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Determination of interaction parameters in peierls-hubbard models describing finite polyenes and polyacetylene
Synthetic Metals, 41-43 (1991) 3471-3475
3471
DETERMINATION OF INTERACTION PARAMETERS IN PEIERLS-HUBBARD MODELS DESCRIBING FINITE POLYENES A N D POLYACETYLENE.
J. Tinka GAMMEL,(a, b) D.K. CAMPBELL,(a) S. MAZUMDAR,(¢) S.N. DIXIT,(d) and E.Y. LOH, Jr.(*,e) (*)Theoretical I)ivision and Center for Nonlinear Studies, Los Alumos National Laboratory, Los Alamos, NM 87545, USA; (b)Physikalisches Institut, Universitiit Bayreuth, Postfach 101251, D-8580 Bayreuth, F.R.G.; (¢)Department of Physics, University of Arizona, Tucson, AZ 85721, USA; (d)Lawrence Livermore National Laboratory, Livermore, CA 94550, USA; (e)Thinking Machines Corporation, 245 First Street, Cambridge, MA 02142-1214, USA.
ABSTRACT We study the extent to which finite polyenes and polyacetylene can be described by the extended Peierls-Hubbard Hamiltonian (ePHH) in an internally consistent manner. Using Lanczos exact diagonalization methods (LEI)M) on small systems, coupled with a novel boundary condition (b.c.) averaging technique, we investigate the excited state spectra of these models for the finite oligomer analogs of tran.q-polyacetylene, (CH)~. We search for sets of parameters describing the electron-phonon (e-p) and electron-electron (e-e) interactions which are consistent with a/l available experimental data on these systems, including the optical gap and bandwidth, the 21Ag state, longitudinal optical phonon frequency W~o, triplet excitations, and the optical absorptions assigned to charged and neutral solitons and to polarons. We conclude that the e-p coupling is substantially weaker than that suggested by purely e-p models and that the e-e interaction parameters are in the intermediate coupling regime (U ~ 2.5t0), consistent with values deduced from the finite polyenes.
INTRODUCTION The availability of a wealth of novel, low-i) materials - e.g., high-temperature superconducting copper oxides, "heavy-fermion" and charge-density wave systems, halogen-bridged transition-metal chains, and organic synthetic metals and superconductors - has recently stimulated the theoretical study of competing e-e and e-p interactions in reduced dimensions. In view of the theoretical difficulties of handling many-body and quantum interactions, much effort has gone into solving "simple" adiabatic tight binding models, and variants of the ePHH have been widely used [1]. In this article, we focus on the 1D ePHH in the context of (CH)ffi. Of the novel, low-I) materials, it has the simplest idealized structure. Thus the prospects of modeling (and understanding) it within the geometrically and quantum chemically simple (single chain, single molecular orbital, nearest neighbor, tight-binding) ePHH are greatest. Nonetheless, the correct parameters for modeling (CH)z remain a matter of much debate [1]. We summarize here our efforts, to date [2], to determine to what extent a single, internally consistent set can be found which reproduces all experimental data - including data on the finite polyenes - to reasonable accuracy, and thus determine to what extent (CH)z, and in general the whole class of novel low-I) materials, can really be modeled by the ePHH. 0379-6779/91/$3.50
© Elsevier Sequoia/Printed in The Netherlands
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Fig. 1 Cleft). Self-cousistent K as a function of Hubbard U for fixed A=0.1, t0=0.5, a=0.5, V=0, and several N with nJT and chain b.c. averaged as discussed in the text. The counter-intuitive increase of K with U can be seen on systems as small as 8-sites. The results for the 12-site ring and JT, nJT and chain (0) b.c.% are also shown. Fig. 2 (right). W~o for the same cases as in Fig. i. Note the L 0 mode frequency tracks K. MODEL AND METHOD
In the context appropriate to (CH)®, the ePHH takes the form [1]
: E ( - t o + oa,)B,,,+, + l
1
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+v E l
(z)
l
