An $\mathrm{\textit{ab-initio}}$ solid state photoluminescence based on frequency constraint: NV$^{-}$ case study.

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 \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, presenting a connection between defects in solid-state and clusters; the first vibrationally resolved \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.