Theory of Nuclear Overhauser Enhancement and C13–1H Dipolar Relaxation in Proton-Decoupled Carbon-13 NMR Spectra of Macromolecules

Abstract
Expressions are presented for the nuclear Overhauser enhancement (NOE), the spin‐lattice relaxation time (T1), and the spin‐spin relaxation time (T2) of a 13C nucleus relaxing by a dipolar interaction with one proton under conditions of complete proton decoupling, and without assuming that the extreme narrowing limit applies. Specific equations are derived for a CH group in a rigid molecule rotating isotropically and also for a C–H group with one degree of internal motion attached to a molecule undergoing isotropic rotational reorientation. Numerical results are presented for T1, T2, and the NOE of a C–H group in a rigid molecule (undergoing purely dipolar relaxation) as a function of the rotational correlation time (τR) and the resonance frequency (ωC). T1 goes through a minimum when τRωC ≈ 0.8 . The NOE varies from the expected value of 2.988 in the extreme narrowing limit to 1.153 when 1/τR is much smaller than the resonance frequency. The numerical results indicate that the signal‐to‐noise ratio in proton‐decoupled 13C Fourier transform spectra of macromolecules in solution may not improve significantly by going to very high magnetic field strengths (such as 51.7 kG), because the increase in basic sensitivity can be offset by a decrease in the NOE and the T2/T1 ratio. The magnitude of the latter two parameters is strongly dependent on τR and the magnetic field strength. Numerical results are also presented for a C–H group with one degree of internal motion. 1/T2 is a monotonically increasing function of τR and τG (the correlation time for internal rotation). The NOE and T1 behave in a more complex manner. The onset of internal rotation may make T1 larger or smaller, depending on the value of τRωC. In the extreme narrowing limit, T1 increases monotonically as τG decreases, reaching a limiting value (when τG≫τR) of 4(1−3 cos2θ)−2T1R, where θ is the angle between the C–H vector and the axis of internal rotation and T1R is the value of T1 in the absence of internal rotation. However, for very slow over‐all reorientation (with respect to the resonance frequency), the onset of internal rotation produces a decrease in T1 until it reaches a minimum and increases again. When τG→ 0 , T1 reaches the same limiting value as in the extreme narrowing case. As expected, in the extreme narrowing limit for τR we get the full NOE of 2.988 regardless of the value of τG. For slow over‐all reorientation, the onset of internal rotation first increases the NOE. As τG gets smaller, the NOE goes through a maximum and then decreases again. As τG→ 0 , the NOE reaches an asymptotic value equal to that in the absence of internal rotation.