  Photoionization cross sections and rate coefficients have been calculated for
all bound vibrational levels of the 1s$\sigma_{\mathrm{g}}$ state of
H$_{2}^{+}$, HD$^{+}$, and D$_{2}^{+}$. The Born-Oppenheimer approximation is
employed in our calculation of vibrationally-resolved photoionization cross
sections. Vibrationally-resolved and local thermal equilibrium photoionization
rate coefficients are presented for photon temperatures less than $50\,000$ K
and are found to be several orders of magnitude larger than previous results in
the literature. Analytic fits for the vibrationally-resolved and local thermal
equilibrium photoionization rate coefficients are provided. Near threshold
oscillations in the vibrationall-resolved photoionization are observed. A
benchmark set of photoionization cross sections are presented. Fixed-nuclei
photoionization cross sections are calculated using two-center true continuum
wave functions and are verified by comparison with previous calculations and
are found to be in excellent agreement in all cases. Data files for our set of
benchmark cross sections, rate coefficients, and fitting parameters for
H$_{2}^{+}$, HD$^{+}$, and D$_{2}^{+}$ are available on Zenodo under an
open-source Creative Commons Attribution license:
https://doi.org/10.5281/zenodo.8304060 .
