Toward a global model of the interactions in low-lying states of methyl cyanide: rotational and rovibrational spectroscopy of the $v_4 = 1$ state and tentative interstellar detection of the $v_4 = v_8 = 1$ state in Sgr B2(N).

New and existing rotational spectra of methyl cyanide were analyzed to extend the global model of low-lying vibrational states and their interactions to $v_4=1$ at 920 cm$^{-1}$. The rotational spectra cover large portions of the 36$-$1439 GHz region and reach quantum numbers $J$ and $K$ of 79 and 16, respectively. Information on the $K$ level structure of CH$_3$CN is obtained from IR spectra. A spectrum of $2\nu_8$ around 717 cm$^{-1}$, analyzed in our previous study, covered also the $\nu_4$ band. The assignments in this band cover 880$-$952 cm$^{-1}$, attaining quantum numbers $J$ and $K$ of 61 and 13, respectively. The most important interaction of $v_4=1$ appears to be with $v_8=3$, $\Delta K=0$, $\Delta l=+3$, a previously characterized anharmonic resonance. We report new analyses of interactions with $\Delta K=-2$ and $\Delta l=+1$, with $\Delta K=-4$ and $\Delta l=-1$, and with $\Delta K=-6$ and $\Delta l=-3$; these four types of interactions connect all $l$ substates of $v_8=3$ in energy to $v_4=1$. A known $\Delta K=-2$, $\Delta l=+1$ interaction with $v_7=1$ was also analyzed, and investigations of the $\Delta K=+1$, $\Delta l=-2$ and $\Delta K=+3$, $\Delta l=0$ resonances with $v_8=2$ were improved, as were interactions between successive states with $v_8\le 3$, mainly through new $v_8\le 2$ rotational data. A preliminary single state analysis of the $v_4=v_8=1$ state was carried out based on rotational transition frequencies and on $\nu_4+\nu_8-\nu_8$ hot band data. A considerable fraction of the $K$ levels was reproduced within uncertainties in its entirety or in part, despite obvious widespread perturbations in $v_4=v_8=1$. We detect rotational transitions of methyl cyanide from within all vibrational states up to $v_4=1$ and $v_4=v_8=1$ tentatively toward the hot molecular core of Sagittarius B2(N) employing the Atacama Large Millimeter Array.