*3.1. One-Particle Problem*

To test the method, we consider the simple problem of positron scattering from a "frozen" hydrogen atom. In this case, the wavefunction depends only on the distance between the positron and the nucleus. Figure 1 contains a schematic diagram of the system with the interparticle distances labeled.

**Figure 1.** A diagram of the positron–hydrogen system with the interparticle distances labeled.

In the frozen-target approximation, the electron moves in the field of the nucleus (considered infinitely massive) and is "fixed" in the ground (1*s*) state of the atom. The dependence of the wavefunction on *r*12 is neglected. Hence, the total wavefunction becomes a product of the 1*s* electron wavefunction and the unknown positron wavefunction, which we denote <sup>Ψ</sup>(*r*), with *r* ≡ *r*2. Considering <sup>Ψ</sup>(*r*) as the wavefunction of the radial motion, the boundary condition at the origin Ψ(0) = 0 is imposed. In this problem, we select a trial wavefunction of the form

$$\Psi(r) = r \sum\_{i=1}^{N} c\_i \exp(-\beta\_i r),\tag{11}$$

where *ci* are variational parameters and *βi* are chosen real exponents. The corresponding basis functions are

$$\varphi\_{i}(r) = r \exp(-\beta\_{i}r), \qquad i = 1, \ldots, N. \tag{12}$$

In the following calculations, the exponents *βi* are chosen as

$$\beta\_i = \beta\_1 \mathbb{Z}^{i-1}, \qquad i = 1, \ldots, N,\tag{13}$$

i.e., forming an even-tempered basis, with *ζ* = 1.5, *N* = 20 and *β*1 = 0.01.

The elements of matrix *Q* are calculated as follows:

$$Q\_{i\bar{j}} = \int\_0^\infty \varrho\_i(r)\varrho\_{\bar{j}}(r)dr = \frac{2}{(\beta\_i + \beta\_{\bar{j}})^3}.\tag{14}$$

Next, the Hamiltonian for the system is considered1. The electrostatic potential of the ground-state hydrogen atom is (see, e.g., Ref. [32], §36, Problem 2):

$$
\hat{\Omega} = \frac{1}{r} + \phi\_c(r) = \left(\frac{1}{r} + 1\right)e^{-2r},
\tag{15}
$$

where *φe*(*r*) is the mean-field potential of the electron cloud and the 1/*r* term accounts for the positron-nucleus interaction.

In addition to *U* , the Hamiltonian of the radial motion of the positron contains its kinetic energy, hence

$$
\hat{H} = -\frac{1}{2}\frac{d^2}{dr^2} + \left(\frac{1}{r} + 1\right)e^{-2r}.\tag{16}
$$

ˆ

The Hamiltonian matrix elements are calculated as follows:

$$H\_{i\bar{j}} = \int\_0^\infty q\_i(r) \left( -\frac{1}{2} \frac{d^2 q\_j(r)}{dr^2} \right) dr + \int\_0^\infty q\_i(r) \left( \frac{1}{r} + 1 \right) e^{-2r} q\_j(r) dr,$$

$$= -\frac{1}{2} \frac{\beta\_i \beta\_j}{(\beta\_i + \beta\_j)^3} + \frac{\beta\_i + \beta\_j + 4}{(\beta\_i + \beta\_j + 2)^3}. \tag{17}$$

The generalized eigenvalue problem (6) for the matrices (14) and (17) was then solved using a simple Python code, and the phase shifts were calculated as described in Section 2.2.

Figure 2 is a plot of *n* against ln *E* for the basis (13). It explains how the phase shifts are found from the pseudostate energy eigenvalues. By construction, for the free-positron eigenvalues *E*(0) *n* (obtained using *H*ˆ = −12 *d*2/*dr*2), the function *g*(ln *E*(0) *n* ) takes integer values, but for the eigenvalues *En* of the positron in the static hydrogen potential, this function takes non-integer values *n* − *δ*/*<sup>π</sup>*, which yield *δ* for specific positron energies *En*.

