- Email: [email protected]
Short Polymer Chains: Geometry and Vibrations
Synthetic Metals 101 (1999) 321-322
Short
Polymer
A. Painelli’, a Dip.
Chim,
Gen.Inorg., bDip.
Chains: L. Del Freoa,
Chim.Anal.,
Chemistry,
Geometry A. Girlandoa,
Chim.
Princeton
and Vibrations
Fis.,
and
Parma
University,
Z.G.
University,
NJ-08544
Soosb I-431
Princeton,
00, Parma,
Italy
US
Abstract with
We present a simple model for an exponential t(R) is applied
Keywords:
Polyacetylene
and
ground state geometry and vibrational to the vibrational properties of PA
derivatives,
Semi-empirical
models
The large polarieability of T-electrons governs the physics of conjugated molecules and polymers. Semiempirical r-electron models are then extremely useful to understand the physical properties of this interesting class of materials. In recent years linear and non-linear spectroscopic data were collected on conjugated polymers and oligomers suggesting that both electron-electron (e-e) and electron-phonon (eph) interactions are important in determining the energy as well as the nature of the excited states [l]. But not enough experimental data, nor theoretical work are available at present as to unambiguously fix reliable parameters for interacting T-electrons coupled to lattice vibrations. The problem of modeling r-electrons simplifies considerably if only ground state (GS) properties are of interest. In fact the GS of conjugated polymers is satisfactorily described by the Hiickel model with effective parameters, implicitely accounting for the effects of e-c interactions [I]. GS properties include equilibrium geometry, static polarizabilities as well as vibrational properties (frequencies, IR and non-resonant Raman (NRR) in t ensities). By an extensive analysis of vibrational data on polyacetylene (PA) we recently extracted detailed information on the PA reference force field (FF) as well as on the e-ph coupling. In particular we were able to prove an exponential dependence of the hopping integral on the bond length [I]: t; = it(, exp (-rR,), with to = 2.5 eV, 7 = 2.1 A-‘. Based on this information we present a simple and internally consistent model for the calculation of GS properties of PA-based systems. This model allows reliable predictions of the geometry and of the vibrational spectra of pristine PA oligomers, as welI as of soliton defects residing on infinite and on short chains. Following a classical work by Longuet-Higgins and Salem [2], the o- force constants K,? is fixed by the chosen t(R) dependence as: KP
= -Arti
- 2r’tipi
(1)
where pi is the a-electron bond-order and A, formally equal to -2dpi/dRi is a constant to be extracted from the experiment. We fix A= 3.2 A-’ to reproduce the alternation S ~0.18 for pristine PA. With this choice the 0379-6779/99/$ - see front matter 0 1999 Elsevier PII: SO379-6779(98)0 1207-7
Science S.A.
and
properties oligomers,
model
of conjugated and to solitons
systems. on finite
A Hiickel chains.
model
calculations.
average displacement of C atoms amounts to N 0.04 A, in agreement with experiment. We then calculate the equilibrium geometries for PA oligomers as well as for fmite chains bearing a soliton defect. The validity of the proposed approach is confirmed by the comparison with available quantum chemistry calculations on short chains: our results agree within 10% with both MNDO results on a 41 site chain [3] and ab initio results on a 21 site chain [4]. The vibrational problem is conveniently dealt with in the Herzberg-Teller approach that naturally distinguishes the reference FF and the contribution to the FF from =electron fluctuations [I]. In particular the contribution from the GS n-electron distribution adds to the Kf in Eq. 1, to define the reference force constants: KF
= A’Yltil
(2)
From a theoretical point of view A’ in the above equation should coincide with A in Eq. (1). However, as often the case with semiempirical models, different values of the parameters are needed to fit different properties. A’ is fixed to 8.30 A-’ by comparison with the reference FF as extracted from the analysis of Raman spectra of pristine PA [l]. In the same spirit we also include an interaction constant for adjacent C-C stretches: KF,=l.67 eV/A’. Following a procedure already described [S] we calculate for PA oligomers the susceptibilities (x) relevant to each reference coordinate: for each chain there is a single mode with a large susceptibility. The corresponding x, normalized to the 200-site, are reported in Fig. 1, upper panel. These susceptibilities compare well with the experimental x (reported in the same figure) as extracted from the analysis of Raman spectra of PA chains with 4 - 24 C atoms [I]. For pristine PA chains the most strongly coupled mode is Raman active: its NRR intensity is reported in Fig.1, lower panel. The intensity increases steeply with the chain length, and in fact it does not show sign of saturation up to 400 sites. The NRR intensity of PA oligomers scale as the 4thpower of the inverse of the optical gap (reported as triangles in the same figure). This behavior reflects the band-gap modulation by the coupled mode in these 1D systems [61.
