Systematic characterization of excited states in π-conjugated polymers

28 November 1997

Chemical Physics Letters 280 Ž1997. 85–90

Systematic characterization of excited states in p-conjugated polymers M. Chandross a , Y. Shimoi b, S. Mazumdar

c,d

a

NCCOSC, RDT & E DiÕision, Code D364, San Diego, CA 92152, USA b Electrotechnical Laboratory, 1-1-4 Umezono, Tsukuba 305, Japan c Department of Physics, UniÕersity of Arizona, Tucson, AZ 85721, USA d Optical Sciences Center, UniÕersity of Arizona, Tucson, AZ 85721, USA Received 15 May 1997; in final form 16 September 1997

Abstract Exact finite chain calculations within a diagrammatic exciton basis give pictorial descriptions of all excited states of p-conjugated polymers. In linear chain polymers the 1B u is an exciton, and the fundamental two-photon states can be broadly classified into triplet–triplet, charge-transfer and singlet–singlet excitations. The mA g , a two-photon state that plays a strong role in nonlinear optics, is a correlated charge-transfer state. The singlet–singlet states occur higher in energy and split into bound biexcitons and a continuum of free two-exciton states. q 1997 Elsevier Science B.V.

It has been demonstrated that the optical 1B u state in p-conjugated polymers is an exciton w1x. The photophysics of these systems is determined by two-photon states of A g symmetry that are strongly dipole-coupled with the 1B u . For example, in the polyŽpara-phenylenevinylenes. ŽPPVs., a low energy ŽLE. A g state at 0.6–0.8 eV above the 1B u has been observed in electroabsorption w2x, two-photon absorption and as a photoinduced absorption ŽPA. in picosecond Žps. spectroscopy w3x. A second high energy ŽHE. A g state, 1.5 eV above the 1B u , also becomes visible as ps PA w2–4x. Understanding the origins of the LE and HE PA and the physical natures of the final states requires systematic characterizations of the excited states Žas one-exciton, band or two-exciton.. We present here such a characterization of the excited states of a linear chain polymer. The characteristics of the excited states for more complicated polymers like the PPVs are identical, with only quantitative differences Žsee below..

A systematic characterization of the excited states has remained elusive because of the intermediate strength electron–electron interactions in p-conjugated polymers w5x. Eigenstates within standard many-body calculations are superpositions of a large number of band or configuration space basis functions for intermediate interactions, and therefore difficult to interpret. This has led to several controversies. Early in the theoretical study of optical nonlinearity in linear chain polyenes, for example, it was shown that for each chain length, a specific A g state Žhereafter the mA g . has unusually large dipole coupling with the 1B u state w6x. The mA g state has been called a biexciton Žbound state of two excitons., based on its relatively high energy in short polyene chain calculations w7x. It has also been argued that the high energy found in short chain calculations is actually a finite size effect, and the mA g in the polymer is an excited one-exciton w8x. The latter conclusion would agree with approximate calcula-

0009-2614r97r$17.00 q 1997 Elsevier Science B.V. All rights reserved. PII S 0 0 0 9 - 2 6 1 4 Ž 9 7 . 0 1 0 6 5 - 8

86

M. Chandross et al.r Chemical Physics Letters 280 (1997) 85–90

tions for long chains w9x. However, the approximate calculations miss the 2A g state that occurs below the 1B u state in linear polyenes due to correlation effects w10x, and thus their relationship with the short chain calculations is not obvious. A different excited state that is considerably higher than the mA g has recently been called a biexciton w11x. However, this work was able to rigorously prove the occurrence of biexcitons only for very strong Coulomb interactions. For realistic intermediate interactions, only qualitative similarities in the dipole moment behavior with the strong interaction case could be presented. Thus, neither the two-exciton character of the proposed biexciton state nor the binding between two excitons could be substantiated for realistic Coulomb interactions. In the present Letter we present a complete characterization of the excited states in p-conjugated polymers within exact many-body short chain calculations that use a molecular exciton basis w12x. Several limiting cases of the exciton basis space have been investigated recently. These involve, Ža. limited configuration interaction ŽCI. studies of a spinless Hamiltonian that artificially decouples single and double excitations w13–15x, or Žb. the limit of unphysically large bond alternation w16x, where all transition dipole couplings are so severely perturbed that identification of the mA g is no longer possible. These are not suitable for our purpose. Our calculations use the complete exciton basis directly, and yield pictorial descriptions of all eigenstates. The limitation to short chains is not a disadvantage, as our focus is not on energies, but on waÕefunctions. As we discuss below, whether a particular finite chain eigenstate evolves into a localized exciton or delocalized bandlike state as the chain length is progressively increased can be determined by careful inspection of the detailed wavefunction in question. This remains true in spite of the strong finite size effects that confine the electrons and holes in these short chains. For example, the location of the electron with respect to the hole in a true delocalized extended state is arbitrary. The signature of such electron–hole delocalization within the exciton basis finite chain calculation is nearly equal contributions to the wavefunction by exciton basis configurations with different electron–hole separations. In contrast, the finite-chain version of a localized exciton state