Here c~. creates an electron in the Wanuier orbital at site 1 with spin ¢; B l , l + l : ~ . ( c ~ c l + l . + c~+z¢cb,); n~,¢=c~¢c~¢; to is the hopping integral with a the e-p coupling describing its distance dependence; A~ is the (adiabatic [14]) displacement of the (CH) units along the axis; and K represents all other costs of distorting the lattice. Coulomb repulsions among electrons are parameterized with the conventional Hubbard U and V. Exact diagonalization studies are made feasible by the LEDM [3], which we have discussed in this context elsewhere [4], but even so one is limited numerically to small system size N < 15, as the basis grows exponentially with N due to the many-body nature of the problem. For such small N, the size and b.c. dependence is large. The importance of the Jahn-Teller/non-Jahn-Teller (JT/nJT) distinction for small N is familiar from the finite cyclic polyenes, where 4N + 2 rings do not dimerize; benzene, e.g., has equal bond lengths. This J T / n J T dichotomy becomes a problem when we wish to extrapolate to the infinite (CH), ]imlt. For U:V=O, one can show that "complex-phase-averaging" [4] - Bloch's theorem - ezactly reproduces the long chain behavior from studying short chains with dhTerent b.c.'s, but it becomes ineffective in the strongly correlated (U~oo) limit for the ~-fllled band systems we are currently considering. In this limit we have found an amplitude b.c. averaging technique effective in reducing finite size effects and improving extrapolation to the infm.ite limit [4]. The value orAL is found by r n i n l m i z ~ the total energy. At fixed K, we use the LEDM iteratively to calculate A~ from the self-consistency condition (SCC) KAc:-a(B~,L+z) . For A l : a o + (--1)eA, one can reverse the question and consider instead A as the independent variable, K (and re) then being determined from the SCC. This procedure fails, of course, when A=0. Except in Fig. 3, where we specifically consider defects, we enforce uniform A since (1) for parameter space searches, this greatly reduces the computation, as no iteration is needed, (2) this avoids "chain end effects" when
3473 extrapolating to the infinite(CH)= 1!mlt,and (3) this agrees with structural data on finitepolyenes [5]. While for strong e-p coupling 4 is reduced by U, it is well known that the Hubbard U counterintuitively initiallyincreases 4 for weak e-p coupling [6].Equivalently,for small fixed 4, U initiallyincreases K [7]. In Fig. I we show K as a function of U for various lattice sizes with :IT, niT, chain, and b.c. averaged (nit plus chain, weighted to reproduce the exact answer at U=O) b.c.'s. Fig. I illustrates the effectsof differentb.c.'son the finitesystems, as well as the convergence of the L E D M with system size. Phase-averaging yields a behavior that is nearly converged to the valued inferred for the infinite case, whereas any single b.c. yields this behavior only after (considerable) extrapolation. Thus b.c. averaging ellmlnates many of the problems in applying the L E D M to extract infinitesystem behavior. To calculate the phonon dispersion relation, we need to evaluate the dynamical matrix Dl,~, = O2E/OyeOyl, [4]. Here we are interested primarily in the L O (q=O longitudinal optical) mode of the uniform ground state, and thus we need only evaluate the total energy E[A] at three different 4 near the equilibrium geometry, requiring minima/additional computation. Studying only the L O mode has the further advantage that it can be calculated for non-equilibrium geometries, such as enforced uniform 4. Fig. 2 shows ~ o as a function of U and system sizefor the same cases as in Fig. 1. Our studies at fixed K show that for the finiterings a~Lo goes soft at finite U, conftrmlng the fact that A vanishes at this U, and that finitesystems do have a "phase transition" to a S D W state. For studies of the optical absorption we need the current operator, Jq,
