  By combining experimental photoluminescence excitation spectroscopy and
calculations based on density functional theory and many-body Green's
functions, the most efficient excitation channels of infra-red (IR) emission
from Mn-Mg$_{k}$ paramagnetic complexes stabilized in GaN:(Mn,Mg) are here
identified. Moreover, a Tanabe-Sugano energy diagram for 3$d^{2}$Mn$ ^{5+}$ is
reconstructed and Mn-Mg$_{3}$ are singled out as the predominant configurations
responsible for the IR emission. The correlation of intensity of the individual
emission lines as a function of temperature and excitation energy, allows
assigning them to well defined and specific optical transitions.
