chao wu, sergey v. malinin, sergei tretiak and vladimir y. chernyak- exciton scattering and...

8/3/2019 Chao Wu, Sergey V. Malinin, Sergei Tretiak and Vladimir Y. Chernyak- Exciton scattering and localization in branche

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Exciton scattering and localization in

branched dendrimeric structures

CHAO WU1, SERGEY V. MALININ1, SERGEI TRETIAK2* AND VLADIMIR Y. CHERNYAK1*1Department of Chemistry, Wayne State University, 5101 Cass Ave, Detroit, Michigan 48202, USA2Theoretical Division, Center for Nonlinear Studies, and Center for Integrated Nanotechnologies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA*e-mail: [email protected]; [email protected]

Published online: 27 August 2006; doi:10.1038/nphys389

-conjugated dendrimers are molecular examples

of tree-like structures known in physics as Bethe

lattices. Electronic excitations in these systems can

be spatially delocalized or localized depending on the

branching topology. Without a priori knowledge of

the localization pattern, understanding photoexcitation

dynamics reflected in experimental optical spectra

is difficult. Supramolecular-like quantum-chemical

calculations quickly become intractable as the molecularsize increases. Here we develop a reduced exciton-

scattering (ES) model, which attributes excited

states to standing waves in quasi-one-dimensional

structures, assuming a quasiparticle picture of optical

excitations. Direct quantum-chemical calculations of

branched phenylacetylene chromophores are used to

verify our model and to derive relevant parameters.

Complex and non-trivial delocalization patterns of

photoexcitations throughout the entire molecular tree

can then be universally characterized and understood

using the proposed ES method, completely bypassing

supramolecular calculations. This allows accurate

modelling of excited-state dynamics in arbitrary

branched structures.

Conjugated polymers and molecular wires are promisingmaterials for a number of technological applications,such as light-emitting, lasing and molecular electronics

applications1,2. Their extended one-dimensional backbonessupport delocalized mobile -electron systems. This resultsin a semiconductor-like electronic structure, which may bemanipulated synthetically. Branching of linear chains producestree-like molecules (dendrimers) that can be used to extenddesirable electronic properties even further3,4. In particular,dendrimers are believed to be promising building blocks forartificial light-harvesting systems58. These considerations arebased on their unique geometric structures known as Cayleytrees or Bethe lattices: as the dendrimer progresses to a highergeneration, the exponentially increasing peripheral terminal

chromophores serve as an antenna matrix to gather the incominglight. The collected energy in the form of photogenerated electronicexcitations is further funnelled to the core, where it can triggerchemical reactions. Cooperative enhancement of nonlinear opticalresponses (such as two-photon absorption) in dendrimers isanother currently active field of research9,10.

Studies investigating artificial light-harvesting functionality indendrimeric structures revealed a fascinating interplay betweenphotoexcitation localization/delocalization patterns and theirenergy-relaxation dynamics. In contrast to fluorescence resonanceenergy transfer from peripheral dyes directly to the core dye,observed in flexible polyarylether backboned dendrimers5, fullyconjugated symmetrical globular phenylacetylene dendrimers7,11

transfer excitations sequentially through the conjugated net12. In

the latter structures, linear oligomers form rigid monodendronslinked via meta-substituted phenyl rings. Meta-substitutions breakthe conjugation leading to efficient localization of photogeneratedexcitations in the linear segments7,1317. The energy gradient is thenautomatically created as the core extends to higher generationswith shorter phenylacetylene segments. The delicate interplayof strong and weak coupling between linear chromophoreshas specific signatures in the temperature-dependent dynamicsand vibrational system relaxation1618. The ortho-substitutionlinkage is less favourable relative to the meta one due to stericinteractions. However, several unsymmetrical dendrimers with fewgenerations were recently synthesized by Peng and co-workers8,19,20.These use both ortho- and para-substituted phenyls as jointsto connect the phenylacetylene segments. As the optical and

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M7-43

M6-33

O6-33O7-43

P7

D9-333

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R4R7 R8R3

R4 R1

R1 R1

R3

R5

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b

R2R6

Figure 1 Supramolecular dendrimers are made up via branching of linear

chains. a, Examples of branching centres. b, Structure and atom labelling of the

para P7, meta M6-33 and M7-43, ortho O6-33 and O7-43 and branched D9-333

molecules considered in this article.

photophysical properties of these systems are significantly differentfrom those of symmetrical branching structures, the fundamentalphotoexcitation dynamics in unsymmetrical dendrimers couldbe different, and the energy-transfer mechanisms are not wellunderstood19,20. Moreover, photoprocesses in compounds usingalternatives to phenyl-based branching centres (for example,nitrogen or phosphorus)9,10 have not been explored in detail,and the universal theoretical understanding encompassing allbranching structures (several examples are shown in Fig. 1a) is yetto be determined.

