- Email: [email protected]
Systematic characterization of excited states in conjugated polymers
ELSEVIER
Synthetic Metals 85 (1997) 1001-1006
Systematic
characterization
of excited
states in conjugated
polymers
M. Chandross”, Y. ShimoibJ, and S. Mazumdaralb, aDepartment
of
Physics and b Optical Sciences Center, University of Arizona, Tucson, AZ 85721, uu ‘ElectrotechnicaI Laboratory, l-1-4 Umezono, Tsukuba 305, Japan.
Abstract We present the first complete systematic characterization of the excited states in conjugated polymers. Our results are relevant for the understanding of the photophysics of these materials. We perform full configuration interaction calculations in an exciton basis within which a long chain polymer is considered as coupled molecular units. Complete pictorial descriptions of all excited states are obtained. In linear chain polymers such as the polyacetylenes and polydiacetylenes the lB, is an exciton, and the fundamental two-photon states can be broadly In the above CT classified into triplet-triplet (TT), charge-transfer (CT) and singlet-singlet (SS) excitations. refers to charge-transfer from one unit to another, and TT and SS are two electron-two hole excitations. In TT the spin angular momenta of two different triplet excitations combine to give an overall singlet, while the individual excitations are singlets in SS. The 2A, is classified as TT. The mA,, an even parity state that plays a strong role in nonlinear optics, is a correlated CT state. The SS states occur higher in energy and for moderate exciton binding split into the biexciton and two-exciton continuum. The calculations can be easily extended to the polyphenylenes, for which the characterization of excited states continues to be possible. These theoretical results are useful in explaining a variety of third order nonlinear optical spectroscopic measurements as well as picosecond photoinduced absorption. lieywords: Semi-empirical sorption spectroscopy
models and model calculations;
1. Introduction The theoretical study of the effects of electron correlations on the energy spectra of n-conjugated molecules has a long history. In the 1950s configuration interaction (CI) was first introduced between singly excited configurations that are degenerate at the Hiickel level, [I-]. Coulomb interactions lift the degeneracies of the Hiickel model and form new eigenstates from linear combinations of the Hiickel wavefunctions. This first order CI proved to be useful in interpreting the optical absorption spectra of a number of conjugated systems [I]. In the 1970s it was observed -that the lowest even parity excited state (2A,) is located energetically below the lowest odd parity state (lB,) [a]. The correlated 2A, was found to be a linear combination of single and double excitations, providing direct evidence of the role of Coulomb interactions in r-conjugated polyenes [351. These results point to a question that, to the best of our knowledge, has not yet been addressed, namely, are 0379-6779/97/%17.00 0 1997 Elsevier Science S.A. All rights reserved PII SO379-6779(96)04248-8
Multiphoton
absorption spectroscopy; Photoinduced
ab-
the double excitations themselves split by Coulomb interactions in a manner similar to the single excitations? The role that Coulomb interactions play in the photophysics of conjugated polymers has remained a difficult problem. Accurate CI calculations are only possible for short chains which have discrete energy levels, and thus the character of a given state is not obvious from energetic considerations alone. However, the short chain wavefunctions gradually evolve into localized exciton or delocalized band states as the chain length is increased, and thus must have the same characteristics as their long chain counterparts. It is then possible, in principle, to classify states by their wavefunctions. This is what is acheived in the present work, using complete CI calculations in an exciton basis (see below)[6,7]. CI calculations are usually performed within delocalized molecular orbital (MO) or localized valence bond (VB) representations where simple physical pictures of excited states are possible in the limits of weak and
M. ChandToss et al. /Synthetic Metals 85 (1997) 1001-I 006
1002
strong Coulomb interactions, respectively. For the region of intermediate Coulomb interactions appropriate for conjugated polymers, however, wavefunctions within both of these representations are complicated superpositions of a large number of basis states, and simple physical interpretations are nearly impossible. Our calculations are performed in a hybrid of the hiI and VB bases which considers an ideal polyacetylene chain to be coupled ethylene units [6,7]. Within this basis, correlated wavefunctions are clearly dominated by a small number of configurations, and a systematic classification of all excited states becomes possible. All wavefunctions can be represented in a simple, pictorial fashion, giving an intuitive understanding of their natures. 2. Theoretical
model
Our theoretical calculations are performed within a Pariser-Parr-Pople type Hamiltonian, H, written as,
+
- Ct2&&2i+l,o
i,o
4i+l,oC2d,o)
su c ni,tni,j+ f c Kj:j(% - l)(nj - I),(1) i %a3 where ci,,(ci+,) creates (annihilates) a n-electron of spin u on carbon atom i, ni,a = c/,~c~,~, and ni = Co ni,a. The parameters tl and t2 (tl > t2) are the nearest neighbor intra- and inter-unit hopping integrals, respectively, and U and xj describe the on-site and intersite Coulomb interactions, respectively. We have performed calculations for a number of different parameterizations of the Coulomb interactions, including the Ohno, Mataga-Nishimoto, and extended Hubbard models. We present results here for calculations within the extended Hubbard model; the conclusions are identical for the other parameterizations. 3. The
exciton
basis
Within the exciton basis approach we visualize polyacetylene as coupled ethylenic dimer units. Each dimer contributes one bonding and one antibonding MO, and the MOs of different units are coupled by Coulomb interactions as well as interunit hopping. The Hamiltonian in Eq. 1 is rewritten in terms of the dimer operators, %,A,, =
2-1’2[C2i-l,a
+
(-1yc2i,o],
(2)
where X = l(2) corresponds to the bonding (antibonding) MO of ethylenic dimer unit i. All many-electron
functions can now be represented by dimer unit basis functions, examples of which are shown in Fig. 1 for the simple case of butadiene (2-units). In Fig. 1 and the following we will represent pairs of singlet spin-bonded singly occupied MOs by a connecting line. xx
;I
pz
-+(;-
g1
z
Cc)
63
W
0-I
Figure 1. Typical exciton basis diagrams for butadiene.
