Astronomical mid-IR spectra show two minor PAH features at 5.25 and 5.7 $mu$m (1905 and 1754 cm$^{rm - 1}$) that hitherto have been little studied, but contain information about the astronomical PAH population that complements that of the major emission bands. Here we report a study involving both laboratory and theoretical analysis of the fundamentals of PAH spectroscopy that produce features in this region and use these to analyze the astronomical spectra. The ISO SWS spectra of fifteen objects showing these PAH features were considered for this study, of which four have sufficient S/N between 5 and 6 $mu$m to allow for an in-depth analysis. All four astronomical spectra show similar peak positions and profiles. The 5.25 $mu$m feature is peaked and asymmetric, while the 5.7 $mu$m feature is broader and flatter. Detailed analysis of the laboratory spectra and quantum chemical calculations show that the astronomical 5.25 and 5.7 $mu$m bands are a blend of combination, difference and overtone bands primarily involving CH stretching and CH in-plane and CH out-of-plane bending fundamental vibrations. The experimental and computational spectra show that, of all the hydrogen adjacency classes possible on PAHs, solo and duo hydrogens consistently produce prominent bands at the observed positions whereas quartet hydrogens do not. In all, this a study supports the picture that astronomical PAHs are large with compact, regular structures. From the coupling with primarily strong CH out-of-plane bending modes one might surmise that the 5.25 and 5.7 $mu$m bands track the neutral PAH population. However, theory suggests the role of charge in these astronomical bands might also be important.
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