Analytical and Numerical Calculations of Diffusion Effects on the Intermolecular Multiple Quantum Coherences in Solution NMR

The intermolecular multiple quantum coherences (iMQCs) that are generated by intermolecular dipolar interactions between distant spins on different molecules, have recently gained considerable attention because their properties are intrinsically different from those of conventional single quantum coherences (SQCs) in solution NMR. This feature allows for a wider range of applications in NMR and MR imaging. Signal attenuation behaviors of the iMQCs induced by the translational diffusion of correlated molecules in the presence of pulsed field gradients should be different from those of SQCs. However, the diffusion effect on the iMQC signals resulting from intermolecular interactions is not yet well-understood. In the prototype CRAZED sequence, which differs from the conventional pulsed gradient spin echo (PGSE) experiment, the molecular diffusion affects the evolution of the iMQCs during both the evolution period t1 and detection period t2. As described elsewhere, during the t1 period, the multiple quantum coherences that correlate with the multi-spins from different molecules should evolve sensitively with the relaxation processes, susceptibility variations, and translational diffusion. Additionally, during the t2 period, diffusion could attenuate the strength of the distant dipolar field (or dipolar demagnetizing field), which is created by the modulated z-magnetization, and can convert the iMQCs into detectable signals, while there is no diffusional effect in the conventional PGSE experiment. Hence, the diffusion effect on the signal attenuation with the dipolar field should be considered for both the evolution and detection periods in the CRAZED-type experiments. Previous studies have analytically revealed the diffusion effects on the iMQC signals for the limited condition. In this paper, we analytically show the evolution behavior of an iMQC signal in a CRAZED-type sequence with molecular diffusion effects, and a comparison with numerical simulations conducted using various diffusion coefficients. To distinguish the diffusion effects during the t1 and t2 periods, the position of the first encoding gradient pulse was varied from the beginning to the end of the t1 period. It should be noted that the other dynamics, except for the distant dipolar field (DDF) and diffusion, were ignored in the analytical and numerical calculations to clearly understand the effects of diffusion on the iMQC evolution. In principle, both the quantum and classical treatments yield the same predictions for the iMQCs signal generated by intermolecular dipolar couplings in solution NMR. The quantum picture retains the individual dipolar couplings for all evolution periods and averages them at the end, while the classical picture averages all dipolar couplings first and then make evolution under that mean field. For evaluating the diffusion effects, the classical approach would be suitable since the modified Bloch equation can easily incorporate the diffusion effects as well as the distant dipolar field. The Bloch equation, which modified to include the distant dipolar field and molecular diffusion process, can be written as