Hyperconjugation

One of the ways alkyl groups stabilize carbocations is via hyperconjugation.  We can use resonance structures to illustrate this in the 2-propyl cation:

Alternatively, MO theory can show us the effects of this partial electron donation.  In the picture below, the methyl groups are sequentially rotated (by 22.5°) through a 90° dihedral rotation.  Note how the contribution of the methyl hydrogens changes as the C-H bonds align (or twist out of alignment) with the empty p orbital at C-2:

Energies are as follows (AM1; solvation with water; kcal/mol): the LUMOs are displayed showing the degree of interaction with the various C-H orbitals.
 

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