(2) The problem of calculating the optical absorption coefficient within the LEDM reduces to finding the spectral weight of J~l¢0) [4,8]. For small systems, the optical spectra will be sparse. Indeed, benzene has only a single absorption peak. Benzene also illustrates the difference between real rings, where q:27r/N corresponds to the physical current, and "periodic b.c.'s" which assumes ~=0 and predicts 2 absorption peaks. In Fig. 3, to illustrate how b.c. averaging yields non-sparse spectra as in the infinite system which agree with expectations based on strong- and weak-coupling arguments [4,8], we sbow a spectrum obtained using several b.c.'s in the range - 1 < z < 1. DETERMINATION OF POLYACETYLENE PARAMETERS To begin our discussion, it is useful to recall how the parameters are determined in the conventional SSH model [9] of (CH)z, which corresponds to (1) with U : V : 0 . Since this is a single-electron theory, it can be solved analytically for the ground state. One finds that the bandwidth is given by W : 4 t 0 , the optical gap by E g : 4 a A , and that the dimerization A is determined by the SCC r k : 2 ( K ' ( 6 ) - E ' ( 6 ) ) / ( 1 - 6 2 ) , where ~ = ~ 4 / ~ , k=Kto/2a 2, and E' and K' are elliptic integrals. Fitting to W :10eV, Eg=I.4eV, and A=0.086• yields t0=2.5eV, a=4.1eV/A, and K=21eV/.~ 2. However, U=O parameters can never reproduce the experimental fact that the 21A~ state lies below the llB~ state [10], and finite a is required if dimerization is to occur at all. Thus the observed optical gap must come from both e-e and e-p interactions, and hence this investigation. The above results suggest that the b.c. averaged results on small systems provide reasonable estimates of the infinite limit. Hence we can use manageable size systems to carry out an exhaustive parameter search. Our procedure is to span the dimensionless parameter space: %=.5, a ~ . 5 , $, u:U/2to, and v : V / 2 t o ; determining k, e g : E J 2 t 0 , w:W/2t~, and fl~o:W~o/(4K/M ). To date, for N=6,8,10,12 and b.c.=-1, 0,1, we have nearly completed a search over 6=.02, .04, ..., .20, v=O, .2, ..., 2., and u=v,v + .2, ...,3., determini-g ground state properties only. The same b.c. averaging scheme as in Fig. 1 was used to determine K. The optical gap was defined to be the lowest value of e~ for the 3
3474
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Fig. 3. OpticM abso~tion spectra obtained using the b.c. averagi,~gapproach on a 10-siterings showing neutrul ($°) and charged (S ±) solitonand triplet(T) abso~tions along with the ~ound state (Eg) for one set of parameters meeting all our sorting criterion, yet far from the "conventional wisdom": t 0 : l . 0 , a : 3 . 2 , K : 4 3 . 9 , U:3.2, and V=0. Note that, to the extent that they can be determined within the LEDM, the soliton and triplet absorptions also do not rule out this set. b.c.'s (usually JT), and appeared to scale with U - V , as predicted by strong-coupling [11]. For U~2V, the gap appeared to depend on U - 2 V as expected from decoupled dimer arguments [4I. The LEDM does not reliably give outer band edges; thus to was determined by where the absorption had fallen to 1% of its mAYimum. Note this can differ considerably from the actual band edge. We also find z0/e~ and fl~o are remarkably insensitive to parameters. To determine the parameters describing (CH)~, we scale each of these d.imension/ess data points to the actual gap and dimerization to determine the dimensiona/parameters. If the bandwidth and w~o then agree with their actual values, we accept these as possible parameters. However, for (CH~, even some relatively basic information is not known precisely. For example, although $ SH used A :0.08C~ for the alternatiug component of the dimerization along the chain, some structural data [12] suggest values as low as 0.022.~. Also, the appropriate value of the bandwidth W is unclear, as experimental spectra never cutoff cleanly. It is "generally agreed" that A:0.03A, W:10eV, and Eg=l.8eV (to correct for 3D effects [10]) are the values to use. The vibration spectra of (CH)~ is complicated by the zig-zag geometry. Though it has an appreciable isotope effect indicating considerable C-H stretch miTi,~, the C=C stretch at PLo----o~Lo/(2~c):1450cm-1 in (CH), is considered to be the mode corresponding most closely to the single LO mode of the theoretical 1D chain [13I. The triplet absorption at ~ = l . 