Supplementary material to: Role of exact exchange in thermally-assisted-occupation density functional theory: A proposal of new hybrid schemes

density functional theory: A proposal of new hybrid schemes

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. Reaction energies of the 30 chemical reactions in the NHTBH38/04 and HTBH38/04 sets (S3).

TABLE S2. Equilibrium geometries of EXTS (S3 to S7).

TABLE S3. Non-hydrogen transfer barrier heights of the NHTBH38/04 set (S7 to S8).

TABLE S4. Hydrogen transfer barrier heights of the HTBH38/04 set (S8 to S9).

TABLE S5. Interaction energies of the S22 set (S9).

TABLE S6. Singlet-triplet energy gap of n-acene (S9 to S10).

TABLE S7. Vertical ionization potential for the lowest singlet state of n-acene (S10 to S11).

TABLE S8. Vertical electron affinity for the lowest singlet state of n-acene (S11).

TABLE S9. Fundamental gap for the lowest singlet state of n-acene (S12).

TABLE S10. Symmetrized von Neumann entropy for the lowest singlet state of n-acene (S12 to S13).

⇤Electronic mail: [email protected]

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TABLES

TABLE S1. Comparison of errors of the reaction energies (in kcal/mol) of the 30 chemical reactions in the NHTBH38/04 and HTBH38/04 sets [1]. For all the TAO-DFT calculations, the optimal ✓ values given in TABLE I are adopted.

KS-DFT TAO-DFT

Reactions Eref B3LYP B3LYP-D3 PBE0 BHHLYP B3LYP B3LYP-D3 PBE0 BHHLYP

H + N2O ! OH + N2 -65.08 3.64 3.78 10.20 -10.47 3.81 3.95 10.52 -9.57

H + FCH3 ! HF + CH³ -26.64 -0.21 0.24 2.60 -0.96 0.26 0.71 3.09 -0.12

H + F2 ! HF + F -103.91 0.22 0.22 0.29 -14.00 0.01 0.01 0.10 -14.21

CH3 + FCl ! CH³F + Cl -52.74 0.20 -0.25 0.08 -5.06 -2.34 -2.79 -1.84 -6.80 F + CH3Cl ! FCH3 + Cl -32.65 -2.16 -2.06 -2.24 -5.81 -1.26 -1.15 -1.27 -4.47 F · · ·CH3Cl ! FCH3· · ·Cl -26.73 0.42 0.42 0.04 -3.83 1.07 1.07 0.82 -2.75

OH + CH3F ! HOCH³+ F -20.11 0.15 -0.01 -0.80 -0.91 0.24 0.08 -0.73 -0.82

OH · · ·CH³F ! HOCH³· · ·F -36.24 -1.53 -1.17 -4.00 -1.93 -1.45 -1.09 -3.93 -1.66

H + N2 ! HN² 3.97 -7.38 -7.41 -7.11 -6.54 -8.44 -8.46 -8.40 -9.03

H + CO ! HCO -19.51 -5.68 -5.80 -5.65 -3.84 -6.89 -7.01 -7.13 -6.68

H + C2H4 ! CH3CH2 -40.03 -1.88 -2.14 -3.42 -4.90 -2.96 -3.22 -4.72 -7.40 CH3 + C2H4 ! CH³CH2CH2 -26.12 2.68 1.02 -4.34 0.09 0.87 -0.79 -6.46 -4.00

HCN ! HNC 15.05 -1.00 -0.98 -1.19 -2.56 -0.81 -0.79 -0.97 -2.23

H + HCl ! H² + Cl -3.0 -1.50 -1.49 4.28 -2.79 -3.51 -3.51 2.93 -3.66

OH + H2 ! H + H²O -16.1 3.25 3.25 -1.23 6.54 3.09 3.09 -1.56 5.58

CH3 + H2 ! H + CH4 -3.2 2.41 2.38 -1.81 2.52 1.80 1.77 -2.51 1.24

OH + CH4 ! CH³ + H2O -12.9 0.84 0.86 0.58 4.01 1.30 1.32 0.95 4.34

OH + NH3 ! H²O + NH2 -9.5 -0.44 -0.43 -0.18 1.57 -0.20 -0.19 -0.02 1.68

HCl + CH3 ! Cl+ CH⁴ -6.2 0.91 0.89 2.47 -0.27 -1.72 -1.74 0.42 -2.42

OH + C2H6! H²O + C2H5 -16.5 -0.18 0.16 -0.38 3.69 0.26 0.59 -0.04 3.90

F + H2 ! HF + H -31.6 1.81 1.81 -1.70 7.63 1.53 1.53 -2.13 6.55

O + CH4 ! OH + CH3 5.6 -3.26 -3.23 -2.38 0.94 -3.01 -2.99 -2.17 1.22

H + PH3 ! PH² + H2 -20.1 -3.71 -3.61 0.33 -3.18 -3.34 -3.23 0.75 -2.45

H + HO ! H² + O -2.4 0.85 0.85 4.19 -3.47 1.22 1.22 4.67 -2.46

H + H2S ! H² + HS -13.8 -2.32 -2.29 2.79 -2.74 -4.24 -4.21 1.49 -3.70

O + HCl ! OH + Cl -0.6 -2.35 -2.34 0.08 0.67 -4.73 -4.73 -1.74 -1.20

NH2 + CH3 ! CH4 + NH -14.4 3.44 3.42 3.03 1.65 3.30 3.28 2.89 1.52

NH2 + C2H5 ! C²H6 + NH -10.8 4.46 4.13 3.99 1.98 4.34 4.01 3.87 1.96

C2H6+ NH2 ! NH³ + C2H5 -7.0 0.26 0.58 -0.20 2.12 0.46 0.78 -0.02 2.22

NH2 + CH4 ! CH³ + NH3 -3.3 1.18 1.19 0.66 2.34 1.40 1.40 0.87 2.56

MSE -0.23 -0.27 -0.03 -1.25 -0.66 -0.70 -0.41 -1.76

MAE 2.01 1.95 2.41 3.63 2.33 2.36 2.63 3.95

rms 2.66 2.61 3.35 4.72 3.05 3.07 3.69 5.00

Max( ) -7.38 -7.41 -7.11 -14.00 -8.44 -8.46 -8.40 -14.21

Max(+) 4.46 4.13 10.20 7.63 4.34 4.01 10.52 6.55

TABLE S2. The 166 experimentally determined equilibrium bond lengths (in Å) of 136 molecules included in EXTS [2].

