\bigr\})$ we have $$\begin{aligned}
 \psi_{L=0}(A; \bigl\{ r_{ij} \bigr\}) = \sum^N_{k=1} C_k exp(-\alpha^{(k)}_{12} r^2_{12} -\alpha^{(k)}_{13} r^2_{13} -\alpha^{(k)}_{23} r^2_{23}
 -\alpha^{(k)}_{14} r^2_{14} -\alpha^{(k)}_{24} r^2_{24} -\alpha^{(k)}_{34} r^2_{34}) \label{gauss}\end{aligned}$$ where $C_k$ are the linear coefficients (or linear variational parameters), and $\alpha^{(k)}_{ij}$ are the varied non-linear parameters. It is clear that for Coulomb systems each of the basis functions in Eq.(\[gauss\]) does not provide the correct long-range asymptotic between particles. Indeed, for the $i$ and $j$ particles the leading term in the asymptotics of the actual atomic wave function must be represented in the form $r^{K}_{ij} \exp(-\beta r_{ij})$. Additional terms for Coulomb (or atomic) systems may include different powers of $log r_{ij}, log \mid log r_{ij} \mid$ and other similar factors. This follows from the general theory of bound states in the few-body Coulomb systems (see, e.g., [@Fock1954] and references therein). It is clear that such terms cannot be derived from the approximate wave function Eq.(\[gauss\]). Based on this conclusion we can predict a number of different bound state properties with the use of variational expansion, Eq.(\[gauss\]). In reality, the variational expansion, Eq.(\[gauss\]), can be used for approximate evaluations of the bound state properties in atomic systems.

Formally, this means that the expectation values determined with 