LONE-PAIR ASSISTED IONIZATION

Due to the negative charge present on the oxygen anion, the nonbonding electrons will be higher in energy. Thus, the HOMO of the molecule are the nonbonding electrons on the negatively charged oxygen.  A pair of non-bonding electrons from the HOMO will attack the LUMO of either the carbon-oxygen bond between the hydroxyl group and the rest of the molecule, or the carbon-oxygen bond between the methoxy group and the remaining portion of the molecule. In either case, the carbon-oxygen sigma antibonding orbital will have electron probability density concentrated around the low-electronegative carbon atom. Since the oxygen non-bonding electrons are near the LUMO carbon, lone-pair assisted ionization will readily occur. Only the addition of the HOMO oxygen non-bonding electrons to the carbon=oxygen sigma bond LUMO attaching the methoxy group to the molecule will drive the reaction forward, and so the oxygen non-bonding electrons will fill the carbon-oxygen  holding the methoxy group to the molecule. As a result, the methoxy group will leave, and the carbonyl pi bond will be restored.

DEPROTONATION

The product from lone-paired assisted ionization is sodium methoxide and carboxylic acid. The proton on the carboxylic acid group is highly acidic, since the deprotonated carboxylic acid group is a resonance stabilized anion. Also, the non-bonding electrons on the methoxy anion are high in energy, and so the HOMO of the methoxy anion is high due to the non-stabilized negative charge concentrated on the oxygen atom. Thus, the methoxy anion oxygen atom is highly basic, and so the acidic proton on the carboxylic acid group will readily transfer to the methoxy anion, resulting in methanol and sodium salt of carboxylate.

(There will be an acidic workup which will protonate the carboxylate.)