Methyl mercaptan (CH$_3$SH) is an important sulfur-bearing species in the interstellar medium, terrestrial environment, and potentially in planetary atmospheres. The aim of the present study is to provide accurate spectroscopic parameters for the most abundant minor isotopolog CH$_3$$^{34}$SH to support radio astronomical observations at millimeter and submillimeter wavelengths. The rotational spectrum of CH$_3$$^{34}$SH, which is complicated by the large-amplitude internal rotation of the CH$_3$ group versus the $^{34}$SH frame, was investigated in the 49$-$510 GHz and 1.1$-$1.5 THz frequency ranges in natural isotopic abundance. The analysis of the spectrum was performed up to the second excited torsional state, and the obtained data were modeled with the RAM36 program. A fit within experimental accuracy was obtained with a RAM Hamiltonian model that uses 72 parameters. Predictions based on this fit are used to search for CH$_3$$^{34}$SH with the Atacama Large Millimeter/submillimeter Array (ALMA) toward the hot molecular core Sgr B2(N2), but blends with emission lines of other species prevent its firm identification in this source.
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