# Quantum and organic chemistry of vitamin D synthesis

I am somewhat confused by certain steps in this mechanism.

To start with, in the conversion of 7-dehydrocholesterol to previtamin D3, how does the UV photon affect the pi-orbital system in such a way as to rearrange the bonds as shown in the diagram? Else, what exactly brings about the change shown?

Similarly, in the thermal isomerisation step from previtamin D3 to Vitamin D3, what quantum effects and other changes cause the rearrangement shown?

Since this is a biological reaction within the human body it also takes place under physiological conditions ($37^\circ C$, 1 atm, pH = 7.4, various intracellular ionic concentrations).

These are pericyclic (i.e. concerted) reactions. They do not need particularly high temperatures, but indeed they can be catalysed by either the right photons or temperature. They are not "reactions" per se, rather just intramolecular rearrangements of electrons over a "chain" of connected atoms.

The first reaction is a photolytically catalysed electrocyclic reaction that results in the opening of the 6-membered ring, they are very common.

The second reaction is a thermally catalysed [1,7] sigmatropic migration of the H. The rate is dependent on the nature of the nucleus that moves, in this case H (D would be slower). Quantum tunneling will indeed enhance the rate of its non-quantum counterpart, but it is not the driving force of the step as you seem to wonder.

As for the reason why these reactions happen in the first place, it's because there is favourable overlap in the transition states between the orbitals of the atoms at the two external ends of the "chain" of atoms interested in each rearrangement.

As you can see below, the HOMO in each case permits favourable overlap (phase-wise and sterics-wise) between the ends of the "chains" in both cases. I couldn't quickly find a picture of a photolytic ring opening but in that case the actual HOMO is a SOMO: one electron from the former HOMO has been promoted (by the photon) to the former LUMO. This will change the overall symmetry properties of the atom chains so that a [1,7] H shift is allowed under photolytic conditions.

Electrocyclic reaction:

Hydride shift:

Suggested reading: "Pericyclic Reactions" by Ian Fleming (OUP, 1998).

• Could you please illustrate how this "favourable overlap" works, preferrably with a diagram of sorts? – James Harrison Sep 12 '15 at 2:38
• Also I wasn't referring to quantum tunneling, I was thinking more about how an MO diagram or frontier pi-orbitals could be used to explain this phenomenon. – James Harrison Sep 12 '15 at 3:09
• @James Harrison: I've put pictures of the FMOs to clarify as requested. I wrote that bit about the tunneling as it was the only reference to quantum effects that could be made in this context and you were asking about quantum effects in the question, apologies if that wasn't clear. – WaTTacK Sep 12 '15 at 13:20
• Thanks very much for adding those in! But one more thing, is rotation still restricted in the C atoms within the conjugated pi system after the 1,7 H shift, or does it otherwise matter that the double bonds seem to be on the same side of the single bond in your FMO diagram, but on different sides in the mechanism in my question? – James Harrison Sep 12 '15 at 13:26
• I see what you mean. I was always given to understand that there is reasonable flexibility around the alkene bonds (at least during the transition state), so that the ends of the conjugated system will be close during the shift, but then readjust themselves soon afterwards to reach a minimum in potential energy - thereby giving the somewhat "straighter" looking π-system that you have in your original picture. Hope this helps. – WaTTacK Sep 12 '15 at 13:37