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We investigate the operation of the chemothermal instability in primordialstar-forming clouds with a suite of three-dimensional, moving-mesh simulations.In line with previous studies, we find that the gas at the centre ofhigh-redshift minihaloes becomes chemothermally unstable as three-bodyreactions convert the atomic hydrogen into a fully molecular gas. Thecompetition between the increasing rate at which the gas cools and theincreasing optical depth to H2 line emission creates a characteristic dip inthe cooling time over the free-fall time on a scale of 100 au. As a result, thefree-fall time decreases to below the sound-crossing time, and the cloud maybecome gravitationally unstable and fragment on a scale of a few tens of auduring the initial free-fall phase. In three of the nine haloes investigated,secondary clumps condense out of the parent cloud, which will likely collapsein their own right before they are accreted by the primary clump. In the otherhaloes, fragmentation at such an early stage is less likely. However, giventhat previous simulations have shown that the infall velocity decreasessubstantially once the gas becomes rotationally supported, the amount of timeavailable for perturbations to develop may be much greater than is evident fromthe limited period of time simulated here.

On the operation of the chemothermal instability in primordial star-forming clouds