Faraday Discuss. Chem. SOC., 1982, 73, 7-17 THE LENNARD-JONES LECTURE Intermolecular Binding BY JOHN A. POPLE Department of Chemistry, Carnegie-Mellon University, Pittsburgh, Pennsylvania 15213, USA. Received 26th April, 1982 1. INTRODUCTION It is most appropriate that this first Lennard-Jones lecture should be given at a Faraday Discussion connected with intermolecular forces. Lennard-Jones fathered the subject through his early work on realistic empirical potentials and their relation to macroscopic properties of gases. The famous " 6-12 potential " u(r) = Ar-12 -Br-6 (1) for the interaction energy of rare-gas atoms has long since been a standard starting point for statistical-mechanical work on all phases of matter.' The potential was based on a combination of long-range interaction energies (initially believed to be proportional to inverse sixth powers of the distance on empirical grounds) and another short-range part introduced to take account of repulsive effects which ultimately prevent close approach of colliding molecules.In addition to his work on empirical force fields, Lennard-Jones played an import- ant role in the development of molecular-orbital theory,2 now widely used for the non- empirical investigation of force fields. In this lecture, I shall emphasize current tech- niques for investigating the interaction energy of closed-shell diamagnetic molecules, using the orbital theory. In the second section, the role of partitioned theories, which attempt separate computation of long- and short-range parts of the potential, will be compared with the general approach treating the interacting molecules as one large system.This will be followed in the third and fourth sections by some illustrative applications to interactions of small molecules, both moderately weak ones such as hydrogen bonds and also strong interactions which lead to complexation into a unified molecule. 2. THEORETICAL METHODS FOR INTERMOLECULAR FORCE FIELDS The history of the application of quantum mechanics to the computation of inter- molecular force fields has proceeded in two broad directions. The first, in part following the philosophy introduced by Lennard-Jones potentials, attempts to par- tition the total interaction into a long-range part, generally. proportional to inverse powers of the intermolecular distance R, and a short-range repulsive part associated with exchange or overlap of the electronic distributions.These contributions have8 THE LENNARD-JONES LECTURE usually been evaluated separately, often by different techniques, and the results added to give a total A - B interaction potential A E A B = A E A B (long-range) + A E A B (short-range). (2) We shall refer to this technique as a " partitioned method." The second general approach, often described as the " supermolecule method," treats the interacting A * - B complex as a single large molecule; it then obtains the interaction potential A E A B by calculating the energy of AB and subtracting the ener- gies of separated A and B molecules: It is essential, of course, that the same theoretical method is used for all three species on the right-hand side of eqn (3).Examples of the application of both of these general techniques will, no doubt, be presented later in this meeting. Some remarks about their relative merits are therefore appropriate. THE PARTITIONED METHOD The long-range part of the partitioned potential (2) can usually be handled by methods which involve only properties of the isolated A and B systems or which avoid computations involving overlapping functions centred on A and B. If the unper- turbed wavefunctions for A and B are YA,'€'B (supposed not to overlap), then the interaction Hamiltonian VAB (the coulomb interaction between the electrons and nuclei of A and the electrons and nuclei of B) may be introduced as a perturbation. If this perturbation is applied to first order, the interaction energy is A E A B (long-range, first-order) = ( Y A Y B ] VAB [ Y A Y B ) .