  The luminescence of fifteen representative Eu$^{2+}$-doped phosphors used for
white-LED and scintillation applications is studied through a Constrained
Density Functional Theory. Transition energies and Stokes shift are deduced
from differences of total energies between the ground and excited states of the
systems, in the absorption and emission geometries. The general applicability
of such methodology is first assessed: for this representative set, the
calculated absolute error with respect to experiment on absorption and emission
energies is within 0.3 eV. This set of compounds covers a wide range of
transition energies that extents from 1.7 to 3.5 eV. The information gained
from the relaxed geometries and total energies is further used to evaluate the
thermal barrier for the $4f-5d$ crossover, the full width at half-maximum of
the emission spectrum and the temperature shift of the emission peak, using a
one-dimensional configuration-coordinate model. The former results indicate
that the $4f-5d$ crossover cannot be the dominant mechanism for the thermal
quenching behavior of Eu$^{2+}$-doped phosphors and the latter results are
compared to available experimental data and yield a 30$\%$ mean absolute
relative error. Finally, a semi-empirical model used previously for
Ce$^{3+}$-doped hosts is adapted to Eu$^{2+}$-doped hosts and gives the
absorption and emission energies within 0.9 eV of experiment, underperforming
compared to the first-principles calculation.
