Second Bound State for the Hydrogen Negative Ion

It is frequently stated that the $1s^2{}_{}{}^1{}_{}{}^{}S$ state is the only bound state of ${\mathrm{H}}^{-}$, although Holøien estimated that the $2p^2{}_{}{}^3{}_{}{}^{}P$ state may also be bound. We show by a variational calculation that the $2p^2{}_{}{}^3{}_{}{}^{}P$ state is definitely bound below the $n=2\mathrm{}$ threshold of hydrogen. The best eigenvalue obtained is -0.125350 a.u.