Excited-State Structure Modifications Due to Molecular Substituents and Exciton Scattering in Conjugated Molecules
Hao Li, Michael J. Catanzaro, Sergei Tretiak, Vladimir Y. Chernyak
EExcited-State Structure Modifications due to MolecularSubstituents and Exciton Scattering in Conjugated Molecules
Hao Li, Michael J. Catanzaro, Sergei Tretiak,
1, 3, ∗ and Vladimir Y. Chernyak † Theoretical Division, Center for Nonlinear Studies,Los Alamos National Laboratory, Los Alamos, NM 87545 Department of Mathematics, Wayne State University, 656 W. Kirby, Detroit, MI 48202 Center for Integrated Nanotechnologies,Los Alamos National Laboratory, Los Alamos, NM 87545 Department of Chemistry, Wayne State University,5101 Cass Avenue, Detroit, MI 48202 (Dated: January 28, 2014)
Abstract
Attachment of chemical substituents (such as polar moieties) constitutes an efficient and con-venient way to modify physical and chemical properties of conjugated polymers and oligomers.Associated modifications in the molecular electronic states can be comprehensively described byexamining scattering of excitons in the polymer’s backbone at the scattering center representingthe chemical substituent. Here, we implement effective tight-binding models as a tool to examinethe analytical properties of the exciton scattering matrices in semi-infinite polymer chains withsubstitutions. We demonstrate that chemical interactions between the substitution and attachedpolymer is adequately described by the analytical properties of the scattering matrices. In partic-ular, resonant and bound electronic excitations are expressed via the positions of zeros and polesof the scattering amplitude, analytically continued to complex values of exciton quasimomenta.We exemplify the formulated concepts by analyzing excited states in conjugated phenylacetylenessubstituted by perylene. a r X i v : . [ phy s i c s . c h e m - ph ] D ec È È ÈÈ È È éé éé éé éééé ‰‰ ‰ é é é í í í - - H z L Im H z L Exciton Band Ω H z L(cid:144) resonant statesbound states bound state Keywords: conjugation, ES approach, electronic excitation, tight-binding model, pery-lene, bound state, resonant state 2he concept of scattering has proven to play an important role in the description of avariety of phenomena in quantum mechanics, condensed matter and quantum field theory,functional analysis, and chemistry. Scattering matrices/operators contain detailed informa-tion on fundamental interactions that control the system dynamics, and are available fromexperimental measurements. In high-energy physics the differential cross-sections, directlyrelated to scattering matrices, are measured and further interpreted using quantum field the-ory with the ultimate goal of revealing the fundamental interactions. In condensed mattertheory, the scattering matrices provide a useful tool of re-summing the short range interac-tion effects, e.g., in the superfluidity theory in the low-density case, where a perturbationtheory can be formulated in terms of the particle-particle scattering matrix that containsall necessary information on the particle-particle interactions [1–3]. The celebrated Fermi-liquid theory allows interpretation of complex dynamics of a strongly interacting systemto be interpreted in terms of quasi-particle spectra and scattering matrices [4, 5]. Chem-ical reactions in gas phase can be conveniently formulated and interpreted as scatteringprocesses with the asymptotic states represented by the reactants and products [6, 7]. Infunctional analysis, scattering theory can be viewed as a tool of analyzing continuous spectraof unbounded (often differential) operators, in particular the projection measures involvedin spectral decompositions [8].Our recent work [9–16] has demonstrated that the quasiparticle picture, coined ExcitonScattering (ES) approach, provides a simple and clear insight into the excited-state electronicstructure in complex conjugated macro-molecules [17–21]. It allows excited electronic statesto be studied in terms of the exciton spectra in infinite polymers and scattering matrices tobe associated with molecular vertices, i.e., termini, joints and branching centers [22–28]. Theexciton scattering properties of molecular vertices can be further described by tight-bindingor equivalently lattice models [29]. This extends the exciton scattering concept to the caseof imperfect molecular geometries aimed at deriving the exciton-phonon Hamiltonian, thusmapping the problem of incoherent energy transfer in branched conjugated structures ontoa much simpler (although still complex) counterpart of incoherent motion of Frenkel-typeexcitons [30, 