- Email: [email protected]
Self localized excitations in conjugated polymers
S~vnthetic Metals, 1 7 (1987) 107 113
1 07
SELF LOCALIZED EXCITATIONS IN CONJUGATED POLYMERS
E. J. MELE and G. W. HAYDEN Department of Physics, Laboratory for Research on the Structure of Matter, University of Pennsylvania, Philadelphia, PA 19104 (U.S.A.)
1. INTRODUCTION The SSH model for polyacetylene [1], and its generalization to nondegenerate ground state polymers [2] have provided a compact description of the self localization of electronic excitations into solitons and polarons in these systems. The analysis for degenerate ground state systems such as polyaeetylene has generated particular interest because of the prediction that photoexcited electron hole pairs are unstable to the spontaneous formation of new quasipartieles which carry the charges of the photocarriers (i.e. 4- e) but spin 0, and a number of experimental investigations have been carried out to probe the properties of these photoexcitations. A number of studies of photoexcitations in finite molecular analogs of these extended systems have also been undertaken, and in this ease a rather different description of the photoexcited properties has evolved [3]. In particular the ordering and symmetry of the slnglet excitations in short polyenes (N < 12) has been interpreted within a model in which electron electron interactions play an essential role, producing a low lying dipole forbidden "neutral" excitation below the one photon absorption edge [3], and lower lying triplet excitations. There has been considerable interest in establishing the connection between these two models. Progress has been difficult because of the necessity of both treating direct electron electron repulsion well beyond the mean field approximation for correctly describing correlated "neutral" excitations in these systems, and the need to examine these phenomena in systems which are sufficiently large to study the self localization of these excitations. In this paper we will briefly review the results of a series of numerical studies which we have carried out which relate these two descriptions of excitations in model conjugated polymers. A more complete presentation of our results is in preparation[4]. The calculations reported here employ a numerical renormalization group method to extract eigenvalues and eigenvectors for the many particle Hamiltonian (approximated in a Hubbard-Peierls model) for N electrons propagating on an N-site polyene, where N can tractably be taken in the range 12< N< 24; results will be presented for N=16. Methods of this type have previously been applied to study the pure Hubbard model[5], we have found that the scheme provides an extremely efficient method for providing accurate description of the low lying eigenstates of the interacting many particle Hamiltonian. This in turn allows us to extensively explore a self consistent geometry optimization for the various electronic excitations in the model. The calculations show that for moderately strong electron electron interactions (which should be appropriate for conjugated polymers like ( C H ) z [6,7]), the system possesses self localized 0379-6779/87/$3.50
© Elsevier Sequoia/Printed in The Netherlands
108
excitations which can be interpreted in terms of the behavior expected in the low U limit of this model. The relaxations of the low lying neutral triplet excitations and the 21A 0 state are obtained and interpreted in a simple physical picture.
