  Population III stars are the first stars in the Universe to form at z=20-30
out of a pure hydrogen and helium gas in minihalos of 10^5-10^6 M$_\odot$ .
Cooling and fragmentation is thus regulated via molecular hydrogen. At
densities above 10^8 cm$^{-3}$, the three-body H2 formation rates are
particularly important for making the gas fully molecular. These rates were
considered to be uncertain by at least a few orders of magnitude. We explore
the impact of new accurate three-body H2 formation rates derived by Forrey
(2013) for three different minihalos, and compare to the results obtained with
three-body rates employed in previous studies. The calculations are performed
with the cosmological hydrodynamics code ENZO (release 2.2) coupled with the
chemistry package KROME (including a network for primordial chemistry), which
was previously shown to be accurate in high resolution simulations. While the
new rates can shift the point where the gas becomes fully molecular, leading to
a different thermal evolution, there is no trivial trend in how this occurs.
While one might naively expect the results to be inbetween the calculations
based on Palla et al. (1983) and Abel et al. (2002), the behavior can be close
to the former or the latter depending on the dark matter halo that is explored.
We conclude that employing the correct three-body rates is about as equally
important as the use of appropriate initial conditions, and that the resulting
thermal evolution needs to be calculated for every halo individually.