31]. The scattering framework in the conjugated molecular systems is differentfrom typical cases usually studied in quantum mechanics, due to discrete, rather than con-tinuous translational symmetry of the asymptotic states. In particular, integer topologicalinvariants, namely winding numbers/topological charges, can be associated with the scat-3ering centers [32]. It is well established in quantum mechanics that analytical propertiesof scattering matrices, more specifically their analytic continuations, provide detailed andimportant information on the underlying potentials, including bound and metastable states.Therefore, analytical properties of exciton scattering on molecular vertices are expectedto provide adequate information of how electronic properties of supramolecular conjugatedstructure are affected by local chemical substituents.In this manuscript, we apply tight-binding models as a tool to study analytical proper-ties of scattering matrices, providing insights into the excited-state electronic structure ofmolecular substituents commonly present in conjugated systems. As an example, Figure 1cdisplays the reflection phase of perylene attached to a semi-infinite phenylacetylene (PA)chain (see Fig. 1a), retrieved from quantum chemistry calculations [15]. Compared to thatin an unsubstituted chain ( φ H ), the plot shows highly non-trivial dependence of the phase onthe exciton quasimomentum k , including 4 resonant features, that are not easy to interpret.Here we show how these features in the reflection phase can be directly ascribed to resonantand bound excited states brought by the substituent.We start with introducing the simplest nearest-neighbor hopping lattice model, wherethe linear segments of a branched conjugated structure are represented by linear chains(graphs) with the same on-site energy ¯Ω and the same hopping constant ¯ J between thenearest neighbors. A molecular vertex is represented by a complete graph, i.e., by a setof fully interconnected sites, with arbitrary on-site energies and hopping constants betweenany pair of sites allowed. The chemical connection between a molecular vertex and theattached linear segment is thus described by introducing the hopping constants betweenthe first lattice site (whose on-site energy is also modified) of the linear segment and anylattice site that belongs to the vertex (e.g., see Fig. 1b). Here we consider the case ofchemical substitution on molecular terminus only (e.g., perylene in Fig. 1a); the generalcase of an arbitrary degree molecular vertex will be analyzed elsewhere. Since the latticesites representing a vertex form a basis set for the vertex tight-binding Hamiltonian, withoutloss of generality we can assume the latter to be diagonal. Therefore, a molecular terminus isdescribed by a set { ω α | α = 1 , . . . , n } of the on-site energies, a vector J = ( J α | α = 1 , . . . , n )of the hopping constants between the vertex sites and the first linear segment site, and themodified on-site energy Ω of the latter, as shown in Fig. 1b. Measuring the energy in theunit of ¯ J and choosing the zero energy level at the middle of the exciton band, without loss4f generality we can set ¯Ω = 0 and ¯ J = 1.Within the described lattice model, the exciton wavefunction on a semi-infinite chainis given by the sets Ψ = (Ψ α | α = 1 , . . . , n ) and ψ = ( ψ j | j = 0 , , . . . ) of its values onthe terminus and the chain respectively. Introducing the multiplicative variable z = e ik that describes the quasimomentum k , we represent the wavefunction on the chain as asuperposition of incoming and outgoing waves ψ j = z − j + r ( z ) z j , j = 0 , , . . . (1)with r ( z ) being the quasimomentum-dependent reflection coefficient at j = 0. The eigen-mode equation adopts a form ψ j − + ψ j +1 = ωψ j , j = 1 , , . . .ω α Ψ α + J α ψ = ω Ψ α , α = 1 , . . . , n (cid:88) α J α Ψ α + ψ = ( ω − Ω ) ψ . (2)Upon substituting Eq. (1) into Eq. (2) we can easily solve the system of linear equationsand obtain the following expressions for the exciton spectrum ω ( z ) = z + z − (3)and reflection coefficient r ( z ) = − z − Ω − F ( ω ) z − − Ω − F ( ω ) , F ( ω ) = (cid:88) α J α ω − ω α , (4)both being represented by meromorphic functions of z in the complex plane. This impliesthat r ( z ) can be also interpreted as a meromorphic function on the projective space C P or aholomorphic function r : C P → C P . These two interpretations originate from viewing C P as a compact complex analytical manifold (of complex dimension 1) topologically equivalentto a sphere C P ∼ = S with the complex structure induced from complex plane