  Hydroxyapatite (HAp) is an important component of mammal bones and teeth,
being widely used in prosthetic implants. Despite the importance of HAp in
medicine, several promising applications involving this material e.g. in
photo-catalysis), depend on how well we understand its fundamental properties.
Among the ones that are either unknown or not known accurately we have the
electronic band structure and all that relates to it, including the band gap
width. We employ state-of-the-art methodologies, including density
hybrid-functional theory and many-body perturbation theory within the GW
approximation, to look at the optoelectronic properties of HAp. These methods
are also applied to the calculation of defect levels. We find that the use of a
mix of (semi-)local and exact exchange in the exchange-correlation functional,
brings a drastic improvement to the band structure. Important side-effects
include improvements in the description of dielectric and optical properties,
not only involving conduction band (excited) states, but also the valence. We
find that the highly dispersive conduction band bottom of HAp originates from
anti-bonding $\sigma^{*}$ states along the $\cdots\textrm{OH-OH-}\cdots$
infinite chain, suggesting the formation of a conductive 1D-ice phase. The
choice of the exchange-correlation treatment to the calculation of defect
levels was also investigated by using the OH-vacancy as testing-model. We find
that donor and acceptor transitions obtained within semi-local DFT differ from
those of hybrid-DFT by almost 2 eV. Such a large discrepancy emphasizes the
importance of using a high-quality description of the electron-electron
interactions in the calculation of electronic and optical transitions of
defects in HAp.