4 e V , and the S±/S ° optical and infrared absorptions have also been identified. We are currently considering how well the inferred parameters match these data on non-uniform geometries. Listed in Table 1 are our results to date for the possible parameters describing (CH)z. We find t0<2.5, as part of the bandwidth comes from U. In Fig. 3, we show, however, that even for a parameter set far from the "conventional wisdom", we obtain optical absorption data, to the extent that they can correctly be determined by the LEDM for non-uniform geometries, in agreement with experiment. C O N C L U S I O N S A N D O P E N ISSUES LEDM with b.c. averaging enables reasonable extrapolations from small systems, for which an exhaustive parameter search can be carried in reasonable time. Our current estimates of the parameters modeling ( C H ) , are shown in the Table 1. Clearly much work remains. The parameter search needs to be repeated on larger systems and the allowed parameters ranges further refined. Dependences of the
3475
inferred parameters on experimental inputs need to be understood, and a prescription for defining the bandwidth needs to be a~eed upon. The amplitude mode formalism for analyzing the LO mode should also be incorporated. The analyses of solitous and triplet optical gap for the inferred parameters need to be performed. Since intrinsic defects can be viewed as 1/N effects, extrapolation to large systems may prove very difficult. Finally, detailed comparisons to finite polyene data, incorporating end ~ o u p effects as necessary, need to be carried out. If the adiabatic ePHH does indeed capture the essential physics, the same parameters must also work for the finite polyenes. We stress that obtaining fits to many different observables with a 8ingle set of physically plausible parameters is crucial to any true understanding of (CH),, or more generally any member of the class of low-D materials. ACKNOWLEDGEMENTS We would like to thank Helmut Biittner, Alberto Girlando, Eugene Mele, Anna Painelli, Doug Scalapino, and Zoltan Soos for valuable discussions. This work was supported by the U.S. DOE-OBES. Table 1: Representative parameters scaled to A=0.03A, W=10eV, E~=I.SeV, and PLO=1450cm -1. N
to(eV)
a(eV/~)
U(eV)
V(eV)
K(eV)
W(eV)
8
1.7 1.4 1.3 1.3 1.2 1.0
4.5 4.3 4.4 4.0 3.4 3.1
4.8 4.0 3.7 4.3 5.8 6.0
1.4 0.6 0.0 1.1 2.4 2.4
52.4 53.1 53.4 51.8 47.6 44.9
11.3 10.4 10.7 11.3 10.5 10.5
i0
1.5 1.4 1.4 1.2 1.2 1.1
4.3 3.9 3.6 4.0 3.2 2.9
4.3 4.5 5.8 4.0 5.7 5.9
0.6 1.1 2.3 0.0 2.4 2.5
53.6 51.4 49.1 50.9 45.5 43.3
11.4 10.3 11.0 10.6 9.6 10.6
REFERENCES 1. For a recent review of the ePHH applied to (CH),, see: D. Baeriswyl, D. K. Campbell, and S. Mazumdar, to be published in Conducting Polymer% H. Kiess, ed., Springer, New York, 1990. 2. Ref. 4b is an earlier "progress report" of this research, and contains somewhat more detail. 3. S. Pissanetsky, Sparse Matrix Technology, Academic, London, 1984. 4. (a) D. K. Campbell, J. T. Gammel, and E. Y. Loh, Jr., to be pub. in the Proc. of the Ann. Adriatico Res. Conf. on Strongly Correl. Electrons, Trieste, Italy, July 18-21, 1989. (b)J. T. Gammel, D.K. Campbell, E.Y. Loh, Jr., S. Mazumdar, and S.N. Dixit, to be pub. in the Proc. of the MRS Syrup. on Elec., Opt., and Mag. Prop. of Org. Solid State Mater.,Boston, MA, Nov. 27- Dec. 2, 1989. 5. R.H. Baug~aman, B.E. Kohler, I.J. Levy, and C. Spaugler, Synth. Metals, 11(1985) 37. 6. e.g., V. Wans, H. Biittner, and J. Volt, Phys. Rev. B, 41 (1990) 9366. 7. G.W. Hayden and Z.G. Soos, Phys. Rev. B, 38 (1988) 6075. 8. P. F. Maldague, Phys. Rev. B, 16, 2437 (1977). 9. W.P. Su, J.R. Schrieffer, and A.J. Heeger, Phys. Rev. Lett., 42 (1979) 1698; Phys. Rev. B, 22 (1980) 2099. 10 B.E. Kohler, C. Span~er, and C. Westertield, J. Chem. Phys., 133 (1964) A171. 11. J. Beraasconi, M.J. Rice, W.R. Schneider, and S. Str~sler, Phys. Rev. B, 12(1975) 1090. 12. J.C.W. Chien, F.E. Karasz, and K. Shimamura, Maluom. Chem. Rapid Comm., 3(1982) 655. 13. E. Ehrenfreund, Z. Vardeny, O. Brafman, and B. Rorovitz, Phys. Rev. B, 36(1987) 1535. 14. A brief s,,mmary, including several references, of expected quantum lattice effects, which we ignore here, may be found in Ref. 1.
