A new mathematical equation relating activation energy to bond angle and distance: A key for understanding the role of acceleration in lactonization of the trimethyl lock system

AM1 semi-empirical molecular orbital and ab initio HF at the 6-31G level calculations for the lactonization processes of 12 different hydroxy acids (1a-1l) which differ in their structural features have been conducted. The calculations obtained reveal the following: (1) The rate-limiting step in the lactonization process is formation of a tetrahedral intermediate and not its collapse as was previously reported. (2) The rate-limiting step in both the acid-catalyzed and uncatalyzed lactonization is composed of two successive steps: approach of the hydroxyl toward the carbonyl carbon until it reaches a distance of 1.4 -1.5 Å, followed by proton transfer from the ether-type oxygen to one of the hydroxyls in the tetrahedral intermediate. Calculations of the activation energies for formation of the tetrahedral intermediate in the 12 hydroxy acids studied indicate: (1) A linear relationship exists between the change in enthalpic energy (E) and the ratio of the attack angle (nucleophilic-oxygen/carbonyl-carbon/αλπηα-carbon) to the distance (nucleophilic-oxygen/carbonyl-carbon) termed α/r; (2) The slope (S) of E vs. α/r plots depend on the nature of the hydroxy acids. Furthermore, plots of S values against the experimental rate values (log k_exp) show a linear correlation with a high correlation coefficient. The combined results suggest that hydroxy acids with low S values have high k_exp values due to enthalpic proximity effects.