with generalized-gradient approximations
Jeng-Da Chai⇤
Department of Physics, Center for Theoretical Sciences, and Center for Quantum Science and Engineering, National Taiwan University, Taipei 10617, Taiwan
References (S2).
TABLE S1. Non-hydrogen transfer barrier heights of the NHTBH38/04 set (S3).
TABLE S2. Hydrogen transfer barrier heights of the HTBH38/04 set (S4).
TABLE S3. Interaction energies of the S22 set (S5).
TABLE S4. Reaction energies of the 30 chemical reactions in the NHTBH38/04 and HTBH38/04 sets (S5 to S6).
TABLE S5. Equilibrium geometries of EXTS (S6 to S9).
TABLE S6. Singlet-triplet energy gap of n-acene (S10).
TABLE S7. Vertical ionization potential for the lowest singlet state of n-acene (S11).
TABLE S8. Vertical electron affinity for the lowest singlet state of n-acene (S12).
TABLE S9. Fundamental gap for the lowest singlet state of n-acene (S13).
TABLE S10. Symmetrized von Neumann entropy for the lowest singlet state of n-acene (S14 to S15).
⇤Electronic mail: [email protected]
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[6] N. Sabbatini, M. T. Indelli, M. T. Gandolfi, and V. Balzani, J. Phys. Chem.86, 3585 (1982).
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[8] J. Hachmann, J. J. Dorando, M. Avilés, and G. K.-L. Chan, J. Chem. Phys.127, 134309 (2007).
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[12] B. Hajgató, M. S. Deleuze, D. J. Tozer, and F. De Proft, J. Chem. Phys.129, 084308 (2008).
TABLES
TABLE S1. Non-hydrogen transfer barrier heights (in kcal/mol) of the NHTBH38/04 set [1].
KS-DFT TAO-DFT
Reactions Eref LDA PBE BLYP BLYP-D LDA PBE BLYP BLYP-D
Heavy-atom transfer reactions
H + N2O ! OH + N2 Vf 18.14 2.69 9.97 8.54 8.19 2.63 9.92 8.48 8.13
Vr 83.22 32.09 52.46 61.67 61.24 37.21 53.40 62.92 62.48
H + FH ! HF + H Vf 42.18 18.48 27.07 26.03 25.76 18.57 27.14 26.10 25.83
Vr 42.18 18.48 27.07 26.03 25.76 18.57 27.14 26.10 25.83
H + ClH ! HCl + H Vf 18.00 2.27 9.64 9.80 9.74 2.14 9.51 9.66 9.60
Vr 18.00 2.27 9.64 9.80 9.74 2.14 9.51 9.66 9.60 H + FCH3 ! HF + CH3 Vf 30.38 13.36 18.51 16.12 15.33 13.36 18.52 16.12 15.33 Vr 57.02 31.72 41.00 42.27 41.33 31.62 40.88 42.14 41.21 H + F2 ! HF + F Vf 2.27 -15.97 -9.86 -11.66 -11.78 -16.13 -9.98 -11.79 -11.91 Vr 106.18 69.40 80.86 82.15 82.02 77.32 82.70 84.15 84.03 CH3 + FCl ! CH3F + Cl Vf 7.43 -11.50 -6.48 -6.97 -8.45 -11.93 -6.78 -7.30 -8.78 Vr 60.17 37.67 41.63 41.76 40.43 46.51 47.37 48.19 46.85 Nucleophilic substitution reactions
F + CH3F ! FCH3 + F Vf -0.34 -12.18 -8.31 -7.90 -8.95 -12.39 -8.51 -8.10 -9.15 Vr -0.34 -12.18 -8.31 -7.90 -8.95 -12.39 -8.51 -8.10 -9.15 F · · ·CH3F ! FCH3· · ·F Vf 13.38 6.55 7.06 6.50 6.84 6.45 6.96 6.40 6.74 Vr 13.38 6.55 7.06 6.50 6.84 6.45 6.96 6.40 6.74 Cl + CH3Cl ! ClCH3 + Cl Vf 3.10 -6.74 -3.80 -3.95 -5.69 -7.07 -4.13 -4.30 -6.03 Vr 3.10 -6.74 -3.80 -3.95 -5.69 -7.07 -4.13 -4.30 -6.03 Cl · · ·CH3Cl ! ClCH3· · ·Cl Vf 13.61 6.74 7.15 5.72 5.55 6.66 7.07 5.65 5.48 Vr 13.61 6.74 7.15 5.72 5.55 6.66 7.07 5.65 5.48 F + CH3Cl ! FCH3 + Cl Vf -12.54 -23.45 -19.52 -19.35 -20.67 -23.59 -19.65 -19.47 -20.79 Vr 20.11 10.10 12.31 13.03 11.62 9.68 11.90 12.61 11.19 F · · ·CH3Cl ! FCH3· · ·Cl Vf 2.89 -1.04 -0.67 -1.65 -1.54 -1.07 -0.70 -1.68 -1.57 Vr 29.62 21.65 21.45 21.03 21.08 21.49 21.29 20.87 20.92 OH + CH3F ! HOCH3+ F Vf -2.78 -14.36 -10.59 -9.77 -11.56 -14.57 -10.74 -9.90 -11.69 Vr 17.33 5.70 9.14 9.07 7.79 5.60 9.06 9.00 7.72 OH · · ·CH3F ! HOCH3· · ·F Vf 10.96 3.39 3.88 3.73 3.17 3.27 3.85 3.73 3.17 Vr 47.20 47.53 43.21 40.24 39.70 47.58 43.28 40.31 39.77 Unimolecular and association reactions
H + N2 ! HN2 Vf 14.69 -2.19 5.19 5.24 5.08 -2.35 5.04 5.09 4.92
Vr 10.72 9.44 9.08 8.51 8.38 9.60 9.19 8.61 8.48
H + CO ! HCO Vf 3.17 -7.57 -1.69 -1.95 -2.25 -7.71 -1.83 -2.09 -2.39
Vr 22.68 26.34 24.66 23.34 23.13 26.54 24.81 23.49 23.28 H + C2H4 ! CH3CH2 Vf 1.72 -5.34 -0.14 -0.69 -1.78 -5.46 -0.27 -0.81 -1.90 Vr 41.75 39.21 40.20 38.11 37.63 39.37 40.35 38.27 37.78 CH3 + C2H4 ! CH3CH2CH2 Vf 6.85 -5.81 1.50 4.72 1.48 -6.13 1.18 4.39 1.15 Vr 32.97 32.95 29.76 24.88 24.13 33.08 29.90 25.02 24.27
HCN ! HNC Vf 48.16 44.83 45.60 46.76 46.94 44.89 45.66 46.81 46.99
Vr 33.11 30.60 30.50 31.68 31.80 30.60 30.50 31.68 31.80
MSE -12.41 -8.52 -8.69 -9.32 -11.93 -8.38 -8.52 -9.15
MAE 12.62 8.62 8.72 9.35 12.15 8.49 8.56 9.19
rms 16.13 10.61 10.27 10.83 15.09 10.28 9.90 10.46
Max( ) -51.13 -30.76 -24.03 -24.16 -46.01 -29.82 -22.03 -22.15
Max(+) 3.66 1.98 0.66 0.45 3.86 2.13 0.81 0.60
TABLE S2. Hydrogen transfer barrier heights (in kcal/mol) of the HTBH38/04 set [1].
