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Structure-Based Theory of FluctuationInduced Energy Transfer in a Molecular Dyad Thomas Renger, Mathias Dankl, Alexander Klinger, Thorben Schlücker, Heinz Langhals, and Frank Müh J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/acs.jpclett.8b02403 • Publication Date (Web): 24 Sep 2018 Downloaded from http://pubs.acs.org on September 26, 2018

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The Journal of Physical Chemistry Letters

Structure-Based Theory of Fluctuation-Induced Energy Transfer in a Molecular Dyad Thomas Renger,∗,† Mathias Dankl,† Alexander Klinger,† Thorben Schlücker,‡ Heinz Langhals,‡ and Frank Müh† Institute of Theoretical Physics, Department of Theoretical Biophysics, Johannes Kepler University Linz, Altenberger Str. 69, 4040 Linz, Austria, and Department of Chemistry, LMU University of Munich, Butenandtstr. 13, D-81377 Munich, Germany E-mail: [email protected]

∗

To whom correspondence should be addressed Institute of Theoretical Physics, Department of Theoretical Biophysics, Johannes Kepler University Linz, Altenberger Str. 69, 4040 Linz, Austria ‡ Department of Chemistry, LMU University of Munich, Butenandtstr. 13, D-81377 Munich, Germany †

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ACS Paragon Plus Environment


Abstract We present a microscopic theory for the description of fluctuation-induced excitation energy transfer in chromophore dimers to explain experimental data on a perylene biscarboximide dyad with orthogonal transition dipole moments. Our non-Condon extension of Förster theory takes into account the fluctuations of excitonic couplings linear and quadratic in the normal coordinates, treated microscopically by quantum chemical/electrostatic calculations. The modulation of the optical transition energies of the chromophores is inferred from optical spectra of the isolated chromophores. The application of the theory to the considered dyad reveals a two-to-three order of magnitude increase of the rate constant by non-Condon effects. These effects are found to be dominated by fluctuations linear in the normal coordinates and provide a structurebased qualitative interpretation of the experimental time constant for energy transfer as well as its dependence on temperature.

6

-1

k

coupling / cm

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

3

0

-3

-6 0

1

2 time / ps

3

Förster resonance energy transfer (FRET) 1–3 between molecules describes a non-radiative transfer of excitation energy from a donor chromophore, that is initially in an electronic excited state, to an acceptor chromophore, initially in the electronic ground state. If the two molecules are sufficiently separated in space, no wavefunction overlap occurs and the electrons stay at their respective molecules. Just the excitation energy is transferred by an electronic transition of the donor to the ground state and a simultaneous excitation of the acceptor. 2


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In standard Förster theory the rate constant kd→a for excitation energy transfer between donor and acceptor is given as kd→a =

2π |V |2 OIα , ~2

where V is the Coulomb coupling between

the transition densities of the chromophores, termed excitonic coupling, and OIα is the overlap integral between the normalized emission lineshape function of the donor and the normalized absorption lineshape function of the acceptor. OIα contains the effect of the intrachromophore electron-vibrational coupling modulating the electronic transition energies of the chromophores. Excitation energy transfer between the donor (with high transition energy) and the acceptor (with low transition energy) involves the transition to excited vibrational states in the ground state potential energy surface (PES) of the donor and in the excited state PES of the acceptor. In this way energy conservation is fulfilled in the transition between the electronic states, and the subsequent intramolecular vibrational relaxation in these PES stabilizes the excitation energy at the acceptor chromophore, giving the transfer a direction. If the excitonic coupling is large compared to the difference in excitation energies between the chromophores and the exciton-vibrational coupling, delocalized excited states are formed. 4 Mediated by the exciton-vibrational coupling, the excitation energy can relax from a quantum mechanically mixed upper “donor-acceptor” exciton state to a lower one. In cases where the energy difference is small and the excitonic and the exciton-vibrational coupling are of similar strength, the exciton-vibrational dynamics gets more complicated and the excited states get partially localized by the nuclear dynamics. 5–8 Most of the natural and artificial light-harvesting aggregates fall into this intermediate coupling regime. 9–13 In the present work we will focus on the weak coupling limit, where the energy difference between donor and acceptor is orders of magnitude larger than the excitonic coupling and the excited states are always localized on the donor or the acceptor molecule. In this case, often a point-dipole approximation (PDA) for the excitonic coupling is valid. The resulting R−6 dependence of the excitation energy transfer rate constant on interchromophore distance R is exploited in FRET experiments to estimate intermolecular distances, based on measured

