Dioxygen binding at copper enzymatic sites is a fundamental aspect of the catalytic activity observed in many biological systems such as the monooxygenases, especially peptidylglycine alpha-hydroxylating monooxygenase (PHM), in which two mononuclear Cu(I) sites are involved. Biomimetic models have been developed: dipods, tripods, and, more recently, functionalized calixarenes. The modelling of calixarene systems, although not unreachable for theory yet, requires, however, a number of preliminary investigations to ensure proper calibrations if relevant description of the metal-ligand interaction at the hybrid quantum mechanical/molecular mechanics levels of theory is the aim. In this paper, we report quantum chemistry investigations on a coherent series of representative cuprous tripodal species characterized by (1) monodentate ligands [Cu(ImH)3]+ (where ImH is imidazole), [Cu(MeNH2)3]+ and [Cu(MeNH2)4]+ , (2) neutral tripodal ligands [CuCH(ImH)3]+, [Cu(tren)]+ [where tren is tris(2-aminoethyl)amine], and [Cu(trenMe3)]+ [where trenMe3 is tris(2-methylaminoethyl)amine] and (3) a hydrido-tris(pyrazolyl)borate [CuBH(Pyra)3]. The structures of these complexes, the coordination mode (eta(2) side-on or eta(1) end-on) of O2 to Cu(I) and the charge transfer from the metal to dioxygen have been computed. For some systems, the coordination by CH3CN and CO is also reported. Beyond results relative to structural properties, an interesting feature is that it is possible to build from computational results only a set of abacuses linking the nu(16O-16O) vibrational frequency of the coordinated O2 molecule to the O-O bond length or to the net charge of the O2 moiety. Such abacuses may help experimentalists in distinguishing between the four possible ways of binding O2 to CuN3 and CuN4 cuprous centres, namely (1) end-on triplet states, (2) side-on triplet states, (3) end-on singlet states and (4) side-on singlet states. These abacuses are extended to three tripods obtained by the substitution of one nitrogen atom by either a phosphorus or a sulphur atom. Moreover, it is shown that any factor favouring pyramidalization at copper favours charge transfer and thus coordination of the incoming O2 moiety. All these allow insight into the coordination mode of O2 and into the charge transfer from Cu(I) in site Cu(M) of PHM.