POTENTIAL ENERGIES FOR ROTATION ABOUT SINGLE BONDS BY KENNETH S . PITZER’ Received 18th January, 1951 The theoretical relationships between the potential energy function for internal rotation and the energy levels and thermodynamic properties are summarized briefly. The height and shape of the potential barrier in ethane are re-examined in relation to all pertinent data. It is concluded that the barrier is very close to the cosine function in shape and from 2750 to 3000 cal./mole in height. The barrier heights for a number of other hydrocarbons are sum- marized and a few comments are made concerning the cause of these barriers. In the last fifteen years a great deal has been learned about the quanti- tative changes in potential energy associated with rotation about single covalent bonds within a molecule.Probably a majority of the examples investigated are hydrocarbons, and conversely all but the simplest hydro- carbon molecules allow internal rotations. Consequently a discussion of this phenomenon is appropriate to a symposium on hydrocarbons. The present contribution will attempt to review principally the theoretical aspects of restricted internal rotation but will also summarize some present best values for potential barriers. The theoretical problems are of two general sox ts. First are the phenomenological inter-relation- ships-the quantum energy levels and the statistical thermodynamic properties in terms of the potential energy function for internal rotation, the pertinent moments of inertia, etc. Second are theories as to the cause of potential barriers in terms of the behaviour of the valence electrons of the molecule.The principal problem involved in relating other properties to potential barriers associated with internal rotation is the calculation of the quantum energy levels for the system. It is more convenient to assume a form of potential function and then calculate energy levels than the reverse. A simple and very useful form is where V,, is the height of the potential peaks above the valleys, 4 is the rotation angle and n is the number of peaks and valleys, all identical, per revolution. Thus, if one or both of the rotating groups are sym- metrical, such as a methyl group, the appropriate value of n is given by symmetry. Nielson 1 first solved the Schrodinger equation with the potential of eqn.(I) and obtained the energy levels for certain cases. The chief complication is the interaction of the internal rotation with the rotation of the entire molecule. Koehler and Dennison treated this matter in further detail and gave solutions for any molecule which can be described as a pair of coaxial symmetrical tops. Actually Koehler and Dennison were particularly interested in methyl alcohol for which their solution is only an approximation. The Schrodinger equation is transformed by the separation of the over-all rotation co-ordinates and yields for a single internal rotation : V = &V,(I - cos n4), * (1) M”(x) + (a + ze cos ~x)M(x) = 0, . * (2) * On temporary leave from Department of Chemistry, University of Cali- fornia, Berkeley.Nielsen, Physic. Rev., 1932, 40, 44.5. Koehler and Dennison, Physic. Rev., 1940, 57, 1006. :Pitzer and Gwinn, J . Chem. Physics, 1942, 10, 428. 66KENNETH S. PITZER 67 with wherein W is the energy of a quantum state (above the zero of the potential curve), I,. is the reduced moment of inertia for the internal rotation and the other symbols are conventional or defined in connection with eqn. ( I ) . This is a relatively well-investigated differential equation, commonly known as the Mathieu equation. The boundary conditions for accept- able solutions depend on the relative moments of inertia of the two parts of the molecule and upon the quantum numbers of over-all rotation. Although these relationships can be fairly simple in special cases, they are so complex in the general case as to preclude presentation here.How- ever, it is important to give the relationship for the reduced moment of inertia for the most general case of eqn. ( z ) , namely, a symmetrical top attached to any rigid frame 3 r, = A ( I - 2 A & ~ / I ~ ) , . * (3) i=l where A is the moment of inertia (about the axis of internal rotation) of the symmetrical top, is the direction cosine between the axis of internal rotation and the ith principal axis of the whole molecule about which the overall moment of inertia is Id. In the special case of two coaxial tops of moment of inertia A , and A s this reduces to the familiar expression I, = A , A , / ( A , + A2). * (4) The general character of the energy levels as a func- tion of potential barrier height is shown in Fig.I. The energy levels lie within the shaded regions, one within each region. The dashed line indicates the potential barrier height. Well below the barrier the levels are simply defined and essentially vibrational. Above the barrier the regions are wide, indicating large interaction with over- all rotation, and the general spacing follou~s the pattern for rotation. Fig. I is strictly a plot of (a + 28) against 13. From eqn. (2) one then notes that the unit of energy on the figure vo FIG. 1 .