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Density matrix renormalisation group calculations of molecular exciton energies in poly(p-phenylene vinylene)
Synthetic
ELSEVIER
Metals 85 (1997)
1155-l 156
Density Matrix Renormalisation Group Calculations of Molecular Exciton Energies in poly(p-phenylene vinylene) Department
of Physics and Centre for Molecuiar
W. Barford and R. J. Bursill* Materials, The University of Sheffield, Sheffield, S3 7RH, United Kingdom.
Abstract Starting from the Pariser-Parr-Pople model of x-conjugated systems, we construct a model of the low lying excitations of poly(p-phenylene vinylene). The model is based on the bonding HOMO and LUMO states of the molecular repeat units. The model is numerically tractable in that it is solved for oligomers of up to 15 units using the density matrix renormalisation group method. The energy of the I’B,- exciton is in good agreement with oligomers, and approaches cu. 2.7 eV. for oligomers of 15 units. Likewise, we predict a 2lA,+ exciton at cu. 2.8 eV, a 13B,+ exciton at 1.6 eV and the singlet exciton binding energy as being 1.4 eV for single chains. We extend this approach to target other absorption bands. For example, we find a localised Frenkel exciton at 5.8 eV, in excellent agreement with the 6 eV absorption peak in ppV. Keywords:
Semi-empirical
models and model calculations.
The observation of electroluminescence in the organic semiconductor poly(p-phenylene vinylene) (PPV) has led to a renewed activity in the physics of Rconjugated systems. The theoretical issues involved with the role and nature of the electronic correlations in the conjugated systems has attracted wide attention [l-4]. The recent development of the density matrix renormalisation group (DMRG) method enables the accurate calculation of ground and excited state energies in one dimensional quantum lattice Hamiltonians. In this paper we present a realistic model of PPV, based on the relevant HOMO and LUMO orbitals of the underlying phenylene and vinylene repeat units, and solve this using the DMRG method. Our starting point is the Pariser-Parr-Pople (P-P-P) model of n-conjugated systems. We take the nearest neighbour hybridisation, tij=t(l - as), where t=2.39 eV for a phenyl bond, 6 is the bond length distortion and The Ohno parameterisation for the a=16 meV/pm. Coulomb interaction is used, with U=ll.l eV. PPV is composed of a sequence of phenylene and vinylene repeat units. Its electronic structure can thus be conveniently expressed in terms of the molecular orbitals (MOs) of these units [5]. The MOs of phenylene are essentially the six MOs of benzene, with four bonding and two non-bonding orbitals. The vinylene unit has a bonding and anti-bonding MO. The low lying excitations of PPV will result from transitions between the HOMO and LUMO states. These states will be predominately constructed from the two bonding HOMO and LUMO states of the phenylene unit and the vinylene MOs. (The two remaining HOMO and LUMO states of the phenylene unit are non-bonding and are not expected to contribute to the low energy physics.) We therefore obtain an effective four band model in terms of these four +----------------------------------------------~---Present Sydney,
address: School of Physics, NSW 2052, Australia.
0379-6779/97/$17.00
0 1997 Elsevier
PII SO379-6779(96)04309-3
University
of New
South
Wales,
Science S.k All rights reserved
MOs. The parameters and many body interactions retained in this model are derived from the underlying P-P-P model, and are shown in table 1. Table 1 Parameters used in the model (eV). Parameter A U V W X
Phenylene 4.78 7.09 4.35 2.24 1.04
unit
Vinylene 5.26 9.37 4.35 2.13 1.72
unit
A is the single particle HOMO-LUMO energy gap, U is the Coulomb repulsion between two electrons on the same repeat unit, V and W are the repulsion between a pair of electrons on nearest and next nearest neighbouring repeat units, respectively, and X is the exchange energy for a pair of electrons in different MOs on the same repeat unit. The nearest neighbour phenylene-vinylene hopping, t, may also be derived from the P-P-P model. However, this value of t is probably underestimated due to the neglect of next nearest neighbour hopping in the P-P-P model. Moreover, additional hybridisation channels, via the neglected states, are not included in our model. We therefore regard this parameter as a phenomenological value, chosen to best fit the vertical transition of the 1 ‘B, exciton in the seven unit oligomer. This gives t=1.2 eV, as opposed to 0.9 eV from the P-P-P analysis. All other parameters remain unadjusted. Except for the trimer result in table 3, the next nearest neighbour interaction, W, is not included in our calculations. The eigenstates of the Hamiltonian are identified by their spatial symmetry under inversion, their spin and Allowed optical their particle-hole symmetry. transitions occur between states of identical spin and
1156
W Ba$ord,
R.J. Bltrsill/S~ntheticiLietals
formalism better reproduces the localization charge carriers along the chain as compared UHF approach.
