**1. Introduction**

In this paper, we present a technique for building compact and simple wave functions of high accuracy, describing two-electron atomic systems such as H<sup>−</sup>, He, Li+, Be2<sup>+</sup> and B3<sup>+</sup> with the *collinear* arrangemen<sup>t</sup> of the particles [1]. The study of mechanism of double photoionization of the helium-like atomic systems by high energy photons [2,3] can serve as an example of possible application (see the details in the next Section).

Methods enabling us to calculate the relevant wave function (WF) and the corresponding non-relativistic energy differ from each other by the calculation technique, spatial variables and basis sets. It is well-known that the *S*-state WF, <sup>Ψ</sup>(*<sup>r</sup>*1,*r*2,*r*12), is a function of three variables: the distances *r*1 ≡ |**<sup>r</sup>**1| and *r*2 ≡ |**<sup>r</sup>**2| between the nucleus and electrons, and the interelectron distance *r*12 ≡ |**r**1 − **<sup>r</sup>**2|, where **r**1 and **r**2 represent radius-vectors of the electrons. We shall pay special attention to the bases that differ from each other both in the kind of the basis functions and in its number (basis size). The Hartree atomic units are used throughout the paper.

It would be useful to give some examples of basis sets intended for describing the relevant *S* states. The correlation function hyperspherical harmonic method (CFHHM) [4,5] employs the basis representing the product of the hyperspherical harmonic (HH) as an angular part, and the *numerical* radial part. The corresponding basis size *N* equals (as a rule) 625. The Pekeris-like method (PLM) [6–8] is used intensively in the current work. The basis size of the PLM under consideration is *N* = 1729 (for the number of shells Ω = 25), and the basis functions can be finally reduced to the form exp(*αr*1 + *β<sup>r</sup>*2 + *<sup>γ</sup><sup>r</sup>*12)*rl*1*rm*2 *<sup>r</sup>n*12, where *α*, *β* and *γ* are the real constants and *l*, *m*, *n* are non-negative integers. Hylleraas [9] (see also [10,11]) was the first who employed the same basis but with *γ* = 0. The authors of Ref. [12] have performed variational calculations on the helium isoelectronic sequence using modification of the basis set that employed by Frankowski and Pekeris [13]. They managed to ge<sup>t</sup> very accurate results using the reduced basis of the size *N* = 230. The

**Citation:** Liverts, E.Z.; Barnea, N. Accurate Exponential Representations for the Ground State Wave Functions of the Collinear Two-Electron Atomic Systems. *Atoms* **2022**, *10*, 4. https://doi.org/10.3390/ atoms10010004

Academic Editors: Anatoli Kheifets, Gleb Gribakin and Vadim Ivanov

Received: 5 November 2021 Accepted: 20 December 2021 Published: 29 December 2021

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variational basis functions of the type exp(*α*˜*r*1 + *β*˜*r*2 + *<sup>γ</sup>*˜*r*12) with complex exponents were used in the works of Korobov [14] (*N* = 1400–2200) and Frolov [15] for *N* = 600–2700 (see also references therein). Application of the Gaussian bases of the size *N* > 100 can be found in the book [16]. The reviews on the helium-like atomic system and the methods of their calculations can be found, e.g., in the handbook [17].

In this paper, we propose a simple method of calculation of the compact but very accurate WFs describing the two-electron atom/ion with collinear configuration. The results and example of application of the relevant technique are presented in the next sections.