Optimized bond lengths are calculated using various methods. For all the TAO-DFT calculations, the optimal ✓ values given in TABLE I are adopted.

KS-DFT TAO-DFT

Molecule Experiment B3LYP B3LYP-D3 PBE0 BHHLYP B3LYP B3LYP-D3 PBE0 BHHLYP

H2N 1.028 1.028 1.028 1.027 1.017 1.027 1.027 1.025 1.015

NO (³⌃ ) 1.258 1.260 1.260 1.250 1.243 1.258 1.258 1.247 1.239

PO (³⌃ ) 1.540 1.542 1.542 1.534 1.520 1.539 1.539 1.531 1.515

BeH⁺ (¹⌃⁺) 1.312 1.314 1.315 1.319 1.305 1.310 1.311 1.315 1.299

CH⁺ (¹⌃⁺) 1.131 1.136 1.136 1.139 1.122 1.135 1.135 1.138 1.120

CN⁺ (¹⌃⁺) 1.173 1.172 1.172 1.170 1.160 1.195 1.195 1.188 1.170

H⁺₃ 0.877 0.880 0.880 0.881 0.873 0.878 0.878 0.879 0.868

H3O⁺ 0.976 0.980 0.980 0.977 0.969 0.979 0.979 0.976 0.967

HCO⁺ 1.097 (CH) 1.095 1.096 1.097 1.088 1.094 1.095 1.095 1.085

1.105 (CO) 1.101 1.101 1.099 1.088 1.100 1.100 1.099 1.086

MgH⁺ (¹⌃⁺) 1.652 1.657 1.657 1.659 1.644 1.653 1.653 1.655 1.641

NO⁺ (¹⌃⁺) 1.063 1.056 1.056 1.054 1.041 1.056 1.056 1.053 1.040

NS⁺ (¹⌃⁺) 1.440 1.427 1.427 1.422 1.405 1.426 1.426 1.421 1.404

OH⁺ (³⌃ ) 1.029 1.037 1.037 1.031 1.021 1.036 1.036 1.030 1.019

SiH⁺ (¹⌃⁺) 1.504 1.515 1.515 1.518 1.500 1.513 1.513 1.517 1.498

NH⁺4 1.021 1.024 1.024 1.022 1.015 1.023 1.023 1.021 1.013

AlCl (¹⌃⁺) 2.130 2.159 2.159 2.144 2.142 2.156 2.157 2.140 2.137

AlF (¹⌃⁺) 1.654 1.679 1.679 1.672 1.659 1.678 1.678 1.671 1.657

AlH (¹⌃⁺) 1.648 1.661 1.661 1.664 1.646 1.656 1.656 1.658 1.635

AlN (³⇧i) 1.786 1.792 1.792 1.788 1.794 1.801 1.801 1.793 1.794

BCl (¹⌃⁺) 1.797 1.719 1.719 1.715 1.707 1.717 1.717 1.713 1.702

BeO (¹⌃⁺) 1.331 1.320 1.320 1.317 1.300 1.329 1.329 1.326 1.310

BeS (¹⌃⁺) 1.742 1.739 1.739 1.736 1.724 1.763 1.764 1.759 1.750

BF (¹⌃⁺) 1.263 1.260 1.260 1.258 1.248 1.260 1.260 1.258 1.247

BF3 1.307 1.312 1.312 1.307 1.298 1.311 1.311 1.306 1.296

BH (¹⌃⁺) 1.232 1.231 1.231 1.240 1.220 1.228 1.228 1.235 1.213

BN (³⇧) 1.281 1.317 1.317 1.315 1.300 1.317 1.317 1.314 1.300

C6H6 1.390 (CC) 1.391 1.391 1.388 1.382 1.390 1.390 1.386 1.380

1.086 (CH) 1.082 1.082 1.083 1.074 1.081 1.080 1.082 1.071

CCl2O 1.177 (CO) 1.173 1.173 1.172 1.161 1.173 1.173 1.172 1.161

1.737 (CCl) 1.753 1.755 1.734 1.732 1.751 1.753 1.731 1.727

CF4 1.315 1.324 1.325 1.316 1.305 1.323 1.324 1.315 1.303

CH2Cl2 1.766 (CCl) 1.780 1.782 1.761 1.762 1.777 1.780 1.758 1.757

1.080 (CH) 1.082 1.083 1.084 1.074 1.081 1.082 1.083 1.072

CH2F2 1.351 (CF) 1.360 1.361 1.350 1.341 1.359 1.360 1.349 1.339

1.084 (CH) 1.090 1.091 1.092 1.081 1.088 1.089 1.091 1.078

CH2O2 (trans) 1.201 (C=O) 1.196 1.196 1.194 1.184 1.196 1.196 1.193 1.183

1.091 (CH) 1.096 1.097 1.098 1.087 1.095 1.095 1.096 1.084

1.340 (CO) 1.343 1.344 1.334 1.325 1.342 1.343 1.333 1.323

0.969 (OH) 0.970 0.970 0.967 0.957 0.969 0.969 0.966 0.955

CH3Cl 1.086 (CH) 1.084 1.085 1.086 1.077 1.083 1.084 1.085 1.074

1.778 (CCl) 1.794 1.796 1.771 1.776 1.791 1.794 1.768 1.771

CH3F 1.086 (CH) 1.089 1.090 1.091 1.082 1.088 1.088 1.089 1.078

1.383 (CF) 1.389 1.390 1.375 1.369 1.388 1.389 1.374 1.367

CH4 1.087 1.088 1.088 1.089 1.081 1.086 1.087 1.087 1.078

CHF3 1.328 (CF) 1.338 1.339 1.329 1.320 1.337 1.338 1.328 1.318

1.091 (CH) 1.089 1.090 1.091 1.080 1.088 1.089 1.090 1.077

Cl2 (¹⌃⁺g) 1.988 2.010 2.010 1.979 1.983 2.012 2.012 1.981 1.986

ClCN 1.629 (CCl) 1.630 1.631 1.620 1.622 1.628 1.629 1.618 1.618

1.160 (CN) 1.153 1.153 1.152 1.138 1.152 1.152 1.151 1.136

ClF 1.628 1.642 1.642 1.621 1.609 1.642 1.642 1.621 1.609

CO (¹⌃⁺) 1.128 1.124 1.124 1.123 1.112 1.124 1.124 1.122 1.111

CO2 1.160 1.159 1.159 1.156 1.145 1.158 1.158 1.155 1.144

CS (¹⌃⁺) 1.535 1.532 1.532 1.529 1.516 1.531 1.531 1.528 1.514

CS2 1.553 1.554 1.554 1.549 1.541 1.553 1.553 1.547 1.539

F2 (¹⌃⁺_g) 1.412 1.394 1.394 1.373 1.357 1.394 1.394 1.372 1.356

