computing collisional ionization equilibrium, as shown in Figure \[figcollision\], have been described by @lykins. It is determined by the balance between collisional ionization from the ground state and recombination by radiative, dielectronic, and three body recombination to all levels of the recombined species.

The photoionization case shown in Figure \[figphoto\] depicts the Active Galactic Nucleus spectral energy distribution (SED), described by @mathews, as a function of the ionization parameter $$\begin{aligned}
U & \equiv & \frac{\Phi_H}{n_H\;c}\ ,\end{aligned}$$ where $\Phi_H$ is the hydrogen-ionizing photon flux, $n_H$ is the density of hydrogen, and $c$ is the speed of light. There is only an indirect relationship between the gas kinetic temperature and the ionization of the gas in this case. Here, the level of ionization is determined by a balance between photoionization by the energetic continuum and the total recombination rate.

The lower panels of Figs. \[figcollision\] and \[figphoto\] show that the amount that the ionization increases due to DR suppression can be large — the ratio can easily exceed 1 dex. Clearly, these results demonstrate that density effects on the ionization balance need to be considered more precisely.

Conclusion
==========

We have investigated the effects of finite densities on the effective DR rate 