Supplementary Material to: Electronic Properties of Zigzag Graphene Nanoribbons Studied by TAO-DFT

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Studied by TAO-DFT

Chun-Shian Wu^{1, 2} and Jeng-Da Chai^{1, 3, 4, ⇤}

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

2Department of Chemistry, National Taiwan University, Taipei 10617, Taiwan

3Center for Theoretical Sciences and Center for Quantum Science and Engineering, National Taiwan University, Taipei 10617, Taiwan

4Physics Division, National Center for Theoretical Sciences (North), National Taiwan University, Taipei 10617, Taiwan

References (S2).

TABLE S1. Singlet-triplet energy gap of GNR[1,n] (S3 to S4).

TABLE S4. Vertical ionization potential for the lowest singlet state of GNR[1–3,n] (S6 to S8).

TABLE S5. Vertical electron aﬃnity for the lowest singlet state of GNR[1–3,n] (S8 to S10).

TABLE S6. Fundamental gap for the lowest singlet state of GNR[1–3,n] (S10 to S12).

TABLE S7. Symmetrized von Neumann entropy for the lowest singlet state of GNR[1–3,n] (S12 to S14).

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[4] Birks, J. B. Photophysics of Aromatic Molecules; Wiley: London, 1970.

[5] Schiedt, J.; Weinkauf, R. Photodetachment Photoelectron Spectroscopy of Mass Selected Anions: Anthracene and the Anthracene-H2O Cluster. Chem. Phys. Lett.1997, 266, 201-205.

[6] Sabbatini, N.; Indelli, M. T.; Gandolfi, M. T.; Balzani, V. Quenching of Singlet and Triplet Excited States of Aromatic Molecules by Europium Ions. J. Phys. Chem.1982, 86, 3585-3591.

[7] Burgos, J.; Pope, M.; Swenberg, Ch. E.; Alfano, R. R. Heterofission in Pentacene-Doped Tetracene Single Crystals. Phys.

Status Solidi B 1977, 83, 249-256.

[8] Hachmann, J.; Dorando, J. J.; Aviles, M.; Chan, G. K. L. The Radical Character of the Acenes: A Density Matrix Renormalization Group Study. J. Chem. Phys.2007, 127, 134309.

[9] Hajgató, B.; Huzak, M.; Deleuze, M. S. Focal Point Analysis of the Singlet-Triplet Energy Gap of Octacene and Larger Acenes. J. Phys. Chem. A2011, 115, 9282-9293.

[10] Mizukami, W.; Kurashige, Y.; Yanai, T. More ⇡ Electrons Make a Diﬀerence: Emergence of Many Radicals on Graphene Nanoribbons Studied by Ab Initio DMRG Theory. J. Chem. Theory Comput 2012, 9, 401-407.

[11] Malloci, G.; Mulas, G.; Cappellini, G.; Joblin, C. Time-Dependent Density Functional Study of the Electronic Spectra of Oligoacenes in the Charge States 1, 0, +1, and +2. Chem. Phys.2007, 340, 43-58.

[12] Deleuze, M. S.; Claes, L.; Kryachko, E. S.; François, J.-P. Benchmark Theoretical Study of the Ionization Threshold of Benzene and Oligoacenes. J. Chem. Phys.2003, 119, 3106.

[13] Hajgató, B.; Deleuze, M. S.; Tozer, D. J.; De Proft, F. A Benchmark Theoretical Study of the Electron Aﬃnities of Benzene and Linear Acenes. J. Chem. Phys.2008, 129, 084308.

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TABLES

TABLE S1. Singlet-triplet energy gap [ET ES] (in kcal/mol) of GNR[1,n] as a function of the ribbon length, calculated using spin-unrestricted KS-LDA [1, 2] and TAO-LDA [3]. The experimental data (uncorrected for zero-point vibrations, thermal vibrations, etc.) are taken from Refs. [4–7], the DMRG data are taken from Ref. [8], the CCSD(T)/CBS data are taken from Ref. [9], and the CAM-B3LYP data are taken from Ref. [10].

