  We develop numerical methods to calculate electron dynamics in crystalline
solids in real-time time-dependent density functional theory employing
exchange-correlation potentials which reproduce band gap energies of
dielectrics; a meta generalized gradient approximation (meta-GGA) proposed by
Tran and Blaha [Phys. Rev. Lett. 102, 226401 (2009)] (TBm-BJ) and a hybrid
functional proposed by Heyd, Scuseria, and Ernzerhof [J. Chem. Phys. 118, 8207
(2003)] (HSE). In time evolution calculations employing the TB-mBJ potential,
we have found it necessary to adopt a predictor-corrector step for stable
time-evolution. Since energy functional is not known for the TB-mBJ potential,
we propose a method to evaluate electronic excitation energy without referring
to the energy functional. Calculations using the HSE hybrid functional is
computationally expensive due to the nonlocal Fock-like term. We develop a
computational method for the operation of the Fock-like term in Fourier space,
for which we employ massively parallel computers equipped with graphic
processing units. To demonstrate significances of utilizing potentials
providing correct band gap energies, we compare electronic excitations induced
by femtosecond laser pulses using the TB-mBJ, HSE, and a simple local density
approximation (LDA). At low laser intensities, electronic excitations are found
to be sensitive to the band gap energy: results using TB-mBJ and HSE are close
to each other, while the excitation of the LDA calculation is more intensive
than the others. At high laser intensities close to a damage threshold, we have
found that electronic excitation energies are similar among the three cases.
