Role of the H$_2^+$ channel in the primordial star formation under
  strong radiation field and the critical intensity for the supermassive star
  formation
  We investigate the role of the H_2^+ channel on H_2 molecule formation during
the collapse of primordial gas clouds immersed in strong radiation fields which
are assumed to have the shape of a diluted black-body spectra with temperature
T_rad. Since the photodissociation rate of H_2^+ depends on its level
population, we take full account of the vibrationally-resolved H_2^+ kinetics.
We find that in clouds under soft but intense radiation fields with spectral
temperature T_rad < 7000 K, the H_2^+ channel is the dominant H_2 formation
process. On the other hand, for harder spectra with T_rad > 7000 K, the H^-
channel takes over H_2^+ in the production of molecular hydrogen. We calculate
the critical radiation intensity needed for supermassive star formation by
direct collapse and examine its dependence on the H_2^+ level population. Under
the assumption of local thermodynamic equilibrium (LTE) level population, the
critical intensity is underestimated by a factor of a few for soft spectra with
T_rad < 7000 K. For harder spectra, the value of the critical intensity is not
affected by the level population of H_2^+. This result justifies previous
estimates of the critical intensity assuming LTE populations since radiation
sources like young and/or metal-poor galaxies are predicted to have rather hard
spectra.