EXCITON-SCATTERING MODEL

We propose the exciton-scattering (ES) model to allow clearinterpretation of photoexcitation delocalization in arbitrarybranched structures. Excitons in infinitely long linear conjugatedchains can be treated as quasiparticles with well-defined momenta.Excited states in finite systems have pronounced node structure andcan be related to the excitations with non-zero momenta21. Forlinear molecules, this correspondence was explored in the contextof electron-energy-loss spectroscopy22,23. The ES approach extendsthese developments to a more general case and naturally attributesexcited states in branched structures to standing waves that resultfrom exciton scattering at the molecular ends, joints and branchingvertices. The ES model rests on the assumption of decoupling

between the centre-of-mass and relative motion of electronholepairs and is asymptotically exact in the limit of theexcitonsize beingshort compared with the linear segment lengths. Any branchedorganic molecule is then treated as a graph 24. The excited states arerepresented by wavefunctionss(x)= as exp(ikxs)+bs exp(ikxs)of the exciton centre-of-mass position xs on the segments. Thesetwo plane waves describe the quantum quasiparticle motion in twopossible directions in a linear segment. The wavevectors are relatedto theexciton frequencies throughthe exciton spectrum (k) in theinfinite-length chains. In a vertex that connects n linear segments,the amplitudes of the outgoing plane waves are related to theincoming waves amplitudes through an n n scattering matrix25(n)rs (). Naturally, the off-diagonal and diagonal elements of describe transmission and reflection processes, respectively. TheFrenkel exciton model seems to be the limiting case in thisapproach, allowing only for complete reflection or transmission.Exciton frequencies and wavefunctions can be found by solving asimple linear problem for the amplitudes as and bs. For a moleculerepresented by a graph with d1 edges (linear segments) we have 2d1independent variables (amplitudes). At each vertex/branching a ofdegree na, we have na linear equations that connect the amplitudesof the outgoing waves in terms of their incoming counterpartsthrough the elements of the scattering matrix (na )a . The number

of linear equations

a na = 2d1 exactly matches the numberof variables, and we arrive at a linear homogeneous 2d1 2d1frequency-dependent problem. The solution gives the transitionfrequencies of the excited states and the shapes of the standingwaves, containing all information on the electronic spectrum ofa macromolecule.

This provides a dramatic reduction in the computationaleffort compared with supramolecular-like quantum-chemicalapproaches and allows swift evaluation of novel organic materialsfor desirable electronic properties. On the other hand, the ES modelbuilds on the high-quality electronic-structure calculations, suchas time-dependent density functional theory26, by requesting theelectronic-structure data on linear segments and relevant verticesas an input. This can be done routinely using standard quantum-

chemical codes and very modest computational resources. Thechosen level of quantum chemistry can be rather arbitrary,however, it should be capable of describing excitonic effects in theexcited states. Excitons are vitally important for the description ofexcited-state structure and photoinduced dynamics in conjugatedchromophores. These quasiparticles are fundamental objects inthe ES approach. Approaches on the basis of the local (ornon-hybrid general gradient) density approximations, HartreeFock, or simpler tight-binding (Huckel) models do not reproducephotoexcited bound states of electrons and holes27, and, thus,should not be used in conjunction with our model.

RESULTS OF QUANTUM-CHEMICAL CALCULATIONS

To illustrate practical applications of the ES theoretical framework,we consider several phenylacetylene oligomers branching atpara-, ortho- and meta-positions at substituted phenyl rings,as shown in Fig. 1b. Similar compounds have recently beenexplored experimentally28. The excited-state structure of thesemolecules was calculated quantum-chemically as described in theMethods section. The computed states are strongly delocalized excitations, which are optically accessible and play a crucial role inenergy-transfer processes. Typically, the lowest-energy (bandgap)transition in such conjugated linear chains has remarkableoscillator strength and characteristically appears as a main peak inthe linear absorption spectra29,30. Figure 2 summarizes our maincomputational results. The contour plots represent the transitiondensity matrices ( )mn (also referred to as electronic normal

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S1 = 2.92 eV, f= 96

1

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S2 = 3.11 eV, f= 0

S1 = 3.01 eV, f= 72

S1 = 3.06 eV, f= 65

S1 = 2.94 eV, f= 30

S1 = 2.96 eV, f =22

S1 = 2.89 eV, f= 49 S2 = 3.05 eV, f= 59 S3 = 3.20 eV, f =11 S8 = 3.35 eV, f= 6

S2 = 3.17 eV, f= 53 S8 = 3.40 eV, f=1 S10 = 3.63 eV, f= 0

S2 = 3.11 eV, f= 58 S4 = 3.31 eV, f= 4 S11 = 3.52 eV, f= 0

S2 = 3.15 eV, f= 19 S9 = 3.55 eV, f= 0 S10 = 3.56 eV, f= 0

S2 = 3.12 eV, f= 26 S7 = 3.37 eV, f= 0 S11 = 3.55 eV, f =0

S3 = 3.32 eV, f= 7

1 12 24 36 48 60

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P7

D9-333

M7-43

M6-33

O7-43

O6-33

Figure 2 Exciton-scattering patterns given by contour plots of transition density matrices from the ground state to excited states of the molecules shown in Fig. 1.

