  Transition metal oxides have been one of the central objects in the studies
of electron correlation effects because of their rich variety of physical
properties mainly depending on the transition metal element. On the other hand,
exploiting the anion degrees of freedom is less popular but can be another
promising way to control properties of strongly correlated materials. In
particular, oxyhydrides offer a unique playground of strongly correlated
low-dimensional electronic structure, where the $s$ orbitals of hydrogen breaks
a chemical bond between the cation $t_{2g}$ orbitals. In this study, we
evaluate the effective interaction, i.e., the screened Coulomb interaction
parameters in low-energy effective models, for vanadium oxyhydrides
Sr$_{n+1}$V$_n$O$_{2n+1}$H$_n$ ($n=1,\infty$) using the constrained
random-phase approximation (cRPA). We find that the effective interaction in
the $t_{2g}$ model, where only the $t_{2g}$ orbitals are explicitly considered,
is strongly screened by the $e_g$ bands compared with that for oxides, because
the $e_g$ bands are much entangled with the $t_{2g}$ bands in the oxyhydrides.
On the other hand, the effective interaction is rather strong in the $d$ model,
where all the vanadium $d$ orbitals are explicitly considered, owing to a large
energy separation between the V-$d$ bands and the anion bands (O-$p$ and
H-$s$), because the O-$p$ states are stabilized by the existence of the
hydrogen atoms. These findings suggest that non-trivial and unique correlation
effects can take place in vanadium oxyhydrides.
