# Electronic Structure Calculations with the Exact Pseudopotential and Interpolating Wavelet Basis

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Interpolating Wavelets on Half-Infinite Interval

#### 2.1. Interpolating Wavelets

#### 2.2. The Basis Set

## 3. Schrödinger Equations of Hydrogen-like Atoms and Helium Atoms in the EPP-Wavelet Basis

#### 3.1. General

#### 3.2. Hydrogen-like Atoms

#### 3.3. Hartree–Fock Equations for Helium Atom

#### 3.4. EPP of Helium Atom

#### 3.5. Total Energy of Helium Atom

## 4. Combination of EPP with Finite Difference Method

## 5. Results

## 6. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

EPP | Exact Pseudopotential Method |

OIW | Ordinary Interpolating Wavelet Method |

FDM | Finite Difference Method |

HF | Hartree–Fock |

## References

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**Figure 1.**Hydrogen 1s orbital eigenenergy relative error. The ${r}_{0}$ is the EPP radius in atomic units and W is the basis size.

**Figure 2.**Relative error of the wavefunction value at the core radius for the hydrogen 1s orbital. Notations as in Figure 1.

**Figure 3.**Hydrogen 2s orbital eigenenergy relative error. Notations as in Figure 1.

**Figure 4.**Relative error of the wavefunction value at the core radius for the hydrogen 2s orbital. Notations as in Figure 1.

**Figure 5.**Hydrogen 2p orbital eigenenergy relative error. Notations as in Figure 1.

**Figure 6.**Relative error of the wavefunction value at the core radius for the hydrogen 2p orbital. Notations as in Figure 1.

**Figure 8.**Relative error of the wavefunction value at the core radius for the 1s orbital of the ground state of the helium atom. Notations as in Figure 1.

**Table 1.**Hartree–Fock total energies of H and He occupation configurations. ${E}_{\mathrm{ref}}$ are chosen reference values. ${E}_{\mathrm{EPP}}$ is the energy given by the EPP–Galerkin-method, ${h}_{\mathrm{EPP}}$ is the grid spacing in the EPP–Galerkin method, ${E}_{\mathrm{OIW}}$ is the energy given by the OIW–Galerkin method, ${E}_{\mathrm{FDM}}$ is the energy given by the Finite Difference Method, and ${N}_{\mathrm{FDM}}$ is the number of grid points in the Finite Difference Method. For EPP, the most accurate results are given. For OIW computations, the number of basis functions is 201 and the finest grid spacing is $0.00625\phantom{\rule{0.277778em}{0ex}}\mathrm{a}.\mathrm{u}$.

System | ${\mathit{E}}_{\mathbf{ref}}/\mathbf{Ha}$ | ${\mathit{E}}_{\mathbf{EPP}}/\mathbf{Ha}$ | ${\mathit{h}}_{\mathbf{EPP}}/\mathbf{a}.\mathbf{u}.$ | ${\mathit{E}}_{\mathbf{OIW}}/\mathbf{Ha}$ | ${\mathit{E}}_{\mathbf{FDM}}/\mathbf{Ha}$ | ${\mathit{N}}_{\mathbf{FDM}}$ |
---|---|---|---|---|---|---|

H 1s | $-0.5\phantom{\rule{3.33333pt}{0ex}}{*}^{\u2020}$ | −0.500000 | 0.075 | −0.50 | −0.49803 | 1001 |

H 2s | $-0.125\phantom{\rule{3.33333pt}{0ex}}{*}^{\u2020}$ | −0.125000 | 0.125 | −0.125 | −0.124741 | 2001 |

H 2p | $-0.125\phantom{\rule{3.33333pt}{0ex}}{*}^{\u2020}$ | −0.125000 | 0.125 | −0.124998 | −0.124995 | 2001 |

He $1{\mathrm{s}}^{2}$ | $-2.8616800{\phantom{\rule{3.33333pt}{0ex}}}^{\u2020}$ [8] | −2.8617 | 0.075 | −2.834868 | −2.839 | 1001 |

$-2.903724\phantom{\rule{3.33333pt}{0ex}}*$ [9] | ||||||

He $1\mathrm{s}2\mathrm{s}\phantom{\rule{0.277778em}{0ex}}{}^{1}\mathrm{S}$ | $-2.147{\phantom{\rule{3.33333pt}{0ex}}}^{\circ}$ [10] | −2.1532 | 0.1 | −2.133579 | −2.132 | 1001 |

$-2.145974\phantom{\rule{3.33333pt}{0ex}}*$ [9] | ||||||

He $1\mathrm{s}2\mathrm{s}\phantom{\rule{0.277778em}{0ex}}{}^{3}\mathrm{S}$ | $-2.171{\phantom{\rule{3.33333pt}{0ex}}}^{\circ}$ [10] | −2.1742 | 0.1 | −2.154536 | −2.155 | 1001 |

$-2.175229\phantom{\rule{3.33333pt}{0ex}}*$ [9] |

^{†}: HF limit; ∘: HF result.

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**MDPI and ACS Style**

Höynälänmaa, T.; Rantala, T.T.
Electronic Structure Calculations with the Exact Pseudopotential and Interpolating Wavelet Basis. *Atoms* **2023**, *11*, 9.
https://doi.org/10.3390/atoms11010009

**AMA Style**

Höynälänmaa T, Rantala TT.
Electronic Structure Calculations with the Exact Pseudopotential and Interpolating Wavelet Basis. *Atoms*. 2023; 11(1):9.
https://doi.org/10.3390/atoms11010009

**Chicago/Turabian Style**

Höynälänmaa, Tommi, and Tapio T. Rantala.
2023. "Electronic Structure Calculations with the Exact Pseudopotential and Interpolating Wavelet Basis" *Atoms* 11, no. 1: 9.
https://doi.org/10.3390/atoms11010009