Calculation of dipolar nuclear magnetic relaxation times in molecules with multiple internal rotations

Abstract

A general method is described for calculating angular autocorrelation functions and thence longitudinal and transverse dipolar nuclear magnetic relaxation times (T ₁ and T ₂) for nuclei in a methylene group of a hydrocarbon chain attached to a spherical molecule. The method is applicable for any value of the rotational diffusion coefficient of the molecule (D ₀), and for any length of chain. Numerical results for autocorrelation functions and for ¹³C, ¹H and ¹⁹F dipolar relaxation times are presented. These show that for atoms more than four carbon-carbon bonds removed from the isotropically tumbling molecule the autocorrelation function is a single exponential and the relaxation times are independent of the motion of the molecule, provided that D_i > D ₀ (where D_i is the diffusion coefficient for rotation about the ith carbon-carbon bond). For the first few methylene groups in the chain, the auto-correlation function is not a single exponential, and the relaxation times show more complex behaviour. In particular, under the condition. where ω is the relevant Larmor frequency for the dipolar interaction under consideration, the following phenomena are predicted for nuclei on the first three or four methylene groups: (i) As the rate of internal motion increases, the spin-lattice relaxation time first decreases, then passes through a minimum and increases to a value determined by the rotational diffusion of the molecule as a whole. This behaviour has been indicated previously by Doddrell et al. [11] for the particular case of a single internal motion. (ii) Even if the spin-lattice relaxation time, T ₁, increases with increasing temperature, indicating that ω²τ_app ² < 1 (where τ_app is the apparent correlation time calculated from the T ₁ value) T ₁ will be longer than T ₂. (iii) Even if, again ω²τ_app < 1 under some circumstances increasing motion can lead to a decrease in T ₁. Thus the proton T ₁ of the first methylene may be longer than that of the second methylene group in the chain. Finally, methods for calculating the diffusion coefficients, D_i , from measured relaxation times, and extensions of the theory to fluorescence polarization and E.S.R. spin-label experiments are briefly discussed.