(4) This is the rigid electronic interaction potential, that is the interaction energy ensuing from the coulomb interaction of unperturbed electronic distributions of the isolated molecules A and B. The expression (4) can be evaluated directly, given the wave- functions YA and YB. Alternatively, if appropriate centres are chosen for A and B, a multipole expansion of VAB may be carried out, giving a series of terms in inverse powers of R, the distance between the molecular centres. For neutral molecules A and B (with the intercentre line along the z-axis), this takes the form (for axial molecules) A i E A B (long-range, first-order) = p&Bfi1(w)a-3 + [1UA@BfiZ(w) + pJ3@Af21(W)]R-4 + * * * ( 5 ) (6) where pA, pB and OA,OB are dipole and quadrupole moments of A and B.Here ~ J W ) = -2 cos 8, cos O2 + sin 8, sin 8, cos (9, - v2) . . . (0,,91),(02,92) being spherical polar coordinates of the axes of A and B relative to the AB line. The moments appearing in the long-range potential (5) may be computed by studies of the isolated molecules A and B or may be measured experimentally. In either case, this expansion provides a physically illuminating picture of the interaction at the largest distances. It was recognized at an early stage that the rigid electrostatic potential (5) failed to give any description of the long-range dispersion forces between spherical systems. To develop a more satisfactory theory, as originally shown by London: it is necessary to introduce the interaction potential VA, to second order. However, non-overlapJ .A . POPLE 9 between A and B wavefunctions may still be assumed. The additional contribution to the long-range potential is then by Rayleigh-Schrodinger perturbation theory. In this expression, Yi,Y!i are the wavefunction and energy for the ith excited state of A and similarly for B. The long-range second-order energy (7) can be further subdivided by separating the double summation The first of these three parts then corresponds to the interaction of the rigid electro- static distribution of A with the induced electronic density change in B; the second part has the converse interpretation. The third part of eqn (8) represents the coupling between the charge-density fluctuations in the two molecules; it represents the dis- persion interaction which exists even for spherical atoms.Each part of the second-order energy expression (8) may be further expanded in inverse powers of R. The leading term in the first part of expansion (9) (for neutral molecules) is then the dipole-induced-dipole interaction, proportional to P A c c B R - ~ where aB is the electronic polarizability of B. The leading term in the third part of eqn (8) is the dipole-dipole contribution to the dispersion interaction, proportional to Higher terms lead to the long-range expansion for spherical atoms AEAB (dispersion) = C6R-6 + C8R-8 + . . .. (9) For molecules, C,,c8 . . . will depend on relative orientations.All the parts of the first- and second-order contributions to the long-range inter- action can be handled by computations on wavefunctions for the isolated systems A and B. The situation is less satisfactory for the short-range part of the composite potential (2). This part depends on the overlap of the A and B wavefunctions. Al- though some progress can be made with an approximate theory involving only elec- tron den~ities,~ full ab initio computations require a study of the compound A - - B system. Some difficulty is encountered because the starting wavefunction YAY, is not fully antisymmetric with respect to all electrons. This can be overcome by appropriate modifications of perturbation t h e ~ r y . ~ However, the computation of the full interaction energy, even in first order, requires the valuation of Here d is a normalized antisymmetrizing operator, H A B is the complete Hamiltonian for A * B; H A , HB are the separate Hamiltonians for isolated A and B; YA,YB are still unmodified wavefunctions for the separate species.The evaluation of eqn (10) gives both the short- and long-range contribution inJirst order only. Thus it includes the long-range parts (4) or ( 5 ) but not the second-order long-range parts (7). A com- mon assumption is that these parts may be combined to give an appropriate full potential (1 1) In practical implementations of the partitioned approach, a choice has to be made for the ground-state wavefunctions YA,YB and for the excited wavefunctions Y&Yd A E A B (full) z A E A B (long + short, first-order) + AEAB (long, second-order).10 THE LENNARD-JONES LECTURE to be used in eqn (7), (10) and (1 1).A reasonable choice is to take ground-state single- determinant Hartree-Fock functions for YA,YB using a finite basis-set expansion for individual molecular orbitals and then construct excited functions Yi,Y$ using virtual orbitals from such a computation. An example of such a study is the (N2)2 potential recently obtained by Berns and van der Avoird.6 THE SUPERMOLECULAR METHOD In the supermolecular method the interaction between the molecules A and B is no longer treated as a perturbation. Rather, the wavefunction YAB for the compound A - B system is calculated for every required relative intermolecular configuration. The interaction energy is then At sufficiently large separations, the wavefunction YAB will become