The two-times Green’s-function formalism is applied to the calculation of the NMR relaxation time T, for systems of hindered-rotating molecules or atomic groups in solids or solid surfaces at very low temperature. The rotation of the molecules is quantummechanically considered as one-phonon induced transitions between the hindered-rotator levels. The nuclear spin system is assumed to be coupled to the bath of thermal phonons through the phonon-rotation interaction. The longitudinal relaxation rate is determined by taking the lowest order approximation for the pole structure of the retarded Green’s function; in this way a multiexponential behavior due to the different nuclear spin species present in the molecule or atomic group is obtained.