F2O 1.405 1.398 1.399 1.377 1.361 1.397 1.398 1.376 1.359

F2S 1.587 1.609 1.609 1.594 1.580 1.608 1.608 1.593 1.578

F2Si 1.590 1.614 1.615 1.607 1.593 1.613 1.614 1.606 1.591

F3HSi 1.562 (SiF) 1.582 1.581 1.576 1.563 1.580 1.580 1.574 1.561

1.447 (SiH) 1.454 1.452 1.458 1.444 1.451 1.450 1.455 1.439

H2 (¹⌃⁺_g) 0.741 0.743 0.743 0.745 0.737 0.741 0.741 0.743 0.732

H2CCCH2 1.076 (CH) 1.083 1.083 1.084 1.075 1.082 1.082 1.082 1.072

1.308 (CC) 1.300 1.300 1.299 1.292 1.299 1.299 1.298 1.289

H2CS 1.611 (CS) 1.606 1.607 1.600 1.592 1.607 1.607 1.600 1.592

1.086 (CH) 1.087 1.088 1.089 1.079 1.086 1.086 1.087 1.076

H2O 0.958 0.961 0.961 0.958 0.950 0.960 0.960 0.957 0.948

H2S 1.336 1.342 1.342 1.341 1.331 1.340 1.340 1.339 1.327

H2Si 1.514 1.522 1.522 1.525 1.509 1.518 1.518 1.519 1.501

HCCCN 1.062 (CH) 1.062 1.063 1.065 1.056 1.061 1.061 1.063 1.053

1.206 (C⌘C) 1.201 1.201 1.200 1.187 1.200 1.200 1.200 1.186

1.376 (CC) 1.368 1.368 1.368 1.371 1.367 1.367 1.366 1.367

1.161 (C⌘N) 1.155 1.155 1.154 1.139 1.154 1.154 1.153 1.139

HCCH 1.061 (CH) 1.062 1.062 1.064 1.055 1.060 1.061 1.062 1.052

1.203 (CC) 1.196 1.196 1.197 1.186 1.195 1.195 1.195 1.183

HCl (¹⌃⁺) 1.275 1.281 1.281 1.278 1.270 1.279 1.279 1.276 1.267

HCN 1.065 (CH) 1.066 1.066 1.068 1.059 1.065 1.065 1.066 1.056

1.153 (CN) 1.146 1.146 1.145 1.133 1.145 1.145 1.144 1.132

HCP 1.066 (CH) 1.070 1.071 1.073 1.063 1.069 1.069 1.071 1.060

1.540 (CP) 1.533 1.533 1.530 1.517 1.532 1.532 1.529 1.516

HF (¹⌃⁺) 0.917 0.922 0.922 0.918 0.910 0.921 0.921 0.917 0.908

HNC 0.994 (NH) 0.997 0.997 0.997 0.988 0.996 0.996 0.995 0.986

1.169 (NC) 1.164 1.164 1.164 1.152 1.164 1.164 1.163 1.151

HOCl 0.964 (OH) 0.967 0.967 0.964 0.954 0.966 0.966 0.963 0.952

1.689 (OCl) 1.699 1.700 1.675 1.665 1.699 1.699 1.674 1.663

LiCl (¹⌃⁺) 2.021 2.022 2.022 2.018 2.012 2.018 2.019 2.013 2.010

LiF (¹⌃⁺) 1.564 1.572 1.572 1.570 1.558 1.569 1.570 1.568 1.554

LiH (¹⌃⁺) 1.596 1.589 1.590 1.598 1.584 1.584 1.584 1.591 1.577

MgO (¹⌃⁺) 1.749 1.739 1.739 1.732 1.722 1.763 1.763 1.755 1.736

MgS (¹⌃⁺) 2.143 2.149 2.149 2.136 2.131 2.199 2.199 2.182 2.177

N2 (¹⌃⁺_g) 1.098 1.091 1.091 1.090 1.078 1.091 1.091 1.089 1.077

N2O 1.127 (NN) 1.121 1.121 1.118 1.101 1.121 1.121 1.118 1.100

1.185 (NO) 1.182 1.183 1.174 1.170 1.182 1.182 1.173 1.169

NaCl (¹⌃⁺) 2.361 2.372 2.376 2.365 2.359 2.372 2.375 2.363 2.363

NaF (¹⌃⁺) 1.926 1.933 1.934 1.929 1.914 1.931 1.932 1.926 1.911

NaH (¹⌃⁺) 1.887 1.876 1.877 1.885 1.871 1.876 1.876 1.884 1.874

NF (³⌃ ) 1.317 1.316 1.316 1.301 1.294 1.315 1.315 1.300 1.292

NH (³⌃ ) 1.036 1.040 1.040 1.038 1.028 1.039 1.039 1.036 1.025

NH3 1.012 0.998 0.998 0.997 0.990 0.997 0.997 0.995 0.987

O2 (³⌃g) 1.208 1.203 1.203 1.192 1.179 1.203 1.203 1.191 1.177

O3 1.272 1.251 1.252 1.236 1.219 1.253 1.253 1.237 1.220

OCS 1.147 (CO) 1.155 1.155 1.152 1.138 1.155 1.155 1.151 1.138

1.561 (CS) 1.563 1.563 1.558 1.555 1.561 1.562 1.556 1.552

P2 (¹⌃⁺_g) 1.893 1.888 1.888 1.880 1.864 1.888 1.888 1.880 1.869

PF (³⌃ ) 1.590 1.609 1.609 1.599 1.586 1.608 1.609 1.598 1.585

PH (³⌃ ) 1.422 1.427 1.427 1.427 1.414 1.425 1.425 1.424 1.410

PH3 1.413 1.418 1.418 1.419 1.406 1.416 1.416 1.416 1.401

PN (¹⌃⁺) 1.491 1.481 1.481 1.478 1.461 1.480 1.480 1.477 1.460

S2(³⌃g) 1.889 1.903 1.903 1.886 1.877 1.902 1.902 1.885 1.878

S2O 1.884 (SS) 1.894 1.895 1.878 1.865 1.898 1.898 1.881 1.871

1.456 (SO) 1.460 1.460 1.451 1.434 1.461 1.461 1.452 1.436

SCS 1.553 1.554 1.554 1.549 1.541 1.553 1.553 1.547 1.539

SiO (¹⌃⁺) 1.510 1.512 1.512 1.507 1.492 1.511 1.511 1.506 1.491

SiS (¹⌃⁺) 1.929 1.938 1.938 1.930 1.918 1.938 1.938 1.930 1.921

SO (³⌃ ) 1.481 1.488 1.488 1.478 1.463 1.487 1.487 1.477 1.461

