Polarons and charge carrier solvation on conjugated carbon chains: A comparative ab initio study
aa r X i v : . [ phy s i c s . c h e m - ph ] M a r Polarons and charge carrier solvation on conjugatedcarbon chains: A comparative ab initio study
M. L. Mayo and Yu. N. Gartstein ∗ Department of Physics, The University of Texas at Dallas, P. O. Box 830688, EC36, Richardson,Texas 75083, USA
E-mail: [email protected] ∗ To whom correspondence should be addressed bstract We study accommodation of an excess charge carrier on long even- N polyynic oligomersC N H due to displacements of the underlying carbon lattice and polarization of the surround-ing solvent in the context of carrier self-localization into a polaronic state. Spatial patternsof bond-length alternation, excess charge and spin densities are compared as derived withHartree-Fock and two hybrid-DFT methods (BHandHLYP and B3LYP) in conjunction withthe polarizable continuum model. Quite distinct resulting pictures of carrier accommodationare found when contributions from different interactions are analyzed. Solvation robustly actsto promote excess charge localization. Introduction
A variety of one-dimensional (1D) semiconductor (SC) nanostructures such as p -conjugated poly-mers (CPs), nanowires, nanotubes, and biological macromolecules attract a great deal of attention.They are interesting scientifically and can be exploited in various technological areas including(opto) electronics, energy harvesting and sensors. The nature and properties of excess chargecarriers on these structures are fundamental for many processes and applications. For purposesof our discussion here, one distinguishes between nearly free band states of excess carriers andself-localized polaronic states. Self-localization and the formation of 1D polarons may occurdue to the (strong) interaction of the electronic subsystem with another subsystem such as dis-placements of the underlying atomic lattice, the mechanism extensively studied for various 1Delectron-phonon systems, and particularly for CPs.
Another, and less explored, implementa-tion of the polaronic effect can take place for 1D SCs immersed in 3D polar media, the situationcommon for applications involving fundamental redox processes in polar solvents. In this case ofwhat could be called charge carrier solvation, the long-range Coulomb interaction is expected to lead to the formation of a localized electronic state on a SC structure surrounded by a self-consistent pattern of the sluggish (orientational) polarization of the solvent. Among consequencesof self-localization, lattice-deformation and solvation-induced polarons can feature new signatures2n the optical absorption due to local intragap electronic levels and substantially reducedmobilities.
In this paper we use three different theoretical levels of ab initio computations to comparepictures of accommodation of an excess electron on a prototypical 1D SC system: long even- N polyynic chains C N H , emerging due to both bond-length adjustments of the carbon lattice andpolarization of the surrounding solvent. The system we study is nearly charge-conjugation sym-metric and similar results are obtained upon an addition of an extra hole. In even- N oligomersthe arising patterns of bond-length modulations are non-topological (polaron-like) being thus rep-resentative of a broad range of CPs. A related comparative study of odd- N polyynic oligomershas been recently published; those feature topological kink-like patterns that can be found inpolymers with the degenerate ground state, trans-polyacetylene being another example. In applying first-principles calculations one expects to be able to elucidate effects due to reor-ganization of valence electrons and many-electron interactions, which can be crucially important inaccommodation of excess charge carriers. Numerous studies at various theoretical levels have beenpublished on the structure of the ground state and excitations in CPs. It is therefore noteworthy that,despite a relatively long history of electron-lattice polarons in CPs (see, e.g., multiple references inRefs. ), some questions have been raised recently regarding applicability of different ab initio frameworks to describe the polaron formation. That pertains to the reported failure of the local-density-approximation and generalized-gradient-approximation DFT (density-functional theory)schemes to detect self-localized charge density distributions in various charged oligomers, whileHartee-Fock (HF), parameterized semi-empirical and possibly hybrid-functional DFT methodshave been reported to lead to charge localization in the middle of oligomers. Quite dif-ferent level-of-theory-dependent magnitudes of the effective electron-phonon coupling have alsobeen found for the dimerized ground states of polymeric systems.
These observations thusbear on an important general issue of a choice of appropriate ab initio methods to faithfullydescribe properties of 1D semiconductors. In our computations for this paper, we compare resultsderived with the GAUSSIAN 03 implemented HF and two popular hybrid-functional DFT meth-3ds (BHandHLYP and B3LYP) in vacuum and solvent environments, the latter represented withinthe framework of the polarizable continuum model.
