Excitation Transport in Molecular Aggregates with thermal motion

Excitation Transport in Molecular Aggregates with thermal motionR. Pant1, a) and S. Wüster1, b)

Department of Physics, Indian Institute of Science Education and Research, Bhopal, Madhya Pradesh 462 023,India

(Dated: 24 February 2020)

Molecular aggregates can under certain conditions coherently transport electronic excitation energy over large distancesdue to dipole-dipole interactions. Here, we explore to what extent thermal motion of entire monomers can guide orenhance this excitation transport. The motion induces changes of aggregate geometry and hence modifies exciton states.Under certain conditions excitation energy can thus be transported by the aggregate adiabatically following a certainexciton eigenstate. While such transport is always slower than direct migration through dipole-dipole interactions, weshow that transport through motion can yield higher transport efficiencies in the presence of on-site energy disorderthan the static counterpart. For this we consider two simple models of molecular motion: (i) longitudinal vibrations ofthe monomers along the aggregation direction within their inter-molecular binding potential and (ii) torsional motion ofplanar monomers in a plane orthogonal to the aggregation direction. We employ a quantum-classical method, in whichmolecules move through simplified classical molecular dynamics, while the excitation transport is treated quantumcoherently using Schrödinger’s equation. For both models we find parameter regimes in which the motion enhancesexcitation transport, however these are more realistic for the torsional scenario, due to the limited motional range in atypical Morse type inter-molecular potential.

I. INTRODUCTION

Molecular aggregates where a large number of organicmolecules assemble into a fairly regular structure can exhibitsignificant excitation energy transport along the structure1,2,which plays a key role in photosynthetic light harvestingprocesses3,4 and has the potential for technological exploita-tion, e.g. in dye-sensitized solar cells5,6 or thin-film opticaland optoelectronic devices7. In all of these, molecular aggre-gates facilitate the absorption of light and subsequent trans-fer of the absorbed energy to a reaction centre8,9 in the formof an electron-hole pair known as exciton. This transfer ofexcitation relies on the long range dipole-dipole interactionsbetween the monomers in the aggregate.

Such dipole-dipole interactions are also a characteristic fea-ture of Rydberg aggregates10, which hence have been pro-posed as quantum simulators for molecular aggregates11,12. InRydberg aggregates, a chain of highly excited Rydberg atomstransports a single energy quantum on spatial- and temporalscales quite different from the molecular context. While theexcitation transfer process in molecular aggregates is typicallystrongly affected by decoherence13–16, it barely is in ultra-coldatoms, as has been experimentally demonstrated17–19.

An idea that naturally arises in Rydberg aggregates, is adi-abatic excitation transport through atomic motion10,20,21. Inadiabatic excitation transport, slow motion of the atoms com-bined with excitation transport via dipole-dipole interactionscan result in efficient and guided transport of the excitationfrom one end of an atomic chain to the other, see schematicin Fig. 1 (a). Based on the analogy between Rydberg- andMolecular aggregates, the question then arises whether adia-batic excitation transport can play a functional role in molecu-lar aggregates, e.g. for light harvesting. Here we report initial

a)Electronic mail: ritesh17@iiserb.ac.inb)Electronic mail: sebastian@iiserb.ac.in

(b)

(a)

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|ei<latexit sha1_base64="lqa/vDgGmDy/4xpQ9H3+1QybH6c=">AAAB8HicbVDLSgNBEJyNrxhfUY9ehgRBEMJuPOgx6MVjBPOQ7BJmJ73JkJnZZWZWWGK+Qg8eFPHq53jL3zh5HDSxoKGo6qa7K0w408Z1J05ubX1jcyu/XdjZ3ds/KB4eNXWcKgoNGvNYtUOigTMJDcMMh3aigIiQQysc3kz91iMozWJ5b7IEAkH6kkWMEmOlhyfwFZF9Dt1i2a24M+BV4i1IuVbyz18mtazeLX77vZimAqShnGjd8dzEBCOiDKMcxgU/1ZAQOiR96FgqiQAdjGYHj/GpVXo4ipUtafBM/T0xIkLrTIS2UxAz0MveVPzP66QmugpGTCapAUnni6KUYxPj6fe4xxRQwzNLCFXM3orpgChCjc2oYEPwll9eJc1qxbuoVO9sGtdojjw6QSV0hjx0iWroFtVRA1Ek0DN6Q++Ocl6dD+dz3ppzFjPH6A+crx8i9pOZ</latexit> Vdd(R)

Vdd(θ)

Vdd(X)

X·X

·XX

YZ

(c)

FIG. 1. (a) Interplay of molecular motion and excitation transferin adiabatic excitation transport. Blue • are molecules or atoms, Xindicates their position and X (green arrow) the corresponding mo-tion. The orange shade represents a selected de-localized excitonstate. This state is always located on the two closest molecules, suchthat if the state is adiabatically followed, the indicated motion en-tails excitation transport. (b) Energy level schematic, with molecularelectronic ground state |g〉 and excited state |e〉 and dipole-dipoleinteractions V dd(R). We also sketch on-site disorder σE of transitionenergies. (c) More detailed sketch of a 1D chain of planar moleculeswith torsional motion along θ and longitudinal motion along X . (bluedisk) molecules, (red large arrow) direction of the transition dipolemoment between |g〉 and |e〉.

explorations of this idea, focussing for now only on a sce-nario where the energy transport remains quantum coherent.The motivation is as follows: quantum adiabatic following inexcitation transport was reported in the atomic case for a com-pletely coherent scenario20. If we cannot find similar featuresin a coherent molecular setting, they are unlikely to be presentin a case with decoherence. We however shall find that adi-

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Excitation Transport in Molecular Aggregates with thermal motion 2

abatic excitation transport persists and thus intend to explorein the future to what extent adiabatic excitation transport cansurvive partial decoherence. Meanwhile, our results shouldalready be applicable to some extent to particularly coherentmolecular aggregates22 such as those reported in23,24.

Three key features, which are the focus of the present arti-cle, change the physics of transport in molecular aggregatescompared to the simpler ultra-cold atomic scenario even ifdipole-dipole interactions remain coherent. These features aresite-to-site energy disorder, inter-molecular binding and ran-dom thermal positions and velocities. Varying all parameterspertaining to these within ranges relevant for molecular aggre-gates, we map out regimes where excitation transport involv-ing molecular motion can yield higher transport efficienciesthan direct dipole-dipole transport in the immobile case, sincemotion counters energy disorder.

For this we set up two different simple models for molec-ular motion in aggregates: (i) Longitudinal motion, in whichmolecules move classically along the direction of aggregationonly, bound to their neighbors through a Morse potential. Thismotion affects the dipole-dipole Hamiltonian through varyingdistances between molecules. (ii) Torsional motion, in whichmolecules at fixed separation can rotate in the plane orthog-onal to the aggregation direction, which affects the dipole-dipole Hamiltonian through varying angles between transitiondipole moments. Both model also involve a quantum degreeof freedom for the electronic state that allows for a single,possibly delocalized electronic excitation. We find that excita-tion transport is more positively affected by motion in the tor-sional model, since for a realistic motional range of moleculeslarger variations of dipole-dipole interactions and hence ex-citon states are accessible through varying angles betweendipole moments. In comparison, during longitudinal motion,variations due to changing separations between monomers aresmaller.