11. FORMALISM The calculations presented here model the electron phonon interaction and the electron electron interaction within the Hubbard-Peierls Hamiltonian If
= ~ [t o+a(u i +1-I//)]
ia,
(c i +l,a+ci ,aq-ci ,a+ci +1,o')
K
)z
+~.. "T (u; +l--U~ U
+.~ T .;,.n;,_~ ! tff
+~
V n~ hi+ ,
i where t0 is a reference hopping amplitude, a is the linear electron phonon coupling term, u i denotes the longitudinal displacement of each backbone carbon atom from a hypothetical equal bond length structure, e i a is the annihilation operator on site i, K represents an unscreened elastic potent'ial, and U and V are the Hubbard on site and off site repulsion integrals. Our strategy will be to seek the set { u i } for which
.= 0
where the brackets denote an expectation value taken in the nth excited many electron state. The difficulty is that we will require that the electronic wavefunctions used in this evaluation properly correlate the electronic "coordinates" in the presence of the two body terms appearing in H. This is based on our expectation that the interesting interaction effects in H will appear only if our approach extends beyond a mean field treatment of U and/or V. Since the number of independent electronic configurations needed to describe N interacting electrons grows notoriously quickly with system size, we require a method for limiting t h e variational freedom in our description of the many particle wavefunctions. To efficiently identify and remove "irrelevant" degrees of freedom in this problem, we have applied a numerical "renormalization group" scheme to this calculation [5]. Our application to the N-site problem proceeds by breaking the N-site Hamiltonian into a series of smaller (typically 2 or 4 site) mutually commuting Hamiltonians by removing "bonds" from the full Hamiltonian. The size of the resulting micro-Harniltonian is chosen to be small enough that a convenient exact numerical diagonalization can he made for states characterized by total spin, S, projection Sz and particle number n. (We allow for all possible fluctuations in these quantities in the microcells). The previously excised bonds are then reintroduced into the calculation, and the cell eigenstates are combined in properly symmetrized products to provide a basis set at the next level of the calculation. The size of the basis set is made manageable by retaining only the lowest lying (in energy) product basis functions. The number that are retained is determined by constraints on storage and CPU time, we have typically found that of order 200 product states per (S, gz, n) manifold provide reasonable results; highly accurate work requires that we improve this basis via second order perturbation theory through the removed states; some technical points relating to the construction of the composite Hamiltonian and the implementation of this perturbation theory are discussed in reference 4. The method
109
can be iterated so that in each "inflation" of the calculation, the system size is doubled (three inflations of the 2-site problem produce the 16 site problem, etc.) It is important to mention that unlike some previous applications of this method [8], no attempt is made to "close" the recursion scheme, i.e. the inflated Hamiltonian does not precisely map onto a renormalized version of the microcell Hamiltonian, but grows increasingly complex as the system size expands. We view the technique as an efficient variational approach, rather than a true scaling theory.
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Figure 1. A comparison of results from an exact diagonalization of the SSH Hamiltonian (U--0) with results for the RG method: (left) ground state total energy as a function of dimerization amplitude, (right) excitation energy of lowest singlet excitation.
A virtue of this composition scheme for the many particle wavefunctions, is that short range correlations are introduced in a very accurate and simple way. In fact this method has its greatest dimculty in building in the correct delocalization associated with the one body hopping terms in the full Hamiltonian. Thus an important test of the method is how well the known spectrum of the noninteraeting (U=0, V--0) Hamiltonian can be recovered. In Figure 1 we present a comparison of the ground state energy, and excitation energy to the first excited state in the noninteracting model, determined by direct diagonalization of the SSH Hamiltonian (exact) and obtained via our renormalization group approximation. The results are plotted as a function of a coordinate representing uniform dimerization of a finite N--16 chain. The approximation is seen to work extremely well; the error in the ground state energy is within the width of the curves plotted, and errors in the excitation energy are less than 0.1 eV over the entire range studied. This demonstrates that our construction of the wavefunction is converging to the correct single determinant picture when the two body interactions are removed. The method is substantially more efficient than a typical CI or valence bond approach; for calibration, the complete calculation for the N--16 system exhausts approximately 7 CPU minutes on an IBM 3081.
