Construction of Multichromophoric Spectra from Monomer Data:Applications to Resonant Energy Transfer

Aurélia Chenu∗ and Jianshu Cao†

Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA

We develop a model that establishes a quantitative link between the physical properties of molec-ular aggregates and their constituent building blocks. The relation is built on the coherent potentialapproximation, calibrated against exact results, and proven reliable for a wide range of parameters.It provides a practical method to compute spectra and transfer rates in multichromophoric sys-tems from experimentally accessible monomer data. Applications to Förster energy transfer revealoptimal transfer rates as functions of both the system-bath coupling and intra-aggregate coherence.

PACS numbers: 36.20.Kd,31.70.Hq,78.66.Qn,82.20.Wt

Molecular self-assembly is used in nature to build com-plex structures and regulate material properties [1], andcan also be exploited for the fabrication of versatilenanostructures [2]. The spatial arrangement of chro-mophoric building blocks strongly influences the elec-tronic distribution [3], enabling a broad range of opticaland transport properties [4]. Nature has mastered thisart, evolving from a very limited number of monomersan impressive diversity of photosynthetic light-harvestingcomplexes [5], which are known to be highly versatile [6]and efficient in absorbing sunlight and transferring thesubsequent excitation [4, 7]. Understanding the sensitiveinterplay between monomer and super-structure (com-posed of monomers), and its influence on the optical,electronic and transport properties is highly desirable forthe synthesis of new materials [3], the design and opera-tion of organic-based devices [8], including solar cells [9],transistors, light-emitting diodes [10], and flexible elec-tronics [11]. Yet despite its fundamental role, the rela-tionship between molecular super-structure and physicalproperties lacks systematic quantitative understanding.In this Letter, we derive such a quantitative method, byestablishing the relation between the aggregate spectraand its constituent monomer building blocks; we furthercalibrate it against exact results, and apply the theoryto the important dynamical process of resonant energytransfer.

Optical excitations of organic compounds involve bothelectronic and vibrational degrees of freedom [12]. Whilethe exciton-phonon interaction is well understood formonomers [13–15], the electronic coupling in super-structures such as multichromophoric (MC) complexesdelocalizes the excitation [16] and therefore requires thetreatment of electron-vibrational coupling, excitonic cou-pling and disorder on an equal footing [17]. The availabletechniques, either exact such as stochastic path integral(sPI) [18] and hierarchical equation of motions (HEOM)[19–21], or approximate such as full-cumulant-expansion(FCE) [22], 2nd-order time convolution [23], time convo-lutionless [24, 25] and other recent developments [26, 27],are computationally expensive and not universally appli-

cable to relate the structure to optical properties. Thesetreatments require microscopic Hamiltonians and thusare not explicit about the structure–spectra relation. Ourapproach establishes such a relation: it allows us to pre-dict the physical properties of complex structures or, con-versely, infer the structure from its measured properties.

While construction of the optical properties is impor-tant in its own right, the spectra also provide addi-tional transport information. The transfer rates betweenweakly coupled excitonic systems can be obtained fromthe overlap of the donor emission and acceptor absorptionspectra using Förster resonant energy transfer (FRET)[28]. The original FRET theory describes the environ-ment through its effect on the monomer spectra. Exten-sions to MC systems [29–32], where the donor/acceptorare composed of coupled chromophores, demonstratedthat the far-field linear spectroscopic line shapes are in-sufficient; rather, the near-field, polarization-resolved ag-gregate spectra are needed to obtain the MCFT (MCFluorescent Transfer) rate in general. Though nonlin-ear spectroscopic experiments [33] could, in principle, beused, the required information is not accessible with cur-rent experimental techniques [32]. The theory developedhere solves this problem, allowing for the construction ofthe aggregate spectra from the experimentally accessiblemonomer spectra.

Our model is based on the coherent potential approx-imation (CPA) [34]: it treats the vibrational couplingexactly at the monomer level and includes all orders ofelectronic coupling, treated exactly up to the second or-der and approximately for higher orders. Benchmarksagainst exact sPI [18] and FCE [22] calculations showthat our model is reliable over a wide range of param-eters. Our theory applies to MCFT and recovers someaspects of the classical treatments [35–38] as a limitingcase. It completes the series of papers quantifying thereliability of different quantum models in MC systems[18, 22, 39].

