  A major challenge in creating a quantum computer is to find a quantum system
that can be used to implement the qubits. For this purpose, deep centers are
prominent candidates, and ab initio calculations are one of the most important
tools to theoretically study their properties. However, these calculations are
highly involved, due to the large supercell needed, and the computational cost
can be even larger when one goes beyond the Kohn-Sham scheme to correct the
band gap problem and achieve good accuracy. In this work, we present a method
that overcomes these problems and provides the optical transition energies as a
difference of Kohn-Sham eigenvalues; and even more, provides a complete and
accurate band structure of the defect in the semiconductor. Despite the
original motivations, the presented methodology is a general procedure, which
can be used to systematically study the optical transitions between localized
levels within the band gap of any system. The method is an extension of the
low-cost and parameter-free DFT-1/2 approximate quasi-particle correction, and
allows it to be applied in the study of complex defects. As a benchmark, we
apply the method to the NV$^-$ center in diamond. The agreement with
experiments is remarkable, with an accuracy of 0.1 eV. The band structure
agrees with the expected qualitative features of this system, and thus provides
a good intuitive physical picture by itself.