Fig. l(a) represents the “noninteracting” ground state of butadiene, which is simply the product wavefunction of the ground states of the individual ethylene units. The single excitations in Figs. l(b) and (c) are easy to understand: these are Frenkel intraunit excitations and charge-transfer (CT) excitations. These are the only types of singly excited diagrams. Double excitations within the exciton basis can involve the same unit, as in Fig. l(d), in which case the excitations are necessarily both singlet. Double excitations can also involve two different units in which case two linearly independent bonding arrangements are possible, as shown in Figs. l(e) and (f). In diagrams (e) and (f) the individual singlet bonds are within a unit and between different units, respectively. As shown in Fig. 2, the linear combination of diagrams (e) and (f) with the ratio 1:2 is equivalent to a triplet-triplet (TT) excitation, in which pairs of triplet excitations on different units combine to give an overall singlet diagram. We refer to diagrams of the types (d) and (e), in which the two excitations are singlets as singlet-singlet or SS excitations. The key to understanding the photophysics of n-conjugated polymers is the distinction between these two classes of double excitations. All eigenstates within the exciton basis are linear combinations of Frenkel, CT, TT and SS basis functions, as well as more complex configurations that are derived from these (note, however, that even these complex functions have pictorial interpretations). The beauty of the exciton basis approach is that the exact eigenstates, in spite of being correlated, can have simple interpretations, as shown below.
4. The
limit
of zero
interunit
charge-transfer
Before examining the wavefunctions of the full Hamiltonian, it is useful to discuss the effects of Coulomb interactions in the limit of zero interunit charge-transfer (i.e. t2 = 0 in Eq. 1). Some of the wavefunctions in this limit have also been discussed recently by other
M. Chandross et al. /SyntheticMetals
1003
85 (1997) 1001-1006 1Af
=
----+0.88y,++++*+t
----++ -0.28+,++++-
-O.ZS,,,?H,
-0.20,,-,, -A+.--
Figure 2. Linear combination of exciton basis diagrams that gives a triplet-triplet excitation. Bonds with arrows denote a triplet spin coupling between the MOs.
authors [9]. In Fig. 3 we show the dominant contributions to the optically relevant wavefunctions in order of increasing energy for 5 units with U = 3tr and V = tl. In Fig. 3 and the following we only show one diagram to represent all diagrams related by mirror-plane and electron-hole symmetries. The superscript (0) on the wavefunctions in Fig. 3 indicates that these are wavefunctions in the limit t2 = 0. The 1Ap) ground state, as expected, consists almost entirely of the noninteracting ground state with small contributions from doubly excited diagrams. The next higher excited state is the 2A$) which is seen to be entirely composed of TT configurations (compare with Fig. 2). The 2Ap) is the lowest of a “band” of TT states below the 1Bi’); the other TT states are similar to the 2Aco) 9 , with the only difference being the seperation between the two triplet excitations. Transitions between the TT states and the lBi”) are strictly forbidden. The lBi”) is a Frenkel exciton in this limit (see Fig. 3), and is strongly dipole coupled to the ground state. There are three such Frenkel excitons for N = 5, with the 1Bp) being the most strongly coupled to the 1Ap). Above the Frenkel exciton are CT states in which the electron and hole are on different units. The CT states in the A, and B, subspaces are degenerate in the limit tz = 0. In Fig. 3, we show the 8Ap) and the 2Bi”), which are the lowest of the six pairs of CT states for the 5-unit chain; higher CT states are similar to these with longer bonds. States at even higher energy are characterized by multiple excitations, and are only optically relevant if they are in the A, subspace, as these states can be reached by excited state absorption from the 1Bp). Within the t2 = 0 limit, besides the ground state only the high energy SS states are dipole coupled to the lB,$‘). Here, the lowest such state is the 20Ap) (see Fig. 3). The 20Ap) is dominated by doubly excited units, and is clearly a SS excitation. The 20Ap) is lower in energy than twice the energy of the lB,, which is the energy of the threshold of the two-exciton continuum in the long chain limit [lo-121. The 2OAp’ is then a bound state of two excitons, or a biexciton. At still higher energy are SS states in which the pairs of singlet excitations are seperated. The lowest of these states is the 58Ap), shown in Fig.
--.+++.++7 +0.541,,,1
--lL?@ " = $0.67++,+++I; 8A@ 9
/
20A@ 9 =
1
+ 0.161 ;+cEr
1
+0.111,st,1
1
+0.55,,1,,
tO.38;;;;I
2B@=0.92,,&, u
tO.61,,+- --+c+c to.49;;-H+ctc+c
to.37;,11;
tO.26;;;;; 58Af
=
+O.Sl,I,,I
tO.49;I;I;
+0.341,,,1
+0.23,+,1&
+0.35;,1;1
Figure 3. Dominant contributions to the normalized wavefunctions of optically relevant states in the limit of t2 = 0. Note that the diagrams are not orthogonal, and thus coefficients greater than unity are possible.
3. The energy of the 58Ap) is almost exactly twice the energy of the lBp), and thus, along with the identification as a SS state, can be classified as the threshold of the two-exciton continuum. Within the t2 = 0 limit, then, a simple picture of the effects of Coulomb interactions has emerged. Within the noninteracting Hiickel model, the single excitations form a band. With the addition of Coulomb interacitons, the single excitations split into Frenkel excitons (the lBi”) in Fig. 3), and CT excitations (the 8Ap) and 2B&O) in Fig. 3), which are conduction band states in the Frenkel limit. The splitting of singly excited states into exciton and band-like states is well known. What is more interesting here is the effect of Coulomb interactions on the doubly excited configurations. The double excitations, which also form a band in the Hiickel limit, split into two broad classes of states. The first are the much discussed TT excitations [13] which occur below the Frenkel exciton and play no role in optical processes in this limit. The second class of states consists of SS excitations, which occur above the threshold of the CT excitations. The SS excitations are further split by Coulomb interactions into bound biexciton or free two-exciton continuum states. These two-exciton states are strongly coupled to the Frenkel exciton, and we expect them to play a strong role in the photophysics
M. Chandross et al. /Synthetic Metals 85 (1997) 1001-I 006
1004
2‘4,: +0.30,,~, -0.27+++++
+0.13,,11, -O.l4,;;cI, +0.12,1,1, to.10,,1,1
-0.13; g5z+yx
Figure 4. The dominant contributions to the exact lB, state of the 5-unit chain within the exciton basis. The contribution of the diagram in which the CT bond is over the entire 5 units is very small (< 0.05).
