  Solar cells based on a light absorbing layer of the organometal halide
perovskite CH$_3$NH$_3$PbI$_3$ have recently reached 15% conversion efficiency,
though how these materials work remains largely unknown. We analyse the
electronic structure and optical properties within the quasiparticle
self-consistent GW approximation. While this compound bears some similarity to
conventional sp semiconductors, it also displays unique features. Quasiparticle
self-consistency is essential for an accurate description of the band
structure: bandgaps are much larger than what is predicted by the local density
approximation (LDA) or GW based on the LDA. Valence band dispersions are
modified in a very unusual manner. In addition, spin orbit coupling strongly
modifies the band structure and gives rise to unconventional dispersion
relations and a Dresselhaus splitting at the band edges. The average hole mass
is small, which accounts for the long diffusion lengths recently observed. The
surface ionisation potential (workfunction) is calculated to be 5.7 eV with
respect to the vacuum level, explaining efficient carrier transfer to TiO$_2$
and Au electrical contacts.
