Polaron Pair versus Bipolaron on Oligothiophene Chains: A Theoretical Study of the Singlet and Triplet States

Abstract

Double positive charges on an isolated conjugated chain: do they remain together forming a single bipolaron (2+) entity (shown in picture) stabilized by strong geometry distortion even in the absence of a counterion or do they repel, yielding two separate polarons (+)? The balance is subtle. Our correlated ab initio and semiempirical calculations show that in oligothiophenes a polaron pair is preferred, for chains containing seven or eight rings and longer. The geometry of π-conjugated oligomers and polymers is very sensitive to the amount of charge the chains carry. Given this strong electron–phonon coupling, it has been shown that excess charge can concentrate and lead to structural distortions extending over a limited section of the chain. Such a self-localization of defects in π-conjugated chains has been amply documented.1 Not only a single charge can be trapped in this way forming a polaron (radical-ion species), but also a double charge is supposed to form a bipolaron (di-ion) structure.1, 2 Direct structural data for charged species are scarce; the form of charge storage has therefore been deduced from various spectral and electrochemical experiments; these results can sometimes lead to different interpretations. In addition to single polarons and bipolarons on isolated chains, π dimers involving two polarons on two chains3, 4 and polaron pairs on a single chain5 have been invoked (it can be noted here that in some instances, spectral assignments considered early on as a signature of bipolarons were later reassigned to polarons6). In the case of a bipolaron, it is useful to discriminate between intrinsic and relative stability. A molecular species is intrinsically stable when the potential energy surface presents a minimum corresponding to its structure. However, such a species can turn out to be unstable relative to a species of the same composition but of a different geometry and/or in a different electronic state. Herein, we ask the question of whether a bipolaron on an isolated oligothiophene chain in the absence of counterions is intrinsically stable and whether it is relatively stable versus a dissociated polaron pair; excluding counterions is especially relevant when this question is addressed to the case of charges injected into semiconducting conjugated polymer or oligomer chains. Hückel theory, in the form of the SSH Hamiltonian, predicts that a bipolaron in nondegenerate ground-state conjugated chains, in polythiophene in particular, is favored over a dissociated pair of polarons; for a doubly charged chain even in the absence of a counterion.7, 8 However, the balance between a bipolaron and a pair of polarons in the theory is subtle and depends on the parameter values of the model Hamiltonians.9 The formation of a bipolaron defect in oligothiophene dications is predicted by semiempirical and ab initio HF calculations;10–12 sufficiently long thiophene chains were shown to support multiple bipolarons.13 However, these results were obtained for the closed-shell configuration. Since the alternative structure corresponding to two dissociated polarons is a diradical, its electronic structure, in the singlet or triplet state, requires an open-shell description (see refs. 5, 14 for discussion). AM1 semiempirical calculations by Tol on cyclic15 and linear14 oligothiophenes with a restricted open-shell Hartree–Fock (ROHF) minimal configuration interaction (CI) wavefunction allowing the description of a polaron-pair structure indicated that a bipolaron is not intrinsically stable and forms only when two polarons are forced to occupy a small place on a chain shorter than seven thiophene rings. It is important to note that, although the singlet and triplet states of a two-polaron biradical are quasidegenerate, the ground state was found to be singlet; thus, the absence of any ESR signal is not necessarily a signature of a bipolaron. Ab initio HF, two-configuration SCF, and DFT calculations16 confirmed the stability of a two-polaron structure for a cyclic 12-thiophene oligomer; it was shown, however, that a bipolaron could form in the presence of a counterion. The stability of a bipolaron relies on the gain in elastic energy caused by the geometric distortion, with regard to the increased electron repulsion due to the creation of a region of concentrated excess charge. Therefore, a proper description of this balance requires taking electron correlation into account. Correlated (MP2) treatments of oligothiophene dications, in the presence or absence of counterions,17, 18 have been up to now limited to chain lengths up to four rings; such chains are too short to allow for a comparison of a single-bipolaron structure versus its dissociated (two polarons on one chain) counterpart. DFT calculations performed in the local density approximation (LDA) on periodic infinite oligothiophene chains suggested that delocalized bipolarons and not polarons are the charge carriers in the dilute limit.19 For a single bipolaron on a finite chain, DFT calculations carried out at both LDA level20 and with a gradient-corrected BLYP functional18 predict that the dication charge and geometry distortion spread over all the available rings of the oligothiophenes, in contrast to the HF results discussed above. However, these DFT results are questionable since a similar delocalization is obtained for a polaron,20, 21 which is in disagreement with a well-localized structure found at the MP2 level.22 Our recent results22 also show that including HF exchange by hybrid DFT increases somewhat the degree of polaron and bipolaron localization in oligothiophenes. Herein, we discuss the results of ab initio HF, hybrid DFT, and MP2 calculations on oligothiophene dications of different chain lengths in their singlet and triplet