The effect of molecular motion on excitation transport hasalso been investigated in25–29. Most of these studies considertransport in the presence of decoherence and none explorethe aspect of adiabaticity, as we do here. In contrast to ourexplicit model for thermal molecular motion, Refs.25,27 con-strain classical harmonic motion to selected harmonic normalmodes. In Ref.26,28 inter-molecular vibrations are consideredquantum mechanically, focussing mainly on one relevant res-onant mode. All articles find an increased transport efficiencyin certain parameter regimes when comparing a static with amobile scenario, in agreement with the results that we shallpresent.

This article is organised as follows: In section II, we intro-duce the features of our molecular aggregate model that arecommon to both scenarios listed above (longitudinal and tor-sional motion), such as dipole-dipole interactions, mechanicalmotion and the quantum-classical propagation scheme that weemploy. The remainder of the article is then arranged in twoparts, in section III we explore motion of monomers alongthe aggregation axis, while in section IV monomers rotate in aplane orthogonal to that axis. Both sections are then structuredsimilarly: We firstly demonstrate in one clear but not nec-essarily realistic scenario how adiabatic excitation transport

would proceed in a molecular setting (section III A and sec-tion IV A), followed by an extensive parameter survey com-paring the transport efficiency with and without motion (sec-tion III C and section IV B). A crucial feature in this surveyis on-site disorder, which we introduce in section III B andthen use also in section IV. Finally the appendices containdetails on our estimates of moments of inertia, appendix A1,single trajectory simulations for the case of longitudinal mo-tion, appendix B, and torsional motion, appendix C, as well asmeasures of adiabaticity, appendix D.

II. EXCITATION TRANSPORT AND MOLECULARMOTION

We model N monomers with mass M and moment of in-ertia I, arranged in a one dimensional (1D) chain along theX direction, where the n’th monomer is located at a definite,classical position Xn. These monomers can be bound to eachother by van-der-Waals forces and/or hydrogen bonds, withinter monomer distances of the order of Angström30. We con-sider each monomer as an electronic two level system withground state |g〉 and first excited state |e〉. The transitiondipole moment µ between these two states is assumed fixedin the Y Z-plane orthogonal to the aggregation direction, andat an angle θ , wrt. the Z axis, see Fig. 1 (c). The distance be-tween monomers shall be large enough to neglect electronicwave function overlap, so that there is no direct exchange ofelectrons between the monomers30. Therefore the only inter-actions capable of excitation energy transfer are Coulomb in-teractions. For the large distances between the monomers, weassume that these can be approximated by the dipole-dipoleinteraction Hamiltonian V (dd)

mn (R), between monomers n andm, which reads

V (dd)mn (R) =

µm ·µn

|Xmn|3−3

(µm ·Xmn)(µn ·Xmn)

|Xmn|5, (1)

where R = (X1,X2, ...,XN ,θ1,θ2, ...,θN)T denotes the collec-

tion of all molecular coordinates, including locations, Xn, andangles, θn, of dipole moments wrt. to an axis orthogonal to theaggregation direction, see Fig. 1 (c). Further, Xmn = Xm−Xnis the separation of monomer m and monomer n and µm =µm(θm) and µn =µn(θn) are their transition dipole moments.We shall assume that there is only a single excitation presentin the aggregate, hence the electronic Hilbert space is spannedby |m〉 = |g...e...g〉, where only the m’th molecule is in theexcited state and all others are in their ground state. We call{|m〉} the diabatic basis. While we included longitudinal co-ordinates Xn and torsional coordinates θn simultaneously in(1), we shall only consider their dynamics one-by-one in thisarticle, in order to separately assess their potential for enhanc-ing transport.

With the above restrictions on degrees of freedom, theHamiltonian of our system can be written as31

H(R) =−N

∑n=1

[h2

2M∂ 2

∂X2n+

h2

2I∂ 2

∂θ 2n

]+Hel(R), (2)

Excitation Transport in Molecular Aggregates with thermal motion 3

where the first term gives the kinetic and rotational energy ofthe molecules and Hel(R) is single exciton Hamiltonian, givenby32

Hel(R) =N

∑m=1

Em|m〉〈m |+ ∑nm;n 6=m

V (dd)mn (R)|m〉〈n |

+

(12 ∑

nm;n 6=mV (B)

mn (R)

)I. (3)

Here Em is the electronic excitation energy at site m, andV (dd)

mn (R) is the matrix element for the dipole-dipole interac-tion between monomer m and n given in Eq. (1), which is re-sponsible for excitation energy transfer. The transition energyEm at each site is different since the influence of the environ-ment could be different for each monomer, see e.g. Ref.33,34.V (B)

mn (R) denotes interactions that do not depend on the elec-tronic state, which we assume to be the case for the inter-molecular binding potential, for simplicity.

To study the dynamics induced by the Hamiltonian (2), con-sider the eigenstate of the electronic part Hel(R)

Hel(R)|ϕs(R)〉=Us(R)|ϕs(R)〉, (4)

where Us(R) defines the s’th potential energy surface for agiven molecular configuration. Us(R) is called the adiabaticpotential energy surface and the corresponding eigenstates arereferred as adiabatic basis states or here Frenkel excitons.Note that Us(R) are supra-molecular energy surfaces. Eachadiabatic state can be written in terms of the diabatic states as,

|ϕs(R)〉=N

∑m=1

fsm(R)|m〉. (5)

Considering motion quantum mechanically for the system (2)would be intractable for all but the smallest systems and alsolikely unnecessary for the answers sought here. Thereforewe use a mixed quantum-classical method, Tully’s surfacehopping35,36, where the motion of molecules is treated clas-sically according to Newton’s equations

M∂ 2

∂ t2 Xm =−∇XmUs(X)−∑n

∇XmV (B)mn (X), (6)

and an ensemble of trajectories is propagated on a specificBorn-Oppenheimer surface Us. The surface index s is al-lowed to stochastically jump in time, to take into account non-adiabatic transitions from one surface to another and the cor-responding change of forces acting on the molecules. Here,we show (6) for the case of longitudinal motion ∂ 2Xm

∂ t2 only, itstorsional version shall be given and used in section IV.

