Faraday Discuss. Chem. Sue., 1982, 73, 45-62 An Intermolecular Potential-energy Surface for (HF)2 BY ANDREW E. BARTON AND BRIAN J. HOWARD Physical Chemistry Laboratory, Oxford University, South Parks Road, Oxford OX1 3QZ Received 18th January, 1982 Molecular-beam spectroscopic data on (HF);? are used to determine the HF * * * HF intermolecular potential-energy surface. The method used in based on the BOARS approximation of Holmgren et al. (J. Chern. Phys., 1977, 67,4414), which is extended to diatom-diatom weakly bound complexes. Molecular constants are calculated for several proposed potential-energy surfaces and compared with experiment. The optimised potential surface is shown to be similar to those derived from ab initio calculations. It is believed accurately to represent the shape of the true surface in the region of the potential minimum and along the path of the tunnelling motion; further information is required before an accurate well depth may be obtained.The molecular-beam radiofrequency and microwave spectra of atom-diatom Van der Waals complexes 1*2 have provided valuable information on the intermolecular potential-energy surface between rare-gas atoms and hydrogen halide molecule^.^ In such complexes the potential surface is basically a function of just two variables, the distance R between the atom'and the centre of mass of the diatom and 0, the angle between the intermolecular axis R and the diatom bond axis; the experimental data are insensitive to the dependence of the potential on the bond length of the diatom, principally because of the high frequency of the vibrational motion of the hydrogen halide compared to that of the other motions within the complex.In this paper we extend the work to diatom-diatom complexes and determine a potential surface for the weakly hydrogen-bonded complex (HF),. In this case the intermolecular potential is a function of four internal coordinates (neglecting the HF bond lengths). This doubling of the degrees of freedom These are defined in fig. 1 . FIG. 1.-Coordinate system for (HF),. adds greatly to the complexity of the potential function to be used and it has been necessary to base our potential on the results of earlier investigations discussed below. Hydrogen bonding is a widely studied interaction both experimentally and the~retically.~ Much of the information about the nature of the binding is obscured, however, because the majority of the studies have been performed in the condensed phases where solvent and lattice effects also exist.In the gas phase relatively few46 P . E . SURFACE FOR (HF)2 studies exist, for example ref. (5)-(8); these are largely restricted to a determination of the structure and an analysis of the harmonic force constants. However, more information is available for (HF),. The microwave and radiofrequency spectra were obtained by Dyke et al.,9 who showed that the dimer was an asymmetric top with an unusual tunnelling motion between the two equivalent hydrogen-bonded structures More recently detailed spectroscopic information on both the K = 0 and K = 1 states of the complex has been obtained lo and these data will be used to determine the potential-energy surface ( K is the angular momentum about the intermolecular axis).In addition a non-linear hydrogen bond is suggested from a crude analysis of the data. Smith has obtained an analysis of the dissociation energy of the dimer (25 4 kJ mol-l) from the temperature dependence of the infrared spectrum, but this estimate may well be too high, an observation which has been made for atom- diatom c~mplexes.'~~'~ Most of the early studies concentrated on the determination of the equilibrium structure of the dimer and the binding energy. A list of these studies is given by P0p1e.l~ Many of these studies have also assumed a linear hydrogen bond for the complex [e.g. ref. (15)], but a full geometry optimisation has indicated the presence of a non-linear hydrogen bond.16 These studies have provided a wide range of equilibrium geometries (0, = 30-80"; 0, = 0-30"; R = 2.5-2.8 A) and dissociation energies (12-33 kJ mol-l).This is largely the outcome of not using Hartree-Fock quality wavefunctions and is a consequence of basis-set superposition errors." Van Duijneveldt has shown that the range of values given above may be significantly reduced by a careful consideration of these errors.'' The effects of electron correlation are also ~ignificant.'