110
How is this picture modified in the presence of electron electron interactions? In Figure 3 we present the analogous spectrum of self localized excitations for the case U=6.0 eV (i.e. U = 2 to), V=0. We judge that this is an appropriate value for the interaction strength from the relative size of the repulsion integrals in the P P P model. (A Monte Carlo study of the ground state of this Hamiltonian also suggests U -- 2t0). Again the ground state is uniformly dimerized, and in fact the calculations reveal a slight U-induced enhancement of the dimerization amplitude, consistent with the results of various previous ground state studies[6-8]. In the singlet excitation spectrum, we also find a state which corresponds to single particle promotion across the Peierls gap (labelled 1Bu ). We see that the charge density in this state drives the formation of a pair of kinks in the bond alternation amplitude, quite similar to the U=0 situation. A quantitative difference is that they are bound slightly more strongly in the presence of U. It is interesting that this excitonic confinement appears in the presence of U only. Grabowski et al. [9] concluded that V would be necessary for confinement of the solitonic exciton; a closer analysis of the present results shows that admixture of higher lying single particle excitations in the presence of U (which are not included in ~]) also mediate an indirect attractive interaction between the charged kinks. erhaps the most important result obtained for this value of the repulsion strength is that we indeed find the lower lying 21Ae state in the singlet excitation spectrum. The relaxed equilibrium lattice confiuration for this excitation is shown in the middle panel; it shows two well defined minima. This result can be understood in the following way. Decomposition of the 2lAg wavefunction shows a relatively large admixture from triplet configurations on either side of the chain, these in turn are coupled to form the overall spin singlet. The triplet excitation is the lowest lying excitation of the chain; it relaxes to form a pair of neutral spin 1/2 kinks (not shown) . The spectrum of low lying neutral excitations is then determined by the interaction between these spin 1/2 kinks. A repulsive interaction leads to dissociation of the pair, so that the neutral triplet and singlet excitations are degenerate, and we should consider the spin 1/2 kinks the fundamental elementary neutral excitations of the chain. For an attractive interaction, in which binding of the spins occurs and the particles overlap, the related singlet combination is found at much higher energy, and a lower lying singlet can be formed by the concerted formation of two of the low lying triplet excitations. In this limit we regard the neutral triplet excitation as the fundamental neutral excitation in the system, and the singlet as a bound complex of these triplets. An attractive interaction, leading to a bound triplet, can be mediated either by a confinement potential (in the case of a nondegenerate ground state polymer), or by system size (in the case of a finite degenerate ground state system). This latter mechanism is apparently operative in the configuration shown in Figure 3. It is interesting to note that this assignment suggests that the energy of the 2lAg excitation should occur at approximately twice the energy of the 3B, excitation, a correlation which was noted in the CI studies of Schulten et al. [10] on short polyenes. We thus interpret the singlet complex in the middle panel of Figure 3 as a bound complex of two triplet excitations; the triplet character of the constituent states is manifest in the depression of the bond alternation amplitude on either side of the chain. It is also interesting to note that even the N=16 system is barely large enough to reveal the nature of these interesting neutral self localized excitations. In fact for this degenerate ground state Hamiltonian, these arguments suggest that the 2lAg excitation decays into an unbound neutral kink pair in the truly extended (large N) system. To investigate this we have studied a more approximate description of the correlated electronic states in this Hamiltonian which is amenable to extrapolation to much longer systems, and observe the dissociation of the 2lAg in the large N limit [4]. Also since these limiting configurations are mixed quantum mechanically, we would expect a continuous evolution from the "bound" regime to the "dissociated" regime as the system size is increased. While these low lying neutral excitation have their analogs in the theory of the Heisberg spin 1/2 chain (to which this model maps in the large U limit) it is quite interesting to note that much of the physics in the presence of moderately strong interactions can be understood in terms of interactions of defects derived from the small U (SSH) description of this system.
111
III. EQUILIBRIUM GEOMETRIES In Figures 2 and 3 we present the equilibrium bond alternation patterns for the ground state and low lying excited states for a model 16 site polyene with fixed end boundary conditions. The ordinate is the staggered displacement field (.4,) and the abcissa labels the bonds. The calculations employ the representative values t o = 3.0 eV, t~ = 8.0 eV/A, K = 68.6 eV/ A2. In Figure 2 we use our computational scheme to recover some well known results from the noninteracting model (U=0,V=0). The ground state is uniformly dimerized, with slight
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Figure 2. Equilibrium lattice co~ figuration for the U=0 SSL model on a 16 site chain for ground state and three lowest singlet excitations,
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4
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14
Figure 3. Equilibrium lattice configuration for the Hubbard Peierls model (U=2t) on a 16 site chain for the ground state and two lowest singlet excitations.