KS-DFT TAO-DFT
Reactions Eref LDA PBE BLYP BLYP-D LDA PBE BLYP BLYP-D
H + HCl ! H2 + Cl Vf 5.7 -3.14 0.56 -2.45 -2.58 -3.68 0.17 -2.89 -3.02
Vr 8.7 -11.27 -1.95 1.84 1.71 -2.70 3.54 7.98 7.86
OH + H2 ! H + H2O Vf 5.1 -18.39 -6.36 -3.49 -3.95 -13.65 -5.61 -2.43 -2.89 Vr 21.2 11.33 13.69 10.39 9.96 10.88 13.44 10.13 9.70
CH3 + H2 ! H + CH4 Vf 12.1 -5.35 3.82 7.14 5.96 -5.58 3.59 6.91 5.73
Vr 15.3 4.94 9.32 7.65 6.61 4.87 9.24 7.58 6.54
OH + CH4 ! CH3 + H2O Vf 6.7 -17.22 -5.65 -2.75 -4.01 -12.41 -4.87 -1.64 -2.92
Vr 19.6 2.21 8.91 10.63 9.24 1.68 8.52 10.24 8.86
H + H2 ! H2 + H Vf 9.6 -2.70 3.64 2.86 2.67 -2.80 3.53 2.76 2.57
Vr 9.6 -2.70 3.64 2.86 2.67 -2.80 3.53 2.76 2.57
OH + NH3 ! H2O + NH2 Vf 3.2 -23.88 -11.94 -9.20 -10.46 -19.22 -11.26 -8.21 -9.47 Vr 12.7 -10.71 -0.71 1.81 0.47 -11.30 -1.14 1.38 0.04 HCl + CH3 ! Cl+ CH4 Vf 1.7 -13.74 -5.94 -3.51 -5.09 -14.16 -6.30 -3.88 -5.46 Vr 7.9 -11.58 -2.95 1.29 -0.15 -2.73 2.73 7.66 6.22 OH + C2H6! H2O + C2H5 Vf 3.4 -20.98 -9.03 -6.12 -7.54 -16.17 -8.26 -5.02 -6.45 Vr 19.9 4.79 10.76 12.36 10.26 4.26 10.37 11.97 9.87 F + H2 ! HF + H Vf 1.8 -24.20 -12.97 -11.61 -11.86 -17.16 -11.52 -9.97 -10.22 Vr 33.4 25.71 25.12 20.70 20.45 24.80 24.73 20.32 20.07
O + CH4 ! OH + CH3 Vf 13.7 -10.69 -0.79 1.44 0.52 -9.95 -1.14 1.20 -0.17
Vr 8.1 -9.27 -0.57 1.68 0.61 -9.32 -0.68 1.50 0.46
H + PH3 ! PH2 + H2 Vf 3.1 -7.35 -1.79 -2.63 -3.14 -7.46 -1.90 -2.73 -3.24 Vr 23.2 9.24 17.91 21.58 20.38 9.02 17.68 21.34 20.15
H + HO ! H2 + O Vf 10.7 -1.69 3.75 1.60 1.48 -1.55 3.84 1.66 1.58
Vr 13.1 -13.40 -1.98 0.86 0.74 -12.63 -2.27 0.69 0.13
H + H2S ! H2 + HS Vf 3.5 -6.73 -1.22 -2.20 -2.59 -7.03 -1.42 -2.48 -2.86
Vr 17.3 -0.86 9.10 14.00 13.36 4.93 12.53 18.07 17.43 O + HCl ! OH + Cl Vf 9.8 -23.13 -10.54 -8.78 -8.86 -19.48 -10.85 -8.75 -9.32 Vr 10.4 -19.55 -7.33 -3.75 -3.83 -7.42 -1.36 3.09 3.00
NH2 + CH3 ! CH4 + NH Vf 8.0 -8.37 0.71 3.57 1.82 -8.62 0.43 3.28 1.53
Vr 22.4 2.16 10.56 13.31 11.66 1.93 10.34 13.09 11.44 NH2 + C2H5 ! C2H6 + NH Vf 7.5 -5.76 2.89 5.94 3.56 -6.02 2.60 5.64 3.26 Vr 18.3 -1.57 7.51 10.59 8.87 -1.81 7.27 10.34 8.62 C2H6+ NH2 ! NH3 + C2H5 Vf 10.4 -9.69 1.44 5.25 3.08 -9.91 1.18 4.99 2.82
Vr 17.4 2.91 9.99 12.72 9.94 2.60 9.68 12.40 9.62
NH2 + CH4 ! CH3 + NH3 Vf 14.5 -6.16 4.39 7.99 6.04 -6.37 4.15 7.75 5.80
Vr 17.8 0.10 7.71 10.35 8.37 -0.20 7.41 10.05 8.06
s-trans cis-C5H8 ! s-trans cis-C5H8 Vf 38.4 24.99 31.19 35.82 34.69 24.88 31.08 35.70 34.58 Vr 38.4 24.99 31.19 35.82 34.69 24.88 31.08 35.70 34.58
MSE -17.90 -9.67 -7.84 -8.89 -16.34 -9.20 -7.25 -8.33
MAE 17.90 9.67 7.84 8.89 16.34 9.20 7.29 8.34
rms 18.92 10.37 8.66 9.52 17.06 9.87 8.24 9.17
Max( ) -32.93 -20.34 -18.58 -18.66 -29.28 -20.65 -18.55 -19.12
Max(+) -7.69 -4.61 -1.56 -2.82 -8.60 -4.77 0.77 0.13
TABLE S3. Interaction energies (in kcal/mol) of the S22 set [2]. The counterpoise corrections are used to reduce the basis set superposition errors. Monomer deformation energies are not included.