~ A decomposition analysis of the binding energy by Morokuma 2o has shown that the multipole (electrostatic) contribution to the energy dominates that of exchange and charge-transfer effects and is largely responsible for the equilibrium angular geometry of the complex.21 Other workers have found similar results using high-quality basis sets.22 Several potential-energy surfaces have been derived from fits to these ab initio calculations.About fifty points are required for even the simplest characterisation of the surface, but the inclusion of more points can be prohibitively expensive especially for calculations with larger basis sets. Yarkony et aZ.23 have mapped out a surface from calculations using a (9s 5p ld/4s lp) basis set and have fitted the points to several simple functional forms of the surface. These ab initio calculations have been used by other workers in studies of the liquid-phase24 and vibrational relax- a t i ~ n . , ~ Both sets of studies have yielded quantities calculated from the potential surface which agree tolerably well with experiment.Other potential surfaces have been derived from different sets of calculations,26-28 but the experimental consequences have not been fully analysed. In this paper we shall use the surface of Alexander and De Pristo 29 derived from Yarkony's ab initio calculation as a starting point for the (HF), potential surface. Many ab initio calculations of (HF), have been undertaken. MOLECULAR-BEAM SPECTRA In order to make use of the spectroscopic constants and to make an accurate comparison between calculated and experimental quantities, it is necessary to reviewA . E . BARTON A N D B . J . HOWARD 47 briefly the spectroscopic parameters derived for (HF),. This information is different from that available for atom-diatom c o m p l e x e ~ .~ * ~ ~ First (HF)2 exhibits the tunnel- ling motion, mentioned earlier, which splits all rotational levels. In addition, spectra for (HF), have been observed in states with non-zero angular momentum, K, about the intermolecular axis. Because a change in K is a major perturbation to the system and results in a significant change in the molecular constants, states of different K are best treated independently. For each value of K, the rotational energy levels are accurately represented by the expression 'F 3{v, + S,[J(J + 1) - K21) (2) where v represents the tunnelling frequency and the symbols 5 label the symmetry of the tunnelling ~ t a t e s . ~ The parameter 6 represents the J-dependent centrifugal distortion of the tunnelling frequency, or alternatively the difference in rotational constants in the two tunnelling states; it is small compared to the tunnelling frequency.(B + C)/2 is the measured rotational constant of the near-prolate asymmetric top and DJ its centrifugal distortion parameter. Separate values of (B + C)/2, v and DJ for each K have been defined since the distortion of these para- meters upon changing the angular momentum K is not negligible 31 and cannot be treated as a perturbation, see below. To eqn (2) we must also add the effects of asymmetry doubling in states with K st 0. Thus for K = 1 we should add 1 B - c * AE' = & [ (7-) J(J + 1) + U2(J + 1)2 . (3) The resultant spectroscopic parameters for (HF)2 are given in table 1.In addition to the rotational parameters mentioned above, Stark measurements have provided the values of the electric dipole in the K = 0 and K = 1 states: paK=' = 2.988 63 (10) D and pUK=l = 2.836 84 (20) D.* These quantities are predominantly the expectation values of the sum of the vector components of the dipole moments of the two HF molecules along the a-inertial axis, i.e. pu,,[(cos 0,) + (cos O,)]. Because of the large contribution of the induced moments and possible charge transfer, care must be taken when attempting to obtain useful structural information from these quantities. However, the contributions to the induced moments from the dipole and quadrupole moments of the neighbouring molecule are calculated to be fairly constant over a fairly large range of angular geometry.For example for O1 = 45-65' and 62 = 0-20', we estimate an induced moment of 0.30 & 0.05 D. This yields a value of (cos 8,) + (cos 0,) of ca. 1.47 (3). The constancy of @induced with respect to change in angular geometry suggests that Ap = paK=O - ,uUK=l is due almost entirely to changes in the vector components of the permanent moments of H F resulting from the change in K. A simple harmonic model suggests that an increase of K from 0 to 1 produces an increase in 0, and O2 of ca. 5 O and IApinducedI < 0.01 D. The sources of information on the shape of the potential surface are as follows: (a) Rotation constant (B + C)/2. This is dominated by the state moment of inertia of the dimer and is a reliable measure of the equilibrium intermolecular distance Re.(b) Dipok moment p. This is determined by the angular geometry of the complex. * D % 3.3356 x C m.48 P . E . SURFACE FOR (HF);! Providing proper account is taken of induction and charge-transfer contributions this provides information on the equilibrium values of O1 and 02. (c) Centrifugal distortion of the dipole moment Ap = ,uo - pr. As mentioned earlier this is dominated by the changes in angular coordinates O1 and O2 resulting from the large centrifugal forces owing to rotation about the a-axis. This provides information on the angular curvature of the potential near the minimum. TABLE 1 .