Computation of the pKa Values of Gallic Acid and Its Anionic Forms in Aqueous Solution: A Self-Similar Transformation Approach for Accurate Proton Hydration Free Energy Estimation

The Gibbs free energies of gallic acid (GA) and its anionic forms in aqueous solution were computed utilizing density functional theory (DFT) at the LSDA, M062X, B3LYP/QZVP levels, in conjunction with the SMD solvation model. The p<i>K</i>_a values corresponding to the four-step deprotonation of GA were determined through a non-linear self-similar transformation expressed as, p<i>K</i>_a = a⋅p<i>K</i>_a(the)^c which establishes a link between theoretical and experimental p<i>K</i>_a values. This approach replaces the previously employed linear relationship, p<i>K</i>_a = a⋅p<i>K</i>_a(the) + b. The proposed model demonstrates high accuracy in reproducing the experimental p<i>K</i>_a1 = 4.16 ± 0.02, p<i>K</i>_a2 = 8.55 ± 0.01, p<i>K</i>_a3 =11.40 ± 0.10, p<i>K</i>_a4 =12.8 ± 0.40 values of GA, with a standard error (SE) of 0.045 and a mean absolute error (MAE) of 0.019 in p<i>K</i>_a unit. Furthermore, it facilitates the precise determination of the Gibbs free energy of the proton hydration, yielding ∆G(H⁺)_aq = 259.4272(75) [kcal mol^-1]. This result conforms acceptably with the experimental value of ∆G(H⁺)_aq = -259.5 [kcal mol^-1].