KS-DFT TAO-DFT
Complex [Symmetry] Eref LDA PBE BLYP BLYP-D LDA PBE BLYP BLYP-D
Hydrogen bonded complexes
(NH3)2 [C2h] -3.17 -5.10 -2.83 -1.78 -3.44 -5.27 -3.00 -1.96 -3.63
(H2O)2 [Cs] -5.02 -7.78 -4.91 -3.99 -4.92 -7.95 -5.09 -4.18 -5.11
Formic acid dimer [C2h] -18.61 -26.82 -18.08 -15.48 -18.50 -27.22 -18.49 -15.89 -18.91 Formamide dimer [C2h] -15.96 -21.82 -14.69 -12.44 -15.63 -22.21 -15.08 -12.85 -16.03 Uracil dimer [C2h] -20.65 -26.15 -18.45 -16.21 -20.21 -26.51 -18.82 -16.59 -20.59 2-pyridoxine·2-aminopyridine [C1] -16.71 -22.77 -15.27 -12.51 -17.28 -23.18 -15.68 -12.94 -17.71 Adenine·thymine WC [C1] -16.37 -21.99 -14.28 -11.42 -16.50 -22.43 -14.73 -11.88 -16.96
MSE -5.13 1.14 3.24 0.00 -5.47 0.80 2.89 -0.35
MAE 5.13 1.14 3.24 0.28 5.47 0.82 2.89 0.37
Dispersion complexes
(CH4)2 [D3d] -0.53 -0.83 -0.08 0.68 -0.31 -0.95 -0.20 0.55 -0.44
(C2H4)2[D2d] -1.51 -2.49 -0.31 1.12 -1.44 -2.71 -0.54 0.89 -1.68
Benzene·CH4 [C3] -1.50 -2.01 -0.01 1.39 -1.23 -2.21 -0.22 1.17 -1.45
Benzene dimer [C2h] -2.73 -2.62 1.90 4.94 -1.93 -3.03 1.49 4.50 -2.36
Pyrazine dimer [Cs] -4.42 -4.46 0.73 3.80 -3.65 -4.86 0.32 3.36 -4.09
Uracil dimer [C2] -10.12 -10.15 -2.69 1.13 -9.68 -10.66 -3.21 0.58 -10.24
Indole·benzene [C1] -5.22 -4.38 2.24 6.32 -3.92 -4.90 1.71 5.76 -4.47
Adenine·thymine stack [C1] -12.23 -11.96 -1.33 4.06 -11.81 -12.63 -2.01 3.35 -12.52
MSE -0.08 4.84 7.71 0.54 -0.46 4.45 7.30 0.13
MAE 0.38 4.84 7.71 0.54 0.54 4.45 7.30 0.27
Mixed complexes
Ethene·ethine [C2v] -1.53 -2.27 -1.16 -0.29 -1.49 -2.41 -1.30 -0.45 -1.64
Benzene·H2O [Cs] -3.28 -4.44 -2.04 -0.49 -3.29 -4.67 -2.27 -0.74 -3.54
Benzene·NH3 [Cs] -2.35 -3.03 -0.92 0.55 -2.18 -3.24 -1.14 0.31 -2.42
Benzene·HCN [Cs] -4.46 -5.85 -2.81 -0.94 -4.65 -6.09 -3.05 -1.19 -4.91
Benzene dimer [C2v] -2.74 -3.06 -0.10 1.86 -2.51 -3.33 -0.38 1.57 -2.81
Indole·benzene T-shape [C1] -5.73 -6.26 -2.02 0.69 -5.67 -6.63 -2.39 0.30 -6.05
Phenol dimer [C1] -7.05 -9.01 -3.86 -1.75 -6.65 -9.34 -4.20 -2.10 -7.00
MSE -0.97 2.03 3.83 0.10 -1.22 1.77 3.55 -0.18
MAE 0.97 2.03 3.83 0.16 1.22 1.77 3.55 0.19
MSE -1.97 2.77 5.05 0.23 -2.30 2.44 4.70 -0.12
MAE 2.08 2.77 5.05 0.33 2.33 2.44 4.70 0.28
rms 3.18 3.89 6.31 0.45 3.40 3.57 5.95 0.37
Max( ) -8.21 0.11 1.03 -0.57 -8.61 -0.07 0.84 -1.00
Max(+) 0.84 10.90 16.29 1.30 0.32 10.22 15.58 0.75
TABLE S4. Comparison of errors (in kcal/mol) of the reaction energies (in kcal/mol) of the 30 chemical reactions in the NHTBH38/04 and HTBH38/04 sets [1].
KS-DFT TAO-DFT
Reactions Eref LDA PBE BLYP BLYP-D LDA PBE BLYP BLYP-D
H + N2O ! OH + N2 -65.08 35.68 22.59 11.96 12.03 30.50 21.60 10.65 10.73
H + FCH3 ! HF + CH3 -26.64 8.28 4.16 0.49 0.63 8.39 4.28 0.62 0.77
H + F2 ! HF + F -103.91 18.54 13.19 10.10 10.10 10.46 11.23 7.97 7.97
CH3 + FCl ! CH3F + Cl -52.74 3.57 4.64 4.01 3.86 -5.71 -1.41 -2.75 -2.89
F + CH3Cl ! FCH3 + Cl -32.65 -0.90 0.82 0.27 0.37 -0.62 1.11 0.57 0.67
F · · ·CH3Cl ! FCH3· · ·Cl -26.73 4.05 4.61 4.05 4.11 4.18 4.73 4.17 4.24
OH + CH3F ! HOCH3+ F -20.11 0.05 0.38 1.27 0.75 -0.07 0.30 1.21 0.70
OH · · ·CH3F ! HOCH3· · ·F -36.24 -7.90 -3.09 -0.27 -0.29 -8.07 -3.19 -0.34 -0.36
H + N2 ! HN2 3.97 -15.60 -7.86 -7.24 -7.28 -15.92 -8.12 -7.49 -7.53
H + CO ! HCO -19.51 -14.40 -6.84 -5.78 -5.87 -14.74 -7.13 -6.07 -6.16
H + C2H4 ! CH3CH2 -40.03 -4.52 -0.31 1.22 0.62 -4.79 -0.59 0.95 0.34
CH3 + C2H4 ! CH3CH2CH2 -26.12 -12.64 -2.14 5.96 3.47 -13.09 -2.60 5.49 2.99
HCN ! HNC 15.05 -0.83 0.06 0.02 0.09 -0.77 0.12 0.08 0.14
H + HCl ! H2 + Cl -3.0 11.13 5.51 -1.29 -1.29 2.01 -0.37 -7.87 -7.87
OH + H2 ! H + H2O -16.1 -13.62 -3.95 2.22 2.19 -8.43 -2.95 3.55 3.51
CH3 + H2 ! H + CH4 -3.2 -7.09 -2.30 2.70 2.55 -7.25 -2.45 2.53 2.39
OH + CH4 ! CH3 + H2O -12.9 -6.53 -1.65 -0.48 -0.36 -1.19 -0.49 1.02 1.13 OH + NH3 ! H2O + NH2 -9.5 -3.67 -1.73 -1.51 -1.43 1.58 -0.63 -0.08 -0.01
HCl + CH3 ! Cl+ CH4 -6.2 4.04 3.21 1.41 1.26 -5.23 -2.83 -5.34 -5.48
OH + C2H6! H2O + C2H5 -16.5 -9.27 -3.29 -1.97 -1.30 -3.93 -2.13 -0.49 0.19
F + H2 ! HF + H -31.6 -18.31 -6.49 -0.72 -0.72 -10.36 -4.65 1.30 1.30
O + CH4 ! OH + CH3 5.6 -7.02 -5.83 -5.84 -5.69 -6.23 -6.06 -5.90 -6.23
H + PH3 ! PH2 + H2 -20.1 3.51 0.40 -4.10 -3.42 3.63 0.52 -3.98 -3.29
H + HO ! H2 + O -2.4 14.11 8.12 3.14 3.14 13.48 8.52 3.37 3.84
H + H2S ! H2 + HS -13.8 7.93 3.48 -2.41 -2.15 1.84 -0.15 -6.74 -6.49
O + HCl ! OH + Cl -0.6 -2.98 -2.61 -4.43 -4.43 -11.46 -8.89 -11.24 -11.71
NH2 + CH3 ! CH4 + NH -14.4 3.87 4.55 4.66 4.56 3.85 4.49 4.59 4.49
NH2 + C2H5 ! C2H6 + NH -10.8 6.62 6.19 6.15 5.49 6.59 6.13 6.09 5.43 C2H6+ NH2 ! NH3 + C2H5 -7.0 -5.61 -1.55 -0.46 0.14 -5.51 -1.50 -0.40 0.20
NH2 + CH4 ! CH3 + NH3 -3.3 -2.96 -0.02 0.94 0.97 -2.87 0.03 1.00 1.04
MSE -0.41 1.08 0.80 0.74 -1.32 0.23 -0.12 -0.20
MAE 8.51 4.39 3.23 3.02 7.09 3.97 3.80 3.67
rms 11.10 6.24 4.37 4.20 9.38 5.97 4.95 4.89
Max( ) -18.31 -7.86 -7.24 -7.28 -15.92 -8.89 -11.24 -11.71
Max(+) 35.68 22.59 11.96 12.03 30.50 21.60 10.65 10.73
TABLE S5. The 166 experimentally determined equilibrium bond lengths (in Å) of 136 molecules included in EXTS [3].
