J. Phys. Chem. 1996, 100, 17327-17333
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Density Functional Study of Polythiophene Derivatives G. Brocks Philips Research Laboratories, Prof. Holstlaan 4, 5656 AA EindhoVen, The Netherlands ReceiVed: July 15, 1996X
Density functional calculations within the Car-Parrinello framework are used to obtain self-consistently the electronic and the geometrical structures of polythiophene derivatives. The relation between the electronic states and quinoid vs aromatic structures is studied on a series of substituted polythiophenes (PTh) and polyisothianaphthenes (PITN). Excellent agreement is found with experimental data, where these are available. Some shortcomings of semiempirical methods in describing π bonds with heteroatoms are revealed. A unique minimal energy structure is obtained for each of the polymers; no other local minima are found. The general rule is that a polymer adapts the structure that leads to the largest possible band gap, which for PTh-like polymers is the aromatic structure and for PITN-like polymers is the quinoid structure. Substitutions do not change the basic geometry and electronic structure of either PTh or PITN but can alter the band gap by several tenths of an electronvolt.
Introduction In the field of semiconducting polymers, polythiophene (PTh) and its derivatives are among the most widely studied materials. Polythiophene 1 is most easily prepared electrochemically or
by oxidative coupling in solution, and it is stable under ambient conditions.1 Although by itself polythiophene is rather intractable, processable and soluble derivatives can be made by attaching alkyl side chains.2 Polyethylenedioxythiophene (PEDOT 2)3 is a derivative that can be doped to a much higher conductivity.4 Such derivatives, like polythiophene itself, have a rather wide band gap of about 2 eV.1 When semiconducting polymers are applied in devices such as diodes and transistors, wide band gaps almost inevitably lead to problems such as finding contacts for injecting electrons that have a sufficiently X
Abstract published in AdVance ACS Abstracts, October 15, 1996.
S0022-3654(96)02106-5 CCC: $12.00
low barrier with respect to the polymer. Over the past decade, a substantial effort has been devoted to finding derivatives with a small band gap. The optical absorption of polymers is usually strongly peaked around the band gap, caused by a density of states typical of quasi one-dimensional systems. Polythiophene with a band gap of 2 eV therefore absorbs strongly in the visible range. When it is heavily doped, its (electronic) structure changes in such a way that the optical absorption is shifted to longer wavelengths.1 Starting from a polymer with a smaller band gap in the neutral state, it is possible to shift the main absorption in the doped state into the infrared. This opens the possibility for the use of these polymers as transparent conductors. One possible route to small band gap polymers is the recently discovered polysquaraines.5,6 An alternative route is formed by a derivative of polythiophene, polyisothianaphthene (PITN 3), which has a rather small band gap of about 1 eV.7 Unfortunately, so far the stability and processability of the PITN material has been rather poor, despite some recent efforts to improve this via new synthetic routes,8 which has prevented the large scale use of this material. One of the problems of PITN, and indeed of semiconducting polymers in general, is their sensitivity to oxidation. When extra electrons are introduced, as in n-type doping, the material becomes highly unstable. These are manifestations of a relatively low ionization potential and electron affinity, respectively. In view of these problems, there has been an increased interest in derivatives with higher electron affinities/ionization potentials.9 Nitrogensubstituted derivatives, such as polythienopyrazine 4 may present a solution to some of the problems scetched above.10 Parallel to the experimental activities, there has been considerable quantum chemical work devoted to polythiophene and its derivatives.11-17 Most of the discussion has been centered around the relationship between the electronic structure, and more specifically the band gap, and the geometrical structure of the polymers.18,19 However, the quantification of this link by means of quantum chemical calculations has followed up till now a rather indirect path. The structures of the polymers are usually obtained by Hartree-Fock11 or semiempirical (typically at the level of MNDO or AM120) calculations on oligomers13-15 or polymers.12 The electronic structure is then obtained using a non-self-consistent one-electron Hu¨ckel-type method, by using either an empirical Hu¨ckel parametrization12,14 or the nonempirical VEH parametrization.21 Such calculations © 1996 American Chemical Society
17328 J. Phys. Chem., Vol. 100, No. 43, 1996
Figure 1. Generic structure and atomic labeling of the polymers discussed in this paper, e.g., the structure with all-carbon atoms corresponds to polyisothianaphthene 3; the structure with nitrogen atoms at positions 6 and 9 corresponds to polythienopyrazine 4a. The two monomers in the unit cell are connected by a screw axis symmetry.
