We present numerical computations and analysis of atomic to molecular (HI-to-H$_2$) transitions in cool ($sim$100 K) low-metallicity dust-free (primordial) gas, in which molecule formation occurs via cosmic-ray driven negative ion chemistry, and removal is by a combination of far-UV photodissociation and cosmic-ray ionization and dissociation. For any gas temperature, the behavior depends on the ratio of the Lyman-Werner (LW) band FUV intensity to gas density, $I_{rm LW}/n$, and the ratio of the cosmic-ray ionization rate to the gas density, $zeta/n$. We present sets of HI-to-H$_2$ abundance profiles for a wide range of $zeta/n$ and $I_{rm LW}/n$, for dust-free gas. We determine the conditions for which H$_2$ absorption line self-shielding in optically thick clouds enables a transition from atomic to molecular form for ionization-driven chemistry. We also examine the effects of cosmic-ray energy losses on the atomic and molecular density profiles and transition points. For a unit Galactic interstellar FUV field intensity ($I_{rm LW}=1$) with LW flux $2.07times 10^7$ photons cm$^{-2}$ s$^{-1}$, and a uniform cosmic-ray ionization rate $zeta=10^{-16}$ s$^{-1}$, an HI-to-H$_2$ transition occurs at a total hydrogen gas column density of $4times 10^{21}$ cm$^{-2}$, within $3times 10^7$ yr, for a gas volume density of $n=10^6$ cm$^{-3}$ at 100 K. For these parameters, the dust-free limit obtains for a dust-to-gas ratio Z$^prime_d lesssim 10^{-5}$, which may be reached for overall metallicities $Z^primelesssim 0.01$ relative to Galactic solar values.
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