Exploring the non-linearity of linear free energy relationships and using machine learning to predict substituent effects The Hammett equation has been a method widely used to analyze linear free energy relationships and the effects substituents produce for a molecule. Through computational analysis using 4 different backbones with 119 substituents discrepancies have been observed that are not easily explained by the original concept of the substituent and reaction constants relating to the molecular energy. When the only property changed is the substituent the energies should exhibit a 1 to 1 relationship regardless of backbone. Instead, non-insignificant variations are observed when comparing the reaction energy, Gibbs free energy and enthalpies. Previous studies1,2 investigating substituent properties and their effects on reaction energies found a strong relationship between the reaction energy and the x-component of the dipole moment attained from the analysis of the electron density. In this work, these relationships were greatly diminished when using a larger set of substituents (119). Machine learning methods were utilized, using 56 substituent property features. Linear regression, random forest, and symbolic regression models were examine. Best results were seen using symbolic regression generating the equation

\(-0.02481276\,e^{\left(-\frac{2}{(\alpha_{zz}^\rho)^2(H_c)^2(\mu_x^c)^2\,e^{\left(j-0.581364556795853\frac{\mu_y^c}{DI(R,H)^2+\alpha_{zx}^c}-\frac{\mu_y^c}{\lambda_{3,c}}\right)}}\right)}\)

\(j=-0.581364556795858\frac{Q_{yz}(R)}{\alpha_{zx}^\rho}-0.581364556795858\,Q_{zz}\lambda_{2,c}\)

Which generated r2 values of 0.8200 for Bicyclo pentane, 0.7223 for bicyclo octane and, 0.7379 for bicyclo undecane, as well as 0.7389, 0.7008 and 0.7076 for the three adamantane orientations used.

References
(1) Smith, A. P. Assessing and Characterizing the Inductive Effect through Silicon Containing Backbones and on Silicon Reactivity, 2014. http://knowledgecommons.lakeheadu.ca/handle/2453/542.
(2) DeGrazia, A. Do QTAIM Substituent Dipole Moment Descriptors Account for the Iductive Effect in Larger Bicyclo-Type Systems?, 2021.