The Spin Dynamics of Heteronuclear Multiple-Spin Systems Formulated in the Extended Strong-Narrowing Limit

Abstract

Employing the perturbation approach directly in the multiply rotating frame, the master equation governing the spin dynamics of scalar-coupled heteronuclear multiple-spin systems has been derived. The resulting equation for the spin-density operator and the matrix form of the relaxation superoperator are relatively simple under the conditions of the “extended” strong-narrowing limit. The extended strong-narrowing limit, which requires that ω^I_iτ_c<< 1 and 2πJ_ijτ_c<< 1 [where I represents one of the spin types in the heteronuclear spin system, ω^I_iis the chemical-shift range of spin type I in radians per second,J_ijis the coupling constantJbetween spiniand spinj(like or unlike) in hertz, and τ_cis the correlation time in seconds per radian], can be applied to heteronuclear spin systems of both small molecules and biopolymers in high-resolution liquid NMR. This newly developed formalism is used to investigate the effect of transverse cross relaxation on the apparent coupling constants in a heteronuclear¹H–¹H–¹³C three-spin system. The calculation shows that, despite the strong dipolar interaction between directly bonded¹H and¹³C, this perturbation on the apparent couplings is trivial. This result is in contrast to the homonuclear proton spin system, where the scalar coupling constants between two protons are significantly modulated if one proton is strongly dipole coupled to another proton. The underlying physical reasons for the different behavior of heteronuclear and homonuclear systems are explained.