Electronic Supplementary Material (ESI) for RSC Advances. This journal is © The Royal Society of Chemistry 2017

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Supplementary material to: Self-consistent determination of the fictitious temperature in thermally-assisted-occupation density functional theory

Chih-Ying Lin,¹ Kerwin Hui,¹ Jui-Hui Chung,¹ and Jeng-Da Chai^{1, 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

References (S2).

FIG. S1. Potential energy curves for the ground state of H2 (S3).

FIG. S2. Occupation numbers of the 1 g orbital for the ground state of H2 as a function of the internuclear distance R (S4).

FIG. S3. Potential energy curves for the ground state of N2 (S4 to S5).

FIG. S4. Occupation numbers of the 3 g orbital for the ground state of N2 as a function of the internuclear distance R (S5 to S6).

FIG. S5. Occupation numbers of the 1⇡ux orbital for the ground state of N2 as a function of the internuclear distance R (S6).

FIG. S6. Torsion potential energy curves for the ground state of twisted ethylene as a function of the HCCH torsion angle (S7).

FIG. S7. Occupation numbers of the ⇡ (1b2) orbital for the ground state of twisted ethylene as a function of the HCCH torsion angle (S8).

TABLE S1. Reaction energies of the 30 chemical reactions in the NHTBH38/04 and HTBH38/04 sets (S8 to S9).

TABLE S2. Non-hydrogen transfer barrier heights of the NHTBH38/04 set (S9 to S10).

TABLE S3. Hydrogen transfer barrier heights of the HTBH38/04 set (S10 to S11).

TABLE S4. Interaction energies of the S22 set (S11 to S12).

⇤Author to whom correspondence should be addressed. Electronic mail: [email protected]

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[1] T. Helgaker, P. Jørgensen, and J. Olsen, Molecular Electronic-Structure Theory, Wiley, New York, 2000.

[2] K. P. Huber and G. Herzberg, in Constants of Diatomic Molecules, Van Nostrand Reinhold, New York, 1979, pp. 412.

[3] D. Sundholm, P. Pyykko, and L. Laaksonen, Mol. Phys., 1985,56, 1411–1418.

[4] M. S. Gordon et al., J. Chem. Phys., 1999,110, 4199–4207.

[5] X. Lopez, M. Piris, J. M. Matxain, F. Ruipérez, and J. M. Ugalde, ChemPhysChem, 2011,12, 1673–1676.

[6] R. G. A. Bone and P. Pulay, Int. J. Quantum Chem., 1992,45, 133–166.

[7] Y. Zhao, B. J. Lynch, and D. G. Truhlar, J. Phys. Chem. A, 2004,108, 2715–2719.

[8] Y. Zhao, N. González-García, and D. G. Truhlar, J. Phys. Chem. A, 2005,109, 2012–2018.

[9] P. Jurečka, J. Šponer, J. Černý, and P. Hobza, Phys. Chem. Chem. Phys., 2006,8, 1985–1993.

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−120

−100

−80

−60

−40

−20 0 20 40 60

1 2 3 4 5 6 7 8 9 10

Relative Energy (kcal/mol)

Internuclear Distance (Å) (θ in mhartree) Reference

θ^* θ=0 θ=7 θ=15 θ=20 θ=40

(a)

−120

−100

−80

−60

−40

−20 0 20 40 60

1 2 3 4 5 6 7 8 9 10

θ^* θ=0 θ=7 θ=15 θ=20 θ=40

(b)

−120

−100

−80

−60

−40

−20 0 20 40 60

1 2 3 4 5 6 7 8 9 10

θ^* θ=0 θ=7 θ=15 θ=20 θ=40

(c)

FIG. S1. Potential energy curves (in relative energy) for the ground state of H2, calculated using spin-restricted (a) TAO-PBE, (b) TAO-BLYP, and (c) TAO-BLYP-D with the ✓^⇤ and system-independent ✓ values. The ✓ = 0 cases correspond to spin- restricted (a) KS-PBE, (b) KS-BLYP, and (c) KS-BLYP-D, respectively. The reference curve is calculated using the CCSD theory (exact for any two-electron system). The zeros of energy are set at the respective spin-unrestricted dissociation limits.

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1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

1 2 3 4 5 6 7 8 9 10

Occupation Number

θ^* θ=0 θ=7 θ=15 θ=20 θ=40

(a)

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2

1 2 3 4 5 6 7 8 9 10

Occupation Number

θ^* θ=0 θ=7 θ=15 θ=20 θ=40

(b)

FIG. S2. Occupation numbers of the 1 gorbital for the ground state of H2as a function of the internuclear distance R, calculated using spin-restricted (a) TAO-PBE and (b) TAO-BLYP/TAO-BLYP-D with the ✓^⇤and system-independent ✓ values. The ✓ = 0 cases correspond to spin-restricted (a) KS-PBE and (b) KS-BLYP/KS-BLYP-D, respectively. The reference data are the FCI NOONs [1].