of n-conjugated 5. Nonzero
polymers.
interunit
charge-transfer
We now present the results of calculations within the full Hamiltonian of Eq. 1. We will show that even though the addition of interunit charge transfer necessarily makes the wavefunctions more complicated, the general classifications of eigenstates as predominantly CT, TT, and SS persisits. We have calculated the exact energies and wavefunctions of the A, and B, subspaces within Eq.(l) for the 4- and 5-unit cases, for various extended Hubbard models and Pariser-Parr-Pople models. The most important states are discussed below for the extended Hubbard model with U = 3t and V = t, where t is the mean hopping parameter. The results with other parameterizations of the Coulomb interactions, such as the Mataga-iYishimoto or Ohno parameters, are similar. We have performed calculations for a wide range of values of the bond alternation parameter S, defined by tl = tc(l + S) and t2 = ts(l - 5). One of the benefits of calculations within the exciton basis is that wavefunctions can be easily interpreted for all 6. We present results here for 5 = 0.1 which is most appropriate for linear chain systems such as polyacetylene. Larger 6 can be used to simulate bond alternation in different systems, such as polysilane and poly(para-phenylene) (PPP) [14]. lA,: The exact lA, is very similar to the 1Ap). The important difference is the additional contributions from diagrams with interunit charge transfer which stabilize the ground state and lower its energy relative to the noninteracting ground state. lB,: The dominant contributions to the lB, for N = 5 are shown in Fig. 4. The largest contributions to the wavefunction come from the Frenkel exciton diagrams and the diagrams with nearest neighbor CT. The contributions of CT diagrams with bonds longer
--+0.12,&X-++
+0.1o,,&Z
-----
-05%+it+t+kit
Figure 5. The dominant contributions to the exact 2A, state of the 5-unit chain within the exciton basis. The wavefunction is still predominantly TT, with small contributions from CT diagrams. The second pair of diagrams corresponds to a TT excitation accompanied by CT.
than nearest neighbor decrease rapidly with the length of the bond. This is a clear signature of the exciton nature of the lB, in the long chain limit. 2A,: The 2A,, shown in Fig. 5, is still predominantly TT (note the nearly 2:l weights of the pairs of diagrams within the brackets), with weak contributions from CT diagrams. This result is in agreement with the earlier work of Tavan and Schulten [13], who demonstrated the TT nature of the 2A, from energetic considerations, and that of Ohmine et al. [15] in a renormalized CI approach. For these parameters, the 3A, and 4A, are also predominantly TT, and are not shown. All TT states have weak dipole couplings with the lB,; as the transition from the lB, to the TT states not only would involve a second excitation, but also spin flips within each unit, a process forbidden in optical excitation. mA, : Previous calculations by several groups have shown that there exists an above gap A, state with unusually large dipole coupling with the lB,. This state is now commonly refeued to as the mAg, where m depends on the parameters of the Hamiltonian and the chain length [16]. The mA, plays a very strong role in the photophysics of n-conjugated polymers, due to its large dipole coupling with the lB,. In the present case m = 5. We find that the mA, is substantially different from all lower A, states, in that it is not TT but predominantly CT. We show the dominant contributions to the mA, in Fig. 6, where it is clear that the mAg is a correlated CT excitation with longer bonds than the lB,. Although a diagram of the type (f) does appear in the wavefunction of the mAs, (see Fig. 6), the corresponding diagram of type (e) is absent, indicating that the mA, has no TT character.
M Chandross et ai. /SyntheticMetals ---
---
5A,:-0.32;-
+0.24;&,+
$0.20;;~~
--
+0.19&
--
-0.16,++,+ x--H
----
85 (1997) 1001-1006 21A, =+o.ZS;;tt+t-;
1005 ----if-o.zo+
+f+
-----+t -
+
+0.34+
+i- +
+t-
-O.Zl,,E, ---0.26+
+0.17;;;=;
++~
--ZSA,=+0.25&++$).
+0.15+c&&
+0.24+
---++-*-+
---0.20; &ye+
--
-$0.18&&
Figure 6. The dominant contributions to the exact mAg state of the 5-unit chain within the exciton basis. The mAg is a CT exciton with bond lengths that are, on average, longer than those in the lB, exciton.
-0.15;1;;1
+o.zO+c.I+c;;-;I:
-O.lG&+~
---
-0.15&p-++I
+O.lF;T&+i-I
-0.15;1~1~
Figure 8. The dominant contributions to the biexciton (21A,) state and to the threshold state of the twoexciton continuum (28A,).
2345678910 A, State
Figure 7. The number of excitations from the exact correlated ground state for the lowest eight excited A, states for N = 5 with S = 0.1, U = 3t and V = t. The 5A, is the mA, for these parameters. Like the 5A,, the SA, is also a CT excitation.
The mA, also has substantial contributions from double excitations, and it is because of this it has also been referred to as the biexciton [17]. This is, however, incorrect. We have calculated the exact expectation value of the number of excitations in all eigenstates. In Fig. 7 we show a plot of this quantity for the lowest eight excited A, states, relative to the number of excitations in the correlated ground state. Note that the number of excitations is fairly large for the 2A, - 4A,, but then decreases sharply at the 5A,, indicating clearly the two excitation TT character of the 2A, - 4A, and the correlated one excitation CT character of the mA,. The S’S states: The true biexciton
should belong to
the class of S’S states, which occur at much higher energy. In the long chain limit, the biexcitons should be lower in energy than the two-exciton continuum states which should occur at 2 x E(lB,) [lo-la], where E(lB,) is the energy of the lB,. In short chains, however, both types of states are higher in energy than 2 x E(lB,). Th is is a finite size effect which is easy to understand. The lB, exciton in Fig. 4 can be seen to have a width of about four units. Finite chains are unable to accomodate two such excitons, and the individual excitons in the finite chain two exciton states have much smaller widths. This raises the relative energies of the two exciton states. What is more important than energies in the present context is that we find distinct SS states which are dominated either by highly localized two exciton diagrams, or by two excitons that are separated from each other. In Fig. 8 we have shown the dominant contributions to the eigenstates that have the characters of the biexciton and two free excitons. The absence of diagrams with seperated exciton pairs in the 21A, (see Fig. 8) is an indication of biexciton formation. Similarly, the absence of diagrams with localized exciton pairs in the 28A, (see Fig. 8) indicates that this state is a two-exciton continuum state. This is the first direct evidence of biexciton binding in conjugated systems; earlier biexciton studies were only able to demonstrate binding in an indirect manner [lo]. We have found similar behavior with the Ohno and Mataga-Nishimoto parameterizations of the Coulomb interactions. 6.