All rights reserved.
322
A. Painelli
et al. I Synthetic
MeraIs
101 (1999)
321-322
chains bearing a charged soliton. The x values are reported in Fig.2, upper panel. -The x values saturate for chains longer than w 100. The infinite chain result is in perfect agreement with the experimental x obtained from the IR spectra of photoexcited PA samples (11. In Fig. 2, lower panel we report the calculated IR intensity of the relevant mode: it increases less dramatically than the NRR intensity with the chain length, and scales as the square of the inverse of the optical gap (triangles in the figure). The superlinear increase of IR intensity with chain length and the early saturation of x vs l/N describes the observed asymmetry of IRAV bands without postulatmg a bimodal or asymmetric distribution for IV. In Fig.3 we report the IRAV spectra as calculated for a distribution of PA segments described by a single gaussian, with fixed center and variable width. In agreement with experiment we get asymmetric lineshapes with long tails in the blue. The width of the calculated band in Fig. 3 decreases with increasing width of the gaussian distribution. This counter-intuitive behavior is understood by noting that a wider distribution gives more weight to long segments whose greater intensity results in a red shift compared to the mean frequency. Short chains are responsible -for the long tai! to the blue. Pristine chains. Upper panel: calculated Fig. 1. (squares) and experimental (stars) x ratioed to the value for the pristine PA, Lower panel: calculated NRR intensity (squares) and the 4-th power of the optical gap (triangles).
2.0
X Fig. 3. Calculated IR spectrum for solitons residing-on chains with a gaussian distribution of lengths. The center of the gaussian is fixed to 29, continuous, long-dashed and short-dashed lines correspond to gaussian widths of 29, 11, 7, respectively. The integrated area decreases for broader distributions since N = 29 solitons already have 75% of the limiting intensity.
1.0 1 .o
I IR 0.5
We thank Princeton,
Fig. 2. Soliton chains: calculated x ratioed to the value for pristine PA. The arrow marks the experimental value (see text). Lower panel: calculated IR intensity (squares) and the square of the optical gap (triangles). The super-linear increase with the chain intensity agrees with the experimental classes of conjugated polymers [7,8]. Similar calculations are performed
length behavior on
of the NRR of several
odd-numbered
NSF-DMR-9530116 MURST and CNR
for support of work in for support of work in Parma,
[ 11 Z.G.Soos, et al., in Handbook of Conducting Polymers, T.A.Skotheim, R.L.Elsenbaumer, and J .R.Reynolc&, Eds.; Marcel Dekker, New York (1998). [ 21 H.C.Longuet-Higgins, and L.Salem, Proc.Roy.Soc.A 251 (1959) 172. [ 3] Y.Mori, Synth.Met., 30 (1989) 227. [ 4] H.D.Villar et al., Phys.Rev.B, 37 (1988) 2520. 151 A.Painelli et al., Phys.Rev.3, 56 (1997) 15100. [ 61 L.Del Free et al., to be published. [ 7] M.Rumi et al., J.Chem.Phys., 106 (1997) 24. [ 81 O.Lazareva and A.N.Shchegolikhn Synth.Met., 84 (1997) 991.
