Supplementary Material to: Electronic and Optical Properties of the Narrowest Armchair Graphene Nanoribbons Studied by

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Supplementary Material to: Electronic and Optical Properties of the Narrowest Armchair Graphene Nanoribbons Studied by

Density Functional Methods

Chia-Nan Yeh,1, Pei-Yin Lee,1, and Jeng-Da Chai1, 2,

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

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

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LIST OF TABLES

S1 Singlet-triplet energy gap EST[in eV] of n-PP as a function of the chain length, calculated using KS-DFT with various density functionals. For comparison, the experimental data [4–6] are taken from the literature. . . 5 S2 Vertical ionization potential IP(1) [in eV] for the lowest singlet state of n-

PP as a function of the chain length, calculated using KS-DFT with various density functionals. For comparison, the experimental data [7] are taken from the literature. . . 6 S3 Vertical ionization potential IP(2) [in eV] for the lowest singlet state of n-

PP as a function of the chain length, calculated using KS-DFT with various density functionals. For comparison, the experimental data [7] are taken from the literature. . . 7 S4 Vertical electron affinity EA(1) [in eV] for the lowest singlet state of n-PP as

a function of the chain length, calculated using KS-DFT with various density functionals. For comparison, the experimental data: Expt1 (vertical EA) [8, 9] and Expt2 (adiabatic EA) [10, 11], are taken from the literature. . . 8 S5 Vertical electron affinity EA(2) [in eV] for the lowest singlet state of n-PP as

a function of the chain length, calculated using KS-DFT with various density functionals. For comparison, the experimental data: Expt1 (vertical EA) [8, 9] and Expt2 (adiabatic EA) [10, 11], are taken from the literature. . . 9 S6 Vertical electron affinity EA(3) [in eV] for the lowest singlet state of n-PP as

a function of the chain length, calculated using KS-DFT with various density functionals. For comparison, the experimental data: Expt1 (vertical EA) [8, 9] and Expt2 (adiabatic EA) [10, 11], are taken from the literature. . . 10

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S8 Fundamental gap Eg(2) [in eV] for the lowest singlet state of n-PP as a function of the chain length, calculated using KS-DFT with various density functionals. For comparison, the experimental data: Expt1 (vertical IP vertical EA) [7–9] and Expt2 (vertical IP adiabatic EA) [7, 10, 11], are taken from the literature. . . 12 S9 Fundamental gap Eg(3) [in eV] for the lowest singlet state of n-PP as a

function of the chain length, calculated using KS-DFT with various density functionals. For comparison, the experimental data: Expt1 (vertical IP vertical EA) [7–9] and Expt2 (vertical IP adiabatic EA) [7, 10, 11], are taken from the literature. . . 13 S10 Optical gap Eopt [in eV] for the lowest singlet state of n-PP as a function of

the chain length, calculated using TDDFT with various density functionals.

For comparison, the experimental data [12, 13] and SAC-CI data [13] are taken from the literature. . . 14 S11 Exciton binding energy Eb(1) [in eV] for the lowest singlet state of n-PP as

a function of the chain length, calculated using KS-DFT and TDDFT with various density functionals. For comparison, the experimental data: Expt1 [(vertical IP vertical EA) Eopt] [7–9, 12, 13] and Expt2 [(vertical IP

adiabatic EA) Eopt] [7, 10–13], are taken from the literature. . . 15 S12 Exciton binding energy Eb(2) [in eV] for the lowest singlet state of n-PP as

a function of the chain length, calculated using KS-DFT and TDDFT with various density functionals. For comparison, the experimental data: Expt1 [(vertical IP vertical EA) Eopt] [7–9, 12, 13] and Expt2 [(vertical IP

adiabatic EA) Eopt] [7, 10–13], are taken from the literature. . . 16 S13 Exciton binding energy Eb(3) [in eV] for the lowest singlet state of n-PP as

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S15 Expectation value of the total spin-squared operatorh ˆS2i for the ground state of cationic n-PP as a function of the chain length, calculated using KS-DFT with various density functionals (at the ground-state geometry of neutral n-PP). 19 S16 Expectation value of the total spin-squared operatorh ˆS2i for the ground state

of anionic n-PP as a function of the chain length, calculated using KS-DFT with various density functionals (at the ground-state geometry of neutral n-PP). 20

I. EXPECTATION VALUE OF THE TOTAL SPIN-SQUARED OPERATOR h ˆS2i

In spin-unrestricted Kohn-Sham density functional theory (KS-DFT), the spatial parts of ↵- and -spin orbitals may be di↵erent. For simplicity, we refer to the spatial parts of

↵- and -spin orbitals as ↵- and -orbitals ( and ). Both ↵- and -orbitals form an orthogonal set, respectively, but ↵- and -orbitals are, in general, not orthogonal to each other:

h i| ji = ij and h i| ji = ij (1)

h i| ji 6= ij (2)

