# Algebraic DVR Approaches Applied to Describe the Stark Effect

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## Abstract

**:**

## 1. Introduction

## 2. Algebraic DVR Methods in 1D Systems

#### 2.1. 3D Harmonic Oscillator

#### 2.2. HO-DVR Approach

#### 2.3. $SU\left(4\right)$-UGA

## 3. Coulombic Potential: Hydrogen Atom

## 4. Stark Effect

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Correlation diagram from the harmonic oscillator ($\kappa =0$) to the Coulomb potential ($\kappa =1$) corresponding to the Hamiltonians (36) and (68), with (77) and (81), respectively. At the right of the figures the principal quantum number for the electron in the H atom, n, is indicated. To simplify only the levels of the Hydrogen system with $L,l=0,1,2$ are included. In the bottom panel a zoom is displayed in order to show the convergence of the levels with different angular momenta. The top level in each group corresponds to angular momentum $L,l=0$. The parameters were taken to be $\u03f5=1$ and $N=4001$ for the $U\left(4\right)$-unitary group approach (UGA), while $N=2001$ for the HO-DVR method.

**Figure 3.**Comparison between exact and calculated radial wave functions ${R}_{21}\left(r\right),{R}_{31}\left(r\right),{R}_{32}\left(r\right)$ for the Hydrogen atom using (75) and (79) with parameters $N=2001$ and $N=4001$ respectively, with $\u03f5=1$. The dash lines correspond to the exact wave functions.

**Figure 4.**Effect of the electric field effect over the subspace $n=2$ with $m=M=0$ in the Hydrogen atom. The black solid circles correspond to spectrum obtained with the $U\left(4\right)$-UGA taking $N=3501$ up to L = 16, while the blue triangles correspond to HO-DVR method taking N = 3001 up to l = 11. Since the calculations were carried out independently, the plotted points do not coincide. The continuous line correspond to the variational approach described in Ref. [32]. $|\overline{E}|$ is the electric field in atomic units.

**Figure 5.**Electric field effect over the $n=2$ levels of Hydrogen atom for $M=m=0$. The black solid circles correspond to the results using the $U\left(4\right)$-UGA method taking N = 3501 and L = 16. The blue triangles correspond to HO-DVR method taking N = 3001 and l = 11.

**Figure 6.**Zoom of the electric field effect over the subspace $n=2$ with $m=M=0$ in the Hydrogen atom provided by the HO-DVR method for $N=1501$ (red triangles) and $N=3001$ (black triangles).

**Figure 7.**Electric field effect over the subspace characterized by the principal quantum number $n\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}3$ with $M=m=0$ in the Hydrogen atom. The solid black dots correspond to the $U\left(4\right)$-UGA method taking N = 3501 up to L = 16, while the blue triangles correspond to the HO-DVR method taking N = 3001 up to l = 11. The continuous lines correspond to the exact results provided by Ref. [32].

**Table 1.**Exact bond energies compared with the energies provided by the HO-DVR method, the basis dimension was taken to be $N=2001$.

Basis Dimension N | |||
---|---|---|---|

2001 | |||

${\mathit{E}}_{\mathit{n}}$ | $\mathit{L}=\mathbf{0}$ | $\mathit{L}=\mathbf{1}$ | $\mathit{L}=\mathbf{2}$ |

−1. | −0.9969 | ||

−0.25 | −0.2496 | −0.2500 | |

−0.1111 | −0.1111 | −0.1111 | −0.1111 |

−0.0625 | −0.0625 | −0.0625 | −0.0625 |

−0.04 | −0.0400 | −0.0400 | −0.0400 |

−0.0278 | −0.0275 | −0.0276 | −0.0276 |

−0.0204 | −0.0173 | −0.0175 | −0.0180 |

**Table 2.**Exact bond energies compared with the energies provided by the diagonalization of (68) taking a total number of bosons $N=4001$ with the $U\left(4\right)$-UGA method.

Total Number of Bosons N | |||
---|---|---|---|

4001 | |||

${\mathit{E}}_{\mathit{n}}$ | $\mathit{L}=\mathbf{0}$ | $\mathit{L}=\mathbf{1}$ | $\mathit{L}=\mathbf{2}$ |

−1. | −1.00330 | ||

−0.25 | −0.25088 | −0.25039 | |

−0.1111 | −0.11237 | −0.11222 | −0.11220 |

−0.0625 | −0.06463 | −0.06457 | −0.06455 |

−0.0400 | −0.04344 | −0.04340 | −0.04336 |

−0.0278 | −0.0330 | −0.0330 | −0.0330 |

−0.0204 | −0.02541 | −0.02555 | −0.02586 |

**Table 3.**Errors of the energies provided by the HO-DVR method. Errors were calculated considering the difference of energies $\left|E\right(N=2051)-E(N=1951\left)\right|$.

Error for $\mathit{L}=0$ | Error for $\mathit{L}=1$ | Error for $\mathit{L}=2$ |
---|---|---|

0.0001 | ||

0.00002 | 9 × 10${}^{-9}$ | |

5 × 10${}^{-6}$ | 3 × 10${}^{-9}$ | 8 × 10${}^{-13}$ |

2 × 10${}^{-6}$ | 1 × 10${}^{-9}$ | 1 × 10${}^{-11}$ |

2 × 10${}^{-6}$ | 5 × 10${}^{-7}$ | 3× 10${}^{-7}$ |

0.00008 | 0.00007 | 0.00005 |

0.0006 | 0.0005 | 0.0005 |

**Table 4.**Errors of the energies provided by the $U\left(4\right)$-UGA method. Errors were calculated considering the difference of energies $\left|E\right(N=4051)-E(N=3951\left)\right|$.

Error for $\mathit{L}=0$ | Error for $\mathit{L}=1$ | Error for $\mathit{L}=2$ |
---|---|---|

0.00007 | ||

0.00002 | 0.00001 | |

0.00003 | 0.00003 | 0.00003 |

0.00005 | 0.00005 | 0.00005 |

0.00009 | 0.00009 | 0.00009 |

0.0001 | 0.0001 | 0.0001 |

0.00008 | 0.00006 | 0.00003 |

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**MDPI and ACS Style**

Bermúdez-Montaña, M.; Rodríguez-Arcos, M.; Lemus, R.; Arias, J.M.; Gómez-Camacho, J.; Orgaz, E.
Algebraic DVR Approaches Applied to Describe the Stark Effect. *Symmetry* **2020**, *12*, 1719.
https://doi.org/10.3390/sym12101719

**AMA Style**

Bermúdez-Montaña M, Rodríguez-Arcos M, Lemus R, Arias JM, Gómez-Camacho J, Orgaz E.
Algebraic DVR Approaches Applied to Describe the Stark Effect. *Symmetry*. 2020; 12(10):1719.
https://doi.org/10.3390/sym12101719

**Chicago/Turabian Style**

Bermúdez-Montaña, Marisol, Marisol Rodríguez-Arcos, Renato Lemus, José M. Arias, Joaquín Gómez-Camacho, and Emilio Orgaz.
2020. "Algebraic DVR Approaches Applied to Describe the Stark Effect" *Symmetry* 12, no. 10: 1719.
https://doi.org/10.3390/sym12101719