  The negative hydrogen ion H$^-$ is, almost without exception, treated in
local thermodynamic equilibrium (LTE) in the modelling of F, G, and K stars,
where it is the dominant opacity source in the visual spectral region. This
assumption rests in practice on a study from the 1960s. Since that work,
knowledge of relevant atomic processes and theoretical calculations of stellar
atmospheres and their spectra have advanced significantly, but this question
has not been reexamined. We present calculations based on a slightly modified
analytical model that includes H, H$_2$, and H$^-$, together with modern atomic
data and a grid of 1D LTE theoretical stellar atmosphere models with stellar
parameters ranging from T$_\mathrm{eff} = 4000$ to 7000~K, $\log{g} = 1$ to 5
cm/s$^2$, and [Fe/H]$=-3$ to 0. We find direct non-LTE effects on populations
in spectrum-forming regions, continua, and spectral lines of about 1-2% in
stars with higher T$_\mathrm{eff}$ and/or lower $\log g$. Effects in models for
solar parameters are smaller by a factor of 10, about 0.1-0.2%, and are
practically absent in models with lower T$_\mathrm{eff}$ and/or higher $\log
g$. These departures from LTE found in our calculations originate from the
radiative recombination of electrons with hydrogen to form H$^-$ exceeding
photodetachment, that is, overrecombination. Modern atomic data are not a
source of significant differences compared to the previous work, although
detailed data for processes on H$_2$ resolved with vibrational and rotational
states provide a more complete and complex picture of the role of H$_2$ in the
equilibrium of H$^-$. In the context of modern studies of stellar spectra at
the percent level, our results suggest that this question requires further
attention, including a more extensive reaction network, and indirect effects
due to non-LTE electron populations.