Monomer spectra.—We introduce the notation and ex-act formulation of monomer spectra using the indepen-dent boson model [14]. Consider a single monomer cou-

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0 Ja

n 20

17

2

pled to a thermal phonon bath, both labeled by n, andcharacterized by the spin-boson Hamiltonian

H(n)0 = H

(n)0S +H

(n)B +H

(n)SB , (1)

where H(n)0S = EnB

†nBn denotes the Hamiltonian of

the electronic system, H(n)B =

∑k ~ωn,kb

†n,kbn,k that

of the bath, and H(n)SB = B

†nBn

∑k gn,k(b

†n,k + bn,k)

their coupling, which is taken linear in the bath co-ordinate. The operator B†n creates an excitation onmonomer n, forming the state B†n |vac〉 = |n〉; b

†n,k cre-

ates a phononic excitation in mode k. The excited stateenergy En = ~ω0n + λn includes the reorganization en-ergy λn =

∑k gn,k/ωn,k. Interaction with the elec-

tric field is treated semi-classically through the system-

radiation interaction Hamiltonian H(n)SR = µ̂n ·Eê, where

µ̂n = ~µngB†n + h.c. denotes the transition dipole moment

operator and ê is a unit vector.Following Sumi [34], we use the retarded Green’s func-

tion [14] associated with the monomer Hamiltonian (1),

G0nn(t) = −i

~Θ(t)e

i~H

(n)0 tBn e

− i~H(n)0 tB†n, (2)

to define the optical spectra—all given in units of en-ergy here. The absorption spectrum is obtained fromthe imaginary part of the Green’s function averaged overthe phonon bath,

〈G0nn(ω)

〉g, where 〈•〉g ≡ TrB [•ρ0g] de-

notes the trace over the bath using the density matrix ofthe system-bath in its ground state, ρ0g. The experimen-tally accessible spectrum includes the dipole transition,

I(n)exp(ω) = (ê · ~µgn) I(n)0 (ω) (~µng · ê), where

I(n)0 (ω) = −2Im

∫ ∞−∞

dt eiωt TrB[G0nn(t)ρ0g

]. (3)

This monomer spectrum can be evaluated exactly asfollows. Assuming a Franck-Condon transition from theground state, the initial state can be taken as the factor-

ized state ρ0g = 1S⊗ρB , where ρB = e−βH(n)B /Tr[e−βH

(n)B ]

is the bath density matrix at equilibrium, with β−1 =kBT . For a harmonic bath at thermal equilibrium, theabsorption lineshape (3) is exactly

I(n)0 (ω) =

2

~Re

∫ ∞0

dt eiωte−iωngte−gn(t), (4)

where ωng ≡ (En − Eg)/~ and Eg denotes the elec-tronic ground state energy. The lineshape functiongn(t) =

∫ t0dτ1∫ τ20dτ2Cn(τ2) is obtained from the bath

auto-correlation function Cn(τ) =1~2 〈H

(n)SB (τ)H

(n)SB (0)〉,

and can be evaluated exactly assuming a Drude spectraldensity, C ′′n(ω) ≡ 2λnΛω/(ω2 + Λ2), with Λ the cutofffrequency [13].

Steady-state emission occurs after the entire system-bath has equilibrated within the single-exciton manifold

and is obtained from〈G0nn(ω)

〉e, with 〈•〉e ≡ TrB [•ρ0e].

The initial state ρ0e entering the averaged Green’s func-tion is not factorized anymore, and the system-bath en-tanglement needs to be considered [22]. Instead of a di-rect calculation, we follow Refs. [18, 25, 29] and usethe detailed balance condition [40] to obtain the emis-sion spectrum from the absorption,

E(n)0 (ω) =

eβ~ω

Z(n)I(n)0 (ω), (5)

where Z(n) = Tr[e−βH(n)0S e−βH

(n)B ] is the monomer parti-

tion function.Multichromophoric spectra.—Consider now a system of

N coupled chromophores described by

H = H0 + V, (6)

where H0 =∑Nn=1H

(n)0 is the sum over N independent

monomers (1), which includes exciton-phonon interac-tion, and V =

∑n 6=m VnmB

†nBm characterizes the inter-

monomer coupling, typically of dipole-dipole nature. Theoperator B†n now denotes excitation of the nth monomerexclusively. Interaction with light is now characterized

by HSR =∑nH

(n)SR . We denote G and G0 the N × N

matrices formed by the Green’s functions associated, re-spectively, with the total Hamiltonian H and the unper-turbed Hamiltonian H0, which matrix elements are

Gnn′(t) = −i

~Θ(t)e

i~HtBn e

− i~HtB†n′ (7a)

G0nn′(t) = −i

~Θ(t)e

i~H0tBn e

− i~H0tB†n′ . (7b)

Note that the diagonal elements of G0 are equal tothe monomer Green’s functions (2). In principle, theMC Green’s function G can be exactly expressed usingthe unperturbed Green’s function G0 and the self en-ergy according to Dysons’ equation [14]. Tracing overthe phonon bath would then provide the MC absorp-tion and emission tensors, I(ω) = −2 Im 〈G(ω)〉g andE(ω) = −2 Im 〈G(ω)〉e, respectively. However, while anexact solution exists for single monomers (1), the inter-monomer coupling V in MC systems (6) tends to delocal-ize the electronic excitation and mix the vibrational andelectronic degrees of freedom. Evaluating the trace thenrequires methods numerically expensive [18], and oftenapproximate [22, 39].

