JOHN G. ASTON 79 RESTRICTED ROTATION IN ETHANE BY L. J. OOSTERHOFF Received 2nd February, 1951 The electrostatic interaction between the methyl groups in ethane is in- vestigated. The basic idea is the same as in the investigation of Lassettre and Dean, but the calculations have been performed along different lines. The charge distributions of the C-H bonds are calculated with wave functions of the same type as those used by Coulson when discussing the dipole moment of this bond. By expanding the reciprocal distance, which occurs in the Coulomb interactions, in a Fourier series it proves possible to calculate directly the cos 34 term in the interaction energy, The cos 64 term proves to be negligible in comparison. The results of the calculations differ in detail from those of Lassettre and Dean, but the general trend is the same.With a purely covalent bond the electrostatic energy of the methyl groups depends only to a negligible extent on the angle of rotation 4. The right order of magnitude of the potential barrier is obtained if an ionic contribution of the type C-H+ is added, leading to a bond dipole moment of 0.7 D with the sense C-H+. Since the relation between the calculated dipole moment and the empirical values as derived, for example, by means of spectroscopy, is still vague, we consider i t premature to conclude that the electrostatic intei actions of bonds with fixed charge distributions are the origin of the potential barrier in ethane. Information about the magnitude of the potential barriers hindering the free rotation about single bonds has resulted from measurements of thermodynamic quantities such as specific heat, entropy, and chemical equilibrium constants.For ethane the specific heat measured at low temperatures, the entropy’ determined with the Fernst theorem and the equilibrium constant of the hydrogenation of ethylene indicate a potential80 RESTRICTED ROTATION barrier of about 2750 cal./mole, as was first suggested by Kemp and Pitzer-l where 4 is the angle of rotation about the C-C bond with 4 = o in the eclipsed or in the staggered configuration. The other terms in the Fourier expansion of the potential energy are usually considered to be negligible. This is in accord with the zig-zag configuration of paraffin molecules in the crystal- line state, with the preponderant occurrence of the rigid isomer of cyclo- hexane, the non-planar structure of cyclopentane,8 etc.Spectroscopic evidence also supports the assumption of a potential barrier of the said order of magnitude with the staggered position as the stable 0ne.4 Various theoretical explanations of the experimentally determined barrier have been suggested starting from a quantum mechanical analysis of the electronic structure of the ethane molecule. The standard methods of treating the electronic structure of saturated molecules like ethane, the H.L.S.P. method or the molecular orbital method, lead to a very small potential barrier.6, Refinements of the H.L.S.P. procedure were introduced by Eyring et aZ.7 Next to the 2s and z p orbitals he considers the influence of 3d and 4forbitals, which proves to be small.More important is the reson- ance of the usual structure with excited structures as, for example, those containing a bond between two hydrogen atoms belonging to different methyl groups, and a double bond between the carbon atoms. According to Eyring et al. these effects may lead to a barrier of the right magnitude, but with the opposed configuration as the stable one. Since now all the experimental evidence is in favour of the staggved form as the more stable, one has to conclude that either a more exact calculation will change the results of Eyring et al., or that still other effects may be large enough to give results in the opposite direction. Another way of attack has been followed by Lassettre and Dean8 and by Oosterhoff.s They calculate the Coulomb interactions of the charge distributions of the two methyl groups.The charge distributions are determined by using for each C-H bond a molecular orbital or valence bond wave function with a certain degree of ionicity. The present paper mainly deals with this method. It differs from the usual H.L.S.P. method by including ionic structures and taking into account the electrostatic interaction of overlap charges in different C-H bonds, which amounts to the same thing as including some of the higher order permutations. A quite different suggestion of Pitzer lo is based on the idea that the length of, for instance, the carbon-carbon bond in ethane is determined by the equilibrium of the interaction of the bonding electrons tending to shorten the bond and the repulsion of the other valence electrons of the carbon atoms.If the charge distributions on the carbon atoms deviate from the cylindrical symmetry, for example, because of a contribution of the d-orbitals to the C-H bonds, the resulting trigonal distributions will a Hassel and Viervoll, Arch. Math. Naturvidenskab., 1944, 47, im. 13. Hassel and Viervoll, Acta Chem. Scand., 1947, I, 149. 