Chemical Physics 481 (2016) 124–132

Contents lists available at ScienceDirect

Chemical Physics

journal homepage: www.elsevier .com/locate /chemphys

Exciton scattering approach for optical spectra calculations in branchedconjugated macromolecules

http://dx.doi.org/10.1016/j.chemphys.2016.08.0330301-0104/� 2016 Elsevier B.V. All rights reserved.

⇑ Corresponding author at: Department of Chemistry, Wayne State University,5101 Cass Avenue, Detroit, MI 48202, United States.

E-mail addresses: serg@lanl.gov (S. Tretiak), chernyak@chem.wayne.edu(V.Y. Chernyak).

Hao Li a, Chao Wub, Sergey V. Malinin c, Sergei Tretiak d,e,⇑, Vladimir Y. Chernyak c

aDepartment of Chemistry, University of Houston, Houston, TX 77204, United Statesb Electronic Structure Lab, Center of Microscopic Theory and Simulation, Frontier Institute of Science and Technology, Xian Jiaotong University, Xian 710054, ChinacDepartment of Chemistry, Wayne State University, 5101 Cass Avenue, Detroit, MI 48202, United Statesd Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, NM 87545, United StateseCenter for Integrated Nanotechnologies, Los Alamos National Laboratory, Los Alamos, NM 87545, United States

a r t i c l e i n f o

Article history:Available online 30 September 2016

Keywords:ExcitonScatteringConjugationDendrimersElectronic excitationsExcited statesBranched structuresES approachScattering phaseTransition dipoles

a b s t r a c t

The exciton scattering (ES) technique is a multiscale approach based on the concept of a particle in a boxand developed for efficient calculations of excited-state electronic structure and optical spectra in low-dimensional conjugated macromolecules. Within the ES method, electronic excitations in molecularstructure are attributed to standing waves representing quantum quasi-particles (excitons), which resideon the graph whose edges and nodes stand for the molecular linear segments and vertices, respectively.Exciton propagation on the linear segments is characterized by the exciton dispersion, whereas excitonscattering at the branching centers is determined by the energy-dependent scattering matrices. Usingthese ES energetic parameters, the excitation energies are then found by solving a set of generalized ‘‘par-ticle in a box” problems on the graph that represents the molecule. Similarly, unique energy-dependentES dipolar parameters permit calculations of the corresponding oscillator strengths, thus, completingoptical spectra modeling. Both the energetic and dipolar parameters can be extracted from quantum-chemical computations in small molecular fragments and tabulated in the ES library for furtherapplications. Subsequently, spectroscopic modeling for any macrostructure within a consideredmolecular family could be performed with negligible numerical effort. We demonstrate the ES methodapplication to molecular families of branched conjugated phenylacetylenes and ladder poly-para-phenylenes, as well as structures with electron donor and acceptor chemical substituents. Time-dependent density functional theory (TD-DFT) is used as a reference model for electronic structure.The ES calculations accurately reproduce the optical spectra compared to the reference quantumchemistry results, and make possible to predict spectra of complex macromolecules, where conventionalelectronic structure calculations are unfeasible.

� 2016 Elsevier B.V. All rights reserved.

1. Introduction: the ES concept

Organic conjugated materials have attracted considerableattention for their technological advantages, interesting photo-physical and photochemical properties, low cost, light weight, flex-ibility, and convenience of solution-based processing techniques[1–8]. The remarkable electronic and optical properties of conju-gated molecules are attributed to delocalized p�electron systemsthat reside on the quasi-one-dimensional molecular backbones,which can be constructed and tuned synthetically [9–13]. There-

fore, synthetic design of conjugated molecular structures targetingspecific optoelectronic properties, requires theoretical understand-ing and numerical modeling of their electronic excitations. How-ever, the excited state electronic structure of conjugatedmolecules is complex due to their low dimensionality, strong elec-tronic correlations and significant electron–phonon coupling [14–17]. Only a few quantum-chemical methodologies that are capableof adequate accounting for electron exchange and correlations, canbe applied to serve the above purpose [18–20]. Alas, even numer-ically efficient methods that satisfy the above requirements (suchas time-dependent density functional theory, TD-DFT [21]), areprohibitively expensive for the excited states computations oflarge systems, due to the unfavorable scaling of the numerical cost[OðN2Þ � OðN5Þ;N with being the number of electron orbitals] [22–24]. On the other hand, even large conjugated structures consist of

H. Li et al. / Chemical Physics 481 (2016) 124–132 125

a limited number of elementary molecular blocks [2,11]. Thisopens an opportunity to apply a multiscale concept for efficientdescription of electronic excitations in a macromolecule by firstcharacterizing the excited states of each building block followedby expressing the electronic properties of the entire structure interms of the building block data. The exciton scattering (ES)approach [16,25] accomplishes this goal by introducing a rigorousalgorithm based on quasi-particle description of electronicexcitations.

Indeed, previous studies in the 1990s interpreted excited elec-tronic states in quasi-one-dimensional organic semiconductors asbound electron–hole pairs (excitons) with an emphasis on the rel-ative motion of electrons and holes [26,27]. Consequently, a simpleand intuitive picture of optical responses emerged from a real-space analysis of the single-electron transition density matricesdefined as

ðnmÞnm ¼ hmjcyncmjgi: ð1ÞHere, jgi (jmi) denotes the ground (excited) state many-electron

wavefunction, cyn (cm) is the Fermi creation (annihilation) operatorof the nth (mth) atomic orbital. Subsequent investigations on elec-tron energy loss spectroscopy (EELS) in linear conjugated oligo-mers [28,29] revealed an importance of the ‘‘center-of-mass”exciton motion related to the diagonal direction in the transitiondensity matrix (Eq. (1)), in contrast to the off-diagonal patterncharacterizing the exciton type.

i1

i2

i 3n=1

n=1

n=2

n=3

n=2

meta-

ortho-

⇐⇒

(a)

(i)

(ii) (iii) (iv)

(vii) (viii) (c)

P10(e)

1 20 40 60 801

20

40

60

80 S1, 3.37 eV

1 20 40 60 801

20

40

60

80 S2, 3.57 eV

1 20 40 61

20

40

60

80 S3, 3.82 eV

Fig. 1. Illustration of the ES approach. (a) structure of a conjugated molecule; (b) excitophenylacetylene (PA) molecules; (d) linear PA molecule consists of 10 repeat units; (e) efrom ground state to excited state in the molecule shown in (d).