usually agree qualitatively but contain quantitative differences.11-15 Density functional theory has also been applied to obtain the electronic structure.16,17 In this paper it is shown that it is possible to calculate both the geometrical and the electronic structure of the polymers simultaneously and fully selfconsistently using density functional methods. The relation between the geometrical and electronic structures of PTh 1, PEDOT 2, PITN 3, and their nitrogen-substituted derivatives 4 is studied in a systematic way. Polymer 5, related to PEDOT, polymer 6, related to the thienopyrazines 4, and polythiadiazole 7 are included for reasons of completeness as is explained in the following sections. Figure 1 illustrates the generic structure of the polymers discussed in this paper. The structure of the polythiophenes shows a distinct bond length alternation (as in most semiconducting polymers). If the bond lengths are such that A1′-A1, A2-A3 > A1-A2, which is the case in the ground state structure of PTh, the structure is aromatic. If A1′-A1, A2A3 < A1-A2, as in PITN, the structure is quinoid. One interesting question is whether the isoelectronic substitution of C-H by N in PITN (compare 3 to 4) changes the structure of the polymer at a qualitative level. For instance, it has been claimed by Quattrochi et al.15 that specific substitution patterns change the structure of PITN from quinoid to aromatic, thereby reducing the band gap drastically. More general, the competition between aromatic vs quinoid structures plays a central role in all the theoretical studies cited above. One often encounters the idea that a quinoid structure reduces the band gap compared to an aromatic structure, but this cannot be true in general, the above cited example providing a counterexample. I will discuss the results of the first principles calculations in the general framework of aromatic vs quinoid structures with the help of a simple generic model that starts from the electronic states of the individual thiophene molecules. As will be shown, the polymers always adapt a structure that leads to a maximum band gap, which is aromatic for (substituted) PTh and quinoid for (substituted) PITN. The basic electronic structure is determined by the topology of the molecules and the connectivity of the molecules in the polymer. The geometry then follows from the “maximum gap” principle. In this sense, the electronic structure determines the geometry rather than the other way around. In the following sections first the computational method will be discussed and subsequently the computational results obtained for the geometrical and electronic structures of the polymers. In the final section these results will be analyzed within the framework mentioned above. Theoretical Section Simultaneous self-consistent optimization of the geometry and the electronic structure is possible within the Car-Parrinello method in which molecular dynamics is combined with an ab initio electronic structure calculation at the level of the local
Brocks density approximation (LDA) to density functional theory.22 Only the valence electrons are treated explicitly and normconserving pseudopotentials are used to represent the ion cores. The (valence) electronic wave functions are expanded in a basis set that consists of plane waves. For technical details regarding the application of this method to polymers the reader is referred to refs 23 and 24; in particular, the factors that determine convergence have been studied in detail in ref 24. In the following, all plane waves have been included in the basis set up to a kinetic energy cutoff of 30 hartrees. A Brillouin zone sampling density of 2k-points per translational unit cell, containing two monomer units, has been used. With these parameters, the typical convergence of bond lengths, for instance, is within 0.01 Å.6,24,25 Structural data of polymers related to PTh are scarce, since the materials are amorphous or at best only partially crystalline.26 The best experimental geometrical data result from X-ray diffraction studies on molecular