approximation[^3]. [Since the surface area of a Strömgren sphere is proportional to $\dot{N}^{2/3}$ where $\dot{N}$ is photoionization rate (see Section \[stroemgren\]), treating a source group as a point source underestimates the surface area of ionized regions. We will explore this effect in our tests. ]{}

        RRB        DRR        CIR            RCRB        DRCR       CICR            CECR        BCR   CCR   CS
  --------------- ----- --------------- --------------- ------ --------------- --------------- ----- ----- -----
   (4), (5), (4)   (2)   (7), (7), (1)   (4), (5), (4)   (3)    (3), (3), (3)   (3), (3), (3)   (4)   (6)   (8)

\

\(1) @abel97; (2) @aldrovandi73; (3) @cen92; (4) @hummer94; (5) @hummer98; (6) @ikeuchi86; (7) @janev87; (8) @agnagn

Non-equilibrium chemistry {#sec:chem}
-------------------------

We solve the non-equilibrium chemistry for e, ${\ensuremath{\mbox{{H$\,${\sc i}}}}}$, ${\mbox{{H$\,${\sc ii}}}}$, ${\mbox{{He$\,${\sc i}}}}$, ${\mbox{{He$\,${\sc ii}}}}$, and ${\mbox{{He$\,${\sc iii}}}}$ implicitly. Note that since we employ the on-the-spot approximation, we use ‘Case B’ recombination coefficients to calculate recombination rates of ${\mbox{{H$\,${\sc ii}}}}$, ${\mbox{{He$\,${\sc ii}}}}$, and ${\mbox{{He$\,${\sc iii}}}}$ throughout this paper.

Using the optical depth obtained by the methods described in Section \[sec:smart\] or \[sec:start\], the photoionization rates of ${\ensuremath{\mbox{{H$\,${\sc i}}}}}$, ${\mbox{{He$\,${\sc i}}}}$, and ${\mbox{{He$\,${\sc ii}}}}$ in each mesh are given by $$\Gamma_i = \sum_\alpha \Gamma_{i, \alpha},$$ where $\Gamma_{i, \alpha}$ denotes the radiative contribution from a radiation 