The concept of the ES method first formulated in Ref. [16] attri-butes a branched conjugated macromolecule (e.g., Fig. 1a) to agraph (Fig. 1b), whose edges and nodes represent the linear seg-ments (composed of the repeat units) and molecular vertices(Fig. 1c), respectively. Exciton motions on such graph form planewaves that scatter at branching centers, joints, and molecular ter-mini. Here both electron and hole move together, and a typical dis-tance between these particles (exciton size) le can be quantified asthe off-diagonal extent of the transition density matrices (see con-tour plots in Fig. 1e). In the simplest case of a perfect infinite linearchain, the exciton quasimomentum k is a well-defined quantumnumber that reflects translational symmetry. Therefore, all elec-tronic excitations can be characterized by the dispersion relationxðkÞ, which connects the exciton frequency (energy) x to itsmomentum. In finite oligomers, electronic states become discrete,and each state features a characteristic standing wave structure inthe diagonal direction of the transition density matrix. For exam-ple, the contour plots of the latter in Fig. 1e correspond to the low-est five excitations in a linear molecule P10 (Fig. 1d). This standingwave pattern is observed in all conjugated macrostructures (e.g.,Fig. 1a), where the exciton size le is small compared to the typicallinear segment length L. Therefore, the proposed quasiparticle rep-resentation, which is asymptotically exact in the long segmentL � le limit, is expected to be adequate. Exciton scattering atbranching centers and molecular termini are further describedusing frequency-dependent n� n scattering matrices CðnÞðxÞ; n

T

T

O

M

αa exp(ikxα)

b exp(-ikxα)

A

B

(b)

(v) (vi)

(ix) (x)

8(d)

0 80 1 20 40 60 801

20

40

60

80 S4, 4.08 eV

1 20 40 60 801

20

40

60

80 S5, 4.34 eV

0.0

0.2

0.4

0.6

0.8

n scattering picture on the quasi-1D graph of the molecule; (c) building blocks ofxciton-scattering patterns given by the contour plots of transition density matrices

126 H. Li et al. / Chemical Physics 481 (2016) 124–132

denoting the vertex degree. These quantities can be viewed as gen-eralized boundary conditions that represent the relevant micro-scopic properties of the building blocks, rather than introducedpurely phenomenologically [30,31]. Matrices CðnÞðxÞ and theexciton dispersion xðkÞ fully determine the exciton motion.Consequently, finding the excited states in a molecule is equivalentto solving a generalized spectral problem for a particle in aquasi-one-dimensional box, which is a 2N � 2N linear systemfor the plane wave amplitudes (two per segment), N being thenumber of linear segments in the molecule under consideration[30,31,25].

All ES parameters described above can be obtained using tradi-tional ‘‘reference” quantum-chemical excited states calculations inrelatively simple molecular fragments associated with the buildingblocks of the original superstructure [30–32,25,33]. Previously,applications of the ES approach has been made to a variety ofmolecular systems. For example, the exciton spectra kðxÞ havebeen extracted for phenylacetylene (PA) oligomers and donor–acceptor molecules [25,31,32,34]. Furthermore, the scatteringmatrices of symmetric double, triple, and quadruple joints havebeen retrieved by utilizing their geometrical symmetries [33]. Tomodel molecular optical spectra, a technique for the transitiondipole calculations within the ES framework has been developed[35,36,34]. It requires the dipolar (optical) ES parameters, transi-tion charge qðxÞ and transition dipole dðxÞ, which can also beretrieved from quantum-chemical computations in molecules(fragments) of moderate size. As reference quantum-chemicalmethodologies, we have been using both semiempirical andab initio techniques. The former applications [32,33,35,34] werebased on the Collective Electronic Oscillator (CEO) method, whichuses the Time-Dependent Hartree–Fock (TDHF) approachcombined with the semiempirical Hamiltonians [27,17]. The latterapplications [25,36] relied on TD-DFT electronic structureapproach [21,37]. Overall, all these studies have shown excellentagreement for the excitation energies and optical spectra bycomparing the results obtained using the ES method with the fullreference quantum-chemical computations. Thus, the ES approachcan be readily applied for a broad variety of macromolecular sys-tems as an extension of the currently available quantum-chemicalmethods for excited state calculations.

In this article we summarize development of the ES theoreticalframework and demonstrate applications of this technique toseveral molecular systems, using modern TD-DFT as a referenceelectronic structure method for excited state calculations in smallmolecular fragments. The manuscript is organized as follows:Section 2 introduces the main ES equations for electronic transitionenergies and dipole moments. Section 3 describes the ES parame-ters extraction techniques illustrated by calculations in representa-tive molecular systems. After tabulating the scattering matrices ofmolecular building blocks, we demonstrate application of the ESmethod to computation of vertical absorption spectra in large con-jugated macromolecules.

2. The ES formalism

Commonly used quantum-chemical calculations of molecularexcited states [37] provide electronic transition energies xm andthe corresponding transition density matrices nm (Eq. (1)). Thetransition dipole moments lm between the ground and excitedstates are further evaluated using the matrix elements of the dipoleoperator l̂ðrÞ in the basis set of atomic orbitals (l̂nm ¼ hnjl̂ðrÞjmi).For example, in the orthogonal basis set lm is given by

lm ¼ Tr l̂nmð Þ: ð2Þ

Both xm (defining positions of resonances) and lm (definingintensities of optical transitions) are necessary to calculate molec-ular optical spectra. Here we formulate a closed set of the ES equa-tions relating xm and lm to the scattering parameters: dispersion

xðkÞ, matrices CðnÞðxÞ, transition charges qðxÞ and transitiondipoles dðxÞ.

A branched conjugated molecule (e.g., Fig. 1a) can be viewed asa graph (Fig. 1b) composed of linear segments consisting of identi-cal repeat units [38–40]. The repeat unit and molecular vertices,including branching centers and termini, are referred to as themolecular building blocks that form the molecular backbones.Fig. 1c shows typical building blocks of the PA molecules. Sincethe number of building block types for a given molecular familyis usually limited, it is possible to characterize each building blockby a tabulated ES parameter library with a limited effort. In an infi-nite, perfectly straight segment, exciton dispersionxðkÞ fully char-acterizes the exciton band, being a property of a repeat unit. In afinite linear chain, exciton motion can be represented by two planewaves that propagate in opposite directions. Their quasimomentahave the same absolute value xðkÞ ¼ xð�kÞ due to the time rever-sal symmetry (in the absence of magnetic fields). Hence, the distri-bution of an electronic excitation is described by the exciton wavefunction consisting of standing waves on each linear segment a

waðxaÞ ¼ aa expðikxaÞ þ ba expð�ikxaÞ; ð3Þwhere xa denotes integer coordinates of repeat units on a segment awith length la. The exciton quasimomentum k resides in a 1D Bril-louin zone 0 6 k 6 2p, with the points k and kþ 2p naturally iden-tified. The standing-wave nature of electronic excitations can bedirectly observed from quantum-chemical computations [16,27].For instance, Fig. 1e displays low-energy electronic excitations fromthe ground state for an example of linear molecule. Here, the con-tour plots of the absolute values of the transition density matricesare prepared according to the index of atoms along the molecularchain. Apparent standing-wave profile can be readily recognizedfrom the diagonal amplitudes, composed from different contribu-tions of molecular repeat units, with respect to the entire lengthof the oligomer. The deviation of the exciton wave function fromthe standing-wave profile is considered as a localized effect at thevertex. We adopt the following normalization condition for the

exciton wave functionsP

aPla

xa¼1waðxaÞw�aðxaÞ ¼ 1.