**Figure 2.** Red circles: values of *n* = 1, 2, ... plotted against ln *E*(0) *n* . Black line: the function *n* = *g*(ln *E*) obtained using cubic-spline interpolation between the free-particle eigenvalue data. Yellow circles: the points on the interpolated curve for the positron energies *En* in the static hydrogen potential. From this, it can be seen that ln *En* corresponds to non-integer ordinates *n* − *δ*/*<sup>π</sup>*.

The phase shifts for low-momentum positrons were compared to those obtained from a numerical solution of the radial Schrödinger equation in the static hydrogen potential (obtained using the codes described in [33]). This comparison is displayed in Figure 3. There is a good agreemen<sup>t</sup> between these sets of data, especially at low momenta *k*, providing evidence that the present variational method allows one to extract the scattering phase shifts from a simple bound-state calculation.

**Figure 3.** Positron–hydrogen *s*-wave scattering phase shifts in the static approximation. Red data points: phase shifts obtained using the present variational method. Black line: data obtained by solving the radial Schrödinger equation using the suite of codes described in [33].

## *3.2. Two-Particle Problem*

In this section, a full two-particle dynamics of positron scattering from a hydrogen atom is considered. Here, the electron is no longer fixed in the 1*s* state of the hydrogen atom and generally, the wavefunction for this system will depend on the distances between all three pairs of particles2, as labeled in Figure 1. A wavefunction of the following form will be considered:

$$\Psi(r\_1, r\_2, r\_{12}) = \sum\_{i=1}^{N} c\_i \exp\left(-\alpha\_i r\_1 - \beta\_i r\_2 + \gamma\_i r\_{12}\right),\tag{18}$$

where the values of coefficients *αi*, *βi* and *γi* are chosen and the constants *ci* are the variational parameters. The integrals to be evaluated in this section are greatly simplified by using the elliptic (Hylleraas [34]) coordinate system (*s* = *r*1 + *r*2, *t* = *r*1 − *r*2, *u* = *r*12), so these coordinates are employed to carry out all of the calculations. The full set of standard integrals used is found in Appendix A. Our calculations begin with a simplified version of (18) using a single value of *αi* = 1 and *γi* = 0, i.e., equivalent to the frozen-target approximation of Section 3.1, gradually building toward the more general case. With each added element of flexibility in the wavefunction, a more accurate solution to the scattering problem is obtained.

Labeling the basis functions

$$\varphi\_i(r\_1, r\_2, r\_{12}) = \exp(-a\_i r\_1 - \beta\_i r\_2 + \gamma\_i r\_{12}),\tag{19}$$

elements of the overlap matrix *Q* are calculated as follows:

$$Q\_{ij} = \int \varrho\_i \, \varrho\_j d\tau,\tag{20}$$

where *dτ* = *π*<sup>2</sup>(*s*<sup>2</sup> − *t*<sup>2</sup>)*uds dt du* is the volume element, and the integration is over 0 ≤ *s* < <sup>∞</sup>, 0 ≤ *u* ≤ *s*, −*u* ≤ *t* ≤ *u*. Substituting *ϕi* from (19) into (20), we find the overlap integral in the form

$$Q\_{i\bar{j}} = \int \exp\left[2(-A\_{i\bar{j}}\mathbf{s} - B\_{i\bar{j}}\mathbf{t} + \Gamma\_{i\bar{j}}\mathbf{u})\right] d\tau\_{\prime} \tag{21}$$

where

$$A\_{ij} = \frac{1}{4}(\alpha\_i + \beta\_i + \alpha\_j + \beta\_j),\tag{22}$$

$$B\_{ij} = \frac{1}{4}(\boldsymbol{\alpha}\_i - \beta\_i + \boldsymbol{\alpha}\_j - \beta\_j),\tag{23}$$

$$
\Gamma\_{i\dot{j}} = \frac{1}{2} (\gamma\_i + \gamma\_{\dot{j}}).\tag{24}
$$

This integral shares its structure with the standard integral ˜*I*1 from Appendix A; hence,

$$Q\_{ij} = \tilde{I}\_1(A\_{ij\prime} \Gamma\_{ij\prime} B\_{ij}).\tag{25}$$

The Hamiltonian operator of the system is given by

$$\hat{H} = -\frac{1}{2}\nabla\_1^2 - \frac{1}{2}\nabla\_2^2 - \frac{1}{r\_1} + \frac{1}{r\_2} - \frac{1}{r\_{12}}.\tag{26}$$