The approximate theory derived here provides an ana-lytical expression in terms of the constituent spectra andstructural properties, allowing for an explicit relation be-tween the optical properties and the structure. Usingthe integral representation of the exponential, the bath-averaged total Green’s function can be expanded, in thetime or frequency domain, as [41]

〈G〉 = 〈G0〉+ 〈G0V G0〉+ 〈G0V G0V G0〉+ . . . , (8)

3

where the 〈•〉 denote the trace over the bath and thesubscript (g or e) characterizing the initial state will bespecified as needed. The 0th order term is simply thediagonal matrix of the monomeric Green’s function (2)with the proper initial state, i.e., 〈G0〉g/e = (〈G0nn〉g/e).The first-order term can be evaluated exactly for the fac-torized initial state, 〈•〉g, because (i) the trace then com-mutes with the bath density operator, (ii) there is oneand only one electronic transition involved, and (iii) theindividual baths are uncorrelated. The trace can thus besplit exactly,

〈G0V G0〉g = 〈G0〉gV〈G0〉g, (9)

where V = (Vnm) is a tensor. For the second-order termin Eq. (8), we neglect the phonon correlations and take〈

G0nn(ω)G0nn(ω)〉g≈〈G0nn(ω)

〉g

〈G0nn(ω)

〉g. (10)

This approximation is analogous to the decouplingscheme used in the single-site dynamical coherent po-tential approximation (CPA) [15] and will be referred assuch. It is the main approximation here, yielding to ourkey result. It allows simplifying all higher orders suchthat the full Green’s function (8) with the initial ground-state density matrix reduces to:

〈G(ω)〉g ≈

〈G0(ω)

〉g

1N −V 〈G0(ω)〉g. (11)

The CPA approach treats the bath coupling exactly atthe monomer level. It is exact up to the second-order ofintra-aggregate coupling V and includes all higher ordersapproximately. As such, our approach is more robustthan other methods derived for weak coupling [42, 43],especially in highly delocalized cases.

The MC absorption tensor is then simply

ICPA(ω) = −2 Im

〈G0(ω)

〉g

1N −V 〈G0(ω)〉g. (12)

The far-field, measurable absorption spectrum, is givenby Iexp(ω) =

∑nn′(ê · ~µn) Inn′(ω) (~µn′ · ê). Note that,

in calculating the full MC tensor, the CPA requiresknowledge of both the real and imaginary parts of themonomeric Green’s function

〈G0(ω)

〉g. The latter is ac-

cessible experimentally through the absorption spectraof the constituent monomers and their transition dipolemoments; the real part is related through the Kramer-Kronig relation [13]. The CPA approach therefore al-lows constructing the MC absorption tensor (12) fromits monomer features, either experimentally or theoret-ically accessible. Also, we show in Ref. [41] that Eq.(12) reduces to the tensor derived from a classical pic-ture of oscillating dipoles [37, 38], when the coupling isrestricted to dipole-dipole interaction. This suggests thatthe CPA approximation (10) is implicit in the classicalelectrostatic treatment of absorption [44].

0

0.05

0.1

0.15

I(ω)

E(ω)

(a)

Iexp(ω) & Eexp(ω) (cm-1)

sPIFCECPA

CPADB

0

0.05

0.1

0.15

-800 -600 -400 -200 0 200 400 600

I(ω)

E(ω)

(b)

ω (cm-1)

FIG. 1. Comparison of models for MC absorption Iexp(ω) andemission Eexp(ω) spectra, for (a) localized and (b) delocalizeddimers. Absorption (12) obtained from the CPA treatmentdeveloped here well matches sPI exact [18] and FCE approx-imate [22] results; emission spectra (13) resemble the sum ofthe individual spectra (not shown here), and are not accuratefor large inter-chromophore couplings (b). Adding detailedbalance, the CPADB (14) provides accurate results over a widerange of couplings. The difference between CPA and CPADBdisplays the influence of the system-bath entanglement, whichis important for emission. Parameters correspond to (a) caseI (V = 20 cm−1, ∆E21 ≡ E2 − E1= 100 cm−1) and (b) caseII (V = 100 cm−1, ∆E21 = 20 cm

−1) in Ref. [22] with λ=100cm−1, Λ=53 cm−1, T=300 K, and ê along ~µ.

Direct application of the CPA (10) with (g → e) yieldsthe emission tensor

ECPA(ω) = −2 Im〈G0(ω)

〉e

1N + V 〈G0(ω)〉e, (13)

where the initial density matrix is the equilibrium statein the first-excited manifold ρ0e. Because of the initialsystem-bath entanglement, the separation of averaging(9) with (g → e) is no longer exact, and the CPA is ap-proximate already in the first order of V for the emissiontensor. Numerical simulations show that the prediction issimilar to the sum of the monomer emission spectra (5),and therefore deviates from the exact solution for strongcoupling [Fig. 1(b)]. Instead of Eq. (13) and similarlyto the monomer treatment (5), we calculate the emissiontensor from the detailed balance (DB), which applies tothe total system as

E(ω) =eβ~ω

Tr[e−βH ]I(ω). (14)

We label this emission tensor by “CPADB” when usingEq. (12) for the absorption tensor I(ω). The normal-ization factor can be obtained either from direct sPI cal-culation [45] or from the absorption spectrum using the

4

mirror property of the spectra, i.e., E(ω0 − λ − ω) =I(ω0 +λ+ω)/Z0S , where Z0S ≡ Tr(e−β

∑nH

(n)0S ) is given

by the monomer system Hamiltonian [13, 18]. The emis-sion spectrum is then Eexp(ω) = ê · (~µT .E(ω).~µ) · ê.

