**5. Conclusions**

The variational method was explored as a means of studying positron–hydrogen scattering and annihilation using square-integrable trial wavefunctions. By setting up and solving a generalized eigenvalue problem, *s*-wave elastic phase shifts and *Z*eff values for the positron–hydrogen system were obtained in good agreemen<sup>t</sup> with benchmark values. Importantly, this was achieved using only a small number of correlation terms in the trial wavefunction, indicating that processes such as virtual positronium formation and polarization of the hydrogen atom can be accounted for using this approach.

Looking forward, this method could facilitate the study of more complex interactions, such as the interaction of positrons with molecules. The key benefit of our approach is the small number of terms required to describe strong electron–positron correlations, meaning that the method is quite economical. To improve upon the current approach, a formal optimization of the trial wavefunction parameters could be performed to increase the accuracy of the calculation. With such a process in place, it would become possible to carry out more complex calculations efficiently.

**Author Contributions:** Methodology, G.G. and S.G.; software, S.G.; writing, G.G .and S.G.; supervision, G.G. All authors have read and agreed to the published version of the manuscript.

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

**Data Availability Statement:** All data are available from the authors upon request.

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

## **Appendix A. Standard Integrals**

This appendix contains results for the six standard integrals employed throughout our calculations. These are evaluated using the elliptic coordinate system: *s* = *r*1 + *r*2, *t* = *r*1 − *r*2 and *u* = *r*12:

$$\begin{split} I\_1(a,b,\emptyset) &= \pi^2 \int\_0^\infty ds \, e^{-2as} \int\_0^s du \, u \epsilon^{2bu} \int\_{-u}^u dt \, e^{-2\xi t} (s^2 - t^2) \\ &= \pi^2 \frac{8a^3 - 13a^2b + 6ab^2 - b^3 + bg^2}{8a^3((a-b)^2 - g^2)^3}, \end{split} \tag{A1}$$

$$\begin{split} I\_2(a,b,\emptyset,\emptyset) &= \pi^2 \int\_0^\infty ds \, s e^{-2as} \int\_0^s du \, e^{2bu} \int\_{-u}^u dt \, e^{-2gt} (u^2 - t^2) \\ &= \pi^2 \frac{5a^2 - 6ab + b^2 - g^2}{8a^2((a-b)^2 - g^2)^3}, \end{split} \tag{A2}$$

$$\begin{split} \tilde{I}\_3(a,b,\emptyset) &= \pi^2 \int\_0^\infty ds \, e^{-2as} \int\_0^s du \, (s^2 - u^2) e^{2bu} \int\_{-u}^u dt \, t e^{-2\emptyset t} \\ &= \pi^2 \frac{\mathcal{g}(-5a^2 + 6ab - b^2 + \mathcal{g}^2)}{8a^3((a-b)^2 - \mathcal{g}^2)^3}, \end{split} \tag{A3}$$

$$\begin{split} f\_1(a,b,\emptyset) &= \pi^2 \int\_0^\infty ds \, s e^{-2as} \int\_0^s du \, u e^{2bu} \int\_{-u}^u dt \, e^{-2gt} \\ &= \pi^2 \frac{(a-b)^2 (4a-b)\mathfrak{g} + b\mathfrak{g}^3}{8a^2 \mathfrak{g} ((a-b)^2 - \mathfrak{g}^2)^3}, \end{split} \tag{A4}$$

$$\begin{split} \mathcal{J}\_2(a,b,\emptyset) &= \pi^2 \int\_0^\infty ds \, e^{-2as} \int\_0^s du \, e^{2bu} \int\_{-u}^u dt \, e^{-2\circ t} (s^2 - t^2) \\ &= -\pi^2 \frac{-5a^2 + 4ab - b^2 + g^2}{8a^3((a-b)^2 - g^2)^2} \, , \end{split} \tag{A5}$$

$$\begin{split} \tilde{J}\_3(a,b,\emptyset) &= \pi^2 \int\_0^\infty ds \, e^{-2as} \int\_0^s du \, u e^{2bu} \int\_{-u}^u dt \, t e^{-2gt} \\ &= -\pi^2 \frac{(a-b)g}{2a((a-b)^2 - g^2)^3} .\end{split} \tag{A6}$$

#### **Appendix B. Parameters for Positron-Scattering Wavefunction Bases**

The positron–hydrogen wavefunctions used in Sections 3.2 and 4.2 all contain the 20 basis functions defined by (13) with *n* = 20, *ζ* = 1.5 and *β*1 = 0.01 (where all *αi* = 1 and *γi* = 0). Additional terms with varying *αi* and nonzero *γi* values were then incorporated to account for electron–positron correlations. The table below contains the sets of quasioptimal values for the parameters obtained using the method described in Section 3.2. These values are displayed for wavefunctions containing one, three, five and nine terms with nonzero *γi* where *NE* denotes the number of nonzero *γi* terms.

**Table A1.** Parameters used for the electron–positron correlation terms, i.e, terms with nonzero *γi*, in positron–hydrogen wavefunctions with *NE* = 1, 3, 5 and 9 such terms.