We further introduce a vector wave function

waðxÞ ¼ raðxÞwaðx;raðxÞÞ; ð4Þwhere raðxÞ is a unit vector in the repeat unit x axis direction, andthe scalar function waðx;raðxÞÞ is the projection of the exciton wavefunction on that axis. In a perfectly straight segment, raðxÞ can bedefined as a sign factor due to raðxÞ ¼ ra. Therefore, the sign ofthe scalar function is altered if the segment orientation is reversed,waðx;�raÞ ¼ �waðx;raÞ. The representation of Eq. (4) is especiallyuseful for complex structures with slightly bent segments (e.g.,loops), where raðxÞ varies slowly as a function of x [30–32,35].

The excitation amplitudes can then be found by solving linearequations, which describe exciton propagation along the linearsegments and exciton scattering at the molecular vertices. Wedenote the outgoing ðþÞ and incoming ð�Þ plane waves on segment

a at vertex b by wðþÞab and w

ð�Þab , respectively. Exciton propagation

along segment a between vertices b and c is described by

wð�Þab ¼ nbaðnca � wðþÞ

ac Þ exp iklað Þ; ð5Þ

where la is the segment a length, nua stands for the unit vector ofthe repeat unit in segment a attached to the vertex u. There aretwo such equations for each segment that correspond to its twoends. At molecular vertex b that connects n linear segments, the

H. Li et al. / Chemical Physics 481 (2016) 124–132 127

exciton scattering is described by a frequency-dependent n� nscattering matrix CbðxÞ, which transforms the incoming to the out-going waves:

wðþÞab ¼ nba

Xb3b

Cb;abðx;nbÞðnbb � wð�Þbb Þ; ð6Þ

nb denoting a set of unit vectors, associated with the branchesattached to vertex b. The off-diagonal elements of Cb that deter-mine the transmission effects at a joint, are orientation-dependentdue to the vector nature of the incoming and outgoing waves. Incontrast, the diagonal elements of Cb related to reflections, areinvariant with respect to the orientation choice.

The ES Eqs. (5) and (6) constitute a frequency-dependent homo-geneous system of linear equations for the plane wave amplitudes

wð�Þab , with the number of equations matching the number of

unknown variables [16,30] that can be referred to as a generalizedspectral problem. Therefore, given the exciton dispersion xðkÞ andthe scattering matrices CðxÞ for all vertices, excitation energiesand exciton wave functions (up to a normalization factor) can beidentified by solving the generalized spectral problem [Eqs. (5)and (6)] [30–32].

To perform the transition dipole calculations we further intro-duce the transition charge and transition dipole parameters ofthe building blocks. An exciton wavefunctionwðxÞ ¼ a expðikxÞ þ b expð�ikxÞ is associated with the transitiondipole distribution and its dual ~wðxÞ ¼ a expðikxÞ � b expð�ikxÞ rep-resents the transition charge distribution. w and ~w are related by

~wðxÞ ¼ 1kdivwðxÞ � 1

kd rðxÞ � w½x;rðxÞð Þ

dxð7Þ

i.e., the ‘‘charge” wavefunction can be viewed as the divergence ofthe ‘‘dipole” counterpart [35].

Far from the scattering centers, the total transition dipole hascontributions from the transition dipoles and charges of the repeatunits, proportional to the amplitudes of the exciton wave functionand its dual counterpart, respectively. This allows to introduce thefrequency-dependent dipole and charge parameters of the repeatunit, that, being weighted with the proper exciton wave function,yield the repeat unit contributions to the total transition dipole.The deviations of transition charges and dipoles from the stand-ing-wave form in the repeat units, located in the close proximityof the scattering centers, can be further ‘‘absorbed” into the vertexcontributions. For any scattering center we can choose the ampli-tudes of either incoming or outgoing waves in the wave functionand its dual to characterize the weights, associated with the vertex,since the outgoing and incoming amplitudes are related to eachother via Eqs. (6) and (7). In the case of incoming waves, the totaltransition dipole can be expressed as the sum of contributions ofall repeat units (the first term) and all vertices (the second term):

l ¼Xa

Xlaxa¼1

qðxÞrðaÞxa~waðxaÞ þ dðxÞwaðxaÞ

� �

þXb

Xa3b

qbaðxÞrb~wð�Þab þ dbaðxÞwð�Þ

ab

� �; ð8Þ

where qðxÞ and dðxÞ (qbaðxÞ and dbaðxÞ) are the frequency-depen-dent transition charge and dipole parameters of the molecularrepeat unit (vertices), respectively. The dipole parameters are rank2 tensors since they provide linear transformations between thetwo vectors, the exciton wave function and the transition dipole.

rðaÞxa and rb denote the positions of the repeat unit xa in segment aand vertex b, respectively. Thus, Eq. (8) provides an expression forthe transition dipole between the ground and excited electronicstate, the latter described by its exciton wavefunction and its dualcounterpart.

3. Application examples

The first step in applying the ES approach to an arbitrary molec-ular structure is to separate the latter into building blocks such aslinear segments and scattering centers (e.g., branching units, joints,termini, kinks, etc.). The definition of building blocks in not uniqueand can be further fine-tuned depending on the accuracy of the ESresults compared to the reference quantum-chemical simulations.The next step is selection of the reference quantum-chemicalapproach (see our discussion in the next sub-section) and perform-ing electronic structure simulations of the subset of the buildingblocks aiming to tabulate the ES parameters as illustrated on thenumerous examples in this section. Formally, this is the mostnumerically costly step, yet it is enormously easier compared tothe complete quantum-chemical supramolecular calculation inthe entire structure. The ES approach itself boils down to solvingEqs. (5) and (6) that results in obtaining transition frequenciesand exciton wavefunctions [25,32,34]. The latter, being substitutedinto Eq. (8) allows the transition dipoles do be computed [35,36].The above procedure is extremely efficient, provided the buildingblocks are already characterized by previously tabulated ES param-eters. The aforementioned step can be repeated as many times asnecessary, for example, to explore excited states in differentmolecular structures composed from the same building blocks.Here we illustrate how these parameters can be extracted fromquantum-chemical computations in molecular fragments of mod-erate size. Absorption spectra of several molecules are further cal-culated using the ES approach and compared to their counterpartsobtained through direct reference quantum-chemicalcomputations.