would have greater contributions from exciton basis functions with short electron–hole separation than those with large electron–hole separation Žsee below.. Thus the characters of the various eigenstates are already discernible in the short chains, even though the actual electron–hole or exciton–exciton separations are not optimal. From such inspections of the wavefunctions we are able to determine the precise natures of the experimentally resolved A g states w2–4x. We are also able to present a direct demonstration of biexcitons within a standard model for p-conjugated polymers. We consider a dimerized linear conjugated chain within the Pariser–Parr–Pople ŽPPP. model w17x, H s Hee q HCT , Hee s U Ý n i ,≠ n i ,x q 12 i

Ý Vi j Ž n i y 1. Ž n j y 1. , i/j

HCT s y Ý t 1 Ž c†2 iy1, s c 2 i , s q c†2 i , s c 2 iy1, s . i,s

y Ý t 2 Ž c†2 i , s c 2 iq1, s q c†2 iq1, s c 2 i , s . .

Ž 1.

i,s

Here c†j, s creates a p –electron of spin s on carbon atom j, n j, s s c†j, s c j, s , and n j s Ýs n j, s . The parameters t 1 and t 2 are the nearest neighbor hopping integrals Ž t 1 ) t 2 ., and U and Vi j correspond to the onsite and intersite Coulomb interactions, respectively. We present results here for the Ohno parameterization w18x of the PPP model: t 1 s t Ž1 q d . s 2.6 eV, t 2 s t Ž1 y d . s 2.2 eV, U s 11.13 eV, and Vi j s UrŽ1 q 0.6117R 2i j .1r2 , where d is the bond-alternation parameter and R i j is the separation ˚ between sites i and j. The results for other in A parameterizations of the PPP model are similar w19x. We rewrite H in terms of the molecular orbitals ŽMOs. of the ethylenic dimer units, a i , l , s s 2y1r2 c 2 iy1, s q Ž y1 .

ly1

c2 i , s ,

Ž 2.

where l s 1Ž2. corresponds to the bonding Žantibonding. MO of the dimer i. In the limit Hee s 0 and t 2 s 0, the ‘‘noninteracting’’ ground state of the chain has all bonding Žantibonding. MOs doubly occupied Žempty.. Nonzero Hee and t 2 lead to CI between single, double, and higher excitations. We represent spin singlet singly occupied pairs of MOs by a bond. For illustration purposes, we show in Fig. 1Ža. a typical exciton basis diagram for a 5-unit Ž10-atom. chain.

M. Chandross et al.r Chemical Physics Letters 280 (1997) 85–90

For Hee / 0 and t 2 / 0 the exact ground state is dominated by the noninteracting ground state, and has nonzero but weak contributions from ‘‘excited’’ configurations Žsee below.. Linear absorption involves a single excitation from the correlated ground state, and nonlinear absorption a second excitation from the correlated optical state. As expected from the nature of the correlated ground state, these eigenstates are dominated by configurations that are singly or doubly excited with respect to the noninteracting ground state, and triple and quadruple excitations make weak contributions. The dominant contributions therefore determine the characters of the excited eigenstates. Double excitations can involve two different bonding and two different antibonding MOs, in which case two linearly independent bonding arrangements are possible. The two diagrams on the left hand side of the equation in Fig. 1Žb. are examples of this. The 2:1 linear combination of the two diagrams shown in Fig. 1Žb. is a triplet–triplet ŽTT. excitation, in which pairs of triplet excitations localized on different units are coupled to form an overall singlet. Before proceeding to the full Hamiltonian, it is instructive to examine the effects of nonzero Hee in the limit of t 2 s 0. Some of the A g states for this limit have been discussed recently w16x. In Fig. 2 we show the dominant contributions to the exact normalized 5-unit wavefunctions of the t 2 s 0 eigenstates most relevant for our discussion, in order of increas. consists almost ing energy. The ground state Ž1AŽ0. g entirely of the noninteracting ground state. The next higher excited state is the 2AŽ0. g , which is clearly TT Ž0. wsee Fig. 1Žb.x. The states 2AŽ0. g through 7A g form a narrow ‘‘band’’ of TT states below the 1BŽ0. u . The 1BŽ0. and the 2BŽ0. are related, and constitute the u u k s 0 and the lowest nonzero k Frenkel excitons,