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energy transfer efficiencies. 3,14–18 Subtleties in the interpretation of these experiments concern the validity of the PDA, 3,18–22 the description of screening/local field corrections of the excitonic coupling 3,23–25 and the treatment of interchromophore conformational motion. 3,26–28 The latter aspect is of particular importance for complexes, where donor and acceptor exhibit an unfavorable equilibrium structure (transition dipole orientation) for energy transfer. 29–34 Due to interchromophore vibrations, the system can reach conformations with larger excitonic coupling enhancing energy transfer. In standard Förster theory, a Condon approximation is applied, which neglects the dependence of the excitonic coupling on the nuclear coordinates and, hence, cannot describe such an enhancement. The present work goes beyond this approximation and was stimulated by recent experiments on molecular dyads (i.e. donor and acceptor are covalently linked), where the two chromophores have orthogonal transition dipole moments, 29,30,32–34 and by theoretical studies on non-Condon effects in excitation energy and electron transfer. 35–37 As a prototypical example, we study a dyad consisting of a benzoperylene donor, a spacer, and a perylene acceptor (Fig. 1). Due to the orthogonal transition dipole moments, the excitonic coupling in point-dipole approximation (PDA) is zero and no excitation energy transfer should occur according to Förster theory. Nevertheless, efficient energy transfer was detected 29,38 with a time constant in the 10 ps time range, 29 where the exact value depends somewhat on the solvent. A numerically exact evaluation of the excitonic coupling with the transition density cube method 19 allowed the authors of the experimental study 29 to go beyond the PDA, but could not explain the fast energy transfer. Electron exchange as the dominant energy transfer mechanism, giving rise to an exponential distance dependence of the rate constant, could be excluded based on measurements on dyads containing spacers with variable length. 29,37 Based on the increase of the rate constant with increasing temperature, further evidence for an earlier hypothesis 38 that energy transfer in the dyad is induced by thermal fluctuations causing a non-orthogonal configuration of transition dipole moments of the chromophores, was obtained. 29 Indeed, some low-frequency normal modes

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were found, along which a modulation of the excitonic coupling was seen. 29 A first theoretical check of this idea was presented by Nalbach et al., 37 who used an empirical model for the fluctuations in excitonic couplings and demonstrated that such a model in principle can explain the experimental data. A simplification of this model, besides the missing microscopic parameterization, is that no fluctuations of the transition energies of the chromophores were included. The latter will turn out to be important for bridging the electronic energy gap between donor and acceptor in the present work, because the modulation of the excitonic (A)

(B)

(C)

Figure 1: (A) Chemical structure of the full perylene biscarboximide dyad with the benzoperylene donor chromophore on the left, the spacer in the middle and the perylene acceptor chromophore on the right. (B) Simplified model of the dyad, where the 1-hexylheptyl (“swallow-tail”) substituents have been truncated to 1-methylethyl groups. (C) Electrostatic potential of the transition density of the donor and the acceptor chromophore, obtained quantum chemically for the simplified model shown in (B) with the HF-CIS method based on a geometry that was optimized with DFT and the B3LYP XC-functional, as described in the text.

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coupling is dominated by interchromophoric vibrations with orders of magnitude smaller frequencies. In the following, a non-Condon extension of Förster theory is presented taking into account the above effects. The theory is based on the Hamiltonian

H=

X

i=d,a

Ei (Q)|iihi| + V (Q)(|diha| + |aihd|) +

X ~ωη

(e)

η

4

(Pη2 + Q2η )

(1)

(g)

of the molecular dyad. In the electronic state |di = |ϕd ϕa i of the dyad, the donor is in its first excited (e) state and the acceptor is in its ground (g) state, whereas in |ai = (g)

(e)

|ϕd ϕa i, the donor is in the ground state and the acceptor is excited. The dependence of the excitation energies and the excitonic coupling on the nuclear coordinates is comprised in Ei (Q) and V (Q), respectively. In the spirit of a normal mode analysis (NMA), the vibrational Hamiltonian is given as a sum over Hamiltonians of uncoupled harmonic oscillators η, with frequency ωη , dimensionless coordinate Qη , and momentum Pη . (i)

We divide the normal coordinates Qη into a subset Qσ of intrachromophoric modes, which are responsible for the modulation of the transition energy Ei of the donor (i = d) and the acceptor (i = a), and a subset of interchromophoric modes Qξ , which modulate the excitonic coupling V . Support for this discrimination, also used in an earlier model study, 36 will be provided later from an analysis of the coupling and transition energy fluctuations of the considered molecular dyad. A linear coupling approximation for Ei (Q) is used

Ei (Q) ≈ Ei (Q0 ) +

X

~ωσ(i) gσ(i) Q(i) σ .

