Supporting information Electronic structure properties of two ...
Supporting information
Electronic structure properties of two-dimensional π-conjugated polymers
Jing Wen†, Ding Luo†, Lin Chengǂ, Kun Zhaoǂ, Haibo Ma†
†
Key Laboratory of Mesoscopic Chemistry of MOE, Collaborative Innovation Center of
Chemistry for Life Sciences, School of Chemistry and Chemical Engineering, Nanjing University, No. 163 Xianlin Avenue, Nanjing 210023, China
ǂ
State Grid Electric Power Research Institute, Wuhan 430074, China
KEYWORDS: density functional theory, two-dimensional, conjugated polymer, HOMO-LUMO gap, frontier molecular orbital.
S1. Methodology tests
I.
Tests for four popular DFT functionals and semi-empirical DFTB
It is well-known that results by density functional theory (DFT) calculations are usually very sensitive to the chosen DFT exchange-correlation functionals. Herein, in order to evaluate the effect of chosen functional for our 2D π-conjugated micro-porous systems, we first performed a benchmark test for calculations by four different DFT functionals as well as a semi-empirical method density-functional based tight-binding (DFTB)
S1
. The tested four DFT functionals include B3LYP
s2
hybrid functional, long-range corrected
functionals (CAM-B3LYP s3, wB97xd s4) and meta-hybrid M06-2X
s5
functionals. Benchmark comparisons
were performed for systems 1.1 and 1.2, and the results of highest occupied molecular orbital (HOMO)-lowest unoccupied molecular orbital (LUMO) gap (HLG), ionization potential (IP) and electron affinity (EA) are shown in Figure S1.
From Figure S1, one may clearly notice that, the most popular B3LYP functional significantly underestimates the values of HLGs when compared with DFT calculations by other functionals, as found in many other theoretical studies
S4, S6-S9
. This indicates again that earlier DFT results by (semi-)local and
standard hybrid functions for 2D conjugated polymers should be carefully revisited by newly developed long-range corrected functionals or other state-of-art functionals. CAM-B3LYP and M06-2X obtained the most similar variation curves for HLGs, while wB97xd predicted HLGs slightly higher than them. For IP and EA, CAM-B3LYP, M06-2x and wB97xd also predicted semi-quantitatively similar changing trends with the increasing oligomer size. Therefore, the long-range corrected CAM-B3LYP functionals is chosen for the calculations of all other systems in the main text of the manuscript. At the same time, it is also shown that the economic semi-empirical DFTB method (using default parameters in G09 software package) also significantly underestimated HLGs and IPs, and overestimated EAs, like B3LYP. Therefore, it can be also noticed that, the semi-empirical DFTB method should be carefully used in conjugated systems although it has significant computational efficiency advantages for large systems.
Our time-dependent density functional theory (TDDFT) calculation with the functional of CAM-B3LYP predicted an extrapolated ( = ∞) optical gap of 4.86 eV for single-chain polymer 1.1, which is in reasonable agreements with the experimental value of 4.0 eV in the solid state S10. This implies again that
functionals like CAM-B3LYP can be used for the (semi-)quantitative investigation of large conjugated systems. In fact, remarkable differences between calculations by B3LYP and those by long-range corrected functionals like CAM-B3LYP have been widely noticed for π-conjugated polymers or oligomers S11-S14 . For example, the slope of the symmetric C=C stretching frequency of oligofurans as a function of inverse length (1/n) is experimentally determined to be 160 cm−1 while it was calculated as 149 and 203 cm−1 for CAM-B3LYP and B3LYP respectively.S11 The polaron binding energy of a model polymer (poly[methyl(phenyl)silylene]) by CAM-B3LYP calculation is 0.23 eV, close to the experimental value 0.29 eV, while B3LYP calculation gives a significantly lower prediction of 0.13 eV.S12 Therefore, long-range corrections or meta-hybrid improvements are highly desired to be used for DFT investigations for both 1D and 2D large conjugated systems.
