Dissecting the hindered rotation of ethane.

The existence of a rotational barrier of ca. 3 kcalmol 1 around the C C single bond in ethane has been known since the early studies of Ebert, Wagner, Eucken and Weigert and Pitzer done in the 1930s. Even the simplest ab initio methods are able to reproduce this fundamental result. However, the physical origin of this hindered rotation is still under controversy in the literature. There are two effects that have been regarded as responsible for the rotational barrier : a larger steric repulsion in the eclipsed conformation and an enhanced stabilization of the staggered conformation due to hyperconjugation. Clearly, the two effects (and possibly some other ones) coexist in ethane, so in order to quantify the magnitude of one of them one must be able to switch off the other in the model calculation. The steric repulsion still remains the most popular explanation of the hindered rotation of ethane. This effect is often understood as the increase in energy that accompanies the antisymmetrization of a wave function originally formed by strictly localized descriptions of two methyl groups brought up to the final ethane geometry where they overlap. This so-called Pauli repulsion is considered to be more important for the eclipsed conformation, where the overlap between the occupied sC H molecular orbitals is larger, thus giving rise to a hindered rotation. The non-orthogonality between the molecular orbitals (MOs) of the two fragments seems to be the main source of controversy of such models. It has been argued that a zerothorder unperturbed system formed by two non-orthogonal sets cannot be put in correspondence with a Hermitian Hamiltonian, raising doubts about any physical argument obtained from a perturbative approach using such a model. (We cannot accept this point of view, which would exclude all the perturbation theories of intermolecular interactions as being illegitimate.) The hyperconjugation effect is due to interactions/delocalizations between electrons of vicinal C H bonds mainly through the C C bond. In an MO picture, this corresponds to favorable two-electron two-orbital interactions between the occupied sC H orbital of one methyl group and the virtual antibonding s*C H orbital of the other. This also involves orbital s*C C. [21] The electron delocalization effect can easily be assessed in valence bond (VB) theory calculations by adding/ removing the appropriate resonance structures from the wave function. In an MO calculation it is not that simple due to the delocalized nature of the MOs. The hyperconjugation effect is estimated by constructing the wave function from sets of MOs localized in methyl moieties. The way such MOs are constructed, in particular whether they are variationally optimized, apparently led to opposite conclusions about the role of the hyperconjugation in the rotational barrier. Another factor to be taken into account is the geometry relaxation that accompanies the internal rotation, and in particular the change in the C C distance. It is somewhat striking that the C C bond is significantly shorter in the staggered conformation than in the eclipsed one although this bond is formally not changing during the rotation. This effect already appears at the simplest minimal basis SCF level of theory: at the STO6G level one gets C C bond lengths equal to 1.535 and 1.545 , respectively. At the same time the effect of geometry relaxation on the height of the barrier is minor. Nonetheless, previous studies concluded that different answers can be obtained for the relative importance of the steric repulsion and the hyperconjugation effect depending on whether the geometry relaxation has been considered or not. Previous energy decomposition analyses relied, in one way or another, on the definition of two methyl fragments. However, in the last years there have been a growing interest in other kinds of energy partitioning schemes, closely related to population analysis and bond-order techniques. Such schemes allow expressing the total energy of a system, exactly or up to a good accuracy, as a sum of atomic and diatomic contributions, as given by Equation (1):