# Many-Body and Single-Body Low-Energy Elastic Positron Scattering by Beryllium Atoms: From Ab Initio to Semiempirical Approaches

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{2}[11], Li

_{2}[12], and N

_{2}[13] as a result of positron impact. Curiously, as far as we know, helium is the only atomic target that has been studied with this method [14,15]. Since beryllium has only four electrons and a simple closed-shell electronic structure, it is the next system in the order of complexity and an excellent candidate for AI many-body calculations.

## 2. Methods and Procedures

#### 2.1. Ab Initio Calculations

#### 2.2. Connecting Polarization, Correlation, and Scattering Length

- (1)
- $R=2.750\phantom{\rule{0.166667em}{0ex}}{a}_{0}$ is representative of the positron–target correlation of the SMC, and from now on, it will be considered a fixed value;
- (2)
- ${\alpha}_{d}=45.60\phantom{\rule{0.166667em}{0ex}}{a}_{0}^{3}$ is representative of the electronic correlation of the target as considered in the SMC.

#### 2.3. Semiempirical Approach

#### 2.4. A Final Remark

_{2}molecules [39], we showed that the SMC provides similar cross sections to those calculated with model correlations when only the ${\alpha}_{d}$ term is considered. Consequently, in this work, we focus on the study of cross sections while considering exclusively the effect of the ${\alpha}_{d}$ polarization. Hyperpolarization effects will be addressed in a future article.

## 3. Results and Discussion

^{2}2s

^{2}(

^{1}S) → 1s

^{2}2s2p (

^{1}P

^{0}) is 2.7 eV [41]. Therefore, the scattering is genuinely elastic for impact energies below ≈2.5 eV.

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

HF | Hartree–Fock |

SMC | Schwinger multichannel method |

CGF | Cartesian Gaussian functions |

FCSVM | Fixed-core stochastic variational method |

ST | Static |

AI | Ab initio |

SEMP | Semiempirical |

MP | Model potential |

SCF | Self-consistent field |

GAMESS | General atomic and molecular electronic structure system |

ICS | Integral cross section |

## Note

1 | With regard to experimental data for alkaline earth atoms, total cross section data are available for Mg [16]. |

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**Figure 1.**Scattering length A as a function of the dipole polarizability ${\alpha}_{d}$ according to Equation (6). The curve was obtained with $R=2.750\phantom{\rule{0.166667em}{0ex}}{a}_{0}$ in order to reproduce the scattering length obtained with the SMC.

**Figure 2.**Phase shifts in radians as a function of the impact energy in eV computed with the SMC and the semiempirical model (SEMP) according to the SMC reference data: $A=20.1$ a${}_{0}$, ${\alpha}_{d}=45.6$ a${}_{0}^{3}$, and $\rho =3.1069$ a${}_{0}$ (see the first line of Table 2). ${\delta}_{0}$, ${\delta}_{1}$, and ${\delta}_{2}$ denote the s-, p-, and d-wave phase shifts, respectively. Some small discrepancies are seen for the p and d waves for energies above 1 eV, while the s wave is well described, as expected.

**Figure 3.**S-wave phase shifts for positron–Be as a function of the dipole polarizability ${\alpha}_{d}$. The values of ${\alpha}_{d}$ are given in ${a}_{0}^{3}$. A yellow line marks ${\delta}_{0}=0$ in order to identify the suppression of the s-wave cross section.

**Figure 4.**The same as Figure 3, but for the p wave. Note that no shape resonance is found for any value of ${\alpha}_{d}$. The legends are the same.

**Table 1.**Basis set of Cartesian Gaussian functions (CGFs) considered to describe the ground state of the Be atom and the positron scattering orbitals in this work. The CGF core was extracted from [23], and functions were added to obtain 15 of each kind. All functions are uncontracted.

Type S | Type P | Type D |
---|---|---|

2155.379 | 7.334782 | 9.180140 |

320.2894 | 1.554565 | 7.180140 |

71.05837 | 0.430418 | 3.080140 |

19.48182 | 0.143014 | 1.556213 |

6.177410 | 0.050654 | 0.955513 |

2.205970 | 0.020871 | 0.758656 |

1.861726 | 0.010000 | 0.525648 |

0.173778 | 0.008000 | 0.102306 |

0.092853 | 0.006000 | 0.019911 |

0.064906 | 0.003000 | 0.003875 |

0.010000 | 0.001000 | 0.000754 |

0.008000 | 0.000800 | 0.000146 |

0.005866 | 0.000100 | 0.000028 |

0.001540 | 0.000050 | 0.000005 |

0.000502 | 0.000010 | 0.000001 |

${\mathit{\alpha}}_{\mathit{d}}/{\mathit{a}}_{0}^{3}$ | $\mathit{A}/{\mathit{a}}_{0}$ | $\mathit{\rho}/{\mathit{a}}_{0}$ |
---|---|---|

45.6 | 20.1 | 3.1069 |

43.0 | 26.9 | 3.1117 |

41.0 | 36.8 | 3.1137 |

39.0 | 60.4 | 3.1154 |

37.0 | 191.6 | 3.1156 |

35.0 | −255.5 | 3.0818 |

**Table 3.**Scattering lengths A and dipole polarizabilities ${\alpha}_{d}$ considered for positron–Be according to previous works. All values are in atomic units.

Reference | ${\mathit{\alpha}}_{\mathit{d}}/{\mathit{a}}_{0}^{3}$ | $\mathit{A}/{\mathit{a}}_{0}$ |
---|---|---|

SMC (present)-AI | 45.60 | 21.12 |

Poveda (2016) et al. [9]-AI | 37.8 | 13.3 |

Reid (2014) et al. [8]-MP | 37.8 | 13.8 |

Bromley et al. (1998) [7]-SEMP | 38.0 | 16.2 |

Szmytkowski (1993) [5]-AI | 45.62 | 67.6 |

Kurtz and Jordan (1981) [2]-MP | 37.8 | not reported |

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**MDPI and ACS Style**

Barp, M.V.; Tenfen, W.; Arretche, F.
Many-Body and Single-Body Low-Energy Elastic Positron Scattering by Beryllium Atoms: From Ab Initio to Semiempirical Approaches. *Atoms* **2023**, *11*, 8.
https://doi.org/10.3390/atoms11010008

**AMA Style**

Barp MV, Tenfen W, Arretche F.
Many-Body and Single-Body Low-Energy Elastic Positron Scattering by Beryllium Atoms: From Ab Initio to Semiempirical Approaches. *Atoms*. 2023; 11(1):8.
https://doi.org/10.3390/atoms11010008

**Chicago/Turabian Style**

Barp, Marcos V., Wagner Tenfen, and Felipe Arretche.
2023. "Many-Body and Single-Body Low-Energy Elastic Positron Scattering by Beryllium Atoms: From Ab Initio to Semiempirical Approaches" *Atoms* 11, no. 1: 8.
https://doi.org/10.3390/atoms11010008