Figure 1 presents the absorption and emission spectrafor the two dimers detailed in Ref. [22], i.e. for weak andstrong inter-chromophore coupling V . It is shown thatthe proposed treatment [Eqs. (12-14)] provides accuratepredictions for both spectra, even for relatively strongcoupling—V/λ = 1 in Fig. 1(b). The CPA with detailedbalance (14) greatly enhances the results over the CPAonly (13), thereby showing the importance of the bath’sfirst-order correlation function when the initial state isthe system-bath entangled density matrix. Comparisonswith the FCE over a wider range of parameters are pre-sented in Ref. [41].

Application to energy transfer rate.—Knowledge of thespectral tensors allows for the determination of the trans-fer rate between a donor (D) and an acceptor (A) aggre-gate using Fermi’s golden rule. We consider a system ofM -coupled donor and N -coupled acceptor chromophoresdescribed by the total Hamiltonian

HAD = HA +HD + JAD, (15)

where the MC Hamiltonian of the donor HD = HD0 +V D and that of the acceptor HA = HA0 + V

A is de-scribed by (6), changing Bn → Dn(An), respectively,and where the inter-chromophore coupling is V D =∑Mm 6=m′ V

Dmm′D

†mDm′ and V

A =∑Nn6=n′ V

Ann′A

†nAn′ .

JAD denotes the coupling between the donor-acceptorchromophores, i.e., JAD =

∑Nn

∑Mm J

ADnmA

†nDm + h.c.

The operators D†m and A†n, respectively, denote excita-

tion of the donor monomer m and the acceptor monomern.

The rate of multichromophoric Förster resonant en-ergy transfer is given by the overlap of the emission andabsorption tensors [22, 25, 32],

k =

∫ ∞−∞

dω

2πTr[JTED(ω)J IA(ω)

], (16)

where the matrix J = (JADnm ) denotes the donor-acceptorcoupling strength. The emission and absorption tensors,respectively ED(ω) = [EDmm′(ω)] and I

A(ω) = [IAnn′(ω)],are the polarization-resolved near-field spectral compo-nents, which are known to be necessary in MC systemsfor significant intra-donor (V D) or intra-acceptor (V A)couplings [32].

Using the derived treatment, specifically the CPA ab-sorption tensor (12) for the acceptor along with theCPADB emission tensor (14) for the donor, the MCFTrate becomes

k ≈∫ ∞−∞

dω

2πTr

[JT

eβ~ω

Tr[e−βHD ]2Im

(〈G0D(ω)〉g

1M −VD 〈G0D(ω)〉g

)×J 2Im

(〈G0A(ω)〉g

1N −VA 〈G0A(ω)〉g

)], (17)

0

0.2

0.4

k (ps-1)

(a) sPIFCECPA

CPADB

0

0.2

0.4

0.6

0.8

1

1 10 100 1000

(b)

λ (cm-1)

FIG. 2. Energy transfer rate (17) for different bath reor-ganization energies λ using the developed CPA and CPADBmodels to calculate absorption (12) and emission (13, 14) ten-sors for localized (a) and delocalized (b) systems. Compari-son with exact sPI results [18] show perfect matching of theCPADB for small electronic coupling (a), and a slight over-prediction of the rate for large coupling (b). The error us-ing only CPA comes from overpredicting the emission ten-sor (cf. Fig. 1). We used JADnm = 10 cm

−1; λD = λA andV Ann′ = V

Dmm′ = V . ∆E

A21 = ∆E

D21 = ∆E21 are as in Fig. (1).

where G0D (G0A) is a M ×M (N ×N) matrix formed bythe Green’s functions of the uncoupled monomers con-stituting the donor (acceptor) aggregate, i.e., defined byEq. (7b) changing Bn → Dn(An). This rate expressiononly requires the monomer bath-averaged Green’s func-tions 〈G0〉g, which includes the system-bath coupling ex-actly at the monomer level and can be evaluated exactlyfor a thermal bath (4) or determined experimentally. Allinfluence from electronic coupling is contained in the ma-trices describing intra-donor VD, intra-acceptor VA andinter donor-acceptor JAD couplings, and not restrictedto dipole-dipole coupling. The rate (17) is exact up tosecond order in the intra-aggregate couplings V and in-cludes all higher orders approximately.

Figure (2) presents the transfer rate for localized anddelocalized donor/acceptor (cases I&II in Ref. [22], re-spectively) for different reorganization energies λ. Com-parison with the exact path-integral calculations showsperfect agreement for the localized case (2a), and a slightoverprediction for highly delocalized MC systems Fig.(2(b).

