The decoupled DFT-$\frac{1}{2}$ method for defect excitation energies

The DFT-$\frac{1}{2}$ method is a band gap correction with GW precision at a DFT computational cost. The method was also extended to correct the gap between defect levels, allowing for the calculation of optical transitions. However, this method fails when the atomic character of the occupied and unoccupied defect levels are similar as we illustrate by two examples, the tetrahedral hydrogen interstitial and the negatively charged vacancy in diamond. We solve this problem by decoupling the effect of the occupied and unoccupied defect levels and call this the decoupled DFT-$\frac{1}{2}$ method for defects.