3471
DETERMINATION OF INTERACTION PARAMETERS IN PEIERLS-HUBBARD MODELS DESCRIBING FINITE POLYENES A N D POLYACETYLENE.
J. Tinka GAMMEL,(a, b) D.K. CAMPBELL,(a) S. MAZUMDAR,(¢) S.N. DIXIT,(d) and E.Y. LOH, Jr.(*,e) (*)Theoretical I)ivision and Center for Nonlinear Studies, Los Alumos National Laboratory, Los Alamos, NM 87545, USA; (b)Physikalisches Institut, Universitiit Bayreuth, Postfach 101251, D-8580 Bayreuth, F.R.G.; (¢)Department of Physics, University of Arizona, Tucson, AZ 85721, USA; (d)Lawrence Livermore National Laboratory, Livermore, CA 94550, USA; (e)Thinking Machines Corporation, 245 First Street, Cambridge, MA 02142-1214, USA.
ABSTRACT We study the extent to which finite polyenes and polyacetylene can be described by the extended Peierls-Hubbard Hamiltonian (ePHH) in an internally consistent manner. Using Lanczos exact diagonalization methods (LEI)M) on small systems, coupled with a novel boundary condition (b.c.) averaging technique, we investigate the excited state spectra of these models for the finite oligomer analogs of tran.q-polyacetylene, (CH)~. We search for sets of parameters describing the electron-phonon (e-p) and electron-electron (e-e) interactions which are consistent with a/l available experimental data on these systems, including the optical gap and bandwidth, the 21Ag state, longitudinal optical phonon frequency W~o, triplet excitations, and the optical absorptions assigned to charged and neutral solitons and to polarons. We conclude that the e-p coupling is substantially weaker than that suggested by purely e-p models and that the e-e interaction parameters are in the intermediate coupling regime (U ~ 2.5t0), consistent with values deduced from the finite polyenes.
INTRODUCTION The availability of a wealth of novel, low-i) materials - e.g., high-temperature superconducting copper oxides, "heavy-fermion" and charge-density wave systems, halogen-bridged transition-metal chains, and organic synthetic metals and superconductors - has recently stimulated the theoretical study of competing e-e and e-p interactions in reduced dimensions. In view of the theoretical difficulties of handling many-body and quantum interactions, much effort has gone into solving "simple" adiabatic tight binding models, and variants of the ePHH have been widely used [1]. In this article, we focus on the 1D ePHH in the context of (CH)ffi. Of the novel, low-I) materials, it has the simplest idealized structure. Thus the prospects of modeling (and understanding) it within the geometrically and quantum chemically simple (single chain, single molecular orbital, nearest neighbor, tight-binding) ePHH are greatest. Nonetheless, the correct parameters for modeling (CH)z remain a matter of much debate [1]. We summarize here our efforts, to date [2], to determine to what extent a single, internally consistent set can be found which reproduces all experimental data - including data on the finite polyenes - to reasonable accuracy, and thus determine to what extent (CH)z, and in general the whole class of novel low-I) materials, can really be modeled by the ePHH. 0379-6779/91/$3.50
© Elsevier Sequoia/Printed in The Netherlands
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Fig. 1 Cleft). Self-cousistent K as a function of Hubbard U for fixed A=0.1, t0=0.5, a=0.5, V=0, and several N with nJT and chain b.c. averaged as discussed in the text. The counter-intuitive increase of K with U can be seen on systems as small as 8-sites. The results for the 12-site ring and JT, nJT and chain (0) b.c.% are also shown. Fig. 2 (right). W~o for the same cases as in Fig. i. Note the L 0 mode frequency tracks K. MODEL AND METHOD
In the context appropriate to (CH)®, the ePHH takes the form [1]
: E ( - t o + oa,)B,,,+, + l
1
E a) + v I
+v E l
(z)
l