Optimized bond lengths are calculated by various methods.
KS-DFT TAO-DFT
Molecule Experiment LDA PBE BLYP BLYP-D LDA PBE BLYP BLYP-D
H2N 1.028 1.037 1.038 1.038 1.038 1.037 1.037 1.038 1.038
NO (3⌃ ) 1.258 1.250 1.267 1.279 1.279 1.250 1.267 1.278 1.278
PO (3⌃ ) 1.540 1.539 1.556 1.563 1.563 1.539 1.556 1.563 1.563
BeH+ (1⌃+) 1.312 1.337 1.330 1.323 1.323 1.336 1.329 1.322 1.322
CH+ (1⌃+) 1.131 1.162 1.153 1.147 1.147 1.162 1.153 1.147 1.147
CN+ (1⌃+) 1.173 1.173 1.181 1.183 1.183 1.203 1.216 1.220 1.220
H+3 0.877 0.906 0.889 0.886 0.886 0.906 0.888 0.885 0.885
H3O+ 0.976 0.990 0.988 0.989 0.990 0.990 0.987 0.989 0.990
HCO+ 1.097 (CH) 1.111 1.104 1.101 1.102 1.111 1.103 1.101 1.102
1.105 (CO) 1.105 1.113 1.113 1.113 1.105 1.113 1.113 1.113
MgH+ (1⌃+) 1.652 1.671 1.673 1.669 1.670 1.669 1.671 1.668 1.669
NO+ (1⌃+) 1.063 1.062 1.070 1.071 1.071 1.062 1.069 1.071 1.071
NS+ (1⌃+) 1.440 1.433 1.444 1.447 1.447 1.433 1.444 1.446 1.446
OH+ (3⌃ ) 1.029 1.055 1.047 1.050 1.050 1.055 1.046 1.050 1.050
SiH+ (1⌃+) 1.504 1.538 1.532 1.527 1.527 1.537 1.532 1.526 1.526
NH+4 1.021 1.035 1.031 1.032 1.033 1.034 1.031 1.031 1.032
AlCl (1⌃+) 2.130 2.136 2.162 2.178 2.179 2.135 2.161 2.178 2.178
AlF (1⌃+) 1.654 1.669 1.693 1.698 1.698 1.669 1.692 1.698 1.698
AlH (1⌃+) 1.648 1.680 1.679 1.674 1.674 1.679 1.678 1.673 1.673
AlN (3⇧i) 1.786 1.769 1.792 1.799 1.799 1.774 1.799 1.810 1.811
BCl (1⌃+) 1.797 1.706 1.729 1.734 1.734 1.706 1.728 1.733 1.733
BeO (1⌃+) 1.331 1.326 1.338 1.339 1.339 1.331 1.345 1.347 1.347
BeS (1⌃+) 1.742 1.739 1.751 1.754 1.754 1.753 1.768 1.775 1.775
BF (1⌃+) 1.263 1.256 1.271 1.272 1.272 1.256 1.271 1.272 1.272
BF3 1.307 1.307 1.322 1.325 1.326 1.307 1.322 1.325 1.326
BH (1⌃+) 1.232 1.256 1.251 1.240 1.240 1.254 1.250 1.239 1.239
BN (3⇧) 1.281 1.323 1.332 1.333 1.333 1.322 1.331 1.332 1.332
C6H6 1.390 (CC) 1.385 1.397 1.401 1.402 1.385 1.397 1.400 1.402
1.086 (CH) 1.094 1.091 1.088 1.089 1.093 1.091 1.088 1.088
CCl2O 1.177 (CO) 1.177 1.185 1.184 1.184 1.177 1.185 1.184 1.184
1.737 (CCl) 1.731 1.754 1.777 1.780 1.730 1.754 1.776 1.779
CF4 1.315 1.318 1.335 1.343 1.344 1.317 1.335 1.343 1.344
CH2Cl2 1.766 (CCl) 1.754 1.779 1.801 1.805 1.754 1.778 1.800 1.804
1.080 (CH) 1.096 1.092 1.088 1.090 1.095 1.091 1.088 1.089
CH2F2 1.351 (CF) 1.348 1.369 1.380 1.382 1.348 1.369 1.380 1.382
1.084 (CH) 1.105 1.100 1.096 1.099 1.105 1.100 1.096 1.098
CH2O2 (trans) 1.201 (C=O) 1.198 1.207 1.208 1.208 1.198 1.206 1.208 1.208
1.091 (CH) 1.111 1.106 1.103 1.104 1.110 1.106 1.103 1.103
1.340 (CO) 1.332 1.353 1.362 1.363 1.332 1.353 1.362 1.363
0.969 (OH) 0.983 0.980 0.981 0.981 0.983 0.979 0.980 0.980
CH3Cl 1.086 (CH) 1.096 1.094 1.091 1.093 1.096 1.093 1.090 1.093
1.778 (CCl) 1.763 1.789 1.815 1.820 1.762 1.788 1.815 1.819
CH3F 1.086 (CH) 1.102 1.099 1.096 1.098 1.102 1.098 1.095 1.098
1.383 (CF) 1.371 1.395 1.409 1.412 1.371 1.395 1.409 1.412
CH4 1.087 1.097 1.096 1.094 1.097 1.097 1.095 1.094 1.096
CHF3 1.328 (CF) 1.330 1.349 1.358 1.359 1.330 1.349 1.357 1.359
1.091 (CH) 1.105 1.100 1.096 1.097 1.105 1.099 1.095 1.097
Cl2 (1⌃+g) 1.988 1.980 2.005 2.040 2.040 1.979 2.005 2.040 2.040
ClCN 1.629 (CCl) 1.612 1.630 1.642 1.643 1.612 1.629 1.642 1.643
1.160 (CN) 1.160 1.167 1.166 1.167 1.160 1.167 1.166 1.167
ClF 1.628 1.625 1.653 1.674 1.675 1.625 1.653 1.674 1.674
CO (1⌃+) 1.128 1.127 1.136 1.136 1.136 1.127 1.136 1.136 1.136
CO2 1.160 1.162 1.171 1.172 1.172 1.161 1.170 1.172 1.172
CS (1⌃+) 1.535 1.533 1.546 1.548 1.548 1.533 1.546 1.548 1.548
CS2 1.553 1.551 1.562 1.567 1.568 1.550 1.562 1.567 1.568
F2 (1⌃+g) 1.412 1.383 1.412 1.431 1.431 1.383 1.411 1.431 1.431
F2O 1.405 1.388 1.417 1.437 1.438 1.388 1.417 1.437 1.438