Fig. 1. Ža. The exciton basis diagram Ž1r2.ŽÝs a†2,2, s a2,1, s .ŽÝs a†5,2, s a 3,1, s .< G :, where < G : is the noninteracting ground state P i a†i1 ≠ a†i1 x <0:, and <0: is the vacuum. Žb. Linear equation showing the origin of the TT exciton basis. The TT diagram shown is a linear combination of functions that have spin z-components Žq1,y1., Ž0,0. and Žy1,q1. on the two units.

87

Fig. 2. The exact N s 5 wavefunctions for PPP–Ohno Hee and t 2 s 0. Each diagram shown represents the complete set of diagrams that would be obtained by applications of mirror-plane and charge-conjugation symmetries.

where k is the center of mass momentum of the exciton in the open chain configuration. Above the Frenkel exciton states there occur charge transfer ŽCT. states, in which an electron has migrated from one unit to another. The CT states in the A g and B u subspaces are degenerate in the limit t 2 s 0. There are six such pairs of states in N s 5; we show in Fig. Ž0. 2 the lowest pair, the 8AŽ0. g and the 4B u . With the Ohno potential, the lowest doubly excited state is the Žsee Fig. 2.. We classify the 18AŽ0. 18AŽ0. g g , consisting of two singlet excitations, as singlet–singlet ŽSS.. In the absence of binding between excitons, the thresh. old of double excitations is exactly 2 = EŽ1BŽ0. u , where EŽ.... is the energy of the state within the . s 2.64 t 1 , parentheses. In the present case, EŽ1BŽ0. u . s 4.21t 1. Thus the 18AŽ0. while EŽ18AŽ0. is a g g bound state of two excitons, i.e., a biexciton. Proceeding similarly, the 36AŽ0. g , with energy 5.41t 1 s . 2 = EŽ1BŽ0. u , is the lowest state with two free excitons. Similar results persist w19x for N s 4 Žcomplete CI. and 6 Žquadruple-CI.. To summarize, at t 2 s 0, the single excitations are split into Frenkel and CT excitations, while linear combinations of double excitations split into the much discussed TT states w10,16,20x, and SS states that occur aboÕe the threshold of the CT states. The SS two-exciton states can be bound or free. We now discuss the complete Hamiltonian. We show that the exact eigenstates can be still characterized as predominantly TT, CT, SS, and combinations of these. We do not show the lowest A g states

88

M. Chandross et al.r Chemical Physics Letters 280 (1997) 85–90 Table 1 The dominant contributions to the lowest three B u states and to the mA g state. Each exciton basis diagram represents the complete set that would be obtained upon application of mirror-plane and charge-conjugation symmetries

Fig. 3. The number of excitations, relative to the exact 1A g state, in the lowest eight excited A g states in N s 5. The dashed line gives the number of excitations in the 3B u state. The inset shows the dipole couplings between the lowest nine A g states and the 1B u state.

explicitly. The 1A g still resembles 1AŽ0. g , with CT diagrams now making nonzero contributions. The exact 2A g is still TT Žaccompanied by weak CT. w10,16,20x, based on the nearly exact 2:1 contributions by pairs of diagrams of the type that describe w x 2AŽ0. g 19 . Similarly, the 3A g , the 4A g and the 5A g , which now occur above the 1B u , are also TT Žsee below.. With Ohno parameters the 6A g is the mA g w7x for the 5-unit chain Žsee inset, Fig. 3.. In Table 1, we have shown the dominant contributions to the mA g and the lowest three B u states. The 1B u has nearly equal contributions from the Frenkel and nearest neighbor CT diagrams, but considerably weaker contribution from diagrams with more distant CT. This is the classic description of the exciton w21x. The 2B u is still the lowest k / 0 version of the 1B u exciton. The 3B u is therefore the lowest B u exciton with greater CT than the 1B u . The compositions of the 3B u and the mA g are nearly identical, with comparable contributions by diagrams with second neighbor and more distant CT. The occurrence of the mA g below the 3B u exciton identifies the mA g as an even parity exciton with greater electron–hole separation than the 1B u . From Table 1, both the 3B u and the mA g have stronger contributions from double and higher excitations than the 1B u and the 2B u . In spite of this, we show that the mA g is not a correlated double excitation, as has been claimed w7x. For each eigenstate x,

