Internal Rotation in Ethane and Substituted Analogs
Rotation about C-C single bonds has been a topic of interest in chemistry since the 1870s . It has been found that the rotation is often hindered by torsional forces arising from interactions among the substituents on the carbon atoms. The potential energy of interaction for the torsional motion of ethane , or substituted ethanes such as , can be approximated by a potential energy of the form[more]
where is the reduced moment of inertia of the two counterrotating methyl groups. This can be put into a standard form of Mathieu's equation,
with the substitutions ( to ), . The physically significant solutions are two families of Mathieu functions. One class is periodic in and transforms according to the representation of the symmetry group ; the other is periodic in and belongs to the doubly degenerate species. If not for tunneling, each energy level would be three-fold degenerate. In reality, a small splitting is observed, which increases with increasing torsional quantum number . The and levels are colored blue and red, respectively. As decreases, the number of discrete levels decreases, as the higher levels are absorbed into the continuum. The actual results using Mathieu functions are quite complicated and the results have been distilled into approximate formulas for the energy levels.[less]
 R. M. Pitzer, "The Barrier to Internal Rotation in Ethane," Accounts of Chemical Research, 16(6), 1983 pp. 207–210. doi:10.1021/ar00090a004.
 L. C. Krisher, "Inversion and Internal Rotation," in Encyclopedia of Physics (R. G. Lerner and G. L. Trigg, eds.), 2nd ed., New York: Wiley-VCH, 1991 pp. 557–558.