crystals of oligomers,27 and only with those data a detailed comparison is possible. The agreement between structures calculated by density functional methods and experimental structures of molecules (oligomers) is usually good;28 for example, bond lengths agree on a scale of 0.01 Å.6 One exception might be the C-C bond length alternation within the thiophene rings. The difference in bond length between C1-C2 and C2-C3 in the central rings of thiophene oligomers is calculated to be 0.02-0.03 Å.29 The experimental values range from 0.03 to 0.06 Å.27 Because of this spread in experimental values, it is hard to assess the possible discrepancy. Comparison with previously calculated geometries of polymers related to PTh11-15 does not provide a clear picture because the semiempirical calculations show a relatively large spread dependent on the particular parameter sets used. Comparing MNDO,12 AM1,13 and PRDDO14 geometries of PITN, for example, one finds differences of 0.05 Å between C-C bond lengths and of 0.07 Å between C-S bond lengths. The STO3G11 geometry of PTh and those obtained with MNDO12 and AM113 show differences on a similar scale. It is not surprising then that the density functional results for the structure of PTh and PITN are within the boundaries of this scale, as can be judged from the results presented in the next section. There is qualitative agreement among all results in the sense that the relative order of bond lengths, the bond length alternation pattern, is the same in all these calculations. For a more discriminating test of the relative accuracy of the computational methods, we have to consider molecules again. One would expect the semiempirical methods to give reasonable results for regular C-C bonding but perhaps to have difficulties with less usual bonds such as the N-N bonds found in polymers 4b, 4d, and 7. To check the reliability in predicting such bonding, the molecules thiophene 8a, 1,3,4-thiadiazole 8b, and 1,2,5-thiadiazole 8c, which have a number of such less common bonds relevant to the polymers discussed in this paper, are chosen to test the computational methods. The calculated geometries of these molecules are presented in Table 1 and compared with the experimental geometries as determined from molecular microwave spectra.30 Table 1 gives the results for three standard semiempirical parameter sets.20 All three parametrizations give good results for thiophene, with perhaps the C-S bond as an exception, which is about 0.04 Å too short for MNDO and AM1. On the other hand, all three parametrizations fail to describe the bonding in 1,3,4-thiadiazole in an adequate way. The experimental structure has short C-N bonds, a sign of significant double bonding character, and a rather long N-N bond. In contrast, the semiempirical structures have C-N and
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J. Phys. Chem., Vol. 100, No. 43, 1996 17329
TABLE 1: Bond Lengths (Å) and Bond Angles (deg) of the Molecules Thiophene 8a, 1,3,4-Thiadiazole 8b, and 1,2,5-Thiadiazole 8ca bond
exptl
MNDO
AM1
PM3
DF
1-2 2-3 3-4 1-2-3 2-3-4 4-5-1
1.714 1.370 1.423 111.5 112.5 92.2
Thiophene 1.679 1.672 1.374 1.377 1.452 1.432 111.9 111.6 111.3 111.5 93.6 93.8
1.725 1.366 1.436 112.1 112.2 91.4
1.696 1.367 1.409 111.3 112.4 92.6
1-2 2-3 3-4 1-2-3 2-3-4 4-5-1
1.721 1.302 1.371 114.6 112.2 86.4
1,3,4-Thiadiazole 1.681 1.710 1.338 1.346 1.322 1.309 112.9 113.2 112.7 113.1 88.8 87.5
1.749 1.330 1.353 113.7 113.1 86.5
1.703 1.314 1.358 114.3 112.1 87.1
1.631 1.328 1.420 106.5 113.5 99.3
1,2,5-Thiadiazole 1.632 1.656 1.319 1.318 1.470 1.476 109.5 107.3 111.8 113.3 97.3 98.9
1-2 2-3 3-4 1-2-3 2-3-4 4-5-1
1.725 1.316 1.455 110.6 113.2 92.4
1.626 1.332 1.405 105.5 114.2 100.4
a All molecules have C 2V symmetry. The second column refers to the experimental geometries.30 The third, fourth, and fifth columns give the structures as calculated using the semiempirical parametrizations MNDO, AM1, and PM3, respectively.20 The sixth column gives the density functional (DF) optimized structures using plane wave kinetic energy cutoffs of 30 hartrees.
