Supporting Information for
Supporting Information for Reliable Prediction with Tuned Range-Separated Functionals of the Singlet-Triplet Gap in Organic Emitters for Thermally Activated Delayed Fluorescence (TADF)
Haitao Sun, Cheng Zhong, and Jean-Luc Brédas*
Physical Science and Engineering Division Solar & Photovoltaics Engineering Research Center King Abdullah University of Science and Technology (KAUST) Thuwal 23955-6900, Kingdom of Saudi Arabia
Corresponding author: [email protected]
S1
Content
1. Introduction of a “Golden proportion” method
S3
2. Table S1. Optimally-tuned ω* values using 6-31G(d) and 6-31+G(d)
S4
3. Table S2. (a) Calculated EVA(S1) values using PCM based on two different Methods; (b) Basis set effects on the calculated EVA(S1) values
S4
4. Table S3. Calculated EVA(S1) and singlet-triplet gap ∆EST using full TDDFT and TDA
S5
5. Table S4. Calculated vertical excitation energies based on the ground-state geometries optimized by LC-ωPBE*, B3LYP, and CAM-B3LYP functionals
S6
6. Table S5. Calculated EVA(S1) for PhCz and NPh3 using high-level CC2 and B2(GP)-PLYP methods
S6
7. Table S5. Calculated EVA(S1) and singlet-triplet gap ∆EST and ∆EST* using other density functionals
S7
8. Table S6. Calculated relaxation energies of λ(S1) and λ(T1)
S8
Scheme S1. Hole distribution of S1 and T1 for CBP and DPA-DPS
S9
9. Table S7. Assignment for the lowest triplet excitation (T1) compared to the experimental ones
S10
10. Figures S1-S3. Histograms of the errors of calculated EVA(S1), ∆EST and ∆EST*, S11-S13
respectively, compared to experimental data
11. Figures S4-S6. Linear correlation coefficient R2 for calculated EVA(S1), ∆EST, and ∆EST*, respectively, between theory and experiment
S14-S16
12. Full names of the molecules investigated in this work
S17
13. References
S18
S2
Introduction of a “Golden proportion” method in the search for the optimal ω value The minimization of J2 in Equation 2b is usually carried out by a fitting method, as illustrated in the left scheme below. Take the PhCz molecule as an example. First evaluations of J2 for ω values from 0.05 to 0.50 Bohr-1 (with intervals of 0.05) are performed. Then, to obtain a more accurate ω value, a finer interval of 0.01 is considered at the bottom of the fitting curve from 0.16 to 0.24. In this case, one needs to perform calculations with at least 20 different ω values in order to obtain the optimal ω with a resolution of 0.01 Bohr-1. The optimal ω using this fitting method is 0.20 Bohr-1. In this work, we have implemented the “Golden proportion” method based on Brent’s algorithm in Python.1 We employed this method to calculate more precise ω values while having to take account of a lesser number of single-point calculations than with the fitting method, as shown in the right scheme below. We have again chosen initial ω values in the range 0.05 ~ 0.50 Bohr-1. First, the “Golden proportion” method generates an initial ω1 value at 0.2219 [=0.50-(0.50-0.05)*0.618] according to the golden ratio (r = 0.618) and the corresponding J2(1). Next, it generates another ω2 at 0.3281 calculated through [=0.50-(0.50-0.2219)*0.618] and J2(2). If J2(2) < J2(1), it searches in the range [ω2, 0.50]; if J2(2) > J2(1), it searches in the opposite direction [0.05, ω2]. This process is repeated until minimization of J2(ω), which corresponds to the optimal ω, is reached. In this way, an optimal ω of 0.2036 is obtained in fewer than 10 steps, with higher precision.
S3
Table S1. Optimally-tuned ω* values for the LC-ωPBE functional with 6-31G(d) and 6-31+G(d) basis sets. Optimal ω* values Compounds
6-31G(d)
6-31+G(d)
PhCz NPh3 CBP α-NPD PIC-TRZ DPA-DPS DTPA-DPS ACRFLCN CC2TA DTC-DPS 2CzPN 4CzPN PXZ-TRZ Spiro-CN 4CzIPN 4CzTPN 4CzTPN-Me
0.2186 0.2135 0.1846 0.1804 0.1485 0.1762 0.1616 0.1835 0.1699 0.1701 0.1866 0.1536 0.1931 0.1759 0.1495 0.1552 0.1501
0.2036 0.1977 0.1734 0.1701 0.1411 0.1663 0.1524 0.1742 0.1598 0.1601 0.1758 0.1455 0.1828 0.1679 0.1422 0.1472 0.1426
Table S2. (a) Calculated EVA(S1) values using PCM(toluene)-TDA-LC-ωPBE* based on two different schemes. EVA(S1) PCM PCM Exp. Value in eV Linear-Response State-Specific (SS) PhCz 4.17 4.17 3.66 NPh3 3.99 3.96 3.74 α-NPD 3.29 Not converged 3.31 ACRFLCN 3.03 3.03 3.05 2CzPN 3.22 3.20 3.19 Spiro-CN 2.73 2.80 2.69 (b) Calculated EVA(S1) values using PCM(toluene)-TDA-LC-ωPBE* with different basis sets. All values are in eV. 6-31G(d) PhCz CBP Spiro-CN ACRFLCN DTC-DPS
4.25 4.00 2.78 3.07 3.67
6-311G(d) 6-31+G(d) 6-311+G(d) 6-311++G(d,p) 4.19 3.96 2.76 3.06 3.62
4.17 3.94 2.73 3.03 3.60 S4
4.15 3.93 2.71 3.03 3.59
4.13 3.93 2.70 3.03 -
Exp. 3.66 3.80 2.69 3.05 3.62
Table S3. Calculated EVA(S1) values and vertical singlet-triplet gap ∆EST and adiabatic ∆EST* using the full TDDFT and TDA at the PCM(toluene)-TDA-DFT/6-31+G(d) level, in comparison to the experimental data. All values are in eV.
