J. Chem. Soc., Faraday Trans. 1, 1988, 84(9), 2959-2966 Calculation of Nuclear Magnetic Resonance van der Waals Chemical Shifts based on a Generalized Polyatomic London Dispersion Theorem John Homer* and Mansur S. Mohammadi Department of Chemical Engineering and Applied Chemistry, Aston University, Aston Triangle, Birmingham B4 7ET A recently published theory for net attractive polyatomic London dispersion forces has been used to calculate 'H n.m.r. gas-to-solution chemical shifts. Good agreement with experimental data was found. The problem of explaining solvent effects on n.m.r. chemical shifts is one of long- standing difficulty. 1v Even the gas-to-solution shifts of isotropic molecules have proved difficult to characterize quantitatively until recently. Homer and Perciva13 have shown that these van der Waals shifts can be explained with reasonable quantitative precision by using a reaction-field approach together with the treatment of a new 'buffeting' intermolecular force.Their approach has been revised recently to produce a generalized theory for polyatomic London dispersion forces that places no direct reliance on empirical scaling factor^.^ This recent theory, which embraces both attractive and repulsive forces between molecules at equilibrium separations, has been used to predict with precision the heats of vaporization for an extensive range of compo~nds.~ The present paper presents an adaptation of the generalized London theory for the calculation of n.m.r. van der Waals chemical shifts. Theoretical It is well known that when corrected for the effects of volume magnetic susceptibility, observed gas-to-solution chemical shifts for isotropic solutes and solvents yield the van der Waals chemical shift,'.ow. The van der Waals screening constant is given by5. ow = - B ( E 2 ) (1) where B is the classical screening coefficient and ( E 2 ) is the mean-square electric field producing 0,. Although the acceptance of eqn (1) is widespread, the methods of calculating ( E 2 ) are numerous [see e.g. ref (2) and references therein]. The recently proposed method of characterizing dispersion forces4 affords a new route to ( E 2 ) . A simple adaptation of this method reveals that ow can be given by ow = - 2BZ, R-%xj (m2)j F(i,j). (2) where xi is the number of atoms of type j in the solvent molecule and 2, is the number of nearest solvent neighbours that surround the solute at infinite dilution in the solvent.lt has been suggested4 that the relationship between the nearest numbers of neighbours for a single (solvent) molecule in its pure liquid and solid (S) phases may be given by It will be assumed that eqn (3) applies to the number of nearest neighbours surrounding a solute molecule at infinite dilution in various solvents. The factor F(i, j ) accounts for the averaging of the inverse sixth power of the distance 2, % zs- 1. (3) 29592960 N.M.R. van der Waals Shifts I I Fig. 1. Schematic definition of parameters relevant to the London dispersion interaction between two C(CH,), molecules. r i , j between atoms i and j in the solute and solvent during random molecular rotation^,^ in terms of the equilibrium intermolecular separation R through (ri,;) = F(i,j) R-6.(4) Whilst an explicit expression for F(i, j ) has been ~ b t a i n e d , ~ the simpler form, obtained numerically, is given4 by F(i,j) = (1 +0.727q- 17.5O9q2+25.550q3)-' ( 5 ) where q = (di+dj)/2R (6) with dt being the distance of the centre of the atom i from the centre of mass of the solute, and dj being the corresponding parameter for atom j in the solvent molecule. The atomic mean-square electric moment required for eqn (2) may be deduced from London's treatment' as where a and I are the polarizability and first ionization potential, respectively. In fact, eqn (7) is applicable strictly to isolated atoms of the inert-gas type. When considering molecules, the polarizability and