammonia system.Investigations of pure water clusters have indicated, among other things, the enhanced structural stability of particular cluster species and the absence of a permanent dipole moment for water clusters (H20),, with 3 < n < 9, both effects indicative of a probable cyclic structure. Table 2 shows TABLE Z?.-FOCUSING OF MOLECULES IN ELECTROSTATIC FIELDS. EVIDENCE OF PERMANENT DIPOLE MOMENT focus do not focus (H20),(3 d n d 9) (NH3), (3 d PI d 6) (HN03)(H,0),(2 G n d 10) (CH30H), (3 d n < 17) (C2H50H),(3 f n d 9) (CD30D), (3 < n d 7) some of the species whose dipole moments have been examined to date.Experi- ments conducted with nitric acid are intended to probe solvation effects and the process of ion-pair formation upon the coclustering of an electrolyte with successively more water molecules.' A similar mechanism of atmospheric aerosol formation has recently been postulated which involves the reaction of a hydrated acidic cation with an appropriate hydrated basic anion following an ion-ion recombination.2 Inform- ation gained from these experiments are relevant to both atmospheric processes and in contributing to our knowledge of solvation and formation of the bulk liquid phase. Recent molecular-orbital calculations indicate that the complex H20(S02) should be fairly strongly bound (ca.10 kcal mol-') with a large dipole moment (ca. 7 D) and have a Lewis acid-base pair structure. Recently, there has been evidence that the kinetics of certain bimolecular reactions may be altered when the reactants are in the presence of another molecule to which one of them can become cl~stered.~ This can possibly have major implications for atmospheric chemistry. In order to account for the observed rapid rate of aerosol formation via the photo-oxidation of SO2 in the presence of H20, Reiss and coworkers postulated the existence of clusters of these specie^.^ In order to confirm the existence of a stable gas-phase H20(S02) adduct, as well as to examine the possibility of existence of higher (H20),(S02), clusters, electrostatic deflection experiments were conducted on cluster species produced by nozzle expansion of a mixture of SO2 and H20.Our results indicate that the H20(S02) species exists and is very polar, as evidenced by its strong refocusing in a quadrupole field. At higher partial pressures of SO2, the126 GENERAL DISCUSSION ion intensity detected as (H2O.SO2)+ did not refocus, strongly suggesting that multi- channel cluster fragmentation may play a dominant role in the ionization of the more highly-clustered species, as it does in atomic cluster^.^ A dominant ionization channel may be (H20)m32(S02)n>2 + e+(H,O-SO,)+ + neutral fragments + 2e. That is, the (H,O.SO,) + ion detected in the high-partial-pressure experiments may arise predominantly from non-polar higher clusters present in the neutral beam and not from the polar (H20.S02) entity.Similar studies currently in progress are intended to compare the sulphur-trioxide-water cluster with its isomer, sulphuric acid. Preliminary evidence indicates the existence of a stable H20(S03) adduct. Recent calculations predict a structure for this species for which the oxygen of the water is bound to the sulphur of the SO3 group. Studies of sulphuric acid clusters have similarly shown that while the monomer possesses a large dipole moment and can be refocused in electrostatic fields, the dimer is non-polar and does not refocus. This suggests that the complex has a head-to-tail configuration, the individual molecular dipole moments cancelling.The presence of significant dimerization among sulphuric acid molecules might retard the rate of nucleation to the condensed (aerosol) state, as the kinetics of clustering is dominated by long-range dipole-dipole and dipole-induced-dipole forces. B. D. Kay, V. Hermann and A. W. Castleman Jr, Chem. Phys. Lett., 1981,80,469. 