Electron-lattice relaxation, and soliton structures and their interactions in polyenes

5 Pages·0.15 MB·English

by  Robert J. Bursill

Electron-lattice relaxation, and soliton structures and their interactions in polyenes.
Robert J. Bursill1 and William Barford2
1School of Physics, University of New South Wales, Sydney, NSW 2052, Australia. 2Department of Physics and Astronomy,
The University of Sheffield, Sheffield, S3 7RH, United Kingdom.
Density matrix renormalisation group calculations of a suitably parametrised model of long
polyenes (polyacetylene oligomers), which incorporates both long range Coulomb interactions and
adiabaticlatticerelaxation,arepresented. The13B+ and21A+statesarefoundtohavea2-soliton
u g
and 4-soliton form, respectively, both with large relaxation energies. The 11B− state forms an
u
exciton-polaron and has a very small relaxation energy. The relaxed energy of the 21A+ state lies
g
below that of the 11B− state. The soliton/anti-soliton pairs are bound.
u
9
9 PACS numbers:71.10.F, 71.20.R, 71.35
9
1

n Electronic interactions in polyenes and polyacetylene [9]usedtheDMRGmethodtostudytherelativestability
a (PA) induce strong spin-density-wavecorrelations in the of bipolarons using the same model.
J
ground state, resulting in low energy spin-flip (or cova- The P-P-P-SHH Hamiltonian is defined as

8 lent) triplet (3B+) excitations. These combine to form
1 even-parity (dipoule-forbidden) singlet (1A+g) excitations. H=−2N−1t Tˆ + 1 N−1∆2+ΓN−1∆
2 Optical(dipole-allowed)transitionstotheodd-paritysin- X i i 4πt0λ X i X i
v glet state (1B−) are essentially ionic in character, re- i=1 i=1 i=1
u N
6 sulting in charge transfer from one site to another. In
+U (n −1/2)(n −1/2)
2 the non-interacting limit the 13B+ and 11B− states are X i↑ i↓
1 degenerate, and the 21A+ state aulways lies huigher in en- i=1
1 g N
1
ergy. However,electroncorrelationscanleadtoareversal
0 + V (n −1)(n −1), (1)
9 of the energetic ordering of the 11B− and 21A+ states. 2X ij i j
u g i6=j
9 Electron-electron correlations in π conjugated systems,
mat/ sPuacrhisears-PPaAr,r-aProepcleon(vPe-nPie-Pnt)lymmodoedle,llwedhibchy itnhceluodnees-blaonndg wishtehree,btoin=d (cid:0)otr0d+er∆o2ip(cid:1)eraantodrTˆoif=the21Pithσ.(cb†i+on1σdc.iσW+ehu.cs.e)
range Coulomb interactions. the Ohno function for the Coulomb interaction: V =
- ij
Electron-phonon interactions in the non-interacting
d U/ 1+βr2, where β = (U/14.397)2 and bond lengths
n limit are described by the SSH model. In the adiabatic q ij
o limit it predicts a wealth of non-linear excitations, in- are in ˚A. The single and double bond lengths used in
c cludingcharged/spinless(S±)andneutral/spin1/2(Sσ) the evaluation of Vij are 1.46 ˚A and 1.35 ˚A, respec-
v: solitons. Itistheinter-playofbothelectron-electronand tively,andthebondangleis1200. Varioussemi-empirical
Xi electron-phononinteractionsinPAwhichleadsto anex- parametrisations exist for t0 and U. We adopt the
tremely rich variety of excitations. To describe these values which are optimal for benzene [10], whose C-C
ar excitations we employ the density matrix renormalisa- bond length of 1.40 ˚A is almost the same as the average
tion group (DMRG) [1] method to solve the P-P-P-SSH bond length in PA thin films, i. e., t0 = 2.539 eV and
model,andutilisetheHellmann-Feynman(H-F)theorem U = 10.06 eV. The dimensionless electron-phonon cou-
to calculate the low-lying excited states and the lattice pling constant, λ, is defined by λ=2α2/πKt0, where K
relaxation associated with them. istheelasticspringconstant(estimatedtobe 46eV˚A−2
Earlierworkonthe solitonicstructureofthe low-lying [11]),andα relatesthe actualdistortionofthe ith.bond
excitations include, a renormalisation group calculation from equilibrium, yi, to ∆i: yi = ∆i/2α. Γ is chosen
of the Hubbard-SSH model of up to 16 sites [2]; a mean- so that the relaxed ground state of an infinite polymer
field study of the Heisenberg-Peierls model [3]; an exact has the same chain length as the unrelaxed state, i. e.,
N−1
diagonalisationofa12siteextendedHubbard-SSHmodel Pi=1 ∆i = 0. This ensures that the average hopping
[4];andastrongcouplingandperturbationcalculationof integralis t0, which is applicable to C-C bond lengths of
theHubbard-SSHmodel[5]. TheDMRGmethodhasre- 1.40˚A.However,thechainlengthispermittedtochange
centlybeenusedbyandYaronetal.[6]andFanoetal.[7] forexcitedstates,andforallthestatesoffiniteoligomers.