HLG (eV)
8
IP (eV)
EA (eV)
8
2
1.1 B3LYP 1.2 B3LYP 1.1 DFTB 1.2 DFTB
6
1
7
1.1 CAM-B3LYP 1.2 CAM-B3LYP
0
4 0.0
1.1 wB97xd 1.2 wB97xd
0.5
1/n
6
1.0 0.0
1.1 M062x 1.2 M062x
0.5
1/n
1.0 0.0
0.5
1/n
1.0
Figure S1. The variations of HLG, IP, and EA of oligomers 1.1 and 1.2 calculated with different methods.
II.
Effect of diffused function in basis set for electron affinity.
Calculation of EA (or the energy of anion) usually requires the usage of a diffused basis set. Therefore in Figure S2 we plotted the EAs by CAM-B3LYP calculations with 6-31+G(d) diffused basis set against those by 6-31G(d) basis set. The excellent linear correlations between the EAs by 6-31G(d) and those by 6-31+G(d) were shown to exist in both 1D and 2D oligomers of both system 1 and 3. This implies that both basis sets can give very similar descriptions for the EAs in conjugated polymers. The slopes of the fitting lines of four groups (1D and 2D oligomers of system 1 and 3) are 0.9488±0.0132, 0.9810±0.0325,
0.9072±0.0013, and 0.8972±0.0089 respectively, and the intercepts are 0.4898±0.0060 eV, 0.5087±0.0145 eV, 0.4424±0.0009 eV, and 0.4329±0.0084 eV respectively. This indicates again that adding a further diffused function to 6-31G(d) basis set just generates a monotonic shift for calculated EA values of all oligomers investigated, and it will not present any qualitative change to our discussion about EA in the main body of our manuscript. For this reason and also in consideration of a good compromise between accuracy and computational efficiency, we adopted 6-31G(d) basis set in our systematic electronic structure
EA (eV) (6-31+G(d))
calculations for anion states.
2
1
1-1D 1-2D 3-1D 3-2D
0 -1
0 1 EA (eV) (6-31G(d))
2
Figure S2. EAs calculated with CAM-B3LYP/6-31G(d) and CAM-B3LYP/6-31+G(d).
εHOMO / eV
-7.0
1.1 1.2
-7.1 -7.2
εLUMO / eV
S2. Energy variations of HOMOs and LUMOs with the increasing oligomer size
0.3 0.0 1.1 1.2
-0.3 -0.6
0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.2 0.4 0.6 0.8 1.0
1/n
1/n 2.1 2.2
-2.0 εLUMO / eV
εHOMO / eV
-5.1 -5.4 -5.7
-6.5 0.0 0.2 0.4 0.6 0.8 1.0 1/n
εLUMO / eV
εHOMO / eV
-6.0
2.1 2.2
-3.0 0.0 0.2 0.4 0.6 0.8 1.0 1/n 0
0.0 0.2 0.4 0.6 0.8 1.0 1/n -5.5 3.1 3.2
-2.5
-1 3.1 3.2
-2 0.0 0.2 0.4 0.6 0.8 1.0 1/n
Figure S3. Energies of HOMO (ε ) and LUMO (ε ) plotted with respect to inverse size (1/n).
S3. DOS for oligomers 1-3
Figure S4. DOS for oligomers 1-3 with largest calculated size.
S4. Solution of Hückel model for 1D systems.
Figure S5. Population distribution of frontier molecular orbitals by a solution of Hückel model for oligomer 1.1 and 3.1.
Figure S6. Fragmentation of monomers in oligomers 2.1 and 3.1.
Table S1. Frontier orbital energies for fragments in oligomers 2.1 and 3.1.
ε /eV
-2.35
1.48
2.06
ε /eV
-6.18
-8.18
-8.92
In order to see the effect of the fluctuation of fragment energies on the delocalization of the wavefunction, we built a model Hückel Hamiltonian (a 18×18 matrix, as shown in Figure S7) in which the off-diagonal terms are approximated as constants and the diagonal elements (αA and αB) have fluctuations. We tuned the value of energy difference △α between αA and αB and the results are depicted in the right panel of Figure S7 with
=0, 0.2, 1.0 respectively. Apparently, the delocalization extent became suppressed with the
increasing fluctuation of diagonal terms △α. This explains the distribution difference of the frontier molecular orbitals in oligomers 2.1 and 3.1 (Figure 2 in the main body of the manuscript).
