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The performance of semiconductor devices can be enhanced by understanding and characterizing the bound states of deep defects. However, simulating these states using conventional $\textit{ab initio}$ techniques, which are limited by small cell volumes, proves challenging due to their extended nature spanning tens of nanometers. Here, we present a semi-$\textit{ab initio}$ method for modeling the bound states of deep defects in semiconductors, applied to the negatively charged nitrogen vacancy (NV$^-$) center in diamond. We employ density functional theory (DFT) calculations to construct accurate potentials for an effective mass model and allowing us to unveil the structure of the bound hole states. We also develop a model to calculate the non-radiative capture cross-sections, which agrees with experiment within one order of magnitude. Finally, we present the first attempt at constructing the photoionization spectrum of NV$^0\rightarrow$ NV$^-$ + bound hole, showing that the electronic transitions of the bound holes can be distinguished from phonon sidebands. This approach can be adapted to other deep defects in semiconductors.