Photophysics
of r-conjugated
polymers
Existing calculations already show that the mA, plays a very strong role in the photophysics of 7rconjugated polymers. Here we have explicitly shown
1006
U Chandross et al. /Synthetic Metals 85 (I 997) 1001-l 006
that the mAg state is predominantly CT (single excitation) in character. This explains the similarity between our earlier conclusions [16] and those obtained within the single CI studies [IS], even though the latter miss the 2A,. The biexciton is also expected to play a strong role in the photophysics, and we expect strong photoinduced absorption (PA) to the lowest biexciton from the optical exciton. In poly(para-phenylene vinylene) (PPV) and related materials, PA to both the mA, [19,20] and the biexciton [21,19] have been seen, while the mA, has been reached also in electroabsorption (EA) [21] and d irect two-photon absorption [22]. We believe that the biexciton has also been seen in PA studies of the polydiacetylenes [23]. The energies of the mA, and biexciton can give independent lower limits for the exciton binding energy. The classification of the mA, as an exciton indicates that the energy seperation between the lB, and the mA, is a lower limit for the binding energy. A lower limit for the biexciton energy is given by E(BX) 2 2 x E(lB,) - B.E., where E(BX) is the energy of the biexciton, E(lB,) is the energy of the lB, ) and B.E. is the exciton binding energy. The experimentally measured energies of the lB, and the biexciton can then also give a lower limit for the exciton binding energy. In PPV both ps PA and direct TPA place the mA, N 0.5 - 0.7 eV above the lB, (note that TPA cannot be explained within the interchain exciton model [19], and must be to the mA,). The high energy ps PA is to the biexciton, which is N 1.4 - 1.7 eV above the lB,. Both experiments give a lower limit for the exciton binding energy of 0.7 eV [7,26]. 7. Conclusion The exciton basis allows for a systematic characterization of all excited states of r-conjugated polymers. The photophysics is dominated by two A, states; (i) the m4, which is an exciton with greater charge-transfer than the lB,, and (ii) the biexciton or bound state of two excitons. For small 6, the 2A, is below the lB,, and is a TT excitation [13,15] that plays a weak role in nonlinear optical processes. For larger 5 the 2A, is above the lB,, and the TT and CT states are strongly mixed. The exciton basis is easily extended to other linear chain polymers, such as the polyphenylenes. The simplest polyphenylene, poly(para-phenylene) can be considered to be a chain of coupled phenyl units [24], allowing detailed wavefunctions analysis similar to the above 1251~ 8. Acknowledgements This work was supported by the NSF (Grant No. ECS-9408810), the AFOSR, and the ONR through the
Center for Advanced Multifunctional Nonlinear Optical Polymers and Molecular Assemblies (CAMP) at the University of Arizona. REFERENCES 1. L. Salem, The Molecular Orbital Theory of Conjugated Systems, (Benjamin, N.Y. 1966). 2. B.S. Hudson and B.E. Kohler, Chem. Phys. Lett. 14, 305 (1972). 3. K. Schulten, I. Ohmine and M. Karplus, 64, 4422 (1976). 4. B.S. Hudson, B.E. Kohler, and K. Schulten in Excited States, Vol. 6, edited by E.C. Lim (Academic Press, 1982)) pp. l- 95. 5. S. Ramasesha and Z.G. Soos, J. Chem. Phys. 80, 3278 (1984). 6. W.T. Simpson, J. Am. Chem. Sot. 73, 5363 (1951). 7. S. Mazumdar and M. Chandross, to appear in Primary Photoexcitations in Conjugated Polymers: Molecular Exciton versus Semiconductor Band Model, edited by N. Serdar Sariciftci (World Scientific, 1996). 8. K. Ohno, Theor. Chim. Acta. 2, 219 (1964). 9. D. Mukhopadhyay, G.W. Hayden, and Z.G. Soos, Phys. Rev. B51, 9476 (1995). 10. F. Guo, M. Chandross and S. Mazumdar, Phys. Rev. Lett., 74, 2096 (1995). 11. K. Ishida, H. Aoki and T. Ogawa, Phys. Rev. B52, 8980 (1995). 12. F.B. Gallagher and F.C. Spano, Phys. Rev. B53, 3790 (1996). 13. P. Tavan and K. Schulten, Phys. Rev. B36, 4337 (1987). 14. Z.G. Soos et al., Phys. Rev. B47, 1742 (1993). 15. I. Ohmine, M. Karplus, and K. Schulten, J. Chem. Phys. 68, 2298 (1978). 16. S.N. Dixit, D. Guo and S. Mazumdar, Phys. Rev. B43, 6781 (1991); 17. P.C.M. McWilliams, G.W. Hayden and Z.G. Soos, Phys. Rev. B43, 9777 (1991). 18. S. Abe, M. Schreiber, W.P. Su and J. Yu, Phys. Rev. B45, 9432 (1992). 19. J.W.P. Hsu et al., Phys. Rev. B49, 712 (1994). 20. S.V. Frolov et al., Bull. Am. Phys. Sot. 41, 435 (1996). 21. J.M. Leng et al., Phys. Rev. Lett. 72, 156 (1994). 22. R. Meyer et al., ibid, 41, 435 (1996). 23. T. Kobayashi, Synth. Metals 50, 565 (1992). 24. M.J. Rice and Yu.N. Gartstein, Phys. Rev. Lett. 73, 2504 (1994). 25. A. Chakrabarti and S. Mazumdar, current proceedings. 26. M. Chandross and S. Mazumdar, current proceedings.
