the use of multi-dimensional gaussoids can be used as a one-sided approximation to the actual bound state properties of atomic systems. The procedure works in the following way. By using relatively large numbers of multi-dimensional gaussoids with carefully optimized non-linear parameters $\alpha^{(k)}_{ij}$ in Eq.(\[gauss\]) we can obtain accurate numerical values of the total energies. Then we continue optimization of the non-linear parameters $\alpha^{(k)}_{ij}$ in Eq.(\[gauss\]). The total energies continue to converge to their limiting values. However, the expectation values of many bound state properties rapidly improve during such an additional optimization. The principal question here is to evaluate the convergence rate of the variational expansion, Eq.(\[gauss\]). In our calculations we have found that the actual convergence rate for the total energies is relatively fast and already for $N = 700 - 1000$ in Eq.(\[gauss\]) we obtain results which can be considered as accurate, or even highly accurate.

The total energies of the ground $2^2S(L = 0)$ and some excited $n^2S(L = 0)$-states ($n \ge 3$) in some three-electron atoms and ions can be found in Table II (in atomic units). Table III contains the bound state properties determined with the use of our best variational wave functions. In such calculations 