In this formalism, the Kohn-Sham (KS) Hamiltonian ˆHs commutes with the total spin projection operator ˆSz, but may not commute with the total spin-squared operator ˆS2. As a result, the KS ground-state wavefunction may not be an eigenfunction of ˆS2. Instead, the expectation value of ˆS2 can be calculated as [assuming that the number of ↵-orbitals (N)

the number of -orbitals (N )]:

h ˆS2i = S(S + 1) + N

N

X

i=1

XN j=1

|h i| ji|2, (3)

where S = (N N )/2, the eigenvalue of ˆSz, can be 0 (singlet), 1/2 (doublet), 1 (triplet), 3/2 (quartet), and so on [1, 2]. Note that the expression takes exactly the same form as

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TABLE S1. Singlet-triplet energy gap EST [in eV] of n-PP as a function of the chain length, calculated using KS-DFT with various density functionals. For comparison, the experimental data [4–6] are taken from the literature.

n LDA PBE BLYP PBE0 B3LYP !B97 !B97X Expt

1 4.23 3.90 3.87 3.84 3.88 3.83 3.88 3.67 [4]

2 3.03 2.86 2.83 2.97 2.95 3.12 3.14 2.94 [5]

3 2.54 2.42 2.39 2.59 2.57 2.79 2.82 2.54 [6]

4 2.29 2.20 2.18 2.45 2.41 2.73 2.75

5 2.15 2.07 2.05 2.37 2.32 2.68 2.70

6 2.06 1.99 1.97 2.34 2.28 2.70 2.72

7 2.00 1.94 1.92 2.32 2.26 2.67 2.69

8 1.95 1.91 1.89 2.31 2.25 2.67 2.69

9 1.92 1.88 1.87 2.30 2.24 2.67 2.69

10 1.90 1.87 1.85 2.30 2.24 2.69 2.71

11 1.89 1.85 1.84 2.30 2.24 2.67 2.69

12 1.88 1.85 1.83 2.30 2.24 2.69 2.71

13 1.87 1.84 1.83 2.30 2.24 2.67 2.69

14 1.86 1.83 1.82 2.30 2.24 2.69 2.71

15 1.86 1.83 1.82 2.30 2.24 2.67 2.69

16 1.85 1.83 1.82 2.30 2.24 2.69 2.71

17 1.85 1.83 1.82 2.30 2.24 2.67 2.69

18 1.85 1.83 1.81 2.30 2.24 2.69 2.69

19 1.85 1.82 1.81 2.30 2.24 2.67 2.69

20 1.84 1.82 1.81 2.30 2.24 2.69 2.71

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TABLE S2. Vertical ionization potential IP(1) [in eV] for the lowest singlet state of n-PP as a function of the chain length, calculated using KS-DFT with various density functionals. For comparison, the experimental data [7] are taken from the literature.

n LDA PBE BLYP PBE0 B3LYP !B97 !B97X Expt

1 9.29 9.00 8.76 9.08 9.01 9.11 9.11 9.25 [7]

2 7.80 7.57 7.31 7.77 7.65 8.01 7.96 8.38 [7]

3 7.12 6.91 6.64 7.18 7.04 7.56 7.50 8.04 [7]

4 6.72 6.52 6.25 6.84 6.69 7.35 7.28 7.80 [7]

5 6.45 6.26 5.99 6.63 6.46 7.24 7.16 7.68 [7]

6 6.25 6.06 5.79 6.47 6.30 7.19 7.10 7.62 [7]

7 6.10 5.92 5.64 6.36 6.17 7.16 7.07

8 5.98 5.80 5.53 6.27 6.08 7.15 7.05

9 5.88 5.70 5.43 6.19 6.00 7.14 7.04

10 5.80 5.62 5.35 6.13 5.93 7.14 7.04

11 5.73 5.55 5.28 6.08 5.87 7.14 7.04

12 5.67 5.49 5.22 6.04 5.83 7.14 7.04

13 5.62 5.44 5.17 6.00 5.78 7.14 7.04

14 5.57 5.39 5.12 5.96 5.75 7.14 7.04

15 5.53 5.35 5.08 5.93 5.71 7.14 7.04

16 5.49 5.32 5.04 5.90 5.68 7.14 7.04

17 5.46 5.28 5.01 5.88 5.66 7.14 7.04

18 5.43 5.25 4.98 5.85 5.63 7.14 7.04

19 5.40 5.23 4.95 5.83 5.61 7.14 7.04

20 5.38 5.20 4.93 5.82 5.59 7.14 7.04

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TABLE S3. Vertical ionization potential IP(2) [in eV] for the lowest singlet state of n-PP as a function of the chain length, calculated using KS-DFT with various density functionals. For comparison, the experimental data [7] are taken from the literature.

n LDA PBE BLYP PBE0 B3LYP !B97 !B97X Expt

1 6.12 5.93 5.65 7.02 6.69 9.44 9.22 9.25 [7]

2 5.44 5.25 4.98 6.20 5.90 8.44 8.24 8.38 [7]

3 5.17 4.98 4.71 5.85 5.58 8.03 7.84 8.04 [7]

