1838
J. Phys. Chem. 1996, 100, 1838-1846
Small Band Gap Semiconducting Polymers Made from Dye Molecules: Polysquaraines G. Brocks* and A. Tol Philips Research Laboratories, Prof. Holstlaan 4, 5656 AA EindhoVen, The Netherlands ReceiVed: August 2, 1995; In Final Form: October 21, 1995X
Small band gaps simplify the use of polymers as semiconductors. Dye molecules can be used to construct semiconducting polymers with small band gaps. We use ab initio calculations to systematically design and study polymers derived from squaraine dyes. The calculated band gaps range from 2.3 eV to as low as 0.2 eV. Simple arguments based upon a Hu¨ckel analysis of the ab initio results enable us to identify the factors that control the size of the band gap. Squaraine polymers synthesized up till now fall into a class in which the band gap (g1.3 eV) essentially reflects the HOMO/LUMO energy difference of the squaraine monomer fragment. We predict that a second class of polymer topology leads to much smaller band gaps. The reason for this is that band formation in the polymer shifts the energy of the highest occupied state of the polymer up with respect to the HOMO of the squaraine monomer, whereas the energy of the lowest unoccupied state of the polymer is fixed at the level of the monomer LUMO because of symmetry. New semiconducting polymers can be designed using such general principles as symmetry and topology.
I. Introduction The promise of application in large-area electrooptical and electronic devices has lead to increasing interest in semiconducting polymers in recent years. The polymers most widely tested up till now have a simple [-A-]n structure, where A is an aromatic fragment like thiophene or pyrrole, or a slightly more complicated [-A-B-]n structure, with A for instance a phenyl and B a vinyl group. These polymers have band gaps in the range 2-3.5 eV.1 They are classified as “wide band gap semiconductors” by comparison with inorganic semiconductors (where Si with a band gap of 1.1 eV sets the standard). While for optical applications in the visible range as in light-emitting diodes2,3 such a wide band gap is suitable, contacting and doping problems arise in much the same way as in inorganic wide band gap semiconductors.4 In practice, semiconducting polymers often have too small an electron affinity, which makes it difficult to find a suitable (metal) contact electrode for injecting electrons into the polymer.5 For a number of applications, such as logic elementsstransistors,6 diodes,7 or (in a heavily doped state) conductorssthere are no a priori restrictions on the band gap. A polymer with a small band gap is bound to have much fewer doping and contacting problems.8 In addition, the optical absorption of polymers is usually strongly peaked around the band gap in a way which reflects the density of states typical of one-dimensional systems.9 Polymers with band gaps in the range 2-3.5 eV therefore absorb strongly in the visible range.1 A much smaller band gap shifts the absorption peak to the infrared. The absorption in the visible region can then be relatively small, especially if the polymer is heavily doped.10 This opens the possibility for the use of semiconducting polymers, after doping, as transparent conductors. The electronic and optical properties of a polymer result from a limited number of states around the highest occupied and the lowest unoccupied levels. Since semiconducting polymers are conjugated molecules, these states are usually π states. The simple monomers, from which the polymers in the examples of the previous paragraph are derived, have their first absorption band in the UV region, and the polymers themselves absorb in the visible region. For instance, λmax ) 235 nm (5.3 eV) for thiophene;11 for polythiophene the onset of absorption is at λ X
Abstract published in AdVance ACS Abstracts, December 15, 1995.
0022-3654/96/20100-1838$12.00/0
Figure 1. Schematic representation of band formation in a polymer. On the left-hand side the HOMO and LUMO energy levels of the monomer are sketched; on the right-hand side the energy bands of the polymer are shown. The band gap Eg depends on the energy separation ∆E between the HOMO and LUMO of the monomer and the bandwidth W(β) resulting from hybridization of these states in the polymer. The actual numbers are for polythiophene.
∼ 590 nm (2.1 eV).10 In simple terms this red shift is the result of the delocalization of electrons and the formation of electronic bands in the polymer,12 which is illustrated schematically in Figure 1. The band gap Eg of the polymer depends on the separation ∆E between the HOMO and LUMO energy levels of the monomers and the bandwidth W(β), which is a function of the hybridization β of the monomer levels in the polymer.13 In searching for small band gap polymers, so far the main focus has been on increasing the bandwidth W(β), by going from aromatic to quinoid structures, for instance.14 However, the most widely used fragments, such as thiophene, pyrrole, phenylene, or vinylene, have too large a HOMO/LUMO gap ∆E, which can only be partially closed by band formation W(β) in the polymer. In order to obtain a polymer with a smaller band gap, it seems therefore obvious that one should start from monomers with a small ∆E, such as can be found among dye molecules.15 The implementation of this idea is not easy, since considering the relatively complicated structure of most dye molecules, it is far from obvious how to construct their polymer equivalents. Recently, one possible route has been introduced by Havinga et al.,16 starting from squaraine and croconaine dyes.15,17 Squaraine dyes are symmetric molecules with two identical donor groups (1,3) coupled to a squaraine four-membered ring; a typical example is shown in 1. The dyes are prepared by condensation of squaric acid with the appropriate donor base.16,18 © 1996 American Chemical Society
Small Band Gap Polymers Depending on the donor, λmax varies from 400 to 800 nm (1.5 to 3 eV; for the example shown in 1, λmax ) 615 nm). The polymer equivalent of 1, as proposed and synthesized by Havinga et al. (shown in 2), is obtained by “doubling” the donor base and applying the same condensation reaction as is used for the squaraine dyes.16 The absorption of the polymer 2 has a λmax ∼ 920 nm (1.35 eV). The absorption onsets of other members of this class of polymers are spread around 1 eV.