Expanding the total wavefunction of the system in the adia-batic basis defined above |ψ(X, t)〉 = ∑

Nm=1 cm|ϕm(X)〉, we

can also obtain the following Schrödinger equation for thecomplex amplitudes cm,

i∂

∂ tcm =Um(X)cm− i

N

∑n=1

dmncn, (7)

where dmn are the non-adiabatic coupling coefficients, whichalso control the probability of stochastic jumps between sur-faces in Tully’s algorithm. They can be written as

dmn =−1M〈ϕm(X(t)) |∇X|ϕn(X(t))〉 · ∂Xm

∂ t. (8)

Besides the movement of the molecules, we are interestedin the exciton dynamics, for which we evolve the total wavefunction in the diabatic basis |ψ(X, t)〉= ∑

Nm=1 cm(t)|m〉, in-

stead of Eq. (7). Its time evolution is thus determined by

i∂

∂ tcm =

N

∑n=1

Hmn[Xmn(t)]cn, (9)

and coupled to (6) through the dependence of the electronicHamiltonian on all molecular positions. Here Hmn[Xmn(t)]is the matrix element for the electronic coupling in Eq. (3),with the dipole-dipole interaction given by Eq. (10) and Xmn =|Xm−Xn|.

While we outlined the formalism jointly here for molec-ular degrees of freedom Xn and θn, the rest of our study isarranged in two parts where we first assume a fixed directionof the molecular transition dipoles, but allow monomers tomove along the aggregation direction, and in a second part fixmolecular positions along the aggregate axis, but allow theirtorsional motion through a plane orthogonal to that axis, andhence varying transition dipole directions. This splitting hasthe objective to clearly determine which degrees of freedomare more conducive for motional enhancement of excitationtransport.

III. EXCITATION TRANSPORT BY LONGITUDINALMOTION

In this section, we consider a chain of identical monomerswith all transition dipole moments µ parallel to each otherand fixed. This is commonly referred to as H-aggregate37,and might be more suitable for excitation transport than J-aggregates with anti-parallel µ, due to the larger excited statelifetime38. The monomers can move along the X axis joiningthem, see Fig. 2 (a). Dipole-dipole interactions Eq. (1) in thiscase are

V (dd)mn (X) =

µ2

|Xmn|3. (10)

The monomers in aggregates are bound by van-der-Waalsforces, which we model with a Morse potential

V (B)mn (X) = De

[e−2α(|Xmn|−X0)−2e−α(|Xmn|−X0)

], (11)

where De is the depth of the well, X0 the equilibrium distanceand α controls the width of the potential. The smaller α , thesofter and wider is the potential, see Fig. 2 (b). To be specific,we choose a mass M = 902330 a.u. and a transition dipole mo-ment µ = 1.12 a.u., roughly matching e.g. carbonyl-bridgedtriaryl-amine (CBT) dyes39.

Excitation Transport in Molecular Aggregates with thermal motion 4

Out[61]=Out[61]= Out[61]= Out[61]= Out[61]= Out[61]=

a X0 X0X0(X0 − a)

X

(a)

0 10 20X [Å]

-2000

-1000

0

1000

2000

V(X)

/k B [K]

(b) = 0.3 Å-1

= 0.9 Å-1

V(dd)

De

X0

(b)

FIG. 2. (a) Sketch of 1D molecular aggregate with off diagonal dis-order, all transition dipoles are parallel. Shaded Gaussians indicatepossible initial thermal position disorder of each monomer. (b) Inter-molecular potentials: Morse potential for α = 0.3 Å−1 (blue dashed)and α = 0.9 Å−1 (red dot-dashed) and the strength of dipole-dipoleinteractions V (dd)

nm (red solid line).

A. Idealized adiabatic excitation transport

In this section, we first elucidate the concept of adiabaticexcitation transport in a clear cut, albeit constructed case.For this, all molecules are initially placed at the equilibriumseparation (X0 = 6Å) of the Morse potential, except the firsttwo molecules, which have a closer separation a = 4Å, seeFig. 2 (a). For this configuration the dipole-dipole interactionbetween the first two molecules is much stronger than betweenany other neighboring molecules in the chain. This results inthe localization of the excitation on these first two molecules,such that the initial electronic aggregate state is to a very goodapproximation the exciton

|ψ(0)〉= 1√2(|m = 1〉+ |m = 2〉). (12)

A second important consequence of the initial condition, isthat the first two molecules very strongly repel each other,since they are deep on the inner repulsive side of the Morsepotential in Fig. 2 (b). Around the geometry choice de-scribed so far, the initial positions and velocities of themolecules are randomized according to the Maxwell Boltz-mann distribution40 at temperature T = 300 K. Finally, weneglect on-site disorder in this section, such that Em = 0 inEq. (3).

The resultant motion of the molecules and the excitationtransfer are shown in Fig. 3, using Eq. (6) coupled to Eq. (9).Initially the two closest molecules strongly repel each other.Molecule 2 moves towards molecule 3, while molecule 1

escapes the chain, since the initial potential energy V M12 (a)

by far exceeds the binding energy De. When molecule 2reaches molecule 3 those two constitute the new closest prox-imity pair. Since the motional time-scale τmov is large com-pared to the characteristic time-scale for dipole-dipole inter-actions τdd ≡ π/V (dd)

12 (a) = 2.7 fs, the system can adiabat-ically follow the exciton quantum state that initially corre-sponds to Eq. (12), which is always localized on the two clos-est molecules. Around t = 0.5 ps, it is hence now localizedon molecule 2 and 3. Since in this close encounter, also themomentum is transferred from molecule 2 to molecule 3, theprocess continues along the chain until molecule 7 escapes itin the end. Just prior to that, at t = 2ps in Fig. 3, the ex-citation has to a large extent been transported to the end ofthe chain, on molecules 6 and 7. We see a small change in

0 1 2 3 4 5 6 7t[ps]

0

0.5

1

P n

12 3 4 5 6

7

(a)

(b)

| cn(t)

|2

FIG. 3. Exemplary adiabatic excitation transport with molecules. (a)(solid white) Mean trajectories of individual molecules, with site in-dex n as indicated. (color shading) Mean electronic excitation prob-abilities or diabatic populations |cn|2 are indicated by color aroundthe respective trajectory. (b) (colored numbered lines) Individual di-abatic populations |cn|2, (black solid) adiabatic population on the ini-tial surface. Binding parameters are α = 0.5 Å−1 and a well depthof De = 2200 K. Results are averaged over 105 trajectories.

the adiabatic population, black line in Fig. 3 (b), which re-duces the transport fidelity. Otherwise, our extreme choice ofinitial conditions has replicated the near perfect transport sce-nario of the atomic case20, where cold atoms are not bound totheir neighbor. A very similar scenario arises if we start fromthe initially localized state |ψ(0)〉 = |1〉. Since this can bewritten as a linear combination of (12) and the correspondinganti-symmetric exciton, both of which are adiabatically trans-ported to the end of the chain as in Fig. 3, also the excitationfrom initial state |1〉 reaches the end of the chain through themotion.

