How does hyperconjugation in alkenes affect bond lengths?

Question

I understand how the diagram below results in hyperconjugation stabilising alkenes.

Essentially, the C-H sigma bonding orbital interacts with the pi bonding orbitals creating two molecular orbitals. It also interacts weakly with the pi* orbital which lowers the energy of psi 1 and 2 stabilising the system.

What I do not understand is the effect this has on bond lengths. We have done a computational investigation in the lab and have found that both the C-H and C-C bonds are weaker, while the C-C bond between the double bond and the alkyl substituent is stronger. I don't understand why.

I have received an explanation for this phenomenon using resonance structures, but I am really looking to understand how to reconcile these observations with the MO diagram above.

Answer 1

The MO interpretation is not really all that complicated: you are allowing the C–H σ orbital to overlap with the C=C π* orbital. This leads to a very slight population of the π* orbital, which is antibonding with respect to the doubly bonded carbons. Therefore, that C=C bond length is slightly lengthened. Likewise, the C–H σ orbital is slightly depopulated, which makes the bond length longer.

At the same time, you are forming a slightly stronger C(sp²)–C(H) bond because these new MOs have bonding interactions between the C–H σ orbital and the p orbital on C(sp²). This is not entirely made clear in the diagram above from Fleming's text, which I have addressed separately here. From the diagram, it seems that there are equal amounts of bonding character (from ψ₁) and antibonding character (from ψ₂) However, the main point which I wrote in that post is that the π* orbital mixes (very slightly) into both ψ₁ and ψ₂. This increases the C–C bonding character in ψ₁ and decreases the antibonding character in ψ₂, such that there is a (very slightly) shorter bond.

The actual text in the book makes this clear, but the diagram does not. I suggest reading the text carefully.