dYAYB where d is the antisymmetrizer.The interaction (12) then becomes identical with the first- order energy (10). If the intermolecular distance is so large that overlap is negligible, the antisymmetrizer has no effect and eventually (12) reduces to the first-order long- range expression (5) ; this is dominated by dipole-dipole interactions for neutral molecules. At shorter distances the supermolecule energy expression (12) takes some account of the repulsive intermolecular forces, since the compound wavefunction is always antisymmetrized and the full Hamiltonian automatically takes account of penetration effects. The supermolecule approach has the further advantage that it may also be used for extremely strong intermolecular interactions which lead to coalescence of AB into a single molecule with indistinguishable fragments.The dimerization of BH3 to give B2H6 with equivalent bridging hydrogens is an example. Deficiencies of the supermolecule approach are mainly associated with limitations on the type of wavefunction used. The great majority of supermolecule calculations in the literature are at the Hartree-Fock level. Thus YAB, YA and YB in eqn (12) are all single-determinant wavefunctions with appropriate optimized coefficients in a basis set expansion. Such wavefunctions will describe long-range multipole-multipole interactions insofar as the multipoles are themselves well described at the Hartree-Fock level. The interaction potential will include multipole-induced-multipole interac- tions, again insofar as the corresponding polarizabilities are well described at this level.However, no account is taken of dispersion forces, since these depend intrinsically on correlation between electrons in separate molecules, an effect completely neglected by the single-determinant wavefunction. Hartree-Fock supermolecule calculations are therefore likely to be most successful for systems where polar, electrostatic interactions dominate. Hydrogen bonding is an example. For the weaker interaction potentials which are dominated by dispersion forces, Hartree-Fock theory is inadequate. It may be noted at this point that the computational effort for a Hartree-Fock super-molecule calculation is comparable to that required for evaluation of the first- order interaction energy (10).Both computations require evaluation of the complete set of four-centre two-electron integrals over all basis functions. This part of a Hartree-Fock ab initio calculation usually dominates the later parts which process the integrals. It would clearly be advantageous to use the supermolecule method with a post Hartree-Fock technique which takes some account of electron correlation. However, not all such methods are suitable for studies of intermolecular force fields. They should have the property of size-con~istency,~ according to which the result of the calculation on the A - - B complex at infinite separation should be the sum of theJ . A . POPLE 1 1 energies calculated separately for A and B. If this condition is not satisfied, the inter- molecular potential calculation by eqn (12) will not tend to zero as the intermolecular distance tends to infinity.The Hartree-Fock technique itself is size-consistent when applied to closed-shell molecules. However, certain simple ways of handling electron correlation fail this test. For example, the method of configuration interaction involving all double sub- stitutions from single-determinant Hartree-Fock reference function is not size- consistent. This is because such a method fails to take account of" unlinked pairs '' of substitutions in the separated molecules A and B. Perhaps the simplest size-consistent post-Hartree-Fock procedure is Marller- Plesset perturbation theory8 carried out fully to any order. This theory handles electron correlation by treating the full Hamiltonian as a perturbation on the Fock Hamiltonian, for which the Hartree-Fock wavefunction is the exact solution.Thus, we introduce a perturbation parameter A and a perturbed Hamiltonian HA = I: + A(H - F) (1 3) where Pis the Fock Hamiltonian and H the full Hamiltonian. The energy correspond- ing to the Hamiltonian HA may then be expanded in powers of A En = E(O) + LE") + R2E'2' + A3E'3' + . . . (14) where explicit expressions for E(O), I?