  We report \textit{ab initio} calculations of the X $^2 \Sigma_{u}^+$, A
$^2\Pi_u$ and B $^2 \Sigma_{g}^+$ states of the Ca$_{2}^+$ dimer. All electron
CAS+MRCI calculations are performed for the X $^2 \Sigma_{u}^+$ and B $^2
\Sigma_{g}^+$ states, while valence CAS+MRCI calculations using an effective
core potential are used to describe the A $^2\Pi_u$ state. A double well is
found in the B $^2 \Sigma_{g}^+$ state. Spectroscopic constants, vibrational
levels, transition moments and radiative lifetimes are calculated for the most
abundant isotope of calcium ($^{40}$Ca). The static dipole and quadrupole
polarizabilities, and the leading order van der Waals coefficients are also
calculated for all three states.