−300

−250

−200

−150

−100

−50 0 50 100 150

1 2 3 4 5 6 7 8 9 10

θ^* θ=0 θ=7 θ=15 θ=20 θ=40

(a)

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−300

−250

−200

−150

−100

−50 0 50 100

1 2 3 4 5 6 7 8 9 10

θ^* θ=0 θ=7 θ=15 θ=20 θ=40

(b)

−300

−250

−200

−150

−100

−50 0 50 100 150

1 2 3 4 5 6 7 8 9 10

θ^* θ=0 θ=7 θ=15 θ=20 θ=40

(c)

FIG. S3. Potential energy curves (in relative energy) for the ground state of N2, calculated using spin-restricted (a) TAO- PBE, (b) TAO-BLYP, and (c) TAO-BLYP-D with the ✓^⇤ and system-independent ✓ values. The ✓ = 0 cases correspond to spin-restricted (a) KS-PBE, (b) KS-BLYP, and (c) KS-BLYP-D, respectively. The reference data ( 228.3 (kcal/mol) at R = 1.098 Å (i.e., the equilibrium bond length)) are the experimental results [2, 3]. The zeros of energy are set at the respective spin-unrestricted dissociation limits.

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2

1 2 3 4 5 6 7 8 9 10

Occupation Number

θ^* θ=0 θ=7 θ=15 θ=20 θ=40

(a)

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1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

1 2 3 4 5 6 7 8 9 10

Occupation Number

θ^* θ=0 θ=7 θ=15 θ=20 θ=40

(b)

FIG. S4. Occupation numbers of the 3 g orbital for the ground state of N2 as a function of the internuclear distance R, calculated using spin-restricted (a) TAO-PBE and (b) TAO-BLYP/TAO-BLYP-D with the ✓^⇤and system-independent ✓ values.

The ✓ = 0 cases correspond to spin-restricted (a) KS-PBE and (b) KS-BLYP/KS-BLYP-D, respectively. The reference data are the NOONs of the MRCI method [4].

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2

1 2 3 4 5 6 7 8 9 10

Occupation Number

θ^* θ=0 θ=7 θ=15 θ=20 θ=40

(a)

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2

1 2 3 4 5 6 7 8 9 10

Occupation Number

θ^* θ=0 θ=7 θ=15 θ=20 θ=40

(b)

FIG. S5. Occupation numbers of the 1⇡ux orbital for the ground state of N2 as a function of the internuclear distance R, calculated using spin-restricted (a) TAO-PBE and (b) TAO-BLYP/TAO-BLYP-D with the ✓^⇤ and system-independent ✓ values. The ✓ = 0 cases correspond to spin-restricted (a) KS-PBE and (b) KS-BLYP/KS-BLYP-D, respectively. The reference data are the NOONs of the MRCI method [4].

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0 10 20 30 40 50 60 70 80 90

0 30 60 90 120 150 180

Torsion Angle (degree)

(θ in mhartree) Reference

θ^* θ=0 θ=7 θ=15 θ=20 θ=40

(a)

0 10 20 30 40 50 60 70 80 90 100

0 30 60 90 120 150 180

θ^* θ=0 θ=7 θ=15 θ=20 θ=40

(b)

0 10 20 30 40 50 60 70 80 90 100

0 30 60 90 120 150 180

θ^* θ=0 θ=7 θ=15 θ=20 θ=40

(c)

FIG. S6. Torsion potential energy curves (in relative energy) for the ground state of twisted ethylene as a function of the HCCH torsion angle, calculated using spin-restricted (a) TAO-PBE, (b) TAO-BLYP, and (c) TAO-BLYP-D with the ✓^⇤ and system-independent ✓ values. The ✓ = 0 cases correspond to spin-restricted (a) KS-PBE, (b) KS-BLYP, and (c) KS-BLYP-D, respectively. The reference data are the CASPT2 results [5]. The zeros of energy are set at the respective minimum energies.