of the to the
85 (1997/
1155-1156
B), it is a common feature of the Raman spectra of aromatic and heteroaromatic systems; 1476-1487 cm-l (line C) which is characteristic of c1or o substituted oligothiophenes and shows intensity enhancement with increasing chain length; and 1035 cm-l (line D), observed in the Raman spectrum for all the oligothiophenes. 3.3. Vibrational
Fig. 1. 6-31G** optimized C-C bond lengths (in the cl,cr’-dimethylsexithiophene in the neutral (filled squares) and in the radical cationic state circles). ROHF method has been used for the shell system. 3.2. Vibrational
spectra
of neutral
A) of state (open open-
molecules
We have used the ab initio 6-31G** level of calculation to determine the infrared and Raman For spectra of the neutral oligothiophenes. comparison purposes we have scaled all the calculated frequencies by a uniform factor of 0.9. All quoted scaled theoretical results are thus the scaled values.
spectra
of oxidized
molecules
In Figure 3 the computed infrared spectra of the radical cation (ROHF/6-31G**) molecules are displayed. Comparing the theoretical infrared spectrum of the neutral oligomers to that of its corresponding radical cations calculated in the same basis set, one immediately recognizes new intensive bands calculated at 1450 (m), 1404 (m), 1376-1290 (s), 1274(w), 1115 (s), 1039 (vs), 967(m), 865 (m) cm-l. The experimental infrared spectra of the 12 doped materials show [4] at least six new experimental infrared bands at 1414 (m), 1327 (s), 1162 (s), 1099 (s), 993 (s) and 946 (m) cm-l, whose intensities increase with increasing doping level. Comparison between calculated and experimental data indicates the existence of a polaron-type defect in the iodine doped materials as has been confirmed by the electronic spectra [5].
1525
DM?T
2 a li!d
E 2 T
1700
1487
;I 1529 1476 16131536 3 z
1500
1300
1038 h
DMQtT
1035 1035
DMTT
1100 Raman
DMBT 900 shift
700
500
300
200
(cm”)
Fig. 2. Theoretical Raman spectra of neutral a,a’-dimethyl oligothiophenes as calculated at the RHF/G-31G** level We restrict our discussion to the most important Raman spectroscopic trends ocurring upon chain elongation. In Figure 2, the computed Raman spectra of DMBT, DMTI’ and DMQtT are displayed. The spectra show very few bands despite the complex chemical structures, which exclusively originate from totally symmetric modes. As in the experimental spectra, for DMBT and DMTT the most important feature is the appearance of only one intense band around 1530 cm-l, this shifts downward by 10 cm-l from the dimer to the tetramer (from 1492 cm-l to 1482 cm-l in the experimental spectra [3]). For longer oligomers a second line around 1470 cm-l becomes stronger with the molecular size. As in the experimental Raman spectra, these two lines show a convergent trend. The theoretical Raman spectra of the neutral molecules show the four characteristic features common to many other classes of oligothiophenes: 1615-1591 cm-l (line A) weak, which show dispersion toward lower frequencies when increasing chain length; 1536-1525 cm-l (line
I
I 1700
1500
1300
1100 Wnvenumber
900
700
500
400
(cm”)
Fig. 3. Theoretical infrared spectra of a,a’-dimethyl oligothiophenes radical cations as calculated at the ROHF/6-31G** level 4. Acknowledgements
This research was financially supported by Direction General de Investigation Cientifica y Tecnica (DGICYT, Spain) through the research project PB93-1244. References
[I] J. Roncali, Chem.Rev. 92, 711 (1992). [23 S. Hotta, K. Waragai, J.Mat.Chem. 1, 835 (1991). [33 V. Hernandez, J. Casado, F. J. Ramirez, G. Zotti, S. Hotta and J. T. Lopez Navarrete, J. Chem. Phys. (in press). [4] J. Casado, F. J. Ramirez, S. Hotta, J. T. Lopez Navarrete and V. Hernandez, Synth. Met. (in press). [5] S. Hotta, K. Waragai, J.Phys.Chem. 97, 7427 (1993).
