Electronic and hyperfine structures of MgH24 with relevance to laser cooling

Abstract

We study the electronic and hyperfine structures of $^{24}\mathrm{MgH}$ molecule, which are of crucial importance for the laser cooling of the molecule. The accurate spectroscopic constants and hyperfine levels are presented for $X^{2}\mathrm{\ensuremath{\Sigma}}^{+}$ and $A^{2}\mathrm{\ensuremath{\Pi}}$ symmetries. The theoretical results are in good agreement with the experimental data. We then provide a scheme for the laser-driven cycling transitions of $A^{2}\mathrm{\ensuremath{\Pi}}_{1/2}\ensuremath{\leftarrow}X^{2}\mathrm{\ensuremath{\Sigma}}_{1/2}$. In addition, we calculate the hyperfine branching ratios for the rate of decay from $|A^{2}\mathrm{\ensuremath{\Pi}}_{1/2},{J}^{\ensuremath{'}}=1/2,+\ensuremath{\rangle}$ to $|X^{2}\mathrm{\ensuremath{\Sigma}}_{1/2},N=1,\ensuremath{-}\ensuremath{\rangle}$ transition. We also provide a sideband modulation scheme to the pumping lasers to match all four hyperfine levels of the $X^{2}\mathrm{\ensuremath{\Sigma}}_{1/2}$ ($v=0$, $N=1$). Finally, we calculate the Zeeman shift of the ($X^{2}\mathrm{\ensuremath{\Sigma}}_{1/2}$, $v=0$, $N=1$) ground state, and the typical hyperfine structure magnetic $g$ factor ${g}_{F}$ and mixed $g$ factors ${g}_{F}^{\mathrm{eff}}$ are also calculated. Our study is expected to be useful for the laser-cooling $^{24}\mathrm{MgH}$ molecule and the interpretation of astronomical spectra involving MgH molecule.