  We complete the formulation of the standard model of first star formation by
exploring the possible impact of $\mathrm{LiH}$ cooling, which has been
neglected in previous simulations of non-linear collapse. Specifically, we find
that at redshift $z\gtrsim 5$, the cooling by $\mathrm{LiH}$ has no effect on
the thermal evolution of shocked primordial gas, and of collapsing primordial
gas into minihaloes or relic HII regions, even if the primordial lithium
abundance were enhanced by one order of magnitude. Adding the most important
lithium species to a minimum network of primordial chemistry, we demonstrate
that insufficient $\mathrm{LiH}$ is produced in all cases considered, about
$[\mathrm{LiH/Li}]\sim 10^{-9}$ for $T\lesssim 100$ K. Indeed, $\mathrm{LiH}$
cooling would only be marginally significant in shocked primordial gas for the
highly unlikely case that the $\mathrm{LiH}$ abundance were increased by nine
orders of magnitude, implying that $all$ lithium would have to be converted
into $\mathrm{LiH}$. In this study, photo-destruction processes are not
considered, and the collisional disassociation rate of $\mathrm{LiH}$ is
possibly underestimated, rendering our results an extreme upper limit.
Therefore, the cooling by $\mathrm{LiH}$ can safely be neglected for the
thermal evolution of Population~III star-forming gas.
