Probable nonexistence ofa Pe3metastable excited state of the positronium negative ion

The ${\mathrm{H}}^{-}$ ion is known to have a metastable $2p^2{}_{}{}^3{}_{}{}^{}P_{}^e{}_{}{}^{}$ triplet excited state. To see if an analog of this state is present in the positronium negative ion ${\mathrm{Ps}}^{-}$ the Coulomb binding energy $E_b$ of the lowest-energy even-parity $L=1\mathrm{}$ configuration of two identical-charge $-e$ fermions of mass $m_1$ plus one spinless particle of mass $m_2$ and charge $+e$ is calculated. We find a value of $E_b$ below the $n=2\mathrm{}$ level of the neutral atom for $\frac{0\le m_2}{M\le 0.17}$ and $\frac{0.90<\mathrm{}{m}_2}{M\le 1}$, where $M=2\mathrm{}{m}_1+m_2$ is the total mass of the ion. However, we cannot say that the ${}_{}{}^3{}_{}{}^{}P_{}^e{}_{}{}^{}$ state exists for ${\mathrm{Ps}}^{-}$, where $\frac{m_2}{M}=\frac{1}{3}$.