The simplicity of our approach [Eq. (17)] allows pre-dicting the transfer rates over a wide range of structuralparameters. Figure (3) shows the rate as a function ofthe reorganization energy and intra-aggregate coupling Vfor systems with different electronic splittings ∆E21. Weclearly see an optimal bath-coupling strength, confirming

5

0.1

0.1

0.1

0.2

0.2

0.3

0.4

100 101 102 103

6 (cm-1)

0

50

100

150

200V

(cm

-1)

0.1

0.1

0.1

0.1 0.2

0.2

0.2

0.3

0.3 0.4

0.4 0

.5

0.6

100 101 102 103

6 (cm-1)

0

50

100

150

200

V (c

m-1

)(a)

V(c

m�

1)

V(c

m�

1)

(b)

� (cm�1)

FIG. 3. Contour map of the transfer rate as functionof the reorganization energy λ = λA = λD and the intra-aggregate coupling strength V = V Ann′ = V

Dmm′ . Panel (a)

exhibits an optimal intra-aggregate coupling strength aroundV ∼ 50cm−1, and panel (b) clearly shows the optimal bathcoupling. The different behaviors between (a) and (b) steam

from different electronic splitting ∆EA(D)12 , respectively, (a)

100 and (b) 20 cm−1.

environment-assisted quantum transport [46]. Interest-ingly, Fig. 3(a) also exhibits an optimal intra-aggregatecoherence (V∼50 cm−1), which was not previously re-ported. The existence of such optimum depends on thesystem configuration, as seen from the comparison withrates in a system with smaller energy gap, Fig. 3(b).While this dependance requires further investigation, ourresults confirm that intra-aggregate couplings can en-hance transfer, which is in line with Refs. [32, 37, 47–49].

In summary, we extended the applicability of the CPAto absorption and emission tensors of multichromophoricsystems, and showed accurate results over a surprisinglywide range of structure parameters. This approach nowallows for a reliable prediction of the MCFT rate, whichreveals that, additionally to optimal environment cou-plings, the intra-aggregate coupling can be optimized toenhance transport. Our treatment identifies the correc-tion terms, and recovers the classical absorption tensoras a limiting case, suggesting that first-order bath corre-lations are neglected classically. Our model could be fur-ther extended to include the off-diagonal bath coupling,introduced through electronic coupling, using, e.g., thetwo-particle dynamical CPA [50].

Beyond fast and reliable characterization of multi-chromophic complexes, a quantitative relation betweenphysical properties and aggregate structure is estab-lished. This straightforward approach is based on spec-troscopic measurements and does not require a micro-scopic Hamiltonian. It allows us to explore a large spaceof structure parameters and optimize the aggregate struc-

ture based on its optical and transport properties. Assuch, we anticipate that it will be a relevant tool to exper-imentally and theoretically describe electronic excitationand excitonic energy transfer.

Acknowledgments.—We thank P. Brumer for interest-ing discussions on the topic and A. del Campo for com-ments on the manuscript. We acknowledge funding fromthe Swiss National Science Foundation (A.C.) and theNSF (Grant No. CHE-1112825).

∗ achenu@mit.edu† jianshu@mit.edu

[1] S. M. Douglas, H. Dietz, and T. Liedl, Nature 459, 414(2009); G. Mayer and M. Sarikaya, Exp. Mech. 42, 395(2002); V. Kos and R. Ford, Cellular and Molecular LifeSciences 66, 311 (2009).

[2] G. Whitesides, J. Mathias, and C. Seto, Science 254,1312 (1991); S. Zhang, Nat. Biotech. 21, 1171 (2003).

[3] U. Scherf and E. List, Adv. Mat. 14, 477 (2002).[4] H. van Amerongen, L. Valkunas, and R. van Gron-

delle, Photosynthetic Excitons (World Scientific, Singa-pore, 2000).

[5] R. J. Cogdell, A. T. Gardiner, H. Hashimoto, andT. H. P. Brotosudarmo, Photochem. Photobiol. Sci. 7,1150 (2008).

[6] X. Hu, A. Damjanović, T. Ritz, and K. Schulten, Proc.Natl. Acad. Sci. U. S. A. 95, 5935 (1998); G. D. Scholes,T. Mirkovic, D. B. Turner, F. Fassioli, and A. Buchleit-ner, Energy Environ. Sci. 5, 9374 (2012).

[7] I. McConnell, G. Li, and G. W. Brudvig, Chem. & Bio.17, 434 (2010).

[8] F. Spano and C. Rivas, Annu. Rev. Phys. Chem. 65, 477(2014).

[9] J.-L. Brédas, J. E. Norton, J. Cornil, and V. Coropceanu,Acc. Chem. Res. 42, 1691 (2009); A. Heeger, Chem. Soc.Rev. 39, 2354 (2010); F. Goubard and F. Dumur, RSCAdv. 5, 3521 (2015).

[10] K. M. U. Sherf, Organic Light Emitting Devices: Syn-thesis, Properties and Applications (New York: Wiley,2006).