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1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

0 10 20 30 40 50 60 70 80 90

Occupation Number

Torsion Angle (degree) (θ in mhartree)

Reference θ^* θ=0 θ=7 θ=15 θ=20 θ=40

(a)

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2

0 10 20 30 40 50 60 70 80 90

Occupation Number

Torsion Angle (degree) (θ in mhartree)

Reference θ^* θ=0 θ=7 θ=15 θ=20 θ=40

(b)

FIG. S7. Occupation numbers of the ⇡ (1b2) orbital for the ground state of twisted ethylene as a function of the HCCH torsion angle, calculated using spin-restricted (a) TAO-PBE and (b) TAO-BLYP/TAO-BLYP-D with the ✓^⇤and system-independent ✓ values. The ✓ = 0 cases correspond to spin-restricted (a) KS-PBE and (b) KS-BLYP/KS-BLYP-D, respectively. The reference data are the half-projected NOONs of the CASSCF method (HPNO-CAS) [6].

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 [7, 8].

with ✓^⇤

Reactions Eref TAO-LDA TAO-PBE TAO-BLYP TAO-BLYP-D

H + N2O ! OH + N2 -65.08 35.68 22.59 11.96 12.03

H + FCH3 ! HF + CH³ -26.64 8.28 4.16 0.49 0.63

H + F2 ! HF + F -103.91 18.54 13.19 10.10 10.10

CH3 + FCl ! CH³F + Cl -52.74 3.56 4.63 4.00 3.85

F + CH3Cl ! FCH³ + Cl -32.65 -0.90 0.82 0.27 0.37

F · · ·CH³Cl ! FCH³· · ·Cl -26.73 3.98 4.41 3.83 3.90

OH + CH3F ! HOCH3+ F -20.11 -0.74 -1.09 -0.41 -0.92

OH · · ·CH3F ! HOCH3· · ·F -36.24 -11.06 -8.12 -5.50 -5.52

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H + CO ! HCO -19.51 -14.40 -6.84 -5.78 -5.87

H + C2H4 ! CH³CH2 -40.03 -4.52 -0.31 1.22 0.62

CH3 + C2H4 ! CH³CH2CH2 -26.12 -12.64 -2.14 5.96 3.47

HCN ! HNC 15.05 -0.83 0.06 0.02 0.09

H + HCl ! H2 + Cl -3.0 11.13 5.51 -1.29 -1.29

OH + H2 ! H + H²O -16.1 -13.62 -3.95 2.22 2.19

CH3 + H2 ! H + CH⁴ -3.2 -7.09 -2.30 2.70 2.55

OH + CH4 ! CH³ + H2O -12.9 -6.53 -1.65 -0.48 -0.36

OH + NH3 ! H2O + NH2 -9.5 -3.67 -1.73 -1.51 -1.43

HCl + CH3 ! Cl+ CH4 -6.2 4.04 3.21 1.41 1.26

OH + C2H6! H²O + C2H5 -16.5 -9.27 -3.29 -1.97 -1.30

F + H2 ! HF + H -31.6 -18.31 -6.49 -0.72 -0.72

O + CH4 ! OH + CH³ 5.6 -7.02 -5.83 -5.84 -5.69

H + PH3 ! PH² + H2 -20.1 3.51 0.40 -4.10 -3.42

H + HO ! H2 + O -2.4 14.11 8.12 3.14 3.14

H + H2S ! H2 + HS -13.8 7.93 3.48 -2.41 -2.15

O + HCl ! OH + Cl -0.6 -2.98 -2.61 -4.43 -4.43

NH2 + CH3 ! CH⁴ + NH -14.4 3.87 4.55 4.66 4.56

NH2 + C2H5 ! C²H6 + NH -10.8 6.62 6.19 6.15 5.49

C2H6+ NH2 ! NH3 + C2H5 -7.0 -5.61 -1.55 -0.46 0.14

NH2 + CH4 ! CH³ + NH3 -3.3 -2.96 -0.02 0.94 0.97

MSE -0.55 0.85 0.56 0.50

MAE 8.63 4.57 3.37 3.19

rms 11.19 6.39 4.47 4.31

Max( ) -18.31 -8.12 -7.24 -7.28

Max(+) 35.68 22.59 11.96 12.03

TABLE S2. Non-hydrogen transfer barrier heights (in kcal/mol) of the NHTBH38/04 set [7, 8].