we define the number of excitations Nx relative to the exact 1A g as

¦Ý a

† i ,2, s

Nx s

i ,2, s

; ¦Ý a

† i ,2, s

a i ,2, s y x

i ,2, s

;

a i ,2, s ,

Ž 3.

0

where in Eq. Ž3. the first term is the expectation value of the number of electrons in the antibonding MOs in the eigenstate in question, and the second term is the same quantity for the 1A g . In Fig. 3 we have plotted the exact Nx for the lowest eight excited A g states. The states 2A g –5A g and 7A g are identifiable as TT from their wavefunctions. The corresponding number of excitations is large, 1.5 for the 3A g and 1.7 for the rest. These numbers are smaller than 2.0 because of admixing with the CT configurations. The number of excitations decreases sharply to 1.25 for the 6A g , indicating clearly that the mA g is the lowest non-TT A g state, or equally correctly, the lowest A g one-excitation with CT character. This characterization is further confirmed from our study of the limit of Hee s 0 ŽHuckel ¨ model. within the exciton basis. The number of excitations corresponding to the noninteracting 2A g

M. Chandross et al.r Chemical Physics Letters 280 (1997) 85–90

Fig. 4. The dominant contributions to the exact 14A g , 19A g and the 25A g states. The localized character of the double excitations in the 14A g state and the two free exciton character of the 19A g and the 25A g states are obvious.

there for d s 0.1 is 1.23 within the exciton basis, eÕen though within the MO basis this number is 1.0. As we discuss elsewhere w19x, this similarity is to be expected, based on the fact that the 2A g is the lowest one-excitation CT state within the Huckel ¨ model. Additionally, Fig. 3 shows that the number of excitations in the lowest B u CT exciton Žthe 3B u . is the same as in the 6A g Žsee Fig. 3.. We have verified that this sudden decrease in the number of excitations at the mA g is a characteristic of all PPP Hamiltonians, independent of parameters w19x. Higher energy excited states can be similarly characterized. The first eigenstate which is neither TT nor CT is the 14A g , the dominant contributions to which are shown in Fig. 4. It is clear that the 14A g is the analog of the 18AŽ0. g of Fig. 2. Also shown are the 19A g and the 25A g , which are similar to the 36AŽ0. g in Fig. 2. The 14A g has almost no contribution from diagrams with separated excitons, while the 19A g and the 25A g have almost no contribution from diagrams containing localized double excitations. This clear demarcation is direct evidence for the bound biexciton character of the 14A g . The two-exciton states in Fig. 4 all occur above 2 = EŽ1B u . in the 5-unit chain. This finite size effect is understandable from Table 1, where the width of 1B u exciton is seen to be about 4 units. The electron–hole separations within individual excitons is therefore much less than optimal in two exciton states of short chains, and this raises their energies relative to the 1B u w11x. In long chains, the lowest two-exciton continuum state should occur at 2 = EŽ1B u ., and the biexciton should occur below this w11,14,15x.

89

The TT characters of the 2A g –5A g explain their weak dipole couplings with the 1B u . Transition to TT configurations from the 1B u requires a spin flip that is not allowed optically and only their weak CT components contribute to the dipole coupling. A transition from the 1B u to the mA g involves the hopping of a hole Želectron. from a bonding Žantibonding. MO to a neighboring bonding Žantibonding. MO that is a highly favorable process analogous to band motion, thus explaining the large dipole coupling of the mA g with the 1B u w6–8x. The calculated dipole moments to the biexciton and the twoexciton continuum w11x, though significant, are actually smaller than what would be expected in the long chain, where individual excitons within two-exciton states acquire greater width. In summary, clear pictorial descriptions of excited states that give physical interpretations of optical processes in linear chain p-conjugated polymers have been obtained. The photophysics is dominated by an exciton with greater electron–hole separation than the 1B u exciton, and a biexciton. The present calculations become applicable to more complicated polymers like PPV if a larger difference between t 1 and t 2 is chosen w22x. Explicit calculations show that differences for large dimerization are only quantitative w19x: CT from one unit to another costs more energy for large dimerization, and this lowers the energy of the 1B u substantially relative to the mA g . Similarly, the biexciton has contributions from both interunit and neighboring intraunit double excitations Žsee Fig. 4.. Increasing dimerization merely redistributes the relative contributions of intra- and interunit double excitations w19x. We therefore assign the LE and HE states in the PPVs w2–4x to the mA g and to the biexciton, respectively. The energy location of the mA g gives a lower bound to the binding energy of the optical exciton, indicating the need for its precise experimental determination.