We will be interested in the carbon-carbonbond-length alternation (BLA) patterns and spatial distributions of excess charge and spin densitiesresulting from accommodation of an extra electron.Our choice of linear polyynic carbon chains as a model system is to a large extent relatedto their structural simplicity that allows us to keep computations on relatively long oligomers (upto N =
100 in this paper) still practicable. In doing so, we attempt to address the intrinsic responseof the system as determined by inherent interactions rather than by the end effects due to the hydro-gen termination. In terms of the polaron formation, one would like to see if the polaron-like spatialstructure is indeed self-maintained, that is, if its spatial extent is not affected by the length of theoligomer. We have explicitly demonstrated in the case of kink-solitons that their resulting spatialextent strongly depends on the computational method used. Our results in this paper also display avery strong method-dependence as will be discussed later. While being a model system, one shouldnote that polyynic chains continue to be the subject of much attention in their own right. A qualitative difference in the electronic structures of polyyne and trans-polyacetylene is that theformer features an extra spatial degree of freedom: p -electron molecular orbitals can have twoindependent orientations perpendicular to the polymer axis. Within the framework of appropri-ately modified Su-Schrieffer-Heeger (SSH) model of conjugated polymers, it was shown that, thanks to the extra degeneracy of p -electron levels, polyynes would possess a particularlyrich family of electron-lattice self-localized excitations. Also, spin-charge relationships for suchexcitations are quite different from trans-polyacetylene’s.In order to distinguish between various effects (due to electron-electron interactions, solventpolarization and carbon lattice adjustments) on resulting charge and spin densities, we study bothsystems with prescribed carbon atom positions (referred to as “rigid geometry” cases) as wellas fully optimized systems, in which carbon atom positions minimize the total energy. Rigidgeometries feature uniform BLA patterns chosen based on the results of optimization in the groundstate of neutral oligomers C N H . The data on neutral oligomers is used as benchmarks against4hich we determine changes in BLA patterns and excess charge distributions in negatively chargedoligomers C N H − . Computations and data processing
All ab initio computations in this study were performed using the GAUSSIAN 03 suite of pro-grams. We have employed pure Hartree-Fock (HF) along with the well-known hybrid -DFTfunctionals BHandHLYP and B3LYP in order to compare results of the different levels of the-ory. Hybrid exchange-correlation density functionals include both local and non-local effectsand are commonly used in studies of electronic properties of CPs. The reader is reminded thatfunctionals used here differ by the amounts of HF exchange included: 50% for BHandHLYP and20% for B3LYP. A rich 6-311++G( d , p ) all electron basis set has been employed throughout. Toinvestigate the effects of solvation in a polar medium, GAUSSIAN 03 offers its implementationof the polarizable continuum model (PCM) described in original publications. For all “in sol-vent” PCM calculations, water has been chosen as a polar solvent with its default parameters inGAUSSIAN 03. In working with long oligomers we met with certain method-dependent limita-tions on obtaining satisfactory levels of convergence. In order to produce reliable results within theHF method, restricted (R) and restricted open (RO) shell wave functions were used. For the hybridDFT methods BHandHLYP and B3LYP fully unrestricted (U) wave functions were utilized. Notethat, within each computational method, we consistently used the same form of the wave function.The subject of our attention in this paper are long C N H chains with even number of carbons N ( N up to 100). It is well established that long even- N neutral oligomers have theirground state with the polyynic structure, that is, they feature an alternating pattern of triple C ≡ Cand single C − C bonds and a gap in the electronic spectrum. We note that wherever possible, weverified that results of our “in vacuum” calculations compare well against previously published ab initio data on even- N neutral and charged C N H systems both in terms of energetics and opti-mized bond lengths. To our knowledge, oligomer lengths explored in our studies (Refs. and5his paper) are appreciably longer than used previously in comparable calculations for chargedsystems.A very convenient and well-known way to characterize the geometry of dimerized poly-meric structures is via bond-length alternation (BLA) patterns d n = ( − ) n ( l n − l n − ) , (1)where l n is the length of the n th carbon-carbon bond defined, e. g., as