  An exact general theory of heteronuclear decoupling is presented for spin-1/2
IS systems. RF irradiation applied to the I spins decouples the S spins by
reversing the sign of antiphase magnetization. A one-to-one correspondence is
demonstrated between the sign of $2S_yI_z$ and the sense of the S-spin coupling
evolution, with magnetization $S_x$ refocused when $\int 2S_yI_z\,dt = 0$. The
exact instantaneous coupling at any time during the decoupling sequence is
readily obtained in terms of the system states, showing that the creation of
two-spin coherence is crucial for reducing the coupling, as required during the
refocusing process. Representative examples from standard cyclic, periodic
composite-pulse and adiabatic decoupling sequences illustrate the decoupling
mechanism. More general aperiodic sequences, obtained using optimal control,
realize the potential inherent in the theory for significantly improved
decoupling. The recently derived equivalence between the dynamics of any
N-level quantum system and a system of classical coupled harmonic oscillators
provides additional insight into the mechanics of decoupling.
