Influence of the First Cation A of A₂SiF₆:Mn⁴⁺ (A = K, Rb, Cs) Phosphors on Their Geometric Structures and the Optical Transition Energies: First-Principles Analysis

First-principles calculations are performed for Mn⁴⁺-doped A₂SiF₆ (where A = K, Rb, and Cs) to investigate the influence of A-site cations on the geometric structures and optical transition energies of the ²E and ⁴T₂ excited states. The geometric structures of the ⁴A₂ ground state and the ²E and ⁴T₂ excited states were successfully modeled using the delta self-consistent field (ΔSCF) method and their optical transition energies were evaluated for Mn⁴⁺-doped A₂SiF₆, which are in good agreement with the experimental values. Notably, the relationships between the zero-phonon line (ZPL) energy of the ⁴T₂ state and the bond length of the Mn⁴⁺-F⁻ bond are investigated in detail. The challenges of computational convergence were effectively addressed through the proposed fractional particle occupancy schemes for modeling the ⁴T₂ state. The computational techniques developed in this study can be applied to other 3d³ activator ion/host matrix combinations, providing valuable insights into their structural and optical properties.