with ✓^⇤

Heavy-atom transfer reactions

H + N2O ! OH + N² V^f 18.14 2.69 9.97 8.54 8.19

V^r 83.22 32.09 52.46 61.67 61.24

H + FH ! HF + H V^f 42.18 18.48 27.07 26.03 25.76

V^r 42.18 18.48 27.07 26.03 25.76

H + ClH ! HCl + H V^f 18.00 2.27 9.64 9.80 9.74

V^r 18.00 2.27 9.64 9.80 9.74

H + FCH3 ! HF + CH³ V^f 30.38 13.36 18.51 16.12 15.33

V^r 57.02 31.72 41.00 42.27 41.33

H + F2 ! HF + F V^f 2.27 -15.97 -9.86 -11.66 -11.78

V^r 106.18 69.40 80.86 82.15 82.03

CH3 + FCl ! CH³F + Cl V^f 7.43 -10.90 -6.01 -6.39 -7.87

V^r 60.17 38.28 42.10 42.35 41.02

Nucleophilic substitution reactions

F + CH3F ! FCH³ + F V^f -0.34 -12.18 -8.31 -7.90 -8.95

V^r -0.34 -12.18 -8.31 -7.90 -8.95

F · · ·CH³F ! FCH³· · ·F V^f 13.38 6.34 6.41 5.72 6.07

V^r 13.38 6.34 6.41 5.72 6.07

Cl + CH3Cl ! ClCH³ + Cl V^f 3.10 -6.73 -3.80 -3.95 -5.69

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Cl · · ·CH Cl ! ClCH · · ·Cl 13.61 6.70 7.07 5.58 5.41

V^r 13.61 6.70 7.07 5.58 5.41

F + CH3Cl ! FCH³ + Cl V^f -12.54 -23.45 -19.52 -19.35 -20.67

V^r 20.11 10.10 12.31 13.03 11.62

F · · ·CH3Cl ! FCH3· · ·Cl V^f 2.89 -1.13 -0.94 -2.01 -1.89

V^r 29.62 21.62 21.38 20.89 20.94

OH + CH3F ! HOCH³+ F V^f -2.78 -15.12 -11.99 -11.35 -13.14

V^r 17.33 5.73 9.21 9.17 7.89

OH · · ·CH3F ! HOCH3· · ·F V^f 10.96 0.23 -1.14 -1.49 -2.05

V^r 47.20 47.53 43.23 40.25 39.71

Unimolecular and association reactions

H + N2 ! HN² V^f 14.69 -2.19 5.19 5.24 5.08

V^r 10.72 9.44 9.08 8.51 8.38

H + CO ! HCO V^f 3.17 -7.57 -1.69 -1.95 -2.25

V^r 22.68 26.34 24.66 23.34 23.13

H + C2H4 ! CH³CH2 V^f 1.72 -5.34 -0.14 -0.69 -1.78

V^r 41.75 39.21 40.20 38.11 37.63

CH3 + C2H4 ! CH³CH2CH2 V^f 6.85 -5.81 1.50 4.73 1.48

V^r 32.97 32.95 29.76 24.88 24.13

HCN ! HNC V^f 48.16 44.83 45.60 46.76 46.94

V^r 33.11 30.60 30.50 31.68 31.80

MSE -12.50 -8.71 -8.89 -9.53

MAE 12.71 8.81 8.93 9.55

rms 16.16 10.75 10.42 10.98

Max( ) -51.13 -30.76 -24.03 -24.15

Max(+) 3.66 1.98 0.66 0.45

TABLE S3. Hydrogen transfer barrier heights (in kcal/mol) of the HTBH38/04 set [7, 8].

with ✓^⇤

H + HCl ! H2 + Cl V^f 5.7 -3.14 0.56 -2.45 -2.58

V^r 8.7 -11.27 -1.95 1.84 1.71

OH + H2 ! H + H²O V^f 5.1 -18.39 -6.36 -3.49 -3.95

V^r 21.2 11.33 13.69 10.39 9.96

CH3 + H2 ! H + CH4 V^f 12.1 -5.35 3.82 7.14 5.96

V^r 15.3 4.94 9.32 7.65 6.61

OH + CH4 ! CH³ + H2O V^f 6.7 -17.22 -5.65 -2.75 -4.01

V^r 19.6 2.21 8.91 10.63 9.24

H + H2 ! H² + H V^f 9.6 -2.70 3.64 2.86 2.67

V^r 9.6 -2.70 3.64 2.86 2.67

OH + NH3 ! H²O + NH2 V^f 3.2 -23.88 -11.94 -9.20 -10.46

V^r 12.7 -10.71 -0.71 1.81 0.47

HCl + CH3 ! Cl+ CH⁴ V^f 1.7 -13.74 -5.94 -3.51 -5.09

V^r 7.9 -11.58 -2.95 1.29 -0.15

OH + C2H6! H²O + C2H5 V^f 3.4 -20.98 -9.03 -6.12 -7.54

V^r 19.9 4.79 10.76 12.36 10.26

F + H2 ! HF + H V^f 1.8 -24.20 -12.97 -11.61 -11.86

V^r 33.4 25.71 25.12 20.70 20.45

O + CH4 ! OH + CH³ V^f 13.7 -10.69 -0.79 1.44 0.52

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H + PH ! PH + H 3.1 -7.35 -1.79 -2.63 -3.14