on the right of the n thcarbon atom (we do not discuss the end bonds to hydrogens). In the infinite dimerized neutralstructure, the dimerization pattern ?? would be uniform, that is, independent of the spatial bondposition n , we denote the magnitude of this equilibrium ground-state pattern as d (this is not tobe confused with a specific site value of d n = ). The double degeneracy of the ground state of theinfinite polymer is in that the uniform pattern ?? can assume either value of + d or − d . For ourpurposes, we retrieve the magnitude d from the middle segment of long even- N neutral oligomersfeaturing a well-established spatially-independent behavior, see illustration in [figure][1][]1, anddisplay that ground-state BLA pattern as positive. As mentioned earlier, the magnitude d is foundto be strongly method-dependent; this dependence is reflected in results shown in [figure][3][]3and in published tabulated numerical data.While exhibiting a spatially-independent behavior in the middle of long neutral oligomers,[figure][1][]1(a) also clearly shows that the end (oligomer termination) effects on the BLA patternare quite substantial. In numerous computations we observed that the “shape” of the end-specificBLA structures stabilizes and becomes easily discernible in long enough oligomers (of course,within a specific computational method). Practically the same persistent shape has been foundin end BLAs of neutral and charged, even- and odd- N oligomers. This fact allows to “elimi-nate” the end effects by comparison and an appropriate “subtraction” procedure, as illustrated in[figure][1][]1(b). Such a procedure of removing the end effects from the finite oligomer data hasbeen used to produced the BLA patterns shown in [figure][3][]3 focussing thus on the changes6 .140.120.100.080.06 d n ( A ) o (a) (0)(-1) Bond index n (b) d n ( A ) o Figure 1: Illustration of end effects in BLA patterns and their elimination: Panel (a) comparesBLA patterns for the neutral (0) and charged (-1) chains with end effects. Panel (b) indicates theBLA pattern after the subtraction of common end BLA structures. Vacuum results are shown withdashed lines and solid lines indicate the resulting BLA pattern upon solvation. Results depictedhere were obtained on the C H chain using the B3LYP method.induced by the accommodation of an excess charge carrier.Very essential for our discussion are the resulting spatial charge and spin distributions over theoligomers, which are naturally related to atomic charges and spins in outputs of ab initio calcula-tions. Importantly, these quantities are calculated from many-electron wave functions and therebyreflect responses due to all electrons in the system. Two procedures, after Mulliken and Löwdin,are widely used for charge population analysis and only Mulliken procedure is available forspins in GAUSSIAN 03. It is well known that calculations of atomic-centric quantities dependon the basis set transformations and sometimes lead to artifacts and spurious results. We havedemonstrated previously that certain cancelation effects take place and Mulliken and Löwdinprocedures lead to more similar results when one is interested in the spatial distribution of the7 .000.500.00-0.50-1.001.000.500.00-0.50-1.00 0 10 20 30 40 500 20 40 60 80 100 Atom index n q n / q m a x r / r s n s m a x (a)(b) Figure 2: Illustrations of the spatial behavior of raw atomic densities scaled with respect to theirmaximum values. Both examples are derived from oligomers with uniform rigid geometries. (a)Excess charge density obtained for C H − in vacuum with the HF method. (b) Spin density ob-tained for C H − in solvent with the BHandHLYP method. excess charge derived as the difference of charge densities on charged and neutral chains. Anexample of Löwdin-derived excess atomic charge densities q n from our current computations isshown in [figure][2][]2(a). [figure][2][]2(b) illustrates an instance of Mulliken-derived atomic spindensities r sn . When used for absolute data values, we keep both densities normalized to unity: (cid:229) n q n = (cid:229) n r sn = . In [figure][2][]2, the data is shown in a scaled form to emphasize its spatial behavior. It is evidentfrom this figure that atomic-centric quantities generally exhibit very strong variations from carbonto the next carbon. It is not clear to what extent these oscillations may result from computational8rtifacts perhaps also related to the presence of bond-oriented density waves in our system. Inorder to eliminate (possibly spurious) high spatial-frequency effects and to focus on longer spatial-range structures, some averaging/smoothing of the atomic data can be applied. Needless to saythat results can thus depend on details of the smoothing procedure. In our earlier work, we haveshown that a simple cell averaging of excess charge data can already lead to quite smooth results,moreover it has