-CALCULATED AND OBSERVED SPECTROSCOPIC CONSTANTS FOR (HF)2 this work Alexander O experimental 6494.9 6532 37 ( 1 7 ) 0.0595 0.061 2 96.2 91.4 - 0.658 (+0.005) (-3.9) 1.068 2.6173 0.1516 0.3714 (- 2.5) 680 1.2 6822 21 0.0665 0.0662 95.5 94.6 - 0.393 (-3.2) 1.362 3.0472 0.4041 (-2.1) (- 0.059) 6494.963 (36) 6528.907 (30) 33.944 (47) 0.059 5(1) I' - 95.207 6(14) 91.189 6(18) 0.002 393 (12) 0.658 697 (3) -4.033 (14) -4.523 8(10) 1.064 286 8(4) 2.988 6(1) 0.151 8(2) a Ref.(29). pseudo CT term. Ref. (10). All values in MHz unless otherwise stated. * ca. 0.60 D is due to the P ( K = 0 ) - p ( ~ = 1). Observed - calculated. (d) Centrgugal distortion constants Dj and DjK. Dj is a measure of how easily the hydrogen bond may be stretched by centrifugal forces and is thus largely deter- mined by the radial curvature of the potential near the minimum. is dominated by the change in intermolecular distance occurring when O1 and O2 are increased owing to the large centrifugal forces resulting from rotation about the a-axis.This provides information on the coupling of the bending and stretching degrees of freedom. (e) TunneZling splittings vo and vl. Quantum-mechanical tunnelling is very sensi- tive to the height, width and shape of the barrier penetrated. Since these quantities are measured in both K = 0 and 1, it is expected that the shape of the potential along the path for tunnelling and in particular the barrier height can be accurately determined. For a known intermolecular distance this is determined by the equilibrium angular structure and to some extent the amplitude of the bending motions. This is expected to reinforce the information mentioned above on the equilibrium geometry and the bending force constants.In all a total of nine separate pieces of information can be used to determine the potential. Most of the information is on the shape of the potential around the ( g ) The asymmetry splitting (B - C).A . E . BARTON A N D B . J . HOWARD 49 minimum in the region sampled by the zero-point vibrational motion. ation is available on the absolute depth of the potential well. Little inform- THEORY The approach used is similar to that used for atom-diatom c o m p l e ~ e s . ~ ~ . ~ ~ ~ We also make the approximation that there is no coupling between the low-frequency vibrations and rotations of the dimer and the stretching vibrations of the diatoms, which are at a very much higher frequency. This in effect treats the diatomic mole- cules as rigid rotors.All molecular constants and potential parameters should be considered as averages over the zero-point vibrational motion of the diatom molecules. HAMILTONIAN Following Dyke et ~ 1 . ~ we write the Hamiltonian for a diatom-diatom complex such as (HF), in terms of a body-fixed axis system: where p is the reduced mass of the complex, which is m H F / 2 ; b, and b, are the rotational constants of the diatoms in the dimer, which will differ little from b H F , the rotational constant of free HF in its ground vibrational state; J i s the total angular momentum of the complex and j, and j2 are the angular momenta of the two HF molecules measured with respect to the molecule-fixed axis (the z-axis is aligned along the inter- molecular vector R).The terms in eqn (4) represent, respectively, the end-over-end rotational kinetic energy of the complex, the vibrational kinetic energy associated with the stretching of the hydrogen bond and finally that associated with the bending of the hydrogen bond. The last term is the intermolecular potential expressed as a function of the internal coordinates defined in fig. 1. Since the potential must be invariant to a complete rotation of an HF molecule by 27c, it is convenient to expand the potential as a sum of products of spherical har- monics which form a suitable complete set of angular functions. Thus v ( R A 0 2 9 ~ ) = 2 V1,grn(R)C1rn(@l~l)C1* - m ( 0 2 ~ 2 ) ( 5 ) Il'm where Clrn(@,9) is the reduced spherical harmonic [4n/(21 + l)]% Yim(@,y) and V,,t,(R) is a suitable function of the intermolecular distance.Polar coordinates 9, and v2 have been included explicitly but V is only a function of 9 = 9, - q2. Because of the symmetry of the potential to tunnelling (equivalent to permuting the two mole- cules) eqn (5) can be ~implified.