Rates of the two-, three-, four-, and five-photon annihilations of the electron-positron pairs are determined numerically for the three-body positronium ion ${\text{Ps}}^{-}\left(e^{-}{e}^{+}{e}^{-}\right)$ and four-body bipositronium “molecule” ${\text{Ps}}_2\left(e^{-}{e}^{+}{e}^{-}{e}^{+}\right)$. The values obtained in our computations are ${\Gamma }_{2\gamma }\left({\text{Ps}}^{-}\right)\approx 2.08048530525\times {10}^9{\text{s}}^{-1}$, ${\Gamma }_{3\gamma }\left({\text{Ps}}^{-}\right)\approx 5.6364151550\times {10}^6{\text{s}}^{-1}$, ${\Gamma }_{4\gamma }\left({\text{Ps}}^{-}\right)\approx 3.075\times {10}^3{\text{s}}^{-1}$, ${\Gamma }_{5\gamma }\left({\text{Ps}}^{-}\right)\approx 5.383{\text{s}}^{-1}$, and ${\Gamma }_{2\gamma }\left({\text{Ps}}_2\right)\approx 4.4385952\times {10}^9{\text{s}}^{-1}$, ${\Gamma }_{3\gamma }\left({\text{Ps}}_2\right)\approx 1.202497\times {10}^7{\text{s}}^{-1}$, ${\Gamma }_{4\gamma }\left({\text{Ps}}_2\right)\approx 6.562\times {10}^3{\text{s}}^{-1}$, ${\Gamma }_{5\gamma }\left({\text{Ps}}_2\right)\approx 11.484{\text{s}}^{-1}$. The four- and five-photon annihilation rates are significantly smaller than the corresponding two- and three-photon annihilation rates known for these systems. We also determine the rates of one- and zero-photon annihilation for the ${\text{Ps}}^{-}$ ion and ${\text{Ps}}_2$ system. The corresponding numerical values are ${\Gamma }_{1\gamma }\left({\text{Ps}}^{-}\right)\approx 3.82491\times {10}^{-2}{\text{s}}^{-1}$, ${\Gamma }_{1\gamma }\left({\text{Ps}}_2\right)\approx 1.94188\times {10}^{-1}{\text{s}}^{-1}$, and ${\Gamma }_{0\gamma }\left({\text{Ps}}_2\right)\approx 2.32197\times {10}^{-9}{\text{s}}^{-1}$.

• Received 3 June 2009

DOI:https://doi.org/10.1103/PhysRevA.80.014502