states. Our interest in the triplets is due to the simple observation that, while a polaron pair can exist as a singlet or a triplet (of which the structures are not expected to differ significantly as long as two polarons are well-separated), the triplets are more easily accessible by ROHF and DFT calculations than the open-shell singlets. It is expected that proper account of electron correlation is crucial for the comparison of similar structures differing significantly in electron-density concentration, such as a polaron pair and a bipolaron. Even in the model formulation of this problem, for one isolated chain, the task is formidable. Herein, we tackle it in different ways. To address dynamical correlation, MP2 geometry optimizations were carried out, whereas to assess the role of nondynamical (quasi-degeneracy) correlation, we performed minimal ab initio and more extended semiempirical CAS–SCF geometry optimizations. Calculations taking into account dynamical and nondynamical correlation simultaneously (CAS–PT or extended MR–CI) would be most relevant but are hardly practical for the systems of current interest. As the starting point for geometry optimizations, the doubly charged oligothiophene molecules from 6T2+ to 10T2+ (denoted as xT with x the number of rings) were taken in the all-trans conformation and kept coplanar, that is, C2v or C2h symmetry for odd- and even-ring oligomers, respectively. No counterions are taken into account. All the computations were performed with the 321G* basis set, because it has been shown previously12 that further extension of the basis set does not lead to any significant changes in the optimized geometries. Furthermore, because our aim is to study long oligomers, extending the basis set appears less important than for smaller molecules. The Gaussian 98 Revision A.723 for ab initio and DFT calculations, and Ampac 6.5524 for semiempirical AM1 calculations, were used in this work. Only HF spin-restricted results can be considered reasonable for both singlets and triplets because of huge spin contamination in the unrestricted HF (UHF) solutions. Typically, for both UHF “triplet” and “singlet” (the latter obtained from the symmetry broken initial guess), S2 is about five instead of correct values of two and zero, respectively. Therefore, singlets were treated as closed-shell species in both HF and MP2 calculations. For the triplets, ab initio ROHF and ROMP2 calculations were performed in single-point calculations (the keyword scf=tight was used in order to obtain the same precision for the energy as in geometry optimizations) as well as semiempirical CAS–SCF. The hybrid DFT calculations were performed with the BHandHLYP functional implemented in Gaussian 98, which is similar to Becke's “half-and-half” functionals.25 Free of excessive parameterization, it takes one half of HF exchange and one half of DFT exchange, including local (LSDA) and gradient-corrected (Becke88) terms, along with the LYP correlation. This choice of functionals is suggested by the excellent results obtained for the structures of polyenes26 and neutral oligothiophenes.27 For the triplets, the DFT calculations were carried out at the unrestricted level, because even though the theoretical status of S2 is not well-defined in DFT (strictly speaking, a Kohn–Sham determinant is not a wavefunction and there is no simple way to determine an operator eigenvalue for the state corresponding to it), the formal spin contamination in unrestricted BHandHLYP (calculated as if the KS determinant were a wavefunction) is rather low (typically, S2 is ca. 2.2, which is close to the value of 2 theoretically required for a triplet). Dynamical electron correlation was taken into account at the spin-restricted MP2 level. The MP2 calculations were conducted with a frozen core, that is, only the inner shells were excluded from correlation (nine lowest pairs of α and β MOs per ring); all the remaining occupied orbitals were included (12 pairs of α and β MOs per ring) as well as all virtual MOs. The geometry was fully optimized for closed-shell singlets, and the vertical triplet energies were calculated with spin-restricted open-shell MP2 (note that open-shell geometry optimizations are not feasible for large molecules with the ROMP2 method in the current version of Gaussian, because analytical gradients are unavailable). For 8T2+, in order to assess if correlating only higher π orbitals is sufficient, a partial MP2(32×32) calculation was also performed, in which 32 higher occupied and 32 lower vacant orbitals were correlated, that is, 4×4 MOs per ring. However, single-reference MP2 calculations may be not optimal for the cases when nondynamical (quasidegeneracy) correlation can be important: while at the Hartree–Fock level, an open-shell singlet wavefunction contains two determinants, it might be a hard task to perform better in the framework of a single reference MP2 ansatz. To address nondynamical correlation separately, we also performed complete active space CAS(n,n) geometry optimizations: minimal ab initio CAS with n=2 and more extended AM1/CAS with n=2, 4, 6, where n is the number of active electrons and orbitals (equal in our case) in usual notation. Reliable information on the extension and location of the charged defects can be obtained from the geometric structure. The geometry changes related to charge self-localization primarily concern the bond-alternation pattern along the backbone.7, 8, 11, 12, 28 Neutral thiophene rings are characterized by a pronounced carbon–carbon bond-length alternation (BLA), which corresponds to the aromaticlike valence bond structure 0 in Scheme 1. Valence bond structures, demonstrating the differences between 0 and 2+. On the other hand, charged oligomers tend to adopt a marked quinoid geometry corresponding to the valence-bond structure 2+ with a reverse BLA pattern. Basically, the distinction