~ The most general intermolecular potential is then V(R,@17@2,~) = 2 2 v11',(R)(Eclrn(@191)C~'-rn(@2~2) + Ci - r n ( @ 1 9 1 ) c i f m ( @ 2 ~ 2 ) I l'<I m a 0 The form of the function Vrlprn(R) will be discussed later. Eqn (4) may be expanded and conveniently rewritten as50 P . E. SURFACE FOR (HF)2 where b(R) = bHF + h2/2pR2. The only good quantum number of the system is the total angular momentum J, but for a complex like (HF)2 with a strongly anisotropic angular potential the projection K of J along the intermolecular axis R is also nearly conserved.CALCULATION OF SPECTROSCOPIC PARAMETERS The solution of the Schrodinger equation in many variables is often most con- veniently obtained by the secular determinant method 35 using a set of product basis functions which span the space of each of the variables involved. However, for a Hamiltonian like that in eqn (7) with three rotational and four internal vibrational variables this can prove computationally very expensive. A far more efficient pro- cedure is to extend the BOARS (Born-Oppenheimer angular radial separation) method of Holmgren et a1.32933 used for atom-diatom complexes, to the case of diatom-diatom complexes. In this we seek an approximate vibrational wave function in which the bending and stretching degrees of freedom are separated in the following manner W b s ( R , 0 l d % , d = Vb(0190299 ;R)xbs(R) (8) where the (Vb(01,02,$9 ;R)} are bending wavefunctions depending parametrically on R.This is analogous to the Born-Oppenheimer separation of electronic and nuclear degrees of freedom in normal m01ecules.~~ Because the molecular-energy levels are represented as a power series in J(J + 1) and because the spectroscopic observables are J-independent quantities, it is convenient to deal initially with a problem that is independent of J and to calculate all J-dependence by perturbation theory. As mentioned earlier, rotation about the intermolecular axis may not be treated as a perturbation and the corresponding quantum number K will be included explicitly.We now define the bending wavefunctions q$(0,,0,,p;R) to be eigenfunctions of the J-independent fixed-R Hamiltonian, Hb, i.e. In the absence of the potential, the solution to the above equation will be essentially that of two free rotors whose wavefunctions are spherical harmonics. It is thus reasonable to expand the bending wavefunction as a sum of products of spherical harmonics VW1,02,V $1 = c CfXR) YJlkl(elY1) YJ2k2(02V2) (10) i where K = kl + k2 and where the symbol j is used to represent all possible quantum numbers jlklJ2k2. The appearance of the coordinates q1 and y2 and not just 9 = q1 - q2 is analogous to the use of redundant angular coordinates in treating electronic angular momenta in linear molecules 37 but this need not worry us as long as we ensure that J, = j l z + j 2 , everywhere. The number of angular basis functions required is very much greater than for the corresponding atom-diatom case,3o mainly because of the increased number of degrees of freedom.However, the number of basis functions used in the problem may be substantially reduced by taking into account the symmetry of the system under the tunnelling motion. Dyke et aL9 have shown that a proper consideration of the per- mutation-inversion symmetry group of (HF)2 leads to an approximate identification ofA . E. BARTON AND B . J . HOWARD 51 the simultaneous permutation-inversion operator Pr2 with the symmetry of the tunnelling motion. Under this operator we obtain (1 1) pi% Y j k ( O l q l ) - yjk(7t - @29q2) = (- l ) j e k yjk(82q2) and similarly for Y j k ( 6 2 q 2 ) .This operator also affects the rotational coordinates. If spherical-top basis functions 9$@K (a’py) are used,34 the effect of the permutation- inversion operator is As a result the symmetrised vibrational functions which multiply 9 $ K ( ~ p y ) are qFf(@,,82,yl;R) = 2 C,Kf (R>E YJlk1(@191) Yj2k2(022032) .i L!z (- 1 ) j 1 - j 2 Y j z k 2 ( 4 9 1 ) Y J l k l ( Q 2 Y 2 ) I (13) where The bending eqn (9) is now solved independently for each symmetry & and for each value of K ; only values of K > 0 need be considered since reflection symmetry ensures degeneracy of &K. For each value of K we have kl + k, = K, and a com- plete set of functions is obtained with kl > k2. Thus kl is allowed to take values from K/2 if K is even and ( K + 1)/2 if K is odd; k, = K - k,. For each value of these quantum numbers ji can take any value 3 Ikil.It is usually necessary to include values of j 2 up to 10 to obtain reasonable convergence of the eigenvalues of the Hamiltonian matrix. The expressions for the matrix elements are given in Appendix 1. Solving eqn (9) at a series of values of R gives a set of effective potential functions Ut*(R) for radial motion. The radial part of the wavefunction zbs(R) in eqn (8) is then given, within the adiabatic appro~imation,~~ by the solution of the one-dimen- sional Schrodinger equation represents the symmetry of the wavefunction under Pf2. (14) ti2 a2 I] - 2m + W ( R ) + rf*(R)]&* (R) = Ei%fs*(R). The term zf*(R) takes into account, to first order, the dependence of t$*(81,02,q;R) on R, and is given by where J dQ represents integration over the angular variables 01,02,q and C&* is the first derivative of C&* with respect to R.A simple method for determining r(R) from the derivatives of the potential was presented in ref. (30) but, because of the large number of basis functions used, it proved expensive here and numerical differenti- ation of C&* was used. The effect of operating the total Hamiltonian in eqn (7) on the zeroth-order vibrational wavefunctions is given by52 P . E. SURFACE FOR (HF)2 where the matrix elements of the