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The first metastable triplet state of HeH$^+$ was found to be present in ionbeam experiments, with its lifetime estimated to be between hundreds ofmilliseconds and thousand of seconds. In this work, we use {\it ab initio}methods to evaluate the radiative lifetimes of the six vibrational levels ofthe $a\ ^3\Sigma^+$ of HeH$^+$. The transition $a\ ^3\Sigma^+ \rightarrow X \^1\Sigma^+$ is spin-forbidden, but acquires intensity through spin-orbitinteraction with the singlet and triplet $\Pi$ states. Large scale CASSCF/MRCIcalculations using an adapted basis set were performed to determine thepotential energy curves of the relevant states of HeH$^+$ as well as the matrixelements of the dipole and spin-orbit operators. The wave functions andenergies of the vibrational levels of the $a\ ^3\Sigma^+$ and $X \ ^1\Sigma^+$states are obtained using a B-spline method and compared to previous works. Wefind that radiative lifetime of the vibrational levels increases strongly with$v$, the lifetime of the $v=0$ state being of 150 s. We also analyze thecontributions from discrete and continuum parts of the spectrum. With such along lifetime, the $a\ ^3\Sigma^+$ state could have astrophysical implications.

Radiative lifetime of the $a\ ^3\Sigma^+$ state of HeH$^+$ from ab initio calculations