between aromatic and quinoid structures allows one to determine the location and extension of the defect. If one plots the three CC bond lengths of each thiophene ring and connects these three points in a graph, a Λ-shaped connection corresponds to an “aromatic” BLA pattern in the rings (the CC bond in the middle is longer than the bonds at both sides of the ring), and a Vshaped connection corresponds to the “quinoid” rings of a charged defect. Inter-ring bonds shorten and adopt a double-bond character in the quinoid structure, which thus also points to the charged defect location. Appearance of a difference in the two CS-bond lengths of a given thiophene ring in a chain also points to its quinoid character, as suggested by the canonical structures in Scheme 1.12, 28 The closed-shell HF/3-21G* singlet structures for the dications are essentially identical to those obtained previously with the same method;12 the closed-shell singlet structures we obtain with the BHandHLYP method are similar. The pattern rings with a more quinoid character and shorter in the with aromatic rings at the chain this corresponds to the presence of a charged defect in the The charge on each ring does not from its value per ring which that the defect is rather and over the in the of chain lengths The that the charges per ring are higher than on the middle of the chain in longer oligomers is with the presence of a bipolaron the in of the pronounced On the basis of these closed-shell calculations, it is expected that, in longer dication chains, a bipolaron would be a HF and BHandHLYP optimized lengths for oligothiophene dications in their closed-shell singlet states HF to singlet the three bond lengths of each thiophene connects all bond The data are for the half of the chains. charge per ring in the oligothiophene dication singlet and triplet structures optimized with different HF closed-shell singlet to singlet bipolaron, ROHF triplet to singlet polaron of the geometry of the oligothiophene dications in the triplet states that the energy of the triplets with to the singlets with chain For chains to or longer than at the ROHF or in BHandHLYP calculations, the triplet in its optimized geometry is more stable than the singlet state. of the and triplet states versus the corresponding optimized singlet states in oligothiophene dications as a of chain The geometry rather similar in ROHF and BHandHLYP optimizations, is predicted to be with to the singlet The thiophene rings are of aromatic in the middle and more quinoid the for the very the CC bonds are in the The charges in the triplets are from the middle the chain the spin pattern is similar to that of the charge The BHandHLYP and ROHF triplet structures are similar to one and to the formation of a pair of This is for the oligomer and the two-polaron structure the ground state at longer chain In the of this the charge found in the middle for the singlet structures might be considered to be a of some of a bipolaron wavefunction at the closed-shell HF level. HF/3-21G* and lengths for oligothiophene dications in their triplet state ROHF to singlet polaron the three bond lengths of each thiophene connects all bond The data are for the half of the chains. spin per ring in oligothiophene dication The two-polaron of the dication triplet state can be in the In we have taken the spin obtained for the structure of the charged a well-defined and we have it on both sides of the the with the spin in the ROHF 10T2+ triplet is a and it is that both occupied molecular orbitals of the 10T2+ triplet and are identical to the and of the and polaron orbitals (denoted and taken as the of in and on both of the chain and the to perform such a comparison from (two per in the 321G* basis in the ROHF 10T2+ triplet in and with the of and MOs in and as in the from two ROHF of as indicated by the The composition of the ROHF of is shown in that this is obtained the that there is no between the and polaron orbitals the of the of the polaron orbitals and their is expected to be Furthermore, in the case of of two their be In the energies of the two triplet are very close to each and their with chain from for 6T2+ to for each of the two electrons of a biradical a different of a set the corresponding singlet and triplet states of the are also at the HF level, with the triplet somewhat more stable due to the exchange correlation into account can be more for the and this even to the of we to this point in the To this we note that optimizations in the triplet state are expected to in the of two of and thus of the excess charges on the chain, to is that the structures with a triplet pair of polarons are lower in energy than the singlet bipolaron structures, when the chain is sufficiently the HF and BHandHLYP of theory, a bipolaron is intrinsically though relatively less stable than a triplet pair of The results in the section that a potential due to a geometric distortion is not to two excess charges together In this up to a more description of electron that is, taking electron correlation into is The geometry optimizations for oligothiophene dications in their closed-shell singlet states at the MP2 structures and charge that significantly with chain and can differ from those of singlets. and optimized lengths in oligothiophene dication singlets. the three bond lengths of each thiophene connects all The data are for the half of the chains. For the MP2 geometry of the four rings is only quinoid with the CC bonds and between the rings more than in the HF and BHandHLYP The charge in the structure also does not significantly from the For and 8T2+, the singlet structures are different from the corresponding HF or BHandHLYP structures, as well as from the 6T2+ MP2 singlet structure above. to two separate polarons at the chain thus the HF or BHandHLYP optimized triplet The two-polaron character is more pronounced for the longer oligomers, with