Deciphering nonlinear optical properties in functionalized hexaphyrins <i>via</i> explainable machine learning

Over the years, several studies have aimed to elucidate why certain molecules show more enhanced nonlinear optical (NLO) properties than others. This knowledge is particularly valuable in the design of new NLO switches, where the ON and OFF states of the switch display markedly different NLO behaviors. In the literature, orbital contributions, aromaticity, planarity, and intramolecular charge transfer have been put forward as key factors in this regard. Based on our previous work on functionalized hexaphyrin-based redox switches, we aim at identifying through explainable machine learning the driving forces of the first hyperpolarizability related to the hyper-Rayleigh scattering (<i>β</i>_HRS) of <i>meso</i>-substituted and/or core-modified [26]- and [30]hexaphyrins. The significant correlation between <i>β</i>_HRS and the HOMO-LUMO energy gap can be further improved by including other orbitals as well as charge-transfer features in a 6-fold cross-validated kernel-ridge-regression model. Our Shapley additive explanations (SHAP) analysis shows that the charge transfer excitation length is more important for 30R systems, whereas the transition dipole moment between the ground and first excited state is one of the main contributors for 26R systems. We also demonstrate that, besides various hexaphyrin-based redox states, the ML model can describe to a large degree the <i>β</i>_HRS response of other hexaphyrins, differing in substitution pattern and topology (26D and 28M).