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Essential excitations in phenylene-based conjugated polymers
Synthetic
ELSEVIER
Essential
excitations
Metals 85 (1997)
1121-1122
in phenylene-based
conjugated
polymers
Aparna Chakrabarti and S. Mazumdar Department of Physics, University of Arizona, Tucson, AZ 85721,USA
Abstract
We have developed a computational approach based on an exciton basis to calculate the excited state spectrum of long chain poly(paraphenylenes). Pictorial descriptions of the excited states relevant in the photophysics of these systems are obtained. Evidence for bound excitons has been found. Keywords: (Semiempirical models and model calculations, Many-body absorption spectroscopy)
1. Introduction
Recent exact finite chain configuration interaction calculations within a diagrammatic exciton basis give pictorial descriptions and thereby natures of all the excited states relevant in the photophysics of 7rconjugated linear chains[l]. Here we present the results of our calculations on poly(paraphenylenes) within the diagrammatic exciton basis. We discuss how far it is correct to visualize these systems as quasi-onedimensional ones.
2. Model We perform the calculations within the extended Hubbard and Pariser-Parr-Pople[2] model Hamiltonians for r-conjugated systems. The form of the Hamiltonian is the same as used in ref.[l]. Since the results for both the Hamiltonians are essentially similar, we report here the results of only the extended Hubbard model calculations. The onsite and nearest neighbor electron interaction parameters are taken as 3.0 and 1.0, respectively, in the units of dimensionlesshopping integral term. We also restrict our discussions to the even-parity states. Poly(paraphenylenes) are modeled as coupled phenyl units within the exciton basis approach[3]. Each phenyl unit contributes only the doubly degenerate HOMO and LUMO levels. The MOs of different units are coupled by the Coulomb interactions and the interunit hopping. 0379-6779/97/$17.00 0 1997 Elsevier PII 30379-6779(96)04292-o
Science S.A All rights reserved
and quasiparticle theories, photoinduced
We perform two sets of calculations. Set 1 deals with four MO levels per unit as described above, whereas Set 2 excludes all the localized levels from the calculations. We believe that the results of Set 2 reproduce the behavior of the excited states in the long chain limit. For linear chains, the optically important excited states consist mainly of single excitations and double excitations. Single excitations are of two types, charge-transfer(CT) excitations and intraunit Frenkel excitons(F). There are two types of double excitations, triplet-triplet and singlet-singlet(SS)[l]. In the case of poly(paraphenylenes) additional intraunit excitations involving both delocalized and localized benzene MO’s are possible giving rise to multiple subclassesof TT, SS and CT within Set 1. 3. Results 3.1. set 1 We find a strong mixing of excitations involving delocalized and localized levels in the excited states. This is because of the finite size of the systems studied here. In linear polyacetylenes, the ordering of the states is : lowest TT < lowest CT < lowest SS[l]. In poly(paraphenylenes), the 2A, is above the lB, and we observe that eigenfunctions are no longer predominantly CT or TT type. We find that the CT states mixed with TT states are followed in energy by the CT states mixed with SS states. Because of space restriction here we re-
1122
port the discussions only.
A. Chakrabarti,
of the results
S. Mazumdar/SyntheticMetals
85 (1997)
BX
of Set 2 calculations
=-O-60
-0.27
Set 2 Theoretical evidence for bound biexcitons in linear chains has been given in ref.[l] and [4]. Here we present evidence for distinct biexciton (BX) state and the threshold state of the two-exciton continuum state (2-EX) in poly(paraphenylenes). Within Set 2 calculations, our results are independent of chain length. Since, however, the number of basis functions increases rapidly with chain length, we present here the results for the 4-unit poly(paraphenylene) open chains. The wavefunctions for BX and 2-EX states are shown in Fig.l(a). The basis functions in Fig. 1 are the delocalized HOMO and LUMO of individual benzene MO’s. Pairs of singlet spin coupled MO’s are connected by a single bond. Note that the state labeled BX has contributions only from exciton basis diagrams with a doubly excited single unit or from diagrams representing two nearest neighbor single excitation, i.e. highly localized double excitations. Furthermore, the nearly 1:1 weights of the two diagrams with nearest neighbor excitations indicate the occurence of the exciton basis valence bond diagram with “crossed bonds” in BX, as indicated by the exact linear relationship in Fig. l(b). In contrast, the state labeled 2-EX has dominant contributions from exciton basis diagrams in which the two excitons are physically separated. The very weak overlap between the two eigenstates is a distinct signature of the bound biexciton character of the BX and the free two-exciton character of 2-EX. It has been claimed that in the presence of moderate to strong exciton-exciton binding, the transition strength of the optical absorption from the optically allowed lowest exciton to the biexciton becomes weaker than that of the transition of the edge state of the twoexciton continuum[4]. For linear chain systems, this conjecture has been verified within three different theoretical models[5]. In Fig. 2 we have shown the relative dipole moments between the lB, exciton and the BX state and the 2-EX state of Fig. l(a). The larger dipole moment of the latter is in agreement with our identification of BX as a bound state of two excitons. To conclude, the universal nature of the absorption from the exciton to the higher two-exciton states has already been established within various potential models for linear chains[5]. We show here that this nature is true for poly(phenylenes) also. This universality is simply the consequence of the one-dimensionality of the systems. In poly(paraphenylene vinylene) (PPV) and derivatives, it has now been convincingly demon-
1121-1122 ---++ +,+.+,m-
+0.15
;;:;
-0.30
;,rr
-+,+-v
-0.41
-+++$?-
-0.11
---+++++
&,I
-0.37
&,I
-0.30
I,,1
,r,T
-0.25
&x
3.2.
