Atomistic Modeling of Rare Earth Ions in Photonic Materials

Rare-earth ions (REIs), especially trivalent lanthanides (Ln ${}^{3+}$ ), are central to photonic technologies due to sharp intra-4f transitions, long lifetimes, and host-insensitive emission. However, modeling REIs remains challenging due to localized 4f orbitals, strong electron correlation, and multiplet structures. This review summarizes atomistic modeling strategies combining quantum chemistry and machine learning (ML). Traditional methods, DFT+ $U$ , hybrid functionals (HSE06), $\mathrm{GW}$ , and DMFT, are benchmarked; for example, hybrid DFT reproduces 4f-5d gaps in Ce ${}^{3+}$ :YAG within 0.1-0.2 eV. Wavefunction methods like CASSCF and CASPT2 capture Stark splittings and transition strengths in Eu ${}^{3+}$ :Y ${}_2$ SiO ${}_5$ . ML models trained on DFT data predict bandgaps with  0.2 eV error and aid inverse design of Ce-doped phosphors with 505 nm emission and 60% retention at 640 K. Unlike prior reviews, this work bridges high-level quantum modeling with ML-driven screening across key applications: upconversion (Yb ${}^{3+}$ -Er ${}^{3+}$ :NaYF ${}_4$ ), lasers (Nd ${}^{3+}$ :YAG), quantum memories (Pr ${}^{3+}$ :Y ${}_2$ SiO ${}_5$ ), and sensors (SrAl ${}_2$ O ${}_4$ :Eu ${}^{2+}$ , Dy ${}^{3+}$ ). Covering over 20 REI-host systems, it integrates insights from DFT, Monte Carlo, MD, and ML potentials. The review thus provides both a methodological guide and a resource for designing next-generation REI-based photonic materials.