V^r 23.2 9.24 17.91 21.58 20.38

H + HO ! H² + O V^f 10.7 -1.69 3.75 1.60 1.48

V^r 13.1 -13.40 -1.98 0.86 0.74

H + H2S ! H2 + HS V^f 3.5 -6.73 -1.22 -2.20 -2.59

V^r 17.3 -0.86 9.10 14.00 13.36

O + HCl ! OH + Cl V^f 9.8 -23.13 -10.54 -8.78 -8.86

V^r 10.4 -19.55 -7.33 -3.75 -3.83

NH2 + CH3 ! CH4 + NH V^f 8.0 -8.37 0.71 3.57 1.82

V^r 22.4 2.16 10.56 13.31 11.66

NH2 + C2H5 ! C²H6 + NH V^f 7.5 -5.76 2.89 5.94 3.56

V^r 18.3 -1.57 7.51 10.59 8.87

C2H6+ NH2 ! NH³ + C2H5 V^f 10.4 -9.69 1.44 5.25 3.08

V^r 17.4 2.91 9.99 12.72 9.94

NH2 + CH4 ! CH3 + NH3 V^f 14.5 -6.16 4.39 7.99 6.04

V^r 17.8 0.10 7.71 10.36 8.37

s-trans cis-C5H8 ! s-trans cis-C⁵H8 V^f 38.4 25.00 31.19 35.82 34.69

V^r 38.4 25.00 31.19 35.82 34.69

MSE -17.90 -9.67 -7.84 -8.89

MAE 17.90 9.67 7.84 8.89

rms 18.92 10.37 8.66 9.52

Max( ) -32.93 -20.34 -18.58 -18.66

Max(+) -7.69 -4.61 -1.56 -2.82

TABLE S4. Interaction energies (in kcal/mol) of the S22 set [9]. The counterpoise corrections are used to reduce the basis set superposition errors. Monomer deformation energies are not included.

with ✓^⇤

Complex [Symmetry] Eref TAO-LDA TAO-PBE TAO-BLYP TAO-BLYP-D

Hydrogen bonded complexes

(NH3)2 [C2h] -3.17 -5.10 -2.83 -1.78 -3.44

(H2O)2 [Cs] -5.02 -7.78 -4.91 -3.99 -4.92

Formic acid dimer [C2h] -18.61 -26.82 -18.08 -15.48 -18.50

Formamide dimer [C2h] -15.96 -21.82 -14.69 -12.44 -15.63

Uracil dimer [C2h] -20.65 -26.15 -18.46 -16.21 -20.21

2-pyridoxine·2-aminopyridine [C¹] -16.71 -22.74 -15.24 -12.48 -17.25

Adenine·thymine WC [C1] -16.37 -21.91 -14.21 -11.33 -16.42

Dispersion complexes

(CH4)2 [D3d] -0.53 -0.83 -0.08 0.68 -0.31

(C2H4)2[D2d] -1.51 -2.49 -0.31 1.12 -1.44

Benzene·CH⁴ [C3] -1.50 -2.01 -0.01 1.39 -1.23

Benzene dimer [C2h] -2.73 -2.63 1.90 4.93 -1.93

Pyrazine dimer [Cs] -4.42 -4.40 0.76 3.82 -3.63

Uracil dimer [C2] -10.12 -10.14 -2.69 1.13 -9.68

Indole·benzene [C¹] -5.22 -4.37 2.26 6.33 -3.91

Adenine·thymine stack [C¹] -12.23 -11.92 -1.30 4.10 -11.77

Mixed complexes

Ethene·ethine [C2v] -1.53 -2.27 -1.16 -0.29 -1.49

Benzene·H2O [Cs] -3.28 -4.44 -2.04 -0.49 -3.30

Benzene·NH³ [Cs] -2.35 -3.03 -0.92 0.55 -2.18

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Benzene dimer [C ] -2.74 -3.06 -0.10 1.86 -2.52

Indole·benzene T-shape [C¹] -5.73 -6.07 -1.86 0.84 -5.52

Phenol dimer [C1] -7.05 -8.99 -3.85 -1.73 -6.64

MSE -1.95 2.78 5.07 0.24

MAE 2.07 2.78 5.07 0.34

rms 3.17 3.90 6.33 0.45

Max( ) -8.21 0.11 1.03 -0.54

Max(+) 0.85 10.93 16.33 1.31