4 5.03 4.84 4.57 5.69 5.41 7.81 7.65 7.80 [7]

5 4.95 4.76 4.49 5.58 5.33 7.70 7.51 7.68 [7]

6 4.90 4.73 4.46 5.52 5.25 7.62 7.43 7.62 [7]

7 4.84 4.68 4.41 5.50 5.22 7.56 7.40

8 4.84 4.65 4.38 5.47 5.20 7.54 7.35

9 4.82 4.65 4.38 5.44 5.17 7.51 7.35

10 4.79 4.63 4.35 5.41 5.17 7.48 7.32

11 4.79 4.63 4.35 5.41 5.14 7.48 7.29

12 4.79 4.63 4.35 5.39 5.14 7.48 7.29

13 4.79 4.60 4.33 5.39 5.12 7.46 7.29

14 4.76 4.60 4.33 5.39 5.12 7.46 7.27

15 4.76 4.60 4.33 5.39 5.12 7.46 7.27

16 4.76 4.60 4.33 5.39 5.12 7.46 7.27

17 4.76 4.60 4.33 5.36 5.12 7.43 7.27

18 4.76 4.60 4.33 5.36 5.12 7.43 7.27

19 4.76 4.60 4.33 5.36 5.12 7.43 7.27

20 4.76 4.60 4.33 5.36 5.09 7.43 7.27

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TABLE S4. Vertical electron affinity EA(1) [in eV] for the lowest singlet state of n-PP as a function of the chain length, calculated using KS-DFT with various density functionals. For comparison, the experimental data: Expt1 (vertical EA) [8, 9] and Expt2 (adiabatic EA) [10, 11], are taken from the literature.

n LDA PBE BLYP PBE0 B3LYP !B97 !B97X Expt1 Expt2

1 -2.16 -2.21 -2.41 -2.26 -2.31 -2.30 -2.32 -1.13 [8] -1.12 [10]

2 -0.52 -0.62 -0.84 -0.74 -0.80 -0.97 -0.97 -0.30 [9] -0.17 [11]

3 0.23 0.12 -0.12 -0.07 -0.12 -0.42 -0.42 0.39 [11]

4 0.68 0.55 0.31 0.32 0.28 -0.15 -0.14 0.66 [11]

5 0.98 0.84 0.59 0.57 0.53 0.00 0.02

6 1.20 1.06 0.80 0.75 0.71 0.09 0.11

7 1.37 1.22 0.96 0.88 0.85 0.14 0.16

8 1.50 1.35 1.09 0.99 0.96 0.17 0.19

9 1.61 1.45 1.19 1.07 1.05 0.18 0.22

10 1.70 1.54 1.28 1.14 1.12 0.20 0.23

11 1.77 1.62 1.36 1.20 1.18 0.20 0.24

12 1.84 1.68 1.42 1.25 1.23 0.21 0.24

13 1.90 1.74 1.47 1.29 1.28 0.21 0.25

14 1.95 1.79 1.52 1.33 1.32 0.22 0.25

15 1.99 1.83 1.57 1.36 1.36 0.22 0.25

16 2.03 1.87 1.61 1.39 1.39 0.22 0.25

17 2.07 1.90 1.64 1.42 1.42 0.22 0.26

18 2.10 1.94 1.67 1.44 1.44 0.22 0.26

19 2.13 1.97 1.70 1.47 1.47 0.22 0.26

20 2.16 1.99 1.73 1.49 1.49 0.22 0.26

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TABLE S5. Vertical electron affinity EA(2) [in eV] for the lowest singlet state of n-PP as a function of the chain length, calculated using KS-DFT with various density functionals. For comparison, the experimental data: Expt1 (vertical EA) [8, 9] and Expt2 (adiabatic EA) [10, 11], are taken from the literature.

n LDA PBE BLYP PBE0 B3LYP !B97 !B97X Expt1 Expt2

1 -5.17 -5.14 -5.33 -4.27 -4.52 -2.04 -2.23 -1.13 [8] -1.12 [10]

2 -2.83 -2.86 -3.10 -2.29 -2.50 -0.57 -0.71 -0.30 [9] -0.17 [11]

3 -1.69 -1.77 -2.01 -1.36 -1.55 0.05 -0.08 0.39 [11]

4 -0.98 -1.09 -1.33 -0.82 -0.98 0.35 0.24 0.66 [11]