Figure S7. Schematic illustration of the model Hückel Hamiltonian (left panel) and the population distribution of frontier molecular orbitals from its numerical solution (right panel).
References (S1) Elstner, M.; Porezag, D.; Jungnickel, G.; Elsner, J.; Haugk, M.; Frauenheim, T.; Suhai, S.; Seifert, G. Self-consistent-charge density-functional tight-binding method for simulations of complex materials properties. Phys. Rev. B 1998, 58, 7260-7268. (S2) Hertwig, R. H.; Koch, W. On the parameterization of the local correlation functional. What is Becke-3-LYP? Chem. Phys. Lett. 1997, 268, 345-351. (S3) Yanai, T.; Tew, D.; Handy, N. C. A new hybrid exchange–correlation functional using the Coulomb-attenuating method (CAM-B3LYP). Chem. Phys. Lett. 2004, 393, 51-57. (S4) Chai, J.-D.; Head-Gordon, M. Long-range corrected hybrid density functionals with damped atom–atom dispersion corrections. Phys. Chem. Chem. Phys. 2008, 10, 6615-6620. (S5) Zhao, Y.; Truhlar, D. G. The M06 suite of density functionals for main group thermochemistry, thermochemical kinetics, noncovalent interactions, excited states, and transition elements: two new functionals and systematic testing of four M06-class functionals and 12 other functional. Theor. Chem. Acc. 2008, 120, 215-241. (S6) Mardirossian, N.; Parkhill, J. A.; Head-Gordon, M. Benchmark results for empirical post-GGA functionals: Difficult exchange problems and independent tests. Phys. Chem. Chem. Phys. 2011, 13, 19325-19337. (S7) Körzdörfer, T.; Parrish, R. M.; Marom, N.; Sears, J. S.; Sherrill, C. D.; Brédas, J.-L Assessment of the performance of tuned range-separated hybrid density functionals in predicting accurate quasiparticle spectra. Phys. Rev. B 2012, 86, 205110. (S8) Adamo, C.; Jacquemin, D. The calculations of excited-state properties with Time-Dependent Density Functional Theory. Chem. Soc. Rev, 2013, 42, 845-856. (S9) Salzner, U.; Aydin, A. Improved Prediction of Properties of π-Conjugated Oligomers with Range-Separated Hybrid Density Functionals. J. Chem. Theory Comput. 2011, 7, 2568-2583. (S10) Musfeldt, J. L.; Reynolds, J. R.; Tanner, D. B.; Ruiz, J. P.; Wang, J.; Pomerantz, M. Luminescent polymers with discrete emitter units. J. Polym. Sci., B: Polym. Phys. 1994, 32, 2395−2404. (S11) Donohoo-Vallett,
P. J.; Bragg, A. E. π‑Delocalization and the Vibrational Spectroscopy of Conjugated Materials:
Computational Insights on Raman Frequency Dispersion in Thiophene, Furan, and Pyrrole Oligomers. J. Phys. Chem. B 2015, 119, 3583-3594 (S12) Nožár, J.; Nešpůrek, S.; Šebera, Jakub. Polaron binding energy in polymers: poly[methyl(phenyl)silylene]. J. Mol. Model. 2012, 18, 623-629. (S13) Jacquemin, D.; Perpète, E. A.; Scalmani, G.; Frisch, M. J.; Kobayashi, R.; Adamo, C. Assessment of the efficiency of long-range corrected functionals for some properties of large compounds. J. Chem. Phys. 2007, 126, 144105. (S14) Krawczyk, P. DFT study of linear and nonlinear optical properties of donor-acceptor substituted stilbenes, azobenzenes and benzilideneanilines. J. Mol. Model. 2010, 16, 659-668 .
