The electronic and geometrical structures of exchange-coupled open-shell systems such as bis(methylene) methane and the chromium–methylene cation were investigated by the molecular orbital and its extended methods: RHF, UHF, UNO-CAS, RMP, UMP, PUMP, APUMP, UCCSD(T), UNO-CI and QCISD(T). The fully optimized geometries of these species by the energy gradients of UMPn solutions are consistent with the experimental values. The total energies of the lowest and highest spin states were calculated in order to elucidate the effective exchange integrals in the Heisenberg model for methylene and bis(methylene) methane. The spin projection is necessary for the UHF-based wavefunctions in the lowest spin state of the species. The binding energies for the chromium cation–methylene double bond were calculated by both spin-unprojected and -projected methods. It was found that the binding energy between Cr+ and 3CH2 by APUHFQCISD(T) is in accord with the experimental value. © 1993 John Wiley & Sons, Inc.