J. Phys. Chem., Vol. 100, No. 5, 1996 1839 structure 3 to the more complicated 2. In view of these two structures, it is appropriate to have the squaraine ring and the vinyl, phenyl, and amine groups as elementary building blocks for constructing our conjugated polymers. With these building blocks we can cover a number of variations in topology. Other (hetero)aromatic fragments could in principle be built into a squaraine polymer,15,17 which will be left for later studies. As a first step, going from 3 to 2, we add vinyl groups on either side of the squaraine ring, leading to polymer 4. By analogy with the terminology used for squaraine dyes we may call this step enlarging of the squaraine chromophore.15 Staying within the same topology, but testing a different chromophore, we replace the vinyl groups by phenyl groups in 5. As an alternative step on the way from 3 to 2, we can first change the amine group to a “double” donor group, which is characteristic of the “Havinga polymers”, as in 7. Again some chemical variation within the same topology can be tested by replacing the phenyl by a vinyl group, as in 6. A combination of these two steps, i.e., enlarging the squaraine chromophore and doubling the donor, brings us from 3 to 2.22 We have performed calculations for all polymers 2-7. As it turns out, polymers 4-7 not only serve to help understand the connection between 3 and 2 but also have interesting properties themselves. In the next section we will discuss the ab initio calculations in some detail. The electronic structures are then presented and analyzed in section III. The general rules obtained from this analysis are summarized in section IV. II. Theoretical Section
Apart from the polymers which have been synthesized in ref 16, there are other possible topologies. Note for instance that the polymer 2 is not obtained by straightforwardly polymerizing the dye molecule 1. The topology has been changed from two “single” donor groups per squaraine ring in the dye molecule to one “double” donor group per squaraine ring in the polymer. Routes to squaraine polymers with other topologies are conceivable. One would like to identify the factors which lead to a small band gap. In this paper we systematically design squaraine polymers using ab initio calculations. Our principal tool will be the Car-Parrinello technique for simultaneously optimizing both the electronic and the geometrical structure of the polymer19 at the level of the local density approximation (LDA) to density functional theory. On the basis of these calculated results, an intuitive chemical model will then be constructed to describe the main features of the polymer band structure, which couples the relevant π states of the polymer to the electronic states of the monomers in the spirit of Figure 1. The aim is to provide a general framework for the influence of the polymer topology and chemical substitutions on the band gap Eg of squaraine polymers. We have recently performed a detailed study of the prototype of squaraine polymers: poly(aminosquaraine) 3, which consists of squaraine rings connected by amine groups.20,21 The calculated band gap of this polymer is only 0.5 eV, whereas that of polymer 2 is much larger, 1.3 eV. In the following we will discuss a number of polymers which will bridge the gap in chemical complexity in a systematic way, going from the simple
The electronic properties of a semiconducting polymer are intimately connected to its geometrical structure.23 In previous studies we found that distortions of the geometry which had a small effect on the total energy had a relatively large effect on the band gap.21,24 It is therefore essential to optimize both the electronic and the geometrical structure. Simultaneous selfconsistent optimization is possible with the Car-Parrinello method, in which geometry optimization and molecular dynamics are combined with an ab initio electronic structure calculation at the level of the local density approximation (LDA) to density functional theory. Only the valence electrons are treated explicitly, and norm-conserving pseudopotentials are used to represent the ion cores. The (valence) electronic wave functions are expanded in a basis set which consists of plane waves. For technical details regarding the application of this method to polymers, we refer to refs 21 and 24; in particular, the factors which determine convergence have been studied in detail in ref 21. In the following, all plane waves have been included in the basis set up to a kinetic energy cutoff of 30 hartrees, with tests up to 40 hartrees. A brillouin zone sampling density has been used which is equivalent to one k point per ∼15 Å along the direction of the polymer axis. The overall reliability of LDA calculations in predicting equilibrium structures of organic molecules is well-established.25 We explicitly test the accuracy of the pseudopotential/plane wave basis set implementation on a squaraine molecule. The equilibrium structure of the squaraine dye 8 has been obtained from X-ray crystal diffraction data.26 In Table 1 we compare the optimized calculated structure with this experimental structure. Note that the agreement is gratifying: the calculated bond lengths are on average within 0.01 Å, and the bond angles are on average within 0.7° of their corresponding experimental values. The systematic errors introduced in the calculation by the finite size of the basis set are assessed by enlarging the plane wave kinetic energy cutoff.21 As shown in Table 1, the changes in the geometry going from a cutoff of 30 hartrees to a 40 hartree
1840 J. Phys. Chem., Vol. 100, No. 5, 1996
Brocks and Tol
TABLE 1: Bond Lengths (Å) and Bond Angles (deg) of the Squaraine Dye Molecule 8a exp
kinetic energy cutoff 30 40
AM1 geometry
1-2 2-3 2′-3 3-4 4-5 5-6 6-7 7-13 13-12 12-4 7-8 8-9 9-10 8-14 14-15 12-11 avb
1.244 1.462 1.460 1.402 1.428 1.359 1.429 1.414 1.369 1.421 1.359 1.462 1.502 1.459 1.517 1.363
Bond Lengths 1.270 1.449 1.462 1.397 1.417 1.363 1.422 1.400 1.385 1.431 1.359 1.453 1.513 1.439 1.515 1.343 0.011
1.251 1.450 1.460 1.396 1.416 1.363 1.422 1.402 1.385 1.432 1.360 1.454 1.506 1.442 1.507 1.342 0.009
1.235 1.477 1.473 1.399 1.424 1.372 1.433 1.412 1.403 1.422 1.389 1.448 1.521 1.448 1.521 1.363 0.011
1-2-3 1-2-3′ 2-3-2′ 3-2-3′ 2-3-4 2′-3-4 3-4-5 4-5-6 5-6-7 6-7-13 7-13-12 13-12-4 12-4-3 12-4-5 6-7-8 13-7-8 7-8-9 7-8-14 8-9-10 8-14-15 9-8-14 4-12-11 13-12-11 avb
135.5 133.1 88.6 91.4 135.3 136.1 120.5 122.2 117.7 117.3 122.0 121.2 122.7 116.3 120.6 122.1 120.8 121.3 113.7 114.1 117.4 123.3 115.5
Bond Angles 136.0 132.0 88.1 91.9 135.3 136.6 120.2 122.6 120.0 118.3 122.0 119.8 122.6 117.2 120.8 120.8 120.4 121.1 112.9 113.0 118.5 123.8 116.4 0.7
136.4 132.1 88.5 91.5 135.6 136.9 120.3 122.5 120.1 118.4 121.8 119.7 122.2 117.4 120.6 120.9 120.3 120.7 113.1 113.3 118.8 123.5 116.8 0.7
135.5 135.6 91.1 88.9 130.6 138.3 118.0 122.8 121.0 117.2 121.3 121.5 125.8 116.2 121.2 121.5 120.0 120.1 116.4 116.5 116.8 125.0 113.5 1.6
a The experimental data are taken from the triclinic crystal structure of ref 26 (Cambridge Crystallographic Data Base REFCODE: VAYSET). The center of the squaraine ring is a point of inversion symmetry. The bond 2′-3 is thus equivalent to 2-3′, etc.; the standard chemical notation of resonance structures (8) is thus misleading. The third and fourth column give the ab initio optimized structures, using the experimental unit cell and plane wave kinetic energy cutoffs of 30 and 40 hartrees, respectively. The fifth column gives the AM1 optimized structure.27 b Average |calc - exp|.