made Mulliken- and Löwdin-derived results nearly identical. The unit cells hereare defined as consisting of pairs of neighboring carbon atoms starting from the oligomer ends (asis common, the chain end cells also include hydrogen atomic charges/spins). We apply the samesummation over the neighboring carbons to generate cell-centric data in this paper. To smoothdata even further, we also apply a simple-minded nearest-neighbor-cell averaging where a cell-specific quantity x i is transformed into h x i i = . x i + . ( x i − + x i + ) . So generated smootheddata on cell-centric charge q i and spin r si densities are used in [figure][3][]3. While the numberof cells (index i ) is of course twice as small as the number of carbons, the procedure preserves thenormalization: (cid:229) i q i = (cid:229) i r si = . While reiterating the caveat on the procedure dependence (especially on some spin data), we men-tion here that the described above averaging produce results with smaller-amplitude variationsthan some other approaches we tried, which is a desirable feature in the absence of other criteriato ascertain the physical validity of the atomic-centric data from computations.As discussed in the Introduction, in addition to the fully relaxed geometries of the underlyingatomic lattice that minimize the total system energy, we also study rigid geometries (RG) with pre-scribed bond lengths. Those have been generated separately for each of the computational methodsused in respective environments (vacuum and solvent). We would first find a fully optimized ge-ometry of the N =
100 neutral oligomer and retrieve the resulting lengths of the C ≡ C and C − Cbonds in its central part (as well as the length of the terminating C − H bonds). We then assignthose bond lengths alternatingly to all neighboring carbon pairs throughout the oligomer. The re-sulting RG structures thus exhibit entirely spatially uniform carbon-carbon dimerization patterns9ithin the confines of terminating hydrogens. We note that there is very little difference, as clearlyseen in [figure][1][]1, in the dimerization amplitude d between neutral chains in vacuum and insolvent. Results and discussion
In this Section we discuss our computational results processed as described above. Our discussionwill revolve around [figure][3][]3 displaying spatial distributions of BLA patterns, excess chargeand spin densities for N =
60 (ROHF computations) and N =
100 (UBHandHLYP and UB3LYP)polyynic oligomers C N H − with an extra electron.It is instructive to first recall how self-trapping of an excess charge carrier (an electron or ahole) is described in a single (quasi)particle picture within a standard 1D continuum adiabaticframework. The self-trapped state corresponds to the localized ground-state wave function y ( x ) in the solution of the non-linear Schrödinger equation: − ¯ h m ¶ y ( x ) ¶ x − Z dx ′ V ( x − x ′ ) | y ( x ′ ) | y ( x ) = E y ( x ) , (2)where x is the coordinate along the structure axis, m the intrinsic effective mass of the carrier, and V ( x ) the effective self-interaction mediated by another subsystem. In the case of a short-rangeelectron-phonon mediation, the self-interaction can be taken local: V ( x ) = g d ( x ) leading thento the widely known exact result y ( x ) (cid:181) / cosh ( gmx / h ) for a continuum 1D polaron. Inthe case of the long-range polarization interaction, the effective V ( x ) behaves as 1 / | x | at largedistances, while the short-range behavior depends on specific system details such as, e.g., theactual transverse charge density distribution and the geometry of the dielectric screening. Inthe absence of the self-interaction, ?? would just yield spatially delocalized plane-wave solutions.Among features of this single-particle description is that the same probability density | y ( x ) | de-termines both spatial distributions of charge and spin of the charge carrier. The other one is thatit involves the adiabatic framework, that is, the mediating subsystem is assumed very slow allow-10 n ( A ) or s i r s i q i q i Bond index n Cell index i vac/solvvac.solv.vac.solv.(a1) (b1) (c1)(a2) (b2) (c2)(a3) (b3) (c3)(a4) (b4) (c4)(a5) (b5) (c5) Figure 3: Processed results for charged C N H − oligomers as arranged in columns by the compu-tational method: (a) ROHF, (b) UBHandHLYP, and (c) UB3LYP. Row 1 displays optimized BLApatterns with end effects eliminated for oligomers in vacuum (dashed lines) and in solvent (solid).The notation is different in rows 2–5, where solid lines show results for fully optimized carbonlattice geometries, while dashed lines (when distinguishable) for rigid geometries with uniformdimerization patterns (see the text). These rows display cell-centric excess charge and spin densi-ties for chains in vacuum (rows 2 and 3) and in the solvent environment (rows 4 and 5).11ng to determine electronic states for its given static configuration, which