[11] E. Orgiu and P. Samoŕı, Adv. Mat. 26, 1827 (2014).[12] Y. Levinson and E. Rashba, Rep. Prog. Phys. 36, 1499

(1973).[13] S. Mukamel, Principles of nonlinear spectroscopy (Ox-

ford University Press, Oxford, 1995) Ch. 8.[14] G. D. Mahan, Many-Particle Physics, 3rd ed. (Kluwer

Academic Publishers, 2000) Ch. 2 and Ch. 4.[15] V. Broude, E. Rashba, and E. Sheka, Spectroscopy of

Molecular Excitons, edited by V. Goldanskii (SpringerSeries in Chemical Physics, 1985).

[16] A. S. Davydov, Theory of molecular excitons (McGraw-Hill, New York, 1962).

[17] A. Chenu and G. Scholes, Annu. Rev. Phys. Chem. 66,69 (2015).

[18] J. M. Moix, J. Ma, and J. Cao, J. Chem. Phys. 142,094108 (2015).

[19] A. Ishizaki and G. R. Fleming, J. Chem. Phys. 130,234111 (2009).

[20] J. Strümpfer and K. Schulten, J. Chem. Phys. 134,

mailto:achenu@mit.edumailto:jianshu@mit.eduhttp://dx.doi.org/10.1038/nature08016http://dx.doi.org/10.1038/nature08016http://dx.doi.org/10.1007/BF02412144http://dx.doi.org/10.1007/BF02412144http://dx.doi.org/10.1007/s00018-009-0064-9http://dx.doi.org/10.1007/s00018-009-0064-9http://dx.doi.org/10.1126/science.1962191http://dx.doi.org/10.1126/science.1962191http://dx.doi.org/10.1038/nbt874http://dx.doi.org/10.1002/1521-4095(20020404)14:73.0.CO;2-9http://dx.doi.org/10.1039/B807201Ahttp://dx.doi.org/10.1039/B807201Ahttp://dx.doi.org/ 10.1073/pnas.95.11.5935http://dx.doi.org/ 10.1073/pnas.95.11.5935http://dx.doi.org/ 10.1039/C2EE23013Ehttp://dx.doi.org/http://dx.doi.org/10.1016/j.chembiol.2010.05.005http://dx.doi.org/http://dx.doi.org/10.1016/j.chembiol.2010.05.005http://dx.doi.org/10.1146/annurev-physchem-040513-103639http://dx.doi.org/10.1146/annurev-physchem-040513-103639http://dx.doi.org/10.1021/ar900099hhttp://dx.doi.org/10.1039/B914956Mhttp://dx.doi.org/10.1039/B914956Mhttp://dx.doi.org/10.1039/C4RA11559Ghttp://dx.doi.org/10.1039/C4RA11559Ghttp://dx.doi.org/10.1002/adma.201304695http://iopscience.iop.org/0034-4885/36/12/001http://iopscience.iop.org/0034-4885/36/12/001http://dx.doi.org/10.1146/annurev-physchem-040214-121713http://dx.doi.org/10.1146/annurev-physchem-040214-121713http://dx.doi.org/ 10.1063/1.4908601http://dx.doi.org/ 10.1063/1.4908601http://dx.doi.org/10.1063/1.3155372http://dx.doi.org/10.1063/1.3155372http://dx.doi.org/10.1063/1.3557042

6

095102 (2011).[21] Y. Jing, L. Chen, S. Bai, and Q. Shi, J. Chem. Phys.

138, 045101 (2013).[22] J. Ma and J. Cao, J. Chem. Phys. 142, 094106 (2015).[23] T. Renger and V. May, Phys. Rev. Lett. 84, 5228 (2000).[24] T. Renger and R. A. Marcus, J. Chem. Phys. 116, 9997

(2002).[25] L. Banchi, G. Costagliola, A. Ishizaki, and P. Giorda, J.

Chem. Phys. 138, 184107 (2013).[26] T.-C. Dinh and T. Renger, J. Chem. Phys. 142, 034104

(2015).[27] A. Gelzinis, D. Abramavicius, and L. Valkunas, J. Chem.

Phys. 142, 154107 (2015).[28] T. Förster, in Moden Quantum Chemistry, Vol. III.,

edited by O. Sinanoǧlu (1965) p. 93.[29] H. Sumi, J. Phys. Chem. B 103, 252 (1999).[30] K. Mukai, S. Abe, , and H. Sumi, J. Phys. Chem. B 103,

6096 (1999).[31] G. D. Scholes, Annu. Rev. Phys. Chem. 54, 57 (2003).[32] S. Jang, M. D. Newton, and R. J. Silbey, Phys. Rev.

Lett. 92, 218301 (2004).[33] G. D. Scholes and G. R. Fleming, J. Phys. Chem. B 104,

1854 (2000).[34] H. Sumi, J. Phys. Soc. Jpn. 36, 770 (1974).[35] H. Kuhn, J. Chem. Phys. 53, 101 (1970).[36] R. R. Chance, A. Prock, and R. Silbey, J. Chem. Phys.