cutoff are small, which proves that the structure is sufficiently converged; in the following we therefore use a cutoff of 30 hartrees. Comparing with the experimental data, we observe in particular that the geometrical features which are discussed in ref 26 are all reproduced by the calculated structure: the planarity (except of the ethyl groups) of the molecule; a short C(7)-N(8) bond, as compared to a normal alkyl-substituted aniline; a long C(2)-O(1) bond, as compared to a normal carbonyl value; a short C(3)-C(4) bond, as compared to a typical sp3-sp2 bond; the four-membered squaraine ring which contains significant double-bonding character; the presence of a quinoidal structure in the six-membered rings. All these geometrical features reflect the extended π-bonding in this molecule. As theoretical reference, the last column of Table 1 gives the optimized structure obtained from a semiempirical
Figure 2. Total energy of the polymer 4 as function of the cell parameter ax in electronvolts per squaraine unit. The triangles represent calculated values; the solid line is a polynomial fit.
AM1 calculation.27 As might be expected, there are differences in the bond lengths on the scale of a few hundredths of an angstrom and differences in bond angles of the order of a few degrees. Compared to the experimental structure, AM1 performs reasonably well, with the exception of the bond lengths C(12)-C(13) and C(7)-N(8), which are ∼0.03 Å too large; this of course also affects the bond angles around these bonds. Note furthermore that, regarding the bond angles in the squaraine ring, both the experimental and the ab initio structure give 2-32′ < 3-2-3′, whereas in the AM1 structure this order is reversed. The bond angles of the ab initio structure are in general closer to the experimental values than those of the AM1 structure. The largest difference between experimental and ab initio structures is in the C-O bond length.28 In contrast to the dye molecule 8, squaraine polymers have only been obtained in amorphous form. Since no information about intermolecular (dis)order is available, we refrain from speculating on the interchain packing and treat the polymer as a one-dimensional chain with translation symmetry along the chain. This approach is useful, since properties like the optical band gap are dominated by the strong one-dimensional character of the polymers.9,29,30 A small technical problem is that using a plane wave basis set implies periodic boundary conditions in three dimensions. A one-dimensional system can however still be modeled in a supercell: the cell parameters ay and az along the second and third directions (perpendicular to the polymer chain) are chosen such that there is no interaction between the polymer chain and its periodic y and z images. Setting ay and az ) 15 a0 turns out to be sufficient; atoms of neighboring chains are then at least 5 Å apart. The cell parameter along the polymer chain direction, ax, is optimized by computing the total energy for a number of cell parameters (with a fixed plane wave kinetic energy cutoff of 30 hartrees) and fitting a polynomial through these points. The results of such a calculation are shown in Figure 2. The optimized structures of polymers 2-7 are given in Table 2. The geometry of the squaraine ring does not vary much from polymer to polymer; bond lengths typically vary by ∼0.02 Å and bond angles by ∼3°. The typical length for a bond from the ring to a neighboring atom is 1.38-1.40 Å in case the neighboring atom is carbon and 1.32-1.34 Å in the case of a nitrogen atom. For polymers 3-5 the bonds on each side of the nitrogen atom are equivalent. (Note that the chemical notation of resonance structures is misleading in this respect.) The geometry of the phenyl ring in polymer 5 differs from that in polymers 2 and 7. In the former the phenyl ring has a definite quinoidal character (similar to the dye molecule in Table 1), whereas the phenyl rings in the latter two are much more
Small Band Gap Polymers
J. Phys. Chem., Vol. 100, No. 5, 1996 1841
TABLE 2: Bond lengths (Å), Optimized Cell Parameter ax (Å), and Bond Angles (deg) of the Squaraine Polymers 2-7a 2 1-2 2-3 2-3′ 3-4 4-5 5-6 6-7 7-8 8-9 9-10 ax 1-2-3 1-2-3′ 2-3-2′ 3-2-3′ 2-3-4 3-4-5 4-5-6 5-6-7 6-7-8 6-7-9 7-8-9 8-9-10
1.268 1.457 1.455 1.378 1.378 1.348 1.392 1.390 1.380 1.398 12.92 134.7 134.6 89.3 90.7 135.5 123.7 126.3 110.8 129.3 116.7 122.0
3b
4
Bond Lengths 1.214 1.247 1.464 1.469 1.472 1.465 1.339 1.390 1.339 1.362 1.358 1.358
4.30
9.05
Bond Angles 139.5 135.9 134.5 135.1 94.0 91.0 86.0 89.0 139.3 136.0 128.7 122.0 122.0 127.4
5
6
7
1.251 1.470 1.458 1.396 1.405 1.365 1.403 1.379 1.379
1.248 1.465 1.467 1.337 1.398 1.345
1.254 1.458 1.449 1.322 1.395 1.386 1.373 1.381
11.48
7.95
9.15
136.4 134.7 91.1 88.9 133.8 118.1 120.1 120.1 119.6 119.9 124.8
137.1 135.0 92.0 88.0 136.8 125.7 124.1
138.0 134.1 92.1 87.9 137.9 124.9 120.7 120.6 119.4 120.0
a The center of the squaraine ring is a point of inversion symmetry for all polymers. Polymers 2,7 and 6 have an additional center of symmetry in the phenyl and vinyl units, respectively; polymers 3-5 have a 2-fold screw axis symmetry along the polymer axis. The standard chemical notation of resonance structures (2-7) is thus misleading for the symmetry. All structures are planar, except for polymer 5, where the torsion angle 6-7-7′-6′ is 46°. b Reference 21.