F2S 1.587 1.598 1.624 1.638 1.639 1.598 1.623 1.638 1.639
F2Si 1.590 1.607 1.629 1.635 1.636 1.607 1.629 1.635 1.636
F3HSi 1.562 (SiF) 1.575 1.595 1.599 1.600 1.575 1.594 1.599 1.600
1.447 (SiH) 1.470 1.468 1.463 1.460 1.469 1.468 1.462 1.460
H2 (1⌃+g) 0.741 0.766 0.751 0.747 0.747 0.766 0.750 0.746 0.746
H2CCCH2 1.076 (CH) 1.095 1.092 1.090 1.091 1.094 1.091 1.089 1.091
1.308 (CC) 1.298 1.308 1.308 1.309 1.298 1.307 1.308 1.309
H2CS 1.611 (CS) 1.601 1.614 1.621 1.622 1.601 1.615 1.622 1.623
1.086 (CH) 1.100 1.097 1.094 1.096 1.100 1.097 1.094 1.096
H2O 0.958 0.971 0.969 0.971 0.971 0.970 0.969 0.971 0.971
H2S 1.336 1.354 1.352 1.352 1.353 1.353 1.351 1.352 1.352
H2Si 1.514 1.538 1.538 1.534 1.533 1.537 1.537 1.533 1.532
HCCCN 1.062 (CH) 1.075 1.071 1.068 1.068 1.075 1.071 1.067 1.068
1.206 (C⌘C) 1.208 1.214 1.212 1.213 1.208 1.214 1.212 1.213
1.376 (CC) 1.355 1.366 1.367 1.368 1.355 1.365 1.367 1.368
1.161 (C⌘N) 1.162 1.170 1.169 1.169 1.162 1.170 1.169 1.169
HCCH 1.061 (CH) 1.074 1.070 1.067 1.068 1.074 1.070 1.067 1.068
1.203 (CC) 1.202 1.207 1.206 1.207 1.201 1.207 1.206 1.206
HCl (1⌃+) 1.275 1.291 1.288 1.291 1.291 1.291 1.288 1.290 1.290
HCN 1.065 (CH) 1.079 1.075 1.072 1.073 1.079 1.075 1.072 1.073
1.153 (CN) 1.151 1.158 1.157 1.158 1.151 1.158 1.157 1.157
HCP 1.066 (CH) 1.083 1.080 1.077 1.078 1.083 1.080 1.076 1.077
1.540 (CP) 1.537 1.547 1.548 1.549 1.536 1.546 1.548 1.548
HF (1⌃+) 0.917 0.932 0.930 0.933 0.933 0.931 0.930 0.933 0.933
HNC 0.994 (NH) 1.009 1.006 1.005 1.006 1.009 1.005 1.005 1.005
1.169 (NC) 1.168 1.176 1.175 1.175 1.167 1.176 1.175 1.175
HOCl 0.964 (OH) 0.978 0.976 0.978 0.978 0.977 0.976 0.977 0.978
1.689 (OCl) 1.678 1.710 1.736 1.736 1.678 1.709 1.736 1.736
LiCl (1⌃+) 2.021 2.004 2.025 2.031 2.041 2.003 2.024 2.030 2.039
LiF (1⌃+) 1.564 1.560 1.583 1.584 1.586 1.559 1.582 1.583 1.586
LiH (1⌃+) 1.596 1.603 1.603 1.595 1.630 1.600 1.600 1.593 1.626
MgO (1⌃+) 1.749 1.726 1.751 1.757 1.757 1.754 1.782 1.789 1.789
MgS (1⌃+) 2.143 2.130 2.155 2.168 2.168 2.170 2.200 2.219 2.220
N2 (1⌃+g) 1.098 1.095 1.103 1.103 1.103 1.095 1.103 1.103 1.103
N2O 1.127 (NN) 1.130 1.138 1.139 1.139 1.130 1.138 1.139 1.139
1.185 (NO) 1.175 1.188 1.197 1.197 1.175 1.188 1.197 1.197
NaCl (1⌃+) 2.361 2.335 2.376 2.387 2.401 2.334 2.374 2.385 2.400
NaF (1⌃+) 1.926 1.908 1.947 1.951 1.956 1.907 1.946 1.950 1.955
NaH (1⌃+) 1.887 1.867 1.890 1.884 1.912 1.865 1.887 1.882 1.909
NF (3⌃ ) 1.317 1.302 1.325 1.341 1.341 1.302 1.325 1.341 1.341
NH (3⌃ ) 1.036 1.054 1.050 1.051 1.051 1.054 1.049 1.050 1.050
NH3 1.012 1.007 1.004 1.005 1.006 1.007 1.004 1.004 1.005
O2 (3⌃g) 1.208 1.203 1.217 1.228 1.228 1.203 1.217 1.228 1.228
O3 1.272 1.250 1.270 1.286 1.286 1.251 1.271 1.287 1.287
OCS 1.147 (CO) 1.160 1.168 1.170 1.170 1.160 1.168 1.170 1.170
1.561 (CS) 1.556 1.568 1.574 1.574 1.555 1.568 1.573 1.574
P2 (1⌃+g) 1.893 1.890 1.904 1.911 1.911 1.890 1.903 1.910 1.910
PF (3⌃ ) 1.590 1.600 1.623 1.633 1.633 1.600 1.623 1.633 1.633
PH (3⌃ ) 1.422 1.442 1.440 1.439 1.439 1.441 1.439 1.438 1.438
PH3 1.413 1.431 1.430 1.428 1.429 1.431 1.430 1.428 1.428
PN (1⌃+) 1.491 1.486 1.498 1.500 1.500 1.486 1.498 1.499 1.499
S2(3⌃g) 1.889 1.896 1.911 1.929 1.929 1.895 1.910 1.928 1.928
S2O 1.884 (SS) 1.889 1.908 1.925 1.926 1.889 1.908 1.926 1.927
1.456 (SO) 1.463 1.478 1.485 1.486 1.463 1.478 1.485 1.486
SCS 1.553 1.551 1.562 1.567 1.568 1.550 1.562 1.567 1.568
SiO (1⌃+) 1.510 1.515 1.529 1.531 1.531 1.514 1.528 1.531 1.531
SiS (1⌃+) 1.929 1.936 1.952 1.960 1.960 1.935 1.951 1.959 1.959
SO (3⌃ ) 1.481 1.489 1.504 1.513 1.513 1.489 1.503 1.513 1.513
SO2 1.431 1.440 1.455 1.461 1.461 1.440 1.455 1.461 1.461
B2H6 1.314 (BH ⇥2) 1.312 1.319 1.317 1.321 1.311 1.319 1.317 1.320
