^{1}

^{1}

^{1}

^{1}

^{2}

^{1}

^{2}

^{3}.

The analytical solutions to a double ring-shaped Coulomb potential (RSCP) are presented. The visualizations of the space probability distribution (SPD) are illustrated for the two- (contour) and three-dimensional (isosurface) cases. The quantum numbers

Since the ring-shaped noncentral potentials (RSNCPs) are used to describe the molecular structure of Benzene as well as the interaction between the deformed nucleuses, they have attracted much attention of many authors [

The plan of this paper is as follows. We present the exact solutions to the system in Section

In the spherical coordinates, the double RSCP is given by

Potential function

Potential function

The Schrödinger equation with this potential is written as (

We are now in the position to consider (

As we know, the SPDs at the position

Taking a series of discrete positions, we may study the values of the respective SPD by numerical calculation. In order to make the graphic resolution better, one takes

The isosurface SPDs with a section plane.

| | | | | |
---|---|---|---|---|---|

2 | 1 | 0 | | | |

3 | 1 | 0 | | | |

3 | 2 | 1 | | | |

4 | 1 | 0 | | | |

4 | 2 | 1 | | | |

4 | 3 | 0 | | | |

4 | 3 | 2 | | | |

5 | 1 | 0 | | | |

5 | 2 | 1 | | | |

5 | 3 | 0 | | | |

5 | 3 | 2 | | | |

5 | 4 | 1 | | | |

5 | 4 | 3 | | | |

6 | 1 | 0 | | | |

6 | 2 | 1 | | | |

6 | 3 | 0 | | | |

6 | 3 | 2 | | | |

6 | 4 | 1 | | | |

6 | 4 | 3 | | | |

6 | 5 | 0 | | | |

6 | 5 | 2 | | | |

6 | 5 | 4 | | | |

The contour of the SPDs in the plane

| | | | | |
---|---|---|---|---|---|

2 | 1 | 0 | | | |

3 | 1 | 0 | | | |

3 | 2 | 1 | | | |

4 | 1 | 0 | | | |

4 | 2 | 1 | | | |

4 | 3 | 0 | | | |

4 | 3 | 2 | | | |

5 | 1 | 0 | | | |

5 | 2 | 1 | | | |

5 | 3 | 0 | | | |

5 | 3 | 2 | | | |

5 | 4 | 1 | | | |

5 | 4 | 3 | | | |

6 | 1 | 0 | | | |

6 | 2 | 1 | | | |

6 | 3 | 0 | | | |

6 | 3 | 2 | | | |

6 | 4 | 1 | | | |

6 | 4 | 3 | | | |

6 | 5 | 0 | | | |

6 | 5 | 2 | | | |

6 | 5 | 4 | | | |

We show the SPDs for various cases

Compared to the cases

We project the SPDs to a plane

To show the isosurface of the SPDs for various RPVs

The SPDs for various RPVs

| | |
---|---|---|

0.01 | | |

1 | | |

5 | | |

10 | | |

20 | | |

50 | | |

70 | | |

90 | | |

99 | | |

Considering given quantum numbers

The isosurface illustration of the state (5, 1, 0) with various values of

| Isosurface illustration | | Isosurface illustration |
---|---|---|---|

0 | | 0 | |

5 | | 5 | |

10 | | 10 | |

25 | | 25 | |

40 | | 40 | |

80 | | 80 | |

The comparison is done for positive and negative

SPDs for different cases of the value of

| Isosurface illustration | Contour illustration |
---|---|---|

0.5 | | |

0 | | |

−0.5 | | |

The analytical solutions to the double RSCP have been obtained and then the visualization of the SPDs for this potential is performed. The contour and isosurface visualizations have been illustrated for quantum numbers

The authors declare that there are no conflicts of interest.

This work is supported by the National Natural Science Foundation of China under Grant no. 11275165, partially by 20180677-SIP-IPN, and by CONACYT, Mexico, under Grant no. 288856-CB-2016. Professor Yuan You acknowledges Jiangsu Overseas Research & Training Program for University Prominent Young & Middle-Aged Teachers and Presidents for support.