contradictions with the known experimental results.

In our analysis below we re-consider all these problems by performing highly accurate calculations of some bound states in three-electron atomic systems. This work has the following structure. In the next Section we discuss the semi-exponential variational expansion in relative coordinates [@Fro2010]. This very compact and accurate variational expansion was introduced in [@Fro2010]. In this study this expansion is applied to determine various bound state properties in the ground $2^2S-$state(s) of the three-electron Li-atom and Be$^{+}$ and B$^{2+}$ ions. The computed expectation values include electron-nucleus and electron-electron delta-functions and cusp-values. In Section III we consider a number of bound (doublet) ${}^2S-$states in different three-electron atoms and ions by using another variational expansion of the wave functions. Here all wave functions are approximated by expansions in six-dimensional gaussoids [@KT]. Note that analytical formulas for all matrix elements which arise in variational calculations of arbitrary $A-$body system in the basis of multi-dimensional gaussoids, where $A \ge 2$ is an arbitrary integer, have been derived in the mid-1970’s (see [@KT] and earlier references therein). Subsequently this variational expansion was successfully applied for bound state computations in many hundreds of nuclear, atomic and molecular systems. In Section III 