charged nucleus. Another our goal is to determine the bound state properties of some of these bound (doublet) states and discuss problems arising during this procedure. By calculating these properties we want to correct mistakes which have been made in earlier papers and continue to propagate in the modern scientific literature.

As is well known in the lowest-order approximation upon the fine structure constant $\alpha (= \frac{e^2}{\hbar c}$) an arbitrary three-electron atom/ion is described by the non-relativistic Schrödinger equation $H \Psi = E \Psi$, where $E < 
0$ and the bound state wave function $\Psi$ has the unit norm. The non-relativistic Hamiltonian $H$ of the three-electron atom/ion is (see, e.g., [@LLQ]) $$\begin{aligned}
 H = -\frac{\hbar^2}{2 m_e} \Bigl[\nabla^2_1 + \nabla^2_2 + \nabla^2_3 + \frac{m_e}{M} \nabla^2_4 
 \Bigr] - \frac{Q e^2}{r_{14}} - \frac{Q e^2}{r_{24}} - \frac{Q e^2}{r_{34}} + \frac{e^2}{r_{12}} 
 + \frac{e^2}{r_{13}} + \frac{e^2}{r_{23}} \label{Hamil}\end{aligned}$$ where $\hbar = \frac{h}{2 \pi}$ is the reduced Planck constant, $m_e$ is the electron mass and $e$ is the electric charge of an electron. In this equation and everywhere below in this study the subscripts 1, 2, 3 designate the three electrons $e^-$, while the subscript 4 denotes the heavy nucleus with mass $M$ ($M \gg m_e$) and positive 