  A prerequisite for laser cooling a molecular anion, which has not been
achieved so far, is the precise knowledge of the relevant transition
frequencies in the cooling scheme. To determine these frequencies we present a
versatile method that uses one pump and one photodetachment light beam. We
apply this approach to C$_{2}^{-}$ and study the laser cooling transitions
between the electronic ground state and the second electronic excited state in
their respective vibrational ground levels, $B ^{2} \Sigma _{u} ^{+}(v=0)
\leftarrow X ^{2} \Sigma _{g} ^{+}(v=0) $. Measurements of the R(0), R(2), and
P(2) transitions are presented, which determine the transition frequencies with
a wavemeter-based accuracy of $0.7\times10^{-3}$ cm$^{-1}$ or 20 MHz. The
spin-rotation splitting is resolved, which allows for a more precise
determination of the splitting constants to $\gamma ' = 7.15(19)\times10^{-3}$
cm$^{-1}$ and $\gamma '' = 4.10(27)\times10^{-3}$ cm$^{-1}$. These results are
used to characterize the ions in the cryogenic 16-pole wire trap employed in
this experiment. The translational and rotational temperature of the ions
cooled by helium buffer gas are derived from the Doppler widths and the
amplitude ratios of the measured transitions. The results support the common
observation that the translational temperature is higher than the buffer gas
temperature due to collisional heating under micromotion, in particular at low
temperatures. Additionally, a rotational temperature significantly lower than
the translational is measured, which agrees with the notion that the mass
weighted collision temperature of the C$_{2}^{-}$-He system defines the
internal rotational state population.