ionization potential for bonded atoms are required to define (m2). Although without rigorous theoretical justification, it has been dem- onstrated recently* that the value of (m2) for bonded atoms may be adequately approximated to the value of (m2) for the inert gas that is closest to the bonded atom in the periodic table.Eqn (2) is strictly applicable to those molecules that rotate more rapidly than they translate in the liquid phase.4 Consequently, eqn (2) will be used to predict ow for molecules expected to fall in this class. To illustrate the implementation of eqn (2) the case of ci, for the lH resonance ow of pure C(CH,), is considered. Relevant features of this system are included in fig. 1. In this case eqn (2) becomes ow = - 2 8 2 , R-6[(m2),1 F(H, C,,) +4(m2),, F(H,C2) + 12(rn2), F(H, H)] (8) wheres (m2)>, = (m'),, and (m'), = (m'),, and 2, from eqn (3) equals 11 because'' Zs = 12.The values of F(H,j) are determined by using di and dj, as defined in fig. 1, in either the explicit expression4 for F ( i , j ) , where applicable, or in the slightly less precise (m2) = 3ccI/2 (7)J . Homer and M . S. Mohammadi 296 1 Table 1. Calculated and experimental values of -cw (ppm) for pure compounds at 30 "C ( Z , , = 11. B = 0.54 x 10l2 cm3 ppm erg-l) compound R / A &',/Aa RIA-J/A H,/kJ mol-' expt12 calc. H2 3.89 4.3 4.9 1 C2H4 4.82 C6H12 6.34 C(CH3), 6.3 Si(CH,), 6.72 Ge(CH,), 6.72 Sn(CH,), 6.94 Pb(CH,), 7.01 CH4 'ZH6 3.89 4.23 4.9 1 4.82 6.27 6.67 6.92 6.78 6.8 6.74 3.27b 4.28" 4.98" 4.67d 6.89d 6.78d 6.70d 6.84d 6.96d - 1.05" 8.1P 1 5.63e 13.53s 30.05e 22.36" 26.91 " 29.76" 33.01" 36.96" - 0.230s 0.279h 0.29 1' 0.203,O. 192 0.229,0.217 0.228,0.205 0.260 0.297,0.3 10 0.358 0.092 0.276 0.324 0.290 0.20i 0.219 0.226 0.260 0.294 0.360 a The volumes used in eqn (9) are at 30 "C except for the first four compounds which are at their melting points. S. Bretsznjder, Prediction of Transport and Other Physical Properties of Fluids (Pergamon Press, Oxford, 1971). R. Reid and T. K. Sherwood, The Properties of Gases and Liquids (McGraw-Hill, New York, 1958). F. H. A. Rummens, W. T. Raynes and H. J. Bernstein, J . Phys. Chem., 1968, 72, 21 11. " Handbook of Physical Chemistry and Physics (CRC Press, Cleveland, Ohio, 53rd edn, 1972-73). Lange's Handbook of Chemistry (McGraw-Hill, New York, 12th edn, 1979).Calculated by extrapolating the gaseous data (-0.482~) to the liquid at its melting point ( V = 33.6 cm3 mol-'). ' The dependence of uw on density for C2H6 and C,H, are given (S. Gordon and B. P. Dailey, J . Chew. Phys., 1961,34, 1084) as -0.488~ and -0.515~ ppm, respectively, from which the entries in the table are deduced. eqn (5); To obtain ow in ppm, B, R and (m2)inert should have units of 10l2 cm3 ppm ergp1, A and In order to proceed further with the general use of eqn (2) it is necessary to have a satisfactory method of deducing R for the liquid state. One way is to make use of the knowledge that whilst the number of nearest neighbours changes on melting, R changes to a smaller extent." Accordingly, where values of R are known from studies of the solid state these may be used in eqn (2).Where solid-state values of R are not known an alternative approach may be used. cm3 erg, respectively. For pure compounds it has been found4.' that R may be obtained from R = 2(0.17 Y,); (9) where Vm is the molecular volume, and the factor of 0.17 accommodates the random packing of particles in a similar manner to that first used by Hertz.12 It has been suggestedg that if the solute is either very much smaller or larger than the solvent, the value of R is obtained from (R,+Rj)/2. Alternatively, if the solute and solvent are of similar size, R is deduced from solvent data, i.e. the solvent cavity size is used for R. Whilst for many molecules eqn (9) proves sati~factory,~?