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In rare-earth semiconductors, wide ligand $p$ and rare-earth 5$d$ bandscoexist with localized, partially filled 4$f$ shells. A simultaneousdescription for both extended and localized states represents a significantchallenge for first-principles theories. Here, we combine an {\it ab initio}dynamical mean-field theory approach to strong local correlations with aperturbative application of the semi-local modified Becke-Johnson exchangepotential to correct the semiconducting gap. We apply this method to calculatethe electronic structure and optical response of the light rare-earthsesquioxides RE$_2$O$_3$ (RE= La, Ce, Pr and Nd). Our calculations correctlycapture a non-trivial evolution of the optical gap in RE$_2$O$_3$ due to aprogressive lowering of the 4$f$ states along the series and their multipletstructure. 2$p$ $-$ 4$f$ hybridization is found to induce a substantial upwardshift for the occupied 4$f$ states occurring within the $p-d$ gap, thusreducing the magnitude of the optical gap. We show that a characteristicplateau observed in the optical conductivity in the Pr and Nd sequioxides rightabove their absorption edge is a fingerprint of 4$f$ states located within the$p-d$ gap.

Combining semi-local exchange with dynamical mean-field theory: electronic structure and optical response of rare-earth sesquioxides