pKa prediction by Hammett and Taft equations

. pKa prediction by Hammett and Taft equations

A more accurate prediction of pKa, but for a small class of compounds, may be made using Hammett equations. The atom typing above reflects the major factors influencing a site's disassociation constant, the atomic species of the site and the very local steric and electronic effects. However no account is made of the longer range electronic (inductive, mesomeric and electrostatic field effects). In 1940, L.P. Hammett demonstrated that the effects on pKa of meta- and para- substitued aromatic compounds (benzoic acids) were linear and additive.

This leads to the Hammett Equation for pKa:
pKa = pKa0 - Rho * Sum(Sigma)

Where Sigma is a constant assigned to a particular substituent, Rho is a constant assigned to the particular acid or base functional group and pKa0 is the pKa of the unsubstituted acid or base. For benzoic acids, pKa0 is 4.20 and Rho is defined to be 1.0.
In the original formulation, two constants need to assigned to each substituent, Sigmameta for meta-substitutions and Sigmapara for para-substitutions. This was soon extended to aliphatic systems by Taft, by introducing Sigmastar (also written Sigma*). Currently over 40 forms of sigma constant have been defined, but many of these corrolate extremely well with each other.
As a worked example, consider the pKa prediction of the compound shown below, 4-chloro-3, 5-dimethylphenol.

The Hammett equation for phenol has pKa0 = 9.92 and Rho = 2.23. The Sigmameta for -CH3 is -0.06 and the Sigmapara is 0.24. Hence the predicted value of the pKa is 9.92 - 2.23*(0.24-0.06-0.06) = 9.70. This compares extremely well with the experimental value of 9.71.
A major benefit of Hammet/Taft equations is their ability to handle special cases.
Tetronic acids (pKa ~3.39):The duck-billed platypus of organic chemistry?

6. Estimation of Sigma Constants
Unfortunately, the Achilles heel of Hammett and Taft based pKa prediction is the dependence upon large databases, both for the substituent constants and for the acid/base functional group under consideration. The functional groups can be supplemented by the atom type based approach described above. Missing sigma constants, however, are a more serious problem. In a recent analysis by Peter Ertl, showed that only 63 of the 100 common substituents (taken from the logpstar database) had measured sigma constants.

One common approach, is to extend the set of known substituents with sigma transmission equations. For example, Sigmastar of -CH2-R can be estimated as 0.41 * the Sigmastar for R. Similarly, -CH=CH- has transmission coefficient 0.51 and -C6H4- has coefficient 0.30. Similar schemes include the Exner-Fiedler method for aliphatic cycles, and the Dewar-Grisdale method for polyaromatic systems. However, this approach cannot be used when a terminal group in unparameterized.

A second approach is to use molecular orbital methods to estimate sigma values when there is no experimental data. Using the strong corrolation between sigmameta and sigmapara and charges calculated with MOPAC's AM1 Hamiltonian, Peter Ertl of Novartis has been able to calculate sigma constants for over 80,000 organic functional groups.