~ it yields spurious values for R for some Group IV tetramethyls (sec table 1).For example, from the liquid-state (20 "C) density of C(CH,), aovalue of 6.67 A is obtained for R using eqn (9), whereas the solid- state value islo 6.21 A. These two values are inconsistent with the suggestion that while 2 changes on melting, R changes to a smaller extent." In view of the success in predicting latent heats of vaporization el~ewhere,~ an alternative way of calculating R is to use the potential-energy expression on which eqn (2) is based to deduce R frop experimental latent heats of vaporization. For C(CH3), this approach yields R = 6.3 A, which agrees well with the solid-state value. This approach to R (given in the data column of table 1) will be used hereafter, and is discussed later.It should be emphasised that whilst this approach is adopted to permit overall consistency, it could be interpreted2962 N.M.R. van der Waals Shifts Table 2. Calculated and experimental values of - a,(ppm) for CH, in several solvents at 30 "C (ZI, = 11, B = 0.54 x cm3 PPm erg-') CH212 5.68 CHBr, 5.83 CBr, 6.32 Br2 4.82 CH,I 5.22 CBrCl, 6.08 CCl, 5.90 CH,Br 5.04 CHCl, 5.67 CH2CI, 5.27 SiCl, 6.40 0.769, 0.767" 0.646, 0.652" 0.594 0.556 0.505, 0.547" 0.533, 0.542" 0.443, 0.4722 0.355, 0.445" 0.407, 0.420" 0.398, 0.407" 0.301, 0.347, 0.765 0.641 0.592 0.502 0.562 0.541 0.540 0.486 0.458 0.527 0.48 1 a Deduced from volumes at 30 "C using eqn (9), and for the pure solvents give the precise latent heats of vaporization. The values of R for the shift calculations are obtained from R = (Rsolvent +R,,,,,e)/2.F. H. A. Rummens, Can. J. Chem., 1976, 54, 254. " W. T. Raynes, J. Chern. Phys., 1969, 51, 3138. to bias the justification of eqn (2). If, however, the values obtained for R from latent heats of vaporization not only permit the calculation of ow but are also in agreement with values obtained for R by other independent means, this will provide some justification for the present theory. The use of eqn (2) depends on an unambiguous and universal value for B. Kromhout and Linder13 have shown quantum mechanically that B = 0.59 x cm3 ppm erg-' for CH;..CH, interactions and Yonemoto14 has similarly shown that B = 0.54 x cm3 ppm erg-' for H;.-H,. In view of the fact that Rummens'~~~ empirical work confirms the latter value, this will be adopted here.The values of di and dj representing actual molecular parameters for the various systems considered here are either given in ref. (4) or have been calculated by the method described therein. Discussion Gas-to-liquid Chemical Shifts Inspection of table 1 reveals satisfactory agreement between the calculated and experimental values of ow for several pure compounds. The fact that for these compounds the values of ow lie within a relatively small screening range is to be expected from the work of Homer and Per~ival.~ They demonstrated that ow should be influenced significantly by the nature of the peripheral atoms of the solvent. For the compounds in table 1 the peripheral atoms are always hydrogen. For solvents with heavier peripheral atoms the present approach, and that of Homer and Per~ival,~ requires that these solvents should give rise to enhanced ow.This is borne out by the data in table 2 which relate to CH, in a range of halogenated solvents. Again the agreement between the calculated and experimental values of ow is generally satisfactory. Data for further mixed solute-solvent systems are presented in table 3. The agreements between calculated and experimental data are again satisfactory. In view of the success of the present approach in predicting pure van der Waals gas- to-solution shifts, it is interesting to consider the case of anisotropic solvents. For these, the (susceptibility-corrected) experimental gas-to-solution shifts can be corrected for van der Waals contributions using eqn (2) to yield estimates of the solvent neighbourJ.Homer and M. S. Mohammadi 2963 Table 3. Calculated and experimental2 values of - a,/ppm at 30 "C for several solute-solvent systems (2, = 1 1, B = 0.54 x cm3 ppm erg-') C6H12 o C(CH3L0 Si(CH31, (R = 6.72 A) CCl, 0 (R = 5.9 A) (R = 6.34 A) solute exptl calc. exptl calc. exptl calc. exptl calc. ( R = 6.3 A) H2 0.485 C2H6 0.305 0.310 0.345 0.370 'GH6 0.394 (R = 6.18 A). 