3.1. Reference electronic structure theory

Any size-consistent quantum-chemical method, capable ofdescribing the bound nature of excitonic states, can be used as areference electronic structure theory for the ES approach [18–20]. Previously, we relied on both semiempirical CEO and ab initioTD-DFT approaches as reference quantum-chemical techniques.Notably, only hybrid DFT kernels or modern long-range correctedfunctionals can adequately describe exciton binding energies[19]. Here, a hybrid functional subjected to Coulomb-attenuatingmethod (CAM-B3LYP) [41,42] and 6-31G basis set were chosenfor all TD-DFT computations, adopted as a reference quantumchemistry approach for excited-state description. The molecularground state geometries have been optimized at the same CAM-B3LYP/6-31G level. All simulations have been performed with theGaussian 09 package [43].

To exemplify the ES modeling, we use molecules based onphenylacetylene (PA) and ladder poly�para�phenylenes (PPP)backbones (see Fig. 2 inset). Excited states in the selected molecu-lar fragments (as detailed below) have been computed with thereference TD-DFT approach, resulting in the vertical excitationenergies, oscillator strengths and transition density matrices. Inthis Account we focus on the lowest energy optically active p-exci-ton band (similar analysis can be applied to any other excitonicband of interest) [30–32]. Calculated electronic states belongingto this band have been singled out by inspecting the correspondingtransition density matrices. Their excitations energies are suffi-cient to compute xðkÞ and CðxÞ scattering sets [25,31,32]. To cal-culate the ES dipole parameters, we recall that, to efficientlyseparate the dipolar contributions of individual building blocks, itis highly desirable to have the basis functions to be bothorthogonal and spatially localized. In contrast, ab initio methodstypically rely on non-orthogonal and frequently over-completeatomic orbital (AO) basis set, and the commonly used Löwdin

Fig. 2. (a) Dispersions of the lowest energy exciton band in phenylacetylene (PA)and ladder poly-para-phenylene (PPP) oligomers (whose linear segments are shownin the insets) derived from the TD-DFT calculations. (b) The reflection phases /T and/X of the unmodified (hydrogen-terminated) and substituted molecular termini.Inset: linear PA molecules substituted by X(X = NH2 and NO2) on one end used toderive the reflection phases.

128 H. Li et al. / Chemical Physics 481 (2016) 124–132

orthogonalization [44] is not adequate due to the loss of the angu-lar momentum symmetries in the original AOs [36]. The NaturalAtomic Orbitals (NAO) constructed from the occupancy-weightedsymmetric orthogonalization procedure preserve atom-like char-acter of angular-symmetry within the molecule [45]. Not surpris-ingly, the NAO transformation is found to be well suited forcalculating the ES dipolar parameters [36] and is readily availablefrom the NBO program [46] built in the major electronic structurecodes such as Gaussian 09 [43]. Here, the transition density matrixnm and dipole matrix l̂ are obtained using the transformationmatrix T from AO to NAO space as

nðNAOÞm ¼ TnðAOÞm Ty; l̂ðNAOÞ ¼ Tl̂ðAOÞTy: ð9ÞConsequently, all dipolar parameters are calculated using the

NAO representation. For example, the transition charge of a build-ing block N is given by qN

m ¼ TrðnNm Þ instead of the ambiguous Mul-liken charge in the AO space.

3.2. ES modeling of simple linear chains

Symmetries possessed by quantum systems simplify the extrac-tion of the ES parameters. Time-reversal symmetry leads to sym-metric exciton dispersion xðkÞ ¼ xð�kÞ and certain symmetry inmatrices CðxÞ. Unitarity, associated with the probability conserva-tion in evolution of quantum systems, is another universal symme-try of the scattering matrices. Finite size linear PA and PPP chainsterminated by neutral (hydrogen atom), donor (–NH2) or acceptor(–NO2) group (see insets in Fig. 2) provide simple illustrativeexamples for the ES theory. Due to the unitarity, a single reflectionphase /TðxÞ parameterizes the scattering matrix CðxÞ ¼ ei/T ðxÞ of amolecular terminus (Fig. 1c(ii)). Therefore, the dispersion xðkÞ andreflection phases /TðxÞ of different termini represent a closed setof the ES parameters for these systems. In linear moleculesterminated by A and B end groups, the wave amplitudes in the

exciton wave function (Eq. (3)) are related by Eq. (6) imposed onthe two molecular termini [25,31]. These two ES equations havenon-trivial solutions only if the following quantization conditionis satisfied:

2kðxÞLþ /AðxÞ þ /BðxÞ ¼ 2pq; ð10Þ

where L is molecular length (the number of repeat units) and q is aninteger that labels the excitation, which is related to the number ofnodes in the exciton wave function (Fig. 1e). /A and /B are thereflection phases of the two termini A and B, respectively. It is worthmentioning that the ES approach does not depend on the choice ofrepeat unit because the choice affects the segment length L thencethe quasimomentum k in Eq. (10) in a consistent way so that thesolutions of the ES equations remain the same.

In the simplest case of linear molecules without chemical sub-stituents, both termini are identical /A ¼ /B ¼ /T ;/T being thereflection phase of the neutral -H terminus. Therefore, Eq. (10) issimplified as kLþ /T ¼ pq, where two functions kðxÞ and /TðxÞshould be evaluated simultaneously. The range of reflection phases/TðxÞ is determined by the definition of q, since /TðxÞ is a periodicfunction of 2p. Therefore, to have p < /T < 2p (considering thehard wall reflection phase /T ¼ p), the integer q is related to thenumber of nodes in the exciton wave function, given by q� 2[30,31]. Accordingly, q can be determined from the transition den-sity matrix (see Fig. 1e). The oligomers suitable for the ES parame-ter extraction should be longer than the exciton size being 2–3repeat units for phenylacetylene excitons (Fig. 1e). Indeed, quan-tum chemistry calculations of 10 linear PA and PPP molecules(Fig. 2a) of different lengths (10–25 repeat units) proved to be suf-ficient to accurately extract kðxÞ and /TðxÞ simultaneously.Namely, using excitation energies and respective transition densitymatrices, the initial values of kðxÞ and /TðxÞ have been derivedfrom two- and four- point numerical extrapolations [31] followedby a piecewise polynomial least-square fit. This procedure pro-duces smooth functions kðxÞ and /TðxÞ tabulated for their futureuse (see Fig. 2).