in turn self-consistentlyresponds to the probability | y ( x ) | (“strong-coupling polarons” ). It should be emphasized thatin realizations of the computational methods used, both displacements of the underlying carbonlattice and solvent polarization constitute such slow subsystems.The single-particle description has the drawback of not explicitly including valence band elec-trons, whose reorganization may be important in the accommodation of the excess carrier. Such areorganization can take place even in electron-lattice models without direct electron-electron inter-actions that have been used to describe non-linear localized excitations in CPs. A nice analysis was in fact given of how polarons of a two-band Peierls dielectric model of CPs evolve into single-particle Holstein polarons in the limit of the “frozen valence band” approximation. Polaronic andsolitonic excitations in these models are also derived within self-consistent adiabatic treatment ofthe underlying lattice. In ab initio computations, reorganization of valence band electrons oc-cur as driven by both the interaction with the bond-length modulations and by electron-electronCoulomb interactions. With many electrons determining the resulting spatial densities, excesscharge and spin distributions should not be necessarily the same. One of the famous instancesof this phenomenon is inverse charge-spin relationships for kink-solitons in trans-polyacetylene (these relationships are different for kinks in polyynes ). We will also mention in this regardseparation of charge and spin degrees of freedoms well known in 1D electronic liquids. Stillone could expect that, for non-topological localized polaronic excitations, excess charge and spindensity distributions would closely follow each other.Turning now to results in [figure][3][]3, we note upfront that, similarly to our observations onkink-solitons, spatial extents of the structures developed as a result of electron accommodationare strongly computational method-dependent. The magnitudes d of the uniform dimerization areconsistent with the notion of the increased effective electron-lattice interactions with the increaseof the amount of the HF exchange in the method functionals. We would be more interested now inqualitative aspects of the results as related to the phenomenon of self-localization.The main reason for our studying rigid geometry (RG) cases has been to try to distinguish ef-12ects due to different interactions. As RGs feature spatially uniform BLA backgrounds, one mightexpect that in vacuum the excess electron would be delocalized over the oligomer showing theparticle-in-the-box type behavior for charge and spin densities. On the other hand, if the latticeis allowed to relax, that could cause, as per a conventional adiabatic polaron picture, spatial lo-calization of excess charge and spin. Vacuum results for the densities are shown in Rows 2 and3 of [figure][3][]3. One indeed observes that RG data displays a delocalized behavior for bothcharge and spin when calculated with hybrid-DFT methods. Not so however when calculated inHF. Hartree-Fock data (panels (a2) and (a3)) display localization of excess charge and spin alreadyin RG, and the degree of localization is only very weakly affected by a quite substantial dip inthe optimized BLA pattern (panel (a1)) when the lattice is allowed to relax. We conclude thatself-localization observed in these results is not caused by the lattice deformation but rather byelectron-electron interactions as treated in HF. The deformation of the BLA pattern then resultsas a response to the developed localized density. We easily confirmed this kind of localizationdue to electron-electron interactions alone in simplified calculations once these interactions aretreated in the HF approximation. One cannot help wondering if this qualitative picture may bejust an artifact of the HF treatment. It is not clear if there are slow (low-frequency) electronicexcitations in this system that exhibits a well-known very substantial HF energy gap. An al-ternative to that picture would be that as a result of Coulomb interactions with the fast electronicsubsystem, the excess charge carrier remains delocalized but with its properties renormalized (see,e.g., calculations of corrections to the DFT quasi-particle energies within the framework of theGW-approximation ). Such a renormalized quasi-particle could then interact with the lat-tice displacements to form an electron-lattice polaron. Perhaps a properly realized time-dependentcomputational scheme could clarify the question of the purely electronic response.While yielding a delocalized behavior in RG, hybrid-DFT computations exhibit other issues inthe context of the polaron formation due to carbon lattice displacements in vacuum. Particularly,as has already been noticed, we do not observe electron self-localization in results obtainedwith