1.184 (BH ⇥4) 1.198 1.196 1.190 1.191 1.198 1.195 1.190 1.191
cyclopropane 1.501 (CC) 1.492 1.509 1.517 1.520 1.492 1.508 1.516 1.520
1.083 (CH) 1.091 1.089 1.087 1.089 1.091 1.089 1.087 1.088
C2 (2⌃+g) 1.268 1.258 1.269 1.268 1.268 1.265 1.274 1.275 1.275
NH (2⇧i) 1.047 1.051 1.051 1.052 1.052 1.051 1.051 1.051 1.051
AlH+(2⌃+) 1.602 1.642 1.635 1.638 1.639 1.640 1.634 1.637 1.637
Cl+2 (2⇧3/2g) 1.892 1.893 1.913 1.941 1.941 1.892 1.913 1.941 1.941
CO+(2⌃+) 1.115 1.114 1.122 1.123 1.123 1.114 1.121 1.123 1.123
HCl+ (2⇧i) 1.315 1.339 1.332 1.334 1.334 1.339 1.332 1.334 1.334
He+2 (2⌃+u) 1.081 1.161 1.176 1.183 1.184 1.161 1.175 1.183 1.183
HF+ (2⇧i) 1.001 1.028 1.021 1.026 1.026 1.028 1.021 1.026 1.026
N+2 (2⌃+g) 1.116 1.107 1.115 1.116 1.116 1.108 1.115 1.117 1.117
NH+(2⇧r) 1.070 1.098 1.090 1.090 1.090 1.098 1.091 1.090 1.090
NH+3 1.014 1.037 1.032 1.032 1.034 1.037 1.032 1.032 1.033
O+2 (2⇧g) 1.116 1.110 1.121 1.127 1.127 1.110 1.121 1.127 1.127
PF+ (2⇧r) 1.500 1.517 1.535 1.541 1.541 1.517 1.535 1.541 1.541
H2O+ (2B1) 0.999 1.019 1.014 1.017 1.017 1.019 1.014 1.016 1.017
AlS (2⌃+) 2.029 2.031 2.049 2.064 2.064 2.052 2.067 2.095 2.096
BeCl (2⌃+) 1.797 1.789 1.806 1.811 1.811 1.788 1.806 1.811 1.811
BeF (2⌃+) 1.361 1.358 1.374 1.376 1.376 1.358 1.373 1.375 1.375
BeH (2⌃+) 1.343 1.361 1.356 1.346 1.346 1.359 1.354 1.345 1.345
BO (2⌃+) 1.205 1.204 1.213 1.213 1.213 1.203 1.212 1.213 1.213
BS (2⌃+) 1.609 1.609 1.620 1.623 1.623 1.609 1.620 1.623 1.623
CCl (2⇧1/2,2⇧3/2) 1.645 1.632 1.657 1.672 1.672 1.630 1.654 1.668 1.668
CF (2⇧r) 1.272 1.263 1.283 1.291 1.291 1.262 1.282 1.289 1.289
CH (2⇧r) 1.120 1.139 1.136 1.132 1.132 1.139 1.135 1.131 1.131
CH3 1.076 1.089 1.086 1.084 1.086 1.089 1.086 1.084 1.086
ClO (2⇧i) 1.570 1.553 1.580 1.600 1.600 1.552 1.576 1.596 1.596
CN (2⌃+) 1.172 1.166 1.174 1.175 1.175 1.168 1.175 1.176 1.176
CP (2⌃+) 1.562 1.555 1.566 1.569 1.569 1.562 1.569 1.576 1.576
H2N (2B1) 1.025 1.039 1.037 1.037 1.037 1.038 1.036 1.037 1.037
HO 0.970 0.986 0.983 0.985 0.985 0.985 0.983 0.985 0.985
HOO 0.971 (OH) 0.991 0.987 0.988 0.988 0.991 0.987 0.988 0.988
1.331 (OO) 1.308 1.333 1.353 1.353 1.311 1.334 1.353 1.353
MgCl (2⌃+) 2.199 2.196 2.225 2.239 2.240 2.195 2.223 2.238 2.239
MgF (2⌃+) 1.750 1.756 1.783 1.788 1.789 1.755 1.782 1.788 1.788
MgH (2⌃+) 1.730 1.753 1.756 1.756 1.757 1.750 1.754 1.754 1.755
NO (2⇧r) 1.151 1.146 1.157 1.161 1.161 1.146 1.157 1.161 1.161
NO2 (2A1) 1.195 1.189 1.203 1.210 1.210 1.189 1.203 1.210 1.210
NS (2⇧r) 1.494 1.491 1.505 1.510 1.510 1.491 1.505 1.510 1.510
OP (2⇧r) 1.474 1.481 1.494 1.498 1.498 1.481 1.495 1.499 1.499
SF (2⇧3/2,2⇧1/2) 1.601 1.601 1.626 1.642 1.642 1.601 1.626 1.641 1.641
SH (2⇧i) 1.345 1.359 1.356 1.357 1.357 1.358 1.356 1.356 1.356
SiCl (2⇧r) 2.058 2.058 2.084 2.104 2.104 2.057 2.082 2.102 2.102
SiF (2⇧r) 1.601 1.617 1.640 1.646 1.646 1.617 1.639 1.646 1.646
SiH (2⇧r) 1.520 1.545 1.544 1.540 1.540 1.544 1.544 1.539 1.539
SiN (2⌃+) 1.572 1.564 1.578 1.582 1.582 1.586 1.597 1.607 1.607
HCO 1.119 (CH) 1.138 1.134 1.133 1.134 1.137 1.134 1.132 1.133
1.175 (CO) 1.172 1.181 1.183 1.184 1.172 1.181 1.183 1.184
MSE 0.004 0.014 0.018 0.019 0.005 0.014 0.019 0.020
MAE 0.013 0.015 0.019 0.020 0.013 0.015 0.020 0.021
rms 0.017 0.019 0.024 0.025 0.017 0.020 0.025 0.026
Max( ) -0.091 -0.069 -0.064 -0.063 -0.091 -0.069 -0.064 -0.064
Max(+) 0.081 0.095 0.103 0.103 0.080 0.095 0.102 0.102
TABLE S6. Singlet-triplet energy gap [ET ES] (in kcal/mol), as a function of the acene length, calculated by various functionals in spin-unrestricted KS-DFT and TAO-DFT. 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], and the CCSD(T)/CBS data are taken from Ref. [9].