SO2 1.431 1.437 1.437 1.429 1.412 1.436 1.436 1.429 1.411

B2H6 1.314 (BH ⇥2) 1.311 1.313 1.314 1.305 1.309 1.310 1.311 1.300

1.184 (BH ⇥4) 1.185 1.186 1.190 1.179 1.183 1.184 1.187 1.174

cyclopropane 1.501 (CC) 1.504 1.505 1.498 1.493 1.503 1.504 1.496 1.490

1.083 (CH) 1.081 1.081 1.082 1.073 1.079 1.080 1.080 1.070

C₂ (²⌃⁺_g) 1.268 1.258 1.258 1.259 1.248 1.262 1.262 1.261 1.249

NH (²⇧i) 1.047 1.041 1.041 1.039 1.029 1.039 1.039 1.038 1.026

AlH⁺(²⌃⁺) 1.602 1.621 1.621 1.617 1.605 1.614 1.614 1.610 1.593

Cl⁺2 (²⇧_3/2g) 1.892 1.908 1.908 1.883 1.877 1.908 1.908 1.882 1.877

CO⁺(²⌃⁺) 1.115 1.108 1.108 1.106 1.093 1.108 1.108 1.105 1.092

HCl⁺ (²⇧i) 1.315 1.323 1.323 1.320 1.310 1.322 1.322 1.318 1.307

He⁺₂ (²⌃⁺_u) 1.081 1.145 1.145 1.132 1.106 1.143 1.143 1.130 1.103

HF⁺ (²⇧i) 1.001 1.013 1.013 1.006 0.997 1.012 1.012 1.004 0.995

N⁺2 (²⌃⁺_g) 1.116 1.105 1.105 1.102 1.092 1.105 1.105 1.103 1.092

NH⁺(²⇧r) 1.070 1.078 1.078 1.076 1.063 1.077 1.077 1.074 1.060

NH⁺3 1.014 1.025 1.025 1.023 1.016 1.024 1.024 1.022 1.013

O⁺₂ (²⇧g) 1.116 1.105 1.105 1.098 1.083 1.105 1.105 1.097 1.082

PF⁺ (²⇧r) 1.500 1.519 1.519 1.511 1.497 1.518 1.518 1.510 1.495

H2O⁺ (²B1) 0.999 1.005 1.005 1.001 0.992 1.004 1.004 1.000 0.990

AlS (²⌃⁺) 2.029 2.045 2.046 2.031 2.029 2.071 2.071 2.049 2.049

BeCl (²⌃⁺) 1.797 1.802 1.802 1.798 1.795 1.800 1.800 1.796 1.791

BeF (²⌃⁺) 1.361 1.364 1.364 1.362 1.352 1.363 1.363 1.361 1.351

BeH (²⌃⁺) 1.343 1.341 1.341 1.349 1.334 1.336 1.336 1.343 1.325

BO (²⌃⁺) 1.205 1.200 1.200 1.199 1.187 1.200 1.200 1.198 1.186

BS (²⌃⁺) 1.609 1.610 1.610 1.607 1.598 1.610 1.611 1.607 1.599

CCl (²⇧1/2,²⇧3/2) 1.645 1.654 1.654 1.642 1.641 1.649 1.649 1.638 1.635

CF (²⇧r) 1.272 1.272 1.272 1.265 1.256 1.271 1.271 1.264 1.254

CH (²⇧r) 1.120 1.122 1.122 1.124 1.110 1.120 1.120 1.122 1.106

CH3 1.076 1.078 1.078 1.079 1.071 1.076 1.077 1.077 1.067

ClO (²⇧i) 1.570 1.576 1.576 1.559 1.560 1.571 1.571 1.554 1.554

CN (²⌃⁺) 1.172 1.162 1.162 1.160 1.146 1.163 1.163 1.160 1.146

CP (²⌃⁺) 1.562 1.553 1.553 1.549 1.537 1.558 1.558 1.552 1.541

H2N (²B1) 1.025 1.027 1.027 1.026 1.016 1.026 1.026 1.024 1.013

HO 0.970 0.974 0.974 0.971 0.961 0.973 0.973 0.969 0.959

HOO 0.971 (OH) 0.975 0.976 0.972 0.960 0.974 0.975 0.971 0.959

1.331 (OO) 1.324 1.324 1.307 1.300 1.323 1.323 1.306 1.298

MgCl (²⌃⁺) 2.199 2.221 2.221 2.208 2.205 2.218 2.219 2.205 2.203

MgF (²⌃⁺) 1.750 1.769 1.769 1.763 1.750 1.768 1.768 1.761 1.747

MgH (²⌃⁺) 1.730 1.742 1.743 1.742 1.729 1.734 1.735 1.733 1.716

NO (²⇧r) 1.151 1.145 1.145 1.139 1.128 1.144 1.144 1.139 1.126

NO2 (²A1) 1.195 1.190 1.190 1.182 1.170 1.190 1.190 1.182 1.169

NS (²⇧r) 1.494 1.490 1.490 1.484 1.471 1.489 1.489 1.483 1.469

OP (²⇧r) 1.474 1.477 1.477 1.471 1.455 1.476 1.476 1.470 1.454 SF (²⇧_3/2,²⇧_1/2) 1.601 1.614 1.614 1.599 1.587 1.612 1.612 1.597 1.585

SH (²⇧i) 1.345 1.346 1.346 1.345 1.335 1.344 1.344 1.343 1.331

SiCl (²⇧r) 2.058 2.083 2.084 2.066 2.067 2.078 2.078 2.060 2.060

SiF (²⇧r) 1.601 1.625 1.625 1.617 1.605 1.623 1.623 1.615 1.602

SiH (²⇧r) 1.520 1.528 1.528 1.531 1.515 1.525 1.525 1.527 1.509

SiN (²⌃⁺) 1.572 1.566 1.566 1.562 1.550 1.575 1.575 1.568 1.555

HCO 1.119 (CH) 1.123 1.123 1.123 1.112 1.120 1.121 1.120 1.107

1.175 (CO) 1.171 1.171 1.168 1.158 1.171 1.171 1.168 1.157

MSE 0.003 0.003 -0.002 -0.012 0.003 0.003 -0.002 -0.014

MAE 0.008 0.008 0.008 0.013 0.008 0.008 0.008 0.015

rms 0.013 0.013 0.012 0.017 0.013 0.014 0.013 0.019

Max( ) -0.078 -0.078 -0.082 -0.090 -0.080 -0.080 -0.085 -0.095

Max(+) 0.065 0.065 0.051 0.025 0.063 0.063 0.049 0.035

TABLE S3. Non-hydrogen transfer barrier heights (in kcal/mol) of the NHTBH38/04 set [1]. For all the TAO-DFT calculations, the optimal ✓ values given in TABLE I are adopted.