UB3LYP – the slight changes in densities due to relaxation (panels (c2) and (c3)) and a13at optimized BLA pattern in the middle of oligomer (panel (c1)) indicate that they are finiteoligomer-size effects – at least within the range of oligomer lengths studied. With UBHandHLYPdata, visually one could probably identify the excess charge density (panel (b2)) in the optimizedsystem as localized. There is however no counterpart to this behavior in the spin density (panel(b3)), which rather looks like a spin-density wave (SDW). Also quite unusual is the shape of thedip in the optimized BLA pattern in panel (b1), quite substantial in magnitude but the spatial extentof which is still likely limited by the oligomer ends rather than being self-consistent.Concluding the survey of the vacuum data in rows 2 and 3, none of the methods we used seemsto have produced results that can be qualified with certainty as self-localization of the excess elec-tron charge and spin densities caused by its interaction with the displacements of the underlyingcarbon lattice.Quite substantial effects take place due to the polarization of the surrounding solvent by theexcess electron. They are easily discernible in [figure][3][]3. One immediately notices in row 1that optimized BLA patterns in the solvent acquire a (much) more spatially localized character.Results for the charge and spin densities in the solvent environment are shown in rows 4 and 5.Using B3LYP computations, we have previously demonstrated that the solvation effect alone can lead to self-localization of excess charge densities. RG data in panels (c4) and (c5) clearly showthat this is the case for both charge and spin densities, even though a complete self-consistency isnot completely achieved yet at this oligomer length. The behavior of spin and charge distributionshere closely follow each other. As HF computations yield localized patterns already in RG, onecannot study the solvation effect here separately. It is however evident from panels in column(a) that solvation results in increased spatial localization of excess charge and spin (all HF dataexhibit a noticeably larger degree of spin localization than that of charge). The BHandHLYP datain column (b) also shows a significant degree of localization of charge density due to solvation(b4), appreciably more than what is caused by the lattice deformation in vacuum (b2). Once again,though, the spin response (b5) in these computations for RG is quite surprising: while, perhaps,exhibiting a localized pattern, it is accompanied by what appears to be a large amplitude and spatial14cale SDW-like response. One, in fact, would not easily discern a localized pattern in the rawatomic data shown in [figure][2][]2(b), and different averaging procedures could lead to differentconclusions. The spin density distribution in this case however does acquire a more delineatedlocalized shape, even if on the background of a smaller SDW-like pattern, when the carbon latticeis allowed to relax. The lattice relaxation does not appear to affect the already well-localized excesscharge density in panel (b4). Very small, and apparently in different directions for charge and spin,is the lattice relaxation effect in the HF data ((a4) and (a5)). The only computational methodthat clearly shows a synergistic amplifying localization effect of solvation and lattice relaxationon both spin and charge densities is B3LYP ((c4) and (c5)). One can however say that whensolvation and lattice relaxation act together, they do cause the localization of excess charge andspin densities in BHandHLYP computations as well (as compared to the delocalized behaviorin the vacuum RG environment). Overall, it is evident that, independently of the computationalmethod, solvation always acts as to boost the degree of localization of excess charge density – theresult quite understandable on simple physical grounds. It is hard to say if the peculiarities of thespin responses (and perhaps related BLA patterns) found in BHandHLYP results are caused bysome computational artifacts or may indeed have a physical origin in different behaviors of chargeand spin degrees of freedom due to many-electron interactions. It would be interesting to see ifsimilar results are obtained in long C n + H − oligomers that have a different electronic symmetry than C n H − systems studied in this paper.A truly self-consistent localized solution should be (practically) oligomer-length independent,which could be straightforwardly verified by comparing all spatial patterns (BLA, charge and spindensity distributions) on oligomers of increasing lengths. That is, for instance, the case when wecompare relevant results obtained in the HF computations for N =
40 and N =
60 oligomers. Onthe other hand, vacuum BLA patterns obtained in hybrid-DFT computations appear to continueto evolve even at our current limit of N =