The first two terms represent the kinetic energy of the electron and positron *T*ˆ. The third and fourth terms describe the interaction of the electron and positron with the nucleus *U*ˆ , and the final term represents the electron–positron interaction *V*ˆ . Hence, the Hamiltonian matrix element *Hij* is considered as the sum of three contributions:

$$H\_{ij} = T\_{ij} + \mathcal{U}\_{ij} + V\_{ij}.\tag{27}$$

In the elliptic coordinates, the expectation value of the kinetic energy takes the form

$$
\langle \Psi | \hat{\mathcal{T}} | \Psi \rangle = \int \left\{ \left( \frac{\partial \Psi}{\partial s} \right)^2 + \left( \frac{\partial \Psi}{\partial t} \right)^2 + \left( \frac{\partial \Psi}{\partial u} \right)^2 \right.
$$

$$
+ \frac{2}{\mathfrak{u}(s^2 - t^2)} \frac{\partial \Psi}{\partial u} \left[ s(u^2 - t^2) \frac{\partial \Psi}{\partial s} + t(s^2 - u^2) \frac{\partial \Psi}{\partial t} \right] \right\} d\tau. \tag{28}
$$

Replacing one of the Ψ by *φi* and the other by *φj*, and mapping the integrals that arise to the set of standard integrals in Appendix A, one obtains

$$T\_{i\bar{j}} = \frac{1}{4} \left[ (a\_i + \beta\_i)(a\_{\bar{j}} + \beta\_{\bar{j}}) + (a\_{\bar{i}} - \beta\_{\bar{i}})(a\_{\bar{j}} - \beta\_{\bar{j}}) + 4\gamma\_i\gamma\_{\bar{j}} \right] I\_1(A\_{i\bar{j}}, \Gamma\_{i\bar{j}}, B\_{i\bar{j}})$$

$$-\gamma\_{\bar{i}} \left[ (a\_{\bar{j}} + \beta\_{\bar{j}})I\_2(A\_{i\bar{j}}, \Gamma\_{i\bar{j}}, B\_{i\bar{j}}) + (a\_{\bar{j}} - \beta\_{\bar{j}})I\_3(A\_{i\bar{j}}, \Gamma\_{i\bar{j}}, B\_{i\bar{j}}) \right]. \tag{29}$$

The matrix element of the electron and positron interaction with the nucleus is

$$
\Delta I\_{i\bar{j}} = \int \varphi\_i \left[ -\frac{2}{s+t} + \frac{2}{s-t} \right] \varphi\_{\bar{j}} d\tau. \tag{30}
$$

This integral is reduced to the standard integrals ˜*J*1 and ˜*J*3 (Appendix A), which gives

$$\mathcal{L}I\_{\vec{i}\vec{j}} = -2[\tilde{f}\_1(A\_{\vec{i}\vec{j}}, \Gamma\_{\vec{i}\vec{j}}, B\_{\vec{i}\vec{j}}) - \tilde{f}\_3(A\_{\vec{i}\vec{j}}, \Gamma\_{\vec{i}\vec{j}}, B\_{\vec{i}\vec{j}})] + 2[\tilde{f}\_1(A\_{\vec{i}\vec{j}}, \Gamma\_{\vec{i}\vec{j}}, B\_{\vec{i}\vec{j}}) + \tilde{f}\_3(A\_{\vec{i}\vec{j}}, \Gamma\_{\vec{i}\vec{j}}, B\_{\vec{i}\vec{j}})].\tag{31}$$

Clearly, the ˜*J*1 terms will cancel here. However, when calculating the free-positron energy eigenvalues, only the first bracketed term on the right-hand side of this equation is required, since the positron–nucleus interaction (second term) is not included in the free-positron Hamiltonian. When both of the Coulomb terms are included, the expression simplifies to

$$\mathcal{U}I\_{\mathrm{ij}} = 4 \sfcorner \!\!\!/ \_3 (\mathcal{A}\_{\mathrm{ij}}, \Gamma\_{\mathrm{ij}}, B\_{\mathrm{ij}}).\tag{32}$$