Short
Polymer
A. Painelli’, a Dip.
Chim,
Gen.Inorg., bDip.
Chains: L. Del Freoa,
Chim.Anal.,
Chemistry,
Geometry A. Girlandoa,
Chim.
Princeton
and Vibrations
Fis.,
and
Parma
University,
Z.G.
University,
NJ-08544
Soosb I-431
Princeton,
00, Parma,
Italy
US
Abstract with
We present a simple model for an exponential t(R) is applied
Keywords:
Polyacetylene
and
ground state geometry and vibrational to the vibrational properties of PA
derivatives,
Semi-empirical
models
The large polarieability of T-electrons governs the physics of conjugated molecules and polymers. Semiempirical r-electron models are then extremely useful to understand the physical properties of this interesting class of materials. In recent years linear and non-linear spectroscopic data were collected on conjugated polymers and oligomers suggesting that both electron-electron (e-e) and electron-phonon (eph) interactions are important in determining the energy as well as the nature of the excited states [l]. But not enough experimental data, nor theoretical work are available at present as to unambiguously fix reliable parameters for interacting T-electrons coupled to lattice vibrations. The problem of modeling r-electrons simplifies considerably if only ground state (GS) properties are of interest. In fact the GS of conjugated polymers is satisfactorily described by the Hiickel model with effective parameters, implicitely accounting for the effects of e-c interactions [I]. GS properties include equilibrium geometry, static polarizabilities as well as vibrational properties (frequencies, IR and non-resonant Raman (NRR) in t ensities). By an extensive analysis of vibrational data on polyacetylene (PA) we recently extracted detailed information on the PA reference force field (FF) as well as on the e-ph coupling. In particular we were able to prove an exponential dependence of the hopping integral on the bond length [I]: t; = it(, exp (-rR,), with to = 2.5 eV, 7 = 2.1 A-‘. Based on this information we present a simple and internally consistent model for the calculation of GS properties of PA-based systems. This model allows reliable predictions of the geometry and of the vibrational spectra of pristine PA oligomers, as welI as of soliton defects residing on infinite and on short chains. Following a classical work by Longuet-Higgins and Salem [2], the o- force constants K,? is fixed by the chosen t(R) dependence as: KP
= -Arti
- 2r’tipi
(1)
where pi is the a-electron bond-order and A, formally equal to -2dpi/dRi is a constant to be extracted from the experiment. We fix A= 3.2 A-’ to reproduce the alternation S ~0.18 for pristine PA. With this choice the 0379-6779/99/$ - see front matter 0 1999 Elsevier PII: SO379-6779(98)0 1207-7
Science S.A.
and
properties oligomers,
model
of conjugated and to solitons
systems. on finite
A Hiickel chains.
model
calculations.
average displacement of C atoms amounts to N 0.04 A, in agreement with experiment. We then calculate the equilibrium geometries for PA oligomers as well as for fmite chains bearing a soliton defect. The validity of the proposed approach is confirmed by the comparison with available quantum chemistry calculations on short chains: our results agree within 10% with both MNDO results on a 41 site chain [3] and ab initio results on a 21 site chain [4]. The vibrational problem is conveniently dealt with in the Herzberg-Teller approach that naturally distinguishes the reference FF and the contribution to the FF from =electron fluctuations [I]. In particular the contribution from the GS n-electron distribution adds to the Kf in Eq. 1, to define the reference force constants: KF
= A’Yltil
(2)
From a theoretical point of view A’ in the above equation should coincide with A in Eq. (1). However, as often the case with semiempirical models, different values of the parameters are needed to fit different properties. A’ is fixed to 8.30 A-’ by comparison with the reference FF as extracted from the analysis of Raman spectra of pristine PA [l]. In the same spirit we also include an interaction constant for adjacent C-C stretches: KF,=l.67 eV/A’. Following a procedure already described [S] we calculate for PA oligomers the susceptibilities (x) relevant to each reference coordinate: for each chain there is a single mode with a large susceptibility. The corresponding x, normalized to the 200-site, are reported in Fig. 1, upper panel. These susceptibilities compare well with the experimental x (reported in the same figure) as extracted from the analysis of Raman spectra of PA chains with 4 - 24 C atoms [I]. For pristine PA chains the most strongly coupled mode is Raman active: its NRR intensity is reported in Fig.1, lower panel. The intensity increases steeply with the chain length, and in fact it does not show sign of saturation up to 400 sites. The NRR intensity of PA oligomers scale as the 4thpower of the inverse of the optical gap (reported as triangles in the same figure). This behavior reflects the band-gap modulation by the coupled mode in these 1D systems [61.