KS-DFT TAO-DFT

Reactions E_ref B3LYP B3LYP-D3 PBE0 BHHLYP B3LYP B3LYP-D3 PBE0 BHHLYP

Heavy-atom transfer reactions

H + N2O ! OH + N2 V^f 18.14 11.37 11.02 13.91 15.61 10.93 10.58 13.35 14.05

V^r 83.22 72.81 72.32 68.79 91.16 72.20 71.71 67.91 88.70

H + FH ! HF + H V^f 42.18 31.01 30.92 33.74 38.36 31.16 31.07 33.89 38.21

V^r 42.18 31.01 30.92 33.74 38.36 31.16 31.07 33.89 38.21

H + ClH ! HCl + H V^f 18.00 12.41 12.34 13.38 16.56 11.85 11.78 12.69 14.94

V^r 18.00 12.41 12.34 13.38 16.56 11.85 11.78 12.69 14.94

H + FCH3 ! HF + CH3 V^f 30.38 21.79 21.25 25.71 29.78 21.70 21.16 25.56 29.02

V^r 57.02 48.64 47.65 49.75 57.38 48.09 47.10 49.11 55.77

H + F2 ! HF + F V^f 2.27 -7.54 -7.72 -4.60 -4.47 -8.02 -8.19 -5.17 -5.58

V^r 106.18 96.16 95.97 99.02 113.44 95.89 95.71 98.65 112.53

CH3+ FCl ! CH3F + Cl V^f 7.43 -1.58 -2.88 0.79 5.68 -3.02 -4.32 -0.85 2.36

V^r 60.17 50.96 50.11 53.45 63.48 52.06 51.21 53.72 61.90 Nucleophilic substitution reactions

F + CH3F ! FCH3+ F V^f -0.34 -3.93 -4.91 -2.67 0.33 -4.81 -5.79 -3.75 -2.05

V^r -0.34 -3.93 -4.91 -2.67 0.33 -4.81 -5.79 -3.75 -2.05 F · · ·CH3F ! FCH3· · ·F V^f 13.38 10.23 10.47 11.69 14.62 10.25 10.48 11.73 14.91 V^r 13.38 10.23 10.47 11.69 14.62 10.25 10.48 11.73 14.91 Cl + CH3Cl ! ClCH3+ Cl V^f 3.10 -0.57 -2.14 0.87 3.10 -1.84 -3.42 -0.66 -0.12 V^r 3.10 -0.57 -2.14 0.87 3.10 -1.84 -3.42 -0.66 -0.12 Cl · · ·CH3Cl ! ClCH3· · ·Cl V^f 13.61 9.22 9.18 11.29 13.07 9.23 9.19 11.30 13.41 V^r 13.61 9.22 9.18 11.29 13.07 9.23 9.19 11.30 13.41 F + CH3Cl ! FCH3 + Cl V^f -12.54 -16.55 -17.81 -15.45 -14.09 -17.21 -18.47 -16.30 -16.24 V^r 20.11 18.26 16.90 19.44 24.36 16.70 15.34 17.62 20.88

F · · ·CH3Cl ! FCH3· · ·Cl V^f 2.89 0.31 0.35 1.66 2.49 0.55 0.59 1.96 3.14

V^r 29.62 26.62 26.66 28.35 33.05 26.22 26.25 27.87 32.62 OH + CH3F ! HOCH3+ F V^f -2.78 -5.83 -7.07 -4.97 -1.35 -6.37 -7.61 -5.72 -3.45 V^r 17.33 14.13 13.05 15.94 19.67 13.50 12.42 15.12 17.48 OH · · ·CH3F ! HOCH3· · ·F V^f 10.96 7.69 7.63 8.82 12.48 7.97 7.91 9.10 12.98 V^r 47.20 45.46 45.04 49.06 50.65 45.66 45.24 49.27 50.88 Unimolecular and association reactions

H + N2 ! HN2 V^f 14.69 7.47 7.36 8.50 11.04 6.81 6.70 7.69 9.28

V^r 10.72 10.88 10.79 11.64 13.61 11.28 11.19 12.12 14.34

H + CO ! HCO V^f 3.17 -0.59 -1.00 0.31 1.46 -1.11 -1.51 -0.33 0.07

V^r 22.68 24.60 24.32 25.47 24.81 25.29 25.00 26.31 26.27

H + C2H4! CH3CH2 V^f 1.72 -0.18 -0.84 0.67 0.49 -0.67 -1.33 0.06 -0.93 V^r 41.75 41.73 41.33 44.12 45.42 42.32 41.92 44.81 46.50 CH3+ C2H4 ! CH3CH2CH2 V^f 6.85 6.00 3.97 4.02 7.78 4.69 2.66 2.48 4.52 V^r 32.97 29.44 29.07 34.48 33.82 29.95 29.58 35.06 34.64

HCN ! HNC V^f 48.16 47.38 47.46 46.37 47.92 47.58 47.67 46.60 48.28

V^r 33.11 33.33 33.39 32.51 35.44 33.34 33.41 32.53 35.45

MSE -4.57 -5.09 -3.13 0.52 -4.88 -5.39 -3.53 -0.51

MAE 4.69 5.19 3.63 2.21 5.08 5.56 4.18 2.75

rms 5.71 6.14 4.63 2.93 6.02 6.49 5.10 3.29

Max( ) -11.17 -11.26 -14.43 -6.74 -11.02 -11.75 -15.31 -7.85

Max(+) 1.92 1.64 2.79 7.94 2.61 2.32 3.63 6.35

TABLE S4. Hydrogen transfer barrier heights (in kcal/mol) of the HTBH38/04 set [1]. For all the TAO-DFT calculations, the optimal ✓ values given in TABLE I are adopted.

