electric (nuclear) charge $Q e$. The notations $r_{ij} = \mid {\bf r}_i - {\bf r}_j \mid = r_{ji}$ stand for the six interparticle distances (= relative coordinates) defined above. In Eq.(\[Hamil\]) and everywhere below in this work we shall assume that $(ij)$ = $(ji)$ = (12), (13), (14), (23), (24), (34). Below only atomic units $\hbar = 1, \mid e \mid = 1, m_e = 1$ are employed. In these units the explicit form of the Hamiltonian $H$, Eq.(\[Hamil\]), is simplified $$\begin{aligned}
 H = -\frac12 \Bigl[ {\bf p}^2_1 + {\bf p}^2_2 + {\bf p}^2_3 + \frac{1}{M} {\bf p}^2_N \Bigr] - \frac{Q}{r_{14}} - 
 \frac{Q}{r_{24}} - \frac{Q}{r_{34}} + \frac{1}{r_{12}} + \frac{1}{r_{13}} + \frac{1}{r_{23}} \label{Hamil1}\end{aligned}$$ where the notations ${\bf p}_i$ designate the momenta of the electrons ($i$ = 1, 2, 3) and the nucleus ($i$ = 4, or $i = N$). In atomic units we have ${\bf p}^{2}_{i} = p^{2}_{i} = -\nabla^{2}_{i}$.

For last fifty years (since the paper [@Lars] was published) the problem of highly accurate calculations of the bound states in three-electron atomic systems has attracted a significant attention. A very nice review of the three-electron atomic problem can be found in [@McW] which also contains all references prior to 1969. 