Recently a new class of mononuclear single molecule magnets (SMMs) on the basis of lanthanide ions has been reported [1]. This class comprises the phthalocyanine double-decker mononuclear complex [(Pc)2Tb]·TBA+ (Pc=dianion of phthalocyanine, TBA+=N(C4H9)4+ [1]). The aim of the present paper is to reveal the mechanisms underlying the SMM behavior of this complex. The results obtained can be summarized as follows: (i) The covalence effects and the tetragonal symmetry of the nitrogen ligand surrounding of the Tb-ion in [(Pc)2Tb]-·TBA+ have been taken into account in calculations of the crystal field. The energies and wave functions of the Stark levels arising from the splitting of the ground 7F6 multiplet of the Tb3+ ion in the ligand field have been computed. On this basis the principal values of the magnetic susceptibility tensor have been found. A reasonable agreement between the calculated and observed magnetic susceptibility has been obtained. (ii) The energies of the low-lying Stark levels of the Tb - ion in [(Pc)2Tb]-·TBA+ were shown to increase with the decrease of the mean value of the z-projection of the total angular momentum, a situation that leads to a barrier for the reversal of magnetization. Thus, the model explains the discovered SMM behavior of the [(Pc)2Tb]-·TBA+ complex arising from the strong single-ion anisotropy associated with the unquenched orbital angular momentum of the terbium ion. (iii) The vibrations of the complex consisting of the terbium ion and the nearest ligand surrounding have been classified according to irreducible representations of the D4d point symmetry group of this complex. The operators of electron-vibrational interaction for the TbIII ion have been derived. The probabilities of all possible non-radiative one-phonon transitions between the Stark levels of the ground 7F6 multiplet of the Tb ion have been evaluated at different temperatures. (iv) The set of master equations for the populations of the Stark levels arising from the ligand field splitting of the ground 7F6 state of the TbIII ion has been solved and on this basis the temperature dependence of the relaxation time of magnetization was estimated. The calculated temperature dependence of the relaxation time of magnetization is in quite good agreement with the experimental one.