# Gaussian Curvature Effects on Graphene Quantum Dots

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

## 3. Results and Discussion

#### 3.1. Curvature Energy

#### 3.2. Regeneration Times

#### 3.2.1. Classical Time

#### 3.2.2. Revival Time

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

DFT | Density Functional Theory |

GGA | Generalized Gradient Approximation |

LDA | Local Density Approximation |

LUMO | Lower Unoccupied Molecular Orbital |

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**Figure 1.**Hexagonal, flat graphene quantum dot used as a starting point for deformation. Image generated with Gaussview 6 [91].

**Figure 2.**The four different geometries considered in this study for the graphene dot with $R=50$ Å and their respective equations: (

**a**) sphere; (

**b**) one-sheet hyperboloid; (

**c**) x-cylinder; (

**d**); y-cylinder. Images generated with GaussView 6 [91].

**Figure 3.**Boundary conditions’ effects on the optimized geometries of an initially spherical quantum dot with $R=40$ Å. Images generated with Gaussview 6 [91]. (

**a**) Fixed surface; (

**b**) fixed edges; (

**c**) fixed vertices.

**Figure 4.**Curvature energy vs. $1/{R}^{2}$ for all four ideal geometries—all atoms forced to lay on the surface—with the flat dot taken as energy origin. Both cylindrical cases give almost identical energies.

**Figure 5.**Curvature energy vs. $1/{R}^{2}$ plots for all geometries, with the flat dot taken as energy origin.

**Figure 6.**View of ${\left|A\left(t\right)\right|}^{2}$ as a function of t for a spherical dot with $R=100$ Å. Analytical values of both regeneration times are shown with dotted lines (orange for classical time, yellow for revival time).

**Figure 7.**Classical time plots as functions of $1/{R}^{2}$ for all four ideal geometries, with numerical values as markers and analytical ones as dotted lines. Both cylinders show near perfect coincidence.

**Figure 8.**Classical time as a function of $1/{R}^{2}$ for different boundary conditions within each geometry, with numerical values as points and analytical ones as dotted lines.

**Figure 9.**Revival time as a function of $1/{R}^{2}$ for all four ideal geometries, with numerical values as points and analytical ones as dotted lines.

**Figure 10.**Revival time as a function of $1/{R}^{2}$ for different boundary conditions within each geometry, with the numerical values as markers and the analytical ones as dotted lines.

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de-la-Huerta-Sainz, S.; Ballesteros, A.; Cordero, N.A. Gaussian Curvature Effects on Graphene Quantum Dots. *Nanomaterials* **2023**, *13*, 95.
https://doi.org/10.3390/nano13010095

**AMA Style**

de-la-Huerta-Sainz S, Ballesteros A, Cordero NA. Gaussian Curvature Effects on Graphene Quantum Dots. *Nanomaterials*. 2023; 13(1):95.
https://doi.org/10.3390/nano13010095

**Chicago/Turabian Style**

de-la-Huerta-Sainz, Sergio, Angel Ballesteros, and Nicolás A. Cordero. 2023. "Gaussian Curvature Effects on Graphene Quantum Dots" *Nanomaterials* 13, no. 1: 95.
https://doi.org/10.3390/nano13010095