Substitutions of conjugated oligomers with polar moieties per-turb the p-electron system and possibly break the electron–holesymmetry, thus representing a useful synthetic approach to tunemolecular electronic and optical properties. Here the effect ofchemical modifications on the optical properties appears via cor-rection of the corresponding ES parameters. Linear oligomers sub-stituted with donor (–NH2) or acceptor (–NO2) group on one side(see Fig. 2b inset) are the simplest cases allowing to extract thereflection phases /XðxÞ for the termini with substituents (i.e.,/A ¼ /T and /B ¼ /X in Eq. (10)). Using previously retrieved kðxÞand /TðxÞ;/XðxÞ can be readily tabulated from the excitationenergies in such molecules. Different reflection phases of –H,–NH2 and –NO2 termini shown in Fig. 2b signify changes in thecorresponding electronic spectra due to substituents [34].

To extract the ES dipole parameters in linear molecules, we firstcalculate the exciton wave functions and their duals from Eqs. (5)–(7) using the tabulated ES energy parameters. The transitioncharges QðxÞ and dipoles lðxÞ of all repeat units can be calculatedfrom the transition density matrices in the NAO basis (see Sec-tion 3.1). The charge parameter qðxÞ of a repeat unit can be foundby comparing the transition charge to the amplitude of the dualwave function QðxÞ=~wðxÞ. Since the transition dipoles of the lowestband excitons are directed along the chain, the dipole tensor of arepeat unit is a scalar dðxÞ determined by the ratio lðxÞ=wðxÞ.The ratios QðxÞ=~wðxÞ and lðxÞ=wðxÞ deviate from the constantswithin several repeat units from the termini, hence the lattershould be excluded from calculations [35]. Obtained using theabove approach functions qðxÞ and dðxÞ (see Fig. 3a) fully charac-terize the transition dipoles of the PA backbone.

Fig. 3. (a) The transition charge qðxÞ and dipole dðxÞ parameters of PA repeat unit.(b) The transition charge parameters qðXÞðxÞ of donor(–NH2)/acceptor(–NO2)substituted and neutral (-H) termini. (c) Same as (b) but the transition dipoleparameters dðXÞðxÞ.

H2N NO2n

0.01

0.050.10

0.501.00

5.0010.00

F

TD DFTn 6n 11

3.5 4.0 4.5

0.01

0.050.10

0.501.00

5.0010.00

Ω eV

F

ES Approachn 6n 11

Fig. 4. Comparison of the optical spectra obtained from quantum-chemical (TDDFT)calculation (top panel) and the ES approach (bottom panel) for molecules shown inthe inset.

H. Li et al. / Chemical Physics 481 (2016) 124–132 129

The transition charge and dipole parameters of molecular ver-tices have internal contributions from all atoms that belong tothe vertex (which can be evaluated from the respective transitiondensity matrices). In addition, mentioned above deviations fromthe standing-wave form in the nearby repeat units (referred to asthe external contributions) should be taken into account as well.Namely, the deviations of transition charge and dipole on therepeat unit x can be described as

DQðxÞ ¼ QðxÞ � qðxÞ~waðxÞ; ð11Þ

DlðxÞ ¼ lðxÞ � dðxÞwaðxÞ: ð12ÞTo accurately evaluate these external contributions, it is suffi-

cient to sum within three repeat units of a given scattering centerin the PA systems [35]. For molecular terminus, a single parameteris sufficient to represent the dipole tensor. Therefore, we can intro-

duce the real-valued parameters qðTÞðxÞ and dðTÞðxÞ, by comparingthe effective transition dipole and charge of the end group to theexciton wave function and its dual, respectively. Following thisidea, the ES dipole parameters of –H, –NH2 and –NO2 termini havebeen extracted, fitted by piecewise polynomials and plotted inFig. 3b and c. As in the case of reflection phase /ðxÞ, modificationof dipole parameters for substituents indicates different opticaltransition intensities in the functionalized oligomers [34].

To this point, the optical properties of the lowest band p-exci-tons for PA backbone and three different end groups have been

characterized by the tabulated ES energy and dipole parameters,which allows to calculate the absorption spectrum of an arbitrarylinear PA molecule terminated by any considered substituent.These calculations are practically instantaneous [25,32,35,36]. Forexample, Fig. 4 shows comparison of the optical spectra calculatedwith the ES approach (completely bypassing the electronic struc-ture numerics) and direct quantum-chemical modeling for two dif-ferent oligomers substituted at the both ends. The transitionenergy discrepancies between the two methods are less than2 meV, whereas the relative differences of transition dipoles ofoptically-allowed states are within 3%. This exemplifies efficiencyand accuracy of the ES modeling.

3.3. Applications of the ES approach to complex branched molecules

To apply the ES approach to complex molecular structures andnetworks, scattering centers of higher degree should be consid-ered. We further characterize symmetric V (meta- and ortho-)and triple Y joints, which structures are shown in Fig. 1c. In thiscase, the joint’s scattering matrix CðxÞ can be diagonalized usingirreducible representations of the vertex symmetry groupG : C ¼ UDUy, where D is the diagonal matrix with elements in aform ei/ [30,31,33]. Therefore, C is parameterized by the scatteringphases /ðxÞ.

CV ðxÞ of the symmetric V joint is a 2� 2 matrix with the ele-ments rðxÞ and tðxÞ, being the complex reflection and transmis-sion amplitudes [25]. The symmetry group of a V joint is C2 withtwo irreducible representations. Therefore, CV can be parameter-ized by two real phases /0 and /1 corresponding to the symmetricand antisymmetric configurations, denoted by subscripts 0 and 1,respectively. The scattering matrix can be diagonalized by

CV ¼ UDUy ¼ r t

t r

� �;

U ¼ 1ffiffi2

p1 11 �1

� �; D ¼ ei/0 0

0 ei/1

!:

ð13Þ

Consequently, rðxÞ and tðxÞ are related to two real energy-dependent phases /0ðxÞ and /1ðxÞ by

r ¼ 12

ei/0 þ ei/1� �

; t ¼ 12

ei/0 � ei/1� �

: ð14Þ

Fig. 5. (a)meta-conjugated (M) and ortho-conjugated (O) V-type vertices. The atomsin the region marked with ‘‘ðþÞ” should be considered as part of the vertex, whereasthose in the region marked with ‘‘ð�Þ” should be ‘‘deducted” from the vertex due tothe overlap of the attached repeat units. The coordinate system for the transitiondipole in the V-joints is shown on the right. (b) Scattering phases of M and O joints,subscripts 0 and 1 represent the two symmetries (symmetric and antisymmetricmodes) of the scattering. (c) The transition charge and dipole parameters of the Mjoint. (d) Same as (c) but of the O joint.

