**3. Results**

The two-exponential representations (excepting the case of *Z* = 1) for the two-particle coalescences only (corresponding to the particular cases *λ* = 0 and *λ* = 1) were reported in Ref. [25]. In the current paper, we calculate the parameters **C***N* and **b***N* of the model WFs, *fN*(*r*) ≡ *fN*(*r*; *λ*) for the number of exponentials 3 ≤ *N* ≤ 6. Our calculations are represented for various *collinear* configurations including in particular the two-particle coalescences and the boundary case *λ* = −1. The results are presented in Tables 1–4 together with the corresponding accuracy estimations *RN* and the sets {*<sup>n</sup>*1, *<sup>n</sup>*2,...,<sup>2</sup>(*<sup>N</sup>* − 1)} of integers included into the integrals (14). It is seen from all tables that the more exponentials generate the higher accuracy of the model WF.

One should note that for *λ* = 0, describing the case of the electron–nucleus coalescence, we were able to calculate the model WFs, *fN*(*r*) represented by three and four exponentials only (*N* = 3, 4). However, at least the case of *N* = 4 shows very high accuracy, which is confirmed by the following. Recall that the integral *RN* characterizes the general accuracy of *fN*(*r*). In order to track changes in accuracy with distance *r* we used the logarithmic function of the form

$$L\_N^{(\lambda)}(r) = \log\_{10}|1 - f\_N(r)/\Phi\_{PLM}(r,\lambda)|.\tag{17}$$

It is seen from Tables 1 and 2 that at least for *λ* = 0 and given *N* the minimal accuracy (represented by maximum *RN*) is demonstrated by the negative ion H<sup>−</sup>, whereas the maximum accuracy (represented by minimum *RN*) is demonstrated by the positive ion B3+. The logarithmic functions *L*(0*N* (*r*) are shown in Figures 1 and 2 for these two-electron ions with boundary (under consideration) nucleus charges *Z* = 1 and *Z* = 5. It is seen that the deviations of the model WF from the PLM WF are practically uniform along the *r*-axis, and that one extra exponential improves accuracy by 1-2 (decimal) orders. Regarding the accuracy of the model WF, *f*4(*r*) we would like to emphasize the following. In Ref. [1] (see Fig. 3(b) therein) it was displayed the logarithmic function L(*r*) of the form (17), which describes the difference between the PLM WF and the CFHHM WF for the *λ* = 0 *collinear* configuration of the H− ion. The so called correlation function hyperspherical harmonic method (CFHHM) [4,5] with the maximum HH indices *Km* = 128 (1089 HH basis functions) was used for calculation of the fully (3-dimensional) WF of the negative ion H<sup>−</sup>. Comparison of the logarithmic estimations *L*(04 (*r*) and the corresponding L(*r*) shows that the model WF *f*4(*r*) is even more close to the PLM WF than the CFHHM WF for all values

of *r*, which indicates the *extremely high accuracy* of the model WF (at least for *λ* = 0 and *Z* = 1) represented by *four exponentials* only. It is seen (see Figure 2) that the accuracy of the model WF *f*4(*r*) for B3<sup>+</sup> is higher by about 2 decimal orders than the 4-exponential WF for H<sup>−</sup>. The logarithmic estimation L(*r*) for B3<sup>+</sup> is not presented in Ref. [1]. However, the relevant calculations show that for this case (*λ* = 0 and *Z* = 5), the model WF *f*4(*r*) is more close to the PLM WF than the CFHHM WF, as well.

**Figure 1.** Negative ion of hydrogen H−(*Z* = 1): (**a**) the WF, <sup>Φ</sup>(*<sup>r</sup>*, 0) at the electron-nucleus coalescence (the *collinear* configuration with *λ* = 0) times *r*; (**b**) the logarithmic estimates L(04 (*r*) and L(03 (*r*) of the difference (see Equation (17)) between the model WF, *f*4(*r*) and the PLM WF (solid curve, blue online), and between the model WF, *f*3(*r*) and the PLM WF (dashed curve, red online), respectively.

**Figure 2.** Ground state of the positive ion of boron B3<sup>+</sup>(*Z* = 5): (**a**) the WF, <sup>Φ</sup>(*<sup>r</sup>*, 0) at the electron– nucleus coalescence (the *collinear* configuration with *λ* = 0) times *r*; (**b**) the logarithmic estimates L(04 (*r*) and L(03 (*r*) of the difference between the model WF, *f*4(*r*) and the PLM WF (solid curve, blue online), and between the model WF, *f*3(*r*) and the PLM WF (dashed curve, red online), respectively.

It was mentioned earlier that the behavior of the two-electron atomic WF near the nucleus is described by the Fock expansion (6), which reduces to expansion (7) for the *collinear* arrangemen<sup>t</sup> of the particles. The most compact model WFs represented by the sum of three or four exponentials were obtained for the case of the electron-nucleus coalescence corresponding to the *collinear* parameter *λ* = 0. Tables 1 and 2 together with Figures 1 and 2 demonstrate the high accuracy of those model WFs. It should be emphasized that the accuracy of *f*4(*r*) for *λ* = 0 is close to the accuracy of the variational PLM WF, <sup>Φ</sup>*PLM*(*<sup>r</sup>*, 0) for all *r* > 0. Furthermore, the relevant calculations show that the model WF *f*4(*r*) mentioned above is, in fact, more accurate than <sup>Φ</sup>*PLM*(*<sup>r</sup>*, 0) in the vicinity of nucleus (*r* → 0). We can argue this because the leading terms of the series expansion of *f*4(*r*) (for *λ* = 0) are more close to the corresponding terms of the Fock expansion than the

ones for <sup>Φ</sup>*PLM*(*<sup>r</sup>*, <sup>0</sup>). Actually, Equations (10) and (11) provide by definition the condition *f*4(0) = 1 and *f* 4(0) = −*Z* + 1/2, corresponding exactly to the Fock expansion. Moreover, it is seen from Equation (9)) that for *λ* = 0 the logarithmic term of the Fock expansion is annihilated because *ζ*0 = 0, and hence *F*(0)/2 = *ξ*0, where we denoted *<sup>F</sup>*(*r*) ≡ <sup>Φ</sup>(*<sup>r</sup>*, <sup>0</sup>). One should notice that *λ* = 0 is, in fact, the single case of the *collinear* arrangemen<sup>t</sup> when the explicit expression for the angular Fock coefficient *ξλ* can be derived in the form [1,22]

$$
\zeta\_0 = \frac{1 - 2E}{12} - Z\left(\frac{3 - \ln 2}{6}\right) + \frac{1}{3}Z^2,\tag{18}
$$

where *E* is the non-relativistic energy of the two-electron atom/ion under consideration. It is seen from Table 5 that (besides *f* 4(0)) the values of *f* 4 (0)/2 is much closer to the theoretical values (18) than *<sup>F</sup>PLM*(0)/2 for all *Z*. These results confirm the above conclusion about the accuracy of the model WF near the nucleus.

**Table 5.** The first and second derivatives of the *collinear* WF with *λ* = 0 at the nucleus. The PLM WF, *<sup>F</sup>*(*r*) ≡ <sup>Φ</sup>*PLM*(*<sup>r</sup>*, 0) at the electron–nucleus coalescence is introduced.


**Author Contributions:** All authors contributed equally. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Acknowledgments:** This work was supported by the PAZY Foundation, Israel.

**Conflicts of Interest:** The authors declare no conflict of interest.
