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J. Phys. Chem. B 1998, 102, 1710-1712
Saturation of the Optical Band Gap and Properties of Five-Membered Heteroaromatic Oligomers Yi Luo* FYSIKUM, UniVersity of Stockholm, Box 6730, S-113 8S Stockholm, Sweden
Kenneth Ruud,† Patrick Norman, Dan Jonsson, and Hans Ågren Institute of Physics and Measurement Technology, Linko¨ ping UniVersity, S-58183 Linko¨ ping, Sweden ReceiVed: December 30, 1997
The saturation behavior of the optical band gap and dynamic (hyper)polarizabilities of five-membered heteroaromatic oligomerssoligothiophene, oligofuran and oligopyrroleshas been studied by means of ab initio response theory in the random phase approximation. The calculated optical band gap for the oligomers and the extrapolated band gap for the polymers are found to be within 0.2 eV of available experimental data. The calculated saturation lengths of the band gaps are in good agreement with experimental determinations. The saturation of the polarizabilities is found to be slower than that of the band gaps, especially when dispersion is taken into account. The calculations indicate that previous experimental observation of fast convergence of the optical properties of oligothiophenes may be due to disorder in the samples.
It is known that the (hyper)polarizabilities of conjugated oligomers increase exponentially with increasing number of monomer units until a linear dependencesa saturation regionsis reached. The oligothiophenes constituted the first system for which the saturation behavior of the band gap, polarizability, and hyperpolarizability was experimentally observed.1 The saturation region was found at about 7 repeat units. Later, much slower convergence was observed experimentally for polyenes;2 to reach saturation of the hyperpolarizability for polyenes in a substituted disordered form, as much as 125 repeat units were required. In general, experimental studies on the saturation behavior of optical properties of oligomers have been quite rare because of the difficulties of synthesis. On the other hand, the development of efficient quantum chemistry methods has enabled calculations of ever larger molecules, and ab initio calculations have now become potential tools for studying saturation. So far, most theoretical studies have been devoted to chain saturation of trans-polyenes employing both semiempirical (see ref 3 and references therein) and ab initio4-8 methods. Lu et al.7 predicted saturation of the hyperpolarizability of the trans-polyenes at approximately 45 repeat units at the Hartree-Fock (HF) level with a 6-31G basis set. A similar result was advocated by Ruud et al.8 who showed that the saturation would occur at about 40 repeat units at the Hartree-Fock level with a 4-31++G derived basis. The difference between those two HF results is probably due to the difference in basis set, geometry, and fitting functions employed. However, it is quite clear that the theoretically predicted saturation occurs at a significantly smaller number of repeat units than that observed experimentally. As shown by Lu et al.,7 the disorder in the experimental samples and other structural defects in the chain would modify the saturation length, a fact which can explain the mismatch between theory and experiment in this respect. † Permanent address: Department of Chemistry, University of Oslo, P.O. Box 1033, Blindern, N-0315 Oslo, Norway.
In this paper, we will examine the saturation behavior of the optical band gap and (hyper)polarizabilities of three different five-membered heteroaromatic oligomers, oligothiophenes (OTH), oligofuran (OFU), and oligopyrrole (OPY), by means of ab initio calculations. For this purpose we use a well-tested level of electronic structure theorysthe time-dependent self-consistent field method (TDSCF), also denoted the random phase approximation (RPA)sand a standard basis set, 6-31++G,9,10 for the determination of the molecular properties. The geometries of all samples are optimized by the AM1 method.11 All ab initio calculations have been performed with a previously described parallel implementation of the RPA equations12 within the DALTON quantum chemistry code.13 The calculations were carried out with 98% slave efficiency on different parallel platforms, including a Cray-T3E and a Cray-Origin 2000 MPP machine, using up to 64 processors in the calculations. The optical band gaps, Eg, of oligothiophenes have been experimentally measured by two different groups1,14 who report very similar results. We have compared our RPA results for the optical band gap with these experimental data in Figure 1. The agreement improves as the chain length increases. It is quite striking that except for the thiophene monomer the RPA energies are within 0.2 eV of the experimental data. These results for the heteroaromatic oligomers thus provide a confirmation of previous observations for trans-polyenes and diphenylpolyenes, namely that RPA provides excellent results for the optical band gap of conjugated oligomers.15-17 We have adopted the fitting function used by Lu et al.7
(
P(N) ) A
)
1 - exp(-BN) 1 + C exp(-DN)
(1)
where P(N) can be optical band gap (Eg), polarizability per monomer unit (R/N), or the second hyperpolarizability per monomer unit (γ/N). The saturation length Nsat is defined to be the length at which the property per monomer unit is within
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Properties of Five-Membered Heteroaromatic Oligomers
J. Phys. Chem. B, Vol. 102, No. 10, 1998 1711
Figure 1. Comparison between RPA and experimental optical band gaps (Eg) of oligothiophenes. Two sets of experimental data reported by two different groups from absorption spectra, Exp. I (filled circle)1 and Exp. II (diamond),14 are shown.