2-EX
=a44 -0.30
Fig. l(a).
to.12
---+-It-H--It
Nature of BX and 2-EX states.
x
Bound excitonic
=
Fig. l(b).
-v -
molecule
-II
Twoc. excitons spatially
bound.
strated[6] that ultrafast photoinduced absorption (PA) occurs from the optical exciton to a low energy state at -0.7eV and a high energy (HE) state at -1.5eV. We assign the HE PA to the transition from the exciton to the lowest BX state. 1.5
Fig. 2. Ratio of dipole moments with the optical exciton * = 0.54.
References 1. M. Chandross, Y. Shimoi and S. Mazumdar, Present proceeding. 2. (a) R. Pariser and R. G. Parr, J. Chem. Phys., 21 (1953) 767. (b) J. A. Pople, Tra& Farad. SOL, 49 (1953) 1375. 3. (a) H. C. Longuet-Higgins and J. N. Murrell, Proc. Phys. Sot., 68 (1955) 601. (b) M. J. Rice and Y. N. Gartstein, Phys. Rev. Lett., 73 (1994) 2504. 4. F. Guo, M. Chandross and S. Mazumdar, Phys. Rev. Lett. , 74 (1995) 2086. 5. S. Mazumdar et. al. , J. Chem. Phys. , 104 (1996) 9292. 6. S. V. Frolov et. QI., Present Proceeding.
ELSEVIER
Essential
excitations
Metals 85 (1997)
1121-1122
in phenylene-based
conjugated
polymers
Aparna Chakrabarti and S. Mazumdar Department of Physics, University of Arizona, Tucson, AZ 85721,USA
Abstract
We have developed a computational approach based on an exciton basis to calculate the excited state spectrum of long chain poly(paraphenylenes). Pictorial descriptions of the excited states relevant in the photophysics of these systems are obtained. Evidence for bound excitons has been found. Keywords: (Semiempirical models and model calculations, Many-body absorption spectroscopy)
1. Introduction
Recent exact finite chain configuration interaction calculations within a diagrammatic exciton basis give pictorial descriptions and thereby natures of all the excited states relevant in the photophysics of 7rconjugated linear chains[l]. Here we present the results of our calculations on poly(paraphenylenes) within the diagrammatic exciton basis. We discuss how far it is correct to visualize these systems as quasi-onedimensional ones.
2. Model We perform the calculations within the extended Hubbard and Pariser-Parr-Pople[2] model Hamiltonians for r-conjugated systems. The form of the Hamiltonian is the same as used in ref.[l]. Since the results for both the Hamiltonians are essentially similar, we report here the results of only the extended Hubbard model calculations. The onsite and nearest neighbor electron interaction parameters are taken as 3.0 and 1.0, respectively, in the units of dimensionlesshopping integral term. We also restrict our discussions to the even-parity states. Poly(paraphenylenes) are modeled as coupled phenyl units within the exciton basis approach[3]. Each phenyl unit contributes only the doubly degenerate HOMO and LUMO levels. The MOs of different units are coupled by the Coulomb interactions and the interunit hopping. 0379-6779/97/$17.00 0 1997 Elsevier PII 30379-6779(96)04292-o
Science S.A All rights reserved
and quasiparticle theories, photoinduced
We perform two sets of calculations. Set 1 deals with four MO levels per unit as described above, whereas Set 2 excludes all the localized levels from the calculations. We believe that the results of Set 2 reproduce the behavior of the excited states in the long chain limit. For linear chains, the optically important excited states consist mainly of single excitations and double excitations. Single excitations are of two types, charge-transfer(CT) excitations and intraunit Frenkel excitons(F). There are two types of double excitations, triplet-triplet and singlet-singlet(SS)[l]. In the case of poly(paraphenylenes) additional intraunit excitations involving both delocalized and localized benzene MO’s are possible giving rise to multiple subclassesof TT, SS and CT within Set 1. 3. Results 3.1. set 1 We find a strong mixing of excitations involving delocalized and localized levels in the excited states. This is because of the finite size of the systems studied here. In linear polyacetylenes, the ordering of the states is : lowest TT < lowest CT < lowest SS[l]. In poly(paraphenylenes), the 2A, is above the lB, and we observe that eigenfunctions are no longer predominantly CT or TT type. We find that the CT states mixed with TT states are followed in energy by the CT states mixed with SS states. Because of space restriction here we re-
1122
port the discussions only.