Here c~. creates an electron in the Wanuier orbital at site 1 with spin ¢; B l , l + l : ~ . ( c ~ c l + l . + c~+z¢cb,); n~,¢=c~¢c~¢; to is the hopping integral with a the e-p coupling describing its distance dependence; A~ is the (adiabatic [14]) displacement of the (CH) units along the axis; and K represents all other costs of distorting the lattice. Coulomb repulsions among electrons are parameterized with the conventional Hubbard U and V. Exact diagonalization studies are made feasible by the LEDM [3], which we have discussed in this context elsewhere [4], but even so one is limited numerically to small system size N < 15, as the basis grows exponentially with N due to the many-body nature of the problem. For such small N, the size and b.c. dependence is large. The importance of the Jahn-Teller/non-Jahn-Teller (JT/nJT) distinction for small N is familiar from the finite cyclic polyenes, where 4N + 2 rings do not dimerize; benzene, e.g., has equal bond lengths. This J T / n J T dichotomy becomes a problem when we wish to extrapolate to the infinite (CH), ]imlt. For U:V=O, one can show that "complex-phase-averaging" [4] - Bloch's theorem - ezactly reproduces the long chain behavior from studying short chains with dhTerent b.c.'s, but it becomes ineffective in the strongly correlated (U~oo) limit for the ~-fllled band systems we are currently considering. In this limit we have found an amplitude b.c. averaging technique effective in reducing finite size effects and improving extrapolation to the infm.ite limit [4]. The value orAL is found by r n i n l m i z ~ the total energy. At fixed K, we use the LEDM iteratively to calculate A~ from the self-consistency condition (SCC) KAc:-a(B~,L+z) . For A l : a o + (--1)eA, one can reverse the question and consider instead A as the independent variable, K (and re) then being determined from the SCC. This procedure fails, of course, when A=0. Except in Fig. 3, where we specifically consider defects, we enforce uniform A since (1) for parameter space searches, this greatly reduces the computation, as no iteration is needed, (2) this avoids "chain end effects" when
3473 extrapolating to the infinite(CH)= 1!mlt,and (3) this agrees with structural data on finitepolyenes [5]. While for strong e-p coupling 4 is reduced by U, it is well known that the Hubbard U counterintuitively initiallyincreases 4 for weak e-p coupling [6].Equivalently,for small fixed 4, U initiallyincreases K [7]. In Fig. I we show K as a function of U for various lattice sizes with :IT, niT, chain, and b.c. averaged (nit plus chain, weighted to reproduce the exact answer at U=O) b.c.'s. Fig. I illustrates the effectsof differentb.c.'son the finitesystems, as well as the convergence of the L E D M with system size. Phase-averaging yields a behavior that is nearly converged to the valued inferred for the infinite case, whereas any single b.c. yields this behavior only after (considerable) extrapolation. Thus b.c. averaging ellmlnates many of the problems in applying the L E D M to extract infinitesystem behavior. To calculate the phonon dispersion relation, we need to evaluate the dynamical matrix Dl,~, = O2E/OyeOyl, [4]. Here we are interested primarily in the L O (q=O longitudinal optical) mode of the uniform ground state, and thus we need only evaluate the total energy E[A] at three different 4 near the equilibrium geometry, requiring minima/additional computation. Studying only the L O mode has the further advantage that it can be calculated for non-equilibrium geometries, such as enforced uniform 4. Fig. 2 shows ~ o as a function of U and system sizefor the same cases as in Fig. 1. Our studies at fixed K show that for the finiterings a~Lo goes soft at finite U, conftrmlng the fact that A vanishes at this U, and that finitesystems do have a "phase transition" to a S D W state. For studies of the optical absorption we need the current operator, Jq,
