Density functional theory study of hydrogen atom abstraction from a series of para-substituted phenols: why is the Hammett σ(p)+ constant able to represent radical reaction rates?

The rate of hydrogen atom abstraction from phenolic compounds by a radical is known to be often linear with the Hammett substitution constant σ(+), defined using the S(N)1 solvolysis rates of substituted cumyl chlorides. Nevertheless, a physicochemical reason for the above "empirical fact" has not been fully revealed. The transition states of complexes between the 2,2-diphenyl-1-picrylhydrazyl radical (dpph·) and a series of para-substituted phenols were determined by DFT (Density Functional Theory) calculations, and then the activation energy as well as the homolytic bond dissociation energy of the O-H bond and charge distribution in the transition state were calculated. The heterolytic bond dissociation energy of the C-Cl bond and charge distribution in the corresponding para-substituted cumyl chlorides were calculated in parallel. Excellent correlations among σ(+), charge distribution, and activation and bond dissociation energies revealed quantitatively that there is a strong similarity between the two reactions, showing that the electron-deficiency of the π-electron system conjugated with a substituent plays a crucial role in determining rates of the two reactions. The results provide a new insight into and physicochemical understanding of σ(+) in the hydrogen abstraction from substituted phenols by a radical.