  In calculations of the ionization state, one is often forced to choose
between the Case A recombination coefficient $\alpha_{\rm A}$ (sum over
recombinations to all hydrogen states) or the Case B recombination coefficient
$\alpha_{\rm B}$ (sum over all hydrogen states except the ground state). If the
cloud is optically thick to ionizing photons, $\alpha_{\rm B}$ is usually
adopted on the basis of the "on-the-spot" approximation, wherein recombinations
to the ground state are ignored because they produce ionizing photons absorbed
nearby. In the opposite case of an optically thin cloud, one would expect the
Case A recombination coefficient to better describe the effective recombination
rate in the cloud. In this paper, I derive an analytical expression for the
effective recombination coefficient in a gas cloud of arbitrary optical
thickness which transitions from $\alpha_{\rm A}$ to $\alpha_{\rm B}$ as the
optical thickness increases. The results can be readily implemented in
numerical simulations and semi-analytical calculations.