(2)

σ

Here, Ei (Q0 ) denotes the transition energy at the equilibrium position of nuclei Q0 in the electronic ground state of the dyad. Using a second-order approximation for the coordinate dependence of the excitonic cou-

6


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pling V (Q) results in

V (Q) ≈ V0 +

X

~ωξ gξ Qξ +

ξ

XX √ √ ~ ωξ ωξ′ gξ,ξ′ Qξ Qξ′ , ξ

(3)

ξ′

where V0 is the excitonic coupling at Q0 . The coupling constants gξ and gξ,ξ′ are obtained from a combination of the transition charges from electrostatic potential method (TrEsp) 21 for the excitonic coupling and a NMA, as described in detail in the supporting information (SI). The result for the linear coupling constant gξ reads 39

gξ =

(ξ)

(d) (a)

1

X

3/2 (2~)1/2 ωξ I,J

qI qJ (0) |RI

−

(0)

(0) RJ | 3

(0)

(RI − RJ ) · (ξ)

AJ

1/2

MJ

(ξ)

−

AI

1/2

MI

!

,

(4)

(ξ)

where ωξ is the frequency of normal mode ξ, AI and AJ are the components of the Ith atom of the donor and the Jth atom of the acceptor, respectively, in the eigenvector of this (d)

normal mode, qI

(a)

and qJ are atomic transition charges of these two atoms, and MI and

MJ are the respective atomic masses. The expression for the quadratic coupling constant gξ,ξ′ , not included in our earlier work, 39 is given in the SI, eq 4. The rate constant kd→a for excitation energy transfer between donor and acceptor in second-order perturbation theory in the excitonic coupling, using also a Markov approximation, is given as 40

kd→a

1 = 2 ~

Z

∞ −∞

o n dt eiωda t trvib Ud† (t)V Ua (t)V Wdeq .

(5)

Here, ~ωda is the energy difference between the minima of the potential energy surfaces of the donor and the acceptor, Ud† (t) denotes the hermitian conjugate of the time-evolution operator of the vibrational degrees of freedom in the initial electronic state |di, while Ua (t) is the time-evolution operator of the vibrations in the final electronic state |ai. We have assumed that after optical excitation of the donor chromophore, the vibrations relax fast compared to excitation energy transfer, as described in eq 5 by the equilibrium statistical 7



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operator of the vibrations in the initial electronic state Wdeq . Since in our model only the intrachromophoric degrees of freedom modulate the transition energies in eq 2, the thermal average in eq 5 can be factorized into an intrachromophoric and an interchromophoric part, as described in detail in the SI, resulting in the rate constant

kd→a

1 = 2 ~

Z

∞

dteiωda t eGd (t)−Gd (0) eGa (t)−Ga (0) F (t) .

(6)

dωJi (ω) (1 + n(ω))e−iωt + n(ω)eiωt

(7)

−∞

Here, Gi (t) =

Z

∞ 0

contains the Bose-Einstein distribution function of vibrational quanta

n(ω) =

1 e~ω/kB T

(8)

−1

and the spectral density Ji (ω) of the intrachromophoric electron-vibrational coupling

Ji (ω) =

X ν

Sν(i) δ(ω − ων(i) ) + Jicl (ω)

(9)

of the donor (i = d) and the acceptor (i = a). For simplicity and to keep the number of parameters small, we have divided Ji (ω) such that a few effective-high frequency modes ν are separated and the remaining modes are treated classically. In doing so, we assume that the vibrational quanta of the latter are small compared to the thermal energy, ~ω > kB T . Si = ν Sν is given as a 8


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(i)

(i)

sum over the Huang-Rhys factors Sν of the high-frequency modes and Eλ =

R∞ o

dω~ωJicl (ω)

is the reorganization energy of the low-frequency (classical) part of the intrachromophoric spectral density. The function F (t) in eq 6, resulting from the thermal average of the interchromophoric degrees of freedom in eq 5, reads (inter)