N-N bonds of almost equal length (MNDO, PM3) or a very short N-N bond and relatively long C-N bonds (AM1), which is qualitatively wrong. Apparently, the strength of the N-N bond with respect to the C-N bonds is overestimated by the semiempirical methods. AM1 seems to be doing worse for the bonding to nitrogen, but for the C-S bond length the AM1 value is closest to experiment. On the whole, neither of the semiempirical methods does much better than the other two. Similar conclusions can be made comparing the calculated semiempirical and experimental structures of 1,2,5-thiadiazole. All semiempirical parameter sets lead to an overestimated strength of the C-N bond with respect to the C-C bond, which shortens the former and lengthens the latter. Best in this respect is PM3, but that gives a N-S bond that is much too long. Without an extensive survey, it is not possible to determine exactly how widespread the shortcomings of the semiempirical methods in describing the bonding to heteroatoms are. They certainly occur in all five-membered 3,4-diazole rings, such as in the 1,3,4-oxadiazoles. There again, the semiempirical methods give qualitatively the wrong bond length alternation pattern compared to experimental results.31 One can observe from Table 1 that the density functional geometries are much closer to experimental results. The typical difference is on the scale of 0.01 Å in bond lengths and 1° in bond angles. Comparing the structures of all molecules in Table 1, we observe that all types of bonding, C-C, C-N, etc., are described with the same accuracy. Since the electronic structure is closely linked to the equilibrium structure, it is important to obtain the latter with sufficient accuracy. Moreover, when a series of structures, such as polymers 1-7, are compared, it is vital to obtain all structures with the same accuracy, which, in view of Table 1, is possible only within a first principles method. Formally, the density functional scheme is suited for calculating ground state properties only.32 The true band structure of a solid, comprising all single particle-like excitations, will thus be different from the band structure obtained from a density functional calculation.33 Nevertheless, the usefulness of the
TABLE 2: Bond Lengths (Å) and Bond Angles (deg) of Polymers 1-7 As Calculated Using a Plane Wave Kinetic Energy Cutoff of 30 hartreesa 3
4a
4b
4cb
4db
6
1-2 2-3 3-4 4-5 1-5 2-6 6-7 7-8 8-9 9-3 1-1′
1.448 1.416 1.448 1.736 1.736 1.395 1.384 1.382 1.384 1.395 1.338
1.429 1.403 1.429 1.754 1.754 1.337 1.341 1.388 1.341 1.337 1.366
1.444 1.404 1.444 1.738 1.738 1.394 1.342 1.334 1.342 1.394 1.368
1.436 1.405 1.431 1.761 1.735 1.337 1.340 1.340 1.388 1.384 1.362
1.447 1.407 1.447 1.758 1.734 1.338 1.341 1.335 1.339 1.396 1.366
1.428 1.415 1.428 1.749 1.749 1.334 1.639 1.639 1.334
Egap (eV)
0.90
0.85
0.56
0.90
0.70
1.39
bond
bond
1.366
1
2
5
7
1-2 2-3 3-4 4-5 1-5 2-6 6-7 7-8 1-1′
1.377 1.407 1.377 1.729 1.729
1.377 1.407 1.377 1.722 1.722 1.381 1.381 1.329 1.421
1.324 1.346 1.324 1.712 1.712
1.448
1.376 1.406 1.376 1.721 1.721 1.370 1.435 1.503 1.416
Egap (eV)
2.16
1.74
2.42
3.9
1.430
a The polymers are grouped into quinoid 3, 4, 6 and aromatic 1, 2, 5, 7 structures. The values for the gap are obtained by dividing the density functional gaps by 0.6, as explained in the text. b The results for head-to-head linkages and a regular alteration of head-tail and tailhead linkages15 are almost identical.
latter for interpreting band structures of inorganic solids has been established empirically over the last few decades: the dispersion of the bands (and thus the bandwidths) tends to agree very well with experimental results, but the band gap is roughly a factor of 2 too small.33 The main discrepancies between the experimentally measured and the density functional band structures can be corrected by simply shifting the density functional conduction bands upward by a constant.34 Furthermore, it is shown in a recent study that for a range of semiconducting polymers the difference between the density functional band gap, as calculated for a single chain, and the measured optical band gaps is very systematic.23 Dividing the calculated values by a factor of 0.6 reproduced the measured values to within 10%. This empirical relation also works well for the recently discovered polysquaraines.5,6 In the following we will use this semiempirical “renormalization” factor of 0.6 to scale the calculated density functional band gaps. As usual, the complete band structure can then be constructed by rigidly shifting the calculated conduction bands upward.33,34 Results Table 2 gives the optimized bond lengths of the polymers 1-7, as calculated using the density functional method described in the previous section. The atom labels refer to the generic structure of Figure 1. The optimized structures of all polymers turn out to be planar. A general classification can be made for the structures of Table 2: polymers 1, 2, 5, and 7 have bond lengths A1′-A1, A2-A3 < A1-A2, which means that their structure is aromatic (or PTh-like). Polymers 4 and 6 have an ordering of bond lengths A1′-A1, A2-A3 > A1-A2, so these structures are quinoid (or PITN-like). Comparing structures 1 and 7, or structures 3 and 4, we observe that the isoelectronic substitutions by nitrogen atoms do not change the qualitative