5 -0.49 -0.60 -0.84 -0.44 -0.57 0.52 0.44

6 -0.14 -0.24 -0.49 -0.16 -0.30 0.60 0.52

7 0.16 0.03 -0.22 0.03 -0.08 0.65 0.57

8 0.38 0.24 0.00 0.19 0.11 0.68 0.60

9 0.57 0.44 0.19 0.33 0.24 0.68 0.63

10 0.73 0.60 0.33 0.44 0.38 0.71 0.63

11 0.87 0.71 0.46 0.54 0.49 0.71 0.63

12 0.98 0.84 0.57 0.63 0.57 0.71 0.63

13 1.09 0.93 0.68 0.71 0.65 0.71 0.63

14 1.17 1.03 0.76 0.76 0.71 0.71 0.63

15 1.25 1.12 0.84 0.82 0.79 0.71 0.63

16 1.33 1.17 0.93 0.87 0.84 0.71 0.63

17 1.39 1.25 0.98 0.93 0.90 0.71 0.63

18 1.44 1.31 1.03 0.98 0.95 0.71 0.63

19 1.50 1.36 1.09 1.01 0.98 0.71 0.65

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TABLE S6. Vertical electron affinity EA(3) [in eV] for the lowest singlet state of n-PP as a function of the chain length, calculated using KS-DFT with various density functionals. For comparison, the experimental data: Expt1 (vertical EA) [8, 9] and Expt2 (adiabatic EA) [10, 11], are taken from the literature.

n LDA PBE BLYP PBE0 B3LYP !B97 !B97X Expt1 Expt2

1 0.84 0.73 0.52 -0.24 -0.11 -2.56 -2.39 -1.13 [8] -1.12 [10]

2 1.80 1.63 1.41 0.79 0.90 -1.33 -1.22 -0.30 [9] -0.17 [11]

3 2.15 2.01 1.77 1.22 1.31 -0.84 -0.73 0.39 [11]

4 2.37 2.20 1.96 1.47 1.52 -0.57 -0.46 0.66 [11]

5 2.48 2.31 2.07 1.61 1.66 -0.41 -0.30

6 2.56 2.39 2.12 1.69 1.74 -0.33 -0.22

7 2.61 2.45 2.18 1.74 1.80 -0.24 -0.14

8 2.64 2.48 2.23 1.80 1.85 -0.19 -0.08

9 2.67 2.50 2.26 1.82 1.88 -0.16 -0.05

10 2.69 2.53 2.26 1.85 1.90 -0.14 -0.03

11 2.72 2.56 2.29 1.88 1.90 -0.11 0.00

12 2.72 2.56 2.29 1.88 1.93 -0.08 0.00

13 2.75 2.59 2.31 1.90 1.93 -0.08 0.03

14 2.75 2.59 2.31 1.90 1.96 -0.08 0.03

15 2.78 2.59 2.31 1.93 1.96 -0.05 0.03

16 2.78 2.59 2.34 1.93 1.96 -0.05 0.05

17 2.78 2.61 2.34 1.93 1.96 -0.05 0.05

18 2.78 2.61 2.34 1.93 1.99 -0.05 0.05

19 2.78 2.61 2.34 1.93 1.99 -0.03 0.05

20 2.80 2.61 2.34 1.96 1.99 -0.03 0.08

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TABLE S7. Fundamental gap Eg(1) [in eV] for the lowest singlet state of n-PP as a function of the chain length, calculated using KS-DFT with various density functionals. For comparison, the experimental data: Expt1 (vertical IP vertical EA) [7–9] and Expt2 (vertical IP adiabatic EA) [7, 10, 11], are taken from the literature.

n LDA PBE BLYP PBE0 B3LYP !B97 !B97X Expt1 Expt2

1 11.45 11.21 11.17 11.34 11.32 11.41 11.43 10.38 [7, 8] 10.37 [7, 10]

2 8.32 8.19 8.15 8.51 8.45 8.97 8.94 8.68 [7, 9] 8.55 [7, 11]

3 6.89 6.80 6.77 7.24 7.16 7.98 7.92 7.65 [7, 11]

4 6.04 5.97 5.94 6.52 6.42 7.49 7.42 7.14 [7, 11]

5 5.47 5.41 5.39 6.05 5.93 7.24 7.15

6 5.05 5.01 4.99 5.72 5.58 7.10 6.99

7 4.74 4.70 4.68 5.47 5.32 7.02 6.91

8 4.48 4.45 4.44 5.28 5.12 6.98 6.86

9 4.28 4.25 4.23 5.13 4.95 6.96 6.83

10 4.11 4.08 4.07 4.99 4.81 6.94 6.81

11 3.96 3.94 3.92 4.89 4.69 6.93 6.80

12 3.83 3.81 3.80 4.79 4.59 6.93 6.79

13 3.72 3.70 3.69 4.71 4.50 6.92 6.79

14 3.63 3.61 3.60 4.63 4.43 6.92 6.79

15 3.54 3.52 3.51 4.57 4.36 6.92 6.79

16 3.46 3.45 3.44 4.51 4.29 6.92 6.78

17 3.39 3.38 3.37 4.46 4.24 6.92 6.78

18 3.33 3.32 3.31 4.41 4.19 6.92 6.78

19 3.27 3.26 3.25 4.37 4.14 6.92 6.78

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TABLE S8. Fundamental gap Eg(2) [in eV] for the lowest singlet state of n-PP as a function of the chain length, calculated using KS-DFT with various density functionals. For comparison, the experimental data: Expt1 (vertical IP vertical EA) [7–9] and Expt2 (vertical IP adiabatic EA) [7, 10, 11], are taken from the literature.