In summary, a pulse combining motion and excitationtransfer can facilitate high fidelity transport of an excitation

Excitation Transport in Molecular Aggregates with thermal motion 5

through a chain. For its kinematic similarity with the popularclass-room tool to demonstrate momentum conservation, theprocess has been likened to Newton’s cradle in Ref.20. Thephysical basis is quantum adiabaticity, which leads to a limi-tation of this technique: To remain adiabatic, we require slowmotion τmov � τdd ≡ π/V (dd)

12 (a), as discussed above, whichmeans the transported excitation energy will always arrive ear-lier if we start in a localized state |ψ(0)〉 = |π1 〉 instead ofEq. (12), and consider an equidistant chain. However, the sit-uation is less clear when on-site energy disorder is present,since localization might then preclude an excitation startingin a localized state to arrive at all.

The example in this section has been chosen to clearly illus-trate the concept of adiabatic excitation transport, but will notlikely be practically useful due to the extreme and thus ther-mally inaccessible initial state for the positions of monomers1 and 2. In the following we much more generally comparethe transport properties of moving and static aggregates in thepresence of on-site energy disorder.

B. Disorder and exciton localization

The on-site energy disorder σE in an aggregate, sketched inFig. 1 (b), arises due to the coupling of the monomers withtheir environment33,34. Since the local environment may bedifferent for each site, this can cause slightly different tran-sition energy shifts En for each monomer. We assume herethat the time-scale for variation of such shifts is slow, so thatthe En are constant throughout the transport process, hence weonly treat static disorder. For the En in Eq. (3), we assume aGaussian distribution41

pE(En−E0) =1√

2πσEe−(En−E0)

2/2σE (13)

where σE is the standard deviation and E0 the unperturbedtransition energy of each molecule. The distribution is as-sumed to be identical for all monomers. For realistic systems,more sophisticated distributions may apply42,43.

Disorder strongly affects the energy level structure andwave-function of exciton states in (4), which in turn influ-ences the excitation transfer. One measure of the impact ofdisorder is the de-localization length of the exciton over theaggregate44,45. For weak disorder the exciton is de-localizedover the entire aggregate, while for strong disorder it becomeslocalized on a smaller number of monomers. This may causeexciton trapping46,47, which is detrimental to excitation trans-port.

We now demonstrate that motion and hence excitationtransport can overcome disorder induced localization in a sim-ple test-case. For this, we take a chain composed of sevenmonomers placed at their equilibrium separations X0, and thensubject to thermal position distributions. We compare a staticand a mobile scenario, both exhibiting an identical realisationof the disorder (13).

The coherent dynamics of excitation transfer for a singletrajectory is shown in Fig. 4. The disorder strength wasσE = 500 cm−1, α = 0.5Å−1 and the temperature T = 300

K. At t = 0 the excitation is injected at the first site (#1, in-put site), hence |c1(0)|2 = 1. Starting from this initial state,we investigate whether the excitation reaches the output site.We see that for the static system with X = 0, the populationreaching the output site remains very small Fig. 4(c). In con-trast, for the dynamic system thermal fluctuations overcomethe disorder and can deliver the excitation to the output sitewith quite high probability. While here we only demonstrated

0 20 40t[ps]

0

0.5

1

|c7|2

10-3

0 20 40t[ps]

0

0.5

1

|c7|2

-1500

0

1500

En [cm-1]

7 6 5 4 3 2 1n

| cn(t)

|2

| cn(t)

|2

(a) (b)

(d)(c)

FIG. 4. Excitation transport can overcome on-site energy disorderthrough motion. (a) Monomer trajectories and excitation probabilityin the immobile case in the same style as for Fig. 3. (b) The same formobile molecules. (c) Population reaching the output site #7 in theimmobile case. (d) The same for the mobile case. The inset showsthe realisation of on-site energy disorder (13) used for both, panel (a)and (b).

this for a specific single realisation of random positions, ve-locities and energy disorder, the latter shown in Fig. 4 (d), wewill nextly confirm that motion can help to overcome disor-der also in the ensemble average. A detailed inspection of theadiabatic populations pn = |cn|2, see (7), shows however thatthe electronic dynamics is only intermittently adiabatic, witha large number of non-adiabatic transitions. We will commenton this again later in this article.

C. Transport efficiency

The single trajectory simulation in the previous section in-dicates that the motion of the molecules in an aggregate canpotentially play a key role in efficient quantum coherent trans-port of excitation energy in the presence of disorder. To ex-plore this more deeply, we now quantify the efficiency of exci-tation transport at different temperatures T and site disorders.We define the efficiency of excitation transfer ετ in terms ofthe maximum probability for the excitation to reach to the out-put site within a time τ , following e.g. Ref.48:

ετ = maxt∈[0,τ]

|〈m = out |ψ(t)〉|2. (14)

The input site is #1 as in section III B.For a given choice of molecular interaction potentials and

temperature, we then calculate the mean efficiency by averag-

Excitation Transport in Molecular Aggregates with thermal motion 6

ing over many different kinematic configurations with the ini-tial positions and velocities of the molecules drawn from theMaxwell Boltzmann distribution as described in section III A.We first calculate the maximum (14) for each realisation andthen average these maxima over the trajectories. We finallycompare two different efficiencies: ε

(motion)τ , the transport ef-

ficiency in the case of mobile molecules and ε(static)τ , the trans-

port efficiency in the case of immobile molecules. Note, thatthe latter case will still include position fluctuations accordingto a thermal distribution, only forces and velocities are set tozero.

The efficiency ε(motion)τ is shown in Fig. 5 (a), after averag-

ing over 5000 random configurations for seven sites, for thechoice τ = 50 ps and output site #7. We consider a range ofdisorder strengths for which the aggregate transitions from de-localized excitons to strongly localized excitons for our choiceof other parameters. We see that the effect of temperature onefficiency is small, within a reasonable range of temperature.This is because for the chosen α , the accessible range of inter-molecular separations does not significantly vary with temper-ature due to the tight potential, see Fig. 2 (b). To assess theimpact of motion on transport for any given case, we finallyresort to the relative efficiency, defined as the ratio

η =ε(motion)τ

ε(static)τ

. (15)

For the same scenario, the relative efficiency η is shown inFig. 5 (b). In the chosen parameter range, we find relative ef-ficiencies of up to about four, which indicates an enhancedtransport in the mobile system compared to the static one.Nextly, we examine how this enhancement is affected by the

140 180 220 260 300T [K]

200

400

600

E[cm

-1]

1*

4* 3*

2*0

0.1

0.2

0.3

0.4

0.5

140 180 220 260 300T [K]

200

400

600

E [cm

-1]

0

1

2

3

4(a) (b)

ϵ(mot

ion)

τ

η

FIG. 5. (a) Transport efficiency ε(motion)T , see Eq. (14), in the case of

mobile molecules using a potential width according to α = 0.5Å−1.Each of the (6× 10) tiles represents a simulation result for on-sitedisorder strength and temperatures indicated on the axes. Labels inthe corners mark cases for which we show exemplary single trajecto-ries in appendix B, Fig. 12. (b) Relative efficiency η as the efficiencyfor the mobile scenario divided by the immobile one, see Eq. (15)

width of the potential well that binds monomers to each other.The width is controlled by the parameter α in the Morse po-tential, (11). For small α , the width of the well is increased,so we expect larger excursions of inter-molecular separationsthan for large α . These may overcome the localization ofexcitons due to the accompanying strong variation of dipole-dipole interaction strengths. The efficiency for the mobile sys-tem is shown in Fig. 6 (a), single trajectories for the four pa-rameter sets in the corners can be found in appendix B. If the

well is narrow, the transport efficiency remains smaller thanfor the case of wider well. This suggests, that larger dynami-cally accessible position deviations of the molecules enhancetransport. Particularly for quite soft inter-molecular bindingcorresponding to α = 0.3 Å−1 in Fig. 6, we see a marked im-pact of motion on excitation transport.