KS-DFT TAO-DFT
n-acene Experiment DMRG/cc-pVDZ DMRG/STO-3G CCSD(T)/CBS LDA PBE BLYP BLYP-D LDA PBE BLYP BLYP-D
2 61.0 61.0 61.5 65.83 65.19 60.88 60.44 60.30 64.77 60.59 60.14 59.99
3 43.1 44.0 45.9 48.21 43.40 40.26 40.19 40.20 43.22 40.12 40.03 40.04
4 29.3 31.9 34.7 33.49 29.06 26.61 26.74 26.82 29.01 26.60 26.71 26.79
5 19.8 23.4 26.7 25.27 19.20 17.20 17.45 17.58 19.60 17.62 17.83 17.95
6 17.5 21.0 17.71 12.16 10.48 10.81 10.95 13.55 11.84 12.11 12.22
8 14.2 9.20 3.07 1.81 2.87 2.37 7.85 6.41 6.65 6.72
10 11.6 4.64 0.53 1.43 1.31 1.31 5.91 4.66 4.82 4.86
12 10.7 1.59 2.17 1.95 1.89 4.82 3.77 3.88 3.91
14 3.47 1.57 3.27 3.18 3.98 3.09 3.19 3.21
16 4.43 2.10 3.86 3.83 3.37 2.60 2.68 2.69
18 2.94 2.29 2.32 2.33
20 2.62 2.00 2.06 2.07
22 2.37 1.80 1.85 1.86
24 2.15 1.64 1.69 1.70
26 1.98 1.51 1.55 1.56
28 1.83 1.39 1.43 1.44
30 1.70 1.29 1.33 1.33
32 1.59 1.21 1.24 1.25
34 1.49 1.13 1.16 1.16
36 1.40 1.07 1.10 1.10
38 1.33 1.01 1.04 1.04
40 1.26 0.95 0.98 0.99
42 1.20 0.91 0.93 0.93
44 1.14 0.86 0.88 0.89
46 1.09 0.83 0.85 0.85
48 1.04 0.79 0.81 0.82
50 1.00 0.76 0.78 0.78
52 0.96 0.73 0.75 0.75
54 0.92 0.70 0.72 0.72
56 0.89 0.66 0.69 0.70
58 0.86 0.65 0.67 0.66
60 0.83 0.63 0.64 0.65
62 0.80 0.61 0.63 0.63
64 0.78 0.59 0.61 0.61
66 0.75 0.57 0.59 0.59
68 0.73 0.55 0.56 0.56
70 0.71 0.54 0.55 0.55
72 0.69 0.52 0.54 0.54
74 0.67 0.51 0.52 0.53
76 0.65 0.50 0.51 0.51
78 0.64 0.48 0.50 0.50
80 0.62 0.46 0.47 0.48
82 0.60 0.46 0.47 0.47
84 0.59 0.45 0.46 0.46
86 0.58 0.44 0.45 0.45
88 0.56 0.42 0.44 0.44
90 0.55 0.41 0.43 0.43
92 0.54 0.41 0.41 0.41
94 0.53 0.40 0.41 0.41
96 0.51 0.39 0.40 0.40
98 0.50 0.38 0.39 0.39
100 0.49 0.38 0.38 0.39
TABLE S7. Vertical ionization potential (in eV) for the lowest singlet state of n-acene as a function of the acene length, calculated by various functionals in spin-unrestricted TAO-DFT. The experimental data are taken from the compilation in Ref.
[10], and the CCSD(T)/CBS data are taken from Ref. [11].
TAO-DFT
n-acene Experiment CCSD(T)/CBS LDA PBE BLYP BLYP-D
2 8.14 8.24 7.81 7.58 7.33 7.32
3 7.44 7.47 7.00 6.79 6.54 6.54
4 6.97 6.95 6.46 6.25 6.00 6.00
5 6.59 6.57 6.07 5.87 5.62 5.62
6 6.43 5.79 5.59 5.34 5.34
8 5.44 5.25 4.99 4.99
10 5.23 5.04 4.78 4.78
12 5.08 4.89 4.63 4.62
14 4.96 4.77 4.51 4.50
16 4.86 4.67 4.41 4.41
18 4.78 4.60 4.33 4.33
20 4.71 4.53 4.27 4.26
22 4.66 4.48 4.21 4.20
24 4.61 4.43 4.16 4.16
26 4.57 4.39 4.12 4.11
28 4.53 4.35 4.08 4.08
30 4.50 4.32 4.05 4.04
32 4.47 4.29 4.02 4.01
34 4.44 4.26 3.99 3.99
36 4.42 4.24 3.97 3.96
38 4.39 4.21 3.95 3.94
40 4.37 4.19 3.93 3.92
42 4.35 4.18 3.91 3.90
44 4.34 4.16 3.90 3.88
46 4.32 4.14 3.88 3.87
48 4.31 4.13 3.86 3.85
50 4.29 4.11 3.85 3.84
52 4.28 4.10 3.84 3.83
54 4.27 4.09 3.82 3.82
56 4.26 4.08 3.81 3.80
58 4.24 4.07 3.80 3.79
60 4.23 4.06 3.79 3.78
62 4.23 4.05 3.78 3.77
64 4.22 4.04 3.77 3.76
66 4.21 4.03 3.76 3.76
68 4.20 4.02 3.76 3.75
70 4.19 4.01 3.75 3.74
72 4.18 4.01 3.74 3.73
74 4.18 4.00 3.73 3.73
76 4.17 3.99 3.73 3.72
78 4.16 3.99 3.72 3.71
80 4.16 3.98 3.71 3.71
82 4.15 3.98 3.71 3.70
84 4.15 3.97 3.70 3.70
86 4.14 3.96 3.70 3.69
88 4.14 3.96 3.69 3.68
90 4.13 3.95 3.69 3.68
92 4.13 3.95 3.68 3.67
94 4.12 3.95 3.68 3.67
96 4.12 3.94 3.67 3.67
98 4.11 3.94 3.67 3.66
100 4.11 3.93 3.67 3.66
TABLE S8. Vertical electron affinity (in eV) for the lowest singlet state of n-acene as a function of the acene length, calculated by various functionals in spin-unrestricted TAO-DFT. The experimental data are taken from the compilation in Ref. [10], and the CCSD(T)/CBS data are taken from Ref. [12].