Each plot shows the probability amplitudes of an electron moving fromone molecular position (horizontal axis) to another (vertical axis) on electronic excitation. The axis

labels represent individual carbon atoms according to the labelling in Fig. 1. The inset in each plot shows the excited state number (for example, S11 is the eleventh singlet

state), its transition frequency and oscillator strength f.

modes) between the ground and electronically excited states, astopographic maps that reflect the single-electron reduced densitymatrix changes on molecular photoexcitation29,31.

To establish a reference point, we start with the linear chain P7.The results are shown in the first row. S1 represents the bandgaptransition and can be associated with k= 0 momentum exciton inthe infinite chain limit. The electronhole pair created on opticalexcitation is delocalized over the entire chain (diagonal in the plot).

The exciton size (maximal distance between electron and hole) isabout 3 repeat units (largest off-diagonal extent of the non-zeromatrix area). The higher-frequency electronic transitions (S2, S3and so on) have characteristic node structure and correspond tothe excitons with non-zero momenta.

The second row in Fig. 2 shows excited states of M7-43.The meta-connection clearly breaks the molecule into two linearsegments. The excited states of the molecule, S1 and S2 (S7 and S11),

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directly correspond to S1 (S2) excitation of the linear oligomers P4and P3, respectively. An electrostatic coupling between S1 and S2of M7-43 can be related to the interaction of transition dipoles ofS1 states of P4 and P3. This leads to shifts in the electronic energylevels, and slightly mixes S1 states of P4 and P3. Owing to vanishingoscillator strengths, this mixing becomes weaker for higher-lyingstates S7 and S11. A very similar situation is observed in the secondmeta-compound M6-33, with the exception that the excitationenergies of its linear segments P3 are the same. Subsequently, thelowest excited states S1 and S2 are coherent plus and minussuperpositions of the S1 (P3) excitations. These are Frenkel-typeexcitons15,32, characteristic for molecular aggregates. The splittingbetween S1 and S2 states of M6-33 measures the coupling strength.Both S1 and S2 states have non-zero oscillator strengths resulting asa vector sum of the corresponding transition dipoles. Thus, meta-based connections effectively break the branching structure into anumber of electrostatically interacting chromophores, which makesit possible to apply reduced Frenkel-like hamiltonian models15,32.

Next, we analyse the properties of ortho-branching. The loweststate S1 in both O7-43 and O6-33 looks very similar to the S1state of the linear chromophore P7. The higher-lying excited statesof O7-43 and O6-33 also have characteristic node structures,irrespective of the position of the ortho-joint along the chain

(compare the fourth and fifth rows with the first row in Fig. 2).This clearly demonstrates that ortho-branching does not preventexciton delocalization and the excitation pattern is similar to that inthe linear chains. However, there are important differences betweenpara- and ortho-connections. The second excited state S2 in bothO7-43 and O6-33 appears as the state with the largest oscillatorstrength in their spectra, whereas it is optically forbidden in thelinear chromophore P7. At the same time, the bandgap state S1 ofO7-43 and O6-33 has a relativelysmall oscillator strength comparedwith that of P7. This redistribution of the oscillator strength amongexcitons in the band is a manifestation of molecular geometry.Compared with the linear chains, where both parts contributenearly evenly to the total oscillator strength of the S1 state, thetransition dipole of the S1 state in the ortho-structures is reduced

due to vector addition of the dipole components on the linearsegments. Albeit the S2 state gains strong oscillator strength forthe same reason. Subsequently, states that are optically forbiddenin the linear chains, can be experimentally detected in the ortho-structures, whose linear absorptions are expected to have severallow-lying peaks. In contrast to meta-compounds, which haveindependent interacting linear chromophores, these states in ortho-molecules appear due to transfer of the oscillator strength to thehigh-lying excitons as a consequence of molecular geometry.

After the methodological study of possible phenyl-based joints,we consider a combined structure D9-333 (Fig. 1b). The bottomrow of Fig. 2 shows transition density matrices of the opticallyrelevant excited states. Unlike previous cases, their interpretationusing the delocalization/localization language14,15 does not look

straightforward. For example, ortho-branching assumes fullydelocalized states, whereas S2 shows clear localization trends.