non-adiabatic perturbation Zna are defined as a function of R by the expression The effects of Zna and the J-dependence in the Hamiltonian may be included using perturbation theory.This J-dependence in the Hamiltonian may be split into a rotational part (1 8) " J 2 z r o t = and a Coriolis part Xna has only matrix elements off-diagonal in bending state. It contributes to the vibrational energy in second and higher orders. CALCULATION OF SPECTROSCOPIC CONSTANTS For a rotationless molecule, the principal molecular quantity derived from molecular-beam spectra of ( HF)2 is the tunnelling frequency, which corresponds to the difference in energy of the pair of adjacent vibrational levels of different permu- tation symmetry Pt2. However, compared with the well depth or the separation of other vibrational levels, the tunnelling frequency is a very small quantity so that in order to fit this parameter accurately an extremely large basis set of angular basis functions is required.Convergence to at least 0.01 cm-' was sought and for this values of kl up to 7 had to be included. The highest corresponding values o f j , was ca. 12 for k , z K/2, diminishing t o j , = 8 for ki = 7. This produced a Hamiltonian matrix of dimension 391 x 391 to be diagonalised, and it is this diagonalisation that is the major contributor to time of computing molecular parameters for a given potential surface. Initial calculations used smaller matrices of dimension 256 x 256 where a convergence to within 0.05 cm-l was obtained. In these initial fits to obtain a potential surface, a radial grid of 11 points was used (2.29-3.29 A), and eqn (14) was solved 38 over a smooth radial curve of 1000 points obtained by cubic-spline interpolation.A finer radial grid of 49 points (2.29-3.49 A) allowed convergence of the rotational constant to ca. 0.1 MHz; for the coarse grid errors were ca. 1 MHz. A smaller number of radial points is required for (HF)2 than for atom-diatom systems, since the well is an order of magnitude deeper and nearly harmonic over the range of the zero-point motion in the ground state. In addition the zeroth-order Born-Oppenheimer wavefunction is a much better approxim- ation tp the total wavefunction than for the atom-diatom cases; this is shown by the absence of peaking in the adiabatic and non-adiabatic correction terms as a function of R. For example z(R) in the ground state reaches a maximum of 1.9 cm-l, whereas for Ar-DCl, z(R) has a sharp peak of height 40 cm-l for the Neilsen-Gordon surface [see ref.(33), fig. 3(a)]. The small value of z(R) for (HF)2 can be attributed to the smooth slow change of the bending wavefunction q$+ with respect to R. As a result Lagrange n-point interpolation formulae 39 could be used to calculate the radial derivatives C,K,*.A . E. BARTON AND B . J . HOWARD 53 In calculating the vibrational energies and tunnelling frequencies the only neglected term in the Hamiltonian is the non-adiabatic correction Z,,, which couples together different bending states. This term can be included by perturbation theory but because of the accuracy of the original Born-Oppenheimer separation such effects provide a negligible correction ((0.01 cm-’) to the tunnelling frequency and were usually neglected.The other spectroscopic parameters were obtained as expectation values of the appropriate operator over the zeroth-order wavefunctions. Where necessary cor- rections were calculated using second-order perturbation theory. The expressions for these observables are the same as in the atom-diatom system except that the states IubK) are replaced throughout by lubK & l}. The rotational constant +(B + C), is the coefficient of J(J + 1) in the rotational energy. The dominant contribution comes from the expectation value of Ifrot ; also there are corrections in second order from the Coriolis coupling of states of K differing by one, and from the non-adiabatic coupling of states. Thus for K = 0 A large number of excited bending and stretching vibrational states are required in the above summation, and as a result the calculation can prove computationally expensive. Instead we have used the theory of Epstein 40 for calculating second-order corrections to energies within the Born-Oppenheimer approximation.If there is a perturbation between different bending states, then treating R as a constant, there is a second-order correction to the energy in state b : where x i b , ( R ) = $V)b*ZrV)bvds2. Then averaging over the stretching vibrational wavefunction gives the second-order correction to the energy J%’ = Sxbs(R)E~”(R)xbS(R)dR. (23) This approximation is obeyed well when the separation of bending curves is large compared with the spacing of stretching vibrational levels. For the lowest level the approximation used should lead to a slight overestimate of the interaction energy, since integrals involving a possibly larger energy denominator, (Eubl - Eooo) have been replaced by a smaller R-dependent denominator Ub(R) - Uo(R).However, since the separation of the curves Ub(R) and Uo(R) vary little with R, the errors in the approxim- ation are probably < 5%. The asymmetry splitting in the K = 1 state is given by a similar coupling of the K = 1 and K = 0 states by the Coriolis Hamiltonian. Thus where I& is the hybrid state 2-*()K = 1 ) & (K= - 1)) and it is the positive combin- ation that is displaced by the Coriolis interaction. Again the summation is simplified by the use of the Epstein approximation and results for the optimised potential are shown in table 3.54 P .E . SURFACE FOR (HF)2 TABLE 2.