the charge per ring the chain The rings are whereas the four middle rings are the bonds are in the middle and adopt double-bond character the chain In the MP2 singlet the character of the dication is even more pronounced than in the ROHF and BHandHLYP Furthermore, no potential minimum corresponding to a geometry can be obtained by MP2 singlet geometry for and the bipolaron appears to be intrinsically To assess the role of the electron correlation in the MP2 it is to the obtained by correlating only the higher π orbitals in a partial MP2 geometry performed for the active space of 32 occupied and 32 that is, four each per ring the molecular geometry is to the HF the two-polaron it from that obtained when only the orbitals are In it has been that, taking into account dynamical electron correlation as as in correlating can be of in theoretical description of molecules. these MP2 which a strong in of polaron-pair structure, be taken with The MP2 singlet geometry (note especially a very long bond of in and charge a strong of the two It is a that, at the of MP2 because of in single-reference MP2 can well molecules with broken bonds and the at long an The value of the MP2 energy increases the theory on of due to of the and Therefore, if a geometry is carried out, a dissociated species can be This could also be the case for a polaron pair on a long the energies in the To whether allowing for a reference is relevant for our problem, we performed initio for 8T2+, AM1 for and CAS–SCF geometry optimizations, thus taking of nondynamical correlation, but a of dynamical correlation The is that the of the CAS wavefunction in longer chains to both single-bipolaron and dissociated polaron-pair structures, which depends on the starting On the other hand, the semiempirical and charge are of the active space ab initio and semiempirical CAS results are in For the state, a CAS that is from a bipolaron geometry up with a bipolaron structure of which the wavefunction is this is the closed-shell On the other hand, if also for the state, is from a polaron-pair a dissociated polaron pair is the structure in the case of ab initio the structure is very similar to the ROHF triplet in the structures of the active space singlet bipolaron to closed-shell lengths and charge per ring for oligothiophene dications and 10T2+ the CAS for the we note that the wavefunction in this latter case strong nondynamical correlation since it contains as two determinants, and for the HF and The values are and both in ab initio and the required for the as a of of the and which are the and of the and polaron The form of this CAS is to the ROHF singlet into account the composition of and as of the and polaron the latter expected to be with the triplet if the two polarons only it the geometry and the charge of the optimized singlet and triplet polaron pair are CAS results that both of singlet species are intrinsically stable and by a potential from each For at the level, the bipolaron is more stable than the pair The bipolaron could thus be out in the MP2 the MP2 on a pair is an important To address the in a polaron we also calculated the ROMP2 vertical energies at the singlet (note that the MP2 related to of the occupied and vacant orbitals do not for that, on from 6T2+ to 8T2+, the MP2 singlet more stable than the ROMP2 vertical For 8T2+, the triplet somewhat lower in The optimized triplet dication structures are of both starting geometry and active space are The charge in these triplets is identical to those in the singlet polaron the same the geometry of the singlet and triplet polaron pairs is somewhat especially in the where the bond pattern in the triplets is less Given different electron in the singlets and triplets, similar electron different geometries. AM1 and ab initio results are in the three dication the most stable in all calculations is the triplet polaron For 8T2+, the singlet bipolaron, higher on the active space and the singlet polaron pair is the the However, the relative stability of the latter increases with the active space the singlet polaron pair is higher than the bipolaron by and only with and respectively. In the composition of the singlet polaron pair wavefunction does with the active space thus, this species is the most correlation and by the active we into account more dynamical correlation. For the pair is more stable than the bipolaron with minimal and its stability with to the latter with and respectively. the same the between the polaron-pair structures with the active space results that, in the absence of a the polaron pair is more stable than the bipolaron in Furthermore, it is that, in longer chains, and with correlation taken into the ground state of this polaron pair be a have theoretically the electronic and geometric structure of oligothiophene isolated chains two positive charges in the absence of HF and DFT calculations predict the formation of a single a bipolaron. In ROHF and DFT calculations that the stability of the optimized triplet structures, which to a dissociated pair of increases with chain for chains of seven or eight the pair the ground-state structure. In MP2 and CAS calculations, the structures corresponding to polaron pairs are obtained by geometry optimizations in the singlet state. A bipolaron though intrinsically is relatively unstable with to a triplet or even pair when a significant amount of electron correlation is taken into account. that these results to an isolated oligomer chain in the absence of and other molecules of the same oligomer are to which our results are such as oligomers in dilute solutions. The of that shorter oligothiophenes, two polarons on a chain and not a this observation is fully with our theoretical also point out a DFT when our was in which the on bipolaron to two-polaron structure in thiophene oligomers is The in is by the for and by the and the for The at is by the the of the and the