  Equilibrium crystal structures, electron band dispersions and band gap values
of layered GaSe and InSe semiconductors, each being represented by four
polytypes, are studied via first-principles calculations within the density
functional theory (DFT). A number of practical algorithms to take into account
dispersion interactions are tested, from empirical Grimme corrections to
many-body dispersion schemes. Due to the utmost technical accuracy achieved in
the calculations, nearly degenerate energy-volume curves of different polytypes
are resolved, and the conclusions concerning the relative stability of
competing polytypes drawn. The predictions are done as for how the equilibrium
between different polytypes will be shifted under the effect of hydrostatic
pressure. The band structures are inspected under the angle of identifying
features specific for different polytypes, and with respect to modifications of
the band dispersions brought about by the use of modified Becke-Johnson (mBJ)
scheme for the exchange-correlation (XC) potential. As another way to improve
the predictions of band gaps values, hybrid functional calculations according
to the HSE06 scheme are performed for the band structures, and the relation
with the mBJ results discussed. Both methods nicely agree with experimental
results and with state-of-the-art GW calculations. Some discrepancies are
identified in cases of close competition between the direct and indirect gap
(e.g., in GaSe); moreover, the accurate placement of bands revealing relatively
localized states is slightly different according to mBJ and HSE06 schemes.