Synthetic Metals 85 (1997) 1001-1006
Systematic
characterization
of excited
states in conjugated
polymers
M. Chandross”, Y. ShimoibJ, and S. Mazumdaralb, aDepartment
of
Physics and b Optical Sciences Center, University of Arizona, Tucson, AZ 85721, uu ‘ElectrotechnicaI Laboratory, l-1-4 Umezono, Tsukuba 305, Japan.
Abstract We present the first complete systematic characterization of the excited states in conjugated polymers. Our results are relevant for the understanding of the photophysics of these materials. We perform full configuration interaction calculations in an exciton basis within which a long chain polymer is considered as coupled molecular units. Complete pictorial descriptions of all excited states are obtained. In linear chain polymers such as the polyacetylenes and polydiacetylenes the lB, is an exciton, and the fundamental two-photon states can be broadly In the above CT classified into triplet-triplet (TT), charge-transfer (CT) and singlet-singlet (SS) excitations. refers to charge-transfer from one unit to another, and TT and SS are two electron-two hole excitations. In TT the spin angular momenta of two different triplet excitations combine to give an overall singlet, while the individual excitations are singlets in SS. The 2A, is classified as TT. The mA,, an even parity state that plays a strong role in nonlinear optics, is a correlated CT state. The SS states occur higher in energy and for moderate exciton binding split into the biexciton and two-exciton continuum. The calculations can be easily extended to the polyphenylenes, for which the characterization of excited states continues to be possible. These theoretical results are useful in explaining a variety of third order nonlinear optical spectroscopic measurements as well as picosecond photoinduced absorption. lieywords: Semi-empirical sorption spectroscopy
models and model calculations;
1. Introduction The theoretical study of the effects of electron correlations on the energy spectra of n-conjugated molecules has a long history. In the 1950s configuration interaction (CI) was first introduced between singly excited configurations that are degenerate at the Hiickel level, [I-]. Coulomb interactions lift the degeneracies of the Hiickel model and form new eigenstates from linear combinations of the Hiickel wavefunctions. This first order CI proved to be useful in interpreting the optical absorption spectra of a number of conjugated systems [I]. In the 1970s it was observed -that the lowest even parity excited state (2A,) is located energetically below the lowest odd parity state (lB,) [a]. The correlated 2A, was found to be a linear combination of single and double excitations, providing direct evidence of the role of Coulomb interactions in r-conjugated polyenes [351. These results point to a question that, to the best of our knowledge, has not yet been addressed, namely, are 0379-6779/97/%17.00 0 1997 Elsevier Science S.A. All rights reserved PII SO379-6779(96)04248-8
Multiphoton
absorption spectroscopy; Photoinduced
ab-
the double excitations themselves split by Coulomb interactions in a manner similar to the single excitations? The role that Coulomb interactions play in the photophysics of conjugated polymers has remained a difficult problem. Accurate CI calculations are only possible for short chains which have discrete energy levels, and thus the character of a given state is not obvious from energetic considerations alone. However, the short chain wavefunctions gradually evolve into localized exciton or delocalized band states as the chain length is increased, and thus must have the same characteristics as their long chain counterparts. It is then possible, in principle, to classify states by their wavefunctions. This is what is acheived in the present work, using complete CI calculations in an exciton basis (see below)[6,7]. CI calculations are usually performed within delocalized molecular orbital (MO) or localized valence bond (VB) representations where simple physical pictures of excited states are possible in the limits of weak and
M. ChandToss et al. /Synthetic Metals 85 (1997) 1001-I 006
1002
strong Coulomb interactions, respectively. For the region of intermediate Coulomb interactions appropriate for conjugated polymers, however, wavefunctions within both of these representations are complicated superpositions of a large number of basis states, and simple physical interpretations are nearly impossible. Our calculations are performed in a hybrid of the hiI and VB bases which considers an ideal polyacetylene chain to be coupled ethylene units [6,7]. Within this basis, correlated wavefunctions are clearly dominated by a small number of configurations, and a systematic classification of all excited states becomes possible. All wavefunctions can be represented in a simple, pictorial fashion, giving an intuitive understanding of their natures. 2. Theoretical
model
Our theoretical calculations are performed within a Pariser-Parr-Pople type Hamiltonian, H, written as,
+
- Ct2&&2i+l,o
i,o
4i+l,oC2d,o)
su c ni,tni,j+ f c Kj:j(% - l)(nj - I),(1) i %a3 where ci,,(ci+,) creates (annihilates) a n-electron of spin u on carbon atom i, ni,a = c/,~c~,~, and ni = Co ni,a. The parameters tl and t2 (tl > t2) are the nearest neighbor intra- and inter-unit hopping integrals, respectively, and U and xj describe the on-site and intersite Coulomb interactions, respectively. We have performed calculations for a number of different parameterizations of the Coulomb interactions, including the Ohno, Mataga-Nishimoto, and extended Hubbard models. We present results here for calculations within the extended Hubbard model; the conclusions are identical for the other parameterizations. 3. The
exciton
basis
Within the exciton basis approach we visualize polyacetylene as coupled ethylenic dimer units. Each dimer contributes one bonding and one antibonding MO, and the MOs of different units are coupled by Coulomb interactions as well as interunit hopping. The Hamiltonian in Eq. 1 is rewritten in terms of the dimer operators, %,A,, =
2-1’2[C2i-l,a
+
(-1yc2i,o],
(2)
where X = l(2) corresponds to the bonding (antibonding) MO of ethylenic dimer unit i. All many-electron
functions can now be represented by dimer unit basis functions, examples of which are shown in Fig. 1 for the simple case of butadiene (2-units). In Fig. 1 and the following we will represent pairs of singlet spin-bonded singly occupied MOs by a connecting line. xx
;I
pz
-+(;-
g1
z
Cc)
63
W
0-I
Figure 1. Typical exciton basis diagrams for butadiene.