('), E(2) . . . may be obtained.8 In practice this series is terminated at some order n and the value of A then set equal to unity. The resulting energy is described as MPn. The MPl energy is identical to Hartree-Fock. Second-order theory (MP2) is the simplest order at which correlation is taken into account, all correlated electron pairs being treated independently at this level.Higher orders of Marller-Plesset (MP3 and MP4) take account of pair-pair interactions, orbital modification by correlation, correlated electron triads and unlinked pairs of pairs. Computer programs have now been developed for application of this theory up to full fourth order (MP4). Supermolecule methods based on Marller-Plesset theory, even at the MP2 level, should be effective techniques for studying intermolecular force fields. In addition to being size-consistent, they take account of electron correlation in the whole super- molecule A * * - B. The contribution of electron correlation to the intermolecular potential is then determined by subtracting the separate electron correlation energies for A and B.Thus, at the MP2 level, the contribution of electron correlation is AEAB (correlation) = E(2)(AB) - lP2)(A) - E(2)(B). (1 5 ) This is added directly to the Hartree-Fock supermolecule potential to give the overall MP2 force field. At sufficiently large distances it can be shown that the second-order energy expression (1 5 ) does include a dispersion-force term proportional to R-6. This term is, in fact, identical to that obtained from the partitioned long-range energy (7) provided that Hartree-Fock functions are used and provided that the energy denomin- ators in (7) are replaced by differences of Fock one-electron energies. It appears, therefore, that MP2 supermolecule studies (with a suitably chosen basis set) should provide a powerful method of generating intermolecular force fields which incorporate all the main long-range effects, including dispersion forces, as well as taking account of effects of electron correlation on the repulsive part of the potential surface.The higher-order MP3 and MP4 levels of theory could be used in a similar manner. Few Marller-Plesset supermolecule intermolecular force fields have yet been published. In the following sections we present some preliminary results in this12 THE LENNARD-JONES LECTURE direction. The procedure used is first to find the global minimum energy on the Hartree-Fock potential surface, a process greatly aided by recently developed analytic gradient techniques. The split-valence plus d-polarization basis set 6-3 lG* is used for these cal~ulations.~ This leads to a set of equilibrium geometries (optimized with respect to intra- and inter-molecular geometrical parameters) together with maximum intermolecular interaction energies (well depths).These Hartree-Fock structures (corresponding to the HF/6-31 G* theoretical model) are then used for single-point calculations with Marller-Plesset theory taken to fourth order in the space of single, double and quadruple substitutions from the Hartree-Fock determinant. This level (MP4SDQ) differs from full fourth-order theory by omission of triple substitutions which are more difficult to calculate. These single-point computations use the larger 6-31G** basis set which also contains polarization functions on hydrogen atoms.The final energies may be denoted by MP4SDQ/6-3 1 G**//HF/6-3 1 G*, where " / / " denotes " at the geometry of." They may be used to make some overall assessment of the role of electron correlation on binding energies. 3. INTERMOLECULAR BINDING BETWEEN FIRST-ROW HYDRIDES A systematic study of interaction potentials between closed-shell molecules should begin with the smallest and simplest such systems. These are H2 and the first-row simple hydrides XH,, where X goes from lithium to fluorine and m is the correspond- ing classical valence. We omit H2, where interaction energies tend to be small, and consider only the interactions involving the molecules LiH, BeH,, BH3, CHI, NH3, OHz and FH. Many of the complexes XH, - . . Y H , from this series have been examined at the HF/6-31G* level in joint work with D.J. DeFrees and P. V. R. Schleyer. The global minima have been determined leading to HF/6-3 1 G* equili- brium binding energies listed in table 1 . The corresponding structures are illustrated TABLE 1 .-HF/6-31G * BINDING ENERGIES OF XH, YH, (kJ rnol-') j 13 27 51 I 23 39 I 25 0 "Based on HF/6-31G* geometries. in fig. 1; detailed lengths and angles are available in the Carnegie-Mellon Quantum Chemistry Archive.lO The energies in table 1 may be partitioned into groups by the dotted lines shown. Interactions with methane are shown by lower level calculations to be generally small (<lo kJ mol-l) except for LiH. These will not be considered further. The com- plexes in the bottom right-hand corner (NH,OH,,FH with NH,,OH,,HF) are the hydrogen-bonded set with interaction energies in the range 13-51 kJ mol-I.AllJ . A . POPLE 13 1 (DZh) **..H*. H - Be '' Be - H * I . . - . H ,H . *. L i ' ' B e - H . . ' H .' .H .. .. ' Li' " B H %> 0 - H * * O p H H 16 (C,) F-H.