J -recoupling patterns arising from two chemically equivalent nuclear spins in magic-angle spinning spectra of solids

A general description of magic-angle spinning (MAS) nuclear magnetic resonance (NMR) spectra arising from a pair of chemically equivalent nuclear spins is presented in terms of average Hamiltonian theory (AHT). In general, the MAS NMR spectra of such a spin-pair exhibit a spinning frequency dependent four-line pattern from which it is possible to extract the indirect spin–spin coupling constant, J, involving the ‘‘equivalent’’ spin pair. Explicit expressions for the spinning frequency dependence of the four-line pattern have been derived using AHT. In principle, correction terms to any order can be included; however, consideration of correction terms up to and including third order appear to be sufficient to interpret the most important features characteristic of J-recoupled spectra involving chemically equivalent spin pairs. The average Hamiltonian theory predicts three different general types of recoupling patterns. The type of recoupling pattern observed for a particular chemically equivalent spin pair is predicted to depend on the relative magnitudes of the indirect homonuclear coupling constant, J, the direct homonuclear dipolar coupling constant, R, the magnitude of the instantaneous chemical shift difference between the ‘‘equivalent’’ spins in frequency units, and the MAS spinning frequency. All reported examples of spinning frequency dependent MAS NMR spectra arising from a pair of chemically equivalent spins can be understood using the theoretical expressions derived here. As an example, we interpret the unusual J-recoupling pattern observed in 31P MAS NMR spectra of Hg(PPh3)2(NO3)2. The recoupling pattern is unusual in that 2J(P,P) is given by the separation of alternate lines in the four-line pattern. Similar unusual J-recoupling patterns were first reported by Eichele, Wu, and Wasylishen.