-POTENTIAL PARAMETERS ~~ ~ this work Alexander 29 this work Alexander 29 Ao/cm-' Bo/A- ' Al/cm- ' BJfi-' A2/cm- ' B2/A - A,/cm- B3/A - A4/cm-' B,/A- 671.7 (10.1) a 4.04 (0.03) 3.5 151 .O (0.3) 1.8 537.0 2.1 0.0 -30.0 (9.0) - 629.6 3.78 3.59 54.07 1.89 2.08 192.3 535.1 224.9 1.49 0.0 - 79.08 - 0.898 0.0 35.29 - 3.855 0.0 44.3 773.31 767.73 358.1 3 393.03 165.85 0.0 79.0 (6.3) 0.0 Uncertainties quoted as one standard deviation. The centrifugal distortion constant D, is the coefficient of J2(J + 1)2 in the expres- sion for the energy. This can be determined by perturbation theory, but it was found convenient to solve the radial eqn (14) adding Zrot to the radial potential in the form of a centrifugal barrier h'J(J+ 1)/2puR2.The equation was solved for a range of values of J (0-10). The eigenvalues were fitted to a power series in J(J + 1). The TABLE 3.-vALUES OF SPECTROSCOPIC PARAMETERS ( K = 0 STATES) FOR (HF)2 + state - state energies/cm- zeroth-order - 1086.9452 non-adiabatic -0.0169 total - 1086.9621 tunnelling frequency/cm- rotational constants/MHz zero th-order 6594.99 non-adiabatic 0.14 total 6496.82 P1 : zeroth-order DJ : zeroth-order/kHz 61.2 coriolis -98.31 (B - C)/MHz b = O b - 8.44 b = 1-5 99.61 b = 6-9 5.02 total 96.19 - 1086.2871 -0.0166 - 1086.3037 0.6584 (&0.005 cm-') 6594.39 0.13 - 101.56 6492.96 (+5 MHz) 1.4332 (~0.001) 61.2 (-I1 kHz) -9.55 98.96 2.02 91.43 ( - I 5 MHz) ' P I is (000 - IP(cosOl) + (cos0,)lOOO + >, see text. Contributions from different bending channels, b, of K = 0 state.constant term corresponded to the vibrational energy ; the coefficient of J(J + 1) gave the zeroth-order rotational constant B and the coefficient of J2(J + 1)' gave D,. In reality there are further higher-order Coriolis corrections to DJ, but these are much smaller than the experimental errors and can be neglected. The last remaining term to be calculated is (cos8,) + (cos~,) in the K = 0 and K = 1 states. In a tunnelling molecule the expectation values of these quantities areA . E . BARTON AND B . J . HOWARD 55 identically zero and the measured dipole moment is really a transition moment between the + and - tunnelling states. Thus (25) (cOs&) + ( ~ 0 ~ 0 2 ) = Spoxoo(~0~4 + cos&)poxoodRdR. Higher-order corrections are not warranted.PARAMETERISATION OF THE POTENTIAL SURFACE It is known that at long range the forces between two HF molecules are dominated by the classical electrostatic forces. The work of Morokuma and coworkers 20*21 suggests that this is still an important contribution to the intermolecular interaction in the region of the potential minimum. To this must be added at least some angle- dependent repulsion. However, since the exchange repulsion depends upon the over- lap of the charge distribution of the two molecules and since the contours of the electron density in an HF molecule do not deviate greatly from ~pherical,~' it is reasonable to assume that the basic features of the repulsive potential can be represen- ted by the first few terms in the series in eqn (6).Thus the potential was assumed to take the form V(R,8,,8,,p) = A,exp [- Bo(R - 2.79)] - A,exp[-&(R - 2.79)] + A,exp[-B,(R - 2.79)]{~0~8, - COS~,} + A2exp[-B2(R - 2.79)](2 cos& cos0,) + Velect. (26) The repulsive terms were assumed to vary exponentially with distance. Also a reference distance of 2.79 A, a first guess at the intermolecular separation, was used in these calculations so that the pre-exponential terms A possessed physically meaningful values. It was found necessary, like Alexander,29 to use two exponential terms to represent the dominant isotropic interaction. The classical electrostatic interaction between two linear molecules was expanded as an inverse power series in intermolecular distance. All terms up to R-5 were included.Written in terms of the permanent multipole moments of HF, Velect can then be written as56 P. E . SURFACE FOR (HF), D 42 and Q = 2.36 (3) D A.43 No value is known for the octupole moment but it was felt necessary to include the dipole-octupole term in the potential since it is formally of the same magnitude as the quadrupole-quadrupole term. We shall, however, treat the octupole moment as an adjustable parameter to add flexibility to the attractive part of the potential. First it assumes a very limited parameterisation of the repulsion. Secondly higher-order electrostatic