-Energy levels for restricted internal rotation. The dashed line shows the top of the potential barrier. is proportional to ( ~ 2 / j , ) . The statistical mechanical formulae for thermodynamic properties involve sums over all the energy levels which, at sufficiently high tem- peratures, are not sensitive to small shifts in individual energy levels.Thus it is commonly sufficient to know that the energy levels for a given molecule are located one within each band shown in Fig. r . The Tables prepared by Gwinn and the writers for the thermodynamic functions are based on this assumption and are limited to the range of temperature,68 ROTATION ABOUT SINGLE BONDS moment of inertia, etc., for which the assumption is valid. While this covers almost all molecules a few exceptions such as methyl alcohol have been found. Halford 4 has given an interesting method for these excep- tional cases wherein a few low energy levels are calculated exactly by the method of Koehler and Dennison 2 and the higher levels are obtained by the much easier but approximate method of Halford6 based on old quantum theory.Treatments have also been given for less symmetrical6 or more com- plex classes of molecules than that of the rigid frame with attached symmetrical top. Although the classical mechanics can be carried out quite rigorously even for rotating groups attached to other rotating groups all unsymmetrical, etc., the quantum mechanics becomes very involved and has been handled only to a low approximation. However, actual molecules in these categories frequently approach classical be- haviour so that the thermodynamic properties can be calculated relatively accurately. Shape of Potential Barrier.-While most of the treatments of restricted internal rotation have been based on the simple cosine formula for poten- tial energy, eqn.(I), it has been realized that this was an assumption, and occasional calculations for other shapes have been made.s Recently Halford 6 s and Blade and Kimball 9 have discussed general methods by which the energy levels can be calculated conveniently for other shapes of potential curves. Halford has considered primarily examples quali- tatively different whereas Blade and Kimball attempted to examine whether eqn. (I) was accurate in simple cases such as ethane. Since some of the conclusions of Blade and Kimball seem to the present author to be misleading, an analysis of the situation for ethane is included here. Blade and Kimball constructed a potential curve from two parabolas fitted smoothly together. The parabola forming the lower part of the potential curve is defined in terms of a cosine barrier with the same curva- ture at the minimum.Specializing our problem to a three-fold sym- metrical barrier as appropriate to ethane, one has for the true cosine barrier equivalent to eqn. (I), v = Q vr (I - cos 34). - ( 5 ) It is important to distinguish between this true cosine barrier height Vgm and the constant for the lower parabola in the two parabola system because the two will be found to have different values for the same set of energy levels. Therefore we write V f z for the quantity defined by Blade and Kimball and their lower parabola becomes The upper parabola is defined in terms of the total potential barrier height as follows, Halford, J. Chem. Physics, 19.50, 18, 444.5Halford, J. Chem. Physics, 1947, 15, 645, and 1948, 16, 410. Pitzer, J. Ckem. Physics, 1946, 14, 239. 7 Kilpatrick and Pitzer, J. Chem. Physics, 1949, 17, 1064. 8 Halford, J. Chem. Physics, 1948, 16, 560. 