1980, 283, 55. P. M. Holland and A. W. Castleman, Jr., J. Photochem., 1981, 16, 347. E. Hamilton Jr and C. Naleway, J. Phys. Chem., 1976, 80, 2037. D. Marvin and H. Reiss, J. Chem. Phys., 1971, 69, 1897. B. D. Kay, Ph.D. Thesis (University of Colorado, 1982) and references therein. (a) N. Lee,Ph.D. Thesis (Yale University, 1976); (6) A. Hermann, E. Schumacher and L. Waste, J. Chem. Phys., 1978,68, 2327. * (a) E. E. Ferguson, personal communication; (b) F. Arnold and P.Fabian, Nature (London), Dr. F. A. Baiocchi and Prof. W. Klemperer (Haruard University) said: In comment- ing upon the very important work presented by Drs. Legon and Millen, we wish to make a few points. The linearity of the structure of C0,HF is well established in our view. We note that the microwave and radio spectrum of both hydride and deuteride were investigated. In addition the complex SCOHF was shown to have a similar structure. More recently Robert Altman and Mark Marshall have completed a structural study of C02HCl (and DCl). This system is also linear. In our view the difference between C0,HF and N20HF remains an intriguing problem. The structure of hydrogen-bonded complexes with oxygen-containing systems appears to be a rich field.We have recently completed radiofrequency and micro- wave spectroscopy of the hydrogen fluoride-formaldehyde system. The structure of the complex is a rigid planar geometry, shown below. The OH bond length is short, 1.74 A. The angle between the CO and HF units is 71", a surprising angle in view of the large dipole moment of both units, since the dipole-dipole interaction is almost zero for this orientation. The electric-dipole-GENERAL DISCUSSION 127 moment components are large, namely pa = 3.75 D and f i b = 1.40 D. This n-bonded complex complements the complexes discussed by Legon and Millen. Finally we wish to point that there is a considerable arbitrariness in fitting Lennard- Jones potentials to weak bonds with the hope of extracting binding energies.We note, since we have also done what Legon and Millen have done, that everyone when fitting hydrides uses the heavy-atom distances in the fitting procedure even though it is the hydrogen that is the actual bonding atom. When molecules involving heavy atoms are considered then the actual bonding (nearest) atoms distance is used in fitting the two parameters of the Lennard-Jones potential. In this sense the present procedure is arbitrary. Mr. A. J. B. Cruickshank (University ofBrisrol) said : Dr. Legon’s paper presents not only zero-point dissociation energies, but also effectively complete specifications of the vibrational modes characterising each hydrogen-bonded dimer. Since the geometry and the intermolecular distances are known, it should be possible to set up the rotation- vibration partition function for each dimer species.Indeed Dr. Legon refers to the partition functions for the dimer and the participating molecules. Is it not possible therefore, given the effective enthalpy of formation, -Do, to calculate the standard Gibbs energy of formation and the association constant, and so the equilibrium con- centration of hydrogen-bonded dimers? Could one thereby make an estimate of the contribution made by these dimers to the gross thermodynamic properties of the gas mixture ? Dr. A. C. Legon and Prof. D. J. Millen (Uniuersity College, London) said: We thank Prof. Klemperer for his comments about the geometry of CO, - - HF and for his presentation of the recently determined structure of H,C=O - - - HF.We note that the latter result fits well into the simple classication scheme presented in our paper. In connection with the use of Lennard-Jones potential-energy functions to describe the radial potential-energy variation in the dimers discussed in our paper, we recognise its parametric nature and limitations. However, as pointed out in introducing the paper, it is a simple function which allows a value of the dissociation energy D, to be determined for such dimers from the ground-state spectroscopic constants alone under the pseudodiatomic approximation. In addition to this advantage, the Lennard-Jones potential, in the one case where comparison with a directly measured value of D, is possible (HCN * * * HF in table 5 of our paper), leads to good agreement and, more- over, the changes in D, along series such as HCN - - * HX and OC - - - HX where X = F, C1, Br (see table 7) are qualitatively in agreement with chemical expectations.In reply to Prof. Berry’s informally-posed question about the isomer CO * - - HF, a careful search was made for its rotational spectrum by using the pulsed-nozzle, Fourier-transform technique but without success. If, as appears likely, this isomer were even 20 cm-l higher in wavenumber than OC - - HF it probably would not be detected at the effective temperature