The ADC( n) approximation scheme to the one-particle Green function is briefly outlined [ADC( n) stands for algebraic diagrammatic construction accurate to nth order of perturbation theory]. The solution of the equations can be done either by a pole search algorithm (PSA) or by the Davidson diagonalisation approach CDDA) whose problems and advantages are discussed. The results obtained for N 2 are compared. The methods are applied to interpret the ionisation spectra of p-quinodimethane, p-benzoquinone and to calculate the lowest ionisation energies of the transition metal diatomics CU2, Ag 2 , Cr2 and M0 2 . The ADC(4) equations are computationally very demanding and additional approximations need to be introduced in order to apply this method to any but the smallest systems. Such a simplified model is presented and applied to N2 and CS 2 .