17330 J. Phys. Chem., Vol. 100, No. 43, 1996 character of the structure, i.e., the structure stays aromatic 1 or quinoid 3. This is in contrast with some of the conclusions in ref 15, where for polymers 4a, 4c, and 4d the nitrogen substitution is predicted to result in a qualitative change in the geometry, from quinoid to aromatic, the latter being lowest in energy. In view of the discussion presented in the previous section, density functional methods are expected to give the more accurate structures. In general, in our geometry optimizations we have found no evidence for even a metastable aromatic structure for polymers 3 and 4, as has been tested explicitly by starting from hypothetical aromatic geometries. So the quinoid structure presents a unique stable minimum. For the other polymers, the aromatic structure is such a unique minimum, with no evidence for a quinoid (metastable) structure. The quinoid structures of polymers 3, 4, and 6, as given in Table 2, are relatively similar. The C-C bond, which connects the monomers, is around 1.34 Å in PITN and 1.36-1.37 Å in the nitrogen-substituted derivatives. The geometry of the fivemembered thiophene ring varies slightly, but the shortest of the C-C bonds is always around 1.41 Å, the longer one is around 1.45 Å, and the C-S bond is around 1.74 Å. Asymmetric substitutions in the ring fused on top of the thiophene ring, as in polymers 4c and 4d, result in a slight asymmetry, but in this ring the C-C distances are always around 1.39 Å; C-N distances are around 1.34 Å, and N-N distances are also around 1.34 Å. It is not clear a priori that polymer 6 belongs to the same class, but its geometry is definitely quinoid. As far as the π electrons are concerned, the sulfur atoms can act as a substitution of two carbon atoms, as proposed by LonguetHiggins.35 The aromatic structures of polymers 1, 2, 5, and 7 are also similar. The length of the shorter C-C bond in the thiophene ring is 1.38 Å and that of the longer one is 1.41 Å; the C-C bond between the thiophene rings is 1.42-1.45 Å. The basic structure is unchanged by adding alkoxy side chains, as in PEDOT 2. The ring fused on top of the thiophene ring is aliphatic; the C-C bond is a single bond of 1.50 Å. The oxygen atoms lie in the plane of the polymer, but one of the carbon atoms is displaced above and the other below the plane. The C-C bond makes an angle of 26° with the plane of the thiophene ring. This structure is consistent with what can be expected from these two carbon atoms being sp3 hybridized. In order to make a comparison with a polymer having a nonaliphatic side chain, which more resembles polymers 3 and 4, the geometry of polymer 5 has also been optimized. It can be obtained from PEDOT by extracting two hydrogen atoms from the two carbon atoms in the alkoxy ring. The structure of this polymer now becomes completely planar (like that of polymers 3 and 4; note, however, that polymer 5 is not isoelectronic with these polymers). One would expect this from the two carbon atoms in the top ring being sp2 hybridized. The structure of polymer 5 is similar to that of PEDOT; the C-C bond of 1.33 Å indicates the expected double bond. Polythiadiazole 7 is isoelectronic with polythiophene. It is therefore not surprising that its structure resembles the (aromatic) structure of polythiophene. The electronic structure of the polymers is closely linked to their equilibrium structures. This is illustrated by their band gaps, which are also given in Table 2. Although the band gaps are spread out over a considerable range, they can be classified into two groups. The polymers with PITN-like structures 3, 4, and 6 have band gaps around 1 eV; the band gaps around 2 eV correspond to PTh-like structures 1, 2, 5, and 7. Since the former group has quinoid structure and the latter an aromatic structure, there seems to be a correspondence between the
Brocks
Figure 2. Dispersion of the calculated electronic bands of polythiophene 1 along the polymer axis. Γ refers to k ) 0; Χ refers to k ) π/az, where az is the translational unit along the polymers axis (two thiophene rings). The zero of energy is chosen at the Fermi level in the middle of the band gap.