0.34 0.47 0.60 0.73 0.87 [Å-1]

200

400

600

E[cm

-1]

1*

4* 3*

2*0

0.1

0.2

0.3

0.4

0.5

0.34 0.47 0.60 0.73 0.87[Å-1]

200

400

600

E[cm

-1]

0

2

4

6

8

10(a) (b)

η

ϵ(mot

ion)

τ

FIG. 6. The same as for Fig. 5, but we vary the width of the inter-molecular binding potential well instead of the temperature, whichis fixed at T = 300K. Small α correspond to a larger width. Singletrajectories for the labels in the corner of (a) are shown in appendix B,Fig. 13

We have chosen the relatively small system size of 7monomers for numerical convenience. Extending this tolarger system of e.g. 13 molecules, we find that the rela-tive efficiency η increases with system size, for otherwiseidentical parameters. This is expected since the localisationlength from energy disorder would remain constant, henceend-to-end transport without motion will be more stronglysuppressed for larger chains. In contrast, the mode of adia-batic excitation transport as in Fig. 3 does not worsen muchfor larger chains, hence the relative improvement provided bymotion could be larger.

While the motivation for the present work and the reasonfor our expectation that motion might aid coherent transportstem from the concept of adiabatic excitation transport dis-cussed in section III A, the results shown so far indicate onlythat motion may have a beneficial effect, but not whether thisis due to adiabatic processes. When looking at individual tra-jectories in more detail, which we show in appendix B, it ap-pears that the quantum dynamics of the excitation containsboth, adiabatic periods as well as non-adiabatic transitions.We however do find, that the parameter space regions with thelargest relative efficiencies η , are more adiabatic than others,see appendix D.

IV. EXCITATION TRANSPORT BY TORSIONALMOTION

In the previous section, we have shown that while longi-tudinal motion of molecules along the aggregation directioncan enhance the efficiency of excitation transport in the pres-ence of disorder, this enhancement is not very large within therange of realistic parameters for molecular interaction, whichare the narrowest potential explored by us. It turns out thatthe situation is improved if the motional degree of freedomis changed from longitudinal motion to torsional motion, ex-plored in this section. We now assume that the separation of

Excitation Transport in Molecular Aggregates with thermal motion 7

the molecules is fixed at 3.4 Å, but the molecules are allowedto rotate around the aggregate X-axis within the Y Z-plane.Any possible relative tilt of the molecules out of this planeis ignored for simplicity.

The dipole-dipole interaction in Eq. (1) with these con-straints can be written as

V (dd)mn (θ) =

µ2

X30

cosθmn, (16)

where θmn = θn − θm is the angle between the direction ofthe transition dipole moments of molecule m and moleculen, shown as red arrows in Fig. 7 (a). The transition dipoleof magnitude µ is assumed spatially fixed in the plane of themolecule which can now rotate round the X-axis.

We assume that the molecules prefer to align at certain an-gles with their neighbors, which can be chemically engineeredfor example by the addition of appropriate side chains23. Todescribe these preferred orientation and torsional excursionsaround them, we employ a potential energy

V (B)mn (θmn) =

V0

2[1− cos(2Kθ (θmn−θ0))], (17)

shown in Fig. 7 (b), where V0 is the height of the potentialbarrier, Kθ determines the spacing of minima and hence thesymmetry, and θ0 determines the equilibrium angle(s) and isfixed by the detailed shape of the molecule. For reasons thatshall become clear shortly, we assume an approximate four-fold symmetry, hence Kθ = 2. The symmetry should not beperfect though, to justify the use of a single excited electronicstate per molecule, and hence a well defined direction for thetransition dipole moment.

In order to allow the potential (17) to control equilibriumorientations of the molecules, and not be overwhelmed bya torque from dipole-dipole interactions in (16), we havechanged the dipole strength to µ = 0.6 a.u., which is abouthalf of the value taken in section III for longitudinal motion.

After fixing the desired symmetry, we nextly adjust thetorsional potential strength V0 such that the angular spread∆θ in thermal equilibrium at T = 300K is ∆θ = 8◦, roughlymatching angular spreads modelled in Ref.39. This results inthe maximum value of the potential V0/kB = 1923K. Whilethis potential in principle still allows full rotations of themolecules, near room temperature molecules will almost ex-clusively perform small torsional oscillations about the min-ima of (17).

We model torsional dynamics analogous to the case of lon-gitudinal motion, except that instead of the positions Xn, theangles θn are allowed to evolve. The classical equations ofmotion for the angular displacement of molecules read

I∂ 2

∂ t2 θm =−∇θmUs(θ)−∑n

∇θmV (B)mn . (18)

See appendix A1 for the our simple estimate of a represen-tative moment of inertia I for dye molecules, based on CBT.As before, the exciton dynamics is obtained by expanding thetotal wavefunction in diabatic states

i∂

∂ tcm =

N

∑n=1

Hmn[θmn(t)]cn, (19)

0 90 180 270 360

0

500

1000

1500

V()/k

B [K]

V(B)

V(dd)

(a)

(b)

θX0

X

Y

Z

FIG. 7. (a) Sketch of the central units in a 1D self-assembled chainof dyes, which can slightly rotate about the aggregation axis. (bluedisk) molecules, (red arrow) direction of the transition dipole mo-ment between |g〉 and |e〉. (b) (solid blue line) Periodic potential(17) for the torsional motion of the molecules in (a), with a periodic-ity of 90◦ due to some assumed approximate four-fold symmetry ofeach molecule. (red dashed) Strength of dipole-dipole interactions(16) for orientation.

where Hmn[θmn(t)] is the matrix element for the electroniccoupling in Eq. (3), with dipole-dipole interaction given byEq. (16).

