Origin of the β₁ parameter describing the nephelauxetic effect in transition metal ions with spin-forbidden emissions

Recently a new nephelauxetic parameter ${\beta }_1=\sqrt{{\left(\frac{C}{C_0}\right)}^2+{\left(\frac{B}{B_0}\right)}^2}$, where B, C (B₀, C₀) are the Racah parameters of the transition metal ions in a crystal (free state), was introduced to rationalize the variations in the energy of the ²E-⁴A₂ and ¹E-³A₂ spin-forbidden emission transitions that occur when ions with the d³ and d⁸ electron configurations are introduced as optically active impurities in a lattice. A critical analysis of the vast experimental data that is available in the archival literature shows that the energies of the above-mentioned transitions are linear functions of the β₁ parameter. In this work, we further confirm this empirical finding and provide a derivation of the observed linear dependence by considering the Tanabe-Sugano matrices for the corresponding electron configurations.