n Experiment DMRG/cc-pVDZ DMRG/STO-3G CCSD(T)/CBS CAM-B3LYP KS-LDA TAO-LDA

2 61.0 61.0 61.5 65.83 64.57 65.19 64.77

3 43.1 44.0 45.9 48.21 43.12 43.40 43.22

4 29.3 31.9 34.7 33.49 28.59 29.06 29.01

5 19.8 23.4 26.7 25.27 18.45 19.20 19.60

6 17.5 21.0 17.71 13.38 12.16 13.55

7 13.40 6.98 9.91

8 14.2 9.20 11.07 3.07 7.85

9 7.00 0.99 6.66

10 11.6 4.64 13.38 0.53 5.91

11 3.65 0.86 5.32

12 10.7 17.99 1.59 4.82

13 2.52 4.38

14 3.50 3.98

15 4.34 3.65

16 4.43 3.37

17 3.14

18 2.94

19 2.77

20 2.62

21 2.49

22 2.37

23 2.25

24 2.15

25 2.06

26 1.98

27 1.90

28 1.83

29 1.76

30 1.70

31 1.64

32 1.59

33 1.54

34 1.49

35 1.45

36 1.40

37 1.36

38 1.33

39 1.29

40 1.26

41 1.23

42 1.20

43 1.17

44 1.14

45 1.11

46 1.09

47 1.07

48 1.04

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51 0.98

52 0.96

53 0.94

54 0.92

55 0.91

56 0.89

57 0.87

58 0.86

59 0.84

60 0.83

61 0.82

62 0.80

63 0.79

64 0.78

65 0.76

66 0.75

67 0.74

68 0.73

69 0.72

70 0.71

71 0.70

72 0.69

73 0.68

74 0.67

75 0.66

76 0.65

77 0.64

78 0.64

79 0.63

80 0.62

81 0.61

82 0.60

83 0.60

84 0.59

85 0.58

86 0.58

87 0.57

88 0.56

89 0.56

90 0.55

91 0.54

92 0.54

93 0.53

94 0.53

95 0.52

96 0.51

97 0.51

98 0.50

99 0.50

100 0.49

TABLE S2. Singlet-triplet energy gap [ET ES] (in kcal/mol) of GNR[2,n] as a function of the ribbon length, calculated using spin-unrestricted KS-LDA [1, 2] and TAO-LDA [3]. For comparison, the DMRG and CAM-B3LYP data are taken from Mizukami et al. [10].

n DMRG/6-31G CAM-B3LYP KS-LDA TAO-LDA

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2 39.66 37.36 35.77 35.61

3 17.53 12.22 15.32 16.53

4 5.30 5.07 3.98 9.03

5 2.31 4.38 0.43 6.77

6 2.08 6.46 0.22 5.45

7 2.77 10.61 0.42 4.23

8 16.83 0.69 3.28

9 1.18 2.68

10 31.13 1.70 2.32

11 1.30 2.07

12 36.67 1.86

13 1.68

14 1.53

15 1.41

16 1.31

17 1.22

18 1.14

19 1.08

20 1.01

21 0.96

22 0.91

23 0.87

24 0.83

25 0.79

26 0.76

27 0.73

28 0.70

29 0.68

30 0.65

31 0.63

32 0.61

33 0.59

34 0.57

35 0.55

36 0.54

37 0.52

38 0.51

39 0.49

40 0.48

41 0.47

42 0.46

43 0.45

44 0.43

45 0.42

46 0.41

47 0.41

48 0.40

49 0.39

50 0.38

TABLE S3. Singlet-triplet energy gap [ET ES] (in kcal/mol) of GNR[3,n] as a function of the ribbon length, calculated using spin-unrestricted TAO-LDA [3]. For comparison, the CAM-B3LYP data are taken from Mizukami et al. [10].

n CAM-B3LYP TAO-LDA

2 24.91 24.04

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5 3.23 5.38

6 8.07 3.80

7 2.82

8 13.38 2.35

9 2.05

10 43.12 1.79

11 1.58

12 60.42 1.42

13 1.29

14 1.19

15 1.09

16 1.01

17 0.95

18 0.89

19 0.84

20 0.79

21 0.75

22 0.71

23 0.68

24 0.65

25 0.62

26 0.60

27 0.57

28 0.55

29 0.53

30 0.51

TABLE S4. Vertical ionization potential (in eV) for the lowest singlet state of GNR[1–3,n] as a function of the ribbon length, calculated using spin-unrestricted TAO-LDA [3]. For GNR[1,n], the experimental data are taken from the compilation in Ref.

[11], and the CCSD(T)/CBS data are taken from Ref. [12].

GNR[1,n] GNR[2,n] GNR[3,n]