6 Blade and Kimball, J. Chem. Physzcs, 1950, IS, 630.CORRIGENDUM To replace Plate I-facing page 82 '' Spcctroscopy and Molecular Structure ", etc., Discumion of the Farady Society, No. g, 1950.KENNETH S. PITZER 69 Blade and Kimball derived the energy levels approximately by the phase integral or W.K.B. method and give graphs from which they may be calculated. In treating ethane, however, they considered only one set of heat capacity data on the gas from gz to 134~ K, ignoring other data in this region, all heat capacity data at higher temperatures, the value of the entropy, and the spectroscopic value for the separation of the first two energy levels.Thus their conclusion that barriers over the range 1550 to 2700 cal./mole were correct for ethane cannot be accepted without further examination. In attempting to obtain the maximum information about the potential barrier in ethane, one should first consider the spectroscopic value recently obtained by Smith lo for the separation between the lowest pair of energy levels. His value is based on two combination bands and in each case the rotational fine structure is verified as correct for the assignment given. Thus there can be little doubt but that his value of 290 cm.-l is essentially correct. Unfortunately this value is not as precise as might be desired. The two bands yielded 287-1 and 294.8 cm.-l and it is even conceivable that the band centres were incorrectly identified l1 by 10 or zocm -l.Thus this energy level separation could be as low as the 275 cm.-l value derived by Kistiakowsky, Lacher and Stitt 1 2 from specific heat data. The entropy of ethane was obtained by Witt and Kemp lS from low temperature heat capacity measurements and is compared in Table I with calculated values using Smith's moments of inertia and vibration frequencies. One should first note that the entropy comparison at 184.1' K concerns primarily the lowest energy level separation. This is indicated by the fact that the value for a 3050 cal./mole cosine barrier, which corresponds to a zgo cm.-l separation of the lowest levels, is only 0.04 cal./deg.mole higher than the value for a zgo cm.-l harmonic oscillator, and similarly for the 2750 cal./mole barrier which has a 275 cm.-l separation of the lowest levels. Consequently it would be fruitless to compare a further variety of barrier shapes at this point. TABLE I.-ENTROPY OF ETHANE AT 184.1" I< (Ideal gas a t I atm. in cal./deg. mole) Experimental . - 49'54fO.I5 Harmonic osc. v = 290 cm.-l. . 0.74 Harmonic osc. 275 cm.-l. . 0.81 Calc. for trans., rot., and vibr. . . 48.69 Expl. for internal rotation . 0.85 Cosine barrier Vo = 3050 cal./mole . 0.78 Cosine barrier 2750 cal./mole . 0.87 One can conclude that the separation of the lowest pair of energy levels is probably in the 275-290 cm.-l range, where the upper limit arises from the entropy value and the lower limit from the spectroscopic data.The principal basis for discussion of the shape of the potential barrier is the heat capacity curve for the gas. Fig. 2 shows the experimental data l4 over the range 90-305' K, together with several calculated curves. The contributions of translation, rotation and vibration have been sub- tracted so that only the contribution of internal rotation is shown in Fig. 2 . The two solid curves are for cosine barriers of the heights indicated, while l o Smith, J. Chem. Physics, 1949, 17, 139. l1 Dr. L. Smith mentioned this possibility in private communication. l2 Kistiakowsky, Lacher and Stitt, J . Chem. Physics, 1939, 7, 289. l3 Witt and Kemp, J . Amer. Chem. SOC., 1937, 59, 273. l4 Hunsmann, 2. 9hysiR. Chem. B, 1938, 39, 23 ; Eucken and Parts, 2.physik. Chem., B 1933,20,184 ; Heuse, Ann. Physik, 1g1g,59,86 ; Kistiakowsky and Nazmi, J. Chem. Physics, 1938, 6, 18 ; Kistiakowsky and Rice, J . Chem. Physics, 1939, 7, 281 ; also ref. (12).70 ROTATION ABOUT SINGLE BONDS the dashed curves are for a series of two parabola barriers of various heights. Each of the latter series has the lowest pair of energy levels spaced at 290cm.-l which corresponds to VEG = 2570 cal./mole. One can see from the upper dashed curve, for V F = 2050 cal./mole, how Blade and Kimball were led to false conclusions by considering only the data of Kistiakowsky, Lacher and Stitt. This curve fits this particular set of data fairly well but is totally unacceptable at higher temperatures. Also the difference between the quantities J$- and Vt: is indicated by their respective values 3050 and 2570, each corresponding to the 290 cm.-l separation of lowest energy levels.I FIG. 2.-The heat capacity for internal rotation in ethane for various assumed potential barriers. The 275-290 cm.