Theory of relaxation of quadrupolar nuclei in deformable molecules in isotropic solutions. Application to DNA

Abstract Expressions are derived for the longitudinal ( R Q 1 ) and transverse ( R Q 2 ) relaxation rates of quadrupolar nuclei in deformable molecules in isotropic solution. These rates are expressed in terms of principal spectral densities, which are all Fourier transforms of the same principal correlation function. These results are specialized to molecules that exhibit mean local cylindrical symmetry and explicitly evaluated for the case when the equilibrium orientation of the principal axis of the electric field gradient (efg) tensor is perpendicular to the local symmetry axis. Proper account is taken of collective twisting and bending deformations, uniform (rigid-body) twisting and tumbling motions, and local angular motions of the efg tensor within its subunit. Analytical expressions for the internal correlation functions, which reflect the local angular motions, are derived for a model in which the efg tensor frame undergoes small amplitudes of overdamped libration in a harmonic potential that governs each of its three rotational degrees of freedom. Sample calculations of ( R Q 1 ) are presented for a range of possible rms angular displacements and relaxation times of the internal motions for the particular case of deuterons at the H8 positions of adenine and guanine in a 12 bp duplex DNA. Somewhat simpler analytical expressions for the long-time limits of the internal correlation functions are derived for the case when the efg tensor frame undergoes an arbitrary amplitude of overdamped libration in an isotropic harmonic deflection potential. A particularly simple expression is obtained when the efg tensor undergoes small amplitudes of motion in a completely isotropic deflection potential with arbitrary orientation. These simpler expressions can be used to calculate R Q 1 and R Q 2 , when the local angular motions relax in a time much less than the inverse Larmor frequency (ω −1 ), and their contribution to the integrated area under the principal correlation function is negligible compared to the Fourier cosine transform of the principal correlation function. Corresponding results for relaxation by pure chemical-shift anisotropy and by pure-dipolar interactions are presented in the present notation.