  We apply density functional theory to investigate interactions between
electronic and vibrational states in crystal defects with multi-mode dynamical
Jahn-Teller (JT) systems. Our focus is on transitions between orbital singlet
and degenerate orbital doublet characterized by $E \otimes (e \oplus e \oplus
\cdots)$ JT coupling, which frequently occurs in crystal defects that are
investigated for applications in quantum information science. We utilize a
recently developed methodology to model the photoluminescence (PL) spectrum of
the negatively charged split nickel-vacancy center (NiV$^{-}$) in diamond,
where JT-active modes significantly influence electron-phonon interactions. Our
results validate the effectiveness of the methodology in accurately reproducing
the observed 1.4 eV PL lineshape. The strong agreement between our theoretical
predictions and experimental observations reinforces the identification of the
1.4 eV PL center with the NiV$^{-}$ complex. This study highlights the critical
role of JT-active modes in affecting optical lineshapes and demonstrates the
power of advanced techniques for modeling optical properties in complex systems
with multiple JT-active frequencies.