3Aston, Schumann, Fink and Doty, J . Amer. Chem. SOC., 1941, 63, 2029. Aston, Fink and Schumann, J . Amer. Chem. SOC., 1943, 65, 341 ; cf. Kilpatrick, Pitzer and Spitzer, J . Amer. Chem. Soc., 1947, 69, 2483. One is led to this value assuming a barrier of the shape v = v, cos 34, * (1) Probably the staggered configuration is the stable one. Kemp and Pitzer, J .Chem. Physics, 1936, 4, 749. * Smith, J . Chem. Physics, 1949, 17, 139. 5 Penney, Proc. Roy. SOC. A , 1934, 144, 166. 7 Gorin, Walter and Eyring, J . Amer. Chem. SOC., 1939, 61, 1876. 8 Lassettre and Dean, J. Chem. Physics, 1948, 16, 151, 553 ; 1949, 17, 317. IOPitzer, J . Amer. Chem. SOC., 1948, 70, 2140. Eyring, J . Amer. Chem. SOC., 1932, 54, 319. Oosterhoff, Thesis (Leyden, 1949).L. J. OOSTERHOFF 81 always give a larger repulsion if the hydrogens of one methyl group line up with those of the other methyl group than in the staggered position. The decreased repulsion in this configuration will allow the C-C bond to shorten a little, thereby increasing its binding energy and stabilizing the staggered form. Without an explicit calculation it is difficult to see whether this effect may account for the magnitude of the experimentally determined barriers.Method of Calculation.-In relating the experimental barrier in ethane to the electrostatic interaction of the charge distributions of the methyl groups, we assume that those parts of the charge distributions which contribute to the barrier do not overlap appreciably. In the calculation of the charge distribution in a C-H bond we make use of the wa.ve €unctions which were used by Coulson 11 in the discussion of the dipole moment of this bond. These functions are either localized two-centre molecular orbitals or electron pair wave functions of the H.L.S.P.-type including ionic contributions. They are built up of a carbon function #t and a hydrogen IS function #h: The carbon function, one of a set of four equivalent wave functions suited to form four tetrahedrally arranged bonds, is where that particular #22p function has been chosen which has rotational symmetry around the line joining the two bonded atoms.h? = 4#28 + * 1 / 3 $ z p The electron pair wave function is s = l$t#hdT* Coulson does not introduce a term with #h(~)#h(z) since the energy of a negative hydrogen ion and a positive carbon ion at the C-H distance of the normal C-H bond is supposed to be very high. Although we do not think this argument very convincing, since the situation indicated by #h(I)#h(2) is rather different from a negative hydrogen ion, we will likewise leave this function out of consideration, mainly because a really satisfactory investigation of the problem along the present lines would anyhow require a much more refined discussion of the individual bonds.The charge distribution of each electron can be considered as a linear combination of the three normalized charge clouds $t8, #&/s and h 2 , and is given by where a#t2 f h!%#hIs + Y#h2, . . ‘ (3) 2 S2 2 AS I I y=y2-* Using localized two-centre molecular orbitals it is possible, by adjusting the coefficients, to make the charge distributions nearly the same. This does not necessarily imply that the energies corresponding to these wave functions are also nearly equal, since the spatial correlations of the two electrons are always different. l1 Coulson, Trans. Faruduy Soc., 1942, 38, 433.82 RESTRICTED ROTATION We are only interested in that part of the Coulomb interaction of the methyl groups which changes on rotation about the C-C bond.Now the sum of the t,htz distributions of the three C-H bonds of one methyl group is a distribution with cylindrical symmetry about the C-C direc- tion. The interaction with these distributions can therefore be left out of the calculation. A t,hh2 charge distribution has spherical symmetry around the hydrogen nucleus. So in a calculation of the magnitude of the Coulomb interaction with this distribution we may reckon as if the total charge is concentrated at the hydrogen nucleus. More complicated are the calculations in which the overlap charges are involved. To a first approximation it is allowable to replace a charge cloud distributed according to $&/S by an equal charge placed in the centre of charge.For a further approximation it is necessary to consider also the higher moments of this distribution. Electrostatic Interaction of Two Charge Clouds .