# Collisional Classical Dynamics at the Quantum Scale

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Classical Trajectory Monte Carlo Method

#### 2.2. Electron Capture

#### 2.3. Ionization

#### 2.4. Multiple-Electron Targets via One-Active-Electron Models

#### 2.5. Explicit Multiple-Electron Descriptions of He

#### 2.5.1. The Heisenberg Core CTMC

#### 2.5.2. The Bohr Atom

#### 2.5.3. The Split-Shell Model

#### 2.5.4. The Dynamical Screening CTMC

#### 2.5.5. The Energy-Bounded CTMC

#### 2.5.6. The Quasi-Classical Møller Approach

#### 2.5.7. The Soft-Core Coulomb Potential Model

#### 2.5.8. The Gaussian Kernel Approximation

#### 2.6. Explicit Multiple-Electron Descriptions of Atoms beyond He: The Sequential Electrons and Independent Electrons CTMC

#### 2.7. Explicit Multiple-Electron Descriptions of Molecules

#### 2.7.1. The H${}_{2}$ Molecule

#### 2.7.2. The Molecular Multicenter—CTMC Approaches

#### 2.7.3. The Classical Overbarrier—CTMC Approach

- –
- The virtual electron energy must overcome the potential saddle barrier.
- –
- A random number $\delta \in (0,1)$ is sorted and compared with the ratio $\Delta t/{T}_{e}$, ${T}_{e}$ being the classical electron orbital period. If $\delta <\Delta t/{T}_{e}$, the electron is created and randomly located within a sphere of a few atomic units of radius centered on the target.

#### 2.7.4. The Dynamical Adaptative CTMC Model

#### 2.8. Line Emission Cross Sections following Charge Exchange in Collisions Involving Highly Charged Projectiles and Multelectronic Targets

- –
- Multiply excited states dominantly stabilized via multiple Auger processes.
- –
- Only two-electron Auger processes are considered.
- –
- Transitions involving electrons in the same shell proceed first. If several electrons are in different shells, the Auger process involves the two electrons which are energetically closer.
- –
- Each Auger transition proceeds with the unit probability to the nearest continuum limit. The decaying electron falls to a well-established n level according to the energy conservation equation.
- –
- If the new configuration still provides a multiple-excited state involving more than two electrons, these rules are applied again until only two electrons remain bound to the projectile.
- –
- If a cascading process leads to an asymmetric double-excited state, the event is characterized as double radiative decay. Otherwise, a final Auger process takes place, and the event is characterized as a single charge exchange.

## 3. Conclusions and Perspectives

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**CTMC total cross section for single electron capture (SC) from H as a function of the projectile charge Z${}_{P}$ at an impact energy of 25 keV/u. A linear fit is incorporated in red to highlight the strict linear dependence on Z${}_{P}$.

**Figure 2.**CTMC n-state-selective charge-exchange cross sections for O${}^{8+}$ + H(1s) collisions. The dashed line indicates the ${n}^{-3}$ scaling expected for the population of high n-values.

**Figure 3.**CTMC (5,l)-state-selective charge-exchange cross sections for O${}^{8+}$ + H(1s) collisions: The dashed line indicates a statistical distribution arbitrarily normalized to the (5,2) cross section.

**Figure 5.**CTMC electron emission events represented in momentum space for 100 keV H${}^{+}$ + H(1s) collisions.

**Figure 6.**Interelectronic distance as a function of one of the electrons’ radial coordinates for the EB-CTMC and HC-CTMC models. The overlapping density plot in green represents the radial 2-particle density predicted by a correlated variational wavefunction.

**Figure 7.**Schematic diagram of Auger decay processes following multiple-electron capture events in an 8-electron sequential CTMC simulation of 4 keV/u Ar${}^{18}$+Ar collisions: (

**a**) Autoionizing multiple capture; (

**b**) double radiative decay. In these figures, the temporal sequence of events runs from left to right. Green arrows represent electrons emitted to the continuum; orange arrows represent electronic transitions to inner discrete levels; and red zigzag arrows represent photons emitted during the de-excitation process.

U${}_{\mathit{i}}$ (a.u.) | ${\mathit{\alpha}}_{\mathit{i}}$ |
---|---|

2 | 0.016 |

1 | 0.0984 |

0.6667 | 0.1923 |

0.5 | 0.2185 |

0.4 | 0.1849 |

0.3333 | 0.1349 |

0.2857 | 0.0920 |

0.25 | 0.0630 |

Z${}_{\mathit{i}}$ | ${\mathit{\alpha}}_{\mathit{i}}$ |
---|---|

0.5 | 0.0724 |

0.625 | 0.07658 |

0.75 | 0.09665 |

0.875 | 0.09230 |

1.0 | 0.16204 |

1.25 | 0.17971 |

1.5 | 0.13019 |

1.75 | 0.08071 |

2.0 | 0.06475 |

2.5 | 0.04493 |

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Otranto, S.
Collisional Classical Dynamics at the Quantum Scale. *Atoms* **2023**, *11*, 144.
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Otranto S.
Collisional Classical Dynamics at the Quantum Scale. *Atoms*. 2023; 11(11):144.
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2023. "Collisional Classical Dynamics at the Quantum Scale" *Atoms* 11, no. 11: 144.
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