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In this work we present theoretical calculations and analysis of the vibronicstructure of the spin-triplet optical transition in diamond nitrogen-vacancycentres. The electronic structure of the defect is described using accuratefirst-principles methods based on hybrid functionals. We devise a computationalmethodology to determine the coupling between electrons and phonons during anoptical transition in the dilute limit. As a result, our approach yields asmooth spectral function of electron-phonon coupling and includes bothquasi-localized and bulk phonons on equal footings. The luminescence lineshapeis determined via the generating function approach. We obtain a highly accuratedescription of the luminescence band, including all key parameters such as theHuang-Rhys factor, the Debye-Waller factor, and the frequency of the dominantphonon mode. More importantly, our work provides insight into the vibrationalstructure of nitrogen vacancy centres, in particular the role of local modesand vibrational resonances. In particular, we find that the pronounced mode at65 meV is a vibrational resonance, and we quantify localization properties ofthis mode. These excellent results for the benchmark diamond nitrogen-vacancycentre provide confidence that the procedure can be applied to other defects,including alternative systems that are being considered for applications inquantum information processing.

First-principles theory of the luminescence lineshape for the triplet transition in diamond NV centre