A. Chakrabarti,
of the results
S. Mazumdar/SyntheticMetals
85 (1997)
BX
of Set 2 calculations
=-O-60
-0.27
Set 2 Theoretical evidence for bound biexcitons in linear chains has been given in ref.[l] and [4]. Here we present evidence for distinct biexciton (BX) state and the threshold state of the two-exciton continuum state (2-EX) in poly(paraphenylenes). Within Set 2 calculations, our results are independent of chain length. Since, however, the number of basis functions increases rapidly with chain length, we present here the results for the 4-unit poly(paraphenylene) open chains. The wavefunctions for BX and 2-EX states are shown in Fig.l(a). The basis functions in Fig. 1 are the delocalized HOMO and LUMO of individual benzene MO’s. Pairs of singlet spin coupled MO’s are connected by a single bond. Note that the state labeled BX has contributions only from exciton basis diagrams with a doubly excited single unit or from diagrams representing two nearest neighbor single excitation, i.e. highly localized double excitations. Furthermore, the nearly 1:1 weights of the two diagrams with nearest neighbor excitations indicate the occurence of the exciton basis valence bond diagram with “crossed bonds” in BX, as indicated by the exact linear relationship in Fig. l(b). In contrast, the state labeled 2-EX has dominant contributions from exciton basis diagrams in which the two excitons are physically separated. The very weak overlap between the two eigenstates is a distinct signature of the bound biexciton character of the BX and the free two-exciton character of 2-EX. It has been claimed that in the presence of moderate to strong exciton-exciton binding, the transition strength of the optical absorption from the optically allowed lowest exciton to the biexciton becomes weaker than that of the transition of the edge state of the twoexciton continuum[4]. For linear chain systems, this conjecture has been verified within three different theoretical models[5]. In Fig. 2 we have shown the relative dipole moments between the lB, exciton and the BX state and the 2-EX state of Fig. l(a). The larger dipole moment of the latter is in agreement with our identification of BX as a bound state of two excitons. To conclude, the universal nature of the absorption from the exciton to the higher two-exciton states has already been established within various potential models for linear chains[5]. We show here that this nature is true for poly(phenylenes) also. This universality is simply the consequence of the one-dimensionality of the systems. In poly(paraphenylene vinylene) (PPV) and derivatives, it has now been convincingly demon-
1121-1122 ---++ +,+.+,m-
+0.15
;;:;
-0.30
;,rr
-+,+-v
-0.41
-+++$?-
-0.11
---+++++
&,I
-0.37
&,I
-0.30
I,,1
,r,T
-0.25
&x
3.2.
2-EX
=a44 -0.30
Fig. l(a).
to.12
---+-It-H--It
Nature of BX and 2-EX states.
x
Bound excitonic
=
Fig. l(b).
-v -
molecule
-II
Twoc. excitons spatially
bound.
strated[6] that ultrafast photoinduced absorption (PA) occurs from the optical exciton to a low energy state at -0.7eV and a high energy (HE) state at -1.5eV. We assign the HE PA to the transition from the exciton to the lowest BX state. 1.5
Fig. 2. Ratio of dipole moments with the optical exciton * = 0.54.
References 1. M. Chandross, Y. Shimoi and S. Mazumdar, Present proceeding. 2. (a) R. Pariser and R. G. Parr, J. Chem. Phys., 21 (1953) 767. (b) J. A. Pople, Tra& Farad. SOL, 49 (1953) 1375. 3. (a) H. C. Longuet-Higgins and J. N. Murrell, Proc. Phys. Sot., 68 (1955) 601. (b) M. J. Rice and Y. N. Gartstein, Phys. Rev. Lett., 73 (1994) 2504. 4. F. Guo, M. Chandross and S. Mazumdar, Phys. Rev. Lett. , 74 (1995) 2086. 5. S. Mazumdar et. al. , J. Chem. Phys. , 104 (1996) 9292. 6. S. V. Frolov et. QI., Present Proceeding.