of ca. 3 K characteristic of the technique. Mr Cruickshank is of course correct in suggesting that, in the case of HCN - * * HF, sufficient information exists to calculate the thermodynamic functions, and in particular the equilibrium constant, for the reaction since in this case a proper value of Do and the values of the quantities necessary to calculate the partition function are available.This is not true in other cases where either information about Do and/or the bending vibrations is lacking. We plan to128 GENERAL DISCUSSION calculate the thermodynamic functions for the association reaction involving HCN and H F. Dr. J. T. Brobjer (University of Sussex) said: I would like to point out that it is possible to predict the lowest-energy geometry for most dimers by using the inform- ation contained in the multipole moments of the monomers. It is relatively well known that for polar molecules it is largely the electrostatic part of the potential that determines the dimer structure. With our multipole-fitted point- charge models we have calculated the equilibrium structure (for any given &,) and our results are in close agreement with experiment, as shown in table 3.TABLE 3 .-COMPARISON OF EXPERIMENTAL DIMER CONFIGURATIONS WITH THOSE PREDICTED BY OUR MULTIPLE-FITTED POINT-CHARGE MODELS ~~ ~ expt.a p c b expt. p.c. expt. p.c. expt. p.c. R' 5.27 5.27 6.4 6.4 6.69 6.57 8.28 8.28 81 - - 6 0 -6 -5 -7 0 0 82 60-70 66 50 62 79.5 91 0 0 HCN - * HF HCN - - * HCl H2 * * * HF N2 * * * HF expt. p.c. expt. p.c. expt. p.c. expt. p.c. R 6.32 6.32 R R R R R R 0, 0 0 0 0 90 90 0 0 8 2 0 0 0 0 0 0 0 0 expt. p.c. expt. p.c. 5 R 5.62 5.63 - 0 -0.7 -4 - 81 02 58 73 46 52 Experimental; ' point charge; ' R expressed in atomic units between centre of mass (note that the point-charge models give 19~ and O2 but not R).Note that a lone-pair model in its simplest form would fail to account for the differences in angles for (HF),, HF * - - HCI, HCl HF and (HCl),. Dr. A. C. Legon and Prof. D. J, Millen (University College, London) said: Dr. Brobjer's proposal is interesting. We note, however, that the multipole-fitted point- charge model requires, first, A,, either from experiment or from reliable ab initio calculation, in which cases the geometry is likely to have been established, and secondly, several terms in the multipole expansion of the electric charge distribution, which are generally not known except for the simplest molecules. We believe that the lone-pair model in its simplest form does not make a sig- nificantly worse prediction of the angles than the point-charge model in the cases of (HF), and HF - - .HC1 (for which experimental angles are known) and in the case ofGENERAL DISCUSSION 129 HCl - - HF we are not aware that experimental values are available. Finally, we take the view that it is the experimental angle that allows us to diagnose the lone-pair direction most directly and thus the result for (HCI), is the best experimental evidence available for a localised orbital description of the HC1 monomer. Dr. B. J. Howard, Dr. C. M. Western and Mr. P. D. Mills (Oxford University) said: We wish to raise the question of the nature of the Van der Waals bond. Traditionally the attractive forces involved have been considered as the natural extension of the long-range electrostatic, induction and dispersion forces to the region of the potential minimum.It has, however, also been shown that the structure of some Van der Waals molecules correlates strongly with a HOMO-LUMO picture of the interaction and, since there must be some orbital overlap between the molecules in this region, that this leads to the possibility of an incipient chemical bond. More recently, electric quad- rupole coupling constants have been measured for quadrupolar rare-gas nuclei in Kr- HC1 and Xe-HF.3 The observed field gradients at the nuclei can be accounted for by an induced electric quadrupole in the rare-gas electron distribution by the neigh- bouring hydrogen halide. It is concluded that there is no need to invoke theories of charge transfer to explain the Van der Waals interactions in these species.We have investigated the radiofrequency and microwave spectra of the Van der Waals molecule Ar-NO using the technique of molecular-beam electric resonance spectroscopy. The molecule is an ideal probe of electronic rearrangement