  We present a systematic study that clarifies validity and limitation of
current hybrid functionals in density functional theory for structural and
electronic properties of various semiconductors and insulators. The three
hybrid functionals, PBE0 by Perdew, Ernzerhof, and Becke, HSE by Heyd,
Sucseria, and Ernzerhof, and a long-range corrected (LC) functional, are
implemented in a well-established plane-wave-basis-set scheme combined with
norm-conserving pseudopotentials, thus enabling us to assess applicability of
each functional on equal footing to the properties of the materials. The
materials we have examined in this paper range from covalent to ionic materials
as well as a rare-gas solid whose energy gaps determined by experiments are in
the range of 0.6 eV - 14.2 eV: i.e., Ge, Si, BaTiO$_3$, $\beta$-GaN, diamond,
MgO, NaCl, LiCl, Kr, and LiF. We find that the calculated bulk moduli by the
hybrid functionals show better agreement with the experiments than the
generalized gradient approximation (GGA) provides, whereas the calculated
lattice constants by the hybrid functionals and GGA show comparable accuracy.
The calculated energy band gaps and the valence-band widths for the ten
prototype materials show substantial improvement using the hybrid functional
compared with GGA. In particular, it is found that the band gaps of the ionic
materials as well as the rare-gas solid are well reproduced by the LC-hybrid
functional, whereas those of covalent materials are well described by the HSE
functional. We also examine exchange effects due to short-range and long-range
components of the Coulomb interaction and propose an optimum recipe to the
short-range and long-range separation in treating the exchange energy.
