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We describe eigenstate approaches for understanding the spectroscopy and dynamics of tunnelling molecular systems from a high resolution frequency domain perspective. Powerful exact and approximate ab initio rovibrational methods are introduced for treating large-amplitude, anharmonic motion, such as is encountered in tunnelling systems. The utility of these methods is illustrated by several examples from the recent literature, including H2O2, CH3, gauche-1,4-butadiene, and S1 C2H2. Detailed insights into the structure and dynamics of these tunnelling systems are revealed by the combination of high accuracy theory with careful analysis of high-resolution experimental spectra.

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