structure and the size of the band gap.18,19 This correspondence will be discussed in the following section. The known experimental band gaps are 2.1 eV (PTh 11), 1.64 eV (PEDOT 24), 1.0 eV (polyisothianaphthene 37), and 0.95 eV (polythienopyrazine 4a10). As can be observed from Table 2, there is good agreement with the calculated values. The calculated band gaps for polymers 4b-7 must be considered predictions. The nitrogen substitutions in PITN can alter the gap by several tenths of an electronvolt. A change in the same order of magnitude is possible by adding alkoxy side chains to PTh. Nitrogen substitution in PTh has a more severe effect, since one is altering the backbone of the polymer. Figure 2 shows the one-dimensional band structure along the polymer chain of PTh. The upper valence bands and the lower conduction bands are formed by π states. The band structure of Figure 2 agrees qualitatively with the results of previous studies.16,17,21 At the Χ point the bands are 2-fold degenerate as a result of the screw axis symmetry that connects the two monomers in the unit cell.36 In principle it is possible to unfold the bands using this symmetry.6,18,37 In order to simplify a comparison with the latter representation, it is customary to denote the two bands that are degenerate at Χ as one band. The topmost valence band ΓΧΓ then has a bandwidth of 3.7 eV, and the lowest conduction band has a width of 3.0 eV. On the basis of general experience with density functional calculations, one expects these values to be close to the experimental ones.33,34,38 The almost flat band that crosses the topmost valence band at around -3 eV corresponds to states that have almost no amplitude on C1 and C4 carbon atoms. Within Hu¨ckel terminology, its dispersion is the result of next nearest neighbor (and even longer range) interactions, and therefore, it is expected to be small. At lower energy, bands of σ symmetry begin to cross the π states, which also happens at energies above the lowest conduction band. The gap is formed at the Γ point in the Brillouin zone. The band structure of PITN is given in Figure 3. Qualitatively, there are similarities with the band structure of PTh. The average density of π states in PITN is larger, but in principle one expects the dispersion of the bands to be similar. The π bands in PITN then overlap more than in PTh, which results in a number of avoided crossings between these bands, as can be observed in Figure 3. This leads to the highest valence band
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(a)
Figure 3. Dispersion of the electronic bands of polyisothianaphthene 3 presented in the same way as in Figure 2.
Figure 4. Dispersion of the electronic bands of polythienopyrazine 4a presented in the same way as in Figure 2.
having a width of 2.2 eV and the lowest conduction band having a width of 1.5 eV. Figure 4 shows the band structure of polythienopyrazine 4a. Qualitatively, the structure of the valence bands resembles that of PITN. In Figure 4, the dispersionless band at -2 eV corresponds to nitrogen lone-pairlike states, which are of course absent in PITN (Figure 3). The dispersionless bands at around -3 eV in both Figures 3 and 4 are similar to the one discussed above for PTh. For PITN, Figure 3, a third π valence band overlaps with the higher ones between -2.5 and -3.5 eV, which results in a pattern of avoided crossings and a mixing of states. In Figure 4, this third band lies at a lower energy and does not overlap with the higher ones, which means that the state at around -3.4 eV corresponds to the bottom of the highest valence band. Thus, the bandwidth of the latter in polythienopyrazine 4a is 3.1 eV. The seemingly large difference between this number and the 2.2 eV found for PITN is somewhat misleading, since the latter number is influenced very strongly by the avoided crossings. Comparing the widths of the ΓΧ part of the topmost valence bands in Figures 3 and 4, one finds 1.7 eV for both, which indicates a similar dispersion in both cases. The dispersion of the lowest conduction band in Figure 4 is only 0.7 eV. Again, this is caused by a strong interaction/avoided crossing with a second
(b)
Figure 5. Schematic view of the wave functions at the Γ point that belong to the highest two valence bands and the lowest conduction band of polythiophene. The black circles represent a positive phase of the function and the white circles a negative phase; the size of the circles represents the local amplitude. The figures are calculated by projecting the wave functions, represented on a basis of plane waves, onto a basis set consisting of localized atomic orbitals. All the states shown here are π states, which are entirely comprised of pz orbitals (where xy denotes the plane of the polymer). The states are numbered starting from the highest occupied polymer orbital (HOPO) and lowest unoccupied polymer orbital (LUPO). The solid box denotes qualitatively the HOMO of a single isolated thiophene molecule, the dashed box denotes the LUMO, and the dotted box the HOMO-1 state.
