included from the variational expansion. Finally, we developed a very effective variational expansion for three-electron atoms and ions which can now be used to construct very compact and accurate wave functions for arbitrary three-electron atom and/or ion. For simplicity, below we consider only the doublet $2^{2}S(L = 0)$-states in the three-electron Li atom and similar Be$^{+}$ and B$^{2+}$ ions. By using our semi-exponential variational expansion we determine bound state properties of these three-electron atomic systems. It should be mentioned that some of these properties have never been determined in earlier studies.

The variational wave function of the doublet $S(L = 0)$-states of the three-electron atom/ion is written in the form $$\begin{aligned}
 \Psi_{L=0} = \psi_{L=0}(A; \bigl\{ r_{ij} \bigr\}) (\alpha \beta \alpha - \beta \alpha \alpha) + \phi_{L=0}(B; \bigl\{ 
 r_{ij} \bigr\}) (2 \alpha \alpha \beta  - \beta \alpha \alpha - \alpha \beta \alpha) \label{psi}\end{aligned}$$ where $\psi_{L=0}(A; \bigl\{ r_{ij} \bigr\})$ and $\phi_{L=0}(B; \bigl\{ r_{ij} \bigr\})$ are the two independent spatial parts (also called the radial parts, or radial functions) of the total wave function. Each of these two radial functions is, in fact, a radial factor (for states with $L = 0$) in front of the corresponding three-electron spin functions $\chi_1 = \alpha 