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Unfolding optical transition weights of impurity materials for first-principles LCAO electronic structure calculations

A method to analyze optical transitions is developed by combining the Kubo-Greenwood formula with the unfolding method to construct an unfolded electronic band structure with optical transition weights, which allows us to investigate how optical transitions are perturbed by imperfections such as impurity, vacancy, and structural distortions. Based on the Kubo-Greenwood formula, we first calculate frequency-dependent optical conductivity based on the first-principles electronic structure calculations using the linear combinations of atomic orbitals. Benefiting from the atomic orbital basis sets, the frequency-dependent optical conductivity can be traced back to their individual components before summations over all of $k$ points and bands. As a result, optical transition weights of the material can be put on the unfolded electronic band structure to show contributions at different $k$ points and bands. This method is especially useful to study the effects of broken symmetry in the optical transitions due to presence of impurities in the materials. As a demonstration, decomposed optical transition weights of a monolayer Si-doped graphene are shown in the electronic band structure.