(2) The problem of calculating the optical absorption coefficient within the LEDM reduces to finding the spectral weight of J~l¢0) [4,8]. For small systems, the optical spectra will be sparse. Indeed, benzene has only a single absorption peak. Benzene also illustrates the difference between real rings, where q:27r/N corresponds to the physical current, and "periodic b.c.'s" which assumes ~=0 and predicts 2 absorption peaks. In Fig. 3, to illustrate how b.c. averaging yields non-sparse spectra as in the infinite system which agree with expectations based on strong- and weak-coupling arguments [4,8], we sbow a spectrum obtained using several b.c.'s in the range - 1 < z < 1. DETERMINATION OF POLYACETYLENE PARAMETERS To begin our discussion, it is useful to recall how the parameters are determined in the conventional SSH model [9] of (CH)z, which corresponds to (1) with U : V : 0 . Since this is a single-electron theory, it can be solved analytically for the ground state. One finds that the bandwidth is given by W : 4 t 0 , the optical gap by E g : 4 a A , and that the dimerization A is determined by the SCC r k : 2 ( K ' ( 6 ) - E ' ( 6 ) ) / ( 1 - 6 2 ) , where ~ = ~ 4 / ~ , k=Kto/2a 2, and E' and K' are elliptic integrals. Fitting to W :10eV, Eg=I.4eV, and A=0.086• yields t0=2.5eV, a=4.1eV/A, and K=21eV/.~ 2. However, U=O parameters can never reproduce the experimental fact that the 21A~ state lies below the llB~ state [10], and finite a is required if dimerization is to occur at all. Thus the observed optical gap must come from both e-e and e-p interactions, and hence this investigation. The above results suggest that the b.c. averaged results on small systems provide reasonable estimates of the infinite limit. Hence we can use manageable size systems to carry out an exhaustive parameter search. Our procedure is to span the dimensionless parameter space: %=.5, a ~ . 5 , $, u:U/2to, and v : V / 2 t o ; determining k, e g : E J 2 t 0 , w:W/2t~, and fl~o:W~o/(4K/M ). To date, for N=6,8,10,12 and b.c.=-1, 0,1, we have nearly completed a search over 6=.02, .04, ..., .20, v=O, .2, ..., 2., and u=v,v + .2, ...,3., determini-g ground state properties only. The same b.c. averaging scheme as in Fig. 1 was used to determine K. The optical gap was defined to be the lowest value of e~ for the 3
3474
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5
I0
Fig. 3. OpticM abso~tion spectra obtained using the b.c. averagi,~gapproach on a 10-siterings showing neutrul ($°) and charged (S ±) solitonand triplet(T) abso~tions along with the ~ound state (Eg) for one set of parameters meeting all our sorting criterion, yet far from the "conventional wisdom": t 0 : l . 0 , a : 3 . 2 , K : 4 3 . 9 , U:3.2, and V=0. Note that, to the extent that they can be determined within the LEDM, the soliton and triplet absorptions also do not rule out this set. b.c.'s (usually JT), and appeared to scale with U - V , as predicted by strong-coupling [11]. For U~2V, the gap appeared to depend on U - 2 V as expected from decoupled dimer arguments [4I. The LEDM does not reliably give outer band edges; thus to was determined by where the absorption had fallen to 1% of its mAYimum. Note this can differ considerably from the actual band edge. We also find z0/e~ and fl~o are remarkably insensitive to parameters. To determine the parameters describing (CH)~, we scale each of these d.imension/ess data points to the actual gap and dimerization to determine the dimensiona/parameters. If the bandwidth and w~o then agree with their actual values, we accept these as possible parameters. However, for (CH~, even some relatively basic information is not known precisely. For example, although $ SH used A :0.08C~ for the alternatiug component of the dimerization along the chain, some structural data [12] suggest values as low as 0.022.~. Also, the appropriate value of the bandwidth W is unclear, as experimental spectra never cutoff cleanly. It is "generally agreed" that A:0.03A, W:10eV, and Eg=l.8eV (to correct for 3D effects [10]) are the values to use. The vibration spectra of (CH)~ is complicated by the zig-zag geometry. Though it has an appreciable isotope effect indicating considerable C-H stretch miTi,~, the C=C stretch at PLo----o~Lo/(2~c):1450cm-1 in (CH), is considered to be the mode corresponding most closely to the single LO mode of the theoretical 1D chain [13I. The triplet absorption at ~ = l . 