F (t) = trvib

n

(vib)

eih0

t/~

(vib)

V e−ih0

t/~

o X V W0eq = Fij (t) ,

(11)

i,j=0,2

that contains contributions in different orders i and j of the first and second excitonic coupling V under the trace in eq 11 in their dependence on the nuclear coordinates (see eq 3). The lowest order term F00 = V02 leads to the standard Förster theory rate constant. All higher orders are non-Condon contributions. The term containing the contribution of the excitonic coupling that is linear in the vibrational coordinates reads F11 (t) = ~2

X ξ

gξ2 ωξ2 (1 + n(ωξ ))e−iωξ t + n(ωξ )eiωξ t ,

(12)

and F01 = F10 = 0. The higher-order contributions F12 and F21 are also zero, and the expressions for the non-zero second-order contributions F02 = F20 and F22 (t) are given in SI eqs 9 and 10, respectively. Finally, the rate constant in eq 6 becomes, using eqs 10, 11 - 12 and SI eqs 9 and 10, (0)

(1)

(2)

kd→a = kd→a + kd→a + kd→a ,

(13)

(s)

where the different contributions kd→a are given as (s) kd→a

=e

−(Sa +Sd )

X X (S (d) )n1 (S (d) )n2 X X (S (a) )n1 (S (a) )n2 n1

n2

1

2

1

2

n1 !

n2 !

n1 !

n2 !

m1

m2

Fn(s) . 1 ,n2 ,m1 ,m2

(14)

Here, we have taken into account two explicit high-frequency intrachromophoric modes per

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chromophore. An inclusion of more modes is straightforward. The contribution with s = 0 in eq 14 is the Förster rate constant resulting from Fn(0) = 1 ,n2 ,m1 ,m2

|V0 |2 Dn1 ,n2 ,m1 ,m2 (ω = 0) , ~2

(15)

with √

π~ Dn1 ,n2 ,m1 ,m2 (ω) = q (d) (a) kB T (Eλ + Eλ )  2  P P (d) (a) (d) (a) ~(ω + ω − n ω − m ω ) − E − E da ν µ λ λ ν ν ν µ   × exp − . (d) (a) 4(Eλ + Eλ )kB T

(16)

The s = 1 term on the r.h.s. of eq 14 contains the contribution from F11 (t) in eq 12 and is obtained from Fn(1) = 1 ,n2 ,m1 ,m2

X

gξ2 ωξ2 ((1 + n(ωξ ))Dn1 ,n2 ,m1 ,m2 (ωξ ) + n(ωξ )Dn1 ,n2 ,m1 ,m2 (−ωξ ))

(17)

ξ

with the n(ωξ ) in eq 8 and the function Dn1 ,n2 ,m1 ,m2 (ω = ±ωξ ) in eq 16. The remaining (2)

contributions, originating from F22 (t), F20 (t) and F02 (t) are comprised in Fn1 ,n2 ,m1 ,m2 that is (0)

given in SI eq 11. The Förster rate constant kd→a can be expressed in terms of the overlap integral

OIα =

Z

(d)

dωDI (ω)Dα(a) (ω)

(18)

(d)

between the normalized fluorescence lineshape function DI of the donor and the normalized (a)

(0)

V2

absorption lineshape function of the acceptor Dα , as expected, kd→a = 2π ~02 OIα . These lineshape functions are obtained in the SI from the general expression, valid for displaced harmonic oscillator potential energy surfaces, 41–43 by applying the above semiclassical treat-

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ment of the intrachromophoric spectral density Ji (ω) (eq 9) and read (i)

(i)

X X (S )n1 (S )n2 ~ (i) 1 2 e−Si Dα/I (ω) = q n ! n (i) 1 2! n1 n2 4πEλ kB T  2  P (i) (i) (i)  ±~(ω10 − ω) − Eλ − ν=1,2 nν ~ων  × exp − , (i) 4Eλ kB T

(19)

(i)

where ~ω10 is the energy difference between the minima of the potential energy surfaces of the excited and the ground electronic state of chromophore i = d, a. (1)

The non-Condon contribution to the rate constant kd→a contains the function Dn1 ,n2 ,m1 ,m2 (±ωξ ) (0)