All rights reserved.
322
A. Painelli
et al. I Synthetic
MeraIs
101 (1999)
321-322
chains bearing a charged soliton. The x values are reported in Fig.2, upper panel. -The x values saturate for chains longer than w 100. The infinite chain result is in perfect agreement with the experimental x obtained from the IR spectra of photoexcited PA samples (11. In Fig. 2, lower panel we report the calculated IR intensity of the relevant mode: it increases less dramatically than the NRR intensity with the chain length, and scales as the square of the inverse of the optical gap (triangles in the figure). The superlinear increase of IR intensity with chain length and the early saturation of x vs l/N describes the observed asymmetry of IRAV bands without postulatmg a bimodal or asymmetric distribution for IV. In Fig.3 we report the IRAV spectra as calculated for a distribution of PA segments described by a single gaussian, with fixed center and variable width. In agreement with experiment we get asymmetric lineshapes with long tails in the blue. The width of the calculated band in Fig. 3 decreases with increasing width of the gaussian distribution. This counter-intuitive behavior is understood by noting that a wider distribution gives more weight to long segments whose greater intensity results in a red shift compared to the mean frequency. Short chains are responsible -for the long tai! to the blue. Pristine chains. Upper panel: calculated Fig. 1. (squares) and experimental (stars) x ratioed to the value for the pristine PA, Lower panel: calculated NRR intensity (squares) and the 4-th power of the optical gap (triangles).
2.0
X Fig. 3. Calculated IR spectrum for solitons residing-on chains with a gaussian distribution of lengths. The center of the gaussian is fixed to 29, continuous, long-dashed and short-dashed lines correspond to gaussian widths of 29, 11, 7, respectively. The integrated area decreases for broader distributions since N = 29 solitons already have 75% of the limiting intensity.
1.0 1 .o
I IR 0.5
We thank Princeton,
Fig. 2. Soliton chains: calculated x ratioed to the value for pristine PA. The arrow marks the experimental value (see text). Lower panel: calculated IR intensity (squares) and the square of the optical gap (triangles). The super-linear increase with the chain intensity agrees with the experimental classes of conjugated polymers [7,8]. Similar calculations are performed
length behavior on
of the NRR of several
odd-numbered
NSF-DMR-9530116 MURST and CNR
for support of work in for support of work in Parma,
[ 11 Z.G.Soos, et al., in Handbook of Conducting Polymers, T.A.Skotheim, R.L.Elsenbaumer, and J .R.Reynolc&, Eds.; Marcel Dekker, New York (1998). [ 21 H.C.Longuet-Higgins, and L.Salem, Proc.Roy.Soc.A 251 (1959) 172. [ 3] Y.Mori, Synth.Met., 30 (1989) 227. [ 4] H.D.Villar et al., Phys.Rev.B, 37 (1988) 2520. 151 A.Painelli et al., Phys.Rev.3, 56 (1997) 15100. [ 61 L.Del Free et al., to be published. [ 7] M.Rumi et al., J.Chem.Phys., 106 (1997) 24. [ 81 O.Lazareva and A.N.Shchegolikhn Synth.Met., 84 (1997) 991.