-The Coulomb interaction of two charge clouds with densities u, and u, is given by the integral L is the distance between a point in the first cloud and a point in the second one. i’ FIG. I. d -Co-ordinate system for the interaction of C-H bonds. The charge distributions which enter into the prcsent calculations always have cylindrical symmetry around a C-H bond. Therefore the centre of charge lies on the line through the carbon and the hydrogen nucleus. For the evaluation of integral (4) it is convenient to consider the reciprocal distance L-l as a function of the distance between the charge centres and the distances between the two above-mentioned points and the charge centres.In Fig. I the two carbon nuclei are shown at a distance d apart. The charge centres are lying at distances Y, and r2 from the carbon nuclei. Each charge centre serves as the origin of a co-ordinate system. The z-axis lies along the C-H line in the sense C-H. The x-axis lies in the plane through the C-C bond and the C-H bond, the positive sense pointing to the other methyl group. The y-axis is fixed by a right- handed co-ordinate system. The two co-ordinate systems can be dis- tinguished by the indices I or 2. + is the angle of rotation around the C-C bond ; + = o if the two C-H bonds lie in one plane. We call L the distance between the points (xl, yl, 2,) and (x2, y2, +), Lo being the distance between the points (0, 0, 0) and (0, 0, 0).Assumrng tetrahedral bond angles we get We will proceed in the following way. 16 9 3 3 9 Lo2 = R2 - -ylv2 cos + ; R2 = y12 + Y~~ + Zr,d + %,d + s l y , + d2. ( 5 )L. J. OOSTERHOFF 83 We will now develop L-l into a power series in x,, yl, 2, and x,, ye, z,. This may conveniently be written as The Greek letters, which have the same meaning as the Latin letters, have been used to bring out explicitly that the differential quotients refer only to the co-ordinates in L. The index o indicates that the differential quotients have to be taken at the origin. The series (6) may be used in (4) on conditions of non-overlapping, etc., which we will assume to be fulfilled. Inserting this series in ( 4 ) and introducing the abbreviations (I, 2) means that the same term as the foregoing has to be added but with the indices I and z interchanged.Ql(Jl2 - +p12) is the only non-zero component of the quadrupole moment of the charge distribution I. Q1 (5,3 - a<,p12) is the only non-zero component of the octupole moment. The dipole moment Qltl and the other components of the quadrupole moment and the octupole moment are zero in consequence of the choice of the origin in the centre of charge and of the cylindrical symmetry of the charge clouds. Fourier Expansion of Reciprocal Distance .-It has already been said that we are only interested in that part of the Coulomb interaction which changes on rotation of the methyl groups around the C-C bond.In consequence of the trigonal symmetry of the methyl groups, the inter- action energy may. be expected to depend on the angle of rotation $ according to provided that the zero of $ has been suitably chosen. This formula suggests that it will be advantageous to insert in the energy expression (7) a Fourier development of the reciprocal distance, for in the resulting development only’ the coefficients of cos 34, cos 64, etc., need to be con- sidered. The other terms cancel when adding together all the interactions between the charge distributions of the two methyl groups. Expanding the reciprocal distance V = V o + QV, cos 34 + QV, cos 64 + . . ., . - (8) 16 = (R8 ---y1y2 cos 4 LO 9 in a Fourier series yields the result : € 0 = I, €, = 2, m = 1 , 2 , . . . .84 RESTRICTED ROTATION The coefficient of cos m$ is closely related to the Legendre functions of the second kind.12 In fact, this coefficient may also be written as In the present calculations the series for the coefficients of cos 3+ and cos 64 proved to converge rapidly, and even the term with cos 64 is very small with respect to the cos 34 term.Inserting the expression (9) in (7) yields the desired development of the Coulomb interaction energy of two charge clouds. The interaction of point charges can be calculated easily. For the computation of the interaction energy of the higher moments the derivatives of Lo1 are re- quired which involves a somewhat lengthy but quite feasible numerical calculation. Charge Distributions .