in that it contains the open-shell molecule NO which possesses both unpaired electron spin and unquenched orbital angular momen tum. NO also possesses suitable unoccupied orbitals for possible HOMO-LUMO charge transfer. In addition the electron distribution can be monitored via the large nuclear hyperfine interactions due to the nitrogen nucleus. Analysis of the rotational spectrum shows the complex to be T-shaped as predicted by molecular-beam scattering data and electron-gas calculations of the inter- molecular potential.However, unlike conventional open-shell asymmetric-top molecules, the orbital angular momentum is only partially quenched, the spin-orbit coupling being significantly larger than the barrier to free orbital motion of the un- paired electron. We determine the vibrationally averaged structure to be near T- shaped with an argon to NO centre-of-mass distance of 3.65 A and with this inter- molecular axis at an angle of 95" to the NO internuclear axis. The barrier to free orbital motions as measured by the difference in energy between the in-plane and out- of-plane z* orbital on NO is ca. 3 cm-l, much less than the total bond energy of ca. 100 ~ m - ' . ~ In addition, the nitrogen nuclear hyperfine structure can be explained by projecting the known hyperfine constants of NO on to the inertial axes of Ar-NO.The observed hyperfine splittings are reproduced to within 1% and we conclude that there is little electron rearrangement. This result together with the unusually small barrier to orbital motion further support the view that the individual molecules retain their identity in a Van der Waals molecule and that there is no need to invoke incipient chemical bonding. S. J. Harris, K. C . Janda, S. E. Novick and W. Klemperer, J. Chem. Phys., 1975,63,881. M. R. Keenan, L. W. Buxton, E. J. Campbell, T. J . Balle and W. H. Flygare, J. Chem.Phys., 1980,73, 3523. L. W. Buxton, E. J. Campbell, M. R. Keenan, T. J. Balle and W. H. Flygare, Chem. Phys., 1981, 54, 173. H. H. W. Thuis, P1z.D. Thesis (Nijmegen, 1979).G. C. Nielson, G. A. Parker and R. T. Pack, J. Chem. Phys., 1977,66,1396. Prof. A. D. Buckingham (Cambridge Uniuersity) said: Dr. Howard raised the matter130 GENERAL DISCUSSION of electron delocalization in Van der Waals molecules containing the nitric oxide radical. He reported that he saw no evidence for electron transfer in the Van der Waals molecule. A very sensitive test of electron delocalization is provided by measurements of the unpaired electron density at a nucleus in the neighbour. This can be achieved by observations of hyperfine splittings that involve a nuclear spin of the diamagnetic neighbouring molecule.' Delocalization can also be detected by measuring the effect of a paramagnetic buffer-gas on the n.m.r. resonance f r e q ~ e n c y .~ ~ ~ Thus for 129Xe ( I = 5) the large downfield shifts proportional to the pressure of 0, and of NO have been interpreted in terms of a contact shift resulting from the over- lap of the 5s atomic orbitals of Xe with the unpaired orbitals of the O2 or NO. Al- though the intermolecular contact shift was readily measured in the case of Xe in O2 and NO, its magnitude is reduced by the smallness of the overlap of the 5s orbital of Xe with the unpaired orbitals of O2 or NO because of " shielding " by the 5p electrons. The effect would presumably be larger in Hg in O2 or NO (199Hg has I = + and an abundance of 16.86%). G . W. Canters, Corvaja and E. de Boer, J . Chern. Phys., 1971,54,3026. C. J. Jameson, A. K. Jameson and S . M. Cohen, Mol. Phys., 1975, 29, 1919.A. D. Buckingham and P. A. Kollman, Mol. Phys., 1972, 23, 65. ' C. J. Jameson and A. K. Jameson, Mol. Phys., 1971, 20, 957. Prof. Z. H. Zhu (Chengdu University, China) and Prof. J. N. Murrell (University of Sussex) said: The dimer of HCN has been examined by microwave spectroscopy and found to have a linear structure.l This contrasts with the structure of the hydrogen halide dimers which, although having a linear hydrogen bond, are non-linear overall. and from thermodynamic measurements an enthalpy of dimerization of -3.3 kcal mol-l was ~ b t a i n e d . ~ Theoretical calculations by Johansson et al.5 have been made on the collinear dimer using SCF theory and an STO-3G basis (there had earlier been calculations with the CNDO method).6 Retaining the monomer geometry they obtained a distance RCN = 3.2 for the hydrogen-bond bridge and a stabilization energy of 3.7 kcal mol-l.They also calculated the energy of a side-by-side (rectangular) structure but found no stable form. In contrast the CNDO calculations found this side-by-side structure to be more stable than the collinear.6 We have re-examined the dimer structure using a larger basis and made a more extensive examination of the potential energy surface. Calculations were first made at The existence of the HCN dimer in the gas phase was postulated as early as 1929 1.053 1.142 2.233 1.059 1.145 H- C-N -----H-C--N FIG. 4.