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

The energy barrier of 2.9 kcal/mole hindering internal rotation in the ethane molecule may be conveniently decomposed into a nuclear—nuclear term which favors the staggered conformer by 5.1 kcal/mole and an electronic energy term of 2.2 kcal/mole favoring the eclipsed form. Through use of the integral Hellmann—Feynman theorem, the electronic energy change may be expressed as ΔE= ∫ ρES(1)H′(1)dτ(1), where ρES(1) is the one-electron transition density between the eclipsed and staggered states and H′ (1) is the difference between the one-electron nuclear—electron attraction operators for the two forms. By expanding both ρES(1) and H′ (1) in Fourier series in terms of the azimuthal angle φ about the carbon—carbon axis, one may obtain the explicit dependence of the electronic energy change on the threefold component ρ3 of the transition density, ΔE≈π ∫ ρ3(r, θ)A3(r, θ)dτr,θ, where A3(r, θ) is the threefold component of the operator H′ (1). This formula shows precisely how the electronic component of the internal-rotation barrier arises from the fact that in both the staggered and eclipsed conformers the electron density is not cylindrically symmetric about the carbon—carbon axis, but contains appreciable threefold components. Analysis of the best available ethane wavefunctions confirms the idea that these threefold density components are most important near the protons.