100 oligomers. The difference between results derivedfor oligomers of different lengths is substantially much smaller for computations in the solventenvironment. This is in agreement with the observation evident from [figure][3][]3 that solvation15enerally leads to more spatially localized patterns, which are therefore much weaker affected bychain ends in our longest oligomers.One can also look at the self-consistency issue from the viewpoint of energetics. [table][1][]1lists some energy parameters derived from the computational outputs for oligomers of differentlengths. A well-defined and physically important quantity characterizing accommodation of anexcess electron is the electron affinity E A , corresponding to the difference of total system energiesfor neutral C N H and negatively charged C N H − chains. [table][1][]1 features electron affinities E v A in vacuum and E s A in solvent environments, all obtained for fully optimized systems. Needlessto note that numerical values in the table are strongly method-dependent. One can easily seethat, within each method, the values of E s A change only little when going to longer oligomers, anindication of the convergence to self-consistent solutions. Vacuum values E v A change by largerquantities, which is contributed to by two factors: vacuum structures are farther away from self-consistency, and the unscreened Coulomb field of the excess charge is long-range (long-rangeelectron polarization effects may be reflected in the illustration of [figure][2][]2(a), see also Ref. ).The solvent environment efficiently screens such long-range Coulomb effects. Table 1: Electron affinities and polarization energies in eV method system E v A E s A U U − C H H H H H H U and U − stored in the solvent polarizationfor, respectively, neutral and charged oligomers, as retrieved from GAUSSIAN 03 outputs. Wepreviously analyzed the evolution of these quantities with oligomer lengths and found that U grows linearly with the length while U − first decreases with the length, achieves its minimumwhen the length becomes larger than the self-consistent size of the self-localized charge carrier,and then continues to increase just as U does. By these features, one can also judge from the table16ata that self-consistency has been practically achieved in ROHF and UBHandHLYP computationsbut longer oligomers would need to be employed to observe explicit self-consistency in UB3LYPcomputations.Looking at the trends displayed in results obtained with two hybrid-DFT methods, we cannotexclude that there may be a range of “customized” hybrid functionals with the “right” mixture ofHF and DFT correlations that would yield results featuring self-localization behavior for an excesscharge carrier consistent with the single-particle picture of ?? . One could then observe that in theabsence of the interaction with a slow subsystem (uniform RG in vacuum) the resulting charge andspin densities are delocalized, while self-localization takes place due to the lattice deformation anddue to solvation as separate effects as well as synergistically. In addition, the resulting localizedcharge and spin densities would largely follow each other even if with small variations due to many-electron interactions. Another conjecture also suggested by our data is that the failure to detect theformation of electron-lattice polarons in pure DFT and B3LYP computations may be related to theunscreened long-range Coulomb field of the excess charge in vacuum. Indeed, we clearly detectedthe formation of lattice kink-solitons in such computations, and electron-lattice polarons canbe thought of as bound states of kink-antikink pairs. The screening of long-range Coulombforces by fast polarization of the surrounding medium could energetically stabilize the polaronformation. Fast polarization here would act only as to reduce the magnitude of Coulomb forces.This screening effect is different from nearly static reorganization of a slow solvent considered inthis paper and demonstrated to be a robust promoter of excess charge localization. Exploring theabove conjectures could be an interesting avenue for further developments.
References (1)
Polarons and excitons ; Kuper, C. G., Whitfield, G. D., Eds.; Plenum: New York, 1963.(2) Appel, J. In
Solid State Physics ; Seitz, F., Turnbull, D., Ehrenreich, H., Eds.; Academic: NewYork, 1968; Vol. 21, p 193. 173) Rashba, E. I. In
Excitons. Selected chapters ; Rashba, E. I., Sturge, M. D., Eds.; North Hol-land: Amsterdam, 1987; p 273.(4) Alexandrov, A. S.; Mott, N.
Polarons and bipolarons ; World Scientific: Singapore, 1995.(5) Lu, Y.
Solitons and Polarons in Conducting Polymers ; World Scientific: Singapore, 1988.(6) Heeger, A. J.; Kivelson, S.; Schrieffer, J. R.; Su, W. P.
Rev. Mod. Phys. , , 781.(7) Barford, W. Electronic and Optical Properties of Conjugated Polymers ; Oxford UniversityPress: Oxford, 2005.(8) Basko, D. M.; Conwell, E. M.