-Calculated energy and geometry of (HCN)'. E = - 185.8056 au; AE = 4.559 kcal mol-' the 6-31 G level using the Gaussian 76 program to find the optimum geometry of the collinear dimer; the results are shown in fig.4. The CN distance of 3.292 A is only a little longer than the value found by Johansson and co-workers but we predict a short- ening by 0.1A of the CN bond length in the H-bond donor molecule. We have con- firmed (fig. 5 ) that this structure is stable to a bending displacement which retains the linear hydrogen bond. Calculations were then made at the geometry found above using the 6-311G** basis (a triple valence basis plus polarization functions on all atoms).8 After correc- tion for basis set superposition error this gave a binding energy of 4.56 kcal mo1-I. With this basis the dipole moment of the HCN monomer is calculated to be 3.21 DGENERAL DISCUSSION 131 -185.54 - 1 85.56 - -1 85.58 - -185.60 - -185.62 - -185.64 -185.66 FIG.5.-Change in energy of (HCN)2 on distortion from the linear structure. which is slightly greater than the experimental value of 2.98 D.9 If we take the major contribution to the dimer energy as being the dipole-dipole energy, then we can make an empirical correction to the binding energy and arrive at the value 3.93 kcal mol-l. It is surprising that the previous ab initio calculations on the side-by-side dimer did not find a stable structure as the dipole-dipole energy is stabilizing for this configur- ,,.,j 70.0 6 5 . 0 &- 1.053 1.144 H - C - N g; I 78.49' I l'e I N - C J H €=-185.8017 a u A€=2.110 kcal mol-' 3.25 3.50 3.75 4.00 4.25 4.50 RIA FIG. 6.-Contours of the side-by-side dimer calculated with the 6-31G basis. Contour a = -185.66015 &, intervals 5 x lo-" Eh.132 GENERAL DISCUSSION ation. We suspected that this was because there was insufficient geometrical flexi- bility in the previous work and to confirm this we have obtained contours on a two- variable surface for the geometry shown in fig.6. With fixed monomer dimensions we found at the 6-31G level the geometry shown. At this geometry, using the 6--311G** basis and superposition correction we obtained a binding energy of 2.11 kcal mol-’. After the empirical dipole moment correction this reduces to 1.82 kcal mo1-’. By symmetry, the side-by-side dimer could not be detected by microwave but it may contribute to non-ideal gas behaviour at high temperatures. A. C. Legon, D. J. Millen and P. J. Mjoberg, Chem. Phys.Lett., 1977,47,589. T. R. Dyke, B. J. Howard and W. Klemperer, J . Chem. Phys., 1972,56, 2442. H. Sinosaki and R. Hara, Tech. Repts. Tohoku Imp. Univ., 1929, 8, 19. W. F. Giauque and R. A. Ruehrwein, J. Am. Chem. SOC., 1939,61,2626. A. Johansson, P. Kollman and S. Rothenberg, Theor. Chim. Acta, 1972,26,97. J . R. HoyIand and L. B. Kier, Theor. Chim. Acta, 1969, 15, 1. ’ W. J. Hehre, R. Ditchfield and J. A. Pople, J. Chem. Phys., 1972, 56, 2257. * R. Krishnan, J. S. Binkley, R. Seeger and J. A. Pople, J . Chem. Phys., 1980,72, 650. Stand., no. 10 (1967). R. D. Nelson, Jr., D. R. Lide, Jr., and A. A. Maryatt, Nut/ Stand. Ref. Data Ser., Nut1 Bur. Dr. J. M. Hutson (Uniuersity of Waterloo) said : Dr. McKellar has observed infrared lines of HD-Ar broadened by predissociation, and has compared the measured widths with preliminary calculations by Corey and LeRoy (see fig.13 of McKellar’s paper). However, the calculations were performed using the secular equation (SE) method for calculating widths, and this has since been found to be unreliable.2 Prof. LeRoy and I have therefore repeated the width calculations using the close-coupling method and the BC3 potential of Carley and L ~ R o ~ . ~ The basis set included all channels with HD rotational quantum number up t o j = 3, and the coupled equations were solved 0.0 1 3 5 7 9 I‘ FIG. 7.