ELSEVIER
Metals 85 (1997)
1155-l 156
Density Matrix Renormalisation Group Calculations of Molecular Exciton Energies in poly(p-phenylene vinylene) Department
of Physics and Centre for Molecuiar
W. Barford and R. J. Bursill* Materials, The University of Sheffield, Sheffield, S3 7RH, United Kingdom.
Abstract Starting from the Pariser-Parr-Pople model of x-conjugated systems, we construct a model of the low lying excitations of poly(p-phenylene vinylene). The model is based on the bonding HOMO and LUMO states of the molecular repeat units. The model is numerically tractable in that it is solved for oligomers of up to 15 units using the density matrix renormalisation group method. The energy of the I’B,- exciton is in good agreement with oligomers, and approaches cu. 2.7 eV. for oligomers of 15 units. Likewise, we predict a 2lA,+ exciton at cu. 2.8 eV, a 13B,+ exciton at 1.6 eV and the singlet exciton binding energy as being 1.4 eV for single chains. We extend this approach to target other absorption bands. For example, we find a localised Frenkel exciton at 5.8 eV, in excellent agreement with the 6 eV absorption peak in ppV. Keywords:
Semi-empirical
models and model calculations.
The observation of electroluminescence in the organic semiconductor poly(p-phenylene vinylene) (PPV) has led to a renewed activity in the physics of Rconjugated systems. The theoretical issues involved with the role and nature of the electronic correlations in the conjugated systems has attracted wide attention [l-4]. The recent development of the density matrix renormalisation group (DMRG) method enables the accurate calculation of ground and excited state energies in one dimensional quantum lattice Hamiltonians. In this paper we present a realistic model of PPV, based on the relevant HOMO and LUMO orbitals of the underlying phenylene and vinylene repeat units, and solve this using the DMRG method. Our starting point is the Pariser-Parr-Pople (P-P-P) model of n-conjugated systems. We take the nearest neighbour hybridisation, tij=t(l - as), where t=2.39 eV for a phenyl bond, 6 is the bond length distortion and The Ohno parameterisation for the a=16 meV/pm. Coulomb interaction is used, with U=ll.l eV. PPV is composed of a sequence of phenylene and vinylene repeat units. Its electronic structure can thus be conveniently expressed in terms of the molecular orbitals (MOs) of these units [5]. The MOs of phenylene are essentially the six MOs of benzene, with four bonding and two non-bonding orbitals. The vinylene unit has a bonding and anti-bonding MO. The low lying excitations of PPV will result from transitions between the HOMO and LUMO states. These states will be predominately constructed from the two bonding HOMO and LUMO states of the phenylene unit and the vinylene MOs. (The two remaining HOMO and LUMO states of the phenylene unit are non-bonding and are not expected to contribute to the low energy physics.) We therefore obtain an effective four band model in terms of these four +----------------------------------------------~---Present Sydney,
address: School of Physics, NSW 2052, Australia.
0379-6779/97/$17.00
0 1997 Elsevier
PII SO379-6779(96)04309-3
University
of New
South
Wales,
Science S.k All rights reserved
MOs. The parameters and many body interactions retained in this model are derived from the underlying P-P-P model, and are shown in table 1. Table 1 Parameters used in the model (eV). Parameter A U V W X
Phenylene 4.78 7.09 4.35 2.24 1.04
unit
Vinylene 5.26 9.37 4.35 2.13 1.72
unit
A is the single particle HOMO-LUMO energy gap, U is the Coulomb repulsion between two electrons on the same repeat unit, V and W are the repulsion between a pair of electrons on nearest and next nearest neighbouring repeat units, respectively, and X is the exchange energy for a pair of electrons in different MOs on the same repeat unit. The nearest neighbour phenylene-vinylene hopping, t, may also be derived from the P-P-P model. However, this value of t is probably underestimated due to the neglect of next nearest neighbour hopping in the P-P-P model. Moreover, additional hybridisation channels, via the neglected states, are not included in our model. We therefore regard this parameter as a phenomenological value, chosen to best fit the vertical transition of the 1 ‘B, exciton in the seven unit oligomer. This gives t=1.2 eV, as opposed to 0.9 eV from the P-P-P analysis. All other parameters remain unadjusted. Except for the trimer result in table 3, the next nearest neighbour interaction, W, is not included in our calculations. The eigenstates of the Hamiltonian are identified by their spatial symmetry under inversion, their spin and Allowed optical their particle-hole symmetry. transitions occur between states of identical spin and
1156
W Ba$ord,
R.J. Bltrsill/S~ntheticiLietals
formalism better reproduces the localization charge carriers along the chain as compared UHF approach.
