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The electronic structures of α-BiFeO3 are calculated by using a full-potential linearized-augmented-plane-wave method. We employed the local-density approximation (LDA) with the modified Becke-Johnson (mBJ) exchange potential and the LDA + U method. The indirect (direct) bandgap of 2.24 (2.44) eV obtained by LDA + U method is in good agreement with an experiment, while the mBJ potential produces the indirect bandgap of 2.55 eV, and the direct bandgap is slightly larger than the indirect one. The discrepancy between the experimental x-ray spectra and the calculated Fe-3d and O-2p density of states were revealed to be due to the effects of the core hole. The core-hole effects are also responsible for the smaller bandgap in x-ray spectroscopy than the optical spectroscopy. The calculated valence-charge density and the bonding character obtained by LDA + U method also provides the stronger ionic character of the compound than the mBJ potential. Although the mBJ method is very efficient one, it is still very time consuming compared to the LDA + U method. The most suitable exchange-correlation potential for α-BiFeO3 is the LDA + U. Therefore, it is better to use the LDA + U method for the electronic-structure calculations of BiFeO3 compound not only for reducing the calculational time but also for better description of bandgaps and some physical properties. From the similar calculations carried out for transition-metal monoxide system it was found that the inadequacy of using the mBJ potential for the description of the localized 3d-states is rather universal.