  We present a theoretical study of vibrational and vibronic properties of a
point defect in the dilute limit by means of first-principles density
functional theory calculations. As an exemplar we choose the negatively charged
nitrogen-vacancy center, a solid-state system that has served as a testbed for
many protocols of quantum technology. We achieve low effective concentrations
of defects by constructing dynamical matrices of large supercells containing
tens of thousands of atoms. The main goal of the paper is to calculate
luminescence and absorption lineshapes due to coupling to vibrational degrees
of freedom. The coupling to symmetric $a_1$ modes is computed via the
Huang-Rhys theory. Importantly, to include a nontrivial contribution of $e$
modes we develop an effective methodology to solve the multi-mode $E \otimes e$
Jahn-Teller problem. Our results show that for NV centers in diamond a proper
treatment of $e$ modes is particularly important for absorption. We obtain good
agreement with experiment for both luminescence and absorption. Finally, the
remaining shortcomings of the theoretical approach are critically reviewed. The
presented theoretical approach will benefit identification and future studies
of point defects in solids.
