Hartree-Fock ab initio calculations have been performed on electronic energy surfaces of the ground and the core hole state species of N2, O2 and NO in order to investigate vibrational structure of their high resolution ESCA spectra. The importance of relaxation effects in predicting geometry changes following core hole photoionization is known and was verified here. As a new feature the influence of relaxation on lineshapes is studied. Both the conventional Franck-Condon analysis and a method for calculating linewidths of photoelectron spectra from the gradient of the hole state energy curve are employed. Two major conclusions are made: (i) Agreement with experiment using the Hartee-Fock scheme can be obtained only if the core holes are treated localized. (ii) The vibrational effects for the two multiplet split components in the paramagnetic core hole ions NO+ and O2+ are different, and these effects are reflected in the ESCA linewidths. Ab initio calculations within the Hartree-Fock formalism have been carried out on the electronic energy surfaces of the ground and the core hole states of N2, O2 and NO. The differences in geometries and force constants between the ground and the ionized species follow a systematic pattern [4-7,15]. The results of these studies are used to compute high resolution ESCA spectra. Two interesting features occur. 1. (i)The agreement with the experimental spectra can only be achieved if the core holes are treated as localized in the Hartree-Fock scheme. This is analogous to recent results on angular distributions of molecular-photoelectrons obtained by Dill et a. [45]. Even though correlation corrections to symmetry restricted calculations can in principle yield good answers, limited configuration interaction treatments can be misleading for transition energies [46,47] and presumably for predicting lineshapes. 2. (ii) The linewidths of the two multiplet-split components in the paramagnetic molecules O2 and NO should not be expected to be the same a priori. This can play an important role in interpretation of high resolution ESCA spectra of some paramagnetic transition metal compounds which also exhibit a substantial exchange induced splitting [47-51]. However, in general, to obtain a better agreement between theory and experiment both more advanced ab initio calculations and more detailed account of anharmonicity in the Franck-Condon analysis are required. Coupled with the development of more advanced instrumentation, such calculations are presently in progress. © 1979.