typical of the values found at these densities in cosmological simulations of Pop. III.1 star formation [see e.g. @2008MNRAS.387.1021G], and account for the fact that the HD/H$_{2}$ ratio is elevated over the cosmological D/H ratio of $2.6 \times 10^{-5}$ [@mol08] owing to the effects of chemical fractionation [@g08]. In the case of D$^{+}$, fractionation is unimportant at our starting temperature, and so we simply set ${\rm D^{+}} / {\rm H^{+}} 
= 2.6 \times 10^{-5}$. In our Pop. III.2 simulations, we adopt the same initial H$^{+}$ and D$^{+}$ abundances, but set $x_{\rm H_{2}} = 3 \times 10^{-3}$ and $x_{\rm HD} = 3 \times 10^{-6}$, following @2008MNRAS.387.1021G. In both the Pop. III.1 and Pop. III.2 simulations, we assumed that all of the helium remained neutral, and set the initial abundances of all of our other tracked species to zero.

Within the BE spheres, we impose a turbulent velocity field that has a power spectrum of $P(k) \propto k^{-4}$. We assume that the turbulence considered here has its origin in gravitationally driven flows that arise as the gas and dark matter virialize in mini-halos (Wise & Abel 2007; Greif et al. 2008; Klessen & Hennebelle 2010). As the gas is compressible in nature 