0.420' 0.443 C6H12 0.267 0.265 C(CH314 0.290 0.307 0.320 Si(CH3)4 0.267 0.299 0.322 0.360 C2H4 0.474 0.470 0.500 0.394 - - - - - 0.278 0.273 - - 0.285 0.419 0.300 - 0.195 0.413 0.210 - 0.225 0.394 0.240 - 0.257 0.310 0.203 - 0.203' 0.201 0.192' - 0.230 0.233 0.220 - 0.187 - 0.233 0.239 0.270 - 0.282 - - 0.383 - 0.378 - 0.400 - - - - - 0.397 - 0.385 - - - 0.407 0.268 0.291 - - 0.243 0.220 - - 0.187 0.223 0.229 0.219 - - 0.217 - - - 0.240 0.216 0.255 - 0.182 0.175 0.185 0.168 0.212 - - - 0.228 0.226 0.205 - a Calculated from its latent heat of vaporization: RL.J = 6.32 A.' F. H. A. Rummens, Can. J. Chem., 1976, 54, 254. anistropy screening, oa. Table 4 represents estimates of oa for four common solvents. It can be seen that the signs and magnitudes of oa are as expected.l*16 Moreover, it can be seen that, in general, the magnitude of oa increases as the size of the solute decreases. l6 Equilibrium Intermolecular Separations Eqn (2) derives from a theorem for a net attractive polyatomic London dispersion potential which has been shown elsewhere4 to arise from the resultant of the attractive and repulsive forces. By using a self-consistent approach it provides a satisfactory explanation of latent heats of vaporization and van der Waals nuclear screening constants.Nevertheless, it has to be acknowledged that its rigorous validification depends principally on the consideration of three factors. The first relates to the mean- square electric moment of bonded atoms and the hypothesis that this may be approximated by the value of this parameter for the corresponding inert-gas atom. Although this hypothesis is without theoretical justification, it has been tested extensively.'. It has been shown that the approach permits the satisfactory prediction of molecular first-ionization potentials and the discrimination between molecular structures. The second and third factors concern the methods of calculating 2, and R.Moelwyn-HughesL7 gives the approximate relationship 2, = w- HFIHS) (10) (1 1) which is equivalent to 2, = Z,Hv/(Hv + HF) where HF, Hs and H , are the heats of fusion, sublimation and vaporization, respectively.Table 4. Estimation of the neighbour-anistropy screening, ca, from experimental gas-to-solution shifts and calculated ow for several anistropic solvents (Z,, = 11 except for CS, for which Z,, = 7,) % CH,CN(R = 4.8 A) k B solute exptl" calc. CJa exptl" calc. =a exptl* calc. c a exptlb calc. 6, 3 CH4 C6H,(R = 6.18 A) C(NO,),(R = 6.57 A) CS,(R = 5.14 A) 0.124 -0.432 0.556 -0.053 -0.580 0.530 -0.583 -0.414 -0.169 -0.416 -0.317 -0.099 -0.452 -0.394 -0.058 0.235 -0.379 0.614 -0.120 -0.447 0.327 0.272 -0.225 0.497 0.043 -0.298 0.341 -0.398 -0.206 -0.192 -0.285 -0.241 -0.044 - -0.300 - C6H6 C(CH,), 0.213 -0.218 0.431 0.028 -0.292 0.264 -0.426 -0.201 -0.225 -0.317 -0.234 -0.083 2 Si(CH,), 0.15 1 - 0.229 0.380 0.035 -0.290 0.255 -0.520 -0.198 -0.322 -0.343 -0.237 -0.106 9 h a F.H. A. Rummens, Can. J . Chem., 1976, 54, 254. W. T. Raynes and M. A. Raza, Mol. Phys., 1969, 17, 157.J . Homer and M. S. Mohammadi 2965 Comparison of the values obtained for 2, using eqn (10) and (1 1) with accepted values for Zs demonstrate the validity of approximation (3). If the methods of calculating 2, and (m2) are accepted it follows that values for R may be deduced from, for example, latent heats of vaporization, and these should compare favourably with corresponding values deduced independently.The ex- perimental values for latent heats of vaporization given in table 1 can be deduced using the values given as R in column 1 (these have been used to calculate