Fig. l(a) represents the “noninteracting” ground state of butadiene, which is simply the product wavefunction of the ground states of the individual ethylene units. The single excitations in Figs. l(b) and (c) are easy to understand: these are Frenkel intraunit excitations and charge-transfer (CT) excitations. These are the only types of singly excited diagrams. Double excitations within the exciton basis can involve the same unit, as in Fig. l(d), in which case the excitations are necessarily both singlet. Double excitations can also involve two different units in which case two linearly independent bonding arrangements are possible, as shown in Figs. l(e) and (f). In diagrams (e) and (f) the individual singlet bonds are within a unit and between different units, respectively. As shown in Fig. 2, the linear combination of diagrams (e) and (f) with the ratio 1:2 is equivalent to a triplet-triplet (TT) excitation, in which pairs of triplet excitations on different units combine to give an overall singlet diagram. We refer to diagrams of the types (d) and (e), in which the two excitations are singlets as singlet-singlet or SS excitations. The key to understanding the photophysics of n-conjugated polymers is the distinction between these two classes of double excitations. All eigenstates within the exciton basis are linear combinations of Frenkel, CT, TT and SS basis functions, as well as more complex configurations that are derived from these (note, however, that even these complex functions have pictorial interpretations). The beauty of the exciton basis approach is that the exact eigenstates, in spite of being correlated, can have simple interpretations, as shown below.
4. The
limit
of zero
interunit
charge-transfer
Before examining the wavefunctions of the full Hamiltonian, it is useful to discuss the effects of Coulomb interactions in the limit of zero interunit charge-transfer (i.e. t2 = 0 in Eq. 1). Some of the wavefunctions in this limit have also been discussed recently by other
M. Chandross et al. /SyntheticMetals
1003
85 (1997) 1001-1006 1Af
=
----+0.88y,++++*+t
----++ -0.28+,++++-
-O.ZS,,,?H,
-0.20,,-,, -A+.--
Figure 2. Linear combination of exciton basis diagrams that gives a triplet-triplet excitation. Bonds with arrows denote a triplet spin coupling between the MOs.
authors [9]. In Fig. 3 we show the dominant contributions to the optically relevant wavefunctions in order of increasing energy for 5 units with U = 3tr and V = tl. In Fig. 3 and the following we only show one diagram to represent all diagrams related by mirror-plane and electron-hole symmetries. The superscript (0) on the wavefunctions in Fig. 3 indicates that these are wavefunctions in the limit t2 = 0. The 1Ap) ground state, as expected, consists almost entirely of the noninteracting ground state with small contributions from doubly excited diagrams. The next higher excited state is the 2A$) which is seen to be entirely composed of TT configurations (compare with Fig. 2). The 2Ap) is the lowest of a “band” of TT states below the 1Bi’); the other TT states are similar to the 2Aco) 9 , with the only difference being the seperation between the two triplet excitations. Transitions between the TT states and the lBi”) are strictly forbidden. The lBi”) is a Frenkel exciton in this limit (see Fig. 3), and is strongly dipole coupled to the ground state. There are three such Frenkel excitons for N = 5, with the 1Bp) being the most strongly coupled to the 1Ap). Above the Frenkel exciton are CT states in which the electron and hole are on different units. The CT states in the A, and B, subspaces are degenerate in the limit tz = 0. In Fig. 3, we show the 8Ap) and the 2Bi”), which are the lowest of the six pairs of CT states for the 5-unit chain; higher CT states are similar to these with longer bonds. States at even higher energy are characterized by multiple excitations, and are only optically relevant if they are in the A, subspace, as these states can be reached by excited state absorption from the 1Bp). Within the t2 = 0 limit, besides the ground state only the high energy SS states are dipole coupled to the lB,$‘). Here, the lowest such state is the 20Ap) (see Fig. 3). The 20Ap) is dominated by doubly excited units, and is clearly a SS excitation. The 20Ap) is lower in energy than twice the energy of the lB,, which is the energy of the threshold of the two-exciton continuum in the long chain limit [lo-121. The 2OAp’ is then a bound state of two excitons, or a biexciton. At still higher energy are SS states in which the pairs of singlet excitations are seperated. The lowest of these states is the 58Ap), shown in Fig.
--.+++.++7 +0.541,,,1
--lL?@ " = $0.67++,+++I; 8A@ 9
/
20A@ 9 =
1
+ 0.161 ;+cEr
1
+0.111,st,1
1
+0.55,,1,,
tO.38;;;;I
2B@=0.92,,&, u
tO.61,,+- --+c+c to.49;;-H+ctc+c
to.37;,11;
tO.26;;;;; 58Af
=
+O.Sl,I,,I
tO.49;I;I;
+0.341,,,1
+0.23,+,1&
+0.35;,1;1
Figure 3. Dominant contributions to the normalized wavefunctions of optically relevant states in the limit of t2 = 0. Note that the diagrams are not orthogonal, and thus coefficients greater than unity are possible.
3. The energy of the 58Ap) is almost exactly twice the energy of the lBp), and thus, along with the identification as a SS state, can be classified as the threshold of the two-exciton continuum. Within the t2 = 0 limit, then, a simple picture of the effects of Coulomb interactions has emerged. Within the noninteracting Hiickel model, the single excitations form a band. With the addition of Coulomb interacitons, the single excitations split into Frenkel excitons (the lBi”) in Fig. 3), and CT excitations (the 8Ap) and 2B&O) in Fig. 3), which are conduction band states in the Frenkel limit. The splitting of singly excited states into exciton and band-like states is well known. What is more interesting here is the effect of Coulomb interactions on the doubly excited configurations. The double excitations, which also form a band in the Hiickel limit, split into two broad classes of states. The first are the much discussed TT excitations [13] which occur below the Frenkel exciton and play no role in optical processes in this limit. The second class of states consists of SS excitations, which occur above the threshold of the CT excitations. The SS excitations are further split by Coulomb interactions into bound biexciton or free two-exciton continuum states. These two-exciton states are strongly coupled to the Frenkel exciton, and we expect them to play a strong role in the photophysics
M. Chandross et al. /Synthetic Metals 85 (1997) 1001-I 006
1004
2‘4,: +0.30,,~, -0.27+++++
+0.13,,11, -O.l4,;;cI, +0.12,1,1, to.10,,1,1
-0.13; g5z+yx
Figure 4. The dominant contributions to the exact lB, state of the 5-unit chain within the exciton basis. The contribution of the diagram in which the CT bond is over the entire 5 units is very small (< 0.05).