KS-DFT TAO-DFT

Reactions E_ref B3LYP B3LYP-D3 PBE0 BHHLYP B3LYP B3LYP-D3 PBE0 BHHLYP

H + HCl ! H2+ Cl V^f 5.7 -0.77 -0.94 2.94 1.71 -2.11 -2.29 1.45 -0.92

V^r 8.7 3.73 3.55 1.66 7.50 4.40 4.22 1.52 5.74

OH + H2 ! H + H2O V^f 5.1 0.51 0.20 -0.02 6.81 -0.24 -0.55 -1.00 4.82

V^r 21.2 13.35 13.05 17.31 16.37 12.76 12.46 16.66 15.34

CH3+ H2! H + CH4 V^f 12.1 8.73 8.21 6.89 11.76 7.84 7.32 5.85 9.62

V^r 15.3 9.52 9.03 11.90 12.43 9.25 8.75 11.56 11.58

OH + CH4 ! CH3 + H2O V^f 6.7 1.92 1.04 1.62 9.60 1.13 0.26 0.58 7.45

V^r 19.6 13.97 13.08 13.94 18.49 12.73 11.84 12.53 16.01

H + H2 ! H2 + H V^f 9.6 4.22 4.17 5.63 6.44 3.86 3.81 5.19 5.53

V^r 9.6 4.22 4.17 5.63 6.44 3.86 3.81 5.19 5.53

OH + NH3! H2O + NH2 V^f 3.2 -2.62 -3.49 -2.27 7.61 -3.72 -4.59 -3.66 4.86

V^r 12.7 7.32 6.44 7.41 15.54 5.97 5.09 5.85 12.68

HCl + CH3 ! Cl+ CH4 V^f 1.7 -1.59 -2.64 -2.49 1.86 -3.13 -4.18 -4.27 -1.55

V^r 7.9 3.70 2.67 1.24 8.32 4.79 3.76 1.51 7.06

OH + C2H6! H2O + C2H5 V^f 3.4 -1.05 -2.18 -1.23 6.87 -1.86 -2.98 -2.29 4.64 V^r 19.9 15.63 14.17 15.66 19.68 14.39 12.93 14.25 17.24

F + H2 ! HF + H V^f 1.8 -6.03 -6.33 -4.90 1.20 -6.78 -7.08 -5.82 -0.54

V^r 33.4 23.76 23.46 28.40 25.17 23.29 22.99 27.92 24.51

O + CH4! OH + CH3 V^f 13.7 6.78 6.14 7.16 15.33 5.97 5.33 6.14 13.32

V^r 8.1 4.44 3.78 3.94 8.79 3.38 2.72 2.71 6.51

H + PH3! PH2+ H2 V^f 3.1 -1.09 -1.44 0.35 1.32 -1.59 -1.93 -0.26 -0.17

V^r 23.2 22.72 22.28 20.12 24.60 21.85 21.41 19.09 22.39

H + HO ! H2+ O V^f 10.7 4.14 3.93 7.13 7.53 3.70 3.49 6.60 6.55

V^r 13.1 5.69 5.48 5.33 13.40 4.88 4.67 4.33 11.41

H + H2S ! H2 + HS V^f 3.5 -0.58 -0.98 1.05 2.00 -1.62 -2.02 -0.01 -0.19

V^r 17.3 15.54 15.11 12.06 18.54 16.42 15.99 12.30 17.32

O + HCl ! OH + Cl V^f 9.8 0.96 0.81 2.22 12.23 -1.01 -1.16 0.01 8.43

V^r 10.4 3.90 3.75 2.74 12.16 4.32 4.16 2.36 10.23

NH2+ CH3 ! CH4 + NH V^f 8.0 6.13 5.25 5.05 10.45 5.11 4.23 3.85 8.10

V^r 22.4 17.09 16.23 16.42 23.20 16.21 15.35 15.36 20.99

NH2+ C2H5 ! C2H6+ NH V^f 7.5 8.20 6.77 6.89 12.13 7.14 5.71 5.65 9.74

V^r 18.3 14.54 13.43 13.70 20.96 13.60 12.50 12.58 18.58 C2H6+ NH2 ! NH3+ C2H5 V^f 10.4 8.73 7.40 7.26 14.85 7.73 6.40 6.04 12.31 V^r 17.4 15.47 13.82 14.45 19.73 14.27 12.62 13.06 17.09

NH2+ CH4 ! CH3 + NH3 V^f 14.5 11.27 10.23 9.87 17.14 10.32 9.27 8.71 14.75

V^r 17.8 13.39 12.34 12.50 18.10 12.22 11.17 11.14 15.48 s-trans cis-C5H8 ! s-trans cis-C5H8 V^f 38.4 38.77 38.58 35.57 44.72 38.30 38.11 35.00 43.40 V^r 38.4 38.77 38.58 35.57 44.72 38.30 38.11 35.00 43.40

MSE -4.48 -5.12 -4.60 0.58 -5.20 -5.84 -5.55 -1.43

MAE 4.56 5.14 4.60 2.48 5.20 5.84 5.55 2.40

rms 5.10 5.62 4.88 3.11 5.79 6.34 5.80 3.15

Max( ) -9.64 -9.94 -7.77 -8.23 -10.81 -10.96 -9.79 -8.89

Max(+) 0.70 0.18 -0.61 6.32 -0.10 -0.29 -1.85 5.00

TABLE S5. Interaction energies (in kcal/mol) of the S22 set [3]. The counterpoise corrections are used to reduce the basis set superposition errors. Monomer deformation energies are not included. For all the TAO-DFT calculations, the optimal ✓ values given in TABLE I are adopted.