aromatic in character. The C-N bond which couples the phenyl ring to the neighboring nitrogen atom is much shorter for polymer 5 than for polymers 2 and 7. Furthermore, in the latter two this C-N bond is much longer than the C-N bond on the other side of the nitrogen atom. All this points to a larger delocalization of the π electrons in polymer 5 and a more localized phenyl fragment in polymers 2 and 7. The vinyl fragments in polymers 4 (and 2) and 6 also have a different geometry. C-C and C-N bond lengths are for example 1.36 and 1.36 Å for polymer 4 and 1.35 and 1.40 Å for polymer 6. The latter thus shows much more of a localized vinyl fragment. Geometry optimization leads to planar equilibrium structures for all polymers, except for polymer 5, where steric hindrance between the ortho hydrogen atoms of the phenyl rings attached to the nitrogen atoms prevents these rings from being in one plane. This situation is similar to that found in polyaniline.31 For some of the other polymers the energy required for distortion to a nonplanar structure is very small. For instance, rotating the phenyl ring of polymer 7 out of plane by 30° costs only 0.02 eV. (The band gap then increases by 0.4 eV, however.) Formally, the density functional scheme is suited to calculate ground state properties (such as the equilibrium structure), and most of its applications to excited states have no rigorous theoretical foundation.32 The true band structure of a solid, which incorporates all quasiparticle (or single particle-like) excitations, will thus in general be different from the band structure as obtained in an LDA calculation.33 Nevertheless, the usefulness of LDA for interpreting band structures of inorganic solids has been established empirically over the past few decades: the dispersion of the bands (and thus the bandwidths) tends to agree very well with experiment, but the band gap is roughly a factor of 2 too small.33 The main discrepancies between the experimentally measured and the
TABLE 3: Band Gaps (in eV) of Polymers 2-7a -Sq′-Nstructure 3 4 5
-Sq′-N-A-N-
band gap 0.5 0.2 1.1
structure
band gap
6 7 2
1.8 2.3 1.3
b
a The numbers are LDA values divided by 0.6 (for a discussion, see text). b Reference 21. c Experiment, 1.4 eV; cf. ref 16.
LDA band structures can be corrected by simply shifting the LDA conduction bands upward by a constant.34 For comparison, Hartree-Fock (HF) calculations on Si produce band widths that are a factor ∼1.5 too large, whereas the band gap is a factor ∼5.5 too large.35,36 Similar trends are observed when comparing the results of LDA and HF calculations on the polymer (SN)n,37,38 where the former are in much better agreement with experimental data. In addition, we have shown in a recent study that for a range of semiconducting polymers the difference between the LDA band gap, calculated for a single chain, and the measured optical band gaps is very systematic.24 Dividing the calculated LDA values by a factor of 0.6 reproduced the measured values to within 0.2 eV. In the following we will use this semiempirical “renormalization” factor of 0.6 to scale the calculated LDA band gaps of squaraine polymers.39 As usual, the complete band structure can be constructed by rigidly shifting the LDA conduction bands upward.33,34 III. Results and Discussion The calculated band gaps for polymers 2-7 are given in Table 3. From the results of ref 21 we estimate the convergence of the LDA band gap to be 0.1 eV. Note that the agreement between the “renormalized” calculated and the experimental16 values for polymer 2 is quite good. The results presented in Table 3 indicate that the polymers can be divided into two classes: polymers 3-5, which have small band gaps, and polymers 2, 6, and 7, which have larger band gaps.40 In the rest of this section a simple chemical model is constructed in order to describe the main features of the band structures of these two classes of polymers. We will strive for a qualitative understanding of the influence of the polymer topology and chemical substitutions on the band gap of these squaraine polymers. It is useful to describe the polymers within the following classification scheme. The polymers with small band gaps have the general topology
[-Sq′-N-]n
(1)
where Sq′ is the squaraine chromophore fragment and N denotes the amine group. The squaraine chromophore Sq′ can have an internal structure, which is represented as
Sq′ ) B-Sq-B
(2)
where Sq is the four-membered squaraine ring and B is an aromatic or a vinyl fragment. An example of such a chromophore is the divinylsquarine fragment 9, which is part of polymer 4. The polymers with larger band gaps have a more complicated topology, given by
[-Sq′-N-A-N-]n
(3)
where Sq′ is the squaraine chromophore fragment, N denotes the amine group, and A is either an aromatic fragment (such as the phenyl ring in 2 and 7) or a vinyl group (in 6). In the following, the class of polymers which have the topology of eq
1842 J. Phys. Chem., Vol. 100, No. 5, 1996
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Figure 4. Calculated energy LDA levels (in electronvolts) of a squaraine ring (a) and a divinylsquaraine fragment (b) 9. The fragments are cut out from the polymers, keeping the geometry fixed and saturating the dangling bonds with hydrogen atoms. Only π states are shown; (a)s denote states which are (anti)symmetric with respect to an inversion point through the center of the squaraine ring (cf. Figure 5). The zero of energy is chosen at the level of the HOMO. Figure 3. (a) Schematic picture of the highest occupied polymer orbital (HOPO) of poly(aminodivinylsquaraine) 4. The calculated polymer wave function is projected onto a local atomic orbital basis; since the HOPO is a π state, only the pz components are nonzero. The symbols represent the local phase and amplitude of the wave function by their color (black:negative, white:positive) and size, respectively. (b) The calculated HOMO of divinylsquaraine in the same representation.
1 will be denoted by -Sq′-N-, whereas polymers with eq 3 topology will be denoted by -Sq′-N-A-N-. The central issue discussed in this section is why the band gaps of polymers with -Sq′-N- topology are small and, in particular, why they are smaller than the band gaps of polymers with -Sq′-N-A-N- topology. As expected, most of the states around the band gap are π states. The highest occupied polymer orbital (HOPO) of polymer 4 is given in Figure 3a as an example. Although this state has a very complicated nodal pattern, it can be classified in a relatively simple way by comparison with the divinylsquaraine fragment 9. The highest molecular orbital (HOMO) of this (isolated) fragment, as calculated using the same ab initio method, is shown in Figure 3b. The HOMO of this fragment is remarkably similar to the corresponding part in the HOPO of the polymer. In fact, qualitatively the HOPO can be described as an antibonding combination between the fragment HOMO and the pz orbitals of the nitrogen atoms (where z is the direction perpendicular to the plane of the polymer). By inspection, the HOPO and LUPO states of all the squaraine polymers can be described on a fragment basis, where the fragments are exactly those used in the classification scheme introduced in the preceding section.41 This statement is quantified by introducing a simple (Hu¨ckel) model based upon fragment states. We start from the molecular orbital levels (MO’s) of individual fragments. Figure 4 shows the calculated π energy levels of two possible Sq′ chromophore fragments: a simple squaraine ring, such as found in polymers 3, 6, and 7, and a divinylsquaraine fragment 9, such as found in polymers 2 and 4. As expected, the larger fragment has a denser energy spectrum and a smaller energy difference between the HOMO and LUMO states. The states labeled by s are symmetric with respect to a inversion point through the center of the squaraine ring; the states labeled by a are antisymmetric. The diphenylsquaraine fragment, such as found in polymer 5, has a slightly denser spectrum than divinylsquaraine, but the calculated HOMO/LUMO energy difference is larger, 1.4 eV. If we divide the HOMO/LUMO energy difference of these Sq′ fragments by the same factor 0.6 which we used to obtain the
Figure 5. Schematic structure of poly(aminodivinylsquaraine) 4. The optimized geometry of 4 has inversion points at the centers of the squaraine rings, labeled by i. The polymer is planar, and the plane of the molecule forms a mirror plane. In addition, we have a 2-fold screw axis along the line connecting the centers of the rings, indicated by the dotted line and labeled by 21. The period associated with the screw axis is labeled by ax.