Acknowledgements The authors acknowledge support from the NSF ŽECS-9408810., the AFOSR and the ONR through the MURI center ŽCAMP. at the University of Arizona.

90

M. Chandross et al.r Chemical Physics Letters 280 (1997) 85–90

References w1x Proceedings Int. Conf. Synth. Metals 1996, Synth. Metals, Vols. 84 and 85, 1997. w2x J.M. Leng, S. Jeglinski, X. Wei, R.E. Benner, Z.V. Vardeny, F. Guo, S. Mazumdar, Phys. Rev. Lett. 72 Ž1994. 156. w3x S.V. Frolov, W. Gellermann, Z.V. Vardeny, M. Ozaki, K. Yoshino, in ref. 1, Synth. Metals 84 Ž1997. 493; see also, R.K. Meyer, R.E. Benner, Z.V. Vardeny, M. Liess, M. Ozaki, K. Yoshino, Y. Ding, T. Barton, ibid 549 Ž1997.. w4x J.W. Blatchford, S.W. Jessen, L.B. Lin, J.J. Lih, T.L. Gustafson, A.J. Epstein, D.K. Fu, M.J. Marsella, T.M. Swager, A.G. MacDiarmid, S. Yamaguchi, H. Hamaguchi, Phys. Rev. Lett. 76 Ž1996. 1513. w5x D. Baeriswyl, D.K. Campbell, S. Mazumdar, in: H. Kiess ŽEd.., Conjugated Conducting Polymers, Springer, Berlin, 1992. w6x J.R. Heflin, K.Y. Wong, O. Zamani-Khamiri, A.F. Garito, Phys. Rev. B 38 Ž1988. 1573. w7x P.C.M. McWilliams, G.W. Hayden, Z.G. Soos, Phys. Rev. B 43 Ž1991. 9777. w8x D. Guo, S. Mazumdar, S.N. Dixit, F. Kajzar, F. Jarka, Y. Kawabe, N. Peyghambarian, Phys. Rev. B 48 Ž1993. 1433.

w9x S. Abe, J. Yu, W.P. Su, Phys. Rev. B 45 Ž1992. 8264. w10x B.S. Hudson, B.E. Kohler, K. Schulten, in: E.C. Lim ŽEd.., Excited States, Vol. 6, Academic Press, 1982. w11x F. Guo, M. Chandross, S. Mazumdar, Phys. Rev. Lett. 74 Ž1995. 2096. w12x W.T. Simpson, J. Am. Chem. Soc. 73 Ž1951. 5363. w13x K. Ishida, H. Aoki, T. Chikyu, Phys. Rev. B 47 Ž1993. 7594. w14x K. Ishida, H. Aoki, T. Ogawa, Phys. Rev. B 52 Ž1995. 8980. w15x F.B. Gallagher, F.C. Spano, Phys. Rev. B 53 Ž1996. 3790. w16x D. Mukhopadhyay, G.W. Hayden, Z.G. Soos, Phys. Rev. B 51 Ž1995. 9476. w17x R. Pariser, R.G. Parr, J. Chem. Phys. 21 Ž1953. 767; J.A. Pople, Trans. Farad. Soc. 49 Ž1953. 1375. w18x K. Ohno, Theor. Chim. Acta. 2 Ž1964. 219. w19x M. Chandross, Y. Shimoi, S. Mazumdar, unpublished. w20x K. Schulten, I. Ohmine, M. Karplus, J. Chem. Phys. 64 Ž1976. 4422. w21x I. Ohmine, M. Karplus, K. Schulten, J. Chem. Phys. 68 Ž1978. 2298. w22x Z.G. Soos, S. Etemad, D.S. Galvao, S. Ramasesha, Chem. Phys. Lett. 194 Ž1992. 341.