  A variational approach is developed for bound state calculations in three-
and four-electron atomic systems. This approach can be applied to determine, in
principle, an arbitrary bound state in three- and four-electron ions and atoms.
Our variational wave functions are constructed from four- and five-body
gaussoids which depend upon the six ($r_{12}, r_{13}, r_{14}, r_{23}, r_{24},
r_{34}$) and ten ($r_{12}, r_{13}, r_{14}, r_{15}, r_{23}, r_{24}, r_{25},
r_{34}, r_{35}$ and $r_{45}$) relative coordinates, respectively. The approach
allows one to operate with the different number of electron spin functions. In
particular, the trial wave functions for the ${}^1S$-states in four-electron
atomic systems include the two independent spin functions $\chi_1 = \alpha
\beta \alpha \beta + \beta \alpha \beta \alpha - \beta \alpha \alpha \beta -
\alpha \beta \beta \alpha$ and $\chi_2 = 2 \alpha \alpha \beta \beta + 2 \beta
\beta \alpha \alpha - \beta \alpha \alpha \beta - \alpha \beta \beta \alpha -
\beta \alpha \beta \alpha - \alpha \beta \alpha \beta$. We also discuss the
construction of variational wave functions for the excited $2^3S$-states in
four-electron atomic systems.