* F, H FIG. 1 .-Structures of dimers involving first-row hydrides.14 THE LENNARD-JONES LECTURE involve single hydrogen bonds, the water dimer structure 16 being typical. It is known that the HF/6-31G* result is in excellent agreement with experiment l1 for the geometry of this species. However, there is evidence that the binding energy is too large because of basis set limitations. Smaller Hartree-Fock values of 15 and 21 kJ mol-1 have been reported in the literature using larger basis sets. In the top right-hand corner of table 1 are complexes representing donor-acceptor interactions between molecules having lone pairs of electrons (NH3, OH2, FH) on the one hand with electron-deficient molecules (LiH, BeH,, BH3) on the other.The interaction energies are considerably stronger (in the range 38-108 kJ mol-I). Finally the top left corner contains the interactions among the electron-deficient molecules (LiH, BeH,, BH3) themselves. These molecules may be considered as both electron acceptors and donors, the hydride end of bonds being electron-rich and thus capable of donating electrons towards the vacant orbital on the other molecule. For the three symmetrical dimers (LiH)2, (BeH2)2 and (BH3)2, complete coalescence to a hydrogen- bridged molecule is predicted. This is, of course, found experimentally for B2H6.The interaction energies here are even stronger, being in the range 85-207 kJ mol-l. TABLE 2.-MP4SDQ/6-3 1 G * * CORRELATION ENERGY CONTRIBUTIONS TO BINDING ENERGIES OF XH, YH, (kJ mol-l) LiH BeH2 BH3 NH3 OH2 FH LiH 6 23 56 BeH2 35 71 BH3 74 NH3 4 4 5 OH2 4 4 FH 5 a Based on HF/6-31 G* geometries. The single-point MP4SDQ/6-3 1 G**//HF/6-31 G* computations can be used to assess the role of electron correlation in modifying the intermolecular binding energy. Table 2 lists the correlation energy contributions at this level; table 3 gives the corres- ponding total binding energies. (The donor-acceptor complexes of NH,, OH2, FH with LiH, BeH,, BH, have not yet been examined at this level). The first conclusion to be drawn from table 2 is that electron correlation increases the strength of hydrogen bonds by a small and rather constant amount of ca.5 kJ mol-l. The contribution of correlation to the binding of electron-deficient complexes in the top left corner, on the TABLE 3.-MP4SDQ/6-31G** BINDING ENERGIES OF XH, YH, (kJ rno1-l) a LiH BeH2 BH3 NH3 OH2 FH LiH 199 191 264 BeH2 137 203 BH3 164 OH2 27 42 FH 30 NH3 17 31 54 Based on HF/6-31G* geometrics.J . A . POPLE 15 other hand, is quite large, particularly for compounds involving boron. The largest correlation effect is for diborane (BH3)2, which can reasonably be attributed to a transition from localized electron-pair bonds in 2BH3 to delocalized three-centre bonds in B2H6 where correlation effects are likely to be larger. The net result, noted in a previous study,12 is that electron correlation contributes nearly half of the total bind- ing of two BH3 molecules.The corresponding treatment of (LiH),, on the other hand, shows that the contribution of electron correlation for this dibridged species is quite small. This is perhaps to be expected as the binding in Li2H2 is a great deal more ionic than in B&&. The above results are discussed for the partial fourth-order (SDQ) Msller-Plesset theory. It is important to know how rapidly this series converges for the intermolecular contribution to the correlation energy. This can be answered, in part, by comparing MP2, MP3 and MP4SDQ contributions to some of the binding energies. These results are listed in table 4. They show that the second-order MP2 level of theory describes effects fairly well, with some small overestimation in all four of these cases.TABLE 4.<ORRELATION ENERGY CONTRIBUTIONS TO BINDING ENERGIES AT VARIOUS ORDERS OF M0LLER-PLESSET THEORY (kJ m0l-l) ~~ molecule HF MP2 MP3 MP4SDQ (BH312 89.5 173.5 167.5 163.7 (NH3)2 13.0 18.0 17.0 16.8 (OH,), 23 .O 28.6 27.1 27.2 WHIZ 25.0 31.2 29.3 29.8 _ _ _ _ ~ ~ ~ ~ ~ ~ a Using the 6-31G** basis and HF/6-31G* geometries. 