Lastly, the matrix element of the electron–positron Coulomb interaction is given by

$$V\_{i\bar{j}} = -\int \varphi\_i \frac{1}{u} \varphi\_{\bar{j}} d\tau. \tag{33}$$

This integral reduces to the standard integral ˜ *J*2 in Appendix A to give

$$V\_{\rm ij} = -\mathbb{J}\_2(\mathcal{A}\_{\rm ij}, \Gamma\_{\rm ij}, B\_{\rm ij}).\tag{34}$$

Combining the results in (29), (32) and (34), an expression for the Hamiltonian matrix element is obtained:

$$\begin{split} H\_{\text{ij}} &= \frac{1}{4} \Big[ (\boldsymbol{a}\_{i} + \boldsymbol{\beta}\_{i})(\boldsymbol{a}\_{\boldsymbol{j}} + \boldsymbol{\beta}\_{\boldsymbol{j}}) + (\boldsymbol{a}\_{i} - \boldsymbol{\beta}\_{i})(\boldsymbol{a}\_{\boldsymbol{j}} - \boldsymbol{\beta}\_{\boldsymbol{j}}) + 4\gamma\_{\boldsymbol{i}}\gamma\_{\boldsymbol{j}}\big] I\_{1}(\boldsymbol{A}\_{\boldsymbol{i}\boldsymbol{j}}, \boldsymbol{\Gamma}\_{\text{ij}}, \boldsymbol{B}\_{\text{ij}}) \\ &- \gamma\_{\boldsymbol{i}} \big[ (\boldsymbol{a}\_{\boldsymbol{j}} + \boldsymbol{\beta}\_{\boldsymbol{j}})I\_{2}(\boldsymbol{A}\_{\boldsymbol{i}\boldsymbol{j}}, \boldsymbol{\Gamma}\_{\text{ij}}, \boldsymbol{B}\_{\text{ij}}) + (\boldsymbol{a}\_{\boldsymbol{j}} - \boldsymbol{\beta}\_{\boldsymbol{j}})I\_{3}(\boldsymbol{A}\_{\boldsymbol{i}\boldsymbol{j}}, \boldsymbol{\Gamma}\_{\text{ij}}, \boldsymbol{B}\_{\text{ij}}) \Big] \\ &+ 4\boldsymbol{\J}\_{3}(\boldsymbol{A}\_{\boldsymbol{i}\boldsymbol{j}}, \boldsymbol{\Gamma}\_{\text{ij}}, \boldsymbol{B}\_{\text{ij}}) - \boldsymbol{\J}\_{2}(\boldsymbol{A}\_{\boldsymbol{i}\boldsymbol{j}}, \boldsymbol{\Gamma}\_{\text{ij}}, \boldsymbol{B}\_{\text{ij}}). \end{split} \tag{35}$$

Note that for *γi* = 0, the matrix element (35) derived using equation (28) is not symmetric, i.e., *Hij* = *Hji*. Hence, *Hij* must be replaced by the symmetrized combination

$$H'\_{i\dot{j}} = \frac{1}{2}(H\_{i\dot{j}} + H\_{\dot{j}i})\_\star \tag{36}$$

before solving the generalized eigenvalue problem (6), which is consistent with the derivation in Section 2.1.

After setting up the Hamiltonian and overlap matrices, the generalized eigenvalue problem (6) is solved for the energy eigenvalues and eigenvectors. The eigenvalues for the free positron (omitting the positron–nucleus and positron–electron interaction terms) and those for the full Hamiltonian are analyzed to extract the phase shifts, as outlined in Section 2.2. It is noted here that solving (6) provides energy eigenvalues of the whole system. Hence, to obtain the positron energies *E*(0) *n* and *En*, the energy of the ground-state hydrogen atom (−0.5 a.u.) must be subtracted from the eigenvalues.