ringing in the bond alternation amplitude due to the finite system size. The lowest electronic excitation corresponds to promotion of an electron across the Peierls gap, this leads to the formation of oppositely charged kinks shown in the second panel. (The kinks are oppositely charged for the singlet excitation, there is a degenerate triplet excitation (not shown) in which the defects are neutral and carry spin 1/2). The next equilibrium singlet also contains a two defect structure, the excitation corresponds to a two particle excitation across the gap between the defect states. This is an even parity singlet, and is degenerate with the odd parity singlet in the infinite chain limit. A higher lying even parity singlet is obtained by single particle excitation deeper into the conduction "band". The electronic wavefunctions in these higher lying states take the form of standing waves reflected from the two boundaries, the equilibrium lattice displacement field shows two well defined dips which reflect the peaks in the perturbed charge density.
112
IV. DISCUSSION We summarize the results of our investigation of the effects of moderately strong electron electron repulsion on the excitation spectrum of an N--16 chain with the following observations: (a) The IB u state which is accessible by one photon absorption from the ground state produces a self localized state which is quite similar to the state obtained in the U=0 limit. The equilibrium configuration produces a pair of oppositely charged solitons, and in the presence of the on site U an attractive interaction between the kinks binds these to form a solitonic exciton, similar to that described by Grabowski et al. [9]. In a nondegenerate ground state polymer, the confinement potential leads to an additional binding of the kinks, and the optical excitation can be characterized as a self trapped singlet exciton. (b) A neutral triplet excitation aB u provides the lowest lying electronic excitation in the system. It relaxes to form a neutral kink antikink pair. (Interestingly this is precisely the lowest lying triplet excitation of the noninteracting SSH model.) In a nondegenerate ground state system, the confinement potential will give rise to a residual binding of this pair; in this case the excitation resembles a self trapped triplet exciton. (c) For moderate repulsion strengths, the lowest singlet excitation is a IA. state. For an extended system with a degenerate ground state this decayYs into two neutral spin 1/2 quasiparticles, which are closely related to the neutral SSH solitons. For a system in which the spins overlap, either by an attractive interaction mediated by the confinement potential in a nondegenerate ground state polymer, or by confinement on a finite (short) chain, the singlet is better characterized as a bound complex of neutral triplet excitations. The properties of this state have been thoroughly explored by application of the P P P Hamiltonian to finite polyenes which provide a well documented example of this latter situation [e.g. ref. 3]. The consequences of these low lying neutral excitations for the observed properties of photoexcitations in conjugated polymers should be interesting to explore. They are likely to play an important role in the relaxation of photoexcitations to the ground state in this system. We believe that it is likely that the recent observation of a triplet exciton in PDA [11] is related to the self trapped triplet excitation discussed above for nondegenerate systems. There has been some speculation that the photoinduced absorption feature at 1.35 eV, is associated with a neutral excitation which is apparently found in variety of conjugated systems[12-13]. There is ongoing discussion about the possibility of intrinsic photoproduction of spin 1/2 centers in polyacetylene, which could be associated with the decay of the ~Ag state.
ACKNOWLEDGEMENTS This work was supported in part by NSF through the MRL program under DMR 84 14640, and under Grant DMR 84-05524. Helpful discussions with S. Mazudmar, F.D.M. Haldane and J. Orenstein are gratefully acknowledged.
REFERENCES 1 2 3 4
W.P. Su, J. R. Schrieffer and A. J. Heeger, Phys. Rev. B, 22 (1980) 2099. S.A. Brazovskii and N. Kirova, JETP Lett., 33 (1981) 4. B. S. Hudson and B. E. Kohler, Synth. Met., 9 (1984) 241. G. W. Hayden and E. J. Mele, Phys. Rev. B (1986), preprint.