62, 2245 (1975).[37] E. N. Zimanyi and R. J. Silbey, J. Chem. Phys. 133,

144107 (2010).[38] S. Duque, P. Brumer, and L. A. Pachón, Phys. Rev.

Lett. 115, 110402 (2015).[39] J. Ma, J. M. Moix, and J. Cao, J. Chem. Phys. 142,

094107 (2015).[40] N. V. Kampen, Stochastic Processes in Physics and

Chemistry (Elsevier, 2007).[41] A. Chenu and J. Cao, Supplementary Material: link

between the CPA and the classical approach and furthercomparison with FCE results.

[42] M. Yang, J. Chem. Phys. 123, 124705 (2005).[43] T. Mančal, V. Balevičius, and L. Valkunas, J. Phys.

Chem. A 115, 3845 (2011).[44] J. Cao and B. J. Berne, J. Chem. Phys. 99, 6998 (1993).[45] J. M. Moix, Y. Zhao, and J. Cao, Phys. Rev. B 85,

115412 (2012).[46] V. Chorošajev, A. Gelzinis, L. Valkunas, and D. Abra-

mavicius, J. Chem. Phys. 140, 244108 (2014).[47] Y. C. Cheng and R. J. Silbey, Phys. Rev. Lett. 96, 028103

(2006).[48] K. Kim, M.-S. Chang, S. Korenblit, R. Islam, E. E. Ed-

wards, J. K. Freericks, G.-D. Lin, L.-M. Duan, andC. Monroe, Nature 465, 590 (2010).

[49] J. Wu, R. J. Silbey, and J. Cao, Phys. Rev. Lett. 110,200402 (2013).

[50] R. Friesner and R. J. Silbey, Chem. Phys. Lett. 93, 107(1982); R. E. Lagos and R. A. Friesner, Phys. Rev. B29, 3045 (1984).

http://dx.doi.org/10.1063/1.3557042http://dx.doi.org/ http://dx.doi.org/10.1063/1.4775843http://dx.doi.org/ http://dx.doi.org/10.1063/1.4775843http://dx.doi.org/10.1063/1.4908599http://dx.doi.org/10.1103/PhysRevLett.84.5228http://dx.doi.org/10.1063/1.1470200http://dx.doi.org/10.1063/1.1470200http://dx.doi.org/10.1063/1.4803694http://dx.doi.org/10.1063/1.4803694http://dx.doi.org/http://dx.doi.org/10.1063/1.4904928http://dx.doi.org/http://dx.doi.org/10.1063/1.4904928http://dx.doi.org/http://dx.doi.org/10.1063/1.4918343http://dx.doi.org/http://dx.doi.org/10.1063/1.4918343http://dx.doi.org/10.1021/jp983477uhttp://dx.doi.org/ 10.1021/jp984469ghttp://dx.doi.org/ 10.1021/jp984469ghttp://dx.doi.org/10.1146/annurev.physchem.54.011002.103746http://dx.doi.org/10.1103/PhysRevLett.92.218301http://dx.doi.org/10.1103/PhysRevLett.92.218301http://dx.doi.org/10.1021/jp993435lhttp://dx.doi.org/10.1021/jp993435lhttp://dx.doi.org/10.1143/JPSJ.36.770http://dx.doi.org/10.1063/1.1673749http://dx.doi.org/10.1063/1.430748http://dx.doi.org/10.1063/1.430748http://dx.doi.org/http://dx.doi.org/10.1063/1.3488136http://dx.doi.org/http://dx.doi.org/10.1063/1.3488136http://dx.doi.org/10.1103/PhysRevLett.115.110402http://dx.doi.org/10.1103/PhysRevLett.115.110402http://dx.doi.org/ 10.1063/1.4908600http://dx.doi.org/ 10.1063/1.4908600http://dx.doi.org/http://dx.doi.org/10.1063/1.2046668http://dx.doi.org/10.1021/jp108247ahttp://dx.doi.org/10.1021/jp108247ahttp://dx.doi.org/10.1063/1.465446http://dx.doi.org/ 10.1103/PhysRevB.85.115412http://dx.doi.org/ 10.1103/PhysRevB.85.115412http://dx.doi.org/http://dx.doi.org/10.1063/1.4884275http://dx.doi.org/10.1103/PhysRevLett.96.028103http://dx.doi.org/10.1103/PhysRevLett.96.028103http://dx.doi.org/10.1103/PhysRevLett.110.200402http://dx.doi.org/10.1103/PhysRevLett.110.200402http://dx.doi.org/10.1016/0009-2614(82)83674-9http://dx.doi.org/10.1016/0009-2614(82)83674-9http://dx.doi.org/10.1103/PhysRevB.29.3045http://dx.doi.org/10.1103/PhysRevB.29.3045

7

Appendix

Derivation of the CPA absorption tensor.— We givehere the details that lead to the total Green’s function(11) used to obtain the absorption tensor (12). We startfrom the definition of the total Green’s function in thetime domain (7a) and write the last exponential usingthe integral representation

e−iH~ t =

∞∫−∞

dω

−i2πe−i(ω+i0+)t

ω − (H0+V )~ + i0+. (S1)

Because the solution for the non-interacting monomersis known, we use the fact that the Hamiltonian splitsinto H = H0 + V and obtain the response function fromiteration of the identity:

1

ω −H=

1

ω −H0+

1

ω −HV

1

ω −H0. (S2)

Using the convolution theorem and transforming in theFourier domain, the total Green’s function becomes

Gnn′(ω) = G0nn′(ω) + G0nn(ω)Vnn′G0n′n′(ω) (S3)+∑m

G0nn(ω)VnmG0mm(ω)Vmn′G0n′n′(ω) + . . . .