Compound
TD DFT
TDA
LC-ωPBE*
LC-ωPBE*
EVA(S1)
PhCz 4.08 NPh3 3.93 CBP 3.89 α-NPD 3.25 PIC-TRZ 3.15 DPA-DPS 3.54 DTPA-DPS 3.42 ACRFLCN 3.03 CC2TA 3.63 DTC-DPS 3.56 2CzPN 3.18 4CzPN 2.53 PXZ-TRZ 2.93 Spiro-CN 2.72 4CzIPN 2.51 4CzTPN 2.30 4CzTPN-Me 2.26 a MAD 0.15 a The MAD values are values.
∆EST
∆EST* EVA(S1) ∆EST ∆EST*
Exp. values
EVA(S1)
∆EST(00)
0.85 0.82 4.17 0.73 0.75 3.66 0.55 0.74 0.87 3.98 0.61 0.74 3.74 0.57 0.79 1.02 3.94 0.66 0.88 3.80 0.71 0.70 0.93 3.29 0.57 0.76 3.31 0.73 0.26 0.10 3.17 0.22 0.11 3.35 0.18 0.65 0.55 3.60 0.60 0.49 3.53 0.52 0.60 0.51 3.48 0.56 0.46 3.47 0.46 0.28 0.23 3.03 0.07 0.02 3.05 0.24 0.46 0.13 3.66 0.35 0.13 3.64 0.20 0.44 0.38 3.60 0.39 0.27 3.62 0.36 0.48 0.38 3.22 0.41 0.24 3.19 0.31 0.13 0.05 2.55 0.13 0.00 2.82 0.15 0.12 0.01 2.94 0.07 0.01 2.73 0.06 0.21 0.01 2.73 0.07 0.01 2.69 0.06 0.12 0.03 2.52 0.12 0.01 2.85 0.10 0.13 0.06 2.32 0.14 0.06 2.61 0.09 0.11 0.08 2.28 0.12 0.09 2.49 0.09 0.11 0.10 0.15 0.07 0.09 calculated with respect to the corresponding experimental
S5
Table S4. Calculated vertical excitation energies (in eV) of lowest singlet excited state EVA(S1), vertical ∆EST and adiabatic ∆EST* based on the ground-state geometries optimized by LC-ωPBE*, B3LYP, and CAM-B3LYP functionals.
TDA-LC-ωPBE*
TDA-LC-ωPBE*
TDA-LC-ωPBE*
@LC-ωPBE*
@B3LYP
@CAM-B3LYP
Exp. Val.
Compound EVA(S1) ∆EST ∆EST* EVA(S1) ∆EST ∆EST* EVA(S1) ∆EST ∆EST* EVA(S1)∆EST(00) PhCz NPh3 CBP α-NPD ACRFLN PXZ-TRZ 2CzPN 4CzTPN MADa
4.17 3.96 3.95 3.27 3.03 2.93 3.21 2.32 0.18
0.74 0.61 0.65 0.57 0.07 0.07 0.42 0.13 0.10
0.75 0.74 0.88 0.76 0.02 0.01 0.24 0.06 0.12
4.17 3.98 3.94 3.29 3.03 2.94 3.22 2.32 0.18
0.73 0.61 0.66 0.57 0.07 0.07 0.41 0.14 0.08
0.75 0.74 0.88 0.76 0.02 0.01 0.24 0.06 0.12
4.21 3.99 4.00 3.34 3.09 2.96 3.26 2.37 0.20
0.73 0.47 0.64 0.55 0.05 0.02 0.40 0.15 0.11
0.75 0.74 0.88 0.76 0.02 0.01 0.24 0.06 0.12
3.66 3.74 3.80 3.31 3.05 2.73 3.19 2.61
0.55 0.57 0.71 0.73 0.24 0.06 0.31 0.09
a
The MAD values are calculated with respect to the corresponding experimental values.