n LDA PBE BLYP PBE0 B3LYP !B97 !B97X Expt1 Expt2

1 11.29 11.07 10.99 11.29 11.21 11.48 11.46 10.38 [7, 8] 10.37 [7, 10]

2 8.27 8.11 8.08 8.49 8.41 9.01 8.95 8.68 [7, 9] 8.55 [7, 11]

3 6.86 6.75 6.72 7.21 7.13 7.97 7.92 7.65 [7, 11]

4 6.01 5.93 5.90 6.50 6.39 7.46 7.40 7.14 [7, 11]

5 5.44 5.36 5.33 6.01 5.90 7.18 7.07

6 5.03 4.98 4.95 5.69 5.55 7.02 6.91

7 4.68 4.65 4.63 5.47 5.31 6.91 6.83

8 4.46 4.41 4.38 5.28 5.09 6.86 6.75

9 4.24 4.22 4.19 5.12 4.93 6.83 6.72

10 4.05 4.03 4.03 4.98 4.79 6.78 6.69

11 3.92 3.92 3.89 4.87 4.65 6.78 6.67

12 3.81 3.78 3.78 4.76 4.57 6.78 6.67

13 3.70 3.67 3.65 4.68 4.46 6.75 6.67

14 3.59 3.56 3.56 4.63 4.41 6.75 6.64

15 3.51 3.48 3.48 4.57 4.33 6.75 6.64

16 3.43 3.43 3.40 4.52 4.27 6.75 6.64

17 3.37 3.35 3.35 4.44 4.22 6.72 6.64

18 3.32 3.29 3.29 4.38 4.16 6.72 6.64

19 3.27 3.24 3.24 4.35 4.14 6.72 6.61

20 3.21 3.21 3.18 4.30 4.05 6.72 6.61

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TABLE S9. Fundamental gap Eg(3) [in eV] for the lowest singlet state of n-PP as a function of the chain length, calculated using KS-DFT with various density functionals. For comparison, the experimental data: Expt1 (vertical IP vertical EA) [7–9] and Expt2 (vertical IP adiabatic EA) [7, 10, 11], are taken from the literature.

n LDA PBE BLYP PBE0 B3LYP !B97 !B97X Expt1 Expt2

1 5.28 5.20 5.14 7.27 6.80 12.00 11.62 10.38 [7, 8] 10.37 [7, 10]

2 3.65 3.62 3.56 5.41 5.01 9.77 9.47 8.68 [7, 9] 8.55 [7, 11]

3 3.02 2.97 2.94 4.63 4.27 8.87 8.57 7.65 [7, 11]

4 2.67 2.64 2.61 4.22 3.89 8.38 8.11 7.14 [7, 11]

5 2.48 2.45 2.42 3.97 3.67 8.11 7.81

6 2.34 2.34 2.34 3.84 3.51 7.95 7.65

7 2.23 2.23 2.23 3.76 3.43 7.81 7.54

8 2.20 2.18 2.15 3.67 3.35 7.73 7.43

9 2.15 2.15 2.12 3.62 3.29 7.67 7.40

10 2.10 2.10 2.10 3.56 3.27 7.62 7.35

11 2.07 2.07 2.07 3.54 3.24 7.59 7.29

12 2.07 2.07 2.07 3.51 3.21 7.56 7.29

13 2.04 2.01 2.01 3.48 3.18 7.54 7.27

14 2.01 2.01 2.01 3.48 3.16 7.54 7.24

15 1.99 2.01 2.01 3.46 3.16 7.51 7.24

16 1.99 2.01 1.99 3.46 3.16 7.51 7.21

17 1.99 1.99 1.99 3.43 3.16 7.48 7.21

18 1.99 1.99 1.99 3.43 3.13 7.48 7.21

19 1.99 1.99 1.99 3.43 3.13 7.46 7.21

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TABLE S10. Optical gap Eopt [in eV] for the lowest singlet state of n-PP as a function of the chain length, calculated using TDDFT with various density functionals. For comparison, the experimental data [12, 13] and SAC-CI data [13] are taken from the literature.

n LDA PBE BLYP PBE0 B3LYP !B97 !B97X Expt SAC-CI

1 7.29 7.22 7.15 7.49 7.39 7.58 7.56 6.20 [12]

2 4.51 4.45 4.40 4.86 4.75 5.26 5.22 5.03 [13] 5.09 [13]

3 3.68 3.64 3.60 4.12 4.01 4.68 4.63 4.46 [13] 4.33 [13]

4 3.21 3.18 3.14 3.72 3.61 4.38 4.32 4.21 [13] 3.90 [13]

5 2.90 2.88 2.85 3.48 3.35 4.21 4.15 4.06 [13] 3.63 [13]

6 2.68 2.67 2.64 3.32 3.19 4.10 4.03 4.01 [13]

