  In the cold neutral medium, high out-of-equilibrium temperatures are created
by intermittent dissipation processes, including shocks, viscous heating, and
ambipolar diffusion. The high-temperature excursions are thought to explain the
enhanced abundance of CH$^{+}$ observed along diffuse molecular sight-lines.
Intermittent high temperatures should also have an impact on H$_2$ line
luminosities. We carry out simulations of MHD turbulence in molecular clouds
including heating and cooling, and post-process them to study H$_2$ line
emission and hot-gas chemistry, particularly the formation of CH$^+$. We
explore multiple magnetic field strengths and equations of state. We use a new
H$_2$ cooling function for $n_{\rm H} \leq 10^5\,{\rm cm}^{-3}$, $T\leq
5000\,{\rm K}$, and variable H$_2$ fraction. We make two important simplifying
assumptions: (i) the ${\rm H}_2/{\rm H}$ fraction is fixed everywhere, and (ii)
we exclude from our analysis regions where the ion-neutral drift velocity is
calculated to be greater than 5 km/s. Our models produce H$_2$ emission lines
in accord with many observations, although extra excitation mechanisms are
required in some clouds. For realistic r.m.s. magnetic field strengths
($\approx 10$ $\mu$G) and velocity dispersions, we reproduce observed CH$^+$
abundances. These findings contrast with those of Valdivia et al. (2017).
Comparison of predicted dust polarization with observations by {\it Planck}
suggests that the mean field $\gtrsim 5 \mu$G, so that the turbulence is
sub-Alfv\'enic. We recommend future work treating ions and neutrals as separate
fluids to more accurately capture the effects of ambipolar diffusion on CH$^+$
abundance.
