Theory of frequency-selective homonuclear dipolar recoupling in solid-state NMR

In solid-state nuclear magnetic resonance, frequency-selective homonuclear dipolar recoupling is key to quantitative distance measurement or selective enhancement of correlations between atoms of interest in multiple-spin systems, which are not amenable to band-selective or broadband recoupling. Previous frequency-selective recoupling is mostly based on the so-called rotational resonance (R²) condition that restricts the application to spin pairs with resonance frequencies differing in integral multiples of the magic-angle spinning (MAS) frequency. Recently, we have proposed a series of frequency-selective homonuclear recoupling sequences called SPR (short for Selective Phase-optimized Recoupling), which have been successfully applied for selective ¹H-¹H or ¹³C-¹³C recoupling under from moderate (∼10 kHz) to ultra-fast (150 kHz) MAS frequencies. In this study, we fully analyze the average Hamiltonian theory of SPR sequences and reveal the origin of frequency selectivity in recoupling. The theoretical description, as well as numerical simulations and experiments, demonstrates that the frequency selectivity can be easily controlled by the flip angle (p) in the (p)_ϕk(p)_ϕk+π unit in the pSPR-N_n sequences. Small flip angles lead to frequency-selective recoupling, while large flip angles may lead to broadband recoupling in principle. The result shall shed new light on the design of homonuclear recoupling sequences with arbitrary frequency bandwidths.