C viastereographic projection C (cid:44) → C P , i.e., C P is obtained from C by adding the infinitepoint.The reflection coefficient satisfies obvious relations r ( z − ) = ( r ( z )) − , r ( z ∗ ) = ( r ( z )) ∗ (5)that reflect unitarity of quantum mechanics combined with time-reversal symmetry.5 direct inspection of Eq. (4) shows that the reflection coefficient can be represented in aform r ( z ) = zP n +1 ( z ) /Q n +1 ( z ), with P n +1 and Q n +1 being polynomials of degree (2 n +1),which means that r ( z ), as a meromorphic function in C P , has 2( n + 1) zeros and 2( n + 1)poles, or equivalently that the map r : C P → C P has degree n A ( r ) = 2( n + 1), hereafterreferred to as the analytical index. Due to the symmetry relations [Eq. (5)], the roots of Q n +1 are inverse to the roots of P n +1 with the roots of each polynomial being either real orcoming in mutually complex conjugated pairs. This implies that the positions of the roots,of say P n +1 , fix r ( z ) up to a multiplicative factor, the latter being fixed by the condition r (1) = − r ( −
1) = −
1] that follows from Eq. (4) [note that generally Eq. (5)also implies r (1) = ±
1, as well as r ( −
1) = ± r ( z ) can be represented in a form r ( z ) = z m (cid:89) j =1 w j z − z − w j n +1 (cid:89) j = m +1 z − z j z j z − , (6)where { z j } and { w j } are the zeros of the polynomials P n +1 and Q n +1 , respectively, locatedinside the circle | z | = 1, hereafter referred to as the circle.We now tie the analytic properties described above together with the underlying chemistryin terms of bound states. According to quantum-mechanical scattering theory and Eq. (1),the bound states in a semi-infinite chain correspond to the poles of r ( z ) located inside thecircle, or equivalently the zeros located outside, so that the energies of the bound states aregiven by ω ( w j ), which implies that w j should be real for all j = 1 , . . . , m . In our earlier work(unpublished result), we have introduced the topological index (winding number) n T ( r ),associated with a vertex. A direct calculation yields n T = (cid:73) | z | =1 dz πi r − drdz = 2 n + 2 − m. (7)Introducing the molecular vertex analytical and topological charges Q A = ( n A − / Q T = ( n T − /
2, respectively, we conclude that in a finite-length linear molecule with L repeat units, represented by L sites within the nearest neighbor lattice model, the numberof exciton states is given by N = L + Q (1)A + Q (2)A , (8)whereas the number of bound states associated with a vertex a is given by m ( a ) = Q ( a )A − Q ( a )T .In a molecule with the lengths of linear segments considerably exceeding the localizationlengths of the bound states, we have the number of exciton states inside the exciton band N = L + Q (1)T + Q (2)T . (9)6his is true since the excitons with energies outside the exciton band are excellently ap-proximated by just the bound states. Note that Eq. (9) reflects one of the statements ofthe index theorem for the case of linear molecules presented in our earlier work [32], andEq. (7) can be interpreted as a relation between the analytical and topological propertiesof a molecular vertex: the topological index is formed from (2( n + 1) − m ) positive and m negative contributions associated with the zeros and the poles, respectively, of r ( z ), locatedinside the circle.The analytical index theorem [Eq. (8)] can be also derived in more general terms bynoting that the ES equations for a linear molecule can be written in a form˜Γ( z ) − , ˜Γ( z ) = r (1) ( z ) r (2) ( z ) z L − , (10)so that the number of its solutions is given by the analytical index n A (˜Γ −
1) (unpublishedresult). We further observe n A (˜Γ −
1) = n A (˜Γ) = n (1)A + n (2)A + 2( L − , (11)and note that there are two unphysical solutions with z = ±
1, whereas each exciton state isrepresented by a pair of symmetry related solutions with mutually inverse values of z . Thisresults in N = ( n A (˜Γ − / −
1, which reproduces Eq. (8).It is worth mentioning that shifting the reference point of reflection by ∆ j results ina factor of z j in r ( z ) [32], and changes of both n A and n T by 2∆ j , which followed bythe change of Q A and Q T by ∆ j ; the latter is compensated in Eqs. (8) and (9) by thecorresponding change of L by − ∆ j . For convenience, we define Q A and Q T using Q A / T =( n A / T − − j ) /