113
5 J. W. Bray and S. T. Chui, Phys. Rev. B, !9 (1979) 4876. 6 J. E. Hirsch and M. Grabowski, Phys. Rev. Lett., 52 (1984) 1713; D. Baeriswyl and K. Maki, Ph_vs. Rev. B, 31 (1985) 6633. 7 8 9 10 11 12
Z. Soos and S. Ramasesha, Phys. Rev. B, 29 (1984) 5410. J. Hirsch, Phys. Rev. B, 22 (1980) 5259. M. Grabowski, D. Hone and J. R. Schrieffer, Phys. Rev. B, 31 (1985) 7850. K. Schulten, I. Ohmine and M. Karplus, J. Chem. Phys., 64 (1976) 4422. L. Robins, J. Orenstein and G. Levi, Phys. Rev. Lett., 56 (1986). W. P. Su, 1986 preprint. 13 W.-K. Wu and S. Kivelson, 1986 preprint. 14 F. Moraes, Y. W. Park and A. J. Heeger, Synth. Met., 13 (1986) 113.
1 07
SELF LOCALIZED EXCITATIONS IN CONJUGATED POLYMERS
E. J. MELE and G. W. HAYDEN Department of Physics, Laboratory for Research on the Structure of Matter, University of Pennsylvania, Philadelphia, PA 19104 (U.S.A.)
1. INTRODUCTION The SSH model for polyacetylene [1], and its generalization to nondegenerate ground state polymers [2] have provided a compact description of the self localization of electronic excitations into solitons and polarons in these systems. The analysis for degenerate ground state systems such as polyaeetylene has generated particular interest because of the prediction that photoexcited electron hole pairs are unstable to the spontaneous formation of new quasipartieles which carry the charges of the photocarriers (i.e. 4- e) but spin 0, and a number of experimental investigations have been carried out to probe the properties of these photoexcitations. A number of studies of photoexcitations in finite molecular analogs of these extended systems have also been undertaken, and in this ease a rather different description of the photoexcited properties has evolved [3]. In particular the ordering and symmetry of the slnglet excitations in short polyenes (N < 12) has been interpreted within a model in which electron electron interactions play an essential role, producing a low lying dipole forbidden "neutral" excitation below the one photon absorption edge [3], and lower lying triplet excitations. There has been considerable interest in establishing the connection between these two models. Progress has been difficult because of the necessity of both treating direct electron electron repulsion well beyond the mean field approximation for correctly describing correlated "neutral" excitations in these systems, and the need to examine these phenomena in systems which are sufficiently large to study the self localization of these excitations. In this paper we will briefly review the results of a series of numerical studies which we have carried out which relate these two descriptions of excitations in model conjugated polymers. A more complete presentation of our results is in preparation[4]. The calculations reported here employ a numerical renormalization group method to extract eigenvalues and eigenvectors for the many particle Hamiltonian (approximated in a Hubbard-Peierls model) for N electrons propagating on an N-site polyene, where N can tractably be taken in the range 12< N< 24; results will be presented for N=16. Methods of this type have previously been applied to study the pure Hubbard model[5], we have found that the scheme provides an extremely efficient method for providing accurate description of the low lying eigenstates of the interacting many particle Hamiltonian. This in turn allows us to extensively explore a self consistent geometry optimization for the various electronic excitations in the model. The calculations show that for moderately strong electron electron interactions (which should be appropriate for conjugated polymers like ( C H ) z [6,7]), the system possesses self localized 0379-6779/87/$3.50
© Elsevier Sequoia/Printed in The Netherlands
108
excitations which can be interpreted in terms of the behavior expected in the low U limit of this model. The relaxations of the low lying neutral triplet excitations and the 21A 0 state are obtained and interpreted in a simple physical picture.