n Experiment CCSD(T)/CBS TAO-LDA TAO-LDA TAO-LDA

2 8.14 8.24 7.81 6.55 5.95

3 7.44 7.47 7.00 5.89 5.46

4 6.97 6.95 6.46 5.58 5.31

5 6.59 6.57 6.07 5.43 5.19

6 6.43 5.79 5.30 5.08

7 5.59 5.19 5.00

8 5.44 5.10 4.94

9 5.33 5.02 4.88

10 5.23 4.96 4.84

11 5.15 4.91 4.80

12 5.08 4.86 4.76

13 5.01 4.82 4.73

14 4.96 4.78 4.70

15 4.91 4.75 4.67

16 4.86 4.72 4.65

17 4.82 4.69 4.63

18 4.78 4.67 4.61

19 4.75 4.64 4.59

20 4.71 4.62 4.57

21 4.68 4.60 4.55

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22 4.66 4.58 4.54

23 4.63 4.56 4.52

24 4.61 4.55 4.51

25 4.59 4.53 4.50

26 4.57 4.52 4.49

27 4.55 4.50 4.47

28 4.53 4.49 4.46

29 4.51 4.48 4.45

30 4.50 4.47 4.44

31 4.48 4.45

32 4.47 4.44

33 4.45 4.43

34 4.44 4.42

35 4.43 4.41

36 4.42 4.41

37 4.40 4.40

38 4.39 4.39

39 4.38 4.38

40 4.37 4.37

41 4.36 4.37

42 4.35 4.36

43 4.35 4.35

44 4.34 4.35

45 4.33 4.34

46 4.32 4.34

47 4.31 4.33

48 4.31 4.32

49 4.30 4.32

50 4.29 4.31

51 4.29

52 4.28

53 4.27

54 4.27

55 4.26

56 4.26

57 4.25

58 4.24

59 4.24

60 4.23

61 4.23

62 4.23

63 4.22

64 4.22

65 4.21

66 4.21

67 4.20

68 4.20

69 4.20

70 4.19

71 4.19

72 4.18

73 4.18

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75 4.17

76 4.17

77 4.17

78 4.16

79 4.16

80 4.16

81 4.15

82 4.15

83 4.15

84 4.15

85 4.14

86 4.14

87 4.14

88 4.14

89 4.13

90 4.13

91 4.13

92 4.13

93 4.12

94 4.12

95 4.12

96 4.12

97 4.11

98 4.11

99 4.11

100 4.11

TABLE S5. Vertical electron aﬃnity (in eV) for the lowest singlet state of GNR[1–3,n] as a function of the ribbon length, calculated using spin-unrestricted TAO-LDA [3]. For GNR[1,n], the experimental data are taken from the compilation in Ref.

[11], and the CCSD(T)/CBS data are taken from Ref. [13].

2 -0.20 -0.48 -0.62 0.74 1.40

3 0.53 0.28 0.28 1.49 1.98

4 1.07 0.82 0.90 1.89 2.21

5 1.39 1.21 1.34 2.10 2.38

6 1.47 1.66 2.27 2.54

7 1.89 2.41 2.65

8 2.06 2.53 2.74

9 2.19 2.62 2.82

10 2.30 2.70 2.89

11 2.39 2.77 2.94

12 2.48 2.83 3.00

13 2.55 2.89 3.04

14 2.61 2.93 3.08

15 2.67 2.98 3.12

16 2.72 3.01 3.15

17 2.77 3.05 3.18

18 2.81 3.08 3.21

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19 2.85 3.11 3.24

20 2.89 3.14 3.26

21 2.92 3.16 3.28

22 2.95 3.19 3.30

23 2.98 3.21 3.32

24 3.00 3.23 3.34

25 3.03 3.25 3.36

26 3.05 3.27 3.37

27 3.07 3.29 3.39

28 3.09 3.30 3.40

29 3.11 3.32 3.42

30 3.13 3.33 3.43

31 3.15 3.34

32 3.16 3.36

33 3.18 3.37

34 3.19 3.38

35 3.21 3.39

36 3.22 3.40

37 3.23 3.41

38 3.25 3.42

39 3.26 3.43

40 3.27 3.44

41 3.28 3.45

42 3.29 3.46

43 3.30 3.47

44 3.31 3.48

45 3.32 3.48

46 3.33 3.49

47 3.33 3.50

48 3.34 3.50

49 3.35 3.51

50 3.36 3.52

51 3.36

52 3.37

53 3.38

54 3.39

55 3.39

56 3.40

57 3.40

58 3.41

59 3.42

60 3.42

61 3.43

62 3.43

63 3.44

64 3.44

65 3.45

66 3.45

67 3.46

68 3.46

69 3.46

70 3.47

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71 3.47

72 3.48

73 3.48

74 3.48

75 3.49

76 3.49

77 3.50

78 3.50

79 3.50

80 3.51

81 3.51

82 3.51

83 3.52

84 3.52

85 3.52

86 3.52

87 3.53

88 3.53

89 3.53

90 3.54

91 3.54

92 3.54

93 3.54

94 3.55

95 3.55

96 3.55

97 3.55

98 3.56

99 3.56

100 3.56

TABLE S6. Fundamental gap (in eV) for the lowest singlet state of GNR[1–3,n] as a function of the ribbon length, calculated using spin-unrestricted TAO-LDA [3]. For GNR[1,n], the experimental data are taken from the compilation in Ref. [11], and the CCSD(T)/CBS data are taken from Refs. [12, 13].