4. INTERMOLECULAR BONDING INVOLVING ACETYLENE For a long time it has been recognized that the n-electrons of unsaturated hydro- carbons may act as electron donors in intermolecular bonding. This suggestion goes back originally to Dewar.13 There have been infrared spectroscopic studies of such complexes both in solution and in low-temperature matrices.Very recently, a ro- tational spectrum of the dimer involving acetylene and hydrogen chloride has been studied by rotational As a further example of the supermolecule approach to these problems? some recent work with Janet E. Del Bene on bonding to acetylene will be discussed. The experimental data on complexes between C2H2 and hydrogen halides have all been interpreted in terms of a " T-type " dimer. This has the proton of the proton- donor molecule directly above the middle of the n-bond of acetylene. These struc- tures (for HF and HCl) have been optimized at the HF/3-21G level, leading to the dimer geometries shown in fig. 2. Calculations of second derivatives of the energy with respect to nuclear coordinates confirm that these structures are indeed local minima in the potential surface.The total distance between the midpoint of the CC bond and the chlorine atom of hydrogen chloride is found to be 3.80 A, in reqsonable agreement with the experimental value of 3.699 A obtained by Legon et aZ.14 The energies of the hydrogen bonds in these systems may be evaluated by single- point calculations at these geometries and comparing with corresponding calculations on separated molecules. The level used is MP4SDQ/6-3 1 G*//HF/3-21 G. This gives equilibrium dissociation energies of 19 kJ mol-' for C2H2 - HF and 12 kJ mo1-I for C2H2 - - HCl. If electron correlation is not included (HF/6-31G*//HF/3-21G), the16 THE LENNARD- JONES LECTURE corresponding values are 17 and 9 kJ mol-'. Clearly, even if the proton acceptor is a non-polar unsaturated molecule such as acetylene, the hydrogen-bond energy is still dominated by the Hartree-Fock contribution, electron correlation corrections adding 2-3 kJ mol-l.An interesting point to investigate is the possible role of acetylene as a proton donor in hydrogen bonds. This is a possibility in view of the known high acidity of acetylenic hydrogen. In fact, a more complete search of the C2H2 HF surface does Y O 5 C H 2.36 , . . . . . , . 19 0.94 H- F H-C-C- 21 \ 1.05 C H 20 FIG. 2.-Structures of dimers involving acetylene. reveal a second potential minimum with the molecules attached H-CEC-H HF (actually distorted from the pure linear arrangement). However, the full correlated energy calculation yields a smaller binding energy than the T-type structure 19.Hydrogen fluoride is a relatively poor proton acceptor in hydrogen bonding. A more promising alternative is to consider hydrogen bonding between acetylene and ammonia in the C,, form 21. A preliminary study of this species indicates quite a strong bonding of 16 kJ mol-l. In fact ammonia is more strongly bound to acetylene in this configuration than it is in one with an NH bond acting as proton donor towards the n-electrons of C2H2. Species such as 21 have not, to my knowledge, yet been characterized by any experimental technique and represent a challenging theoretical prediction. 5. CONCLUSION The application of ab initio quantum mechanics to intermolecular binding has clearly come a long way in recent years. I believe that the supermolecule approach, using underlying techniques which make allowance for electron correlation, will become a standard procedure and will probably replace some features of the par- titioned approach. Given such a theoretical technique, it becomes possible to predict structures and binding energies for a large number of dimeric systems ranging from those with weak dispersion-type binding to those with extremely strong bonds leading essentially to new chemical bonds. In this sense, we are moving towards a unified theory of intra- and inter-molecular binding.J . A . POPLE 17 This research was supported by the National Science Foundation (grant no. CHE 8 1-0 1 06 1-0 1). J. 0. Hirschfelder, C. F. Curtiss and R. B. Bird, Molecular Theory of Gases and Liquids (Wiley, New York, 1974). F. London, 2. Phys. Chem., Teil By 1930,11, 222. R. G. Gordon and Y . S. Kim, J. Chem. Phys., 1972,56, 3122. D. N. Chipman, J. D. Bowman and J. 0. Hirschfelder, J. Chem. Phys., 1973,59,2830. R. M. Berns and A. van der Avoird, J. Chem. Phys., 1980,72, 6107. ' J. E. Lennard-Jones, Trans. Faraday Soc., 1929, 25, 668. ' J. A. Pople, J. S. Binkley and R. Seeger, Int. J. Quant. Chem. Symp., 1976, 10, 1. * C. Mraller and M. S . Plesset, Phys. Rev., 1934,56, 618. P. C. Hariharan and J. A. Pople, Theor. Chim. Acta, 1973, 28, 213. lo R. A, Whiteside, M. J. Frisch, J. S. Binkley, D. J. DeFrees, H. B. Schlegel, K. Raghavachari and J. A. Pople, Carnegie-Mellon Quantum Chemistry Archive (Carnegie-Mellon University, Pittsburgh, 1981). l1 T. R. Dyke and J. S. Muenter, J. Chem. Phys., 1974,70,2929. l2 R. Ahlrichs, Theor. Chim. Acta, 1974, 35, 59. l3 M. J. S . Dewar, J. Chem. Soc., 1946,406. l4 A. C. Legon, P. D. Aldrich and W. H. Flygare, J. Chem. Phys., 1981, 75, 625.