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I have some confusion about perturbation theory, and just how it fits into the hard-hard/soft-soft interactions within organic reactions. I have come across a formula for perturbation that is proportional to an overlap constant squared but is inversely proportional to the energy difference of the orbitals in question. This suggests that for orbitals of poor energy match, there is least perturbation, and vice versa for good orbital energy matching. How does this translate into a hard/hard and soft/soft interpretation? I am not looking for a quantitative answer because my chemistry is not advanced enough!

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What you are referring to as perturbation theory sounds like a very crude approximation to the 'charge-transfer' or CT intermolecular interaction. In quantum chemistry lingo, the CT interaction is the mixing of occupied orbitals on one molecule/fragment with the virtual orbitals on another fragment (this is the 2-body CT interaction). This permits the transfer of electron density from one fragment to another (charge-transfer).

Instead of redoing the HF (variational) calculations with the new orbitals included, perturbation theory is instead used to calculate the nth order correction as a result of occ(a)+vir(b) mixing. Differencing the HOMO on fragment A with the LUMO on fragment is just a way to quickly approximate the more complicated (and expensive) perturbation theory expressions for the CT iteraction.

So this seems to make sense in the context of HSAB theory and Lewis acids where acid/base exchange electron pairs and thus participate in a strong charge-transfer interaction.

*EDIT: I'd like to add that the orbital energy of the LUMO orbital (in fact, all virtual orbitals) is sensitive to the basis-set size. When a large enough basis-set is used, the LUMO can always be forced to near 0.00. Orbital energies, at the end of the day, are not nature.

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