-Comparison of observed and calculated widths for S1(0) lines of HD-Ar broadened by predissociation. A, A, N-branch, J’ = I’ + 2; 0 , 0, T-branch J‘ = I’ - 2 (closed symbols, observed; open symbols, calculated).GENERAL DISCUSSION 133 using the method of DeVogelaere.Level energies and widths were calculated by fitting the S-matrix eigenphase sum to a Breit-Wigner formula with a quadratic background term.4 The calculated level widths are compared with the experimental widths in fig. 7: note that states with I’ > 7 can also predissociate by tunnelling through the centrifugal barrier, and were not considered here. The agreement between the observed and calculated level widths is very good. This confirms that the observed broadening is due to predissociation by internal rotation, and is additional evidence for the validity of the Carley-LeRoy BC3 potential for H,-Ar. J. E. Grabenstetter and R. J. LeRoy, Chem. Phys., 1979, 42, 41. R. J. LeRoy and J. S. Carley, Adu. Chem. Phys., 1980,42, 353. A. U. Hazi, Phys. Rev. A , 1979,19,920; C . J. Ashton, M. S. Child and J. M. Hutson, J. Chem. Phys., 1982, to be published. * R. J. LeRoy, G. C. Corey and J. M. Hutson, Faraday Discuss. Chem. Sac., 1982,73, 339. Dr. G. G. Baht-Kurti and Mr. I. F. Kidd ( University of Bristol) said : We would like photon energy/lO-’ cm-I FIG. 8.-Total photofragmentation cross section for the process Ar-H2(v = 0 , j = 0) ”y, [Ar-Hz(u = 0 , j = 2)]* _j Ar + H,.134 GENERAL DISCUSSION to report that we are in the process of performing calculations of photofragmentation cross-sections for the H,-Ar van der Waals molecule, which is one of the systems examined experimentally by Dr. McKellar. The theory underlying our calculations has been described in previous publications.' -3 The cross-sections which we calculate are of the form where yJMt is the original H,-Ar bound wavefunction, ~,Y-(~~W) is a scattering wave- function in which k specifies the direction of the Ar-H, relative motion and vimj are the H2 vibrational and rotational quantum numbers. The cross-section given above is integrated over all scattering directions, summed over final mj and averaged over initial Mi quantum numbers. Further details of the theory are available from pub- lished paper~.I-~ To illustrate the type of results we are currently able to obtain we present in fig. 8 and 9 below two different types of cross-section. In fig. 8 we show a total photofragmentation cross-section (i.e. summed over all -12 -10 - 8 - 6 - 4 -2 0 2 4 6 8 10 12 photon energy/10-3 cm-' hv FIG. 9.-Partial photofragmentation cross section for the process Ar-H2(u = 0,j = 0) [Ar-H2(v = 1 , j = 4)]* - Ar + H2(u = 1 , j = 0).GENERAL DISCUSSION 135 10 a 6 4 d I 2 2 X x .3 U ?v 2 w c -; E ._ M ." - 1 - 1 - 1 - 1 1 -10 -8 -6, - 4 - 2 0 2 4 6 8 1 real S matrix x lo-' FIG. 10.-Argand diagram of diagonal SI1 matrix element over energy range corresponding to fig. 8. final states of H,) centred around a photon energy of 360.2031 em-'. The processes corresponds to the excitation of a metastable rotationally excited state of the H, within the Ar-H, Van der Waals molecule Ar-H2(yZi Lv_ [Ar-H, v = o j = 2 ] * --+ Ar + H,. ground state Fig. 9 shows a partial photofragmentation cross-section corresponding to the produc- tion of H2 in its u = 1, j = 0 state. This cross-section is centred around the higher photon energy of 5285.0002 cm-I and corresponds to the processes Ar-H2(3Zt)hY- [Ar-H,(Yzi)]* ---+ Ar + H2('jZ$). ground state Of particular interest is the fact that the lineshape of this absorption cross-section is not symmetric. This can be interpreted as originating from a Fano-type interaction between the metastable bound state and the c o n t i n ~ u m . ~ In fig. 10 we plot an Argand diagram of a diagonal element of the S matrix which arises in our calculations over the energy range corresponding to that of fig. 8. As may readily be seen, the phase of136 GENERAL DISCUSSION the S matrix element changes by 2n over this small energy range, thus displaying a characteristic resonance behaviour. M. Shapiro, J. Chem. Phys., 1972,56, 2582. G. G. Balint-Kurti and M. Shapiro, Chem. Phys., 1981, 61, 137. U. Fano, Phys. Rev., 1961,124, 1866. * I. F. Kidd, G . G. Baht-Kurti and M. Shapiro, Faraday Discuss. Chem. SOC., 1981, 71, 287.