A. Idealized adiabatic excitation transport

As in section III A, we begin with the basic demonstrationthat molecular torsion can cause transport in principle. We as-sume that the orientations of all molecules are at the equilib-rium of the torsional potential Eq. (17), with angle betweenadjacent axes of θ0 = 70◦. Now if we decrease the angu-lar separation between the direction of the dipole momentsof first two molecules the dipole-dipole interaction betweenthese two molecules is stronger and the excitation will againget localised on them as in (12). The angular distribution andthe angular velocity distribution of each monomer is again aGaussian with standard deviation σθ and σω respectively, dis-order is set to zero, Em = 0. The coherent dynamics of excita-tion transport is now obtained by solving Eq. (18) and Eq. (19)

Excitation Transport in Molecular Aggregates with thermal motion 8

as a coupled system.The resultant excitation transport is show in Fig. 8, aver-

aged over 105 trajectories. The dislocation at the end willcause a repulsive torque on molecule 2 which causes it to ro-tate its axis towards that of molecule 3, increasing the dipole-dipole interaction between those two and carrying the excita-tion with it, as in section III A. Again the scheme proceedsover several sites. Compared to the scenario of section III A,

(a)

(b)| c

n(t)|2

FIG. 8. Exemplary adiabatic excitation transport by molecularrotations. (a) (solid white) Mean orientation angles of individualmolecules, with site index n as indicated. (color shading) Mean elec-tronic excitation probabilities or diabatic populations |cn|2 are indi-cated by color around the respective trajectory. (b) (colored num-bered lines) Individual diabatic populations |cn|2, (black solid) adia-batic population on the initial surface. Results are averaged over 105

trajectories.

we see a larger reduction of adiabatic population as the blackline in Fig. 8 (b), with a corresponding larger drop of the fi-delity of transport.

B. Transport efficiency

In this section we explore to what extent torsional motion ofthe molecules during energy transport can overcome disorder.We again take a chain of seven molecules, however now welet the dipole moment of each molecule point perpendicularto each other, i.e. θ0 = π/2 in Fig. 7 (a). We chose this anglesince dipole-dipole interactions at the precise equilibrium po-sition vanish, which will necessarily enhance the relative im-pact of random molecular rotations near the equilibrium orien-tation on dipole-dipole interactions and hence exciton states.

Fluctuations of on-site-energies are again described byEq. (13). The transport of excitation for a single trajectoryat room temperature for both, the mobile and the static system

is shown in figure Fig. 9. As before the initial state of exci-tation at time t = 0 is |c1|2 = 1. The standard deviation fororientation angles at room temperature is taken to be σθ = 8◦.We see as in section III B, that in a case where for the staticsystem the excitation is almost completely localized on theinput site, the mobile system manages to transfer 80% of theexcitation energy to the output site. Therefore also torsionalmotion of the molecules can help to combat localization andguide the transfer of excitation along the chain. The effect of

0 20 40 60 80t[ps]

0

0.005

0.01

|c7|2

0 20 40 60 80t[ps]

0

0.5

1

|c7|2

-5000500

En [cm-1]

7654321n

| cn(t)

|2

| cn(t)

|2

(a) (b)

(c) (d)

FIG. 9. Excitation transport through torsional motion can overcomedisorder. (a) Monomer orientations and excitation probability in theimmobile case in the same style as for Fig. 3. (b) The same formobile molecules. (c) Population reaching the output site #7 in theimmobile case. (d) The same for the mobile case. The inset in (d)shows the selected single realisation of on-site energy disorder.

site disorder and temperature on efficiencies and relative ef-ficiencies is shown in Fig. 10, using the same definitions asin section III C. Here the potential (17) is fixed and the effi-ciency is obtained by averaging over 5000 random configura-tions of seven sites. We have re-adjusted the range of disorderstrengths to cover the regime from de-localized excitons tostrongly localized excitons for the different setting here. Incontrast to results in section III C, there is now a small in-crease in efficiency with temperature for the mobile system.We see an even more significant increase in the relative trans-port efficiency Fig. 10b, compared to the case of longitudinalmotion. Nextly, we fix the temperature at T = 300 K and varythe width of the potential well by changing V0 in the potential(17) and hence σθ . The results are shown in Fig. 11. As in theearlier section on longitudinal motion, we recover the schemethat wider potentials allowing a larger range of motion giverise to larger relative efficiencies of excitation transport.

An inspection of the adiabaticity of individual trajectories,shown in appendix C, again shows a mixture of adiabatic andnon-adiabatic contributions to the dynamics, as we had seenfor the scenario with longitudinal motion along the aggregateaxis. We also again find the general trend that regions inparameters space with high relative efficiency co-incide withthose showing more adiabaticity, see appendix D.

Excitation Transport in Molecular Aggregates with thermal motion 9

140 180 220 260 300T [K]

100

200

300E [c

m-1

](a)

1* 2*1*

4* 3*

2*0

0.1

0.2

0.3

0.4

0.5

140 180 220 260 300T [K]

100

200

300

E [cm

-1]

0

1

2

3

4

(a) (b)

ϵ(mot

ion)

τ

η

FIG. 10. Variation of transport efficiency with disorder strength andtemperature for torsional motion of molecules, for a well width ofσθ = 8◦. (a) Absolute efficiency ε

(motion)T in the mobile case. La-

bels in the corners mark cases for which we show exemplary singletrajectories in appendix C, Fig. 14. (b) Relative efficiency η as theefficiency for the mobile scenario divided by the immobile one.

5 7 9 11 13 [Degree]

100

200

300

E[c

m-1

]

1*

4* 3*

2*0

0.1

0.2

0.3

0.4

0.5

5 7 9 11 13 [Degree]

100

200

300

E[cm

-1]

0

1

2

3

4

5

6(a) (b)

ϵ(mot

ion)

τ

η

FIG. 11. Variation of transport efficiency with disorder strength andwell width for torsional motion of molecules at T = 300 K. (a) Ab-solute efficiency ε

(motion)T in the mobile case. Labels in the corners

mark cases for which we show exemplary single trajectories in ap-pendix C, Fig. 15. (b) Relative efficiency η as the efficiency for themobile scenario divided by the immobile one.

V. CONCLUSIONS AND OUTLOOK

We have explored how mechanical motion of the moleculesin an aggregate affects the efficiency of excitation transportcompared to an immobile scenario. We find a motion-inducedenhancement of the excitation transfer efficiency over staticconfiguration in the presence of on-site energy disorder forboth, longitudinal motion of molecules along the aggregateaxis and rotational or torsional motion of them in the planeorthogonal to that axis, in small systems of up to N = 7monomers. We conclude that a strong connection betweenthe motion of the molecules and the coherent propagation ofthe electronic excitation has the potential to increase the effi-ciency of excitation transfer significantly.

While a possible cause of enhancement of transport can befound in adiabatic excitation transport20 as demonstrated insection III A and section IV A, for most of the inspected pa-rameter space the situation is less clear, with dynamics ex-hibiting both: periods where eigenstates are adiabatically fol-lowed, interspersed with sudden non-adiabatic transitions. Forthe most general electronic dynamics when the excitation stateis a superposition of excitons and can thus exhibit transportfrom site to site also in the absence of any motion, a clear linkof transport with adiabaticity or its absence is conceptuallychallenging, but would be of interest in the future.

In this article, we have ignored decoherence, for example

caused by intramolecular vibrations, although decoherenceand vibrations frequently play a crucial role in transport49–58.In the next step of this exploration, we will thus extendthe quantum dynamics calculation for excitation transport toan open-quantum-system technique, such as non-Markovianquantum state diffusion15,59–61, which will be coupled to clas-sical time-dependent trajectory for the molecules as in thiswork. Earlier research using simpler models for motion andenergy disorder than the present work suggests that motioncan enhance transport even in the presence of decoheringenvironments25,28.

