# Matrix Methods for Solving Hartree-Fock Equations in Atomic Structure Calculations and Line Broadening

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## Abstract

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## 1. Introduction

## 2. The Hartree-Fock Method for Atomic Structure Calculations

- (1)
- Assume a wavefunction for the state of interest (hydrogenic is good enough for this step).
- (2)
- Determine a mean-Coulomb field acting on electron i based on the wavefunctions of the other electron.
- (3)
- Solve the one-electron Schrödinger equation for electron i in its mean-Coulomb field to generate a new set of wavefunctions.
- (4)
- Repeat steps (2) and (3) until convergence is achieved.

## 3. Finite Difference Matrix to Solve the Schrödinger Equation for One-Electron Atom

## 4. Matrix Form of the Hartree-Fock Equation

## 5. Extension to Free-Electron Wavefunctions

## 6. Application to Spectral Line Broadening

#### 6.1. Atomic Structure

#### 6.2. Electron-Atom Collisions

## 7. Summary

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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1 | |

2 | This method fails to capture some correlation effects because the potential in which the electron is moving is defined to be a mean field of the other electrons, which is to say that it does not account for the strong repulsion when the two electrons are close to each other. |

3 | If the two wavefunctions are assumed to be orthogonal, then these terms vanish. |

4 | In fact, radial solutions of three-electron problems can be solved on a laptop with the help of sparse matrix eigenvalue solvers. Any system larger than this would require the use of a supercomputer. |

5 | |

6 | This may be a dangerous approximation because some (non-negligible) processes need to be considered when orthogonality is not guaranteed. |

**Figure 2.**The ortho-helium wavefunctions compared with Cowan’s 1s 2s wavefunction. The new calculations with exact exchange are given in red, while Cowan’ calculation are in dot-dashed black lines.

**Figure 3.**The $l=0$ partial wave of the free-electron wavefunction under different approximations. Black dot dashed is the plane wave, dotted blue is using an LDA to approximate the exchange correlations, and solid red is the exact exchange treatment. The states are reacting to the presence of a hydrogen atom in the $2s$ state.

n | $\mathbf{\Delta}\mathit{r}=0.5$ | $\mathbf{\Delta}\mathit{r}=0.2$ | $\mathbf{\Delta}\mathit{r}=0.1$ | $\mathbf{\Delta}\mathit{r}=0.05$ | Exact |
---|---|---|---|---|---|

1 | −0.94427 | −0.99019 | −0.99751 | −0.99937 | −1.00000 |

2 | −0.24621 | −0.24938 | −0.24984 | −0.24996 | −0.25000 |

3 | −0.11035 | −0.11098 | −0.11108 | −0.11110 | −0.11111 |

4 | −0.06218 | −0.06237 | −0.06240 | −0.06241 | −0.06250 |

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**MDPI and ACS Style**

Gomez, T.; Nagayama, T.; Fontes, C.; Kilcrease, D.; Hansen, S.; Montgomery, M.; Winget, D. Matrix Methods for Solving Hartree-Fock Equations in Atomic Structure Calculations and Line Broadening. *Atoms* **2018**, *6*, 22.
https://doi.org/10.3390/atoms6020022

**AMA Style**

Gomez T, Nagayama T, Fontes C, Kilcrease D, Hansen S, Montgomery M, Winget D. Matrix Methods for Solving Hartree-Fock Equations in Atomic Structure Calculations and Line Broadening. *Atoms*. 2018; 6(2):22.
https://doi.org/10.3390/atoms6020022

**Chicago/Turabian Style**

Gomez, Thomas, Taisuke Nagayama, Chris Fontes, Dave Kilcrease, Stephanie Hansen, Mike Montgomery, and Don Winget. 2018. "Matrix Methods for Solving Hartree-Fock Equations in Atomic Structure Calculations and Line Broadening" *Atoms* 6, no. 2: 22.
https://doi.org/10.3390/atoms6020022