An $\mathrm{\textit{ab-initio}}$ effective solid state photoluminescence by frequency constraint of cluster calculation

Measuring the photoluminescence of defects in crystals is a common experimental technique for analysis and identification. However, current theoretical simulations typically require the simulation of a large number of atoms to eliminate finite size effects, which discourages computationally expensive excited state methods. We show how to extract the room-temperature photoluminescence spectra of defect centres in bulk from an $\mathrm{\textit{ab-initio}}$ simulation of a defect in small clusters. The finite size effect of small clusters manifests as strong coupling to low frequency vibrational modes. We find that removing vibrations below a cutoff frequency determined by constrained optimization returns the main features of the solid state photoluminescence spectrum. This strategy is illustrated for an NV$^{-}$ defect in diamond, presenting a connection between defects in solid state and clusters; the first vibrationally resolved $\mathrm{\textit{ab-initio}}$ photoluminescence spectrum of an NV$^{-}$ defect in a nanodiamond; and an alternative technique for simulating photoluminescence for solid state defects utilizing more accurate excited state methods.