Figure 3. Length dependence of polarizability, R(-ω;ω)/N, for oligopyrroles (filled circle), oligofurans (diamond), and oligothiophenes (triangle) at (a) static limit and (b) wavelength 532 nm.
Figure 2. Length dependence of optical band gap for oligopyrroles (filled circle), oligofurans (diamond), and oligothiophenes (triangle).
95% of the limiting value. The values of the monomers are not included in these fittings. The optical band gaps of OTH, OFU, and OPY are shown in Figure 2. In general, the ordering of the oligomers in terms of the band gap is OPY > OFU > OTH. The optical band gap for the corresponding polymers can be found to be 3.09, 2.76, and 2.67 eV for polypyrrole (PPY), polyfuran (PFU) and polythiophene (PTH), respectively. The predicted Eg is in good agreement with the measured band gap of 3.2,18 3.0, and 2.52.7 eV20,21 for PPY, PFU, and PTH, respectively. The saturation
length Nsat is about 7, 8, and 8 units for OPY, OFU, and OTH, respectively. The calculated saturation length of the oligothiophenes is in good agreement with experimental observations.1,14 Unfortunately, we have not found any experimental band gap data for oligopyrroles and oligofurans. However, on the basis of the excellent agreements obtained for OTH, we are inclined to believe that the RPA band gap results for OPY and OFU should be close to that which would have been measured in experiment. Figure 3 shows polarizabilities, R(-ω;ω)/N, at the static limit and at a wavelength λ ) 532 nm plotted against the number of repeat units, N. It is interesting to note that in the static limit one can observe a saturation behavior for all compounds. However, for the dynamic polarizabilities, only OPY displays such a trend. This indicates that the saturation lengthsalso denoted the correlation lengthsfor the polarizability depends on the frequency. The saturation length for the static R/N is found to be about 9 repeat units for all three oligomers. The limiting numbers for R/N are 143.9, 144.1, and 183.5 au for OPY, OFU, and OTH. The dynamic polarizability has also been fitted by the fitting function. It shows that the largest calculated values for OPY(N ) 9), OFU(N ) 9), and OTH(N ) 8) are about 91%, 84%, and 68% of the limiting values, respectively. The saturation length has been estimated to be 11, 12, and 14 repeat units for OPY, OFU, and OTH, respectively. This shows that the saturation length NRsat at finite frequencies should be larger than that at the static limit. It was found experimentally that the saturation length of R(-ω;ω) at the wavelength 632.8 nm for oligothiophenes might be around 7 repeat units,1 which is even shorter than what is calculated here for the static value, 9 repeat units. However, it should be noted that the measured values for R of OTH in ref 1 is about one order of magnitude
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Luo et al. nm, respectively. For the oligothiophenes, the experimental saturation length for ESHG at 1064 nm is found to be the same as that for the polarizability: 7 repeat units.1 Our calculations do not support this observation. Another experimental measurement for oligothiophenes up to 6 repeat units shows no sign of saturation.14 We find the order OPY > OFU > OTH for the optical band gap and OPY < OFU < OTH for the (hyper)polarizabilities. Finally, the ordering in terms of the saturation lengths is found to be the same as the order for the (hyper)polarizabilities. Acknowledgment. This work has received support from NSC at Linko¨ping University and the Research Council of Norway (Program for Supercomputing) through grants of computer time. References and Notes
Figure 4. Length dependence of hyperpolarizability, γ(-2ω;0,ω,ω)/ N, for oligopyrroles (filled circle), oligofurans (diamond), and oligothiophenes (triangle) at (a) static limit and (b) wavelength 1064 nm.
smaller than those reported by Zhao, Singh, and Prasad14 which showed no saturation up to 6 repeat units. Therefore, we believe that disorder in the sample might be responsible for the fast convergence reported by Thienpont et al.1 Much slower convergence is found for the hyperpolarizability, γ(-2ω;0,ω,ω)/N; see Figure 4. The saturation length for the hyperpolarizabilities is found to be more dependent on the frequency. Therefore, it is not reliable to use the value of the static saturation length to explain the experimental results that are obtained at finite frequencies. The large differences found between theory and experiment for polyenes might therefore in part be due to the reason that the theory was for the static value and the experiment was for dynamic value. Only for OPY, the results for N ) 9 seem to be close to saturation. Using the above specified fitting function, we have found the saturation length to be 10 and 11 for the static and electric field induced second-harmonic generation (ESHG) at the wavelength 1064
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