  Context. Rigorous modeling of the \ion{Ba}{ii} ${\lambda}4554$ formation is
potentially interesting since this strongly polarized line forms in the solar
chromosphere where the magnetic field is rather poorly known. Aims. To
investigate the role of isotropic collisions with neutral hydrogen in the
formation of the polarized \ion{Ba}{ii} ${\lambda}4554$ line and, thus, in the
determination of the magnetic field. Methods. Multipole relaxation and transfer
rates of the $d$ and p-states of \ion{Ba}{ii} by isotropic collisions with
neutral hydrogen are calculated. We consider a plane parallel layer of
\ion{Ba}{ii} situated at the low chromosphere and anisotropically illuminated
from below which produces linear polarization in the ${\lambda}4554$ line by
scattering processes. To compute that polarization, we solve the statistical
equilibrium equations for \ion{Ba}{ii} levels including collisions, radiation
and magnetic field effects. Results. Variation laws of the relaxation and
transfer rates with hydrogen number density $n_{\textrm {\scriptsize H}}$ and
temperature are deduced. The polarization of the ${\lambda}4554$ line is
clearly affected due to isotropic collisions with neutral hydrogen although the
collisional depolarization of its upper level $^2P_{3/2}$ is negligible. This
is because the alignment of the metastable levels $^2D_{3/2}$ and $^2D_{5/2}$
of the \ion{Ba}{ii} are vulnerable to collisions. At the height of formation of
the ${\lambda}4554$ line where $n_{\textrm {\scriptsize H}} \sim 2 \times
10^{14}$ cm$^{-3}$, we find that the neglecting of the collisions induces
inaccuracy of $\sim$ 25% on the calculation of the polarization and $\sim$ 35 %
inaccuracy on microturbulent magnetic field determination.