forces make a significant, perhaps lo%, contribution to the energy in the region of the minimum. However, these higher terms vary rapidly with angle and their contribution to experimental observables appears to be insignificant for all except the tunnelling frequency, and here the dipole-octupole term may give the required flexibility.Thirdly no account has been taken of the induction and dispersion forces. The former are calculated to be much smaller than the neglected electrostatic contribution to the interaction, and the dispersion interaction is expected to be dominated by the angle- independent isotropic contribution. This has little affect on molecular constants except the well-depth, and can be mimicked in the region of the potential minimum by the two isotropic exponential terms in the potential. Finally no attempt has been made to represent valence and charge-transfer contributions to the interaction. Although these terms are probably not small, the energy-partition technique of Morokuma suggests that they have little angle dependence and can be largely included in the isotropic term in the potential.The potential in eqn (26) is similar to that derived by Alexander and DePristo from ab initio calculations. They wrote the potential 29 in a coupled representation suitable for scattering calculations. However, transforming to a decoupled representation as we have used (see Appendix 2), their potential can be written in the form A number of criticisms can be made of this assumed potential. V(R,O1,Bl,p) = Aoexp[--Bo(R - 2.79)] - A,exp[--B3(R - 2.79)] + {A,exp[-B,(R - 2.79)] - A,exp[-B4(R - 2.79)]} (cos0, - cos0,) - (A,exp[-B,(R - 2.79)] - A,exp[-B,(R - 2.79)]} (2cos0, cos0, - sine, sine, cosp) + {[A6(2.79/R) - A,]exp[--Bti(R 2.79)I)f(COSQi, COSe2) + Veiect (28) where f(cose,, case,) = cosel (3cos2e2 - 1) - cos0, (3c0s281 - 11, the angle dependence of the leading dipole-quadrupole term.Alexander neglected the quad- rupole-quadrupole and dipole-octupole terms as well as the higher-order contributions to the electrostatic potential. These potential parameters are listed in table 2. RESULTS AND DISCUSSION The values for the spectroscopic parameters calculated from the optimised and Alexander potentials are given in table 1 ; these can be readily compared with the experimental values. A limited parameterisation of the potential permits a quite accurate fit to the experimental quantities, and we believe that this indicates that the shape of the potential in the region of the minimum has been accurately determined, In addition the inclusion of the theoretical electrostatic interaction means that the potential is asymptotically correct at long range. The slightly large calculated values of (B + C)/2 indicate an over-short equilibrium separation.The slightly The Alexander potential also does remarkably well.A . E . BARTON AND B . J . HOWARD 57 180 120 0 . 3 60 0 60 120 180 W" FIG. 2.-Contour plot of the optimised potential for (HF)2 showing the tunnelling path. Contours are at 100 cm-' intervals above the minimum. The centre-of-mass separation is Re (table 4) and p = 180" (fig. 1). The positions of the potential minima are indicated by dots. 180 120 60 0 60 120 180 @,lo FIG. 3.-Contour plot of the Alexander potential.Contours are the same as for fig. 2, and the centre-of-mass separation is again fixed at Re (table 4) and @ = 180".58 P. E. SURFACE FOR (HF)2 incorrect values determined for the tunnelling frequency and the dipole moment indicate some errors in the angle dependent part of the potential. In fig. 2 and 3 may be compared contour plots of the optimised and Alexander potentials as a function of 0, and O2 at the equilibrium intermolecular distance. The equilibrium angular structures are 8, = 67", 82 = 12" and 8, = 56", 8, = 17", respectively, and it is this FIG. 4.-Radial potential for the optimised surface for (HF),. v, = 180" and &,Oz are fixed at the values given in table 4. The contributions to the potential are as follows: Vmm = isotropic inter- action, vmd = term involving cos 0, - cos &, vdd = term involving cos O1 cos &, and V,,,, = long range multipole attraction. change in 61 that is basically responsible for the large value of the dipole moment calculated for the Alexander potential.The slightly more linear geometry ensures a longer path for the tunnelling motion. This, combined with a higher barrier, is largely responsible for the low calculated value of the tunnelling frequency for the K = 0 state. In order to obtain a good fit to the K-dependence of the dipole moment it was necessary to include a term of the form of the dipole-octupole interaction in the attractive potential. Note that at Ro = 2.79 A, the value of p/Ri of 79 cm-I is equivalent to an octupole moment of 1.45 D A2.This is very close to the ab initio values of Cade and Huo 44 (I .87 D A2) and of Maillard and Silvi 45 (1.78 D A2 at the SCF level and 1.699 D A2 from a CI calculation). Thus this term in the attractiveA . E. BARTON AND B . J . HOWARD 59 potential appears to possess approximately the physical meaning we have attributed to it. Any discrepancy between the observed and expected values of the dipole- octupole parameter is probably owing to the effects of suppressed higher multipole terms. Regarding the form of the calculated potential it may be seen that the repulsive forces are dominated by the isotropic (angle-independent) terms and this appears to support our view of a nearly spherical HF molecule. This is demonstrated in fig. 4, where the radial dependence of the potential at the equilibrium values of el, B2 and 9 200c l0OC 3 I n C 2 LL u -loo( -200( 1 I I 1 I I I 2 4 6 RIA FIG.5.