Within Hartree-Fock theory, the first vertical ionization energy (VIE) of a closed-shell system can be calculated from the HOMO eigenvalue (Koopmans' theorem). Since the change in the number of electrons and the correlation energy are not correctly evaluated in a single-determinant description, remarkable errors can be found. Please look at this thread for the extension of Koopmans' theorem to DFT:
VIEs can be calculated as the difference between the total electronic energies calculated for the starting compound (at its optimized geometry) and the corresponding cation (at the same geometry as the starting compound).
The adiabatic IE (AIE) can be calculated by taking into account the geometry relaxation of the cation. Hence, the ZPE-corrected total electronic energy of the fully relaxed cation should be considered in this case.