  A first-principles computational study is presented of various phases of
LiAlO$_2$.The $\beta$ and $\gamma$ tetrahedral phases are found to be very
close in energy with the $\gamma$ phase having the lowest energy. The
octahedral $\alpha$ phase is a high-pressure phase and the transition pressure
from the $\gamma$ and $\beta$ phases to $\alpha$ is determined to be about 1
GPa. The electronic band structures are determined using the quasiparticle
self-consistent (QS) $GW$ method. The effective masses of the band edges and
the nature of the band gaps are presented. The lowest energy $\gamma$ phase is
found to have a pseudodirect gap of 7.69 eV. The gap is direct at $\Gamma$ but
corresponds to a dipole forbidden transition. The imaginary part of the
dielectric function and the absorption coefficient are calculated in the
long-wavelength limit and the random phase approximation, without local field
or electron-hole interaction effects for each phase and their anisotropies are
discussed. Si doping on the Al site is investigated as a possible $n$-type
dopant in $\gamma$-LiAlO$_2$ using a 128 atom supercell corresponding to 3.125
\% Si on the Al sublattice in the generalized gradient approximation and a
smaller 16 atom cell with 25 \% Si in the QS$GW$ approximation. The Si is found
to significantly perturb the conduction band and lower the gap but a clearly
separated deep donor defect level is not found. However, the donor binding
energy is still expected to be relatively deep, of order a few 0.1 eV in the
hydrogenic effective mass approximation.