Table S5. Calculated vertical excitation energies of the lowest singlet excited state EVA(S1) for PhCz and NPh3 at different levels. All values are in eV. PhCz NPh3 TDA-LC-ωPBE*/6-31+G(d) 4.17 3.98 TDA-B2-PLYP/def2-TZVPa 3.98 3.94 a TDA-B2GP-PLYP/def2-TZVP 4.15 4.13 b RI-CC2//def2-TZVP/SVP 3.98 3.94 Experiment 3.66 3.74 a 2 b From ref . ‘Resolution-of-identity’ RI-CC2 calculations were carried out with Turbomole,3 using the def2-TZVP basis set and SVP auxiliary basis set; the solvent model is not included in these calculation since the solvent effect has been shown to be very limited.2
S6
Table S6. Calculated EVA(S1) (in eV) and both vertical ∆EST and adiabatic ∆EST* singlet-triplet gaps (in eV).a TDA PBE Compound PhCz NPh3 CBP α-NPD PIC-TRZ DPA-DPS DTPA-DPS ACRFLCN CC2TA DTC-DPS 2CzPN 4CzPN PXZ-TRZ Spiro-CN 4CzIPN 4CzTPN 4CzTPN-Me MADc
B3LYP
EVA(S1) ∆EST ∆EST* EVA(S1) ∆EST ∆EST* EVA(S1) ∆EST ∆EST*
3.55 3.22 2.89 3.29 2.12 2.76 2.68 1.74 2.08 2.52 2.19 1.79 1.34 1.20 1.84 1.61 1.51 0.95
0.48 0.26 0.19 1.11 0.05 0.27 0.25 0.01 0.02 0.13 0.19 0.09 0.03 0.00 0.08 0.10 0.07 0.17
0.50 0.32 0.40 0.01 0.10 0.12 0.09 0.02 0.05 -0.24 -0.37 -0.17 0.04 0.00 -0.12 -0.09 0.05 0.28
3.97 3.73 3.56 3.01 2.88 3.38 3.31 2.46 3.02 3.22 2.77 2.41 2.19 1.95 2.44 2.18 2.12 0.39
M06HF Compound PhCz NPh3 CBP α-NPD PIC-TRZ DPA-DPS DTPA-DPS ACRFLCN CC2TA DTC-DPS 2CzPN 4CzPN PXZ-TRZ
M062X
EVA(S1) ∆EST 4.39 4.23 4.21 3.69 3.82 3.92 3.87 3.48 4.07 3.97 3.58 3.26 3.21
0.67 0.56 0.64 0.62 0.40 0.59 0.58 0.23 0.38 0.39 0.42 0.28 0.06
0.62 0.44 0.44 0.37 0.08 0.47 0.45 0.01 0.06 0.24 0.26 0.15 0.03 0.01 0.12 0.12 0.10 0.10
0.64 0.59 0.70 0.48 0.09 0.32 0.33 0.02 0.06 -0.01 -0.12 -0.10 0.03 0.00 -0.07 -0.04 0.06 0.16
4.39 4.23 4.21 3.69 3.82 3.92 3.87 3.48 4.07 3.97 3.58 3.26 3.21 3.09 3.19 2.96 2.97 0.43
CAM-B3LYP
0.67 0.56 0.64 0.62 0.40 0.59 0.58 0.23 0.38 0.39 0.42 0.28 0.06 0.15 0.16 0.18 0.16 0.09
0.69 0.69 0.83 0.80 0.13 0.53 0.57 0.18 0.13 0.46 0.38 0.35 0.00 0.00 0.25 0.25 0.15 0.10
ωB97XD
∆EST* EVA(S1) ∆EST ∆EST* EVA(S1) ∆EST ∆EST*
0.73 0.74 0.88 1.01 0.55 0.60 0.74 0.86 0.47 0.79 0.86 0.82 0.17
4.38 4.27 4.27 3.74 3.88 3.97 3.92 3.57 4.13 4.04 3.62 3.30 3.34 S7
0.88 0.78 0.91 0.89 0.70 0.76 0.75 0.57 0.70 0.59 0.59 0.40 0.19
0.86 0.91 1.05 1.07 0.19 0.67 0.73 0.49 0.17 0.64 0.52 0.34 0.00
4.42 4.35 4.36 3.82 3.96 4.05 3.99 3.76 4.23 4.17 3.75 3.40 3.55
0.83 0.78 0.90 0.97 0.70 0.76 0.73 0.68 0.71 0.63 0.63 0.40 0.33
0.82 0.90 1.06 1.11 0.32 0.73 0.77 0.59 0.28 0.76 0.60 0.45 0.00
Spiro-CN 4CzIPN 4CzTPN 4CzTPN-Me MADc
3.09 3.19 2.96 2.97 1.04
0.15 0.16 0.18 0.16 0.25
0.66 0.52 0.55 0.28 0.34
3.20 3.25 3.00 3.01 0.49
0.39 0.27 0.21 0.20 0.26
0.03 0.21 0.25 0.16 0.19
3.50 3.34 3.10 3.13 0.61
0.61 0.28 0.23 0.22 0.30
0.12 0.29 0.19 0.31 0.24
a
All the calculations are performed at the PCM(toluene)-TDA/6-31+G(d) level. Geometries of the S1 states are optimized using the implemented TD-DFT gradients at the CAM-B3LYP/6-31G(d) level considering the toluene solvent. Geometries of the T1 states are assessed by spin-relaxed open-shell optimazation at the UCAM-B3LYP/6-31G(d) level considering the toluene solvent. b Experimental data are taken from ref 4. c The MAD values are calculated with respect to the corresponding experimental values.
