The rigorous treatment of relaxation for the dipolar-multipolar AX spin system (I = 1/2, S > 1/2) in the presence of the dipolar I-S coupling, anisotropy chemical shift and quadrupolar interaction of S spin is proposed. The calculations of the spin evolution under the relaxation Hamiltonian ate based on the second-order time-dependent perturbation theory and ate carried out in the operator representation. For this task the double commutator identities of the type [[LS.',A~]A_~~] and [[/.S~",A~]A u~,] are derived. The first-order differential equati0ns for the evolution of longitudinal twospin order I~~, z-magnetization of S spin S~' and coherences (I._S~) in the spin system IS with scalar coupling between spin 1/2 and quadrupolar spin S > 1/2 were obtained. These equations ate used to get equations for the evolutions of each component of the multiplet structure of spin L The imaginary part of the cross-correlation spectral density function and indirect spin-spin coupling Hamiltonian are taken into account. Equations for the longitudinal components of the I spin spectrum in the presente of cross-correlation effects were obtained also. Longitudinal and transverse relaxation times and cross-relaxation times in the presence of cross-correlation D-CSA, Q-CSA, Q-D were analyzed.
In the frame of the density matrix formalism the lineshapes and dynamic frequency shifts in the multiplet structure of a spin I = 1/2 scalar coupled to a quadrupolar nucleus with a spin S >-1 undergoing dipolar (D), quadrupolar (Q) and chemical shift anisotropy (CSA) relaxation are studied. Analytic expressions for the cross-correlation spectral densities JQ-csn JD-Q JD-csA and also for the real and imaginary Redfield elements of the relaxation matrix were obtained in the general case of noncoincident and nonaxial Q, D, and CSA tensors. On the assumption that CSA and Q interactions are comparable in magnitude the contributions of cross-correlation terms CSA-Q, CSA-D, Q -D in the linewidth and dynamic shifts of the multiplet pattern of a spin I = 1/2 were analyzed in a wide interval of correlation times (10 -12_l0 -7 s) for a spin system 13 C-"B and 31 P-59Co as an example.
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