**1. Introduction**

The aim of this paper is to explore a numerically frugal method of including important electron–positron correlations in the calculations of positron (*e*<sup>+</sup>) scattering from atoms and molecules. A good understanding of positron interactions with matter is a crucial element in the development of current and future applications of antimatter [1,2]. It is also important for tests of quantum electrodynamics [3] and fundamental experiments with antihydrogen [4].

Since the prediction [5] and discovery [6] of the positron's existence, many experimental and theoretical studies have been focussed on revealing the nature of its interactions with atoms and molecules [7]. Measurements and calculations show that low-energy positron interaction with atoms and molecules is characterized by strong electron–positron correlations. The first of these correlation effects is polarization of the target electron distribution by the positron. It gives rise to the attractive polarization potential with the asymptotic form −*αe*2/2*r*4, where *α* is the dipole polarizability of the target, *e* is the charge of the projectile (positron), and *r* is the distance between the positron and the target. This polarization potential is similar to that which affects electron scattering.

The second correlation effect, which is specific to positrons, is *virtual positronium formation*. Positronium (Ps) is a light hydrogen-like atom that consists of an electron and a positron. Ps has a binding energy of *E*Ps = 6.8 eV. For positron energies *ε* > *EI* − *E*Ps, where *EI* is the ionization energy of the target, Ps formation is an important ionization channel in positron collisions. For targets with *EI* > *E*Ps and *ε* < *EI* − *E*Ps, the Ps formation channel is closed. However, atomic electrons can still tunnel from an atom or molecule to the positron to form a Ps-like state temporarily. This effect makes a distinct and sizeable attractive contribution to the interaction of low-energy positrons with atoms and molecular targets. At the same time, this contribution makes positron scattering and annihilation calculations particularly challenging.

Amusia and co-workers [8] were probably the first to recognize the importance of virtual Ps formation. They were able to incorporate this effect and gauge its magnitude using

**Citation:** Gregg, S.K.; Gribakin, G.F. Calculation of Low-Energy Positron-Atom Scattering with Square-Integrable Wavefunctions. *Atoms* **2022**, *10*, 97. https://doi.org/ 10.3390/atoms10040097

Academic Editor: Yew Kam Ho

Received: 1 September 2022 Accepted: 18 September 2022 Published: 22 September 2022

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many-body theory calculations for He. (Very recently, this approach was used for positron scattering from atoms with half-filled valence shells [9].) A more accurate approximation for the Ps formation contribution to the positron–atom correlation potential [10] enabled predictions of positron binding to neutral atoms [11] and reliable calculations of positron scattering from noble gas atoms [12]. Ultimately, a consistent *ab initio* method for calculating the Ps formation contribution was developed and tested [13]. It provided a complete and accurate picture of positron scattering and annihilation from noble-gas atoms [14], and has now been generalized to molecular calculations that can yield high-quality predictions of positron–molecule binding energies [15].

Many-body theory allows one to identify the virtual Ps formation contributions with a particular class of diagrams that contribute to the positron–target correlation potential. When other approaches are used, the physical effect of virtual Ps formation is still present, but it manifests itself in a different way. In single-center convergen<sup>t</sup> close-coupling calculations of positron scattering from hydrogen, one observes it as slow convergence with respect to the maximum orbital momentum of the electron and positron states used [16]. This is also seen in configuration–interaction calculations of positron–atom bound states [17]. Such high-angular-momentum states are needed to describe an electron–positron pair (Ps) localized some distance away from the atomic nucleus. This "problem" is immediately removed, however, when a two-center approach is used, in which functions that depend on the electron–positron distance (and hence, describe Ps) are included in the expansion of the wavefunction [18]. It was also seen in Kohn-variational calculations [19,20] that the inclusion of such "virtual Ps" terms in the wavefunction yields significant improvements in the convergence of the scattering phase shifts and a pronounced enhancement of the positron annihilation rate at energies just below the Ps formation threshold. Finally, when the Schwinger multichannel method is used for positron scattering from molecules [21], calculations are significantly improved by adding basis states on extra centers placed away from the atomic nuclei [22]. In this case, such centers help to describe Ps formed virtually outside the molecule. Such "ghost" centers are also used in the most sophisticated manybody theory calculations of positron–molecule binding to enable the accurate description of virtual Ps formation [15].