KS-DFT TAO-DFT

Complex [Symmetry] Eref B3LYP B3LYP-D3 PBE0 BHHLYP B3LYP B3LYP-D3 PBE0 BHHLYP Hydrogen bonded complexes

(NH3)2 [C2h] -3.17 -2.19 -3.11 -2.77 -2.56 -2.79 -3.71 -3.46 -3.91

(H2O)2 [Cs] -5.02 -4.49 -5.23 -4.92 -4.96 -5.10 -5.84 -5.63 -6.30

Formic acid dimer [C2h] -18.61 -17.29 -19.61 -18.86 -18.90 -18.71 -21.04 -20.52 -21.96 Formamide dimer [C2h] -15.96 -13.97 -16.54 -15.27 -15.32 -15.31 -17.88 -16.83 -18.19 Uracil dimer [C2h] -20.65 -17.87 -21.20 -19.16 -19.46 -19.16 -22.48 -20.64 -22.21 2-pyridoxine·2-aminopyridine [C1] -16.71 -13.72 -17.63 -15.35 -14.57 -15.19 -19.09 -17.05 -17.79 Adenine·thymine WC [C1] -16.37 -12.82 -17.05 -14.50 -13.94 -14.46 -18.69 -16.40 -17.60

MSE 2.02 -0.55 0.81 0.97 0.82 -1.75 -0.58 -1.64

MAE 2.02 0.57 0.88 1.05 0.88 1.75 0.58 1.64

Dispersion complexes

(CH4)2 [D3d] -0.53 0.40 -0.52 -0.04 0.16 -0.01 -0.93 -0.51 -0.78

(C2H4)2[D2d] -1.51 0.51 -1.61 -0.35 -0.04 -0.27 -2.39 -1.25 -1.88

Benzene·CH4 [C3] -1.50 0.82 -1.43 -0.08 0.33 0.08 -2.17 -0.92 -1.41

Benzene dimer [C2h] -2.73 3.84 -2.19 1.80 3.06 2.31 -3.72 0.03 -0.76

Pyrazine dimer [Cs] -4.42 2.56 -3.71 0.52 1.57 1.05 -5.22 -1.22 -2.15

Uracil dimer [C2] -10.12 -0.86 -10.03 -3.29 -2.76 -2.76 -11.92 -5.49 -7.26

Indole·benzene [C1] -5.22 4.81 -3.74 2.02 3.69 2.82 -5.72 -0.28 -1.27

Adenine·thymine stack [C1] -12.23 1.46 -11.01 -2.14 -0.95 -1.04 -13.51 -5.04 -6.93

MSE 6.48 0.50 4.59 5.41 5.06 -0.92 2.95 1.98

MAE 6.48 0.53 4.59 5.41 5.06 0.92 2.95 2.13

Mixed complexes

Ethene·ethine [C2v] -1.53 -0.64 -1.70 -1.18 -0.94 -1.15 -2.21 -1.76 -2.13

Benzene·H2O [Cs] -3.28 -1.21 -3.53 -2.21 -1.86 -2.02 -4.34 -3.14 -3.68

Benzene·NH³ [Cs] -2.35 -0.08 -2.44 -1.03 -0.64 -0.87 -3.22 -1.93 -2.44

Benzene·HCN [C^s] -4.46 -1.93 -4.68 -3.28 -2.92 -2.81 -5.55 -4.28 -4.94

Benzene dimer [C2v] -2.74 1.02 -2.67 -0.28 0.30 0.03 -3.67 -1.43 -2.07

Indole·benzene T-shape [C¹] -5.73 -0.50 -5.43 -2.36 -1.52 -2.07 -6.99 -4.23 -5.32

Phenol dimer [C1] -7.05 -2.94 -7.19 -4.18 -4.05 -4.19 -8.45 -5.64 -6.96

MSE 2.98 -0.07 1.80 2.22 2.01 -1.04 0.68 -0.06

MAE 2.98 0.18 1.80 2.22 2.01 1.04 0.74 0.39

MSE 3.95 -0.02 2.50 2.98 2.74 -1.22 1.10 0.18

MAE 3.95 0.43 2.52 3.01 2.76 1.22 1.49 1.42

rms 5.17 0.59 3.62 4.22 3.98 1.37 2.40 1.98

Max( ) 0.53 -1.00 -0.25 -0.29 -0.10 -2.43 -1.91 -3.35

Max(+) 13.69 1.48 10.09 11.28 11.19 -0.40 7.19 5.30

TABLE S6. Singlet-triplet energy gap [ET ES] (in kcal/mol) of n-acene as a function of the acene length, calculated using various hybrid functionals in spin-unrestricted KS-DFT and TAO-DFT. For all the TAO-DFT calculations, the optimal ✓ values given in TABLE I are adopted. 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 Expt. DMRG/cc-pVDZ DMRG/STO-3G CCSD(T)/CBS B3LYP B3LYP-D3 PBE0 BHHLYP B3LYP B3LYP-D3 PBE0 BHHLYP

2 61.0 61.0 61.5 65.83 62.65 62.64 62.59 62.96 60.58 60.57 60.17 58.79

3 43.1 44.0 45.9 48.21 41.78 41.81 41.31 41.40 40.44 40.47 39.81 38.77

4 29.3 31.9 34.7 33.49 27.70 27.75 26.99 26.50 27.27 27.31 26.60 25.74

5 19.8 23.4 26.7 25.27 17.87 17.92 16.95 18.41 18.74 18.78 18.10 17.35

6 17.5 21.0 17.71 10.93 10.97 10.90 15.01 13.31 13.34 12.71 11.99

7 13.40 7.31 7.33 8.23 14.23 9.89 9.92 9.34 8.60

8 14.2 9.20 5.78 5.78 7.31 15.02 7.73 7.75 7.21 6.43

9 7.00 5.35 5.34 7.39 16.89 6.30 6.32 5.81 4.99

10 11.6 4.64 5.57 5.56 8.12 18.48 5.30 5.31 5.08 3.99

11 3.65 6.24 6.23 9.35 18.65 4.56 4.57 4.12 3.27

12 10.7 7.25 7.23 10.76 18.57 3.99 4.00 3.97 2.73

13 3.53 3.54 3.62 2.30

14 3.15 3.16 2.78 1.95

15 2.85 2.86 2.51 1.70

16 2.61 2.61 2.28 1.59

17 2.39 2.40 2.09 1.35

18 2.22 2.22 1.93 1.21

19 2.08 2.08 1.78 1.12

20 1.95 1.95 1.67 1.02

21 1.83 1.83 1.57 0.94

22 1.73 1.73 1.49 0.88

23 1.64 1.64 1.39 0.75

24 1.56 1.56 1.33 0.77

25 1.49 1.49 1.24 0.70

26 1.42 1.42 1.15 0.65

27 1.36 1.36 1.14 0.66

28 1.31 1.30 1.10 0.64

29 1.25 1.26 1.08 0.61

30 1.21 1.21 1.03 0.56

TABLE S7. Vertical ionization potential (in eV) for the lowest singlet state of n-acene as a function of the acene length, calculated using various hybrid functionals in spin-unrestricted TAO-DFT. For all the TAO-DFT calculations, the optimal ✓ values given in TABLE I are adopted. 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 Expt. CCSD(T)/CBS B3LYP B3LYP-D3 PBE0 BHHLYP