band gaps of Table 3, we see immediately that polymers of -Sq′-N-A-N- topology have a band gap which is near the Sq′ fragment HOMO/LUMO energy difference. In contrast, polymers of -Sq′-N- topology have a band gap which is much smaller than the HOMO/LUMO energy difference of the corresponding fragment (i.e., as small as 10% of that energy difference). We now make the simplifying assumption that only the frontier orbitals of the fragment, i.e. the HOMO and LUMO states, are important for describing the states around the highest occupied (HOPO) and lowest unoccupied (LUPO) polymer orbitals. In other words, we ignore for the moment the contribution of the other π states of the fragment. In view of the energy level spectra of Figure 4, we expect this assumption to be reasonable at least qualitatively over a region of a few electronvolts around the band gap of the polymers. The validity of this assumption has to be checked by comparing the energy levels resulting from the Hu¨ckel model with the ab initio results. A. Small Band Gaps. In the polymers of -Sq′-Ntopology Sq′ fragments are linked via the nitrogen atoms of the amine groups. In a Hu¨ckel model one can then construct the π electronic states of the polymer as a linear combination of the pz orbital on the nitrogen atoms and the frontier (HOMO/ LUMO) states of the fragments. Symmetry plays a key role in our description. The first symmetry element we use is the point of inversion in the center of the squaraine ring, already used in Figure 4 to label the fragment states. A second symmetry element is a 2-fold screw axis along the polymer axis. Both these symmetry elements are illustrated in Figure 5 on the example of polymer 4. These two symmetry elements (inver-
Small Band Gap Polymers
Figure 6. An MO diagram for polymer states which are antisymmetric (A) with respect to screw axis symmetry. In this schematic representation, the screw axis is along the Sq′-N-Sq′-N line and maps one Sq′ unit on the other. The HOMO state of the Sq′ fragment is antisymmetric a with respect to the inversion point. To illustrate its symmetry, the state is represented schematically as a simple π orbital, but one has to bear in mind that the HOMO has a complicated internal nodal structure, as shown in Figure 3. The HOMO forms bonding/ antibonding combinations with the pz orbital on the nitrogen atom. The LUMO state of the Sq′ fragment is symmetric s with respect to the inversion point. Again, a schematic π* representation is chosen; the internal nodal structure is also much more complicated. The LUMO does not interact with the nitrogen pz orbital and forms a nonbonding state in the polymer (i.e., the amplitude of the nitrogen pz orbital in this state is exactly zero; a nonzero value would give a polymer state of neither s nor a symmetry and canceling bonding/antibonding interactions).
sion point and screw axis) are used to characterize the π states of the polymer. Figure 6 shows an MO diagram for polymer states which are antisymmetric (A) with respect to screw axis symmetry and which are labeled as (anti)symmetric (a)s with respect to the inversion point. The Sq′ HOMO and LUMO are represented schematically as simple π and π* states, respectively, but one has to bear in mind that these states have a complicated internal nodal structure, as shown for example in Figure 3. Only the HOMO state of the fragment, which is antisymmetric a with respect to the inversion point symmetry, can form bonding/antibonding combinations with the nitrogen pz orbital. The LUMO state of the fragment, which is symmetric s, does not interact with nitrogen pz orbitals, and it becomes a nonbonding state in the polymer. In other words, there is no combination of nitrogen pz orbitals possible which is both symmetric with respect to the inversion center s and antisymmetric with respect to the screw axis (A). In a similar way, there is no possible combination of nitrogen pz orbitals with both a inversion and (S) screw axis symmetry. So for polymer states which are symmetric (S) with respect to the screw axis symmetry, it is the LUMO state of the fragment, with s inversion symmetry, which forms bonding/antibonding combinations with the nitrogen pz orbital, whereas the HOMO state of the fragment, having a inversion symmetry, becomes a nonbonding state of the polymer. The MO diagrams of both the antisymmetric (A) and symmetric (S) cases of screw axis symmetry cases are illustrated on the right-hand and left-hand side of Figure 7, respectively. The size of the bonding/antibonding splittings is determined by calculation; see below. These two MO diagrams serve as a starting point for constructing the complete band structure of the Hu¨ckel model. The full helical symmetry group of the polymer includes all multiples of the screw axis operation. The electronic states of
J. Phys. Chem., Vol. 100, No. 5, 1996 1843
Figure 7. Schematic MO diagram of the interactions between the frontier orbitals of the Sq′ fragment and the pz orbital of the nitrogen atom at k ) 0 on the left-hand side and k ) π/ax on the right-hand side. The states at k ) (π/ax), 0 are (anti)symmetric (A), (S), with respect to screw axis symmetry. Because of symmetry, there is no interaction between the HOMO of the Sq′ fragment and the pz state of nitrogen atom at k ) 0. For the same reason, there is no interaction between the LUMO of the Sq′ and the pz at k ) π/ax. The bands represented by the dashed lines interpolate between these two extremes and comprise the full band structure within the Hu¨ckel model.