Another significant extension to bridge the gap betweenthese model calculations and realistic molecular systems,would be to treat energy transport beyond the dipole-dipoleapproximation. For the relevant case where the intermolec-ular distance is comparable to the size of the molecules,higher multipole transitions play a significant role62–64. Shortrange excitonic couplings may significantly deviate from thedipolar form showing an exponential distance dependence65.Since adiabatic transport discussed here relies on significantchanges of the excitonic Hamiltonian with molecular posi-tions, it might be enhanced by such effects.

For a final confirmation for realistic and technologicallyrelevant settings, such as dye-sensitised light harvestingtechnology5,6, we can replace the simple classical point par-ticle motion of the present article (6) by full fledged molec-ular dynamics simulations evolving all the nucleii, and theevolution of electronic states by the simple matrix model (9)used here with time-dependent density functional theory ofthe many electron aggregate system in what is known as QM-MM schemes.

Finally, since this work was inspired by ideas that have orig-inated in a quantum simulation context with cold atoms20, alsoother features discovered in that context might be portable toa molecular setting. One prominent one is the use of coni-cal intersections66 as switches for coherence and direction ofexcitation transport67–69.

ACKNOWLEDGMENTS

It is a pleasure to thank Alexander Eisfeld, Jan-MichaelRost and Varadharajan Srinivasan for helpful comments. Wealso thank the Max-Planck society for financial support underthe MPG-IISER partner group program. The support and theresources provided by Centre for Develoment of AdvancedComputing (C-DAC) and the National Supercomputing Mis-sion (NSM), Government of India are gratefully acknowl-edged. RP is grateful to the Council of Scientific and Indus-trial Research (CSIR), India, for a Shyama Prasad Mukher-jee (SPM) fellowship for pursuing the Ph.D (File No. SPM-07/1020(0304)/2019-EMR-I).

AIP PUBLISHING DATA SHARING POLICY

The data that support the findings of this study are availablefrom the corresponding author upon reasonable request.

Excitation Transport in Molecular Aggregates with thermal motion 10

Appendix A: Estimate of dye-moelcule moment of inertia androtational potential

For example, in the supramolecular assembly of Ref.23

that provided some guidance for our simple model of tor-sional motion in section IV, a single monomer consist ofa CBT core attached via amide linker with 4-(5-hexyl-2,2′-bithiophene)naphtalimide (NIBT). Therefore the total massof the monomer is the sum of masses of these constituents,which amounts to M = 1009.03 amu. We simply assume thatthis mass is distributed uniformly in a square disc, with sidelength a = 50 Å. The moment of inertia of a disc with massdensity ρ is

I =16

ρa4 (A1)

For a uniform mass distribution ρ = M/(a2) we then find amoment of inertia of 2.95×109 a.u.

The maximum of the potential barrier V0 is obtained by tak-ing the Taylor expansion of Eq. (17) about θmn = θ0, and thendefining a target angular width ∆θ using the thermal equipar-tition theorem

V0Kθ (∆θ)2 =kBT

2(A2)

We assume a typical angular spread ∆θ = 8◦ at temperature300K similar to Ref.39, from which we obtain V0/kB = 1923K.

To obtain σθ at a different temperature T after this initial

allocation, we again use Eq. (A2), σθ =√

kBT2KθV0

. The dis-tribution in angular velocity also relies on the equipartitiontheorem, yielding a width σω =

√kBT/I.

Appendix B: Single Trajectories for longitudinal motion

To provide more intuitive access to the ensemble averagedresults in the main text, section III C, we now additionally pro-vide some individual trajectories along the edges of the inves-tigated parameter space.

We see in the bottom panels (1* and 2*) of Fig. 12, thatfor small energy disorder σE , the localization effect is weak.Thus even in the absence of molecular motion the excitationquite likely and rapidly reaches the output site, a result thatno longer can be much improved upon by motion. In contrast,the upper panels (3* and 4*) with strong disorder show quitelocalised excitons, where for example panel #3 then showsdynamics where this disorder has been overcome by thermalmotion.

By controlling the width of the inter-molecular bindingwell, we can control the amplitude of excursions of themolecules. Single trajectories here show how the probabil-ity of excitation transport is influenced by changing the widthas well as the site disorder. We see in Fig. 13 that even forlarge site disorder the excitation energy can reach the outputsite with probabilities of more than 50% in mobile case.

Further analysis of adiabatic populations reveals that theexciton dynamics is rarely purely adiabatic but typically also

TABLE I. Parameters for single trajectory data in Fig. 12. Positionsrefer to the tags in Fig. 5

.Positions Parameters#1 σE = 100 cm−1 and T = 120 K#2 σE = 100 cm−1 and T = 300 K#3 σE = 600 cm−1 and T = 300 K#4 σE = 600 cm−1 and T = 120 K

(1*) (2*)

(4*)

| cn(t)

|2

| cn(t)

|2

| cn(t)

|2

| cn(t)

|2

(3*)

FIG. 12. Single trajectory time evolution of the position of indi-vidual molecules together with exciton dynamics for α = 0.5 Å−1.The numbering refers to tags in Fig. 5 and the four different parame-ter sets given in table I. Top panels show molecular coordinates anddiabatic populations |cn(t)|2 in the same style as Fig. 3 (a). Bottompanels show the corresponding adiabatic populations |cn(t)|2, withlegend included for case (4∗). Surfaces are numbered from the high-est energy exciton state to the lowest. Colored arrows refer to specificdynamical events discussed in the text.

non-adiabatic at many instances. To assess this further, weshow adiabatic populations for all the trajectories in Fig. 12and Fig. 13. From these we can deduce that the evolu-tion shows a mixture of two types of dynamics: Firstly adi-abatic periods, during which adiabatic populations |cn(t)|2stay nearly constant. Secondly, prominent intermittent non-adiabatic transitions, at which adiabatic populations showquite sudden significant changes. If we manually relate eitherof these two features in the bottom panels, with the excita-tion probability of each molecule shown in the top panels, we

TABLE II. Parameters for single trajectory data in Fig. 13. Positionsrefer to the tags in Fig. 6

Positions Parameters#1 σE = 100 cm−1 and α = 0.30 Å−1

#2 σE = 100 cm−1 and α = 0.87 Å−1

#3 σE = 600 cm−1 and α = 0.87 Å−1

#4 σE = 600 cm−1 and α = 0.30 Å−1

Excitation Transport in Molecular Aggregates with thermal motion 11

(1*) (2*)

(3*)(4*)

| cn(t)

|2

| cn(t)

|2

| cn(t)

|2

| cn(t)

|2

FIG. 13. Single trajectory time evolution of the position of individ-ual molecules together with excitation dynamics, in the same styleas Fig. 12, but here we fix the temperature at 300K. The numberingrefers to tags in Fig. 6 and the four different parameter sets given intable II. Colored arrows refer to specific dynamical events discussedin the text.

can identify two qualitatively different phenomena: (i) Duringa period of adiabaticity, the excitation exhibits a slow trans-fer from one molecule to another, see Fig. 13 panel (4*), attimes indicated by yellow arrows near t = 2 ps and t = 19 ps.This corresponds to adiabatic transport as discussed in sec-tion III A. (ii) Exactly coinciding with a sudden non-adiabatictransition, the excitation transfers from one molecule to an-other, see Fig. 13 panel (4*), around t = 42 ps and t = 48 ps,marked with red arrows. This should rather be referred to asnon-adiabatic excitation transport.