of n-conjugated 5. Nonzero
polymers.
interunit
charge-transfer
We now present the results of calculations within the full Hamiltonian of Eq. 1. We will show that even though the addition of interunit charge transfer necessarily makes the wavefunctions more complicated, the general classifications of eigenstates as predominantly CT, TT, and SS persisits. We have calculated the exact energies and wavefunctions of the A, and B, subspaces within Eq.(l) for the 4- and 5-unit cases, for various extended Hubbard models and Pariser-Parr-Pople models. The most important states are discussed below for the extended Hubbard model with U = 3t and V = t, where t is the mean hopping parameter. The results with other parameterizations of the Coulomb interactions, such as the Mataga-iYishimoto or Ohno parameters, are similar. We have performed calculations for a wide range of values of the bond alternation parameter S, defined by tl = tc(l + S) and t2 = ts(l - 5). One of the benefits of calculations within the exciton basis is that wavefunctions can be easily interpreted for all 6. We present results here for 5 = 0.1 which is most appropriate for linear chain systems such as polyacetylene. Larger 6 can be used to simulate bond alternation in different systems, such as polysilane and poly(para-phenylene) (PPP) [14]. lA,: The exact lA, is very similar to the 1Ap). The important difference is the additional contributions from diagrams with interunit charge transfer which stabilize the ground state and lower its energy relative to the noninteracting ground state. lB,: The dominant contributions to the lB, for N = 5 are shown in Fig. 4. The largest contributions to the wavefunction come from the Frenkel exciton diagrams and the diagrams with nearest neighbor CT. The contributions of CT diagrams with bonds longer
--+0.12,&X-++
+0.1o,,&Z
-----
-05%+it+t+kit
Figure 5. The dominant contributions to the exact 2A, state of the 5-unit chain within the exciton basis. The wavefunction is still predominantly TT, with small contributions from CT diagrams. The second pair of diagrams corresponds to a TT excitation accompanied by CT.
than nearest neighbor decrease rapidly with the length of the bond. This is a clear signature of the exciton nature of the lB, in the long chain limit. 2A,: The 2A,, shown in Fig. 5, is still predominantly TT (note the nearly 2:l weights of the pairs of diagrams within the brackets), with weak contributions from CT diagrams. This result is in agreement with the earlier work of Tavan and Schulten [13], who demonstrated the TT nature of the 2A, from energetic considerations, and that of Ohmine et al. [15] in a renormalized CI approach. For these parameters, the 3A, and 4A, are also predominantly TT, and are not shown. All TT states have weak dipole couplings with the lB,; as the transition from the lB, to the TT states not only would involve a second excitation, but also spin flips within each unit, a process forbidden in optical excitation. mA, : Previous calculations by several groups have shown that there exists an above gap A, state with unusually large dipole coupling with the lB,. This state is now commonly refeued to as the mAg, where m depends on the parameters of the Hamiltonian and the chain length [16]. The mA, plays a very strong role in the photophysics of n-conjugated polymers, due to its large dipole coupling with the lB,. In the present case m = 5. We find that the mA, is substantially different from all lower A, states, in that it is not TT but predominantly CT. We show the dominant contributions to the mA, in Fig. 6, where it is clear that the mAg is a correlated CT excitation with longer bonds than the lB,. Although a diagram of the type (f) does appear in the wavefunction of the mAs, (see Fig. 6), the corresponding diagram of type (e) is absent, indicating that the mA, has no TT character.
M Chandross et ai. /SyntheticMetals ---
---
5A,:-0.32;-
+0.24;&,+
$0.20;;~~
--
+0.19&
--
-0.16,++,+ x--H
----
85 (1997) 1001-1006 21A, =+o.ZS;;tt+t-;
1005 ----if-o.zo+
+f+
-----+t -
+
+0.34+
+i- +
+t-
-O.Zl,,E, ---0.26+
+0.17;;;=;
++~
--ZSA,=+0.25&++$).
+0.15+c&&
+0.24+
---++-*-+
---0.20; &ye+
--
-$0.18&&
Figure 6. The dominant contributions to the exact mAg state of the 5-unit chain within the exciton basis. The mAg is a CT exciton with bond lengths that are, on average, longer than those in the lB, exciton.
-0.15;1;;1
+o.zO+c.I+c;;-;I:
-O.lG&+~
---
-0.15&p-++I
+O.lF;T&+i-I
-0.15;1~1~
Figure 8. The dominant contributions to the biexciton (21A,) state and to the threshold state of the twoexciton continuum (28A,).
2345678910 A, State
Figure 7. The number of excitations from the exact correlated ground state for the lowest eight excited A, states for N = 5 with S = 0.1, U = 3t and V = t. The 5A, is the mA, for these parameters. Like the 5A,, the SA, is also a CT excitation.
The mA, also has substantial contributions from double excitations, and it is because of this it has also been referred to as the biexciton [17]. This is, however, incorrect. We have calculated the exact expectation value of the number of excitations in all eigenstates. In Fig. 7 we show a plot of this quantity for the lowest eight excited A, states, relative to the number of excitations in the correlated ground state. Note that the number of excitations is fairly large for the 2A, - 4A,, but then decreases sharply at the 5A,, indicating clearly the two excitation TT character of the 2A, - 4A, and the correlated one excitation CT character of the mA,. The S’S states: The true biexciton
should belong to
the class of S’S states, which occur at much higher energy. In the long chain limit, the biexcitons should be lower in energy than the two-exciton continuum states which should occur at 2 x E(lB,) [lo-la], where E(lB,) is the energy of the lB,. In short chains, however, both types of states are higher in energy than 2 x E(lB,). Th is is a finite size effect which is easy to understand. The lB, exciton in Fig. 4 can be seen to have a width of about four units. Finite chains are unable to accomodate two such excitons, and the individual excitons in the finite chain two exciton states have much smaller widths. This raises the relative energies of the two exciton states. What is more important than energies in the present context is that we find distinct SS states which are dominated either by highly localized two exciton diagrams, or by two excitons that are separated from each other. In Fig. 8 we have shown the dominant contributions to the eigenstates that have the characters of the biexciton and two free excitons. The absence of diagrams with seperated exciton pairs in the 21A, (see Fig. 8) is an indication of biexciton formation. Similarly, the absence of diagrams with localized exciton pairs in the 28A, (see Fig. 8) indicates that this state is a two-exciton continuum state. This is the first direct evidence of biexciton binding in conjugated systems; earlier biexciton studies were only able to demonstrate binding in an indirect manner [lo]. We have found similar behavior with the Ohno and Mataga-Nishimoto parameterizations of the Coulomb interactions. 6.