of the to the
85 (1997/
1155-1156
B), it is a common feature of the Raman spectra of aromatic and heteroaromatic systems; 1476-1487 cm-l (line C) which is characteristic of c1or o substituted oligothiophenes and shows intensity enhancement with increasing chain length; and 1035 cm-l (line D), observed in the Raman spectrum for all the oligothiophenes. 3.3. Vibrational
Fig. 1. 6-31G** optimized C-C bond lengths (in the cl,cr’-dimethylsexithiophene in the neutral (filled squares) and in the radical cationic state circles). ROHF method has been used for the shell system. 3.2. Vibrational
spectra
of neutral
A) of state (open open-
molecules
We have used the ab initio 6-31G** level of calculation to determine the infrared and Raman For spectra of the neutral oligothiophenes. comparison purposes we have scaled all the calculated frequencies by a uniform factor of 0.9. All quoted scaled theoretical results are thus the scaled values.
spectra
of oxidized
molecules
In Figure 3 the computed infrared spectra of the radical cation (ROHF/6-31G**) molecules are displayed. Comparing the theoretical infrared spectrum of the neutral oligomers to that of its corresponding radical cations calculated in the same basis set, one immediately recognizes new intensive bands calculated at 1450 (m), 1404 (m), 1376-1290 (s), 1274(w), 1115 (s), 1039 (vs), 967(m), 865 (m) cm-l. The experimental infrared spectra of the 12 doped materials show [4] at least six new experimental infrared bands at 1414 (m), 1327 (s), 1162 (s), 1099 (s), 993 (s) and 946 (m) cm-l, whose intensities increase with increasing doping level. Comparison between calculated and experimental data indicates the existence of a polaron-type defect in the iodine doped materials as has been confirmed by the electronic spectra [5].
1525
DM?T
2 a li!d
E 2 T
1700
1487
;I 1529 1476 16131536 3 z
1500
1300
1038 h
DMQtT
1035 1035
DMTT
1100 Raman
DMBT 900 shift
700
500
300
200
(cm”)
Fig. 2. Theoretical Raman spectra of neutral a,a’-dimethyl oligothiophenes as calculated at the RHF/G-31G** level We restrict our discussion to the most important Raman spectroscopic trends ocurring upon chain elongation. In Figure 2, the computed Raman spectra of DMBT, DMTI’ and DMQtT are displayed. The spectra show very few bands despite the complex chemical structures, which exclusively originate from totally symmetric modes. As in the experimental spectra, for DMBT and DMTT the most important feature is the appearance of only one intense band around 1530 cm-l, this shifts downward by 10 cm-l from the dimer to the tetramer (from 1492 cm-l to 1482 cm-l in the experimental spectra [3]). For longer oligomers a second line around 1470 cm-l becomes stronger with the molecular size. As in the experimental Raman spectra, these two lines show a convergent trend. The theoretical Raman spectra of the neutral molecules show the four characteristic features common to many other classes of oligothiophenes: 1615-1591 cm-l (line A) weak, which show dispersion toward lower frequencies when increasing chain length; 1536-1525 cm-l (line
I
I 1700
1500
1300
1100 Wnvenumber
900
700
500
400
(cm”)
Fig. 3. Theoretical infrared spectra of a,a’-dimethyl oligothiophenes radical cations as calculated at the ROHF/6-31G** level 4. Acknowledgements
This research was financially supported by Direction General de Investigation Cientifica y Tecnica (DGICYT, Spain) through the research project PB93-1244. References
[I] J. Roncali, Chem.Rev. 92, 711 (1992). [23 S. Hotta, K. Waragai, J.Mat.Chem. 1, 835 (1991). [33 V. Hernandez, J. Casado, F. J. Ramirez, G. Zotti, S. Hotta and J. T. Lopez Navarrete, J. Chem. Phys. (in press). [4] J. Casado, F. J. Ramirez, S. Hotta, J. T. Lopez Navarrete and V. Hernandez, Synth. Met. (in press). [5] S. Hotta, K. Waragai, J.Phys.Chem. 97, 7427 (1993).