11. FORMALISM The calculations presented here model the electron phonon interaction and the electron electron interaction within the Hubbard-Peierls Hamiltonian If
= ~ [t o+a(u i +1-I//)]
ia,
(c i +l,a+ci ,aq-ci ,a+ci +1,o')
K
)z
+~.. "T (u; +l--U~ U
+.~ T .;,.n;,_~ ! tff
+~
V n~ hi+ ,
i where t0 is a reference hopping amplitude, a is the linear electron phonon coupling term, u i denotes the longitudinal displacement of each backbone carbon atom from a hypothetical equal bond length structure, e i a is the annihilation operator on site i, K represents an unscreened elastic potent'ial, and U and V are the Hubbard on site and off site repulsion integrals. Our strategy will be to seek the set { u i } for which
.= 0
where the brackets denote an expectation value taken in the nth excited many electron state. The difficulty is that we will require that the electronic wavefunctions used in this evaluation properly correlate the electronic "coordinates" in the presence of the two body terms appearing in H. This is based on our expectation that the interesting interaction effects in H will appear only if our approach extends beyond a mean field treatment of U and/or V. Since the number of independent electronic configurations needed to describe N interacting electrons grows notoriously quickly with system size, we require a method for limiting t h e variational freedom in our description of the many particle wavefunctions. To efficiently identify and remove "irrelevant" degrees of freedom in this problem, we have applied a numerical "renormalization group" scheme to this calculation [5]. Our application to the N-site problem proceeds by breaking the N-site Hamiltonian into a series of smaller (typically 2 or 4 site) mutually commuting Hamiltonians by removing "bonds" from the full Hamiltonian. The size of the resulting micro-Harniltonian is chosen to be small enough that a convenient exact numerical diagonalization can he made for states characterized by total spin, S, projection Sz and particle number n. (We allow for all possible fluctuations in these quantities in the microcells). The previously excised bonds are then reintroduced into the calculation, and the cell eigenstates are combined in properly symmetrized products to provide a basis set at the next level of the calculation. The size of the basis set is made manageable by retaining only the lowest lying (in energy) product basis functions. The number that are retained is determined by constraints on storage and CPU time, we have typically found that of order 200 product states per (S, gz, n) manifold provide reasonable results; highly accurate work requires that we improve this basis via second order perturbation theory through the removed states; some technical points relating to the construction of the composite Hamiltonian and the implementation of this perturbation theory are discussed in reference 4. The method
109
can be iterated so that in each "inflation" of the calculation, the system size is doubled (three inflations of the 2-site problem produce the 16 site problem, etc.) It is important to mention that unlike some previous applications of this method [8], no attempt is made to "close" the recursion scheme, i.e. the inflated Hamiltonian does not precisely map onto a renormalized version of the microcell Hamiltonian, but grows increasingly complex as the system size expands. We view the technique as an efficient variational approach, rather than a true scaling theory.
"
T
• -
r
~
-
-T ....
T....
F--
i
I00 50
075
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t OOG
uo (~)
J .008 010
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i
000
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~ _ __ ~___ a___ 004
006
008
OlO
uo (~)
Figure 1. A comparison of results from an exact diagonalization of the SSH Hamiltonian (U--0) with results for the RG method: (left) ground state total energy as a function of dimerization amplitude, (right) excitation energy of lowest singlet excitation.
A virtue of this composition scheme for the many particle wavefunctions, is that short range correlations are introduced in a very accurate and simple way. In fact this method has its greatest dimculty in building in the correct delocalization associated with the one body hopping terms in the full Hamiltonian. Thus an important test of the method is how well the known spectrum of the noninteraeting (U=0, V--0) Hamiltonian can be recovered. In Figure 1 we present a comparison of the ground state energy, and excitation energy to the first excited state in the noninteracting model, determined by direct diagonalization of the SSH Hamiltonian (exact) and obtained via our renormalization group approximation. The results are plotted as a function of a coordinate representing uniform dimerization of a finite N--16 chain. The approximation is seen to work extremely well; the error in the ground state energy is within the width of the curves plotted, and errors in the excitation energy are less than 0.1 eV over the entire range studied. This demonstrates that our construction of the wavefunction is converging to the correct single determinant picture when the two body interactions are removed. The method is substantially more efficient than a typical CI or valence bond approach; for calibration, the complete calculation for the N--16 system exhausts approximately 7 CPU minutes on an IBM 3081.