7 2.53 2.52 2.50 3.21 3.08 4.03 3.96

8 2.42 2.41 2.38 3.13 2.99 3.98 3.91

9 2.33 2.32 2.31 3.07 2.94 3.94 3.87

10 2.26 2.26 2.25 3.03 2.89 3.91 3.84

11 2.21 2.21 2.20 2.99 2.86 3.91 3.83

12 2.17 2.17 2.16 2.97 2.83 3.88 3.81

13 2.14 2.14 2.13 2.95 2.81 3.86 3.79

14 2.11 2.11 2.10 2.93 2.79 3.84 3.78

15 2.09 2.09 2.08 2.92 2.78 3.84 3.76

16 2.07 2.07 2.06 2.90 2.77 3.83 3.75

17 2.05 2.05 2.04 2.90 2.76 3.82 3.74

18 2.04 2.04 2.03 2.89 2.75 3.81 3.74

19 2.03 2.03 2.02 2.88 2.74 3.81 3.74

20 2.01 2.02 2.01 2.88 2.74 3.80 3.73

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TABLE S11. Exciton binding energy Eb(1) [in eV] for the lowest singlet state of n-PP as a function of the chain length, calculated using KS-DFT and TDDFT with various density functionals. For comparison, the experimental data: Expt1 [(vertical IP vertical EA) Eopt] [7–9, 12, 13] and Expt2 [(vertical IP adiabatic EA) Eopt] [7, 10–13], are taken from the literature.

n LDA PBE BLYP PBE0 B3LYP !B97 !B97X Expt1 Expt2

1 4.16 3.99 4.02 3.85 3.93 3.83 3.87 4.18 [7, 8, 12] 4.17 [7, 10, 12]

2 3.81 3.74 3.76 3.65 3.70 3.71 3.72 3.65 [7, 9, 13] 3.52 [7, 11, 13]

3 3.21 3.15 3.16 3.12 3.15 3.30 3.29 3.19 [7, 11, 13]

4 2.83 2.79 2.80 2.80 2.81 3.11 3.09 2.93 [7, 11, 13]

5 2.57 2.54 2.54 2.57 2.57 3.03 3.00

6 2.37 2.34 3.35 2.41 2.39 3.00 2.96

7 2.21 2.18 2.18 2.27 2.25 2.99 2.95

8 2.07 2.04 2.05 2.15 2.12 3.00 2.95

9 1.95 1,92 1.93 2.06 2.02 3.01 2.96

10 1.84 1.82 1.82 1.97 1.92 3.03 2.97

11 1.75 1.73 1.73 1.89 1.84 3.02 2.97

12 1.66 1.64 1.64 1.82 1.76 3.04 2.98

13 1.59 1.57 1.57 1.76 1.70 3.06 3.00

14 1.52 1.50 1.50 1.70 1.63 3.08 3.01

15 1.45 1.44 1.44 1.65 1.58 3.08 3.02

16 1.40 1.38 1.38 1.61 1.53 3.09 3.03

17 1.34 1.33 1.32 1.56 1.48 3.10 3.04

18 1.29 1.28 1.28 1.52 1.44 3.10 3.04

19 1.25 1.23 1.23 1.49 1.40 3.11 3.04

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TABLE S12. Exciton binding energy Eb(2) [in eV] for the lowest singlet state of n-PP as a function of the chain length, calculated using KS-DFT and TDDFT with various density functionals. For comparison, the experimental data: Expt1 [(vertical IP vertical EA) Eopt] [7–9, 12, 13] and Expt2 [(vertical IP adiabatic EA) Eopt] [7, 10–13], are taken from the literature.

n LDA PBE BLYP PBE0 B3LYP !B97 !B97X Expt1 Expt2

1 4.00 3.85 3.85 3.80 3.82 3.91 3.89 4.18 [7, 8, 12] 4.17 [7, 10, 12]

2 3.77 3.66 3.69 3.63 3.65 3.75 3.73 3.65 [7, 9, 13] 3.52 [7, 11, 13]

3 3.17 3.10 3.12 3.09 3.12 3.29 3.29 3.19 [7, 11, 13]

4 2.81 2.75 2.76 2.78 2.79 3.07 3.08 2.93 [7, 11, 13]

5 2.54 2.48 2.48 2.54 2.55 2.97 2.93

6 2.35 2.31 2.31 2.37 2.36 2.92 2.88

7 2.15 2.13 2.13 2.26 2.23 2.88 2.87

8 2.05 2.00 2.00 2.15 2.09 2.88 2.84

9 1.91 1.89 1.88 2.05 1.99 2.89 2.85

10 1.79 1.77 1.78 1.95 1.90 2.86 2.86

11 1.71 1.71 1.70 1.88 1.80 2.87 2.83

12 1.64 1.61 1.62 1.79 1.74 2.89 2.86

13 1.56 1.54 1.52 1.73 1.65 2.88 2.87

14 1.48 1.45 1.47 1.70 1.62 2.91 2.86

15 1.42 1.39 1.41 1.65 1.55 2.91 2.88

16 1.36 1.36 1.34 1.61 1.51 2.92 2.89

17 1.32 1.29 1.30 1.54 1.46 2.90 2.90

18 1.28 1.25 1.26 1.49 1.41 2.91 2.90

19 1.24 1.21 1.22 1.47 1.39 2.91 2.87

20 1.20 1.19 1.17 1.42 1.32 2.92 2.88

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TABLE S13. Exciton binding energy Eb(3) [in eV] for the lowest singlet state of n-PP as a function of the chain length, calculated using KS-DFT and TDDFT with various density functionals. For comparison, the experimental data: Expt1 [(vertical IP vertical EA) Eopt] [7–9, 12, 13] and Expt2 [(vertical IP adiabatic EA) Eopt] [7, 10–13], are taken from the literature.