Figure 8. Parameters of the Hu¨ckel model used to interpret the band structure of -Sq′-N- topology polymers. The parameters are the energy levels of the Sq′ fragment HOMO and LUMO, RH and RL, the energy level of the nitrogen pz, and the nearest-neighbor hybridization of the fragment HOMO and LUMO with the nitrogen pz, βH and βL. In the actual fits to the ab initio bands both RH and RL are fixed; RL is fixed by the position of the lowest unoccupied polymer orbital (LUPO) because of symmetry (discussion: see text). The HOMO/LUMO energy difference ∆RHL ) RL - RH is fixed at the value of the corresponding fragment; cf. Figure 4.
the polymer and their corresponding energy levels can be labeled with k,12 where k assumes all values between -π/ax and π/ax, with ax the translation associated with the screw axis operation;42 cf. Figure 5. States which are symmetric (S) with respect to screw axis symmetry belong to k ) 0, whereas antisymmetric (A) states belong to k ) π/ax.12,42 A complete band structure, consisting of the set of energy levels En(k) (where n ) 1, 2, 3 labels the bands; cf. Figure 7), forms an interpolation between these two extreme cases, as shown by the dashed lines in Figure 7. In the ground state all levels of the lower two bands are filled with electrons, whereas the uppermost band is empty. The lowest possible transition energy or band gap is found at k ) π/ax, as the difference between the uppermost and the middle curve.43 We have argued that the ab initio band structure can be interpreted in terms of a simple Hu¨ckel model based upon fragment frontier orbitals. The parameters of the Hu¨ckel model, which contains only nearest-neighbor interactions (and no overlap matrix), are indicated in Figure 8. The parameters can be obtained by fitting to the ab initio bands. The band structures of polymers 3-5 around the band gap are qualitatively quite similar; an example of the ab initio and the fitted bands is given for polymer 4 in Figure 9. (For polymer 3, see ref 21.) We observe that the fit is surprisingly good in view of the simplicity of the model.
1844 J. Phys. Chem., Vol. 100, No. 5, 1996
Figure 9. Band structure of poly(aminodivinylsquaraine) 4 at energies around the band gap. Bands below the Fermi energy EF are occupied, and those above EF are unoccupied; EF is arbitrarily chosen as the zero of energy. Only the π bands around EF are shown; the solid lines give the ab initio LDA results, and the dashed lines denote the Hu¨ckel fit, obtained by optimizing the three parameters RN, βH, and βL. This figure corresponds with the schematic Figure 7.
The parameters of the model can be used to describe the effects of chemical substitutions. Within the model the band gap in these polymers is between antibonding state of the fragment HOMO and the nitrogen pz orbital and the nonbonding fragment LUMO; cf. Figure 7. The size of the band gap is determined by the HOMO/LUMO energy difference ∆RHL ) RL - RH versus the hybridization βH of the HOMO with the nitrogen pz and the position of the nitrogen pz energy level RN. In principle, we could tune the band gap: a smaller HOMO/ LUMO energy difference ∆RHL, a larger hybridization βH, and/ or an upward shift of the nitrogen pz level RN relative to the fragment HOMO all decrease the band gap (to a point where the HOPO and LUPO levels of the polymer cross: then the band gap would increase again). Experimentally, the only thing one can do is to change the chromophore (as we did in the set of polymer 3-5). The effects of varying chromophores are very subtle, since all parameters ∆RHL, βH, and RN are affected simultaneously. For instance, comparing the squaraine and divinylsquaraine fragments, we see that the HOMO/LUMO energy difference ∆RHL in the latter is smaller (cf. Figure 4), which by itself would decrease the band gap. Also, the amplitude of the HOMO wave function on the individual carbon atoms of the fragment is smaller, simply because in divinylsquaraine the wave function is spread out over more carbon atoms. But this in turn leads to a smaller hybridization βH with the nitrogen pz in the polymer, because it is proportional to the amplitude of the wave function on the carbon atom which is directly coupled to the nitrogen.44 A smaller βH by itself would increase the band gap. So enlarging the chromophore leads to βH and ∆RHL working oppositely on the band gap, and only a detailed calculation can decide which of the two will win. Polymer 5 has a larger band gap than either 3 or 4, because in this case both βH and ∆RHL work unfavorably: ∆RHL for the diphenylsquaraine fragment is relatively large (cf. Figure 1), whereas βH is small. The latter is the result not only of the amplitude effect for larger chromophores discussed above but also of the steric hindrance around the nitrogen atom, which introduces the torsion discussed in the preceding section. The torsion decreases the overlap between the fragment states and the nitrogen pz orbital, which decreases βH. In summary the small band gap in the polymers of -Sq′N- topology is due to the following factors. The HOMO/
Brocks and Tol
Figure 10. Schematic representation of the HOMO/LUMO energy levels of the Sq′ and A fragments and the pz energy levels of the N atom, all of which belong to the same unit cell of the polymer. States that are (anti)symmetric (a)s are represented in the same way as in Figure 6. The two N atom pz states can be combined to symmetric and antisymmetric combinations. The parameters used for the Hu¨ckel fit are in principle as in Figure 8. In practice, we only fit the parameters βL and βA indicated explicitly in this figure; βA denotes the hydridization of the A fragment HOMO and LUMO with the nitrogen pz. (All dotted lines indicate the same parameter βA.) All energy level positions R are fixed relative to a common origin R0, which is also a fit parameter.
LUMO energy difference of the squaraine fragment ∆RHL is quite small to start with. The bonding/antibonding interaction between the HOMO of the fragment and the nitrogen pz orbital shifts the HOPO of the polymer up with respect to the HOMO of the fragment. Since there is no such interaction for the LUMO of the fragment (at the same k point), the LUPO of the polymer is at the same level as the LUMO. In detail, the size of the band gap depends upon the parameters of the individual chromophores. B. Breaking the Symmetry. The Sq′ and A fragments in the -Sq′-N-A-N- polymers are two completely different units not related by any symmetry. This obviously means that there is no screw axis connecting the two nitrogen atoms. The absence of screw axis symmetry, as compared to the polymers discussed in the previous section, introduces an extra degree of freedom which has very large consequences for the electronic structure. A simple Hu¨ckel model in the spirit of the previous section now requires the HOMO/LUMO levels of both the Sq′ and the A fragments and the nitrogen pz orbitals of the amine fragments. The band structure can then be set up in much the same way as in the previous section. Again we start with a description of those polymer states which are relevant for understanding the band gap. The fragment levels are illustrated in Figure 10, where (a)s labels again states that are (anti)symmetric with respect to an inversion center. Note in particular that the LUMO of the A fragment is much higher in energy than the LUMO of the Sq′ fragment. The dye fragment Sq′ absorbs in the visible, whereas the A fragment (vinyl or phenyl) absorbs in the UV; a large part of this difference is reflected in the position of the LUMO. Since there is no screw axis symmetry which maps Sq′ onto A, then obviously also the nitrogen atoms are not connected by screw axis symmetry. We have therefore the freedom to combine the pz states of the two nitrogen atoms in linear combinations which are (anti)symmetric (a)s with respect to the inversion center. The π states of the polymer can now be constructed conceptually in a three-state process, which is illustrated by MO diagrams in Figure 11. First, we switch on the interaction between the three antisymmetric a states, i.e., the HOMO’s of the Sq′ and the A fragments and the antisymmetric combination of the nitrogen pz orbitals, shown schemati-
Small Band Gap Polymers
J. Phys. Chem., Vol. 100, No. 5, 1996 1845
Figure 12. Band structure of polymer 6 at energies around the band gap. Bands below the Fermi energy EF are occupied, and those above EF are unoccupied. Only the π bands around EF are shown; the solid lines give the ab initio LDA results, and the dashed lines denote the Hu¨ckel fit (obtained by optimizing the parameters βL, βA, and the energy level origin R0). The polymer states shown in Figure 11 are found at k ) 0 (only the uppermost four states are shown).