Phys. Rev. Lett. , , 098102.(9) Gartstein, Y. N. Phys. Lett. A , , 377.(10) Mayo, M. L.; Gartstein, Y. N. Phys. Rev. B , , 073402.(11) Mayo, M. L.; Gartstein, Y. N. J. Chem. Phys. , , 134705.(12) Ye, A.; Shuai, Z.; Kwon, O.; Brédas, J. L.; Beljonne, D. J. Chem. Phys. , , 5567.(13) Ussery, G. L.; Gartstein, Y. N. J. Chem. Phys. , , 014701.(14) Conwell, E. M.; Basko, D. M. J. Phys. Chem. B , , 23603.(15) Gartstein, Y. N.; Ussery, G. L. Phys. Lett. A , , 5909.(16) Mayo, M. L.; Gartstein, Y. N. J. Chem. Phys. , , 064503.(17) Meisel, K.; Vocks, H.; Bobbert, P. Phys. Rev. B , , 205206.(18) Martin, R. M. Electronic Structure ; Cambridge University Press: Cambridge, 2004.(19) Moro, G.; Scalmani, G.; Cosentino, U.; Pitea, D.
Synthetic Metals , , 165.(20) Geskin, V. M.; Dkhissi, A.; Brédas, J. L. Int. J. Quant. Chem. , , 350.1821) Zuppiroli, L.; Bieber, A.; Michoud, D.; Galli, G.; Gygi, F.; Bussac, M. N.; André, J. J. Chem.Phys. Lett. , , 7.(22) Yang, S.; Kertesz, M. J. Phys. Chem. A , , 9771.(23) Yang, S.; Kertesz, M. J. Phys. Chem. A , , 146.(24) Louie, S. G. Conceptual Foundations of Materials. A Standard Model for Ground- andExcited-State Properties , Amsterdam, 2006; p 9.(25) Frisch, M. J. et al.
Gaussian 03, Revision D.01 , 2004, Gaussian, Inc., Wallingford, CT, 2004.(26) Leach, A.
Molecular Modelling: Principles and Applications ; Prentice Hall: EnglewoodCliffs, NJ, 2001.(27) Tomasi, J.; Mennucci, B.; Cammi, R.
Chem. Rev. , , 2999.(28) Szafert, S.; Gladysz, J. A. Chem. Rev. , , 4175.(29) Szafert, S.; Gladysz, J. A. Chem. Rev. , , PR1.(30) Horný, L.; Petraco, N. D. K.; Pak, C.; Schaefer, H. F. J. Am. Chem. Soc. , , 5861.(31) Horný, L.; Petraco, N. D. K.; Schaefer, H. F. J. Am. Chem. Soc. , , 14716.(32) La Magna, A.; Deretzis, I.; Privitera, V. Eur. Phys. J. B , , 311.(33) Kertesz, M.; Yang, S. Phys. Chem. Chem. Phys. , , 425.(34) Rice, M. J.; Bishop, A. R.; Campbell, D. K. Phys. Rev. Lett. , , 2136.(35) Rice, M. J.; Phillpot, S. R.; Bishop, A. R.; Campbell, D. K. Phys. Rev. B , , 4139.(36) Mennucci, B.; Tomasi, J. J. Chem. Phys. , , 5151.(37) Cancès, E.; Mennucci, B.; Tomasi, J. J. Chem. Phys. , , 3032.1938) Cossi, M.; Barone, V.; Mennucci, B.; Tomasi, J. Chem. Phys. Lett , , 253.(39) Cossi, M.; Scalmani, G.; Rega, N.; Barone, V. J. Chem. Phys. , , 43.(40) Karpfen, A. J. Phys. C: Solid State Phys. , , 3227.(41) Fan, Q.; Pfeiffer, G. V. Chem. Phys. Lett , , 472.(42) Rashba, E. I. Optika i Spektroskopiya , , 75, 88.(43) Holstein, T. Ann. Phys. , , 325.(44) Campbell, D. K.; Bishop, A. R.; Fesser, K. Phys. Rev. B , , 6862.(45) Giamarchi, T. Quantum Physics in One Dimension ; Clarendon Press: Oxford, 2004.(46) Giuliani, G. F.; Vignale, G.