the values of ow). Note that the values given as R,,,, from eqn (9) are generally in good agreement with those calculated from latent heats of -Japorization. More importantly, the values, given as RL.J, that have been deduced from viscosity data, using the Lennard-Jones potential, are in good agreement with those deduced here. It appears, therefore, that the theorem for a net attractive polyatomic London dispersion p~tential,~ from which eqn (2) derives, is suitable for explaining the properties of molecular systems at equilibrium molecular separations. This in itself raises the important question concerning the reason why the Lennard-Jones potential requires an R-12-dependent term to achieve the same ends as that on which the present work is based, and which does not require such a term.Although necessarily based on speculation, it is interesting to address this question. The potential function at the heart of this work is based on the assumption that at equilibrium intermolecular separations, the contribution from overlap repulsion is negligible and that the longer-range dipolar repulsions are accommodated in the net attractive dispersion term. The evidence so far suggests that if bonded-atom properties and molecular rotation [through F(i,j)] are properly accounted for, it becomes unnecessary to account separately for an overlap repulsion term based on, say, R-12.The choice of alternative powers of n in the general R-"-dependent repulsion term by some workers could be taken to indicate that this is an empirical scaling term that has become necessary, simply because the attractive R-6 term has not been characterized properly. Of course, at very small intermolecular separations where overlap is important it will be necessary to introduce an additional R-" repulsion term. Conclusions The polyatomic London dispersion potential presented el~ewhere,~ and used to characterize latent heats of vaporization, is shown here to characterize n.m.r. van der Waals chemical shifts. It appears, therefore, that this may be an authentic intermolecular potential function that is capable of more extensive use.The satisfactory use of the explicit polyatomic London theory in predicting ow when using values of R deduced from experimental heats of vaporization and a value of B = 0.54 x lo-'' cm3 ppm erg-' that has been proposed independentlyl43 l5 again indicates the consistency of the present approach. We are grateful to Prof. W. R. McWhinnie for the provision of facilities and M. S. M. thanks Evode Ltd (Stafford) and the International Family Service for financial support. References 1 J. Homer, Appl. Spectrosc. Rev., 1975, 9, 1 . 2 F. H. A. Rummens, van der Waals Forces in NMR Intermolecular Shielding, Vol. 10, NMR Basic 3 J. Homer and C. C. Percival, J. Chem. Soc., Faraday Trans. 2, 1984, 80, 1. 4 J. Homer and M. S. Mohammadi, J . Chem. SOC., Furuday Trans. 2, 1987, 83, 1957. 5 T. W. Marshall and J. A. Pople, Mol. Phys., 1958, 1, 199. 6 M. J. Stephen, Mol. Phys., 1958, 1, 223. 7 F. London, Trans. Faraday SOC., 1937, 33, 8. Principles and Progress, ed. P. Diehl, E. Fluck and R. Kosfield (Springer-Verlag, Hiedleberg, 1976).2966 N.M.R. van der Waals Shifts 8 J. Homer and M. S. Mohammadi, J. Chem. SOC., Faraday Trans. 2, 1987, 83, 1975. 9 M. S. Mohammadi, Ph.D. Thesis (Aston University, 1986). 10 A. H. Mones and B. Post, J . Chem. Phys., 1952, 20, 755. 11 E. A. Moelwyn-Hughes, Physical Chemistry (Pergamon Press, Oxford, 2nd revised edn, 1961). 12 P. Hertz, Math. Ann., 1909, 67, 387. 13 R. A. Kromhout and B. Linder, J. Magn. Reson., 1969, 1, 450. 14 T. Yonemoto, Can. J. Chem., 1966, 44, 223. 15 F. H. A. Rummens, Mol. Phys., 1971, 21, 535. 16 J. Homer and D. L. Redhead, J. Chem. Soc., Faraday Trans. 2, 1972, 68, 1049. 17 E. A. Moelwyn-Hughes, Chemical Statics and Kinetics of Solutions (Academic Press, New York, 1971). Paper 71966; Received 1st June, 1987