  The density functional theory (DFT)+U method based on the linear response
(LR) theory was applied to investigate the electronic structures of Co-based
ternary full Heusler alloy Co$_2Y$Si for exploring half-metallic (HM)
ferromagnets with a wide HM gap. The LR-based DFT+U calculations tend to obtain
a reasonable correlation parameter for $Y$ site, while the correlation of Co
site misleads to the unphysical ground state due to the overestimated parameter
value that arises from the delocalized electronic structure of Co. Furthermore,
we found that the HM gap of Co$_2$MnSi originates from Co $e_u$ orbital in the
conduction state and Co-Mn hybridizing $t_{2g}$ orbital in the valence state
around the Fermi energy. This means that the HM gap is a tunable property by
selecting the $Y$ element and/or mixing several elements into the $Y$ site
through $t_{2g}$ atomic-orbital coupling. Our LR-based DFT+U method was
extended to other ternary Co$_2Y$Si and quaternary Co$_2$($Y$,Mn)Si. We found
that Co$_2$(Ti$_{0.25}$,Mn$_{0.75}$)Si and Co$_2$(Fe$_{0.25}$,Mn$_{0.75}$)Si
show HM nature, with the Fermi energy being at almost the center of the
minority band gap, which leads to high thermal stability.
