Time-dependent chemical evolution during cloud formation: H$_2$-regulated chemistry in diffuse molecular cloud

We investigate the chemical evolution of a forming molecular cloud behind an interstellar shock wave. We conduct three-dimensional magnetohydrodynamics simulations of the converging flow of atomic gas, including a simple chemical network and tracer particles that move along the local velocity field. Then we perform detailed chemical network calculations along the trajectory of each tracer particle. The diffuse part of forming molecular clouds is CO-poor; i.e., H$_2$ and CO abundances do not correlate. In diffuse regions of $n_\mathrm{H}\lesssim 10^{3}\,\mathrm{cm^{-3}}$, we find that the abundances of hydrocarbons and oxygen-bearing molecules are determined by steady-state chemistry reflecting the local H$_2$ abundance, which is determined by the gas density along the trajectory. In denser regions, the abundances are affected by water ice formation, which changes the elemental abundance of carbon and oxygen (i.e., C/O ratio) in the gas phase. Assuming quasi-steady-state chemistry given the abundances of major molecules (e.g., H$_2$) from the simple network, we derive analytic solutions for molecular abundances, which reproduce the calculation results. We also calculate the molecular column densities based on the spatial distribution of tracer particles and their molecular abundances, and compare them with observations of diffuse molecular clouds. We find that the column densities of CH, CCH, and OH are linearly correlated with those of H$_2$, which supports the empirical relation used in the observations. On the other hand, the column density of HCO$^+$ shows non-linear dependence on the H$_2$ column density, reflecting the difference in HCO$^+$ formation paths in CO-poor and CO-rich regions.