to solve the P-P-P model for linear and cyclic polyenes, Theremainingparameter,λ, ischosensothatthe model
respectively. Jeckelmann [8] studied the metal-insulator fitstheverticalexcitationenergiesofthe11Bu− and21A+g
transitionindopedPAbysolvingtheextendedHubbard- states of hexatriene in the gas phase [12]. A choice of
SSHwiththeDMRGmethod. Likewise,Kuwabaraetal. λ = 0.115 and Γ = 0.602 gives 4.965 eV and 5.212 eV,
1
comparedtothe experimentalvaluesof4.96eVand5.21 This always lies lower than E0-0(11B−), which implies
u
eV, for the 11B− and 21A+ states, respectively. that a vertical photo-excitation to the 11B− state will
u g u
The equilibrium values of the bond length distortion decaytothe21A+ state. Finally,theexperimentalvalues
g
are determined by the H-F condition that, of E0-0(11B−) and E0-0(21A+) for N = 10 and 14 are
u g
shown[15]. The21A+ valuesareingoodagreementwith
∂E({∆ }) g
∂∆ i =0⇒∆i =2πt0λhhTˆii−Γi. (2) our calculation. The 11Bu− values are ca. 0.3 eV lower
i than our predictions. The experimental results for N =
H possesses spatial reflection, particle-hole and spin- 8–14 have been analysed by Kohler [15]. For the 21A+
g
flip symmetries. Symmetrised eigenstates of H are con- statetheempiricalrelationE0-0(21A+)=0.96+20.72/N
g
structed by an efficient process which makes use of the was derived, in good agreement with the photoinduced
fact that the block symmetry operators commute with absorption result ca. 1.1 eV for polyacetylene thin films.
the density matrix at all stages of the calculation and However,Fig.1suggeststhatanalgebraicfitting formis
which has been tested by making comparisons with ex- incorrect—thetruescalingbehaviourisexponential,and
axt results [13,14]. canonlybeseenbyconsideringsufficientlylargesystems.
The calculation of the relaxed energy of a given state In Fig. 2 we plot as a function of bond index from
for a given chain length is as follows: (1) The eigenstate the center of the chain, the normalised staggered bond
is calculated for an initial choice of {t } by building up dimerisation, defined as, δ ≡ (−1)i(t −t¯)/t¯, where t¯is
i i i
the lattice to the target chain size using the infinite lat- the average value of t in the middle of the chain [16].
i
tice algorithm of the DMRG method. (2) At the target Note that the 13B+ and 21A+ states undergo consider-
u g
chain size the H-F condition (2) is repeatedly applied able bond distortion, whereas the 11B− state and the
u
until the {t } have converged. (3) Using the new values charged state (denoted E ) show a weak polaronic dis-
i g
of {t }, steps 1 and 2 are repeated. The procedure is tortion of the lattice. The oscillatory behaviour of δ in
i i
succesfully terminatedwhenthe energieshaveconverged the polaronic distortions indicates a local expansion of
aftersuccessivelatticeandH-Fiterations. Itisnecessary the lattice.
to sweep through the lattice after each set of H-F itera- Wefitthe13B+,11B− andchargedstatetoa2-soliton
u u
tions to ensure that the electronic states and the lattice form [5,18],
geometry have convergedsimultaneously.
