ABSTRACT
Suppose we know the pKa for HA. Then we can write the pKa for the unknown HB as follows:
pKa(HB) = pKa(HA) + ΔpKa (3.1)
Here ΔpKa is the difference in pKa of the two related molecules. Following equation 2.1, pKa = ΔGaq/RTln(10), so ΔpKa is given by:
ΔpKa = ΔΔGaq/RTln(10) (3.2)
where
ΔΔGaq = ΔGaq(HB) – ΔGaq(HA) (3.3)
Using equations 2.1-2.3 twice, once for HA and once for HB, we have:
ΔGaq(HA) = Ggas(H+) + Ggas(A-) – Ggas(HA) + ΔGsol(H+) + ΔGsol(A-)
– ΔGsol(HA) (3.4)
and
ΔGaq(HB) = Ggas(H+) + Ggas(B-) – Ggas(HB) + ΔGsol(H+) + ΔGsol(B-)
– ΔGsol(HB) (3.5)
so that equation 3.3 becomes
ΔΔGaq = Ggas(B-) – Ggas(A-) – Ggas(HB) + Ggas(HA) + ΔGsol(B-) –
ΔGsol(A-) – ΔGsol(HB) + ΔGsol(HA) (3.6)
Therefore for relative pKa calculations, combining equations 3.1, 3.2, and 3.6 the following equation can be used to calculate the pKa of HB when the pKa of HA is known:
pKa(HB) = pKa(HA) + [Ggas(B-) – Ggas(A-) – Ggas(HB) + Ggas(HA)
+ ΔGsol(B-) – ΔGsol(A-) – ΔGsol(HB) + ΔGsol(HA)]/RTln(10) (3.7)
Relative pKa calculations can be highly accurate, within a few tenths of a kcal/mol, as errors in both the gas-phase and solution-phase values tend to cancel [7]. Numerous highly accurate relative pKa values have been reported in the literature. For instance, a comprehensive study on a large set of carboxylic acids, alcohols, phenols, and amines using B3LYP for the gas-phase and PCM for the solution-phase calculations spanned over 16 orders of experimental pKa values, and resulted in standard deviations of 0.37, 0.40, and 0.52 pKa units for carboxylic acids, alcohols and phenols, and amines, respectively [150]. Nino and co-workers report highly accurate relative pKa calculations for aminopyridines [120], while Brown and Mora-Diez published moderate relative results for protonated benzimidazoles [38]. A computational study of the acid dissociation of esters
and lactones highlights the case study of diketene [151]. Another study reveals that while absolute predictions of four stepwise protonation constants for nitrilotripropanoic acid fail completely, the relative calculations using B3LYP and PCM are accurate within 1 pKa unit [152]. Borg and Durbeej have determined ground-and excited-state relative pKa calculations for the phytochromobilin chromophore, revealing the rings that are the strongest acids in the ground and bright first excited singlet states [153,154].
