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In the framework of the study of helium-like atomic systems possessing thecollinear configuration, we propose a simple method for computing compact butvery accurate wave functions describing the relevant $S$ state. It is worthnoting that the considered states include the well-known states of theelectron-nucleus and electron-electron coalescences as a particular case. Thesimplicity and compactness imply that the considered wave functions represent alinear combinations of few single exponentials. We have calculated such modelwave functions for the ground state of helium and the two-electron ions withnucleus charge $1 \leq Z \leq 5$. The parameters and the accompanyingcharacteristics of these functions are presented in tables for number ofexponential from 3 to 6. The accuracy of the resulting wave functions areconfirmed graphically. The specific properties of the relevant codes by WolframMathematica are discussed. An example of application of the compact wavefunctions under consideration is reported.

Accurate exponential representations for the ground states of the collinear two-electron atomic systems