The first term on the l.h.s. is only non-zero for δnn′ .Hence we see that the total Green’s function onlydepends on the monomeric function G0 in-between theinteractions in that all functions are diagonal in thesystem basis. Taking the thermal average over thephonon bath with an initial ground state and using theapproximation (10) yields to the matrix representation(8), from which we can obtain the total absorptiontensor (12).

Link to the classical MCFT.— In a recent study[38], the MCFT was derived from classical electrody-namics. Using the picture of classical oscillating dipoles,the MCFT rate (16) was recast into the overlap of anequivalent emission and absorption spectra, defining

EDmm′(ω) = (pDm(ω))

∗ (pDm′(ω)) (S4a)

IAnn′(ω) = ω Im[χ−1A (ω)/�0 − Φ

A]−1nn′ , (S4b)

where pDm(ω) denotes the polarization of the m-th donormolecule. χ(ω) is the polarizability tensor and relatesthe polarization to the total electric field (external andinduced) as pn(ω) = �0χn(ω)E(rn, ω). The dipolesare coupled through dipole-dipole coupling, denoted bythe Φ matrix, such that E(rn, ω) = Eext(rn, ω) +∑nn′ Φnn′pn′(ω). Considering that only the donor com-

plex is excited by the external field, the classically-definedabsorption spectra (S4b) from [38] takes the form

ICED(ω) = ωIm

[�0χA(ω)

1−ΦA�0χA(ω)

](S5)

This matches the derived absorption tensor used in (17),up to a normalizing factor and a factor ω arising forthe different definitions between susceptibility and ab-sorption coefficient, taking 〈G0A(ω)〉 → �0χA(ω) andVA → ΦA, i.e. when we restrict the electronic couplingin our model Hamiltonian (6) to dipole-dipole interac-tion.

Emission, in turn, is a purely quantum phenomena.We look here at the classically defined emission spectrato see how it compares with the quantum definition de-rived in this paper. First-order response theory allowsto obtain the classical polarization from the total field.Solving the system of linear equations yields

p(ω) =�0χ(ω)

1− Φ�0χ(ω)·Eext(ω). (S6)

The classically defined emission spectra (S4a) thendepends on the polarization of the acceptor through thesusceptibility tensor. This is well understood by thePurcell effect, according to which the polarization ofa dipole in a medium depends on scattered electricalfield. However, it is different from the model developedhere, where the donor emission spectrum can be definedindependently of the acceptor state, and where thedonor-acceptor coupling only enters in the rate equation.

Further results.— We present here further compar-ison of the CPA approximation with the FCE [22] forthe absorption (Fig. S1) and emission spectra (Fig. S2),and for the rate (Fig. S3), respectively given by Eqs.(12), (14) and (17). We define the ‘relative difference’ (in%) between the experimental spectra obtained from themethod ‘X’ and the exact spectra as obtained reliablyfrom the FCE as, for absorption,

�X =

∫dω |IFCE(ω)− IX(ω)|∫

dω IFCE(ω), (S7)

and equivalently for the emission with I(ω)→ E(ω). Theelectronic energy splitting is (a) ∆E12 = 100 cm

−1 and(b) ∆E12 = 20 cm

−1 in all figures below. Very goodagreement is found for all the tested regimes, with lessthat 2% error between the FCE and CPADB results.

8

FIG. S1. Relative difference �CPA (%) of the CPA methodcompared to the FCE model for the absorption spectra calcu-lated from the tensor (12) as a function of the reorganizationenergy λ and the intra-aggregate coupling Vnn′ = V showingvery good agreement. (a) ∆E12 = 100 cm

−1 and (b) ∆E12 =20 cm−1.

FIG. S2. Relative difference �CPADB (%) of the CPADBmethod compared to the FCE model for the emission spectracalculated from the tensor (14) as a function of the reorga-nization energy λ and the intra-aggregate coupling Vnn′ = Vshowing very good agreement. (a) ∆E12 = 100 cm

−1 and (b)∆E12 = 20 cm

−1.

FIG. S3. Relative difference �CPADB (%) of the CPADBmethod compared to the FCE model for the transfer rate (17)as a function of the reorganization energy λ and the intra-aggregate coupling Vnn′ = V showing very good agreement.(a) ∆E12 = 100 cm

−1 and (b) ∆E12 = 20 cm−1.

Construction of Multichromophoric Spectra from Monomer Data: Applications to Resonant Energy TransferAbstract Acknowledgments References Appendix