Photoionization cross sections have been computed using correlated initial and final target states obtained with standard ab initio techniques and using a numerical representation of the continuum scattering orbitals. Here, core ionization of small molecular systems will be focused on. A brief review of previous results obtained using these methods is given. New results are then given for the N 1s ionization of NO and the N 1s ionization of N₂. In the study of NO, the spin-coupling scheme which best describes the shape-resonant state and the energy dependence of the branching ratios for the singlet and triplet final ion states will be considered. In the study of N₂ core ionization, the effects of the coupling of nearly degenerate core levels and the effects of correlation on the importance of interchannel coupling will be considered.