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

In the framework of the study of helium-like atomic systems possessing the collinear configuration, we propose a simple method for computing compact but very accurate wave functions describing the relevant $S$ state. It is worth noting that the considered states include the well-known states of the electron-nucleus and electron-electron coalescences as a particular case. The simplicity and compactness imply that the considered wave functions represent a linear combinations of few single exponentials. We have calculated such model wave functions for the ground state of helium and the two-electron ions with nucleus charge $1 \leq Z \leq 5$. The parameters and the accompanying characteristics of these functions are presented in tables for number of exponential from 3 to 6. The accuracy of the resulting wave functions are confirmed graphically. The specific properties of the relevant codes by Wolfram Mathematica are discussed. An example of application of the compact wave functions under consideration is reported.