# TAO-DFT Study on the Electronic Properties of Diamond-Shaped Graphene Nanoflakes

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## Abstract

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## 1. Introduction

## 2. Computational Details

## 3. Results and Discussion

#### 3.1. Singlet-Triplet Energy Gap

#### 3.2. Vertical Ionization Potential/Electron Affinity as Well as Fundamental Gap

#### 3.3. Symmetrized von Neumann Entropy

#### 3.4. Active Orbital Occupation Numbers

#### 3.5. Real-Space Representation of Active Orbitals

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Singlet-triplet energy gap of n-pyrene ((

**a**) n = 2–15 and (

**b**) n = 4–15), calculated by spin-unrestricted TAO-LDA.

**Figure 3.**Vertical ionization potential of ground-state n-pyrene, calculated by spin-unrestricted TAO-LDA.

**Figure 4.**Vertical electron affinity of ground-state n-pyrene, calculated by spin-unrestricted TAO-LDA.

**Figure 6.**Symmetrized von Neumann entropy of ground-state n-pyrene, calculated by spin-unrestricted TAO-LDA.

**Figure 7.**Active orbital occupation numbers (H−9, H−8, …, H, L, …, L+8, and L+9) of ground-state n-pyrene, calculated by spin-restricted TAO-LDA. HOMO/LUMO is denoted as H/L for brevity.

**Figure 8.**Real-space representation of H−2 (2.000), H−1 (2.000), H (1.998), L (0.002), L+1 (0.000), and L+2 (0.000) of ground-state 2-pyrene, calculated by spin-restricted TAO-LDA. Here the isovalue is 0.02 e/Å${}^{3}$. The occupation numbers of active orbitals are given in parentheses, and HOMO/LUMO is denoted as H/L for brevity.

**Figure 9.**Real-space representation of H−2 (2.000), H−1 (1.976), H (1.669), L (0.334), L+1 (0.021), and L+2 (0.000) of ground-state 4-pyrene, calculated by spin-restricted TAO-LDA. Here the isovalue is 0.02 e/Å${}^{3}$. The occupation numbers of active orbitals are given in parentheses, and HOMO/LUMO is denoted as H/L for brevity.

**Figure 10.**Real-space representation of H−2 (1.905), H−1 (1.524), H (1.161), L (0.852), L+1 (0.478), and L+2 (0.080) of ground-state 6-pyrene, calculated by spin-restricted TAO-LDA. Here the isovalue is 0.02 e/Å${}^{3}$. The occupation numbers of active orbitals are given in parentheses, and HOMO/LUMO is denoted as H/L for brevity.

**Figure 11.**Real-space representation of H−2 (1.404), H−1 (1.141), H (1.028), L (0.992), L+1 (0.895), and L+2 (0.576) of ground-state 8-pyrene, calculated by spin-restricted TAO-LDA. Here the isovalue is 0.02 e/Å${}^{3}$. The occupation numbers of active orbitals are given in parentheses, and HOMO/LUMO is denoted as H/L for brevity.

**Figure 12.**Real-space representation of H−2 (1.133), H−1 (1.040), H (1.038), L (1.011), L+1 (0.981), and L+2 (0.872) of ground-state 10-pyrene, calculated by spin-restricted TAO-LDA. Here the isovalue is 0.02 e/Å${}^{3}$. The occupation numbers of active orbitals are given in parentheses, and HOMO/LUMO is denoted as H/L for brevity.

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**MDPI and ACS Style**

Huang, H.-J.; Seenithurai, S.; Chai, J.-D. TAO-DFT Study on the Electronic Properties of Diamond-Shaped Graphene Nanoflakes. *Nanomaterials* **2020**, *10*, 1236.
https://doi.org/10.3390/nano10061236

**AMA Style**

Huang H-J, Seenithurai S, Chai J-D. TAO-DFT Study on the Electronic Properties of Diamond-Shaped Graphene Nanoflakes. *Nanomaterials*. 2020; 10(6):1236.
https://doi.org/10.3390/nano10061236

**Chicago/Turabian Style**

Huang, Hong-Jui, Sonai Seenithurai, and Jeng-Da Chai. 2020. "TAO-DFT Study on the Electronic Properties of Diamond-Shaped Graphene Nanoflakes" *Nanomaterials* 10, no. 6: 1236.
https://doi.org/10.3390/nano10061236