Table S7. Calculated relaxation energies for lowest singlet excited states λ(S1) and lowest triplet excited states λ(T1). All values are in eV. The labels L, M, and S correspond to the molecules with relatively large, medium, and small singlet-triplet gaps. Compound PhCz NPh3 L CBP α-NPD Mean PIC-TRZ DPA-DPS DTPA-DPS ACRFLCN M CC2TA DTC-DPS 2CzPN 4CzPN Mean PXZ-TRZ Spiro-CN S 4CzIPN 4CzTPN 4CzTPN-Me Mean Mean for all
S8
λ(S1) 0.08 0.05 0.18 0.06 0.09 0.21 0.12 0.16 0.15 0.29 0.19 0.31 0.13 0.20 0.17 0.49 0.10 0.08 0.17 0.20
λ(T1) 0.10 0.17 0.40 0.26 0.23 0.10 0.01 0.06 0.09 0.07 0.07 0.14 0.00 0.07 0.12 0.43 0.00 0.00 0.14 0.14
0.17
0.13
Scheme S1. Hole (green) and electron (blue) distributions of S1 and T1 for CBP (isovalue=0.001) and CC2TA (isovalue=0.0006).
S9
Table S8. Assignments of the nature of the lowest triplet excitations (T1) at the LC-ωPBE* level compared to the experimental ones. Cal(S1) CT PhCz CT NPh3 CT CBP CT α-NPD CT PIC-TRZ CT DPA-DPS CT DTPA-DPS CT ACRFLCN CT CC2TA CT DTC-DPS CT 2CzPN CT 4CzPN CT PXZ-TRZ CT Spiro-CN CT 4CzIPN CT 4CzTPN CT 4CzTPN-Me a From ref 4. b Mixing some CT character.
S10
Cal(T1) LE LE LE LE CT LE LE LE LEb LE LEb CT CT LE CT CT CT
Exp (T1)a LE LE LE LE LE LE LE LE LE LE LE LE CT LE LE CT CT
Figure S1. Histograms of the errors for the calculated singlet excitation energies (EVA(S1)) compared to experimental data.
S11
Figure S2. Histograms of the errors for the calculated vertical singlet-triplet gaps (∆EST) compared to experimental data.
S12
Figure S3. Histograms of the errors for the calculated adiabatic singlet-triplet gaps (∆EST*) compared to experimental data.
S13
Figure S4. Linear correlation coefficient R2 for the EVA(S1) energies between theory and experiment.
S14
Figure S5. Linear correlation coefficient R2 for the vertical ∆EST gaps between theory and experiment.
S15
Figure S6. Linear correlation coefficient R2 for the adiabatic ∆EST* gaps between theory and experiment.
S16
Full names of the molecules investigated in this work. PhCz: phenylcarbazole; CBP: 4,4'-bis(carbazol-9-yl)-p-biphenyl; NPh3: triphenylamine; α-NPD: N,N'-diphenyl-N,N'-bis(1-naphthyl)-1,10-biphenyl-4,4'-diamine; PIC-TRZ:12,12'-(6-([1,1'-biphenyl]-4-yl)-1,3,5-triazine-2,4-diyl)bis(11-phenyl-11,12 -dihydroindolo[2,3-a]carbazole); DTC-DPS: 4,4'-sulfonylbis(N,N-bis(4-(tert-butyl)phenyl)aniline); Spiro-CN:2',7'-bis(di-p-tolylamino)-9,9'-spirobi[fluorene]-2,7-dicarbonitrile; DPA-DPS: 4,4'-sulfonylbis(N,N-diphenylaniline); DTPA-DPS: 4,4'-sulfonylbis(N,N-bis(4-(tert-butyl)phenyl)aniline); ACRFLCN:10-phenyl-10H-spiro[acridine-9,9'-fluorene]-2',7'-dicarbonitrile; CC2TA: 9,9''-(6-phenyl-1,3,5-triazine-2,4-diyl)bis((9H-3,9'-bicarbazole)); PXZ-TRZ: 10-(4-(4,6-diphenyl-1,3,5-triazin-2-yl)phenyl)-10H-phenoxazine 2CzPN: 4,5-di(9H-carbazol-9-yl)phthalonitrile ; 4CzIPN: 2,4,5,6-tetra(9H-carbazol-9-yl)isophthalonitrile; 4CzTPN: 2,3,5,6-tetra(9H-carbazol-9-yl)terephthalonitrile; 4CzTPN-Me: 2,3,5,6-tetrakis(3,6-dimethyl-9H-carbazol-9-yl)terephthalonitrile.
S17
References (1) Brent, R. P. (1973), "Chapter 4: An Algorithm with Guaranteed Convergence for Finding a Zero of a Function", Algorithms for Minimization without Derivatives, Englewood Cliffs, NJ: Prentice-Hall, ISBN 0-13-022335-2. (2) Moral, M.; Muccioli, L.; Son, W. J.; Olivier, Y.; Sancho-García, J. C. J. Chem. Theory Comput., 2015, 11, 168-177. (3) Ahlrichs, R.; Bär, M.; Häser, M.; Horn, H.; Kölmel, C. Chem. Phys. Letters, 1989, 162, 165-169. (4) Huang, S.; Zhang, Q.; Shiota, Y.; Nakagawa, T.; Kuwabara, K.; Yoshizawa, K.; Adachi, C. J. Chem. Theory. Comput., 2013, 9, 3872-3877.