#### 3.2.1. Reproducing the Frozen-Target Results.

In the first instance, the frozen-target problem is revisited to check that the twoparticle code produces the same results as the one-particle code. The dependence of the wavefunction on *r*12 is eliminated by setting *γi* = 0. In addition, the electron is fixed in the ground state by setting *αi* = 1. These restrictions will subsequently be lifted to allow the electron to move and to account for the electron–positron correlations. As before, describing the positron requires a wide range of exponents *β<sup>i</sup>*, which are defined as in (13).

Taking all of this into account, we have the following wavefunction:

$$\Psi(r\_1, r\_2) = \sum\_{i=1}^{N} c\_i \exp[-r\_1 - \beta\_i r\_2]. \tag{37}$$

Figure 4 shows that the corresponding phase shifts match those from the one-particle problem 3.1. Figure 4 also shows the phase shifts over a larger range of positron momenta. As expected, at large projectile energies, the scattering phase shift tends to zero. Note also that the phase shift is negative at all energies. This is a consequence of the positron–atom interaction being repulsive in the frozen-target approximation.

**Figure 4.** Positron–hydrogen s-wave scattering phase shifts in the static approximation calculated using a two-particle model (red circles), plotted over two contrasting ranges of *k*. The meaning of the black curve is the same as in Figure 3. Agreement with Figure 3 can be noted.

#### 3.2.2. Variation of *α*: Radial Correlations.

To probe the effect of electron–positron radial correlations, the wavefunction is augmented by including extra terms with *αi* = 1. Physically, this adjustment allows the incident positron to cause displacement of the atomic electron in the radial direction due to the attraction between the two particles. Inclusion of a term, or several terms, in the wavefunction with *αi* = 0.5 will facilitate this type of distortion. The wavefunction in this case is written as

$$\Psi(r\_1, r\_2) = \sum\_{i=1}^{N} c\_i \exp[-\alpha\_i r\_1 - \beta\_i r\_2]. \tag{38}$$

For now, the restriction on the *γi* parameters remains in place, in that *γi* = 0 and the dependence of the wavefunction on the electron–positron distance is neglected. Note that this approximation corresponds to the so-called Temkin–Poet model that was used earlier to test electron–hydrogen and positron–hydrogen scattering [35].

Firstly, a single term is added with *αi* = 0.5 and *βi* = 0.4. This value of *βi* is chosen because we expect radial correlations to be important when the positron is close to the hydrogen atom. At such distances, it is most likely to attract the electron sufficiently to cause significant distortion. Terms with *αi* = 0.5 and a full range of *βi* values are not immediately introduced because we aim to achieve good accuracy with as few correlation terms as possible. Hence, extra terms are introduced individually to test their importance: if a notable change in the phase shift is seen by adding a particular term, the term is retained and used in the basis. If not, the term is discarded and a different choice is made.

Phase shifts were obtained for various sets of parameters, using up to 20 additional terms. It was found that overall, the effect on the phase shift from adding these terms is small. In Figure 5, a set of results is displayed for a calculation with just three additional terms in the basis, which were found to generate a close-to-maximum shift from the frozen-target results (with *αi* = 0.5 and *βi* = 0.2, 0.4, 1.0).

A comparison of Figure 5 with Figure 4 shows that the phase shifts have become slightly less negative due to the addition of the extra terms. This means that electron– positron correlations make the positron–atom interaction less repulsive than in the frozentarget case. However, the overall effect of allowing for the radial correlations between the electron and positron remains very small.

**Figure 5.** Phase shifts obtained by addition of three terms with *αi* = 1 to the basis. Black line: numerical solution of the radial Schrödinger equation in the static potential of the hydrogen atom included for comparison.

