  We report $ab$ $initio$ band diagram and optical absorption spectra of
hexagonal boron nitride ($h$-BN), focusing on unravelling how the completeness
of basis set for $GW$ calculations and how electron-phonon interactions (EPIs)
impact on them. The completeness of basis set, an issue which was seldom
discussed in previous optical spectra calculations of $h$-BN, is found crucial
in providing converged quasiparticle band gaps. In the comparison among three
different codes, we demonstrate that by including high-energy local orbitals in
the all-electron linearized augmented plane waves based $GW$ calculations, the
quasiparticle direct and fundamental indirect band gaps are widened by
$\sim$0.2 eV, giving values of 6.81 eV and 6.25 eV respectively at the $GW_0$
level. EPIs, on the other hand, reduce them to 6.62 eV and 6.03 eV respectively
at 0 K, and 6.60 eV and 5.98 eV respectively at 300 K. With clamped crystal
structure, the first peak of the absorption spectrum is at 6.07 eV, originating
from the direct exciton contributed by electron transitions around $K$ in the
Brillouin zone. After including the EPIs-renormalized quasiparticles in the
Bethe-Salpeter equation, the exciton-phonon coupling shifts the first peak to
5.83 eV at 300 K, lower than the experimental value of $\sim$6.00 eV. This
accuracy is acceptable to an $ab$ $initio$ description of excited states with
no fitting parameter.