Electronic structure properties of two-dimensional π-conjugated polymers
Jing Wen†, Ding Luo†, Lin Chengǂ, Kun Zhaoǂ, Haibo Ma†
†
Key Laboratory of Mesoscopic Chemistry of MOE, Collaborative Innovation Center of
Chemistry for Life Sciences, School of Chemistry and Chemical Engineering, Nanjing University, No. 163 Xianlin Avenue, Nanjing 210023, China
ǂ
State Grid Electric Power Research Institute, Wuhan 430074, China
KEYWORDS: density functional theory, two-dimensional, conjugated polymer, HOMO-LUMO gap, frontier molecular orbital.
S1. Methodology tests
I.
Tests for four popular DFT functionals and semi-empirical DFTB
It is well-known that results by density functional theory (DFT) calculations are usually very sensitive to the chosen DFT exchange-correlation functionals. Herein, in order to evaluate the effect of chosen functional for our 2D π-conjugated micro-porous systems, we first performed a benchmark test for calculations by four different DFT functionals as well as a semi-empirical method density-functional based tight-binding (DFTB)
S1
. The tested four DFT functionals include B3LYP
s2
hybrid functional, long-range corrected
functionals (CAM-B3LYP s3, wB97xd s4) and meta-hybrid M06-2X
s5
functionals. Benchmark comparisons
were performed for systems 1.1 and 1.2, and the results of highest occupied molecular orbital (HOMO)-lowest unoccupied molecular orbital (LUMO) gap (HLG), ionization potential (IP) and electron affinity (EA) are shown in Figure S1.
From Figure S1, one may clearly notice that, the most popular B3LYP functional significantly underestimates the values of HLGs when compared with DFT calculations by other functionals, as found in many other theoretical studies
S4, S6-S9
. This indicates again that earlier DFT results by (semi-)local and
standard hybrid functions for 2D conjugated polymers should be carefully revisited by newly developed long-range corrected functionals or other state-of-art functionals. CAM-B3LYP and M06-2X obtained the most similar variation curves for HLGs, while wB97xd predicted HLGs slightly higher than them. For IP and EA, CAM-B3LYP, M06-2x and wB97xd also predicted semi-quantitatively similar changing trends with the increasing oligomer size. Therefore, the long-range corrected CAM-B3LYP functionals is chosen for the calculations of all other systems in the main text of the manuscript. At the same time, it is also shown that the economic semi-empirical DFTB method (using default parameters in G09 software package) also significantly underestimated HLGs and IPs, and overestimated EAs, like B3LYP. Therefore, it can be also noticed that, the semi-empirical DFTB method should be carefully used in conjugated systems although it has significant computational efficiency advantages for large systems.
Our time-dependent density functional theory (TDDFT) calculation with the functional of CAM-B3LYP predicted an extrapolated ( = ∞) optical gap of 4.86 eV for single-chain polymer 1.1, which is in reasonable agreements with the experimental value of 4.0 eV in the solid state S10. This implies again that
functionals like CAM-B3LYP can be used for the (semi-)quantitative investigation of large conjugated systems. In fact, remarkable differences between calculations by B3LYP and those by long-range corrected functionals like CAM-B3LYP have been widely noticed for π-conjugated polymers or oligomers S11-S14 . For example, the slope of the symmetric C=C stretching frequency of oligofurans as a function of inverse length (1/n) is experimentally determined to be 160 cm−1 while it was calculated as 149 and 203 cm−1 for CAM-B3LYP and B3LYP respectively.S11 The polaron binding energy of a model polymer (poly[methyl(phenyl)silylene]) by CAM-B3LYP calculation is 0.23 eV, close to the experimental value 0.29 eV, while B3LYP calculation gives a significantly lower prediction of 0.13 eV.S12 Therefore, long-range corrections or meta-hybrid improvements are highly desired to be used for DFT investigations for both 1D and 2D large conjugated systems.
