Although have dealt with the π-complex formed by protonation of PhNHOPh in several posts, there was one aspect that I had not really answered; what is the most appropriate description of its electronic nature? Here I do not so much provide an answer, as try to show how difficult getting an accurate answer might be.
In an earlier post, I had shown how an in-phase combination of the HOMO of the anion 1 with the LUMO of the cation 2 led to an occupied molecular orbital for the complex (below, left). An out-of-phase combination of these two gives instead the LUMO of the π-complex (below, right). It might seem as if a pair of electrons would like to occupy the first of these, and indeed a wavefunction constructed on this basis (using this occupancy as the single reference; indeed only state) resulted in the conclusion that the complex was aromatic. The diatropicity (~magnetic aromaticity) was strongest in the region between the two stacked rings, and the individual rings themselves had lost their local aromaticity. One might then infer that a wavefunction constructed by populating the LUMO below would in fact rearrange the aromaticity, returning this property back to the two individual rings. There would in fact be apparently nothing to keep the rings stacked, and so in the limit this wavefunction would correspond to a biradical (both states would be degenerate and hence have one electron in each).
So where along this spectrum of possible interpretations does a more realistic wavefunction settle? To answer this question, we must optimise the self-consistent field describing the electronic structure using BOTH electronic configurations (as a first approximation). This method is known as a multi-reference configuration interaction or CASSCF (complete active space self-consistent field) approach. Technically, the variation principle of minimising the electronic energy is applied to all the eigenvalues of the CI matrix, and not just the single reference state as is normally done. The simplest approach to the molecule above is to consider the active space as just the two orbitals above (the inactive space is represented by all the remaining doubly occupied and unoccupied orbitals), resulting in a CI determinant with three solutions (two electrons in the HOMO, two electrons on the LUMO and one electron in each). One would then use the energy computed from this multi-reference solution to re-optimise the geometry of the π-complex. This would then surely be a “better” description of the wavefunction for this molecule. Or would it? Well, this is what happened when I tried.
What I did learn is that the balance between a mostly single-reference description of the wavefunction (occupancy of 2.0 in the HOMO above) and a multi-reference description (occupancy of 1.0 in the HOMO, 1.0 in the LUMO) is a fine one, and that balance can be perturbed by other effects, such as how one describes the correlation promoting π-stacking of the rings. And to be fair, I have not yet even found out if a CASSCF(2,2) is good enough. Perhaps it should be CASSCF(10,10), since we do have (at least) ten electrons that could populate our active space?
Of course, there are many solutions to the above problem (and some might even solve the analytical first derivatives limitation noted above). So if any reader of this blog has knowledge/expertise of this type of calculation, it would be wonderful to know what the answer is for protonated PhNHOPh. It is such an innocent molecule, and yet it seems such a challenge to properly compute its geometry (and aromaticity).
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