4 e V , and the S±/S ° optical and infrared absorptions have also been identified. We are currently considering how well the inferred parameters match these data on non-uniform geometries. Listed in Table 1 are our results to date for the possible parameters describing (CH)z. We find t0<2.5, as part of the bandwidth comes from U. In Fig. 3, we show, however, that even for a parameter set far from the "conventional wisdom", we obtain optical absorption data, to the extent that they can correctly be determined by the LEDM for non-uniform geometries, in agreement with experiment. C O N C L U S I O N S A N D O P E N ISSUES LEDM with b.c. averaging enables reasonable extrapolations from small systems, for which an exhaustive parameter search can be carried in reasonable time. Our current estimates of the parameters modeling ( C H ) , are shown in the Table 1. Clearly much work remains. The parameter search needs to be repeated on larger systems and the allowed parameters ranges further refined. Dependences of the
3475
inferred parameters on experimental inputs need to be understood, and a prescription for defining the bandwidth needs to be a~eed upon. The amplitude mode formalism for analyzing the LO mode should also be incorporated. The analyses of solitous and triplet optical gap for the inferred parameters need to be performed. Since intrinsic defects can be viewed as 1/N effects, extrapolation to large systems may prove very difficult. Finally, detailed comparisons to finite polyene data, incorporating end ~ o u p effects as necessary, need to be carried out. If the adiabatic ePHH does indeed capture the essential physics, the same parameters must also work for the finite polyenes. We stress that obtaining fits to many different observables with a 8ingle set of physically plausible parameters is crucial to any true understanding of (CH),, or more generally any member of the class of low-D materials. ACKNOWLEDGEMENTS We would like to thank Helmut Biittner, Alberto Girlando, Eugene Mele, Anna Painelli, Doug Scalapino, and Zoltan Soos for valuable discussions. This work was supported by the U.S. DOE-OBES. Table 1: Representative parameters scaled to A=0.03A, W=10eV, E~=I.SeV, and PLO=1450cm -1. N
to(eV)
a(eV/~)
U(eV)
V(eV)
K(eV)
W(eV)
8
1.7 1.4 1.3 1.3 1.2 1.0
4.5 4.3 4.4 4.0 3.4 3.1
4.8 4.0 3.7 4.3 5.8 6.0
1.4 0.6 0.0 1.1 2.4 2.4
52.4 53.1 53.4 51.8 47.6 44.9
11.3 10.4 10.7 11.3 10.5 10.5
i0
1.5 1.4 1.4 1.2 1.2 1.1
4.3 3.9 3.6 4.0 3.2 2.9
4.3 4.5 5.8 4.0 5.7 5.9
0.6 1.1 2.3 0.0 2.4 2.5
53.6 51.4 49.1 50.9 45.5 43.3
11.4 10.3 11.0 10.6 9.6 10.6
REFERENCES 1. For a recent review of the ePHH applied to (CH),, see: D. Baeriswyl, D. K. Campbell, and S. Mazumdar, to be published in Conducting Polymer% H. Kiess, ed., Springer, New York, 1990. 2. Ref. 4b is an earlier "progress report" of this research, and contains somewhat more detail. 3. S. Pissanetsky, Sparse Matrix Technology, Academic, London, 1984. 4. (a) D. K. Campbell, J. T. Gammel, and E. Y. Loh, Jr., to be pub. in the Proc. of the Ann. Adriatico Res. Conf. on Strongly Correl. Electrons, Trieste, Italy, July 18-21, 1989. (b)J. T. Gammel, D.K. Campbell, E.Y. Loh, Jr., S. Mazumdar, and S.N. Dixit, to be pub. in the Proc. of the MRS Syrup. on Elec., Opt., and Mag. Prop. of Org. Solid State Mater.,Boston, MA, Nov. 27- Dec. 2, 1989. 5. R.H. Baug~aman, B.E. Kohler, I.J. Levy, and C. Spaugler, Synth. Metals, 11(1985) 37. 6. e.g., V. Wans, H. Biittner, and J. Volt, Phys. Rev. B, 41 (1990) 9366. 7. G.W. Hayden and Z.G. Soos, Phys. Rev. B, 38 (1988) 6075. 8. P. F. Maldague, Phys. Rev. B, 16, 2437 (1977). 9. W.P. Su, J.R. Schrieffer, and A.J. Heeger, Phys. Rev. Lett., 42 (1979) 1698; Phys. Rev. B, 22 (1980) 2099. 10 B.E. Kohler, C. Span~er, and C. Westertield, J. Chem. Phys., 133 (1964) A171. 11. J. Beraasconi, M.J. Rice, W.R. Schneider, and S. Str~sler, Phys. Rev. B, 12(1975) 1090. 12. J.C.W. Chien, F.E. Karasz, and K. Shimamura, Maluom. Chem. Rapid Comm., 3(1982) 655. 13. E. Ehrenfreund, Z. Vardeny, O. Brafman, and B. Rorovitz, Phys. Rev. B, 36(1987) 1535. 14. A brief s,,mmary, including several references, of expected quantum lattice effects, which we ignore here, may be found in Ref. 1.