S18
Haitao Sun, Cheng Zhong, and Jean-Luc Brédas*
Physical Science and Engineering Division Solar & Photovoltaics Engineering Research Center King Abdullah University of Science and Technology (KAUST) Thuwal 23955-6900, Kingdom of Saudi Arabia
Corresponding author: [email protected]
S1
Content
1. Introduction of a “Golden proportion” method
S3
2. Table S1. Optimally-tuned ω* values using 6-31G(d) and 6-31+G(d)
S4
3. Table S2. (a) Calculated EVA(S1) values using PCM based on two different Methods; (b) Basis set effects on the calculated EVA(S1) values
S4
4. Table S3. Calculated EVA(S1) and singlet-triplet gap ∆EST using full TDDFT and TDA
S5
5. Table S4. Calculated vertical excitation energies based on the ground-state geometries optimized by LC-ωPBE*, B3LYP, and CAM-B3LYP functionals
S6
6. Table S5. Calculated EVA(S1) for PhCz and NPh3 using high-level CC2 and B2(GP)-PLYP methods
S6
7. Table S5. Calculated EVA(S1) and singlet-triplet gap ∆EST and ∆EST* using other density functionals
S7
8. Table S6. Calculated relaxation energies of λ(S1) and λ(T1)
S8
Scheme S1. Hole distribution of S1 and T1 for CBP and DPA-DPS
S9
9. Table S7. Assignment for the lowest triplet excitation (T1) compared to the experimental ones
S10
10. Figures S1-S3. Histograms of the errors of calculated EVA(S1), ∆EST and ∆EST*, S11-S13
respectively, compared to experimental data
11. Figures S4-S6. Linear correlation coefficient R2 for calculated EVA(S1), ∆EST, and ∆EST*, respectively, between theory and experiment
S14-S16
12. Full names of the molecules investigated in this work
S17
13. References
S18
S2
Introduction of a “Golden proportion” method in the search for the optimal ω value The minimization of J2 in Equation 2b is usually carried out by a fitting method, as illustrated in the left scheme below. Take the PhCz molecule as an example. First evaluations of J2 for ω values from 0.05 to 0.50 Bohr-1 (with intervals of 0.05) are performed. Then, to obtain a more accurate ω value, a finer interval of 0.01 is considered at the bottom of the fitting curve from 0.16 to 0.24. In this case, one needs to perform calculations with at least 20 different ω values in order to obtain the optimal ω with a resolution of 0.01 Bohr-1. The optimal ω using this fitting method is 0.20 Bohr-1. In this work, we have implemented the “Golden proportion” method based on Brent’s algorithm in Python.1 We employed this method to calculate more precise ω values while having to take account of a lesser number of single-point calculations than with the fitting method, as shown in the right scheme below. We have again chosen initial ω values in the range 0.05 ~ 0.50 Bohr-1. First, the “Golden proportion” method generates an initial ω1 value at 0.2219 [=0.50-(0.50-0.05)*0.618] according to the golden ratio (r = 0.618) and the corresponding J2(1). Next, it generates another ω2 at 0.3281 calculated through [=0.50-(0.50-0.2219)*0.618] and J2(2). If J2(2) < J2(1), it searches in the range [ω2, 0.50]; if J2(2) > J2(1), it searches in the opposite direction [0.05, ω2]. This process is repeated until minimization of J2(ω), which corresponds to the optimal ω, is reached. In this way, an optimal ω of 0.2036 is obtained in fewer than 10 steps, with higher precision.
S3
Table S1. Optimally-tuned ω* values for the LC-ωPBE functional with 6-31G(d) and 6-31+G(d) basis sets. Optimal ω* values Compounds
6-31G(d)
6-31+G(d)
PhCz NPh3 CBP α-NPD PIC-TRZ DPA-DPS DTPA-DPS ACRFLCN CC2TA DTC-DPS 2CzPN 4CzPN PXZ-TRZ Spiro-CN 4CzIPN 4CzTPN 4CzTPN-Me
0.2186 0.2135 0.1846 0.1804 0.1485 0.1762 0.1616 0.1835 0.1699 0.1701 0.1866 0.1536 0.1931 0.1759 0.1495 0.1552 0.1501
0.2036 0.1977 0.1734 0.1701 0.1411 0.1663 0.1524 0.1742 0.1598 0.1601 0.1758 0.1455 0.1828 0.1679 0.1422 0.1472 0.1426
Table S2. (a) Calculated EVA(S1) values using PCM(toluene)-TDA-LC-ωPBE* based on two different schemes. EVA(S1) PCM PCM Exp. Value in eV Linear-Response State-Specific (SS) PhCz 4.17 4.17 3.66 NPh3 3.99 3.96 3.74 α-NPD 3.29 Not converged 3.31 ACRFLCN 3.03 3.03 3.05 2CzPN 3.22 3.20 3.19 Spiro-CN 2.73 2.80 2.69 (b) Calculated EVA(S1) values using PCM(toluene)-TDA-LC-ωPBE* with different basis sets. All values are in eV. 6-31G(d) PhCz CBP Spiro-CN ACRFLCN DTC-DPS
4.25 4.00 2.78 3.07 3.67
6-311G(d) 6-31+G(d) 6-311+G(d) 6-311++G(d,p) 4.19 3.96 2.76 3.06 3.62
4.17 3.94 2.73 3.03 3.60 S4
4.15 3.93 2.71 3.03 3.59
4.13 3.93 2.70 3.03 -
Exp. 3.66 3.80 2.69 3.05 3.62
Table S3. Calculated EVA(S1) values and vertical singlet-triplet gap ∆EST and adiabatic ∆EST* using the full TDDFT and TDA at the PCM(toluene)-TDA-DFT/6-31+G(d) level, in comparison to the experimental data. All values are in eV.