TAO-DFT
n-acene Experiment CCSD(T)/CBS LDA PBE BLYP BLYP-D
2 -0.20 -0.48 -0.62 -0.72 -0.94 -0.94
3 0.53 0.28 0.28 0.16 -0.07 -0.07
4 1.07 0.82 0.90 0.77 0.52 0.52
5 1.39 1.21 1.34 1.20 0.95 0.95
6 1.47 1.66 1.52 1.26 1.26
8 2.06 1.92 1.66 1.65
10 2.30 2.15 1.90 1.89
12 2.48 2.33 2.07 2.06
14 2.61 2.46 2.20 2.20
16 2.72 2.57 2.31 2.30
18 2.81 2.66 2.40 2.39
20 2.89 2.73 2.47 2.46
22 2.95 2.79 2.53 2.52
24 3.00 2.84 2.58 2.58
26 3.05 2.89 2.63 2.62
28 3.09 2.93 2.67 2.66
30 3.13 2.97 2.71 2.70
32 3.16 3.00 2.74 2.73
34 3.19 3.03 2.77 2.76
36 3.22 3.06 2.79 2.79
38 3.25 3.08 2.82 2.81
40 3.27 3.10 2.85 2.83
42 3.29 3.12 2.87 2.85
44 3.31 3.14 2.88 2.87
46 3.33 3.16 2.90 2.89
48 3.34 3.18 2.91 2.90
50 3.36 3.19 2.93 2.92
52 3.37 3.21 2.94 2.93
54 3.39 3.22 2.95 2.95
56 3.40 3.23 2.97 2.96
58 3.41 3.24 2.98 2.97
60 3.42 3.25 2.99 2.98
62 3.43 3.26 3.00 2.99
64 3.44 3.27 3.01 3.00
66 3.45 3.28 3.02 3.01
68 3.46 3.29 3.03 3.02
70 3.47 3.30 3.04 3.03
72 3.48 3.31 3.04 3.04
74 3.48 3.32 3.05 3.04
76 3.49 3.32 3.06 3.05
78 3.50 3.33 3.07 3.06
80 3.51 3.34 3.07 3.06
82 3.51 3.34 3.08 3.07
84 3.52 3.35 3.08 3.08
86 3.52 3.36 3.09 3.08
88 3.53 3.36 3.10 3.09
90 3.54 3.37 3.10 3.09
92 3.54 3.37 3.11 3.10
94 3.55 3.38 3.11 3.10
96 3.55 3.38 3.12 3.11
98 3.56 3.39 3.12 3.11
100 3.56 3.39 3.13 3.12
TABLE S9. Fundamental gap (in eV) for the lowest singlet state of n-acene as a function of the acene length, calculated by various functionals in spin-unrestricted TAO-DFT. The experimental data are taken from the compilation in Ref. [10], and the CCSD(T)/CBS data are taken from Refs. [11, 12].
TAO-DFT
n-acene Experiment CCSD(T)/CBS LDA PBE BLYP BLYP-D
2 8.34 8.72 8.43 8.30 8.27 8.26
3 6.91 7.19 6.73 6.63 6.61 6.61
4 5.90 6.13 5.56 5.49 5.48 5.48
5 5.20 5.37 4.73 4.67 4.67 4.67
6 4.96 4.13 4.08 4.08 4.08
8 3.38 3.33 3.33 3.33
10 2.93 2.89 2.89 2.89
12 2.60 2.56 2.56 2.56
14 2.34 2.31 2.30 2.31
16 2.13 2.11 2.10 2.10
18 1.97 1.94 1.94 1.94
20 1.83 1.80 1.80 1.80
22 1.71 1.68 1.68 1.68
24 1.60 1.58 1.58 1.58
26 1.52 1.49 1.49 1.49
28 1.44 1.42 1.41 1.42
30 1.37 1.35 1.34 1.35
32 1.30 1.29 1.28 1.28
34 1.25 1.23 1.23 1.23
36 1.20 1.18 1.18 1.18
38 1.15 1.13 1.13 1.13
40 1.11 1.09 1.09 1.09
42 1.07 1.05 1.05 1.05
44 1.03 1.02 1.01 1.01
46 1.00 0.98 0.98 0.98
48 0.96 0.95 0.95 0.95
50 0.94 0.92 0.92 0.92
52 0.91 0.90 0.89 0.89
54 0.88 0.87 0.87 0.87
56 0.86 0.85 0.85 0.85
58 0.84 0.82 0.82 0.82
60 0.81 0.80 0.80 0.80
62 0.79 0.78 0.78 0.78
64 0.77 0.76 0.76 0.76
66 0.76 0.75 0.75 0.75
68 0.74 0.73 0.73 0.73
70 0.72 0.71 0.71 0.71
72 0.71 0.70 0.70 0.70
74 0.69 0.68 0.68 0.68
76 0.68 0.67 0.67 0.67
78 0.67 0.66 0.66 0.66
80 0.65 0.64 0.64 0.64
82 0.64 0.63 0.63 0.63
84 0.63 0.62 0.62 0.62
86 0.62 0.61 0.61 0.61
88 0.61 0.60 0.60 0.60
90 0.59 0.59 0.59 0.59
92 0.58 0.58 0.58 0.58
94 0.58 0.57 0.57 0.57
96 0.57 0.56 0.56 0.56
98 0.56 0.55 0.55 0.55
100 0.55 0.54 0.54 0.54
TABLE S10. Symmetrized von Neumann entropy for the lowest singlet state of n-acene as a function of the acene length, calculated by various functionals in spin-restricted TAO-DFT.
TAO-DFT
n-acene LDA PBE BLYP BLYP-D
2 0.00 0.00 0.00 0.00
3 0.03 0.03 0.03 0.03
4 0.15 0.15 0.15 0.15
5 0.40 0.39 0.39 0.39
6 0.75 0.73 0.73 0.72
8 1.34 1.32 1.32 1.31
10 1.67 1.67 1.67 1.66
12 1.98 1.99 2.00 1.99
14 2.36 2.37 2.38 2.37
16 2.75 2.76 2.78 2.77
18 3.13 3.15 3.17 3.15
20 3.50 3.53 3.55 3.54
22 3.88 3.91 3.94 3.92
24 4.26 4.30 4.32 4.31
26 4.63 4.68 4.71 4.69
28 5.01 5.06 5.09 5.08
30 5.39 5.44 5.48 5.46
32 5.77 5.83 5.87 5.85
34 6.14 6.21 6.25 6.23
36 6.52 6.59 6.64 6.62
38 6.90 6.98 7.03 7.00
40 7.27 7.36 7.41 7.39
42 7.65 7.74 7.80 7.77
44 8.03 8.13 8.18 8.16
46 8.41 8.51 8.57 8.54
48 8.78 8.89 8.96 8.93
50 9.16 9.27 9.34 9.31
52 9.54 9.66 9.73 9.70
54 9.92 10.04 10.11 10.08
56 10.29 10.42 10.50 10.47
58 10.67 10.80 10.89 10.85
60 11.05 11.19 11.27 11.23
62 11.42 11.57 11.66 11.62
64 11.80 11.95 12.04 12.00
66 12.18 12.34 12.43 12.39
68 12.56 12.72 12.81 12.77
70 12.93 13.10 13.20 13.16
72 13.31 13.48 13.59 13.55
74 13.69 13.87 13.97 13.93
76 14.06 14.25 14.36 14.31
78 14.44 14.63 14.74 14.70
80 14.82 15.01 15.13 15.08
82 15.20 15.40 15.51 15.47
84 15.57 15.78 15.90 15.85
86 15.95 16.16 16.29 16.24
88 16.33 16.55 16.67 16.62
90 16.70 16.93 17.06 17.01
92 17.08 17.31 17.44 17.39
94 17.46 17.69 17.83 17.78
96 17.84 18.08 18.22 18.16
98 18.21 18.46 18.60 18.54
100 18.59 18.84 18.99 18.93