130 H. Li et al. / Chemical Physics 481 (2016) 124–132

The symmetry group of a Y joint is D3 with six elements includ-ing the rotations by �2p=3 and three reflections. Similarly, accord-ing to its two irreducible representations: the identity operationhas zero angular momentum m ¼ 0, and rotations result in sym-metric and antisymmetric reflections with m ¼ �1 [33]. Diagonal-ization of CY can be performed as

CY ¼ UDUy;

U ¼ 1ffiffi3

p1 1 11 expð2pi=3Þ expð�2pi=3Þ1 expð�2pi=3Þ expð2pi=3Þ

0B@

1CA;

D ¼expði/SÞ 0 0

0 expði/PÞ 00 0 expði/PÞ

0B@

1CA:

ð15Þ

Following the above scheme, CY are parameterized by two realenergy-dependent phases /SðxÞ and /PðxÞ, associated with theirreducible representations.

The above derivation shows that each scattering phase in sym-metric joints is equivalent to the reflection phase of a terminus thatrepresents the joint for transition density matrix of the corre-sponding symmetry. These phases can be extracted from the quan-tum-chemical results that possess the required symmetries usingEq. (10). Namely, we calculated symmetric PA fragments of V andY types that have identical arm length L (6 repeat units) in eachmolecule, to preserve the joint symmetry. Using these electronicstructure results for the lowest excitonic band and Eq. (10), thescattering phases of meta- (/M

0 ðxÞ and /M1 ðxÞ), ortho- (/O

0 ðxÞ and/O

1 ðxÞ) and Y (/SðxÞ and /PðxÞ) joints have been further tabulated.Figs. 5a and 6a display the resulting functions after polynomial fits.

The number of the relevant ES dipole parameters describing amolecular joint is related to the symmetry as well. The transitiondipole parameters of planar joints always appear in pairs (projec-tions on two principle axes x and y, see Fig. 5a and Fig. 6b inset).The parameter extraction for a molecular vertex is similar to thatfor the terminus [35,33]. Real-valued parameters are retrievedwith respect to the standing waves instead of plane waves andcan be related to the parameters in Eq. (8). The external contribu-tions should be carefully evaluated. For example, in V and Y joints,the contribution from the region marked with ‘‘-” (due to therepeat unit overlap) in the phenyl ring of the linkage should besubtracted from the one from the region marked with ‘‘+” [asshown in Fig. 5a and Fig. 6b inset] [35,33].

The ES dipole parameters for the V joints are transition charge qand two real dipole parameters dx and dy. Here the symmetry ofelectronic excitations is defined as the symmetry of the x projec-tion of the transition dipole distribution. Therefore, q can only beextracted from the data on the antisymmetric modes, since thesymmetry of the exciton dual wave function is opposite to thatof the exciton wave function [35]. Thereby, the charge parametersof the meta- (qMðxÞ) and ortho- (qOðxÞ) joints have been calcu-lated by comparing the transition charge to the sum of the ampli-tudes of the dual wave functions on both sides of the joint, usingantisymmetric excitations. Furthermore, the respective dipole

parameters dMy ðxÞ and dM

x ðxÞ (dOx ðxÞ and dO

y ðxÞ) are retrieved usingthe sum of the values of the wave functions and their duals at bothsides of the joint. The results are shown in Fig. 5. The ES dipoleparameters for symmetric Y joint are transition charge qYðxÞ anddipole dYðxÞ (due to symmetry dY

x ¼ dYy ¼ dY ) [33]. These quantities

have been extracted using similar to the V-joint procedure and theresulting functions are plotted in Fig. 6.

At this juncture, our ES library is rich enough to predict opticalspectra in large complicated PA based molecular structures andnetworks. For example, the absorption spectrum of a dendrimer

has been calculated under the ES framework and compared tothe quantum-chemical results in Fig. 6c. We notice that the ESspectrum accurately reproduces the result of the reference methodeven in macromolecules containing many short segments, compa-rable in length to the exciton size.

4. Conclusions

In this Account, we reviewed the formalism of the Exciton Scat-tering (ES) approach developed for spectroscopic calculations inbranched conjugated macromolecules, where the electronic excita-tions can be viewed as quasiparticles (excitons) that reside on thecorresponding quasi-1D molecular graphs [16]. The ES methodol-ogy has been illustrated using various phenylacetylene basedmolecules including structures with donor/acceptors substituentswith TD-DFT being the reference quantum chemistry method.Exciton dispersion and scattering matrices of commonly used

Fig. 6. (a) The scattering phases of symmetric Y-type joint. (b) The transition charge and dipole parameters of Y joint. Insets: The overlapping atoms from the attached repeatunits are labeled with ‘‘(–)”; the coordinate system for the transition dipole is shown on the right (dY

x ¼ dYy ¼ dY ). (c) Comparison of the absorption spectra obtained from TD-

DFT calculations (top panel) and the ES approach (bottom panel) for dendrimer shown in the inset.

H. Li et al. / Chemical Physics 481 (2016) 124–132 131

molecular vertices have been extracted, which allows efficient andnumerically inexpensive calculations of the excitation energies andexciton wave functions. The transition charge and dipole parame-ters of aforementioned molecular building blocks are also charac-terized. The retrieved ES parameters allow for fast calculations ofabsorption spectra in molecules that consist of the characterizedbuilding blocks, no matter how large or complex they are. Beingspecific, the ES approach reduces the computer time scale to sec-onds compared to days or even weeks when conventional quan-tum-chemical methods are used. The results accuratelyreproduce the outcome of time-consuming supramolecular TD-DFT calculations. This became possible because the ES modelingrelies solely on the pre-tabulated library of scattering parameters,completely bypassing quantum-chemical computations. The accu-racy of the ES simulations can be further improved by refining theES parameters using high-quality spectroscopic data for smallmolecules representing building blocks [30,31]. Calculations ofthe ES parameters for complex asymmetric vertices are possibleas well using the respective exciton wave functions (in additionto the excitation energies) [31,32].

Overall, the ES multiscale approach fills the gap between theneed for the accurate characterization of optical spectra in conju-gated macrostructures and restrictions of general quantum-chem-ical methods due to high numerical cost. The ES methodology isthus complementary, rather than alternative, to the commonlyused quantum-chemical methods and is likely to substantiallyextend their capability to compute excited-state electronic struc-ture in much larger systems, including supramolecules, super-structures and molecular networks. Moreover, developingextensive sets of scattering parameters for various molecularbuilding block families brings forward a great potential of the ESapproach for real-time computational design of molecular struc-tures with desirable electronic and optical properties.