110
How is this picture modified in the presence of electron electron interactions? In Figure 3 we present the analogous spectrum of self localized excitations for the case U=6.0 eV (i.e. U = 2 to), V=0. We judge that this is an appropriate value for the interaction strength from the relative size of the repulsion integrals in the P P P model. (A Monte Carlo study of the ground state of this Hamiltonian also suggests U -- 2t0). Again the ground state is uniformly dimerized, and in fact the calculations reveal a slight U-induced enhancement of the dimerization amplitude, consistent with the results of various previous ground state studies[6-8]. In the singlet excitation spectrum, we also find a state which corresponds to single particle promotion across the Peierls gap (labelled 1Bu ). We see that the charge density in this state drives the formation of a pair of kinks in the bond alternation amplitude, quite similar to the U=0 situation. A quantitative difference is that they are bound slightly more strongly in the presence of U. It is interesting that this excitonic confinement appears in the presence of U only. Grabowski et al. [9] concluded that V would be necessary for confinement of the solitonic exciton; a closer analysis of the present results shows that admixture of higher lying single particle excitations in the presence of U (which are not included in ~]) also mediate an indirect attractive interaction between the charged kinks. erhaps the most important result obtained for this value of the repulsion strength is that we indeed find the lower lying 21Ae state in the singlet excitation spectrum. The relaxed equilibrium lattice confiuration for this excitation is shown in the middle panel; it shows two well defined minima. This result can be understood in the following way. Decomposition of the 2lAg wavefunction shows a relatively large admixture from triplet configurations on either side of the chain, these in turn are coupled to form the overall spin singlet. The triplet excitation is the lowest lying excitation of the chain; it relaxes to form a pair of neutral spin 1/2 kinks (not shown) . The spectrum of low lying neutral excitations is then determined by the interaction between these spin 1/2 kinks. A repulsive interaction leads to dissociation of the pair, so that the neutral triplet and singlet excitations are degenerate, and we should consider the spin 1/2 kinks the fundamental elementary neutral excitations of the chain. For an attractive interaction, in which binding of the spins occurs and the particles overlap, the related singlet combination is found at much higher energy, and a lower lying singlet can be formed by the concerted formation of two of the low lying triplet excitations. In this limit we regard the neutral triplet excitation as the fundamental neutral excitation in the system, and the singlet as a bound complex of these triplets. An attractive interaction, leading to a bound triplet, can be mediated either by a confinement potential (in the case of a nondegenerate ground state polymer), or by system size (in the case of a finite degenerate ground state system). This latter mechanism is apparently operative in the configuration shown in Figure 3. It is interesting to note that this assignment suggests that the energy of the 2lAg excitation should occur at approximately twice the energy of the 3B, excitation, a correlation which was noted in the CI studies of Schulten et al. [10] on short polyenes. We thus interpret the singlet complex in the middle panel of Figure 3 as a bound complex of two triplet excitations; the triplet character of the constituent states is manifest in the depression of the bond alternation amplitude on either side of the chain. It is also interesting to note that even the N=16 system is barely large enough to reveal the nature of these interesting neutral self localized excitations. In fact for this degenerate ground state Hamiltonian, these arguments suggest that the 2lAg excitation decays into an unbound neutral kink pair in the truly extended (large N) system. To investigate this we have studied a more approximate description of the correlated electronic states in this Hamiltonian which is amenable to extrapolation to much longer systems, and observe the dissociation of the 2lAg in the large N limit [4]. Also since these limiting configurations are mixed quantum mechanically, we would expect a continuous evolution from the "bound" regime to the "dissociated" regime as the system size is increased. While these low lying neutral excitation have their analogs in the theory of the Heisberg spin 1/2 chain (to which this model maps in the large U limit) it is quite interesting to note that much of the physics in the presence of moderately strong interactions can be understood in terms of interactions of defects derived from the small U (SSH) description of this system.