HLG (eV)
8
IP (eV)
EA (eV)
8
2
1.1 B3LYP 1.2 B3LYP 1.1 DFTB 1.2 DFTB
6
1
7
1.1 CAM-B3LYP 1.2 CAM-B3LYP
0
4 0.0
1.1 wB97xd 1.2 wB97xd
0.5
1/n
6
1.0 0.0
1.1 M062x 1.2 M062x
0.5
1/n
1.0 0.0
0.5
1/n
1.0
Figure S1. The variations of HLG, IP, and EA of oligomers 1.1 and 1.2 calculated with different methods.
II.
Effect of diffused function in basis set for electron affinity.
Calculation of EA (or the energy of anion) usually requires the usage of a diffused basis set. Therefore in Figure S2 we plotted the EAs by CAM-B3LYP calculations with 6-31+G(d) diffused basis set against those by 6-31G(d) basis set. The excellent linear correlations between the EAs by 6-31G(d) and those by 6-31+G(d) were shown to exist in both 1D and 2D oligomers of both system 1 and 3. This implies that both basis sets can give very similar descriptions for the EAs in conjugated polymers. The slopes of the fitting lines of four groups (1D and 2D oligomers of system 1 and 3) are 0.9488±0.0132, 0.9810±0.0325,
0.9072±0.0013, and 0.8972±0.0089 respectively, and the intercepts are 0.4898±0.0060 eV, 0.5087±0.0145 eV, 0.4424±0.0009 eV, and 0.4329±0.0084 eV respectively. This indicates again that adding a further diffused function to 6-31G(d) basis set just generates a monotonic shift for calculated EA values of all oligomers investigated, and it will not present any qualitative change to our discussion about EA in the main body of our manuscript. For this reason and also in consideration of a good compromise between accuracy and computational efficiency, we adopted 6-31G(d) basis set in our systematic electronic structure
EA (eV) (6-31+G(d))
calculations for anion states.
2
1
1-1D 1-2D 3-1D 3-2D
0 -1
0 1 EA (eV) (6-31G(d))
2
Figure S2. EAs calculated with CAM-B3LYP/6-31G(d) and CAM-B3LYP/6-31+G(d).
εHOMO / eV
-7.0
1.1 1.2
-7.1 -7.2
εLUMO / eV
S2. Energy variations of HOMOs and LUMOs with the increasing oligomer size
0.3 0.0 1.1 1.2
-0.3 -0.6
0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.2 0.4 0.6 0.8 1.0
1/n
1/n 2.1 2.2
-2.0 εLUMO / eV
εHOMO / eV
-5.1 -5.4 -5.7
-6.5 0.0 0.2 0.4 0.6 0.8 1.0 1/n
εLUMO / eV
εHOMO / eV
-6.0
2.1 2.2
-3.0 0.0 0.2 0.4 0.6 0.8 1.0 1/n 0
0.0 0.2 0.4 0.6 0.8 1.0 1/n -5.5 3.1 3.2
-2.5
-1 3.1 3.2
-2 0.0 0.2 0.4 0.6 0.8 1.0 1/n
Figure S3. Energies of HOMO (ε ) and LUMO (ε ) plotted with respect to inverse size (1/n).
S3. DOS for oligomers 1-3
Figure S4. DOS for oligomers 1-3 with largest calculated size.
S4. Solution of Hückel model for 1D systems.
Figure S5. Population distribution of frontier molecular orbitals by a solution of Hückel model for oligomer 1.1 and 3.1.
Figure S6. Fragmentation of monomers in oligomers 2.1 and 3.1.
Table S1. Frontier orbital energies for fragments in oligomers 2.1 and 3.1.
ε /eV
-2.35
1.48
2.06
ε /eV
-6.18
-8.18
-8.92
In order to see the effect of the fluctuation of fragment energies on the delocalization of the wavefunction, we built a model Hückel Hamiltonian (a 18×18 matrix, as shown in Figure S7) in which the off-diagonal terms are approximated as constants and the diagonal elements (αA and αB) have fluctuations. We tuned the value of energy difference △α between αA and αB and the results are depicted in the right panel of Figure S7 with
=0, 0.2, 1.0 respectively. Apparently, the delocalization extent became suppressed with the
increasing fluctuation of diagonal terms △α. This explains the distribution difference of the frontier molecular orbitals in oligomers 2.1 and 3.1 (Figure 2 in the main body of the manuscript).