Figure 11. π states of the polymer of eq 3 topology constructed in a three-stage process. In (a) we switch on the interaction between the antisymmetric a states of Figure 10. In addition, in (b) we switch on the interaction between the LUMO of the Sq′ fragment and the s combination of the nitrogen pz orbitals. Finally, in (c) we switch on the interaction with the LUMO of the A fragment. The levels are on scale for the example of polymer 6. In each diagram, the interaction considered is highlighted. The numbers represent the band gaps for each stage of the process. The final value of 1.1 eV corresponds with the calculated ab initio LDA value.
cally in Figure 11a. This situation resembles the one represented in Figure 6, but now the antibonding combination gives the LUPO of the polymer, whereas the HOPO is formed by the LUMO of the Sq′ fragment. So far, this would again lead to a small band gap. However, the absence of screw axis symmetry now allows for an interaction between the LUMO of the Sq′ fragment, which has s inversion symmetry, and the s combination of the nitrogen pz orbitals. This is shown in Figure 11b, where for the moment we ignore the interaction with the LUMO of the A fragment. The hybridization and the resulting bonding/ antibonding splitting is large. The LUPO of the polymer now results from the antibonding combination of the interaction, which means it is now shifted upward considerably. Since both the LUPO and the HOPO are antibonding combinations which involve respectively the LUMO and the HOMO of the Sq′ fragment, the band gap of the polymer reflects the HOMO/ LUMO energy difference of the fragment. As the third and final step, we now allow for the interaction with the LUMO of the A fragment, which shifts the LUPO down again. However, as long as this LUMO is much higher in energy than the other
levels, the shift is relatively small. This is the situation illustrated by Figure 11c for polymer 6. The analysis presented in Figure 11 is for polymer states at k ) 0, where in this case the band gap is found. The complete band structure of polymer 6 is shown in Figure 12, where again we observe quite a good agreement between the ab initio (solid lines) and the fitted results (dashed lines). The situation for polymers 2 and 7 is qualitatively similar; the relatively large band gap of the latter is also the result of the torsion discussed in section II. The LUPO is shifted down additionally if we lower the LUMO energy level of the A fragment. The more we do this, the more the states of the A fragment become energetically symmetric with respect to the states of the Sq′ fragment. Perfect symmetry is of course obtained if we use a Sq′ fragment instead of the A fragment, but then we are back to the -Sq′-N- class of polymers, which have small band gaps, as we have shown. In conclusion, we expect that the band gaps of polymers with -Sq′-N-A-N- topology reflect the HOMO/LUMO energy differences of the corresponding Sq′ fragments. The polymer HOPO and LUPO states are both antibonding combinations of the Sq′ fragment HOMO and LUMO, respectively. In lowest order, the difference in hybridization only has a small perturbative effect on the band gap as compared to the fragment HOMO/LUMO energy difference, provided that there is a large asymmetry in the relative position of the A and Sq′ energy levels. In a more symmetric positioning of these levels, a smaller band gap is obtained. In this sense, the -Sq′-Npolymer topology is a perfect symmetry limit of the more general -Sq′-N-A-N- topology.45 IV. Conclusions Polymerization of (squaraine) dye molecules provides a fruitful way to make polymers with small band gaps. We have made a systematic theoretical study of this class of polymers, using Car-Parrinello techniques to optimize both the electronic and the geometrical structure of the polymer, finding band gaps ranging from 2.3 eV to as low as 0.2 eV. A simplified Hu¨ckel analysis of the calculated π bands around the gap reveals the principles determining the size of the band gaps. First, the squaraine chromophore fragment has a HOMO/LUMO energy difference which is unusually small for such a small molecule.
1846 J. Phys. Chem., Vol. 100, No. 5, 1996 Second, we have shown that the polymers separate into two classes. In one class, which contains polymers 6, 7, and polymers synthesized in ref 16, such as 2, the band gap reflects the HOMO/LUMO energy difference of the corresponding squaraine chromophore fragment. Band formation in the polymer due to hybridization of these states shifts both HOMO and LUMO up in energy by similar amounts. In addition, we have presented a simpler and more symmetric polymer topology, examples of which are polymers 3-5, which leads to very small band gaps. The main reason for this is that hybridization in the polymer results in an upward shift of the highest occupied level as compared to the HOMO of the fragment (as before), but now the lowest unoccupied state of the polymer is pinned at the position of the LUMO of the fragment by symmetry. The latter is a prediction, since no attempts to synthesize a polymer with this topology have yet been made; we see, however, no a priori reason why such attempts should be unsuccessful and hope to stimulate new synthetical work in this direction. Acknowledgment. It is a pleasure to thank E. E. Havinga, P. J. Kelly, and P. van de Weijer for their careful reading of the manuscript. References and Notes (1) Band gaps in polymers are identified with the onset of optical absorption due to electronic excitations. An excitation energy of 2 (3.5) eV corresponds to an absorption wavelength of 620 (350) nm. (2) Burroughes, J. H.; Bradley, D. D. C.; Brown, A. R.; Marks, R. N.; Mackay, K.; Friend, R. H.; Burns, P. L.; Holmes, A. B. Nature 1990, 347, 539. (3) Gustafsson, G.; Cao, Y.; Treacy, G. M.; Klavetter, F.; Colaneri, N.; Heeger, A. J. Nature 1993, 365, 629. (4) For an overview see, for example: Proceedings of the 6th International Conference on II-VI Compounds and Related Optoelectronic Materials; Nurmikko, A., Yao, T., Ruth, R., Eds.; J. Cryst. Growth, in press. (5) Greenham, N. C.; Moratti, S. C.; Bradley, D. D. C.; Friend, R. H.; Holmes, A. B. Nature 1993, 365, 628. (6) Garnier, F.; Yassar, A.; Hajlaoui, R.; Horowitz, G.; Deloffre, F.; Servet, B.; Ries, S.; Alnot, P. J. Am. Chem. Soc. 1993, 115, 8716. (7) Lous, E. J.; de Leeuw, D. M. Synth. Met. 1994, 65, 45. (8) As for the problem cited above: it is easier to have a polymer with high electron affinity if the band gap is small. (9) Hagler, T. W.; Pakbaz, K.; Voss, K. F.; Heeger, A. J. Phys. ReV. B 1991, 44, 8652. (10) Chung, T.-C.; Kaufman, J. H.; Heeger, A. J.; Wudl, F. Phys. ReV. B 1984, 30, 702. (11) Mason, S. F. In Physical Methods in Heterocyclic Chemistry; Katrizky, A. R., Ed.; Academic: New York, 1963; Vol II, p 1. (12) Hoffmann, R. Solids and Surfaces: A Chemist’s View of Bonding in Extended Structures; VCH: New York, 1988. (13) Using Hu¨ckel terminology, there will be a set of β’s which describe the hybridization interaction between each pair of levels. (14) Andre´, J.-M.; Delhalle, J.; Bre´das, J.-L. Quantum Chemistry Aided Design of Organic Polymers; World Scientific: Singapore, 1991. (15) Fabian, J.; Hartmann, H. Light Absorption of Organic Colorants; Springer-Verlag: Berlin, 1980; p 191. (16) Havinga, E. E.; ten Hoeve, W.; Wijnberg, H. Polym. Bull. 1992, 29, 119; Synth. Met. 1993, 55-57, 299. (17) Forster, M.; Hester, R. E. J. Chem. Soc., Faraday Trans. 1 1982, 78, 1847. (18) Sprenger, H. E.; Ziegenbein, W. Angew. Chem., Int. Ed. Engl. 1968, 7, 530.