-l range for the lowest energy level separation, which was found probable above, receives further support from the data near IOOO K in Fig. 2. The cosine barriers fit the somewhat scattered data in the range around 150O K as well as is possible but yield slightly low values near 300° K. However, the difference at the higher temperatures is within the limit of the various errors involved. The best of the two parabola curves, that for VEK = 2830, is no net improvement being too low in the 150-2ooo K range while yielding slightly better fit near 300' K.Thus one can conclude that the potential barrier for ethane probably has a height in the range 2750-3000 cal./mole and is very close to the cosine function in shape. The deviation indicated is that of a slight lowering and flattening of the potential peaks as compared to the re- mainder of the curve. Only recently has the work of Smith l o provided strong evidence from data on ethane itself that the stable or minimum energy configuration is the staggered one (point group DM). However, the general pattern of evidence from substituted ethanes had long indicated this result. Potential Barriers for Various Hydrocarbons .-There are assembled in Table I1 most of the available values for potential barriers to internal rotation in hydrocarbons.The cosine shape of barrier is ordinarily as- sumed. In cases of several equivalent rotations, the barriers are assumed independent and equal. Where this assumption seems doubtful theKENNETH S. PITZER resulting value is indicated as an average, (av.). Ranges of uncertainty are given where the completeness of the data justified it-in other cases somewhat larger uncertainties must be allowed. Most of the barriers in Table I1 are for methyl group rotations under circumstances in which the three-fold symmetry of the methyl group applies also to the potential function. In toluene and approximately in m- and p-xylene the combination of two-fold symmetry of the phenyl group with the three-fold symmetry of the methyl yields a six-fold sym- metry for the potential barrier.Styrene has a two-fold symmetry for the vinyl against phenyl rotation while I : 3-butadiene has two equal potential peaks but the cis and trans minima are different. TABLE 11.-VALUES FOR POTENTIAL BARRIERS TO INTERNAL ROTATION IN VARIOUS HYDROCARBONS Substance Ethane Propane . isoButane . neoPentane . Propylene . isuButene . trans-2-Butene . cis-2-Butene Toluene . m and p-Xylene . o-Xylene . I : 3-Butadiene . Styrene. . Dimethylacetylene Vo, cal./mole I950 450 bV.1 500 & 500 500 f 500 2000 (av.) 5000 trans to peak { 2575 cis to peak } 2200 0 Source Present paper Pitzer,ls also Kemp and Egan 16 Pitzer and Kilpatrick,17 also Aston et a1.18 Pitzer and Kilpatrick,l7 also Aston and Messerly 19 Kilpatrick and Pitzer 20 Kilpatrick and Pitzer,20 also Kilpatrick and Pitzer 2O Kilpatrick and Pitzer,20 also Aston, et aLZ1 Pitzer and Scott 2a Pitzer and Scott 22 Pitzer and Scott 22 Aston, Szasz, Wooley and Brickwedde 21 Pitzer, Guttman and Westrum 2a Kistiakowsky and Osborne et ~ 1 .2 ~ Aston et aL21 Internal rotation about the middle bond in n-butane is related to the r : 3-butadiene case but somewhat more complex. Of the three potential minima for the n-butane, one is in the trans position while the other two are enantiomorphs called gauche or skew forms. The full details of this potential curve for n-butane are not known yet. Assuming that the potential barriers €or the end methyl group rotations are similar to those in propane, one finds the general magnitude of the height of a potential peak above adjoining valleys to be in the 3000-4000 cal.range.26 Also l6 Pitzer, J . Chem. Physics, 1944, 12, 310. 16 Kemp and Egan, J . Amer. Chem. SOC., 1938, 60, 1521. 17 Pitzer and Kilpatrick, Chem. Rev.. 1946, 39, 435. 18 Aston, Kennedy and Schumann, J . Amer. Chem. SOC., 1940, 62, 2059. 19 Aston and Messerly, J . Amer. Chem. SOC., 1936, 58, 2354. 2 0 Kilpatrick and Pitzer, J . Res., Nat. Bur. Stand., 1946, 37, 163 ; also 21 Aston, Szasz, Woolley and Brickwedde, J . Chern. Physics, 1946, 14, 67. 22 Pitzer and Scott, J . Amer. Chem. SOC., 1943, 65, 803. 23 Pitzer, Guttman and Westrum, Jr., J . Amer. Chem. Soc., 1946, 68, 2209. 21 Kistiakowsky and Rice, J . Chem. Physics, 1940, 8, 618. 26 Osborne, Garner and Yost, J . Amer. Chem. Soc., 1941, 63, 3492. 26Pitzer, J .Chem. Physics, 1940, 8, 711 ; Ind. Eng. Chem., 1944, 36, 829. 