-Radial potential for the Alexander surface for (HF)2. O1 and 0, are fixed at 50" and 15" respectively and (p = 180". The contributions to the curve are labelled in the same way as in fig. 4. is decomposed into its various components. The results for the Alexander potential are very similar except there is a slightly deeper well, see fig. 5. Another important observation is the large amount of zero-point motion possessed by the complex. As well as the bending motion corresponding to tunnelling between the two equivalent hydrogen-bonded configurations, the stretching motion has a large effect on the spectroscopic parameters. For example the equilibrium inter- molecular distance, Re, differs greatly from the effective distance in the ground vib- rational state.For the optimised potential we obtain Re = 2.675 A, which should be compared with the value of 2.79 A derived by Dyke et aL9 from a simple analysis of60 P. E . SURFACE FOR (HF)2 the rotational constant. Since this latter value corresponds to some average inter- molecular distance in the ground vibrational state it can be seen that care must be taken when comparing with the results of ab initio calculations, which should corres- pond to the minimum of the potential. To facilitate a direct comparison between our optimised potential and the results of other calculations we have listed the harmonic force constants V,, = a2V/ax8y in table 4.In general the agreement is remarkably good. The differences in V,, occur TABLE 4.-HARMONIC FORCE CONSTANTS AND EQUILIBRIUM STRUCTURES FOR (HF), this work Alexar der Lischka a Curtiss (6-3 1 G * *) 67.5 12 2.67 5 1705.7 (4.87) 0.188 0.074 0.107 - 0.066 0.01 7 -0.010 0.004 5 118.5 2.70 302 (0.86) 55.5 17 2.56 1998.5 (5.71) 0.200 0.069 0.100 - 0.061 0.027 -0.010 0.009 2 110 2.56 382 (1.09) 56 7 2.80 1330 (3.8) 0.13 0.048 0.15 0.008 2 0.000 01 5 123 2.80 -350 (1) - 0.054 -0.008 4 64 8 2.78 1645 (4.7) 0.20 0.079 0.17 - 0.000 12 lls7p2d/6slp SCF calculation; ref. (28). This is the angle relative to the centres-of-mass of Values Vee, Vvv; units of mdyn A. VRe; units of mdyn. elbar = 180 - elbar; bar = tunnelling barrier geometry. h Z classical the HF units. For the angle relative to the F-F axis subtract 0.9" for 0, and add 0.9" for 0,.in brackets in kcal mol-'. barrier height. Units of mdyn A-1 (1 dyn = 10-5N). because of the large fractional differences in the equilibrium values of 0,; as 0, tends to zero, Vvv must of necessity also tend to zero. There is also quite good agreement in the values of the well depth. We unfortunately have no way of accurately assessing our values. However, assuming we have correctly included the long-range electro- static interaction and from an estimate of the neglected terms (dispersion and induction forces) we expect our value to be accurate to &20%, giving a value of 20.5 & 4 kJ mol-l. Unfortunately little or no information is available from molecular-beam data on the changes in the average bond lengths of the HF units that occur on dimer form- ation.Estimates from ab initio calculations suggest that the change in bond length is small (0.002-0.004 A).28 Spectroscopic data are also available for (DF),, but these were not included in our least-squares fits. This is because the tunnelling frequency is only 0.053 cm-l in K = 0 states, beyond the accuracy of our calculations. In addition because of the smaller value for the rotational constant of DF a larger number of rotational basis functions are required to obtain eigenfunctions of similar quality to those of (HF),. As a consequence a prohibitively large amount of computer time would be required to fit both sets of data, and it is believed that very little additional information on the potential surface would be obtained.A .E . BARTON A N D B . J . HOWARD 61 In conclusion we have shown that the Born-Oppenheimer separation method may be extended to diatom-diatom systems but with a large increase in computation time. The molecular-beam data on (HF)2 contain sufficient information to determine a potential-energy surface which should be accurate around the potential minimum. Finally the derived potential surface almost certainly confirms a non-linear hydrogen bond. 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