Brocks and Tol (19) Car, R.; Parrinello, M. Phys. ReV. Lett. 1985, 55, 2471. (20) Tol, A. J. Chem. Phys. 1994, 100, 8463. (21) Brocks, G. J. Chem. Phys. 1995, 102, 2522. (22) Aliphatic groups such as the methyl bridges in the five-membered rings in 2 do not contribute directly to the π states and are unimportant in discussing the electronic structure around the band gap of the polymer. (23) Heeger, A. J.; Kivelson, S.; Schrieffer, J. R.; Su, W.-P. ReV. Mod. Phys. 1988, 60, 781. (24) Brocks, G.; Kelly, P. J.; Car, R. Synth. Met. 1993, 55-57, 4343. (25) Anzhelm, J.; Wimmer, E. J. Chem. Phys. 1991, 96, 1280. (26) Bernstein, J.; Goldstein, E. Mol. Cryst. Liq. Cryst. 1988, 164, 213. (27) Dewar, M. J. S.; Zoebisch, E. G.; Healey, E. F.; Stewart, J. P. P. J. Am. Chem. Soc. 1985, 107, 3902. Program obtained from: Stewart, J. P. P. MOPAC (V6.0); A General Molecular Orbital Package, QCPE 455; Quantum Chemistry Program Exchange: Indiana University, Bloomington, IN. (28) The C-O bond length could in principle be improved by optimizing the oxygen pseudopotential. Cf.: Laasonen, K.; Car, R.; Lee, C.; Vanderbilt, D. Phys. ReV. B 1991, 43, 6796. (29) Vogl, P.; Campbell, D. K. Phys. ReV. B 1990, 41, 12797. (30) Gomes da Costa, P.; Dandrea, R. G.; Conwell, E. M. Phys. ReV. B 1993, 47, 1800. (31) Bre´das, J.-L.; Quattrocchi, C.; Libert, J.; MacDiarmid, A. G.; Ginder, J. M.; Epstein, A. J. Phys. ReV. B 1991, 44, 6002. (32) Jones, R. O.; Gunnarson, O. ReV. Mod. Phys. 1989, 61, 689. (33) Godby, R. W.; Schlu¨ter, M.; Sham, L. J. Phys. ReV. B 1988, 37, 10159. (34) Baraff, G. A.; Schlu¨ter, M. Phys. ReV. B 1984, 30, 3460. This procedure is often referred to as a scissors operator, because one can think of it as cutting the band structure through the band gap and moving the conduction bands rigidly upward. (35) Ohkoshi, I. J. Phys. C 1985, 18, 5415. (36) Gygi, G.; Baldereschi, A. Phys. ReV. B 1986, 34, 4405. (37) Oshiyama, A.; Kamimura, H. J. Phys. C 1981, 14, 5091. (38) Causa, M.; Dovesi, R.; Pisani, C.; Roetti, C.; Saunders, V. R. J. Chem. Phys. 1988, 88, 3196. (39) Even for the dye molecule 8, this procedure works reasonably well. Dividing the calculated LDA HOMO/LUMO gap of 1.2 eV by 0.6 gives a fair agreement with experiment. See e.g.: Dirk, C. W.; et al. J. Am. Chem. Soc. 1995, 117, 2214. (40) Bear in mind that the out-of-plane torsion in polymer 5 increases the band gap as compared to completely planar structures; cf. ref 31. (41) In some cases we find in our calculation oxygen lone-pair-like n levels close to HOPO π level; cf. ref 21. In order to avoid complicating the discussion, we will only discuss states of π symmetry. Since the n-π* transition is forbidden, n states will not appear in the optical spectrum at a low energy. However, they could in principle act as trap states upon p-doping of the polymer. (42) For the helical symmetry group, see e.g.: Springborg, M. Int. ReV. Phys. Chem. 1993, 12, 241 and references therein. (43) The full translational unit cell for the polymers with screw axis symmetry is -Sq′-N-Sq′-N- with cell parameter 2ax. Labeling the states with the pure translational quantum number K means folding the bands in Figures 7 and 9 around k ) π/2ax. The gap is then found at K ) 0. (44) The reason that βL > βH is simply that the amplitude of the LUMO on the carbon atom of the fragment coupled to the nitrogen atom is much larger than of the HOMO. (45) In ref 16 it was speculated that the small band gaps of squaraine polymers could be the results of donor/acceptor properties of the individual fragments. That type of reasoning relies on charge transfer between the energy levels of the fragments, but one has to assume that the fragment states are not completely rehybridized (and thus remixed and reordered) in the polymer. However, as we have shown here, such hybridization effects are very large here and mask the charge transfer effects. Therefore, it seems that the donor/acceptor concept is not the key to understanding the electronic structure.
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