conduction band, which for PITN, Figure 3, lies at a higher energy. Since the gap in all these polymers is formed at the Γ point in the Brillouin zone, we will consider these π states in more detail. Figure 5 presents a schematic view of the wave functions of PTh at the Γ point, which belong to the highest two valence bands and the lowest conduction band. The occupied states are labeled HOPO-n, where HOPO stands for highest occupied polymer orbital and the unoccupied states are labeled LUPO+n, where LUPO stands for lowest unoccupied polymer orbital. Trying to classify these states, one finds that these states can be constructed from three different patterns of nodes on the thiophene molecules, as indicated by boxes in Figure 5. The patterns of the HOPO and HOPO-3 states correspond to the highest occupied molecular orbital (HOMO) of the single isolated thiophene molecule. In fact the complete highest valence band, cf. Figure 2, between those two states can be constructed qualitatively from the HOMO of the thiophene molecule. The bottom of the band (HOPO-3) corresponds to the all-bonding combination, whereas the top of the band (HOPO) corresponds to the antibonding combination. The HOPO-1 and HOPO-2 states obviously correspond to the almost flat band around -3 eV in Figure 2. The lowest conduction band can be constructed from bonding/antibonding combinations of the LUMO of the thiophene molecule, with the bottom (LUPO) corresponding to the bonding and the top (LUPO+1) to the antibonding combination. Figure 6 gives a schematic view of the wave functions that belong to the highest valence band and the lowest conduction band of PITN. Again, qualitatively, these states can be considered as being constructed from the HOMO and LUMO states of the isothianaphthene molecule, as indicated by the boxes in Figure 6. The picture is less clear than in the case of PTh, since the bottom (top) of the valence (conduction) band is perturbed by interaction with a
17332 J. Phys. Chem., Vol. 100, No. 43, 1996
Brocks
Figure 8. Formation of the polymer states at Γ from the HOMO and LUMO of the individual isothianaphthene molecules. The lowest (highest) states, indicated by dashed lines, are perturbed by interactions with lower (higher) π states. The value of 3.6 eV corresponds to the optical absorption of the molecule.41
Figure 6. Schematic view of the wave functions at the Γ point of polyisothianaphthene in the same representation as in Figure 5.
Figure 7. Formation of the polymer states at Γ from the HOMO and LUMO of the individual thiophene molecules. The numbers are derived from the band gap and bandwidths. The latter, which are the result of the hybridization of the molecular states in the polymer, are insufficiently large to close the HOMO/LUMO gap. The HOMO-1 state leads to a dispersionless band in the polymer.
lower (higher) band, as discussed above. This affects the HOPO-1 and LUPO+1 states. Discussion and Conclusions In this section the results for the (electronic) structure of polymers 1-7, as presented in the previous section, will be discussed in terms of a general framework. For the electronic states this discussion will be restricted to the π states around the Fermi level as shown in Figures 5 and 6. According to the qualification of the previous section, the simple MO diagram of Figure 7 represents the ordering of states at the Γ point starting from the molecular HOMO/LUMO for PTh. The values for the bandwidths and band gap can be derived from Figure 2. One can use these numbers to obtain an estimate for the HOMO/ LUMO gap in the thiophene molecule of 5.4 eV. This number agrees rather well with the experimental value for the absorption of the thiophene molecule of 5.36 eV39 (the fact that the agreement is that good must be considered accidental of course). Qualitatively, the ordering of electronic states will be the same for any reasonable geometry, even a hypothetical quinoid geometry, but quantitatively their energies will of course depend on it. In general, one would expect the energy of a system to go down if the occupied electronic levels become lower in energy. A simple picture emerges if one assumes that a
dominating role is played by the states near the Fermi level, in the spirit of the celebrated Su-Schrieffer-Heeger model.40 Conceptually, we can take a structure with more or less equal C-C bonds as a starting point. One can lower the energy of the top of the valence band by enlarging the bonding character in the HOMO (cf. Figure 5). Given the nodal character of the wave function belonging to this state, this can be accomplished by shortening the C1-C2 and C3-C4 bonds with respect to the C1′-C1 and C2-C3 bonds, in other words, by adopting an aromatic structure. Considering Figure 5, one can observe that at the same time an aromatic structure increases the antibonding character of the state at the bottom of the conduction band (LUPO). In other words this state is shifted upward in energy and the band gap is thus enlarged. If we adopt the quinoid structure instead, the state at the top of the valence band (HOPO) would become more antibonding, and thus, the total energy would increase. The state at the bottom of the conduction band would become more bonding; its energy would decrease, and this would lead to a lower gap. One can thus rationalize that for PTh the structure with maximum band