-The charge densities u have been calculated using Slater functions for the carbon $2s and $21zp functions which figure in $t.Coulson and Duncanson l3 have shown, in a discussion of the momentum distribution in CH,, that there are reasons for increasing Slater's screening constants (I for hydrogen, 1.625 for carbon 2s and z p ) by a factor w = 1.1. Following this suggestion we get the set of functions (in atomic units) $5 is a IS hydrogen function. 0 = 1'1 The length of a carbon-hydrogen bond is taken as 1.093 A. of the overlap integral S = 0,636. I,ht$h,/s with respect to the centre of charge are The value The moments of the overlap charge The distance of the charge centre from the carbon nucleus is 0-794 A. The moments of the +t2 distribution are - - } (13) 5 2 = 0-213 A2, 53 =- 0.082 A3, p2 = 0-307 A2, 5p2 = 0.021 A3, 5 2 - 1 2 p 2 = 0.059 Hi2, 5 3 - $ 5 ~ 2 = - 0.114 A 3 .- - The distance of the centre of the +t2 distribution from the carbon nucleus is 0-370 A. Re s w l t s The quantities 01, B, y measure the chance tor one electron t o be found in one of the three charge distributions I,hr2, $,I,hh/S or In Fig. 2 they are plotted as a function of A, which is a measure of the degree cjf ionicity of the C-H bond. The bond dipole moment p is plotted against h in Fig. 3. This dipole moment was calculated, according t o Coulson, for a charge distribution consisting of the positive charge of the hydrogen nucleus, an equal charge at the carbon nucleus and the charge cloud of the two electrons involved in the formation of the bond. In order t o facilitate comparison of our results with those of Lassettre and Dean the only non-zero components of the quadrupole moment P and the octupole 12 Whittaker and Watson, Modern Analysis (Cambridge, 1946), 4th edn., 13 Coulson and Duncanson, Proc.Camb. Phil. SOC., 1942, 38, 100. p. 316.L. J. OOSTERHOFF 85 moment 0 of the bond are also plotted in Fig. 3. by the formulae The three moments are defined 1 p = ZQ + 5ud7, s The summations refer to the two equal positive charges of the nuclei. integrations extend over the charge cloud of the two electrons. the co-ordinate system has been chosen midway between the two nuclei. units are based on a unit of charge equal to 10-l~ electrostatic unit and the as a unit of length. The The origin of The 1 2 3 4 , - A FIG. 2.-a, p, and y as a function FIG.3.-Bond dipole moment of the ionicity A. p, quadruple moment P and octupole moment 0 as a function of the ioni- city A. The results of the calculation of the electrostatic interaction of the methyl groups are plotted in Fig. 4. The dotted line a gives the value of the barrier height if the charge clouds had been concentrated as point charges a t their centres of charge ; b and c represent the additional interactions, caused by the quadru- pole moment and the octupole moment of the overlap charges. The total inter- action, including the three effects, is given by the curve marked V,. The co- efficient of 4 cos GI$, viz. V,, proved to be negligible as compared with V 3 (about one thousandth of V,). * It is gratifying that the main contribution to the cos 34 term in the inter- action energy results from the interaction of the point charges.The con- tribution of the terms involving the quadrupole moments is very low but the contribution of the octupole moments is rather high. So we do not feel sure that the hgher moments will give a negligible contribution. We do not expect, however, that the values of V , given in the Table will change very much if they are calculated exactly, using the same charge distributions. The values of V,, calculated according to the method of Lassettre and Dean,8 and indicated by V8’ are plotted as a dashed line in Fig. 4. In this method the charge distributions in the G - - H bonds are not split up into separate charge clouds. The interaction of the bonds is calculated using the dipole momcnt, quadrupole moment, etc., of the C-H bond, as defined by (14).From the calculation of the electrostatic interaction for a great number of values of 