Under the SQM level, the minimum in the free energy profile locates at CV ≈ 0.6 Å, which is about 0.1 Å larger than the minimum under the DFT level. This indicates that in the optimal structure under the SQM level, the proton prefers to be closer to one of the nitrogen atoms, while under the DFT level the proton is more diffusive. This can be explained by the difference in the basis sets used in the SQM and DFT calculations. PM6 used the minimum basis set, while a larger basis set 6-31G(d) was used in the DFT calculations. Both of the profiles show a reaction free energy barrier of about 1.2 kcal/mol. This picture is different from the results in a recent study by Zhou and Wang, in which the reaction is barrierless. The disagreement comes from the difference in the Hamiltonians. In their studies, all the atoms were quantum particles, while in the current study only the DMANH molecule was treated quantum mechanically. Quantum-mechanically described water molecules can stabilize the delocalized proton, and thereby lowers the reaction barrier. In the current work, we are not aiming at reproducing the results of a full-QM PIMD model. However, the central idea developed in the current work can be applied to a full-QM PIMD model, if a proper reference Hamiltonian can be found. This is beyond the interest of the current work. For the CV to the right side of the minimum, the potential of mean force increases very fast w.r.t the CV under the DFT level than under the SQM level, which indicates a higher probability of a shared proton between the nitrogen atoms under the DFT level.