2 8.14 8.24 7.52 7.52 7.60 7.34

3 7.44 7.47 6.74 6.74 6.82 6.55

4 6.97 6.95 6.20 6.20 6.28 6.02

5 6.59 6.57 5.82 5.82 5.91 5.63

6 6.43 5.55 5.55 5.63 5.36

7 5.35 5.35 5.44 5.16

8 5.20 5.20 5.28 5.00

9 5.07 5.07 5.16 4.87

10 4.97 4.97 5.07 4.77

11 4.89 4.89 4.97 4.68

12 4.81 4.81 4.92 4.61

13 4.75 4.75 4.86 4.54

14 4.69 4.69 4.78 4.49

15 4.64 4.64 4.73 4.43

16 4.59 4.59 4.68 4.39

17 4.55 4.55 4.64 4.35

18 4.51 4.51 4.60 4.31

19 4.48 4.48 4.57 4.28

20 4.45 4.45 4.54 4.25

21 4.42 4.42 4.51 4.22

22 4.39 4.39 4.48 4.19

23 4.36 4.36 4.46 4.17

24 4.34 4.34 4.43 4.14

25 4.32 4.32 4.41 4.12

26 4.30 4.30 4.39 4.10

27 4.28 4.28 4.37 4.08

28 4.26 4.26 4.35 4.07

29 4.24 4.24 4.34 4.05

30 4.23 4.23 4.32 4.03

TABLE S8. Vertical electron affinity (in eV) for the lowest singlet state of n-acene as a function of the acene length, calculated using various hybrid functionals in spin-unrestricted TAO-DFT. For all the TAO-DFT calculations, the optimal ✓ values given in TABLE I are adopted. 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 Expt. CCSD(T)/CBS B3LYP B3LYP-D3 PBE0 BHHLYP

2 -0.20 -0.48 -0.83 -0.83 -0.77 -1.05

3 0.53 0.28 0.04 0.04 0.11 -0.17

4 1.07 0.82 0.64 0.64 0.72 0.43

5 1.39 1.21 1.07 1.07 1.15 0.86

6 1.47 1.38 1.38 1.46 1.17

7 1.61 1.61 1.70 1.41

8 1.79 1.79 1.87 1.59

9 1.93 1.93 2.01 1.73

10 2.04 2.04 2.12 1.85

11 2.14 2.14 2.23 1.95

12 2.22 2.22 2.30 2.03

13 2.30 2.30 2.37 2.10

14 2.36 2.36 2.45 2.17

15 2.42 2.42 2.51 2.23

16 2.47 2.47 2.56 2.28

17 2.52 2.52 2.61 2.33

18 2.56 2.56 2.65 2.37

19 2.60 2.60 2.69 2.41

20 2.63 2.63 2.72 2.44

21 2.67 2.66 2.76 2.47

22 2.70 2.69 2.79 2.50

23 2.72 2.72 2.82 2.53

24 2.75 2.75 2.84 2.56

25 2.77 2.77 2.87 2.58

26 2.80 2.80 2.89 2.61

27 2.82 2.82 2.91 2.63

28 2.84 2.84 2.93 2.64

29 2.86 2.85 2.95 2.66

30 2.87 2.87 2.97 2.68

TABLE S9. Fundamental gap (in eV) for the lowest singlet state of n-acene as a function of the acene length, calculated using various hybrid functionals in spin-unrestricted TAO-DFT. For all the TAO-DFT calculations, the optimal ✓ values given in TABLE I are adopted. 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 Expt. CCSD(T)/CBS B3LYP B3LYP-D3 PBE0 BHHLYP

2 8.34 8.72 8.35 8.35 8.37 8.39

3 6.91 7.19 6.70 6.70 6.70 6.73

4 5.90 6.13 5.56 5.56 5.56 5.59

5 5.20 5.37 4.75 4.75 4.75 4.78

6 4.96 4.17 4.17 4.17 4.19

7 3.74 3.74 3.74 3.75

8 3.41 3.41 3.41 3.41

9 3.15 3.15 3.15 3.14

10 2.93 2.93 2.95 2.92

11 2.75 2.75 2.75 2.74

12 2.59 2.59 2.62 2.58

13 2.45 2.45 2.49 2.44

14 2.33 2.33 2.33 2.31

15 2.22 2.22 2.22 2.21

16 2.12 2.12 2.12 2.12

17 2.03 2.03 2.03 2.02

18 1.95 1.95 1.95 1.94

19 1.88 1.88 1.88 1.87

20 1.81 1.81 1.81 1.81

21 1.75 1.75 1.75 1.74

22 1.69 1.69 1.69 1.69

23 1.64 1.64 1.64 1.63

24 1.59 1.59 1.59 1.59

25 1.55 1.55 1.54 1.54

26 1.50 1.50 1.50 1.49

27 1.46 1.46 1.46 1.46

28 1.42 1.42 1.42 1.42

29 1.39 1.39 1.39 1.38

30 1.36 1.36 1.36 1.35

TABLE S10. Symmetrized von Neumann entropy for the lowest singlet state of n-acene as a function of the acene length, calculated using various hybrid functionals in spin-restricted TAO-DFT. For all the TAO-DFT calculations, the optimal ✓ values given in TABLE I are adopted.

TAO-DFT

n B3LYP B3LYP-D3 PBE0 BHHLYP

2 0.10 0.10 0.13 0.35

3 0.26 0.26 0.31 0.62

4 0.50 0.50 0.58 0.96

5 0.81 0.81 0.90 1.33

6 1.14 1.14 1.23 1.72

7 1.45 1.45 1.55 2.10

8 1.74 1.74 1.85 2.47

9 2.01 2.01 2.14 2.83

10 2.28 2.27 2.42 3.19

11 2.54 2.54 2.70 3.54

12 2.81 2.80 2.98 3.90

13 3.07 3.07 3.26 4.26

14 3.35 3.34 3.55 4.62

15 3.62 3.61 3.83 4.97

16 3.89 3.89 4.11 5.33

17 4.16 4.16 4.40 5.69

18 4.43 4.43 4.68 6.05

19 4.70 4.70 4.97 6.41

20 4.97 4.97 5.25 6.77

21 5.24 5.24 5.54 7.12

22 5.51 5.51 5.82 7.48

23 5.78 5.78 6.11 7.84

24 6.05 6.05 6.39 8.20

25 6.32 6.32 6.67 8.56

26 6.59 6.59 6.96 8.91

27 6.86 6.86 7.24 9.27

28 7.13 7.13 7.53 9.63

29 7.40 7.40 7.81 9.99

30 7.67 7.67 8.10 10.35