Bound state properties, positron annihilation and hyperfine structure of the four-body positronium hydrides

Bound state properties of the ground (bound) ${}^{1}S(L = 0)-$state(s) in the four-body positronium hydrides ${}^{1}$HPs, ${}^{2}$HPs (DPs), ${}^{3}$HPs (TPs) and MuPs are determined and investigated. By using numerical data from our computations of these four-body systems we have determined a number of different annihilation rates for each of these positronium hydrides and evaluated the hyperfine structure splitting. The properties of the ground (bound) states in the four-body exitonic ${}^{M}h^{+} e^{-}_{2} e^{+}$ complexes, where $M \ge 1$ and $M \le 1$, have also been evaluated numerically. The neutral four-body systems ${}^{M}h^{+} e^{-}_{2} e^{+}$ with $M \gg 1$ are similar to the HPs hydrides. In particular, each of these states has only one bound state. We also discuss applications of the exponential and semi-exponential variational expansions for accurate, bound state computations of the four-body positronium hydrides.