It can be seen from the above that a well-converged positron scattering calculation should either include a large number of wavefunction terms centered on the nuclei or include terms with explicit dependence on the electron–positron distance. The first approach is more straightforward numerically but may lead to very large basis sizes. The second one is more economical but with an added complexity of dealing with a multicenter problem. It is the latter approach that we want to explore, aiming to include as few correlation terms as strictly necessary to obtain good-quality scattering and annihilation data.

In this paper, the scattering and annihilation of positrons is explored for the positron– hydrogen system through use of the variational method with square-integrable trial wavefunctions. Important correlation effects, including that of virtual Ps formation, are accounted for by including functions which depend on the electron–positron distance. The variational method is set up as a generalized eigenvalue problem. From this, elastic *s*-wave phase shifts *δ* and the annihilation parameter *Z*eff for the *e*+-H system are calculated at energies below the Ps formation threshold. Good agreemen<sup>t</sup> with benchmark values of both the phase shifts [23–25] and the annihilation parameter [26] is achieved using only a small number of terms in the wavefunction. By providing evidence that this method is a valid approach to the problem, avenues for future research are opened in which more complex matter–antimatter interactions may be explored. It should be added that the positron–hydrogen system has long been used as a testbed for various calculation methods, with many accurate results available at both low and high energies (see, e.g., Refs. [27–29]).

The paper is structured as follows. In Section 2, we set up the generalized eigenvalue problem which is employed to solve the scattering problem and show how to obtain the scattering phase shifts from bound-state calculations. In Section 3, the method is applied to elastic *s*-wave scattering of a positron from a hydrogen atom. Three sets of phase shift results are presented, beginning with a simple model and progressing toward more detailed descriptions of the system. In Section 4, the annihilation parameter *Z*eff is calculated using the same trial wavefunctions as in Section 3. Section 5 summarizes the work and indicates its future applications.

#### **2. Scattering as a Bound-State Problem**

In this section, we recap how a simple variational method can be used to calculate *s*-wave elastic scattering phase shifts for scattering from an atomic target.

## *2.1. Generalized Eigenvalue Problem*

The method begins with the choice of a trial wavefunction. Consider a system in the state |Ψ- expanded in terms of a set of linearly independent square-integrable basis functions {|*ϕi*-}*Ni*=<sup>1</sup>which, in general, are neither normalized nor orthogonal:

$$\left| \Psi \right> = \sum\_{i=1}^{N} c\_i \left| \varphi\_i \right> . \tag{1}$$

This basis is chosen at the beginning of the problem, and the coefficients *ci* are the variational parameters.

Central to the problem is the minimization of the energy functional,

$$
\langle E \rangle = \langle \Psi | \hat{H} | \Psi \rangle \,, \tag{2}
$$

with respect to the parameters *ci*, whilst holding Ψ|Ψ- = 1. The minimum energy calculated using a trial wavefunction provides an upper bound on the exact ground-state energy of the system.

The normalization constraint is imposed during the minimization through use of a Lagrange multiplier *E*. At the minimum (or a stationary point), we require

$$\frac{\partial}{\partial \mathcal{L}\_k} \left[ \langle \Psi | \hat{H} | \Psi \rangle - E(\langle \Psi | \Psi \rangle - 1) \right] = 0, \qquad \qquad k = 1, \ldots, N. \tag{3}$$

Substituting the expansion of |Ψ- from (1) into (3) gives a system of *N* linear equations. Assuming that the *ci* values are independent of each other, performing partial differentiation with respect to a particular *ck* yields the following:

$$\sum\_{j} c\_{j} \left< \varphi\_{k} \right| \hat{H} \left| \varphi\_{j} \right> + \sum\_{i} c\_{i} \left< \varphi\_{i} \right| \hat{H} \left| \varphi\_{k} \right> - E \left( \sum\_{j} c\_{j} \left< \varphi\_{k} \middle| \varphi\_{j} \right> + \sum\_{i} c\_{i} \left< \varphi\_{i} \middle| \varphi\_{k} \right> \right) = 0. \tag{4}$$

