Quasiparticle self-consistent <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>G</mml:mi><mml:mi>W</mml:mi></mml:mrow></mml:math> band structures and high-pressure phase transitions of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>Li</mml:mi><mml:mi>Ga</mml:mi><mml:msub><mml:mi mathvariant="normal">O</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>Na</mml:mi><mml:mi>Ga</mml:mi><mml:msub><mml:mi mathvariant="normal">O</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow></mml:math>

Quasiparticle self-consistent $GW$ calculations are presented for the band structures of $\mathrm{Li}\mathrm{Ga}{\mathrm{O}}_{2}$ and $\mathrm{Na}\mathrm{Ga}{\mathrm{O}}_{2}$ in the orthorhombic $Pna{2}_{1}$ tetrahedrally coordinated crystal structures, which are closely related to the wurtzite structure of ZnO. Symmetry labeling of the bands near the gap is carried out, and effective-mass tensors are extracted for the conduction-band minimum and crystal-field split valence-band maxima (VBM) at $\mathrm{\ensuremath{\Gamma}}$. The gap is found to be direct at $\mathrm{\ensuremath{\Gamma}}$ and is 5.81 eV in $\mathrm{Li}\mathrm{Ga}{\mathrm{O}}_{2}$ and 5.46 eV in $\mathrm{Na}\mathrm{Ga}{\mathrm{O}}_{2}$. Electron-phonon coupling zero-point normalization is estimated to lower these gaps by about $0.2\ifmmode\pm\else\textpm\fi{}0.1$ eV. Optical response functions are calculated within the independent-particle long-wavelength limit, and they show the expected anisotropy of the absorption onsets due to the crystal-field splitting of the VBM. The results show that both materials are promising candidates as ultrawide-gap semiconductors with wurtzite-based tetrahedrally bonded crystal structures. Direct transitions from the lowest conduction band to higher bands, relevant to $n$-type doped material and transparent conduction applications, are found to start only above 3.9 eV and are allowed for only one polarization, and several higher band transitions are forbidden by symmetry. Alternative crystal structures, such as $R\overline{3}m$ and a rocksalt-type phase with a tetragonally distorted $P4/mmm$ spacegroup, both with octahedral coordination of the cations, are also investigated. They are found to have higher energy but about 20% smaller volume per formula unit. The transition pressures to these phases are determined, and for $\mathrm{Li}\mathrm{Ga}{\mathrm{O}}_{2}$ they are found to be in good agreement with experimental studies. The $R\overline{3}m$ phase also has a comparably high but slightly indirect band gap, while the rocksalt-type phase is found to have a considerably smaller gap of about 3.1 eV in $\mathrm{Li}\mathrm{Ga}{\mathrm{O}}_{2}$ and 1.0 eV in $\mathrm{Na}\mathrm{Ga}{\mathrm{O}}_{2}$.