The role of neutral hydrogen in setting the abundances of molecular
  species in the Milky Way's diffuse interstellar medium. II. Comparison
  between observations and theoretical models
  We compare observations of HI from the Very Large Array (VLA) and the Arecibo
Observatory and observations of HCO$^+$ from the Atacama Large
Millimeter/submillimeter Array (ALMA) and the Northern Extended Millimeter
Array (NOEMA) in the diffuse ($A_V\lesssim1$) interstellar medium (ISM) to
predictions from a photodissociation region (PDR) chemical model and
multi-phase ISM simulations. Using a coarse grid of PDR models, we estimate the
density, FUV radiation field, and cosmic ray ionization rate (CRIR) for each
structure identified in HCO$^+$ and HI absorption. These structures fall into
two categories. Structures with $T_s<40~\mathrm{K}$, mostly with
$N(\mathrm{HCO^+})\lesssim10^{12}~\mathrm{cm^{-2}}$, are consistent with modest
density, FUV radiation field, and CRIR models, typical of the diffuse molecular
ISM. Structures with spin temperature $T_s>40~\mathrm{K}$, mostly with
$N(\mathrm{HCO^+})\gtrsim10^{12}~\mathrm{cm^{-2}}$, are consistent with high
density, FUV radiation field, and CRIR models, characteristic of environments
close to massive star formation. The latter are also found in directions with a
significant fraction of thermally unstable HI. In at least one case, we rule
out the PDR model parameters, suggesting that alternative mechanisms (e.g.,
non-equilibrium processes like turbulent dissipation and/or shocks) are
required to explain the observed HCO$^+$ in this direction. Similarly, while
our observations and simulations of the turbulent, multi-phase ISM agree that
HCO$^+$ formation occurs along sightlines with
$N(\mathrm{HI})\gtrsim10^{21}~\mathrm{cm^{-2}}$, the simulated data fail to
explain HCO$^+$ column densities
$\gtrsim\rm{few}\times10^{12}~\mathrm{cm^{-2}}$. Since a majority of our
sightlines with HCO$^+$ had such high column densities, this likely indicates
that non-equilibrium chemistry is important for these lines of sight.