Finally, proposed quasiparticle scattering idea bridges tradi-tional molecular description (spatially confined wavefunctions,orbitals) with solid-state physics picture (periodic delocalizedwavefunctions, quasiparticles), providing a universal frameworkclosely related to the conventional condensed matter quasiparticledescription of many excitation processes and dynamics such asexcitons, polarons, spins, plasmons, etc. Consequently, the ES

methodology can be naturally augmented by adding exciton-elec-tron, exciton-phonon and electron–phonon scattering processesoccurring at the interfaces, chemical defects and impurities. Forexample, to incorporate electron–phonon coupling and conforma-tional disorder in ’soft’ electronic materials, it is possible toreformulate the scattering equations in terms of an effectivetight-binding (lattice) model, which is a Hamiltonian describingthe parent scattering problem [47].

The tight-binding models can be build in an efficient way bystudying the topological properties of the scattering matrices,namely the integer-valued topological charge, or equivalentlywinding number associated with the corresponding scatteringmatrix [48]. The topological charge provides useful informationon how many cites in the related tight-binding model are neededto adequately describe a scattering center. Studying analytical(geometrical) properties of the scattering matrices, namely theanalytical continuations of CðkÞ with k being the quasimomentum,from the Brillouin zone, represented by a unit circle in the complexplane c (or its compactified version, known as the projective spaceCP1) provide useful microscopic insights into the chemical proper-ties of molecular substituents [49], connected to polymer back-bones. The positions of poles (or zeros) of CðkÞ in CP1 describethe energies and spectral widths of the bound and resonant statesthat occur in a conjugated molecule due to the presence of the sub-stituent, with the above energies sometimes being very differentform the energetics of an uncoupled fragment. Treating a localizedmolecular geometry distortion, e.g., a bond-alternation stretch oran around-a-bond rotation as a scattering center for electronicexcitations (excitons), allows the dependence of the tight-bindingmodel parameters (on-site energies and hopping constants) to beextracted from quantum chemistry computations [50], which leadseventually to an exciton-phonon Hamiltonian. This allows effectsof incoherent energy transfer in a branched conjugated moleculeto be treated at the computational cost of a Frenkel-exciton systemcounterpart.

So far we have been considering polymer backbones with therepeat units that can be more or less adequately described by asingle site in the respective tight-binding model. In the case ofmultiple sites in a repeat unit, or/and longer range hopping (e.g.,next to nearest-neighbor) analytical properties of the scattering

132 H. Li et al. / Chemical Physics 481 (2016) 124–132

matrices, as well as exciton spectra show more complexity. Theycan be still efficiently studied, and, according to our preliminarystudies (the results will be published elsewhere), the Scatteringmatrices can be still represented as meromorphic functions onhigher Riemann surfaces, or using more mathematical languageon algebraic (complex-analytical) curves of higher genus g.

Acknowledgement

This material is based upon work supported by the NationalScience Foundation under Grant No. CHE-1111350. We acknowl-edge support of Directed Research and Development Funds, Centerfor Integrated Nanotechnology and Center for Nonlinear Studies atLos Alamos National Laboratory (LANL). LANL is operated by LosAlamos National Security, LLC, for the National Nuclear SecurityAdministration of the U.S. Department of Energy under contractDE-AC52-06NA25396.

References

[1] A.J. Heeger, Semiconducting polymers: the third generation, Chem. Soc. Rev. 39(2010) 2354–2371.

[2] D. Gust, T.A. Moore, A.L. Moore, Mimicking photosynthetic solar energytransduction, Acc. Chem. Res. 34 (2001) 40–48.

[3] H. Sirringhaus, T. Kawase, R.H. Friend, T. Shimoda, M. Inbasekaran, W. Wu, E.P.Woo, High-resolution inkjet printing of all-polymer transistor circuits, Science290 (2000) 2123–2126.

[4] S.R. Forrest, The path to ubiquitous and low-cost organic electronic applianceson plastic, Nature 428 (2004) 911–918.

[5] A. Murphy, J. Frechet, Organic semiconducting oligomers for use in thin filmtransistors, Chem. Rev. 107 (2007) 1066–1096.

[6] P.F. Barbara, A.J. Gesquiere, S.J. Park, Y.J. Lee, Single-molecule spectroscopy ofconjugated polymers, Acc. Chem. Res. 38 (2005) 602–610.

[7] D.L. Wang, X. Gong, P.S. Heeger, F. Rininsland, G.C. Bazan, A.J. Heeger,Biosensors from conjugated polyelectrolyte complexes, Proc. Nat. Acad. Sci.USA 99 (2002) 49–53.

[8] S. Gunes, H. Neugebauer, N. Sariciftci, Conjugated polymer-based organic solarcells, Chem. Rev. 107 (2007) 1324–1338.

[9] J.C. Bolinger, M.C. Traub, J. Brazard, T. Adachi, P.F. Barbara, D.A.V. Bout,Conformation and energy transfer in single conjugated polymers, Acc. Chem.Res. 45 (2012) 1992–2001.

[10] T.G. Goodson, Time-resolved spectroscopy of organic dendrimers andbranched chromophores, Ann. Rev. Phys. Chem. 56 (2005) 581–603.

[11] T.G. Goodson, Optical excitations in organic dendrimers investigated by time-resolved and nonlinear optical spectroscopy, Acct. Chem. Res. 38 (2005) 99–107.

[12] U.H.F. Bunz, Poly(arylene etynylene)s: From Synthesis to Application, Adv.Polym. Sci. 177 (2005) 1–52.

[13] S.-C. Lo, P. Burn, Development of dendrimers: macromolecules for use inorganic light-emitting diodes and solar cells, Chem. Rev. 107 (2007) 1097–1116.

[14] E. Collini, G.D. Scholes, Coherent intrachain energy migration in a conjugatedpolymer at room temperature, Science 323 (2009) 369–373.

[15] G.D. Scholes, G. Rumbles, Excitons in nanoscale systems, Nature Mat. 5 (2006)683–696.

[16] C. Wu, S.V. Malinin, S. Tretiak, V.Y. Chernyak, Exciton scattering andlocalization in branched dendrimeric structures, Nature Phys. 2 (2006) 631–635.

[17] M. Schulz, S. Tretiak, V.Y. Chernyak, S. Mukamel, Size scaling of third-order off-resonant polarizabilities. electronic coherence in organic oligomers, J. Am.Chem. Soc. 122 (2000) 452–459.

[18] A. Ruini, M.J. Caldas, G. Bussi, E. Molinari, Solid state effects on exciton statesand optical properties of PPV, Phys. Rev. Lett. 88 (2002) 206403–206404.

[19] K.I. Igumenshchev, S. Tretiak, V.Y. Chernyak, Excitonic effects in a time-dependent density functional theory, J. Chem. Phys. 127 (2007) 1–10.

