We have theoretically investigated the double core-hole (DCH) states of the open-shell molecules and examined the possibility of DCH spectroscopy by means of X-ray two-photon photoelectron spectroscopy (XTPPS). Energies of many DCH states were obtained by the CASSCF calculations and the generalized intra- and inter-atomic relaxation energies were evaluated. We show that XTPPS can extract these quantities by the measurement of single and double core-hole ionization potentials. We discuss the influence of chemical environment on the DCH states with two holes at the same atomic site and at two different atomic sites. (C) 2010 Elsevier B.V. All rights reserved.