#### 3.2.3. Nonzero *γ*: Effect of Angular Correlations.

In this section, the flexibility of the wavefunction is increased further by allowing for nonzero values of *γi*, so that the wavefunction takes the most general form

$$\Psi(r\_1, r\_2, r\_{12}) = \sum\_{i=1}^{N} c\_i \exp\left(-\alpha\_i r\_1 - \beta\_i r\_2 + \gamma\_i r\_{12}\right). \tag{39}$$

The addition of *<sup>r</sup>*12-dependence to the wavefunction allows for much stronger correlation between the positron and the electron. Physically, these terms account for effects such as virtual Ps formation and polarization of the atom by the positron. In particular, setting *γi* = −0.5 corresponds to the ground-state Ps wavefunction, allowing the calculation to account for the effect of virtual positronium formation. Note that formation of "real", free Ps is not possible in the chosen energy range, as the incident positron momenta are kept below the Ps formation threshold.

In general, terms with any values of *βi* and *γi* may be used, provided that

$$
\beta\_i - \gamma\_i \ge 0,
$$

to ensure that <sup>Ψ</sup>(*<sup>r</sup>*1,*r*2,*r*12) → 0 for *r*2 → ∞.

Taking all of this into consideration, an initial basis was set up identically to that in the frozen-target problem with all values of *γi* = 0 and *αi* = 1. The nonzero *γi* terms were added one by one. Quasi-optimal values of the parameters for these extra terms were selected by completing the calculations for different sets of exponents and keeping the term which caused the largest upward change in the phase shifts overall. In the present approach, larger phase shifts are obtained when the energy eigenvalues *En* are lower relative to *E*(0) *n* . In a variational calculation, lower energy eigenvalues are obtained when better wavefunctions are used. Hence, it is correct to assume that the best possible choice of terms to add to the basis is that which yields the largest values for the phase shifts. Physically, including electron–positron correlations allows for positron attraction to the atom, increasing the value of the phase shift. Small adjustments are made to all three parameters near the optimum to ensure the best possible value of each parameter, correct to two decimal places.

For a single correlation term, the optimal values of *α*, *β* and *γ* were found to be *α* = 0.80, *β* = 0.04 and *γ* = −0.54. Once the first term had been optimized, a second was added. The values of the parameters for this term were also selected in the manner described above. With each additional term, an improvement (i.e., increase) in the phase shift values is seen.

This process of adding an individual term may be continued for as many terms as required to reach a desired level of accuracy. However, our aim was to achieve good accuracy using as few terms as possible. Hence, the process was terminated after including a maximum of nine additional terms, yielding a basis with a total of 29 functions.

Figure 6 is an overview of the phase shifts obtained using one, three, five and nine additional terms. These results are also shown in Table 1. Details of the parameters used in these calculations can be found in Appendix B. The results obtained here can be compared to the accurate phase shifts, such as those calculated by Schwartz [23] or, later, by Humberston et al. [24]. In Figure 6, the frozen-target phase shifts are also plotted as a lower bound, while the accurate results from a Kohn variational calculation by Humberston et al. [24] provide an upper bound. It is remarkable that including a single well-chosen correlation term with *α* = 0.80, *β* = 0.04 and *γ* = −0.54 provides about 80% of the increase in the phase shift with respect to the uncorrelated frozen-target result. Adding the next few correlation terms brings the variational phase shift to within 0.01 rad of the benchmark result.

**Figure 6.** Positron–hydrogen *s*-wave scattering phase shifts obtained using various numbers of terms with *γi* = 0 in the wavefunction (colored circles, blue: one term, green: three terms, yellow: five terms, red: nine terms). Details of the parameters used in each wavefunction can be found in Appendix B. Pink circles are the Kohn variational calculations of Humberston et al. [24], which are connected by the dashed line to guide the eye. The black solid line is the result of the frozen-target approximation.

In Figure 7, the final phase shifts obtained using all nine correlation (i.e., nonzero *γi*) terms are displayed. Compared with an interpolation of the Kohn variational results of Humberston et al. [24], agreemen<sup>t</sup> is seen to within 8 × 10−<sup>3</sup> rad. In Table 2, the present results obtained with nine correlation terms are shown alongside the results of Ref. [24] interpolated to the same momentum values. This level of agreemen<sup>t</sup> provides evidence that the method employed here is a valid approach to the positron scattering problem, and that it is possible to obtain good-quality scattering data from a bound-state-type calculation that contains only a small number of correlation terms in the wavefunction.