APPLICATION OF EXCITON-SCATTERING MODEL

The ES approach allows a uniform interpretation of all molecularexamples considered (Fig. 1b). The ES system of linear equationsadopts the form:

bs exp(ik ls)=(1)()as exp(ik ls); as =

r

(n)sr ()br.

where ls are the linear segment lengths; n = 2 for M and Omolecules and n=3 for the D branched structure. In the latter caser, s=1,2,3 correspond to the right, left and upper linear segments,

respectively. A qualitatively correct zero-order approximationfor the scattering matrix can be readily constructed. Themolecular end is a vertex with n = 1. Vanishing of the excitonicwavefunction at the ends implies full reflection (1) =1. Thecentral vertex of M molecules is described by a 2 2 scatteringmatrix

(meta)12 =(meta)21 = 0, (meta)11 =(meta)22 =1, which accounts

for the full reflection at meta-connection. The full transmissionof ortho-vertex in O compounds implies:

(ortho)12 = (ortho)21 = 1,

(ortho)

11 =

(ortho)

22 =0. This assignment immediately rationalizes

similarity in the shapes and energies between the S2 (M6-33) andS2 (O6-33), or S10 (M6-33) and S10 (O6-33) states (see Fig. 2):they have the same excitonic wavefunction. The small differencescan be attributed to deviations in the scattering matrices from thezero-order approximation due to, for example, steric interactions.

We further illustrate our ES model for the D9-333 molecule.The analogous rationale can be applied to derive the scatteringmatrix for an arbitrary branching centre. Similar properties ofpara- and ortho-joints (to the zero-order approximation) implysymmetry of the scattering matrix (3)rs with respect to interchangeof branches 2 (left) and 3 (upper) of D9-333 (red labels in Fig. 1b),that is,

(3)22 = (3)33 , (3)23 = (3)32 , (3)12 = (3)13 and (3)21 = (3)31 .

This leaves us with five independent components of the scatteringmatrix:

(3)11 ,

(3)22 ,

(3)23 ,

(3)12 and

(3)21 . Excitonic wavefunctions can

be classified by their parity with respect to interchange of branches2 and 3. Odd exciton modes vanish on branch 1, they are fullycharacterized by the amplitudes a2 and b2 that can be found bysolving a linear problem in an effective linear chain that representsbranch 2. The scattering amplitudes at the left and right ends are(1)=1 and ()=(3)22 (3)23 , respectively. Even exciton modesare characterized by the amplitudes as and bs with s = 1,2. Theycan be found by solving a linear problem in an effective linearchain that represents branches 1 and 2 with a joint in the centre.The joint is described by an effective 2 2 scattering matrix withthe elements

(+)21 =

2

(3)21 ,

(+)12 =

2

(3)12 ,

(+)22 = (3)22 +(3)23

and (+)

11 = (3)11 . The paraortho-joints symmetry is confirmedby vanishing of the S2 mode in D9-333 on the right branchand its striking similarity to the S1 mode in M6-33 in terms

of shape, energy and oscillator strength. The similarity in theenergy and oscillator strength means that the modes have similarwavefunctions, which implies () = (3)22 (3)23 =1. The evenmodes S1, S3 and S8 of D9-333 have 0, 1 and 2 nodes in the effectivelinear molecule, as clearly seen from Fig. 2. They are similar interms of the shapes and energies to the S1, S2 and S8 modes,respectively, of the O6-33 molecule. This implies for the zero-orderapproximation

(+)11 = (+)22 = 0 and (+)12 = (+)21 = 1. Finally, we

arrive at (3)

11 0, (3)12 (3)21 1/

2, (3)

22 1/2 and (3)23 1/2,which completes the scattering matrix form. The differences inthe corresponding energies and irregularities in the node positionsin D9-333 are attributed to the deviation of the scattering matrix(3)rs from its simple zero-order approximation, which turns out tobe small. Strong differences in the oscillator strengths are due to

different geometries of these molecules.In conclusion, we establish stable and intuitively clear

relationships between the structure of optical excitations inbranched organic molecules and the underlying moleculargeometry. The excitation localization/delocalization concept ofmolecular aggregates, previously successfully used for meta-conjugated dendrimers7,1416, fails in the case of general branchedstructures, which involve various combinations of joints andbranching5,9,17. The proposed ES model provides simple, yetquantitatively correct, microscopic insight into electronic spectrain such quasi-one-dimensional organic structures with complextopology35. It is conceptually reminiscent of the graph theory24

and the Huckel model applications33,34 to chemical reactivity. TheES approach is based on very common-life concepts, such as wave

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