The accuracy of the DMRG implementation has been δi =δ¯{1+tanh(2x0/ξ)[tanh((i−x0)/ξ)
checked in a number of ways. First, the method has − tanh((i+x0)/ξ)]}. (3)
been compared with exact results in the non-interacting
(U = 0) limit. The convergence of the ground state en- The 21A+g state, however, evidently requires a 4-soliton
ergywithsuperblockHilbertspacesize(SBHSS)isshown [5,18] fit of the form,
inTableIfortheN =102sitesystemwithvariouslattice
geometries. The total energy converges to within 0.005 δi =δ¯{1+tanh(2x0/ξ)[tanh((i−xd−x0)/ξ)
eV which is sufficient for energy gaps, which are of the − tanh((i−xd+x0)/ξ)+tanh((i+xd−x0)/ξ)
orderof1eV,toberesolvedtowithin1%orbetter. This − tanh((i+xd+x0)/ξ)]}. (4)
represents the DMRG at its least accurate, as the addi-
tionofcorrelationsimprovesconvergence,ascanbe seen These functions give good fits to the relaxed geometries
inTableII,wherewepresenttheDMRGconvergencefor of the N = 102 site system, as shown in Fig. 2. The
the ground state energy and a number of energy gaps. differenceinenergybetweenusingthefitsandtheactual
Using the ground state geometry, the vertical energies relaxedgeometryisaround0.01eV.The4-solitoncharac-
(Ev) ofthe 13B+,11B− and21A+ states arecalculated. terof21A+ state indicates the stronginter-playbetween
u u g g
These, as well as the relaxed energies (E0-0), are shown electron-lattice relaxation and electron-electron correla-
in Fig.1 as functions of1/N. We firstnote that the ver- tions in polyenes, for, as indicated earlier, this state has
ticalenergiesofthe11B− and21A+ statesareveryclose, a considerable triplet-triplet character.
u g
withacrossingatshortchains,andagainforlongchains. Fig.3depictstheconvergenceofthevariousfittingpa-
In the thermodynamic limit Ev(11B−) < Ev(21A+). rametersas a function ofN. The 13B+ and21A+ states
u g u g
This largeN crossinghas also been observedin the U-V converge rapidly with N, whereas the 11B− state shows
u
dimerised Hubbard model [14,17]. strongfinite-size effects, and the coherencelength ξ only
Therelaxationenergyofthe11B− stateismodest(ca. begins to converge at around N = 102. The fact that
u
0.3 eV) and has not converged (i. e. it is still rapidly the soliton structures converge with N leads to two im-
decreasing) for N = 102. By contrast, the relaxation portant observations: First, the soliton/anti-soliton pair
energies of the 13Bu+ and 21A+g states are substantial, are bound, because if they were not their separation x0
being ca. 0.5 eV and 1.0 eV, respectively, and converge would increase with N. Second, the soliton structures
rapidlywithN. Wehavealsocalculatedtheenergyofthe are pinned in the middle of the lattice. This is a conse-
21A+ stateusingtherelaxedgeometryofthe11B− state. quence ofthe classicaladiabatictreatmentofthe lattice,
g u
2
and is one of the reasons why the energy curves flatten R. J. B. was supported by the Australian Research
off rapidly as N →∞. Council. W.B.gratefullyacknowledgesfinancialsupport
To further investigate the soliton/anti-soliton interac- fromtheRoyalSocietyandtheGordonGodfreyBequest
tions, adiabatic potential energy curves [5] (i. e. the en- of the UNSW. We thank E. Jeckelmann and Y. Shimoi
ergy as a function of soliton separation, x0) are plotted for useful discussions. The calculations were performed
in Fig. 4. Our results differ qualitatively from previous at the New South Wales Center for Parallel Computing.
approximatecalculations[5]in that the 11B− and13B+
u u
are bound—the potentials have a minimum and are at-
tractive for large x0. This attractive soliton/anti-soliton
interaction implies much stronger binding for the 21A+
g
statethanthatobtainedin[5],wherethebinding energy
wasfoundtobeca.0.05eV.Itshouldbenoted,however,
∗
Email address: [email protected];
that [5] uses a Hamiltonian with short ranged (on-site)
w.barford@sheffield.ac.uk
interactions, with the strength chosen so as to fit the
[1] S.R.White,Phys.Rev.Lett.69, 2863(1992); Phys.Rev.
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ily use openboundary conditions. Foranevensite chain (1986).
there is one more short bond than there are long bonds. [3] J. Takimoto and M. Sasai, Phys. Rev.B 39, 8511 (1989).
This means that the ground state is non-degenerate, as [4] J. T. Gammel and D. K. Campbell, Synth.Met. 55, 4638
it is energetically unfavourable to swap long and short (1993).
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Therˆoleplayedbyboundaryconditionsissubtleandim- [6] D. Yaron, E. E. Moore, Z. Shuai, J. J. Bredas, J. Chem.