**Table 1.** Results from positron–hydrogen *s*-wave phase shift calculations with wavefunctions containing *NE* = 1, 3, 5 and 9 terms with nonzero *γi* to describe electron–positron correlations (see Appendix B). Phase shift values are shown for the first 11 eigenvalues *En*.

**Figure 7.** Positron–hydrogen *s*-wave scattering phase shifts obtained using nine nonzero *γi* terms in the wavefunction with various *αi*, *βi* and *γi* values (red circles). Details of the parameters used in this wavefunction can be found in Appendix B. Pink circles connected by the dashed line are calculations of Humberston et al. [24].

**Table 2.** Results from positron–hydrogen *s*-wave phase shift calculations with nine *γi* = 0 terms in the wavefunction (see Appendix B) for the first 11 energy eigenvalues *En*. The phase shifts of Humberston et al. [24] *δ*H interpolated to the same values of *k* are also shown.


#### **4. Calculation of the Annihilation Parameter**

In this section, the quality of the variational wavefunctions constructed as described in Section 3 is probed by calculating the normalized annihilation rate, *Z*eff. *Z*eff is the effective number of electrons available to the positron for annihilation [36]. For a positron incident on the hydrogen atom, it is given by [13]:

$$Z\_{\rm eff} = \iint \delta(\mathbf{r}\_1 - \mathbf{r}\_2) |\Psi\_k(\mathbf{r}\_1, \mathbf{r}\_2)|^2 d^3 r\_1 d^3 r\_2. \tag{41}$$

where the wavefunction is normalized to the incident positron plane wave, i.e., <sup>Ψ</sup>*k*(*<sup>r</sup>*1,*r*2) *ψ*<sup>1</sup>*s*(*<sup>r</sup>*1) exp(*i<sup>k</sup>* · *<sup>r</sup>*2), or, for *s*-wave positron scattering, <sup>Ψ</sup>*k*(*<sup>r</sup>*1,*r*2) *ψ*<sup>1</sup>*s*(*<sup>r</sup>*1) sin(*kr*2 + *<sup>δ</sup>*)/*kr*2.

Carrying out the integration over *r*1 and renaming *r*2 ≡ *r* gives

$$Z\_{\rm eff} = \int \left| \Psi\_k(\mathbf{r}, \mathbf{r}) \right|^2 d^3 \mathbf{r}. \tag{42}$$

In this integral, |<sup>Ψ</sup>*k*(*<sup>r</sup>*,*<sup>r</sup>*)|<sup>2</sup> is the electron–positron contact density localized near the atom. Unlike the scattering phase shifts which characterize the wavefunction at large positron distances, the *Z*eff parameter probes the wavefunction at small positron–atom separations. Here, the bound-state-type variational wavefunction Ψ is proportional to the true continuous spectrum wavefunction Ψ*k*, i.e., we have

$$
\Psi(r\_1, r\_2) = \frac{A}{\sqrt{4\pi}} \Psi\_k(r\_1, r\_2),
\tag{43}
$$

where *A* is a normalization constant. For *s*-wave scattering, the normalization constant is obtained from the energy eigenvalue spectrum as (see Ref. [31] for details)

$$A^2 = \frac{2\sqrt{2E}}{\pi} \frac{dE}{dn}.\tag{44}$$

The wavefunctions Ψ generated by solving the generalized eigenvalue problem (as outlined in Section 3) are automatically normalized to unity. Hence, to achieve the correct normalization, these wavefunctions must be divided by *A* when calculating *Z*eff from (42). The value of *A* is calculated for each eigenfunction by substituting the corresponding energy eigenvalue into (44). The derivative *dE*/*dn* is evaluated using the function *n* = *g*(ln *E*) from Section 3 and the fact that

$$\frac{dE}{dn} = E \frac{d \ln E}{dn}.\tag{45}$$

This facilitates a more accurate calculation of the derivative than that obtained by directly calculating *dE*/*dn*.

The annihilation parameter is first calculated in the frozen-target approximation, after which the two-particle problem is considered to include electron–positron correlations.