Compound
TD DFT
TDA
LC-ωPBE*
LC-ωPBE*
EVA(S1)
PhCz 4.08 NPh3 3.93 CBP 3.89 α-NPD 3.25 PIC-TRZ 3.15 DPA-DPS 3.54 DTPA-DPS 3.42 ACRFLCN 3.03 CC2TA 3.63 DTC-DPS 3.56 2CzPN 3.18 4CzPN 2.53 PXZ-TRZ 2.93 Spiro-CN 2.72 4CzIPN 2.51 4CzTPN 2.30 4CzTPN-Me 2.26 a MAD 0.15 a The MAD values are values.
∆EST
∆EST* EVA(S1) ∆EST ∆EST*
Exp. values
EVA(S1)
∆EST(00)
0.85 0.82 4.17 0.73 0.75 3.66 0.55 0.74 0.87 3.98 0.61 0.74 3.74 0.57 0.79 1.02 3.94 0.66 0.88 3.80 0.71 0.70 0.93 3.29 0.57 0.76 3.31 0.73 0.26 0.10 3.17 0.22 0.11 3.35 0.18 0.65 0.55 3.60 0.60 0.49 3.53 0.52 0.60 0.51 3.48 0.56 0.46 3.47 0.46 0.28 0.23 3.03 0.07 0.02 3.05 0.24 0.46 0.13 3.66 0.35 0.13 3.64 0.20 0.44 0.38 3.60 0.39 0.27 3.62 0.36 0.48 0.38 3.22 0.41 0.24 3.19 0.31 0.13 0.05 2.55 0.13 0.00 2.82 0.15 0.12 0.01 2.94 0.07 0.01 2.73 0.06 0.21 0.01 2.73 0.07 0.01 2.69 0.06 0.12 0.03 2.52 0.12 0.01 2.85 0.10 0.13 0.06 2.32 0.14 0.06 2.61 0.09 0.11 0.08 2.28 0.12 0.09 2.49 0.09 0.11 0.10 0.15 0.07 0.09 calculated with respect to the corresponding experimental
S5
Table S4. Calculated vertical excitation energies (in eV) of lowest singlet excited state EVA(S1), vertical ∆EST and adiabatic ∆EST* based on the ground-state geometries optimized by LC-ωPBE*, B3LYP, and CAM-B3LYP functionals.
TDA-LC-ωPBE*
TDA-LC-ωPBE*
TDA-LC-ωPBE*
@LC-ωPBE*
@B3LYP
@CAM-B3LYP
Exp. Val.
Compound EVA(S1) ∆EST ∆EST* EVA(S1) ∆EST ∆EST* EVA(S1) ∆EST ∆EST* EVA(S1)∆EST(00) PhCz NPh3 CBP α-NPD ACRFLN PXZ-TRZ 2CzPN 4CzTPN MADa
4.17 3.96 3.95 3.27 3.03 2.93 3.21 2.32 0.18
0.74 0.61 0.65 0.57 0.07 0.07 0.42 0.13 0.10
0.75 0.74 0.88 0.76 0.02 0.01 0.24 0.06 0.12
4.17 3.98 3.94 3.29 3.03 2.94 3.22 2.32 0.18
0.73 0.61 0.66 0.57 0.07 0.07 0.41 0.14 0.08
0.75 0.74 0.88 0.76 0.02 0.01 0.24 0.06 0.12
4.21 3.99 4.00 3.34 3.09 2.96 3.26 2.37 0.20
0.73 0.47 0.64 0.55 0.05 0.02 0.40 0.15 0.11
0.75 0.74 0.88 0.76 0.02 0.01 0.24 0.06 0.12
3.66 3.74 3.80 3.31 3.05 2.73 3.19 2.61
0.55 0.57 0.71 0.73 0.24 0.06 0.31 0.09
a
The MAD values are calculated with respect to the corresponding experimental values.
