Astrophysical molecular spectroscopy is an important method of searching for new physics through probing the variation of the proton-to-electron mass ratio, $mu$, with existing constraints limiting variation to a fractional change of less than 10$^{-17}$/year. To improve on this constraint and therefore provide better guidance to theories of new physics, new molecular probes will be useful. These probes must have spectral transitions that are observable astrophysically and have different sensitivities to variation in the proton-to-electron mass ratio. Here, we concisely detail how astrophysical observations constrain the set of potential molecular probes and promising sensitive transitions based on how the frequency and intensity of these transitions align with available telescopes and observational constraints. Our detailed investigation focuses on rovibronic transitions in astrophysical diatomic molecules, using the spectroscopic models of 11 diatomics to identify sensitive transitions and probe how they generally arise in real complex molecules with many electronic states and fine structure. While none of the 11 diatomics investigated have sensitive transitions likely to be astrophysically observable, we have found that at high temperatures (1000 K) five of these diatomics have a significant number of low intensity sensitive transitions arising from an accidental near-degeneracy between vibrational levels in the ground and excited electronic state. This insight enables screening of all astrophysical diatomics as potential probes of proton-to-electron mass variation, with CN, CP, SiN and SiC being the most promising candidates for further investigation for sensitivity in rovibronic transitions.
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