111
III. EQUILIBRIUM GEOMETRIES In Figures 2 and 3 we present the equilibrium bond alternation patterns for the ground state and low lying excited states for a model 16 site polyene with fixed end boundary conditions. The ordinate is the staggered displacement field (.4,) and the abcissa labels the bonds. The calculations employ the representative values t o = 3.0 eV, t~ = 8.0 eV/A, K = 68.6 eV/ A2. In Figure 2 we use our computational scheme to recover some well known results from the noninteracting model (U=0,V=0). The ground state is uniformly dimerized, with slight
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14
Figure 3. Equilibrium lattice configuration for the Hubbard Peierls model (U=2t) on a 16 site chain for the ground state and two lowest singlet excitations.
ringing in the bond alternation amplitude due to the finite system size. The lowest electronic excitation corresponds to promotion of an electron across the Peierls gap, this leads to the formation of oppositely charged kinks shown in the second panel. (The kinks are oppositely charged for the singlet excitation, there is a degenerate triplet excitation (not shown) in which the defects are neutral and carry spin 1/2). The next equilibrium singlet also contains a two defect structure, the excitation corresponds to a two particle excitation across the gap between the defect states. This is an even parity singlet, and is degenerate with the odd parity singlet in the infinite chain limit. A higher lying even parity singlet is obtained by single particle excitation deeper into the conduction "band". The electronic wavefunctions in these higher lying states take the form of standing waves reflected from the two boundaries, the equilibrium lattice displacement field shows two well defined dips which reflect the peaks in the perturbed charge density.
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IV. DISCUSSION We summarize the results of our investigation of the effects of moderately strong electron electron repulsion on the excitation spectrum of an N--16 chain with the following observations: (a) The IB u state which is accessible by one photon absorption from the ground state produces a self localized state which is quite similar to the state obtained in the U=0 limit. The equilibrium configuration produces a pair of oppositely charged solitons, and in the presence of the on site U an attractive interaction between the kinks binds these to form a solitonic exciton, similar to that described by Grabowski et al. [9]. In a nondegenerate ground state polymer, the confinement potential leads to an additional binding of the kinks, and the optical excitation can be characterized as a self trapped singlet exciton. (b) A neutral triplet excitation aB u provides the lowest lying electronic excitation in the system. It relaxes to form a neutral kink antikink pair. (Interestingly this is precisely the lowest lying triplet excitation of the noninteracting SSH model.) In a nondegenerate ground state system, the confinement potential will give rise to a residual binding of this pair; in this case the excitation resembles a self trapped triplet exciton. (c) For moderate repulsion strengths, the lowest singlet excitation is a IA. state. For an extended system with a degenerate ground state this decayYs into two neutral spin 1/2 quasiparticles, which are closely related to the neutral SSH solitons. For a system in which the spins overlap, either by an attractive interaction mediated by the confinement potential in a nondegenerate ground state polymer, or by confinement on a finite (short) chain, the singlet is better characterized as a bound complex of neutral triplet excitations. The properties of this state have been thoroughly explored by application of the P P P Hamiltonian to finite polyenes which provide a well documented example of this latter situation [e.g. ref. 3]. The consequences of these low lying neutral excitations for the observed properties of photoexcitations in conjugated polymers should be interesting to explore. They are likely to play an important role in the relaxation of photoexcitations to the ground state in this system. We believe that it is likely that the recent observation of a triplet exciton in PDA [11] is related to the self trapped triplet excitation discussed above for nondegenerate systems. There has been some speculation that the photoinduced absorption feature at 1.35 eV, is associated with a neutral excitation which is apparently found in variety of conjugated systems[12-13]. There is ongoing discussion about the possibility of intrinsic photoproduction of spin 1/2 centers in polyacetylene, which could be associated with the decay of the ~Ag state.
ACKNOWLEDGEMENTS This work was supported in part by NSF through the MRL program under DMR 84 14640, and under Grant DMR 84-05524. Helpful discussions with S. Mazudmar, F.D.M. Haldane and J. Orenstein are gratefully acknowledged.
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