Systematic design and evaluation of multiple-pulse experiments in nuclear magnetic resonance spectroscopy using a semi-continuous Baker–Campbell–Hausdorff expansion

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Abstract
We show that an explicit solution to a semi-continuous analog to the Baker–Campbell–Hausdorff (BCH) problem can be derived by an appropriate combination of the Magnus and BCH expansions. The resulting semi-continuous BCH (scBCH) expansion forms a valuable tool for solving the time-dependent Schrödinger equation for Hamiltonians with complicated, piecewise continuous time dependence. Such Hamiltonians are typical in multiple-pulse coherent spectroscopy. Using the scBCH expansion we derive a number of general formulas, including relations for permuted pulse sequences. These formulas simplify calculation of the effective Hamiltonian for advanced multiple-pulse experiments and allow for evaluation of this to considerably higher order than is possible using the Magnus expansion. This is important for the detailed analysis and systematic design of multiple-pulse experiments which emphasize some interactions while effectively suppressing others. The scBCH expansion is applied to problems of homonuclear dipolar decoupling in solid-state NMR and broadband heteronuclear decoupling in liquid-state NMR. Improved high-order pulse sequences for on- and off-resonance decoupling are introduced and existing recursive expansion strategies are evaluated within the presented theoretical framework.