n LDA PBE BLYP PBE0 B3LYP !B97 !B97X Expt1 Expt2

1 -2.01 -2.02 -2.00 -0.23 -0.58 4.42 4.06 4.18 [7, 8, 12] 4.17 [7, 10, 12]

2 -0.86 -0.83 -0.83 0.56 0.25 4.51 4.25 3.65 [7, 9, 13] 3.52 [7, 11, 13]

3 -0.66 -0.68 -0.66 0.50 0.26 4.19 3.94 3.19 [7, 11, 13]

4 -0.54 -0.54 -0.53 0.50 0.29 4.00 3.79 2.93 [7, 11, 13]

5 -0.42 -0.43 -0.43 0.50 0.32 3.90 3.66 6 -0.34 -0.33 -0.30 0.52 0.32 3.84 3.61 7 -0.30 -0.29 -0.27 0.55 0.35 3.78 3.58 8 -0.21 -0.23 -0.23 0.55 0.35 3.75 3.52 9 -0.18 -0.17 -0.19 0.55 0.36 3.73 3.53 10 -0.17 -0.16 -0.15 0.54 0.37 3.71 3.51 11 -0.14 -0.14 -0.13 0.54 0.38 3.68 3.46 12 -0.10 -0.10 -0.09 0.54 0.38 3.68 3.48 13 -0.10 -0.12 -0.11 0.54 0.37 3.67 3.47 14 -0.09 -0.10 -0.09 0.55 0.36 3.70 3.46 15 -0.10 -0.08 -0.06 0.54 0.38 3.67 3.47 16 -0.08 -0.06 -0.07 0.55 0.39 3.68 3.46 17 -0.06 -0.07 -0.06 0.53 0.40 3.66 3.47 18 -0.05 -0.05 -0.04 0.54 0.38 3.67 3.47 19 -0.04 -0.05 -0.03 0.55 0.39 3.65 3.47

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TABLE S14. Expectation value of the total spin-squared operatorh ˆS2i for the lowest triplet state of n-PP as a function of the chain length, calculated using KS-DFT with various density functionals.

n LDA PBE BLYP PBE0 B3LYP !B97 !B97X

1 2.0131 2.0301 2.0269 2.0724 2.0542 2.1037 2.0900

2 2.0065 2.0169 2.0157 2.0596 2.0415 2.1245 2.1000

3 2.0048 2.0130 2.0122 2.0552 2.0373 2.1407 2.1082

4 2.0037 2.0102 2.0096 2.0518 2.0337 2.1609 2.1231

5 2.0030 2.0083 2.0079 2.0518 2.0327 2.1791 2.1363

6 2.0024 2.0069 2.0066 2.0495 2.0307 2.1753 2.1332

7 2.0021 2.0059 2.0056 2.0494 2.0299 2.1887 2.1428

8 2.0018 2.0051 2.0049 2.0479 2.0287 2.1892 2.1433

9 2.0015 2.0045 2.0043 2.0482 2.0285 2.1901 2.1437

10 2.0014 2.0040 2.0039 2.0471 2.0279 2.1800 2.1362

11 2.0012 2.0036 2.0035 2.0478 2.0279 2.1904 2.1438

12 2.0011 2.0033 2.0032 2.0475 2.0275 2.1799 2.1363

13 2.0010 2.0031 2.0030 2.0476 2.0277 2.1901 2.1439

14 2.0010 2.0029 2.0028 2.0471 2.0275 2.1801 2.1362

15 2.0009 2.0027 2.0027 2.0476 2.0276 2.1901 2.1439

16 2.0008 2.0026 2.0026 2.0476 2.0274 2.1802 2.1364

17 2.0008 2.0025 2.0025 2.0476 2.0276 2.1901 2.1439

18 2.0008 2.0025 2.0024 2.0476 2.0276 2.1802 2.1439

19 2.0007 2.0024 2.0024 2.0476 2.0276 2.1901 2.1439

20 2.0007 2.0024 2.0024 2.0476 2.0276 2.1801 2.1365

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TABLE S15. Expectation value of the total spin-squared operator h ˆS2i for the ground state of cationic n-PP as a function of the chain length, calculated using KS-DFT with various density functionals (at the ground-state geometry of neutral n-PP).