Photophysics
of r-conjugated
polymers
Existing calculations already show that the mA, plays a very strong role in the photophysics of 7rconjugated polymers. Here we have explicitly shown
1006
U Chandross et al. /Synthetic Metals 85 (I 997) 1001-l 006
that the mAg state is predominantly CT (single excitation) in character. This explains the similarity between our earlier conclusions [16] and those obtained within the single CI studies [IS], even though the latter miss the 2A,. The biexciton is also expected to play a strong role in the photophysics, and we expect strong photoinduced absorption (PA) to the lowest biexciton from the optical exciton. In poly(para-phenylene vinylene) (PPV) and related materials, PA to both the mA, [19,20] and the biexciton [21,19] have been seen, while the mA, has been reached also in electroabsorption (EA) [21] and d irect two-photon absorption [22]. We believe that the biexciton has also been seen in PA studies of the polydiacetylenes [23]. The energies of the mA, and biexciton can give independent lower limits for the exciton binding energy. The classification of the mA, as an exciton indicates that the energy seperation between the lB, and the mA, is a lower limit for the binding energy. A lower limit for the biexciton energy is given by E(BX) 2 2 x E(lB,) - B.E., where E(BX) is the energy of the biexciton, E(lB,) is the energy of the lB, ) and B.E. is the exciton binding energy. The experimentally measured energies of the lB, and the biexciton can then also give a lower limit for the exciton binding energy. In PPV both ps PA and direct TPA place the mA, N 0.5 - 0.7 eV above the lB, (note that TPA cannot be explained within the interchain exciton model [19], and must be to the mA,). The high energy ps PA is to the biexciton, which is N 1.4 - 1.7 eV above the lB,. Both experiments give a lower limit for the exciton binding energy of 0.7 eV [7,26]. 7. Conclusion The exciton basis allows for a systematic characterization of all excited states of r-conjugated polymers. The photophysics is dominated by two A, states; (i) the m4, which is an exciton with greater charge-transfer than the lB,, and (ii) the biexciton or bound state of two excitons. For small 6, the 2A, is below the lB,, and is a TT excitation [13,15] that plays a weak role in nonlinear optical processes. For larger 5 the 2A, is above the lB,, and the TT and CT states are strongly mixed. The exciton basis is easily extended to other linear chain polymers, such as the polyphenylenes. The simplest polyphenylene, poly(para-phenylene) can be considered to be a chain of coupled phenyl units [24], allowing detailed wavefunctions analysis similar to the above 1251~ 8. Acknowledgements This work was supported by the NSF (Grant No. ECS-9408810), the AFOSR, and the ONR through the
Center for Advanced Multifunctional Nonlinear Optical Polymers and Molecular Assemblies (CAMP) at the University of Arizona. REFERENCES 1. L. Salem, The Molecular Orbital Theory of Conjugated Systems, (Benjamin, N.Y. 1966). 2. B.S. Hudson and B.E. Kohler, Chem. Phys. Lett. 14, 305 (1972). 3. K. Schulten, I. Ohmine and M. Karplus, 64, 4422 (1976). 4. B.S. Hudson, B.E. Kohler, and K. Schulten in Excited States, Vol. 6, edited by E.C. Lim (Academic Press, 1982)) pp. l- 95. 5. S. Ramasesha and Z.G. Soos, J. Chem. Phys. 80, 3278 (1984). 6. W.T. Simpson, J. Am. Chem. Sot. 73, 5363 (1951). 7. S. Mazumdar and M. Chandross, to appear in Primary Photoexcitations in Conjugated Polymers: Molecular Exciton versus Semiconductor Band Model, edited by N. Serdar Sariciftci (World Scientific, 1996). 8. K. Ohno, Theor. Chim. Acta. 2, 219 (1964). 9. D. Mukhopadhyay, G.W. Hayden, and Z.G. Soos, Phys. Rev. B51, 9476 (1995). 10. F. Guo, M. Chandross and S. Mazumdar, Phys. Rev. Lett., 74, 2096 (1995). 11. K. Ishida, H. Aoki and T. Ogawa, Phys. Rev. B52, 8980 (1995). 12. F.B. Gallagher and F.C. Spano, Phys. Rev. B53, 3790 (1996). 13. P. Tavan and K. Schulten, Phys. Rev. B36, 4337 (1987). 14. Z.G. Soos et al., Phys. Rev. B47, 1742 (1993). 15. I. Ohmine, M. Karplus, and K. Schulten, J. Chem. Phys. 68, 2298 (1978). 16. S.N. Dixit, D. Guo and S. Mazumdar, Phys. Rev. B43, 6781 (1991); 17. P.C.M. McWilliams, G.W. Hayden and Z.G. Soos, Phys. Rev. B43, 9777 (1991). 18. S. Abe, M. Schreiber, W.P. Su and J. Yu, Phys. Rev. B45, 9432 (1992). 19. J.W.P. Hsu et al., Phys. Rev. B49, 712 (1994). 20. S.V. Frolov et al., Bull. Am. Phys. Sot. 41, 435 (1996). 21. J.M. Leng et al., Phys. Rev. Lett. 72, 156 (1994). 22. R. Meyer et al., ibid, 41, 435 (1996). 23. T. Kobayashi, Synth. Metals 50, 565 (1992). 24. M.J. Rice and Yu.N. Gartstein, Phys. Rev. Lett. 73, 2504 (1994). 25. A. Chakrabarti and S. Mazumdar, current proceedings. 26. M. Chandross and S. Mazumdar, current proceedings.