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We report $ab$ $initio$ band diagram and optical absorption spectra ofhexagonal boron nitride ($h$-BN), focusing on unravelling how the completenessof basis set for $GW$ calculations and how electron-phonon interactions (EPIs)impact on them. The completeness of basis set, an issue which was seldomdiscussed in previous optical spectra calculations of $h$-BN, is found crucialin providing converged quasiparticle band gaps. In the comparison among threedifferent codes, we demonstrate that by including high-energy local orbitals inthe all-electron linearized augmented plane waves based $GW$ calculations, thequasiparticle 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 respectivelyat 0 K, and 6.60 eV and 5.98 eV respectively at 300 K. With clamped crystalstructure, the first peak of the absorption spectrum is at 6.07 eV, originatingfrom the direct exciton contributed by electron transitions around $K$ in theBrillouin zone. After including the EPIs-renormalized quasiparticles in theBethe-Salpeter equation, the exciton-phonon coupling shifts the first peak to5.83 eV at 300 K, lower than the experimental value of $\sim$6.00 eV. Thisaccuracy is acceptable to an $ab$ $initio$ description of excited states withno fitting parameter.

The influence of high-energy local orbitals and electron-phonon interactions on the band gaps and optical spectra of hexagonal boron nitride