Table S5. Calculated vertical excitation energies of the lowest singlet excited state EVA(S1) for PhCz and NPh3 at different levels. All values are in eV. PhCz NPh3 TDA-LC-ωPBE*/6-31+G(d) 4.17 3.98 TDA-B2-PLYP/def2-TZVPa 3.98 3.94 a TDA-B2GP-PLYP/def2-TZVP 4.15 4.13 b RI-CC2//def2-TZVP/SVP 3.98 3.94 Experiment 3.66 3.74 a 2 b From ref . ‘Resolution-of-identity’ RI-CC2 calculations were carried out with Turbomole,3 using the def2-TZVP basis set and SVP auxiliary basis set; the solvent model is not included in these calculation since the solvent effect has been shown to be very limited.2
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Table S6. Calculated EVA(S1) (in eV) and both vertical ∆EST and adiabatic ∆EST* singlet-triplet gaps (in eV).a TDA PBE Compound PhCz NPh3 CBP α-NPD PIC-TRZ DPA-DPS DTPA-DPS ACRFLCN CC2TA DTC-DPS 2CzPN 4CzPN PXZ-TRZ Spiro-CN 4CzIPN 4CzTPN 4CzTPN-Me MADc
B3LYP
EVA(S1) ∆EST ∆EST* EVA(S1) ∆EST ∆EST* EVA(S1) ∆EST ∆EST*
3.55 3.22 2.89 3.29 2.12 2.76 2.68 1.74 2.08 2.52 2.19 1.79 1.34 1.20 1.84 1.61 1.51 0.95
0.48 0.26 0.19 1.11 0.05 0.27 0.25 0.01 0.02 0.13 0.19 0.09 0.03 0.00 0.08 0.10 0.07 0.17
0.50 0.32 0.40 0.01 0.10 0.12 0.09 0.02 0.05 -0.24 -0.37 -0.17 0.04 0.00 -0.12 -0.09 0.05 0.28
3.97 3.73 3.56 3.01 2.88 3.38 3.31 2.46 3.02 3.22 2.77 2.41 2.19 1.95 2.44 2.18 2.12 0.39
M06HF Compound PhCz NPh3 CBP α-NPD PIC-TRZ DPA-DPS DTPA-DPS ACRFLCN CC2TA DTC-DPS 2CzPN 4CzPN PXZ-TRZ
M062X
EVA(S1) ∆EST 4.39 4.23 4.21 3.69 3.82 3.92 3.87 3.48 4.07 3.97 3.58 3.26 3.21
0.67 0.56 0.64 0.62 0.40 0.59 0.58 0.23 0.38 0.39 0.42 0.28 0.06
0.62 0.44 0.44 0.37 0.08 0.47 0.45 0.01 0.06 0.24 0.26 0.15 0.03 0.01 0.12 0.12 0.10 0.10
0.64 0.59 0.70 0.48 0.09 0.32 0.33 0.02 0.06 -0.01 -0.12 -0.10 0.03 0.00 -0.07 -0.04 0.06 0.16
4.39 4.23 4.21 3.69 3.82 3.92 3.87 3.48 4.07 3.97 3.58 3.26 3.21 3.09 3.19 2.96 2.97 0.43
CAM-B3LYP
0.67 0.56 0.64 0.62 0.40 0.59 0.58 0.23 0.38 0.39 0.42 0.28 0.06 0.15 0.16 0.18 0.16 0.09
0.69 0.69 0.83 0.80 0.13 0.53 0.57 0.18 0.13 0.46 0.38 0.35 0.00 0.00 0.25 0.25 0.15 0.10
ωB97XD
∆EST* EVA(S1) ∆EST ∆EST* EVA(S1) ∆EST ∆EST*
0.73 0.74 0.88 1.01 0.55 0.60 0.74 0.86 0.47 0.79 0.86 0.82 0.17
4.38 4.27 4.27 3.74 3.88 3.97 3.92 3.57 4.13 4.04 3.62 3.30 3.34 S7
0.88 0.78 0.91 0.89 0.70 0.76 0.75 0.57 0.70 0.59 0.59 0.40 0.19
0.86 0.91 1.05 1.07 0.19 0.67 0.73 0.49 0.17 0.64 0.52 0.34 0.00
4.42 4.35 4.36 3.82 3.96 4.05 3.99 3.76 4.23 4.17 3.75 3.40 3.55
0.83 0.78 0.90 0.97 0.70 0.76 0.73 0.68 0.71 0.63 0.63 0.40 0.33
0.82 0.90 1.06 1.11 0.32 0.73 0.77 0.59 0.28 0.76 0.60 0.45 0.00
Spiro-CN 4CzIPN 4CzTPN 4CzTPN-Me MADc
3.09 3.19 2.96 2.97 1.04
0.15 0.16 0.18 0.16 0.25
0.66 0.52 0.55 0.28 0.34
3.20 3.25 3.00 3.01 0.49
0.39 0.27 0.21 0.20 0.26
0.03 0.21 0.25 0.16 0.19
3.50 3.34 3.10 3.13 0.61
0.61 0.28 0.23 0.22 0.30
0.12 0.29 0.19 0.31 0.24
a
All the calculations are performed at the PCM(toluene)-TDA/6-31+G(d) level. Geometries of the S1 states are optimized using the implemented TD-DFT gradients at the CAM-B3LYP/6-31G(d) level considering the toluene solvent. Geometries of the T1 states are assessed by spin-relaxed open-shell optimazation at the UCAM-B3LYP/6-31G(d) level considering the toluene solvent. b Experimental data are taken from ref 4. c The MAD values are calculated with respect to the corresponding experimental values.