Since the matrix elements of the Hamiltonian are real (assuming real basis functions |*ϕi*-), the first two sums in (4) are identical and hence may be combined. Similarly, the scalar product of any two basis functions in our problem is real, and the second pair of sums may also be combined. This results in the following equation:

$$\sum\_{i} \underbrace{\langle \varrho\_{k} | \hat{H} | \varrho\_{i} \rangle}\_{H\_{ki}} c\_{i} = E \sum\_{i} \underbrace{\langle \varrho\_{k} | \varrho\_{i} \rangle}\_{Q\_{ki}} c\_{i\prime} \tag{5}$$

where *Hki* and *Qki* are the elements of matrices *H* and *Q*, respectively. Hence, (5) takes the form of a matrix equation:

$$H\mathbf{c} = E\mathbf{Q}\mathbf{c},\tag{6}$$

where the vector *c* contains the *ci* values.

The eigenvalues *En* of the generalized eigenvalue problem (6) are energy eigenvalues of the system with Hamiltonian *H* ˆ , with the state |Ψ- defined by coefficients *ci*, i.e., the elements of the corresponding eigenvector. For a system that has a few bound states or no bound states at all, most of the energy eigenvalues will lie in the continuum. The corresponding states |Ψ-, often called *pseudostates*, will not be the true states of the system that represent scattering states. However, it is possible to use the energies and wavefunctions of the pseudostates to determine important properties of the scattering states, e.g., phase shifts or (for positrons) the normalized annihilation rate *Z*eff.

In this work, the generalized eigenvalue problem is solved using Python's eigh function [30] which, given matrices *H* and *Q*, provides the energy eigenvalues and normalized eigenvectors of the system.

## *2.2. Scattering Phase Shifts*

Once the energy eigenvalues have been calculated, they can be used to find the phase shifts *δ*, e.g., for *s*-wave scattering. This assumes that the target is spherically symmetric and states |Ψ- have zero total angular momentum. The method used by Gribakin and Swann in [31] is implemented here. Firstly, the eigenvalue problem is solved for a free particle, i.e., using a chosen basis-state expansion but neglecting the interaction between the projectile and the target in *H* ˆ . The free-particle energy eigenvalues are denoted *E*(0) *n* and increase monotonically with *n*. Hence, it is possible to introduce an invertible function *f*(*n*) such that

$$f(n) = E\_n^{(0)}\tag{7}$$

for *n* = 1, ... , *N*. Next, the eigenvalue problem is solved with the full Hamiltonian using the same basis. The corresponding eigenvalues *En* are shifted with respect to those in equation (7), which can be written as

$$E\_{\rm nl} = f\left(n - \frac{\delta}{\pi}\right),\tag{8}$$

where *δ* is the phase shift [31].

Rearranging (8), the phase shifts may be extracted as a function of the energy eigenvalues *En*:

$$\delta = \left[ n - f^{-1}(E\_{\mathfrak{u}}) \right] \pi. \tag{9}$$

With the introduction of a phase shift *δ*, the function *f*(*n*) must now be defined for real values of *n*, and equivalently, its inverse must be defined for values of energy other than the free-particle energy eigenvalues. The value of *f* −<sup>1</sup>(*E*) for these intermediate values of energy may be found by interpolating between the free-particle energy eigenvalues. However, for particular bases, e.g., those using even-tempered exponents, the function *f*(*n*) varies rapidly and is difficult to interpolate accurately. In this case, a new function *g*(ln *E*) can be defined, such that *f* −<sup>1</sup>(*E*) ≡ *g*(ln *<sup>E</sup>*); this function changes more slowly, making accurate interpolation possible. In the problems that follow, this interpolation is completed using a cubic spline fit to the data for integer *n*. The phase shift may then be calculated as

$$\delta = \left[ n - \lg(\ln E\_n) \right] \pi. \tag{10}$$

In the following analysis, the phase shift will be considered as a function of momentum *k* = √2*E* rather than energy *E*, where we use atomic units and assume that the projectile (positron) has a unit mass and set *E* = 0 at the continuum threshold.

#### **3. Calculation of Elastic** *s***-Wave Positron–Hydrogen Phase Shifts**

In this section, the variational method is used to calculate elastic scattering phase shifts for a positron scattering from a hydrogen atom. The calculation is restricted to *s*-wave scattering which dominates at low positron energies.