A consistent classification of the general time evolutionwith respect to these two features, that can also be employedin the ensemble averages, is an interesting challenge that wedefer to future work.

Appendix C: Single Trajectories for torsional motion

Similar to the longitudinal motion scenario, we now illus-trate the effect of temperature and site disorder on excitationtransport with individual simulation trajectories, but now wefix the molecular positions and allow instead a rotation of themolecular transition dipole axes.

Fig. 14 shows the single trajectories at different tempera-tures for fixed width of the potential well. When the energydisorder is small for example the bottom panels of Fig. 14, thelocalization effect is weak and similar to longitudinal cradlethe excitation rapidly reaches the output site with high arrivalprobability even in the absence of motion. In contrast, oncewe increase the energy disorder towards the upper panels, theexcitation is more localized in the immobile system. Then, forthe mobile system we see a clear transport of the exciton to theoutput site. Nextly, we provide single trajectories for differ-ent widths of the potential in Fig. 15. For wide wells (panel

TABLE III. Parameters for single trajectory plots in Fig. 14. Posi-tions refer to the tags in Fig. 10

Positions Parameters#1 σE = 100 cm−1 and T = 120 K#2 σE = 100 cm−1 and T = 300 K#3 σE = 300 cm−1 and T = 300 K#4 σE = 300 cm−1 and T = 120 K

(1*) (2*)

(3*)(4*)

| cn(t)

|2

| cn(t)

|2

| cn(t)

|2

| cn(t)

|2

FIG. 14. Single trajectory time evolution of the angular positionand exciton dynamics for σθ = 8◦. The numbering refers to tags inFig. 10 (a) and the four different parameter sets given in table III.Even for large site disorder the excitation energy transfer to the out-put site is clearly seen. Colored arrows refer to specific dynamicsevents discussed in the text.

#3) and high on-site disorder, when almost 90% of the exci-ton is localized on the first site for immobile molecules, wesee that the excitation is reaching the output site with morethan 80% probability if motion is included. As in the caseof longitudinal transport, we see a mixture of adiabatic andnon-adiabatic transport processes. A relatively clear trajectorycontaining both is shown in Fig. 15 panel (3∗): The evolutionis non-adiabatic near 20 ps but then it is fairly adiabatic from20 to 85 ps. Exactly at the moment of non-adiabaticity, the toppanel shows how the excitation migrated from molecule 1 tomolecule 7. Fast oscillations in diabatic populations can be at-tributed to beating from a superposition of exciton states, cre-ated by the non-adiabatic transition. However note the longertime scale variations which shifts most of the excitation prob-

TABLE IV. Parameters for single trajectory plots in Fig. 15. Posi-tions refer to the tags in Fig. 11

Positions Parameters#1 σE = 100 cm−1 and σθ = 4◦

#2 σE = 100 cm−1 and σθ = 13◦

#3 σE = 300 cm−1 and σθ = 13◦

#4 σE = 300 cm−1 and σθ = 4◦

Excitation Transport in Molecular Aggregates with thermal motion 12

(4*)

| cn(t)

|2

(1*)

| cn(t)

|2

(3*)

| cn(t)

|2

(2*)

| cn(t)

|2

FIG. 15. Single trajectory time evolution of the angular position andexciton dynamics similar to Fig. 14, but here we fix the temperatureat 300K. The numbering refers to tags in Fig. 11 (a) and the fourdifferent parameter sets given in table III. We can see a clear transportof excitation even for large site disorder. Colored arrows refer tospecific dynamics events discussed in the text.

ability back from site 7 to site 1 around t = 25 ps and thenback again around t = 45 ps. Neither event is accompanied bya significant change in adiabatic populations, hence we wouldclassify these changes as adiabatic transport.

Appendix D: Survey of adiabaticity

While a rigorous classification of transport into adiabatic ornon-adiabatic is left for future studies, a simple comparisonof the global level of adiabaticity between different regionsof parameter space can be gained from our quantum-classicalpropagation algorithm itself:

The molecules move on a single adiabatic potential energysurface m, which may be changed via sudden jumps to anothersurface n by non-adiabatic couplings Eq. (8) between surfacem and surface n. A simple estimate of non-adiabaticity is thusprovided by the mean number of allowed70 jumps21,35,71. Themean number of jumps per trajectory is shown for longitudinalmotion in Fig. 16 and for torsional motion in Fig. 17, for thesame two cuts through parameter space discussed in the mainarticle. We find a larger number of jumps at high tempera-tures or narrow width of the potential well. This is expectedsince either involve faster motion of molecules, which directlyincreases all non-adiabatic couplings in Eq. (8). When wecompare Fig. 16 with Fig. 5 and Fig. 6, we find that regionsof high relative efficiency coincide with those of less allowedjumps and thus more adiabatic dynamics. This could be a hintthat motion of molecules is indeed more helpful if adiabatic,but this would have to be verified more rigorously in the fu-ture. Similar conclusions can be drawn from for the case oftorsional motion in Fig. 17. For narrow width of the poten-tial well or high temperature, the number of allowed jumps islarge due to the faster angular vibration of molecules. In con-

0.34 0.47 0.60 0.73 0.87 [Å-1]

200

400

600

E[cm

-1]

0

5

10

15

20

25

Jum

ps

140 180 220 260 300T [K]

200

400

600

E[cm

-1]

0

5

10

15

20

Jum

ps

(a) (b)

FIG. 16. Mean number of allowed surface jumps in Tully’s al-gorithm, corresponding to non-adiabatic transitions, for longitudinalmotion. (a) We vary temperatures and on-site disorder strengths. (b)We vary the width of the inter-molecular binding potential well andon-site disorder strengths.

trast, there are fewer jumps when we decrease temperatureor increase the width of the well or on-site disorder strength.Again, in the region of high relative efficiency in Fig. 10 andFig. 11 the transport is more adiabatic than in other regions.

5 7 9 11 13 [Degree]

100

200

300

E[cm

-1]

0

10

20

30

40

Jum

ps

140 180 220 260 300T [K]

100

200

300

E [cm

-1]

0

5

10

15

20

25

Jum

ps

(a) (b)

FIG. 17. Mean number of allowed jumps similar to Fig. 16 fortorsional motion. (a) For different temperatures and (b) for differentwidths of the well, as well as in both cases varied energy disorderstrength.

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