4, Lassettre and Dean derive the formula (converted to the units used in this investigation), With our method we derive the formula V3’ (cal./mole) = zg7.5p2 + 339.9 p P + ~ I I - ~ P ~ . . - (15) V,’ (cal./mole) = zg6-0p2 + 343-9 p P + 315.9 P2 - 736.7 PO, (16) * The limiting value of V , for A 3 00 is 4706 cal./mol.86 RESTRICTED ROTATION which agrees very well with (15) except that the dipole-octupole interaction has been included. This latter term is only of importance for high values of A corresponding to a high degree of ionicity of the bond. From Fig. 4 i t can be seen that the agreement between the values of V 3 and V3' is not very satisfactory, but the general trend is the same.Discussion From the graph of V , in Fig. 4 one might conclude that the electro- static interaction of the methyl groups can be used as an explanation of the potential barrier in ethane. In that case one has to assume a large ionicity of the C-H bond. In chemistry the C-H bond is usually con- sidered to be a typical covalent bond and, on the analogy of covalent diatomic molecules, one is inclined to attribute at most a small dipole moment to this bond. Coulson 11 pointed out, however, that the relation between bond character and dipole moment of the C-H bond is quite different (cf. Fig. 3). A small dipole moment is connected with a mixture of the covalent wave function and an ionic wave function of the type C-H+.FIG. +-Potential barrier in ethane as a function of ionicity A. a, contribution of charge clouds concentrated a t their charge centres ; b, con- tribution of quadrupole moments ; c. contribution of octupole moments ; V,, total height of potential barrier ; V,', height of barrier according to (16). The main arguments in favour of a small value of the dipole moment of the C-H bond, say smaller than 0.7 D, have, for the greater part, been derived either from an analysis of measured dipole moments of molecules or from the intensities in vibration spectra (cf. Gent 1 4 ) . A closer examination of the meaning of the bond dipole moments, derived from these sources, makes it doubtful whether they refer to the same quantities as the moments calculated according to Coulson.Therefore we do not think that a covalent character of the C-H bond can be excluded. Lassettre and Dean * assume the value of 0.4 D of the C-H bond dipole moment to be correct. With a small dipole moment the main contribution to the potential barrier calculated with (15) or (16) comes l4 Gent, Quart. Rev., 1948, 2, 383.L. J. OOSTERHOFF 87 from the quadrupole-quadrupole interaction. The calculated value of the potential barrier being too small, Lassettre and Dean have reversed their reasoning and derive an empirical value of the quadrupole moment from the known value of the potential barrier. The ratio of the empirical quadrupole moment t o the calculated one is about the same as the ratio between the accurate value of the quadrupole moment of the hydrogen molecule and that calculated with the crude molecular orbital method. Along these lines Lassettre and Dean come to the conclusion that the quadrupole-quadrupole interaction generally can be considered as the origin of the potential barriers hindering rotation. In view of the results represented in Fig. 4 we do not think that the method of calculation according to formula (15) or (16) is satisfactory. Therefore, since in our opinion it is at present not justifiable to use the bond dipole moment as a guide in selecting the most appropriate value of A, we prefer the conclusion that it may be useful to reckon with the possibility that the potential barrier in ethane is due to electrostatic interaction of C-H bonds, which have a rather high degree of ionicity. On the other hand, it should be borne in mind that the C-H bond may be a nearly covalent bond. In any case a more refined analysis of the electronic structure of ethane is necessary in order to arrive at a definite conclusion. The author wishes to express his thanks to Prof. H. A. Kramers for his interest and advice during the course of this research, to Mr. J. H. Kruizinga for his assistance in several mathematical derivations and for his indispensable help in carrying out many of the numerical cal- culations, and to Mr. J. A. van der Heiden fclr his help in preparing the figures. This paper is published by permission of the Management of the N.V. de Bataafsche Petroleum Maatschappij, The Hague. Koninklijke /Shell-Laboratoriuun, A m s tevd am.