Theory of the Origin of the Internal-Rotation Barrier in the Ethane Molecule. II

In the previous paper, the total energy difference ΔW hindering internal rotation in the ethane molecule was shown to comprise a nuclear—nuclear repulsion energy difference plus an electronic energy change ΔE, which may be computed from the integral Hellmann—Feynman formula, ΔE=∫ρES(1)H′(1)dv(1). Here ρES(1) is the first-order spinless transition density between the eclipsed and staggered states, and H′(1) is the difference between the one-electron electron-nuclear attraction operators for the two forms. It was shown how ρES(1) and H′(1) both could be expanded in Fourier series in the electronic azimuthal angle φ about the CC axis, so that ΔE becomes the sum of contributions from successive Fourier components of the transition density. In the present paper this analysis is continued. The threefold and ninefold components of ρES(1) are accurately computed from the LCAO—SCF wavefunctions of Pitzer and Lipscomb, thus eliminating several mathematical approximations made in the previous paper. The threefold contribution to the rotation barrier is found to be ΔE3≈−2.4 kcal/mole, the ninefold contribution ΔE9≈0.0 kcal/mole. The net barrier is ΔW≈2.4 kcal/mole. Contributions to ΔE3 from localized and corresponding orbital expansions of ρES(1) are evaluated, and contour maps are given of ρ3(r,θ) and ρ3(r,θ)A3(r,θ) over the whole molecule. From the orbital analyses and the density maps, the following conclusions are drawn: (1) The CC and carbon iS orbitals contribute insignificantly to ΔE. (2) Most of ΔE arises from CH orbitals. (3) The quantity ΔE can be interpreted as mainly arising from regions around protons; regions near the CC axis are unimportant. (4) The effects of changes of localized orbitals on rotation are probably not negligible, but their precise evaluation is particularly difficult. The Pitzer—Lipscomb energy difference, 3.3 kcal/mole, is different from the integral Hellmann—Feynman result. A method for bringing the two methods into concordance is described, and it is reasoned that if this were carried through the common result would probably be very near the experimental value, 2.9 kcal/mole. Various other theoretical questions are discussed, and a number of suggestions are made for further research on the internal-rotation problem.