Table S7. Calculated relaxation energies for lowest singlet excited states λ(S1) and lowest triplet excited states λ(T1). All values are in eV. The labels L, M, and S correspond to the molecules with relatively large, medium, and small singlet-triplet gaps. Compound PhCz NPh3 L CBP α-NPD Mean PIC-TRZ DPA-DPS DTPA-DPS ACRFLCN M CC2TA DTC-DPS 2CzPN 4CzPN Mean PXZ-TRZ Spiro-CN S 4CzIPN 4CzTPN 4CzTPN-Me Mean Mean for all
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λ(S1) 0.08 0.05 0.18 0.06 0.09 0.21 0.12 0.16 0.15 0.29 0.19 0.31 0.13 0.20 0.17 0.49 0.10 0.08 0.17 0.20
λ(T1) 0.10 0.17 0.40 0.26 0.23 0.10 0.01 0.06 0.09 0.07 0.07 0.14 0.00 0.07 0.12 0.43 0.00 0.00 0.14 0.14
0.17
0.13
Scheme S1. Hole (green) and electron (blue) distributions of S1 and T1 for CBP (isovalue=0.001) and CC2TA (isovalue=0.0006).
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Table S8. Assignments of the nature of the lowest triplet excitations (T1) at the LC-ωPBE* level compared to the experimental ones. Cal(S1) CT PhCz CT NPh3 CT CBP CT α-NPD CT PIC-TRZ CT DPA-DPS CT DTPA-DPS CT ACRFLCN CT CC2TA CT DTC-DPS CT 2CzPN CT 4CzPN CT PXZ-TRZ CT Spiro-CN CT 4CzIPN CT 4CzTPN CT 4CzTPN-Me a From ref 4. b Mixing some CT character.
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Cal(T1) LE LE LE LE CT LE LE LE LEb LE LEb CT CT LE CT CT CT
Exp (T1)a LE LE LE LE LE LE LE LE LE LE LE LE CT LE LE CT CT
Figure S1. Histograms of the errors for the calculated singlet excitation energies (EVA(S1)) compared to experimental data.
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Figure S2. Histograms of the errors for the calculated vertical singlet-triplet gaps (∆EST) compared to experimental data.
S12
Figure S3. Histograms of the errors for the calculated adiabatic singlet-triplet gaps (∆EST*) compared to experimental data.
S13
Figure S4. Linear correlation coefficient R2 for the EVA(S1) energies between theory and experiment.
S14
Figure S5. Linear correlation coefficient R2 for the vertical ∆EST gaps between theory and experiment.
S15
Figure S6. Linear correlation coefficient R2 for the adiabatic ∆EST* gaps between theory and experiment.
S16
Full names of the molecules investigated in this work. PhCz: phenylcarbazole; CBP: 4,4'-bis(carbazol-9-yl)-p-biphenyl; NPh3: triphenylamine; α-NPD: N,N'-diphenyl-N,N'-bis(1-naphthyl)-1,10-biphenyl-4,4'-diamine; PIC-TRZ:12,12'-(6-([1,1'-biphenyl]-4-yl)-1,3,5-triazine-2,4-diyl)bis(11-phenyl-11,12 -dihydroindolo[2,3-a]carbazole); DTC-DPS: 4,4'-sulfonylbis(N,N-bis(4-(tert-butyl)phenyl)aniline); Spiro-CN:2',7'-bis(di-p-tolylamino)-9,9'-spirobi[fluorene]-2,7-dicarbonitrile; DPA-DPS: 4,4'-sulfonylbis(N,N-diphenylaniline); DTPA-DPS: 4,4'-sulfonylbis(N,N-bis(4-(tert-butyl)phenyl)aniline); ACRFLCN:10-phenyl-10H-spiro[acridine-9,9'-fluorene]-2',7'-dicarbonitrile; CC2TA: 9,9''-(6-phenyl-1,3,5-triazine-2,4-diyl)bis((9H-3,9'-bicarbazole)); PXZ-TRZ: 10-(4-(4,6-diphenyl-1,3,5-triazin-2-yl)phenyl)-10H-phenoxazine 2CzPN: 4,5-di(9H-carbazol-9-yl)phthalonitrile ; 4CzIPN: 2,4,5,6-tetra(9H-carbazol-9-yl)isophthalonitrile; 4CzTPN: 2,3,5,6-tetra(9H-carbazol-9-yl)terephthalonitrile; 4CzTPN-Me: 2,3,5,6-tetrakis(3,6-dimethyl-9H-carbazol-9-yl)terephthalonitrile.
S17
References (1) Brent, R. P. (1973), "Chapter 4: An Algorithm with Guaranteed Convergence for Finding a Zero of a Function", Algorithms for Minimization without Derivatives, Englewood Cliffs, NJ: Prentice-Hall, ISBN 0-13-022335-2. (2) Moral, M.; Muccioli, L.; Son, W. J.; Olivier, Y.; Sancho-García, J. C. J. Chem. Theory Comput., 2015, 11, 168-177. (3) Ahlrichs, R.; Bär, M.; Häser, M.; Horn, H.; Kölmel, C. Chem. Phys. Letters, 1989, 162, 165-169. (4) Huang, S.; Zhang, Q.; Shiota, Y.; Nakagawa, T.; Kuwabara, K.; Yoshizawa, K.; Adachi, C. J. Chem. Theory. Comput., 2013, 9, 3872-3877.
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