  (Screened) hybrid functionals are being used more and more for solid-state
calculations. Usually the fraction alpha of Hartree-Fock exchange is kept fixed
during the calculation, however there is no single (universal) value for alpha
which systematically leads to satisfying accuracy. Instead, one could use a
property of the system under consideration to determine alpha and in this way
the functional would be more flexible and potentially more accurate. Recently,
it was proposed to use the static dielectric constant epsilon for the
calculation of alpha [Shimazaki and Asai, Chem. Phys. Lett. 466, 91 (2008) and
Marques et al., Phys. Rev. B 83, 035119 (2011)]. We explore this idea further
and propose a scheme where the connection between epsilon and alpha is
optimized based on experimental band gaps. epsilon, and thus alpha, is
recalculated at each iteration of the self-consistent procedure. We present
results for the band gap and lattice constant of various semiconductors and
insulators with this procedure. In addition, we show that this approach can
also be combined with a non-self-consistent hybrid approximation to speed up
the calculations considerably, while retaining an excellent accuracy in most
cases.