Phys.108, 7451 (1998).
portant,asrealsystems,suchasoligomersandpolymers
[7] G. Fano, F. Ortolani, L. Ziosi, J. Chem. Phys. 108, 9246
with disorder, have a finite conjugation length.
(1998).
Another consideration is the adequacy of the soliton
[8] J. Jeckelmann, Phys. Rev.B 57, 11 838 (1998).
fits used in generating the adiabatic energy curves. We [9] M. Kuwabara, Y. Shimoi, S. Abe, J. Phys. Soc. Jpn. 67,
consider the generalised potential energy curves where,
1521 (1998).
for a given x0, we allow ξ to vary so as to minimise the [10] R.J.Bursill,C.Castleton,W.Barford,Chem.Phys.Lett.
energy. Resultsforthe11B− stateareincludedinFig.4, 294, 305 (1998).
u
whichshowthatrelaxingξ yieldsasubstantialreduction [11] E. Ehrenfreund, Z. Vardeny, O. Barfman, B. Horovitz,
in the energy, implying weak soliton/anti-soliton bind- Phys.Rev. B 36, 1535 (1987).
ing. However,the energyreductionforthe 13B+ state is [12] W.M.Flicker,O.A.MosherandA.Kuppermann,Chem.
insignificant over the range of x0 values plottedu, indicat- Phys.Lett. 45, 492 (1977).
[13] M. Boman, R.J. Bursill andW. Barford, Synth.Met. 85,
ing that there is a strongerbinding of the solitons in the
21A+ and 13B+ states. A further generalisation of the 1059 (1997).
g u [14] M. Boman and R. J. Bursill, Phys. Rev. B 57, 15 167
soliton fits would be to consider multiple soliton/anti-
(1998).
soliton pairs. [15] B. E. Kohler, J. Chem. Phys. 88, 2788 (1988).
Finally, we note the consequences of our results for
[16] We have checked the DMRG convergence of the δi and
the interpretation of experiments. Our results for small found that, for the N =102 site lattice, they are resolved
polyenesareingoodagreementwithexperiment—theen- to within 1% or less.
ergydifferenceofca.0.3eVforthe11B− statecanprob- [17] Z.Shuai,J.L.Bredas,S.K.PatiandS.Ramasesha,Phys.
u
ablybeexplainedbysolvationeffects[19],supportingthe Rev.B 56, 9298 (1997).
notionthatthecovalent21A+ stateislesspolarisedthan [18] D.K.CampbellandA.R.Bishop,Nucl.Phys.B200,297
g
the ionic 11B−. In the bulk limit the 11B− and 21A+ (1982).
u u g
[19] E. Moore, B. Gherman B. and D. Yaron, J. Chem. Phys.
energies are ca. 0.8 eV higher than data from linear [20]
106, 4216 (1997).
and2-photon[21]absorptionandthirdharmonicgenera-
[20] Relaxation in Polymers (p 174), ed. by T. Kobayashi.
tion[22]experimentsonPAthinfilms. Thisimpliesthat
World Scientific (Singapore) 1993.
there are more substantial energy decreases due to sol- [21] C. Halvorson and A. J. Heeger, Chem. Phys. Lett. 216,
vation and aggregation (interchain hopping and eximer
488 (1993).
formation) effects. Such effects must be investigated via [22] W. S.Fann, et al.,Phys. Rev.Lett. 62, 1492 (1989).
coupledchaincalculations. Also,the neglectofquantum [23] R. H. McKenzie and J. W. Wilkins, Phys. Rev. Lett. 69
fluctuationsintheadiabatictreatmentofthelattice[23], 1085 (1992).
leading to the pinning of the solitonstructures, will con-
tribute to this energy difference. A full treatment must
includedynamicalphonons. Suchatreatmentwouldalso
increase our understanding of the soliton confinement.
