*4.1. One-Particle Calculation*

In the one-particle frozen-target calculation, the wavefunction only depends on the positron radial coordinate *r*. The trial wavefunction (11) is employed for this calculation. The *ci* values calculated previously are used again here, but each eigenfunction is divided by *A* to fulfill the normalization condition (43).

In Figure 8, the eigenfunctions obtained using the present method are divided by *A* and multiplied by √*k*/*<sup>π</sup>*, and they are compared to the true continuous-spectrum radial function *Pk*(*r*) obtained by solving the radial Schrödinger equation [33] and normalized as

$$P\_k(r) \simeq \frac{\sin(kr + \delta)}{\sqrt{\pi k}}.\tag{46}$$

For small *r*, these were found to be in good agreement. As the energy eigenvalues increase, the range of *r* over which the wavefunctions closely match decreases. However, there is always a very good match at *r* ∼ 1 a.u., which dominates in the calculation of *Z*eff.

**Figure 8.** Eigenfunctions obtained from solving the generalized eigenvalue problem (6) in the static approximation (red) and those obtained using the atomic codes [33] (black) for *n* = 2, 4, 6 and 8.

The following integral is used to evaluate *Z*eff in this problem [13]:

$$Z\_{\rm eff} = \frac{1}{A^2} \int\_0^\infty P\_{1s}^2(r) \Psi^2(r) r^{-2} dr,\tag{47}$$

where *<sup>P</sup>*1*s*(*r*) = 2*re*<sup>−</sup>*<sup>r</sup>* is the ground-state radial wavefunction of the hydrogen atom. Substituting (11) into (47) gives the following expression for the annihilation parameter:

$$Z\_{\rm eff} = \frac{8}{A^2} \sum\_{i,j=1}^{N} \frac{\mathbf{c}\_i \mathbf{c}\_j}{(2 + \alpha\_i + \alpha\_j)^3}. \tag{48}$$

Figure 9 shows the corresponding frozen-target *Z*eff values and compares them with those obtained using true continuous-spectrum positron states in the same approximation [13,33]. Apart from some "noise" related to inaccuracies in the calculation of the normalization constant *A* at low energies, there is a good general agreemen<sup>t</sup> between the two sets of results.

**Figure 9.** Red circles: values of *Z*eff obtained in the one-particle model with variational wavefunctions from (48). Black line: *Z*eff data obtained for the frozen-target model from atomic codes [13,33].

## *4.2. Two-Particle Calculation*

In this section, the annihilation parameter is calculated for the two-particle problem. Firstly, the frozen-target results are reproduced by the two-particle code, using a wavefunction with the form of (39) with all *αi* = 1 and *γi* = 0. After verifying that the results match those from the one-particle calculation, these restrictions on the *αi* and *γi* values are lifted, and the full correlated wavefunction is used, subject to (40). Here, it is noted that *r*12 = 0 in the *Z*eff calculation, so exp(*<sup>γ</sup>ir*12) = 1 for each of the *γi*. In addition, *r*1 = *r*2 ≡ *r* is required, and the wavefunction takes the form

$$\Psi(r,r,0) = \sum\_{i=1}^{N} c\_i \exp[- (\alpha\_i + \beta\_i)r]. \tag{49}$$

As before, the *ci* coefficients are calculated such that the eigenfunctions are normalized to unity. Hence, the normalization must be corrected to satisfy (43) using the value of *A* for each eigenfunction.

In this case, evaluation of the integral (42) with the wavefunction (49) yields the following expression for *Z*eff:

$$Z\_{\rm eff} = \frac{16\pi^2}{A^2} \int\_0^\infty |\Psi(r, r, 0)|^2 r^2 dr = \frac{32\pi^2}{A^2} \sum\_{i, j = 1}^N \frac{c\_i c\_j}{(\alpha\_i + \alpha\_j + \beta\_i + \beta\_j)^3}. \tag{50}$$

The coefficients *ci* in this calculation differ from those in the one-particle calculation by a factor of 1/2*<sup>π</sup>*. Taking this into account, the equivalence of the one- and two-particle results can be verified by setting *αi* = *αj* = 1 in (50) to recover the result from the frozen-target approximation (48). Here, however, the focus is on including terms in the wavefunction to describe electron–positron correlations.

When incorporating correlation terms, the same sets of *αi*, *βi* and *γi* parameters are used in the basis as for the phase shift calculations in Section 3.2.3 (the values of which are listed in Appendix B). The accuracy of the *Z*eff calculation does not increase monotonically with the number of correlation terms included the wavefunction, unlike the case of the phase shifts determined by the energy eigenvalues alone. This is shown in Figure 10, where the overall results obtained using one correlation term are more accurate than those obtained using three correlation terms. However, the most accurate set of values

was obtained for the wavefunction with the maximum (nine) correlation terms included. The *Z*eff values from [26] are used as benchmark values to evaluate the accuracy of our calculation. The results from the final calculation are displayed in Table 3 alongside those from [26] interpolated to the same values of momentum.

**Table 3.** *Z*eff values obtained using a wavefunction with nine nonzero *γi* terms (Appendix B) to describe electron–positron correlations and the benchmark results from [26] (*Z*eff,H) interpolated to the same values of momentum.


**Figure 10.** *Z*eff values obtained in the two-particle model using one (green), three (yellow), five (pink) and nine (red) terms in the wavefunction to describe correlations between the electron and positron. Black circles and line: results from [26] and their interpolation using cubic splines.

Overall, good agreemen<sup>t</sup> between our results and the benchmark values is found in the range *k* = 0.1–0.6 a.u. using a trial wavefunction with nine nonzero *γi* terms. Apart from two data points, the accuracy of our variational calculation is better than 10%, which is evidence of the good quality of the wavefunction that incudes only a small number of correlation terms.
