A presentation of the complex polarization propagator in the restricted open-shell self-consistent field approximation is given. It rests on a formulation of a resonant-convergent, first-order polarization propagator approach that makes it possible to directly calculate the X-ray absorption cross section at a particular frequency without explicitly addressing the excited states. The quality of the predicted X-ray spectra relates only to the type of density functional applied without any separate treatment of dynamical relaxation effects. The method is applied to the calculation of the near K-edge X-ray absorption fine structure spectra of allyl and copper phthalocyanine. Comparison is made between the spectra of the radicals and those of the corresponding cations and anions to assess the effect of the increase of electron charge in the frontier orbital. The method offers the possibility for unique assignment of symmetry-independent atoms. The overall excellent spectral agreement motivates the application of the method as a routine precise tool for analyzing X-ray absorption of large systems of technological interest.