2 8.34 8.72 8.43 5.82 4.55

3 6.91 7.19 6.73 4.40 3.48

4 5.90 6.13 5.56 3.69 3.11

5 5.20 5.37 4.73 3.32 2.81

6 4.96 4.13 3.04 2.54

7 3.69 2.78 2.34

8 3.38 2.57 2.19

9 3.13 2.40 2.06

10 2.93 2.26 1.95

11 2.76 2.14 1.85

12 2.60 2.03 1.76

13 2.46 1.94 1.69

14 2.34 1.85 1.62

15 2.23 1.77 1.55

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16 2.13 1.70 1.50

17 2.05 1.64 1.44

18 1.97 1.58 1.40

19 1.89 1.53 1.35

20 1.83 1.48 1.31

21 1.76 1.44 1.27

22 1.71 1.39 1.24

23 1.65 1.35 1.20

24 1.60 1.32 1.17

25 1.56 1.28 1.14

26 1.52 1.25 1.11

27 1.47 1.22 1.09

28 1.44 1.19 1.06

29 1.40 1.16 1.04

30 1.37 1.13 1.02

31 1.33 1.11

32 1.30 1.09

33 1.27 1.06

34 1.25 1.04

35 1.22 1.02

36 1.20 1.00

37 1.17 0.98

38 1.15 0.97

39 1.13 0.95

40 1.11 0.93

41 1.09 0.92

42 1.07 0.90

43 1.05 0.89

44 1.03 0.87

45 1.01 0.86

46 1.00 0.84

47 0.98 0.83

48 0.96 0.82

49 0.95 0.81

50 0.94 0.80

51 0.92

52 0.91

53 0.89

54 0.88

55 0.87

56 0.86

57 0.85

58 0.84

59 0.82

60 0.81

61 0.80

62 0.79

63 0.78

64 0.77

65 0.77

66 0.76

67 0.75

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69 0.73

70 0.72

71 0.72

72 0.71

73 0.70

74 0.69

75 0.69

76 0.68

77 0.67

78 0.67

79 0.66

80 0.65

81 0.65

82 0.64

83 0.63

84 0.63

85 0.62

86 0.62

87 0.61

88 0.61

89 0.60

90 0.59

91 0.59

92 0.58

93 0.58

94 0.58

95 0.57

96 0.57

97 0.56

98 0.56

99 0.55

100 0.55

TABLE S7. Symmetrized von Neumann entropy for the lowest singlet state of GNR[1–3,n] as a function of the ribbon length, calculated using spin-restricted TAO-LDA [3].

n GNR[1,n] GNR[2,n] GNR[3,n]

2 0.00 0.08 0.29

3 0.03 0.59 1.18

4 0.15 1.21 1.44

5 0.40 1.45 1.68

6 0.75 1.65 2.15

7 1.08 1.98 2.63

8 1.34 2.38 3.00

9 1.52 2.74 3.38

10 1.67 3.07 3.79

11 1.82 3.40 4.21

12 1.98 3.74 4.61

13 2.16 4.08 5.01

14 2.36 4.42 5.41

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15 2.55 4.76 5.82

16 2.75 5.09 6.22

17 2.94 5.43 6.62

18 3.13 5.77 7.03

19 3.31 6.11 7.43

20 3.50 6.45 7.84

21 3.69 6.79 8.24

22 3.88 7.12 8.64

23 4.07 7.46 9.05

24 4.26 7.80 9.45

25 4.44 8.14 9.86

26 4.63 8.48 10.26

27 4.82 8.82 10.66

28 5.01 9.15 11.07

29 5.20 9.49 11.47

30 5.39 9.83 11.88

31 5.58 10.17

32 5.77 10.51

33 5.95 10.85

34 6.14 11.19

35 6.33 11.52

36 6.52 11.86

37 6.71 12.20

38 6.90 12.54

39 7.09 12.88

40 7.27 13.22

41 7.46 13.55

42 7.65 13.89

43 7.84 14.23

44 8.03 14.57

45 8.22 14.91

46 8.41 15.25

47 8.59 15.58

48 8.78 15.92

49 8.97 16.26

50 9.16 16.60

51 9.35

52 9.54

53 9.73

54 9.92

55 10.10

56 10.29

57 10.48

58 10.67

59 10.86

60 11.05

61 11.23

62 11.42

63 11.61

64 11.80

65 11.99

66 12.18

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68 12.56

69 12.74

70 12.93

71 13.12

72 13.31

73 13.50

74 13.69

75 13.88

76 14.06

77 14.25

78 14.44

79 14.63

80 14.82

81 15.01

82 15.20

83 15.38

84 15.57

85 15.76

86 15.95

87 16.14

88 16.33

89 16.52

90 16.70

91 16.89

92 17.08

93 17.27

94 17.46

95 17.65

96 17.84

97 18.02

98 18.21

99 18.40

100 18.59