3
TABLEI. Convergenceoftheground stateenergy(in eV) 5 Ev(21A+g)
asafunctionoftheSBHSSfortheN =102sitesysteminthe
non-interacting case (U =0) for three geometries defined by
V) Ev(11B–)
thesolitonform(3),takingξ=4.03and(i)x0 =0(uniformly e 4 u
dimerised geometry), (ii) x0 = ∞ (uniformly dimerised with p( E0-0(11Bu–)
olomnegtrayndwisthhoartkbinokn/dasnrtei-vkeirnskedp)a,iranpdlac(ieidi)1x/04=and243.8/74(oaf tghee- yga 3 Ev(13B+u)
wayalongthelattice). misthenumberofstatesretainedper g
block. ner E0-0(21A+g)
E 2
m SBHSS x0 =0 x0=∞ x0=24.87
75 5920 −332.57146 −330.205 −331.333 E0-0(13B+)
u
100 9384 −332.57217 −330.403 −331.366 1
0.025 0.05 0.075 0.1 0.125 0.15
150 22392 −332.57257 −330.426 −331.407
1/N
200 37512 −332.57271 −330.439 −331.422
FIG.1. Energy gaps for the11B+ (solid lines) 21A+ (dot-
230 52312 −332.57272 −330.446 −331.428 u g
tedlines)and13B+(dashedlines)statesasafunctionof1/N.
270 72392 −332.57273 −330.448 −331.430 u
Vertical/0-0transitionsareindicatedbythin/thicklines. Ex-
EXACT — −332.57276 −330.452 −331.434 perimental 0-0 energies of the 11B− (diamonds) and 21A+
u g
(triangles) states for polyenes in hydrocarbon solution [15].
Theempirical fittingform E0-0(21A+)=0.96+20.72/N, de-
g
rived in [15] from the 8–14-site oligomer data is also plotted
TABLEII. Convergenceofthegroundstate(11A+)energy
andverticaland0-0transitionenergiesofthe21A+agnd11B− (dot-dashed line).
g u
statesasafunctionoftheSBHSSfortheN =102sitesystem.
SBHSS 11A+g 21A+g 21A+g(0-0) 11Bu− 11Bu−(0-0) d
15844 −509.6330807 2.8927 1.8051 2.7719 2.6785 0.08 i 13B+
u
25492 −509.6330971 2.8795 1.8008 2.7650 2.6483
0.06
36312 −509.6331002 2.8764 1.7972 2.7617 2.6392 0.07 d i+d i+1
54916 −509.6331009 2.8744 1.7963 2.7605 2.6345 0.04 2
E
67240 −509.6331010 2.8737 1.7959 2.7601 2.6336 0.02 0.06 g 11B–u
0 21A+ 0.05
g
–0.02
–0.04 0 2 6 8 10 12 14
Bond index, i
5 10 15 20 25 30
Bond index, i
FIG. 2. The geometries (δi as a function of bond index i
from the center of the lattice) of various excitations: 11B−
u
(open diamonds), 13B+ (filled diamonds), 21A+ (stars) and
u g
the charged state, Eg, (open triangles), for the N = 102
site system. The solid lines are fits to the 2-soliton form (3)
(and the 4-soliton form (4) for the 21A+). The inset shows
g
the two-point averages ((2i+1)/2,(δi+δi+1)/2) for the po-
laronic Eg and 11Bu− states, which are well described by the
2-soliton fits.
4
1/N
0.01 0.02 0.03 0.04 0.05 0.06 0.07
x (21A+) 5
d g
x (21A+)
12 g 4.5
4
10
3.5
x (11B– ) 3
8 u
x (11B–) x0 (21A+g) 2.5
6 0 u x (13B+) 2
0 u
4
x (13B+)
u
0.02 0.03 0.04 0.05 0.06 0.07
1/N
FIG. 3. The convergence of the soliton fitting parameters
x0 and ξ for the11Bu− and 13Bu+ states with thelattice size,
N. The inset shows x0, ξ and xd for the21A+g state.
3.5
3
2.5 x.
V) ela
e 2 r
(
rgy1.5 miss.
e e
En0.15 vertical 0-0
2 4 6 8 10 12 14
x
0
FIG.4. Potential energy curves (solid lines) for the 11B−
u
(diamonds),21A+ (stars) and13B+ (triangles) states forthe
g u
N =102 site lattice. The dashed curves are the correspond-
ing ground state (11A+) potential energies. For the 11B−
g u
and 13B+ cases, the curves are generated using the soliton
u
pair form (3) with the fitted values of ξ (4.01 and 12.12, re-
spectively) andvaryingx0. Forthe21A+g case thecurvesare
generated using the 4-soliton form (4) with the fitted value
ξ=4.93 andvaryingx0. xd ischosensothattheratioxd/x0
remains fixed at its fitted value of 1.35. The solid diamonds
are the values of the 11B− energy when ξ is also allowed to
u
vary. The energies of the vertical, 0-0 and emission transi-
tions, and the relaxation energy can be read off from this
plot. This is illustrated using arrowed vertical lines for the
21A+ state.
g
5