gap has the lowest energy and that it is aromatic. The same conclusion also holds for the polymers 2 (PEDOT), 5, and 7 (polythiadiazole), since these have the same basic electronic structure as PTh. The question remains in what sense PITN and related polymers are different. Again, the discussion will be restricted to the π states at Γ around the Fermi level as shown in Figure 6. The corresponding MO diagram is given in Figure 8. The state at the bottom of the valence band roughly corresponds to the bonding combination of the HOMO, but in contrast with PTh, the state at the top of the valence band now corresponds to the bonding combination of the LUMO. The antibonding combination of the HOMO now becomes the bottom of the conduction band, and the antibonding combination of the LUMO roughly corresponds the top of the conduction band. Compared to the MO diagram for PTh, Figure 7, the two levels around the Fermi level have crossed. More generally, the two bands around the Fermi level in the band structure of PITN, Figure 3, show an avoided crossing. The main reason for the difference as expressed by Figures 8 and 7 is that the separation between the HOMO and LUMO states in isothianaphthene is much smaller than in thiophene, which is indicated by the absorption of the two molecules, 3.5841 vs 5.36 eV.39 If the bandwidths resulting from bonding/antibonding combinations of these states in the polymers are similar, a level crossing is far more likely to occur in PITN than in PTh. Again, the ordering of the electronic states in PITN is the same for any reasonable geometry, but the total energy will of course be lowest for one particular structure. The energy of the state at the top of the valence band can be lowered by increasing its bonding character. This can be accomplished by shortening the C1′-C1 and C2C3 bonds with respect to C1-C2 and C3-C4 bonds (cf., Figure 6a, or in other words by making the structure quinoid. At the
Polythiophene Derivatives same time one increases the antibonding character of the state at the bottom of the conduction band (cf., Figure 6c), which means that the total gap will be enlarged. The aromatic structure would increase the antibonding in the state at the top of the valence band and the bonding in the state at the bottom of the conduction band, which would increase the total energy and decrease the gap. The level crossing discussed above thus ensures that the quinoid structure has the lowest energy in PITN and the maximum band gap. The isoelectronic substitutions by nitrogen in polymers 4 (and in a more indirect sense by sulfur in polymer 6) do not change the basic electronic structure of the individual molecules and the basic MO diagram. One therefore expects their band structures to be similar to that of PITN and their structures to be quinoid. In conclusion, density functional methods within the CarParrinello framework have been used to calculate the electronic and equilibrium structures of polythiophene derivatives from first principles. The relation between the electronic states and the quinoid vs aromatic structure of the polymers is studied on a set of substituted polythiophenes (PTh) and polyisothianaphthenes (PITN). Comparison of the results with those of previous semiempirical calculations reveals shortcomings of the latter in describing the π bonding to heteroatoms. In general, a unique structure with minimal energy is found for the polymers with no other local minima. The general rule is that a polymer adapts the structure that leads to the largest possible band gap. For PTh-like polymers 1, 2, 5, and 7 this is the aromatic structure, and for PITN-like polymers 3, 4, and 6 it is the quinoid structure. One of the key ingredients distinguishing between these two classes is the size of the HOMO/LUMO splitting in the single molecules. If it is large, like in thiophene, band formation in the polymers does not lead to a level crossing and the highest occupied polymer orbital (HOPO) is stabilized by the aromatic geometry. If it is much smaller, like in isothianaphthene, a level crossing in the polymer is the result and the HOPO is stabilized by the quinoid geometry. Substitutions do not change the basic geometry and electronic structure of either PTh or PITN but can alter the band gap by several tenths of an electronvolt. Acknowledgment. I thank A. Tol for his contribution and stimulating discussions and D. M. de Leeuw for his careful reading of the manuscript. References and Notes (1) (a) Kobayashi, M.; Chen, J.; Chung, T.-C.; Moraes, F.; Heeger, A. J.; Wudl, F. Synth. Met. 1984, 9, 77. (b) Chung, T.-C.; Kaufman, J. H.; Heeger, A. J.; Wudl, F. Phys. ReV. B 1984, 30, 702, and references therein. (2) Assadi, A.; Svensson, C.; Wilander, M.; Ingana¨s, O. Appl. Phys. Lett. 1988, 53, 195. (3) Jonas, F.; Schrader, L. Synth. Met. 1991, 41-43, 831. (4) Havinga, E. E.; Mutsaers, C. M. J.; Jenneskens, L. W. Chem. Mater. 1996, 8, 769. (5) (a) Havinga, E. E.; ten Hoeve, W.; Wijnberg, H. Polym. Bull. 1992, 29, 119. (b) Havinga, E. E.; ten Hoeve, W.; Wijnberg, H. Synth. Met. 1993, 55-57, 299. (6) Brocks, G.; Tol, A. J. Phys. Chem. 1996, 100, 1838. (7) (a) Wudl, F.; Kobayashi, M.; Heeger, A. J. J. Org. Chem. 1984, 49, 3382. (b) Kobayashi, M.; Colaneri, N.; Boysel, M.; Wudl, F.; Heeger, A. J. J. Chem. Phys. 1985, 82, 5717.
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