Supplementary Information to: TAO-DFT Study on the Electronic Properties of Diamond-Shaped Graphene Nanoﬂakes

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Electronic Properties of Diamond-Shaped Graphene Nanoflakes

Hong-Jui Huang,^{1, ∗} Sonai Seenithurai,^{1, ∗} and Jeng-Da Chai^{1, 2, †}

1Department of Physics, National Taiwan University, Taipei 10617, Taiwan

2Center for Theoretical Physics and Center for Quantum Science and Engineering, National Taiwan University, Taipei 10617, Taiwan

∗ These authors contributed equally to this work.

† Corresponding author. E-mail: [email protected]

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LIST OF TABLES

S1 Singlet-triplet energy gap E_ST (in kcal/mol) of n-pyrene, calculated by spin- unrestricted TAO-LDA. . . 3 S2 Vertical ionization potential IP_v (in eV), vertical electron affinity EA_v (in eV),

fundamental gap E_g (in eV), and symmetrized von Neumann entropy S_vN of ground-state n-pyrene, calculated by spin-unrestricted TAO-LDA. . . 3 S3 Active orbital occupation numbers (HOMO−9, HOMO−8, ..., HOMO,

LUMO, ..., LUMO+8, and LUMO+9) of ground-state n-pyrene, calculated by spin-restricted TAO-LDA. For brevity, HOMO/LUMO is denoted as H/L. 4

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TABLE S1. Singlet-triplet energy gap EST (in kcal/mol) of n-pyrene, calculated by spin- unrestricted TAO-LDA.

n EST

2 50.07

3 23.50

4 10.15

5 4.71

6 2.67

7 1.80

8 1.36

9 1.09

10 0.91

11 0.78

12 0.69

13 0.62

14 0.55

15 0.51

TABLE S2. Vertical ionization potential IPv (in eV), vertical electron affinity EAv (in eV), fundamental gap E_g(in eV), and symmetrized von Neumann entropy S_vNof ground-state n-pyrene, calculated by spin-unrestricted TAO-LDA.

n IP_v EA_v E_g S_vN

2 7.01 0.25 6.76 0.02

3 5.99 1.43 4.55 0.28

4 5.43 2.11 3.32 1.03

5 5.14 2.49 2.65 1.92

6 4.97 2.73 2.24 2.86

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7 4.86 2.90 1.96 3.81

8 4.77 3.03 1.74 4.78

9 4.70 3.13 1.57 5.77

10 4.65 3.21 1.43 6.77

11 4.61 3.29 1.32 7.79

12 4.57 3.35 1.22 8.83

13 4.54 3.40 1.14 9.88

14 4.51 3.45 1.06 10.96

15 4.49 3.49 1.00 12.06

TABLE S3. Active orbital occupation numbers (HOMO−9, HOMO−8, ..., HOMO, LUMO, ..., LUMO+8, and LUMO+9) of ground-state n-pyrene, calculated by spin-restricted TAO-LDA. For brevity, HOMO/LUMO is denoted as H/L.

n 2 3 4 5 6 7 8 9 10 11 12 13 14 15

H−9 2.000 2.000 2.000 2.000 2.000 2.000 2.000 2.000 1.999 1.995 1.987 1.982 1.977 1.951 H−8 2.000 2.000 2.000 2.000 2.000 2.000 2.000 1.999 1.994 1.991 1.987 1.981 1.927 1.805 H−7 2.000 2.000 2.000 2.000 2.000 2.000 1.998 1.997 1.994 1.991 1.970 1.891 1.731 1.521 H−6 2.000 2.000 2.000 2.000 2.000 1.999 1.998 1.997 1.991 1.952 1.837 1.638 1.420 1.252 H−5 2.000 2.000 2.000 2.000 2.000 1.999 1.998 1.985 1.921 1.760 1.530 1.324 1.189 1.123 H−4 2.000 2.000 2.000 2.000 2.000 1.997 1.973 1.870 1.657 1.417 1.241 1.145 1.108 1.096 H−3 2.000 2.000 2.000 2.000 1.995 1.950 1.789 1.533 1.310 1.176 1.112 1.086 1.086 1.091 H−2 2.000 2.000 2.000 1.990 1.905 1.673 1.404 1.223 1.133 1.095 1.077 1.080 1.075 1.083 H−1 2.000 1.999 1.976 1.821 1.524 1.280 1.141 1.073 1.040 1.058 1.071 1.066 1.072 1.055 H 1.998 1.938 1.669 1.351 1.161 1.071 1.028 1.019 1.038 1.035 1.052 1.061 1.040 1.022 L 0.002 0.062 0.334 0.660 0.852 0.946 0.992 1.002 1.011 1.016 0.995 0.977 0.982 1.001 L+1 0.000 0.001 0.021 0.170 0.478 0.744 0.895 0.971 0.981 0.963 0.949 0.961 0.964 0.980 L+2 0.000 0.000 0.000 0.008 0.080 0.299 0.576 0.771 0.872 0.918 0.942 0.940 0.961 0.955 L+3 0.000 0.000 0.000 0.000 0.004 0.040 0.181 0.434 0.670 0.820 0.899 0.938 0.932 0.925 L+4 0.000 0.000 0.000 0.000 0.000 0.002 0.021 0.107 0.306 0.550 0.738 0.845 0.892 0.912 L+5 0.000 0.000 0.000 0.000 0.000 0.001 0.001 0.012 0.064 0.207 0.431 0.645 0.792 0.872

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L+7 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.003 0.005 0.007 0.024 0.090 0.234 0.438 L+8 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.005 0.007 0.011 0.015 0.059 0.166 L+9 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.004 0.011 0.015 0.020 0.040