1947, 38, 191.72 ROTATION ABOUT SINGLE BONDS the gauche or skew minima are known to be about 750-1000 cal./mole above the trans minimum. This latter value came first from the writer’s statistical-thermodynamic treatment of a series of n-paraffins,2s but has recently been measured by Szasz, Sheppard and Rank2’ from the shift of the relative intensity of Raman spectral lines for skew and trans forms with change of temperature. This spectroscopic method has been applied as yet only to the liquid state for n-butane, which leaves the uncertainty of the difference in heat of vaporization of the trans and skew forms. However, data in the gas phase have been obtained for some halogen derivatives using the infra-red instead of the Raman spectrum.Assum- ing that the necessary intensities can be had, gas phase data should be obtainable for hydrocarbons also. The skew-trans energy difference in n-butane has found interesting application in the methyl substituted cyclohexanes where similar geo- metrical situations arise.28 These results also tend to verify the value given above. More complex hydrocarbons have, of course, additional possibilities of internal rotational minima corresponding to non-equivalent configura- tions with different energies. The writer has preferred the term tautonzers (rather than isomers) in describing these configurations since they are in rapid reversible equilibrium with one another. Cause of Potential Barriers.-There have been a number of investiga- tions directed to the explanation of these potential barriers in terms of electron or valence theory.In the present degree of advance of quantum mechanical calculations an exact solution is impossible and attempts at approximation have as often as not yielded the wrong sign or a totally wrong order of magnitude. However, one can say that there is every reason to believe that the actual single bond itself, i.e. the pair of electrons and the orbital describing their motion, does not resist rotation appreciably. The writer has made perturbation calculations for the ethane case and has found that the maximum potential energy differences associated with the carbon-carbon bond itself are several orders of magnitude smaller than the observed barrier.Thus the observed phenomena must be associated with the electron pairs attaching other groups, i.e. the C-H bond electrons in ethane. Of course, where the attached groups are large enough there may also be other interactions between them. stated in 1938 that the potentials were due to “ a n interaction of electron pairs forming single bonds on adjacent polyvalent atoms ”. Today we can add with certainty that the interaction is repulsive, i.e. the staggered configuration is the stable one, and that there is doubt as to the limitation to single bonds on the adjacent atoms. Otherwise this statement summarizes the current position on this question. Of the more detailed calculations that of Lassettre and Dean 5 0 is probably the most interesting and plausible.They attempt to account for the barriers by an electrostatic model recognizing quadrupole as well as dipole moments for the bonds. In hydrocarbons the dipole moments are small but, if the bonding electrons are largely concentrated between the nuclei, the quadrupole moments could be large enough to account for the observed barriers. In extending their work to more complex molecules, Lassettre and Dean found peculiar unsymmetrical configur- ations to have minimum energy in isobutane and neopentane. Thus these molecules would appear to deserve further study. Kistiakowsky, Lacher and Ransom 27 Szasz, Sheppard and Rank, J . Chem. Physics, 1948, 16, 704; see also 28 Beckett, Pitzer and Spitzer, J . Amer. Chem. Soc., 1947, 69, 2488. 29 Kistiakowsky, Lacher and Ransom, J . Chem. Physzcs, 1938, 6, goo. 80 Lassettre and Dean Jr., J . Chew. Physics, 1949, 17, 317. J . Chem. Physics, 1949, 17, 86.KENNETH S. PITZER 73 It seems likely to the writer that the true explanation of these potential barriers may be more complex than any of the treatments offered as yet in that more than one effect is of importance. The same sort of “ van der Waals ” repulsive forces arise between pairs of electrons in different atoms or bonds within a molecule as between different molecules. It has been shown that this effect is too small to account for the full barrier in ethane if only the regions describable as the hydrogen atoms are con- sidered. However, an additional contribution will arise from these same C-H bond electron pairs in the regions near the carbon nuclei and this term is very difficult to calculate. However, these effects can be shown to be of the right order of magnitude. Thus this “van der Waals” type of repulsive forces and the quadrupole electrostatic effects probably each contribute substantially to the net barrier. U.S. Atomic Energy Commission, Washington DC.