Figure S7. Schematic illustration of the model Hückel Hamiltonian (left panel) and the population distribution of frontier molecular orbitals from its numerical solution (right panel).
References (S1) Elstner, M.; Porezag, D.; Jungnickel, G.; Elsner, J.; Haugk, M.; Frauenheim, T.; Suhai, S.; Seifert, G. Self-consistent-charge density-functional tight-binding method for simulations of complex materials properties. Phys. Rev. B 1998, 58, 7260-7268. (S2) Hertwig, R. H.; Koch, W. On the parameterization of the local correlation functional. What is Becke-3-LYP? Chem. Phys. Lett. 1997, 268, 345-351. (S3) Yanai, T.; Tew, D.; Handy, N. C. A new hybrid exchange–correlation functional using the Coulomb-attenuating method (CAM-B3LYP). Chem. Phys. Lett. 2004, 393, 51-57. (S4) Chai, J.-D.; Head-Gordon, M. Long-range corrected hybrid density functionals with damped atom–atom dispersion corrections. Phys. Chem. Chem. Phys. 2008, 10, 6615-6620. (S5) Zhao, Y.; Truhlar, D. G. The M06 suite of density functionals for main group thermochemistry, thermochemical kinetics, noncovalent interactions, excited states, and transition elements: two new functionals and systematic testing of four M06-class functionals and 12 other functional. Theor. Chem. Acc. 2008, 120, 215-241. (S6) Mardirossian, N.; Parkhill, J. A.; Head-Gordon, M. Benchmark results for empirical post-GGA functionals: Difficult exchange problems and independent tests. Phys. Chem. Chem. Phys. 2011, 13, 19325-19337. (S7) Körzdörfer, T.; Parrish, R. M.; Marom, N.; Sears, J. S.; Sherrill, C. D.; Brédas, J.-L Assessment of the performance of tuned range-separated hybrid density functionals in predicting accurate quasiparticle spectra. Phys. Rev. B 2012, 86, 205110. (S8) Adamo, C.; Jacquemin, D. The calculations of excited-state properties with Time-Dependent Density Functional Theory. Chem. Soc. Rev, 2013, 42, 845-856. (S9) Salzner, U.; Aydin, A. Improved Prediction of Properties of π-Conjugated Oligomers with Range-Separated Hybrid Density Functionals. J. Chem. Theory Comput. 2011, 7, 2568-2583. (S10) Musfeldt, J. L.; Reynolds, J. R.; Tanner, D. B.; Ruiz, J. P.; Wang, J.; Pomerantz, M. Luminescent polymers with discrete emitter units. J. Polym. Sci., B: Polym. Phys. 1994, 32, 2395−2404. (S11) Donohoo-Vallett,
P. J.; Bragg, A. E. π‑Delocalization and the Vibrational Spectroscopy of Conjugated Materials:
Computational Insights on Raman Frequency Dispersion in Thiophene, Furan, and Pyrrole Oligomers. J. Phys. Chem. B 2015, 119, 3583-3594 (S12) Nožár, J.; Nešpůrek, S.; Šebera, Jakub. Polaron binding energy in polymers: poly[methyl(phenyl)silylene]. J. Mol. Model. 2012, 18, 623-629. (S13) Jacquemin, D.; Perpète, E. A.; Scalmani, G.; Frisch, M. J.; Kobayashi, R.; Adamo, C. Assessment of the efficiency of long-range corrected functionals for some properties of large compounds. J. Chem. Phys. 2007, 126, 144105. (S14) Krawczyk, P. DFT study of linear and nonlinear optical properties of donor-acceptor substituted stilbenes, azobenzenes and benzilideneanilines. J. Mol. Model. 2010, 16, 659-668 .