2, so that the analytical and topological charges (integers) are the intrinsicproperties of molecular vertices and independent of the reflection point, and the length L should not change with respect to the choice of the reflection position.In summary, within a nearest-neighbor hopping lattice model a molecular terminus isdescribed by the reflection coefficient r ( z ), characterized by its topological charge Q T andanalytical charge Q A ≥ Q T , and represented by a meromorphic function on C P in a formgiven by Eq. (6). It is fully determined by the positions of its ( Q A − Q T ) poles inside the circlethat all lie on the real axis, and ( Q A + Q T +2) zeros, referred to as resonances, inside the circlewhich either belong to the real axis or come in mutually complex conjugated pairs. Statedequivalently, using a term “tight-binding model with nearest neighbor hopping” is equiva-lent to approximating the scattering coefficient r ( z ) with a meromorphic function, defined in7 P , since there is a one-to-one correspondence between such meromorphic functions (withthe certain simple properties, implied by fundamental quantum mechanical symmetries, andexplicitly described earlier in the text) and the sets of parameters of tight-binding models ofa certain class, namely, referred to as nearest-neighbor hopping. Therefore, hereafter we willnot make a distinction between the terms “tight-binding model with nearest neighbor hop-ping” and “scattering coefficients represented by meromorphic functions in C P ”. Havingsaid that, we would like to note that dealing with a tight-binding model as a tool has certainadvantages by providing a simple and intuitive physical insight, as well as opening the wayto efficiently account for exciton-phonon interactions, as proposed in our earlier work [29].An interesting situation occurs when a pair ( z j , z ∗ j ) of resonances lies close to the cir-cle. Introducing a natural notation z j = (1 − δ j ) e ik j with δ j (cid:28)
1, we can represent thecontribution of the above pair of resonances to the reflection coefficient [Eq. (4)] in a form r j ( z ) ≈ z − (1 − δ j ) e ik j z − (1 + δ j ) e ik j z − (1 − δ j ) e − ik j z − (1 + δ j ) e − ik j , (12)where we have applied the approximations (1 − δ j ) − ≈ δ j and (1 − δ j ) ≈ k j ∼ δ j around k = ± k j we have r j ≈
1, i.e.,it does not contribute to the reflection coefficient, whereas in the above regions r ( z ) showsresonant behavior, where the scattering phase φ = − i ln r acquires a contribution of 2 π overa narrow region ∆ k j ∼ δ j . In a simplified scenario where Ω = 0, n = 1 and m = 0, onecan find that δ ≈ J /
2, which indicates that the sharp resonant feature is attributed to theweak coupling between the terminal site and the chain. These resonances, hereafter referredto as phase kinks, correspond to the resonant states, i.e., excited states on the substituent,weakly coupled to the exciton band in the polymer chain.In the aforementioned tight-binding model, the m lattice sites, representing the graph ofthe terminus, related to the bound states simultaneously result in m poles of r ( z ) outsideof the circle, i.e., m zeros located inside the circle. Taking into account a “trivial” zero of r ( z ) at z = 0 and a zero on real axis close to z = 0, which is attributed to the small valueof Ω , there are 2( n − m ) zeros close to and inside of the circle come in mutually complexconjugated pairs which are associated to the ( n − m ) resonant states within the excitonband. Stated differently, the ( Q A − Q T ) bound states are directly associated with the polesinside the circle on the real axis, whereas the resonant states correspond to the Q T pairs of8omplex conjugated zeros of r ( z ) located close to the circle.To illustrate the above formalism, we calculate electronic excitations in linear pheny-lacetylene molecules with and without perylene substituent (Fig. 1a) using standard quan-tum chemical (QC) methodology. The length of the PA molecules varies from 5 to 35 repeatunits with increments of 5 repeat units. As we have shown before [10, 16], any approachfor excited-state computation, that can adequately describe exciton properties (includingthe binding energy and the exciton size), can be used as a reference QC method in theES approach. Here the ground state geometries have been optimized at the semiempiricalAustin Model 1 (AM1) level [33] using the Gaussian09 package [34]. We then applied thecollective electronic oscillator (CEO) method [35–37], which is based on the time-dependentHartree-Fock (TDHF) theory combined with the semiempirical INDO/S (intermediate ne-glect of differential overlap parameterized for spectroscopy) Hamiltonian [38], to computethe excitation energies, transition dipoles, and transition density matrices. The lowest ex-citon band has been singled out by inspecting the structures of transition density matricesin