[20] M. Rohlfing, S.G. Louie, Optical excitations in conjugated polymers, Phys. Rev.Lett. 82 (1999) 1959–1962.

[21] S. Tretiak, V. Chernyak, Resonant nonlinear polarizabilities in the time-dependent density functional (TDDFT) theory, J. Chem. Phys. 119 (2003) 8809–8823.

[22] M. Caricato, G.W. Trucks, M.J. Frisch, A comparison of three variants of thegeneralized davidson algorithm for the partial diagonalization of large non-hermitian matrices, J. Chem. Theory Comput. 6 (2010) 1966–1970.

[23] S. Tretiak, C.M. Isborn, A.M.N. Niklasson, M. Challacombe, Representationindependent algorithms for molecular response calculations in time-dependent self-consistent field theories, J. Chem. Phys. 130 (2009) 054111.

[24] V.Y. Chernyak, M. Schulz, S. Mukamel, S. Tretiak, E.V. Tsiper, Krylov-apacealgorithms for time-dependent Hartree-Fock and density functionalcomputations, J. Chem. Phys. 113 (2000) 36–43.

[25] C. Wu, S.V. Malinin, S. Tretiak, V.Y. Chernyak, Multiscale modeling of electronicexcitations in branched conjugated molecules using an exciton scatteringapproach, Phys. Rev. Lett. 100 (2008) 057405.

[26] S. Mukamel, S. Tretiak, T. Wagersreiter, V. Chernyak, Electronic coherence andcollective optical excitations of conjugated molecules, Science 277 (1997)781–787.

[27] S. Tretiak, S. Mukamel, Density matrix analysis and simulation of electronicexcitations in conjugated and aggregated molecules, Chem. Rev. 102 (2002)3171–3212.

[28] V. Chernyak, S.N. Volkov, S. Mukamel, Exciton coherence and electron energyloss spectroscopy of conjugated molecules, Phys. Rev. Lett. 86 (2001) 995–998.

[29] V. Chernyak, S.N. Volkov, S. Mukamel, Electronic structure-factor, densitymatrices, and electron energy loss spectroscopy of conjugated oligomers, J.Phys. Chem. A 105 (2001) 1988–2004.

[30] C. Wu, S.V. Malinin, S. Tretiak, V.Y. Chernyak, Exciton scattering approach forbranched conjugated molecules and complexes, I. Formalism. J. Chem. Phys.129 (2008) 174111.

[31] C. Wu, S.V. Malinin, S. Tretiak, V.Y. Chernyak, Exciton scattering approach forbranched conjugated molecules and complexes. ii. extraction of the excitonscattering parameters from quantum-chemical calculations, J. Chem. Phys. 129(2008) 174112.

[32] C. Wu, S.V. Malinin, S. Tretiak, V.Y. Chernyak, Exciton scattering approach forbranched conjugated molecules and complexes. III. Applications, J. Chem.Phys. 129 (2008) 174113.

[33] H. Li, C. Wu, S.V. Malinin, S. Tretiak, V.Y. Chernyak, Exciton scattering onsymmetric branching centers in conjugated molecules, J. Phys. Chem. B 115(2011) 5465–5475.

[34] H. Li, C. Wu, S.V. Malinin, S. Tretiak, V.Y. Chernyak, Excited states of donor andacceptor substituted conjugated oligomers: a perspective from the excitonscattering approach, J. Phys. Chem. Lett. 1 (2010) 3396–3400.

[35] H. Li, S.V. Malinin, S. Tretiak, V.Y. Chernyak, Exciton scattering approach forbranched conjugated molecules and complexes. IV. Transition dipoles andoptical spectra, J. Chem. Phys. 132 (2010) 124103.

[36] H. Li, V.Y. Chernyak, S. Tretiak, Natural atomic orbital representation foroptical spectra calculations in the exciton scattering approach, J. Phys. Chem.Lett. 3 (2012) 3734–3739.

[37] C.J. Cramer, Essentials of Computational Chemistry, Wiley, West Sussex,England, 2002.

[38] A.J. Berresheim, M. Muller, K. Mullen, Polyphenylene nanostructures, Chem.Rev. 99 (7) (1999) 1747–1786.

[39] V. Percec, M. Glodde, T.K. Bera, Y. Miura, I. Shiyanovskaya, K.D. Singer, V.S.K.Balagurusamy, P.A. Heiney, I. Schnell, A. Rapp, H.W. Spiess, S.D. Hudson, H.Duan, Self-organization of supramolecular helical dendrimers into complexelectronic materials, Nature 419 (2002) 384–387.

[40] E.L. Spitler, C.A. Johnson, M.M. Haley, Renaissance of annulene chemistry,Chem. Rev. 106 (2006) 5344–5386.

[41] T. Yanai, D.P. Tew, N.C. Handy, A new hybrid exchange-correlation functionalusing the Coulomb-attenuating method (CAM-B3LYP), Chem. Phys. Lett. 393(2004) 51–57.

[42] P.A. Limacher, K.V. Mikkelsen, H.P. Lüthi, On the accurate calculation ofpolarizabilities and second hyperpolarizabilities of polyacetylene oligomerchains using the CAM-B3LYP density functional, J. Chem. Phys. 130 (2009)194114.

[43] M.J. Frisch et al., Gaussian 09 Revision A.01. Gaussian Inc., Wallingford CT,2009.

[44] P.O. Löwdin, On the non-orthogonality problem connected with the use ofatomic wave functions in the theory of molecules and crystals, J. Chem Phys.18 (1950) 365–375.

[45] A.E. Reed, L.A. Curtiss, F. Weinhold, Intermolecular interactions from a naturalbond orbital; donor-acceptor viewpoint, Chem. Rev. 88 (1988) 899–926.

[46] E.D. Glendening, A.E. Reed, J.E. Carpenter, F. Weinhold, NBO Version 3.1.[47] H. Li, S.V. Malinin, S. Tretiak, V.Y. Chernyak, Effective tight-binding models for

excitons in branched conjugated molecules, J. Chem. Phys. 139 (2013) 064109.[48] M.J. Catanzaro, T. Shi, S. Tretiak, V.Y. Chernyak, Counting the number of excited

states in organic semiconductors systems using topology, J. Chem. Phys. 142(2015) 084113.

[49] H. Li, M.J. Catanzaro, S. Tretiak, V.Y. Chernyak, Excited-state structuremodifications due to molecular substituents and exciton scattering inconjugated molecules, J. Phys. Chem. Lett. 5 (2014) 641–647.

[50] T. Shi, H. Li, S. Tretiak, V.Y. Chernyak, How geometric distortions scatterelectronic excitations in conjugated molecules, J. Phys. Chem. Lett. 5 (2014)3946–3952.