Prediction of 4f²−4f¹5d¹ transition energy of Pr³⁺ in fluorides based on first-principles calculations and machine learning

The 4f²−4f¹5d¹ transition energy of Pr³⁺ in fluorides are utilized for various optical materials such as solid-state lasers, phosphors, and scintillators. Therefore, it is important to predict such energies of unknown materials for theoretical design of novel optical materials. In this study, we tried to predict the 4f²−4f¹5d1 transition energies of Pr³⁺ in fluorides based on first-principles calculations and machine learning. The first-principles calculations were performed based on the relativistic discrete variational multi-electron (DVME) method using the model clusters composed of the central Pr³⁺ and the anions closer than the nearest cation. Although the calculated 4f²−4f¹5d¹ transition energies of Pr³⁺ in fluorides showed a relatively good correlation with the experimental ones, the calculated values tend to be overestimated by ca. 2 eV. In order to improve the accuracy of the prediction, we used the calculated transition energies as an attribute for machine learning. As a result, the regression formula to predict the 4f²−4f¹5d¹ transition energy of Pr³⁺ in fluorides has been derived by machine learning using the calculated 4f²−4f¹5d¹ transition energy as well as some other electronic and structural parameters as the attributes. The accuracy of the prediction was significantly improved compared to the simple first-principles calculations.