real-space [12]. The exciton spectrum k ( ω ) and the reflection phases of unmodified andperylene-substituted termini have been extracted within the ES approach [15]. As shownin our previous studies, physically similar results can be expected using other model QCtechniques for excited state calculations such as time-dependent density functional theory(TDDFT) [16]. The ES phase of perylene and the excited-state data provide sufficient inputfor a corresponding tight-binding model [29]: the nearest-neighbor tight-binding parametersin an infinite linear chain, ¯Ω and ¯ J , can be found from the exciton spectrum k ( ω ) andEq. (3).We are now in a position to describe the excited-state chemical properties of peryleneattached to a PA chain in terms of analytical properties of the reflection phase obtainedwith a chosen model of QC. The molecular vertex that represents perylene (Fig. 1a) has thecharges Q A = 7 and Q T = 4, which yields Q A − Q T = 3 bound states and Q T = 4 resonantstates. Using the model depicted in Fig. 1b, we parameterized the tight-binding graph thatrepresents the perylene terminus by fitting its reflection phase. Specifically, in the nearest-neighbor model of the linear chain, extracted and tabulated quantities include ¯Ω = 3 .
492 eV,¯ J = − .
288 eV, on-site energy Ω = 3 .
488 eV of the first site in the segment, and parametersof the terminus given in Table. I. The actual scattering phase, approximated by the abovelattice model, can be viewed as the addition of sharp resonant features (Eq. (12)) on top of9he simplified model, in which the hopping constants J α ( α = 2 , ,
6) have been set to zero,due to the weak couplings between the corresponding states of the terminus and the chain(Fig. 1b).The structure of resonances of r ( z ) is schematically shown in Fig. 2. We obtain 3 poles onthe real axis corresponding to the 3 bound states, as well as the 3 zeros (in blue) associatedwith the poles; a “trivial” zero at z = 0 and another in the vicinity (both in green); and Q T = 4 pairs of resonances (in red) close to the circle that represent the 4 resonant states,shown as scattering phase kinks in Fig. 1c. As a result, the sharp resonances have beenaccurately reproduced in terms of k (Fig. 1c). In addition, energies of the bound states ina semi-infinite chain located outside of the exciton band, can be easily found using Eq. (3).Although the nearest-neighbor lattice models are less accurate in terms of exciton spectrum[29], the bound state energies have been qualitatively well reproduced (Table. II).Thus far, we have described the electronic excitations in perylene-attached PA chain(independent of the length of the polymer) by characterizing the analytical and topologicalproperties of the corresponding exciton scattering matrix. The aforementioned tight-bindingmodel, which relies upon the resonances between the states of perylene and the excitonband of the semi-infinite PA chain, is constructed by inspecting electronic excited states inperylene-substituted PA molecules. Stated differently, the tight-binding graph representingthe perylene substitution (Fig. 1b) is determined by the numbers of the bound and resonantexcitations that we observed in the substituted linear molecules, without really performingQC analysis on the molecule of perylene. Although the above tight-binding model excellentlycharacterizes electronic excitations in such substituted molecules, the QC calculations foundonly 6 excitations (not 7) in perylene. Of these, only 5 of 6 states (states 1, 2, 3, 4, and7 in Fig. 3) are considered to be relevant to the resonances in perylene-substituted PAmolecules (hereafter referred to as perylene-P L , L being the length of the attached linearsegment). This observation indicates that the effect of perylene substitution on the electronicexcitations can not be exactly interpreted as just resonances between states of perylene andof the chain; the analytical structure of the scattering coefficients can have substantialdifferences compared to the one predicted based on just resonances. Such modification of r ( z ) reflects the fact that perylene is “chemically” bonded rather than merely “physically”resonated with the linear segment.Indeed, by careful examination of the transition density matrices of bound and resonant10tates, it is found that all these excitations continuously extend into the first repeat unit ofthe attached PA segment. In particular, states 5 and 6 are highly localized on the first repeatunit of the chain rather than on the perylene (see Fig. 3), and do not match any excitation inthe isolated perylene. Comparing excitation