n LDA PBE BLYP PBE0 B3LYP !B97 !B97X

1 0.7530 0.7567 0.7556 0.7661 0.7656 0.7842 0.7782

2 0.7520 0.7551 0.7545 0.7730 0.7652 0.8231 0.8051

3 0.7512 0.7532 0.7529 0.7691 0.7624 0.8242 0.8066

4 0.7508 0.7522 0.7520 0.7661 0.7602 0.8298 0.8106

5 0.7506 0.7516 0.7514 0.7635 0.7585 0.8313 0.8117

6 0.7505 0.7512 0.7511 0.7613 0.7572 0.8313 0.8115

7 0.7504 0.7510 0.7509 0.7595 0.7561 0.8304 0.8104

8 0.7503 0.7508 0.7507 0.7580 0.7552 0.8294 0.8092

9 0.7503 0.7507 0.7506 0.7569 0.7545 0.8285 0.8081

10 0.7502 0.7506 0.7505 0.7560 0.7539 0.8279 0.8073

11 0.7502 0.7505 0.7505 0.7553 0.7535 0.8274 0.8068

12 0.7502 0.7505 0.7504 0.7547 0.7532 0.8272 0.8064

13 0.7502 0.7504 0.7504 0.7543 0.7529 0.8271 0.8062

14 0.7502 0.7504 0.7503 0.7539 0.7526 0.8270 0.8060

15 0.7501 0.7503 0.7503 0.7536 0.7524 0.8269 0.8060

16 0.7501 0.7503 0.7503 0.7533 0.7523 0.8269 0.8059

17 0.7501 0.7503 0.7503 0.7531 0.7521 0.8269 0.8059

18 0.7501 0.7503 0.7503 0.7529 0.7520 0.8269 0.8059

19 0.7501 0.7503 0.7502 0.7527 0.7519 0.8269 0.8059

20 0.7501 0.7502 0.7502 0.7526 0.7518 0.8270 0.8059

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TABLE S16. Expectation value of the total spin-squared operator h ˆS2i for the ground state of anionic n-PP as a function of the chain length, calculated using KS-DFT with various density functionals (at the ground-state geometry of neutral n-PP).

n LDA PBE BLYP PBE0 B3LYP !B97 !B97X

1 0.7526 0.7556 0.7556 0.7595 0.7577 0.7709 0.7681

2 0.7515 0.7536 0.7535 0.7660 0.7612 0.7984 0.7882

3 0.7510 0.7527 0.7525 0.7652 0.7602 0.8073 0.7955

4 0.7507 0.7520 0.7518 0.7638 0.7591 0.8145 0.8010

5 0.7506 0.7515 0.7514 0.7622 0.7579 0.8174 0.8032

6 0.7504 0.7512 0.7511 0.7606 0.7568 0.8180 0.8035

7 0.7504 0.7510 0.7509 0.7591 0.7559 0.8174 0.8028

8 0.7503 0.7508 0.7507 0.7579 0.7551 0.8164 0.8017

9 0.7503 0.7507 0.7506 0.7568 0.7544 0.8154 0.8006

10 0.7502 0.7506 0.7506 0.7560 0.7539 0.8146 0.7997

11 0.7502 0.7505 0.7505 0.7553 0.7535 0.8140 0.7990

12 0.7502 0.7505 0.7504 0.7547 0.7531 0.8135 0.7984

13 0.7502 0.7504 0.7504 0.7543 0.7529 0.8133 0.7980

14 0.7502 0.7504 0.7504 0.7539 0.7526 0.8131 0.7978

15 0.7501 0.7504 0.7503 0.7536 0.7524 0.8129 0.7976

16 0.7501 0.7503 0.7503 0.7533 0.7523 0.8128 0.7975

17 0.7501 0.7503 0.7503 0.7531 0.7521 0.8128 0.7974

18 0.7501 0.7503 0.7503 0.7529 0.7520 0.8127 0.7973

19 0.7501 0.7503 0.7503 0.7527 0.7519 0.8127 0.7973

20 0.7501 0.7503 0.7502 0.7526 0.7518 0.8127 0.7973

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FIG. S1. Comparison of the singlet-triplet energy gap and expectation value of the total spin- squared operatorh ˆS2i for the lowest triplet state of n-PP as a function of the chain length, calcu- lated using KS-DFT with the !B97 and !B97X functionals.

[1] P. O. L¨owdin, Phys. Rev. 1955, 97, 1474.

[2] J. Wang, A. D. Becke, V. H. Smith Jr., J. Chem. Phys. 1995, 102, 3477.

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[9] A. Modelli, G. Distefano, D. Jones, Chem. Phys. 1983, 82, 489.

[10] J. C. Rienstra-Kiracofe, G. S. Tschumper, H. F. Schaefer, S. Nandi, G. B. Ellison, Chem. Rev.

2002, 102, 231.

[11] T. Nakamura, N. Ando, Y. Matsumoto, S. Furuse, M. Mitsui, A. Nakajima, Chem. Lett. 2006, 35, 888.

[12] N. C. Handy, D. J. Tozer, J. Comput. Chem. 1999, 20, 106.

[13] B. Saha, M. Ehara, H. Nakatsuji, J. Phys. Chem. A 2007, 111, 5473.

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