structures between resonant states in perylene-PA molecules and phenylethynyl perylene (denoted by perylene- P P P r ( z ), which can be conveniently analyzed using tight-binding (lattice)models, provides complete characterization of effects of terminal chemical substitutions inconjugated molecules on electronic excited-state structures. This approach can be applied toany chemical substitution in conjugated polymers, which can be treated as molecular vertexand, thus, characterized by exciton scattering matrix within the ES approach. By inspect-ing the structure of excited states, an effective tight-binding model has been formulated toincorporate the coupling between molecular substituent and the attached molecule. Thetight-binding graph representing the substituent provides an analytical expression for theassociated scattering matrix. By comparing the scattering matrix of the tight-binding modelwith the ES counterpart, we parameterize the tight-binding model and completely obtain theanalytical property of the exciton scattering. Conducting the analytic continuation of thescattering amplitude to complex values of the exciton quasimomentum, the modifications of11lectronic excitations are distinguished in terms of just positions of poles and zeros of thescattering amplitude that provide sufficient information on appearing bound and resonantstates, respectively. Furthermore, delicate description of the interaction between chemicalsubstituent and conjugated polymer have been attained by inspecting detailed excited-stateelectronic structures. As a consequence, corresponding tight-binding model can be built ina way not only of “phenomenological” exactness but also of “chemical” consistency.Starting from quantum-chemical data, processed by the ES method as a bridge, a chemicalsubstitution on a conjugated polymer, in terms of electronic excitations, can be straightfor-wardly represented by properly constructed tight-binding model. In return, the analyticalproperty of exciton scattering, that adequately describes how electronic excitations in con-jugated molecules are affected by the chemical substitution, can be effectively characterized.In other words, the chemical substitution effect on electronic states is fully determined by theexcited-state properties of the substituent and the polymer, as well as their couplings, whichhave been characterized as tight-binding parameters and can provide quick and intuitiveguidance for applying chemical substitutions in molecular design of organic semiconductorswith desired optoelectronic properties. The described interactions between the substituentand the polymer chain play an important role in dynamical processes involving exciton-phonon couplings, e.g., incoherent energy transfer and charge transportation. Taking intoaccount the simplicity of tight-binding models and their nature resting on the resonancebetween the substituent and the chain, the ES analysis and tight-binding representationscan be feasibly applied to photoinduced dynamics in conjugated macromolecular systems.This material is based upon work supported by the National Science Foundation underGrant No. CHE- 1111350. We acknowledge support of the U.S. Department of Energythrough the Los Alamos National Laboratory (LANL) LDRD Program. LANL is operatedby Los Alamos National Security, LLC, for the National Nuclear Security Administrationof the U.S. Department of Energy under contract DE-AC52-06NA25396. We acknowledgesupport of Center for Integrated Nanotechnology (CINT) and Center for Nonlinear Studies(CNLS) at LANL. ∗ [email protected] † [email protected]
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312 0.028 0.143quantum chemistry − .
263 0.035 0.175TABLE II. Bound state energies (in eV) relative to nearby exciton band edge predicted by thenearest-neighbor tight-binding model and from QC computations. Negative value indicates thestate below the exciton band, whereas positive values correspond to states above the band. resonance No. 1 2 3 4 5 6 7perylene-P25perylene N/Aperylene inperylene-P25perylene-P1perylene-P1inperylene-P25
FIG. 3. Electronic excitations, related to bound and resonant states, given by the contour plotsof the transition density matrices from the ground state to excited states of perylene-substitutedlinear PA molecule with 25 repeat units (denoted by perylene-P25), perylene, and phenylethynylperylene (perylene-P1). The axis labels represent indices of carbon atoms starting from perylene(1 to 20) and along the polymer chain. The inset of each plot shows the electronic mode number,the excitation energy Ω and the oscillator strength f ..