(eqs 14, 16 and 17), whereas the Förster rate constant kd→a is proportional to Dn1 ,n2 ,m1 ,m2 (0) (eqs 14, 15 and 16). The physical interpretation of these facts is that the nuclei are frozen during the electronic transition in Förster theory, whereas absorption or emission of vibrational quanta ~ωξ is allowed to accompany the transfer in a non-Condon process. If ~ωξ is much smaller than the electronic energy gap between the donor and the acceptor, the resonance is established mainly by the intrachromophoric vibrational degrees of (1)

freedom and we may approximate the Dn1 ,n2 ,m1 ,m2 (±ωξ ), entering the Fn1 ,n2 ,m1 ,m2 in eq 17, by Dn1 ,n2 ,m1 ,m2 (0). Using, in addition, a high-temperature approximation for the n(ωξ ) ≈

kB T ~ωξ

(eq 8) in eq 17 we obtain (f) kB T ~2

≈ 2Eλ Fn(1) 1 ,n2 ,m1 ,m2

Dn1 ,n2 ,m1 ,m2 (0) ,

(20)

(f)

with the reorganization energy Eλ of the coupling fluctuations, defined as (f)

Eλ =

X

gξ2 ~ωξ .

ξ

11


(21)


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(1)

The rate constant kd→a in this classical high-temperature approximation then is obtained as (f)

(1,cl)

kd→a = 4π

Eλ kB T OIα , ~2

(22) (0)

with the overlap integral in eq 18. The enhancement factor κ = kd→a /kd→a of the rate (1,cl)

constant by the non-Condon effects in the present approximation kd→a ≈ kd→a is obtained (f)

as κ = 2Eλ kB T /V02 . Eqs 13-17 for the quantum rate constant and the simple classical expression in eq 22 represent the main theoretical results of the present work. In the following, the above theory is applied to the perylene biscarboximide dyad in Figure 1. Our structure-based calculation of the interchromophoric spectral density of the coupling fluctuations has been developed previously to obtain the spectral density of pigment-protein complexes. 39 Whereas in the case of pigment-protein complexes the NMA had to be based on a classical force field, a fully quantum chemical (QC) NMA is possible for the present dyad. Geometry optimization and NMA are performed with density functional theory (DFT) using the B3LYP exchange-correlation (XC) functional 44 and a 6-31(.,+)G* basis set employing the program QChem. 45 To simplify the QC computations, the 1-hexylheptyl (swallow-tail) substituents (Fig. 1A) were truncated to 1-methylethyl groups (Fig. 1B). Transition densities of monomeric chromophores (including the spacer) were computed employing HF-CIS 46 or time-dependent density functional theory (TD-DFT) in the random phase approximation (RPA) 47 with various XC-functionals including B3LYP, 44 , BHHLYP, 48 B65LYP, 49 and CAM-B3LYP 50 , using a 6-31G** basis set. The transition dipole moment for the transition from the ground to the first excited state varies for the different QC methods between 6.7 D and 7.6 D for the donor and between 8.9 D and 9.5 D for the acceptor (SI Table 1). From extinction coefficients in chloroform solvent, we estimated transition dipole moments of 7.9 D and 10.2 D for donor and acceptor, respectively, using the expression by Knox and Spring, 51 as described in the SI. Atomic transition charges were obtained from a fit of the QC transition densities on a 3D grid using CHELPBOW. 52 The transition charges, obtained

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with the different QC methods, are rescaled such that the experimental transition dipole moment results, leading to slight variations of the excitonic coupling V0 with respect to the (QC)

values V0

obtained with the original transition charges (SI, Table 1).

In Figure 2A, the square of gξ describing the fluctuations of excitonic couplings linear in the normal mode coordinates are shown, using the (rescaled) transition charges calculated (A)

0.02 0.02 0.015

2

0.015

gξ

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0.01

0.01 0.005

0.005

0 0

0 0

10

1000

20

30

2000 -1 ω ξ / cm

40

50

3000

(B)

Figure 2: (A): Square of linear coupling constants gξ (eq 4) as a function of the normal mode frequency ωξ . The inset gives an enlarged view on the coupling constants for small frequencies. (B): Second-order coupling constants gξ,ξ′ (SI eq 4) as a function of the normal mode frequencies ωξ and ωξ′ in the low-frequency region. The values of gξ,ξ′ at higher frequencies are practically zero.

13



with the HF-CIS method. The largest coupling constants are obtained for the first three normal modes with vibrational frequencies of 10.3, 12.6 and 20.0 cm−1 . An illustration of these normal modes and the resulting fluctuations in the excitonic coupling is given in the SI (including videos of the molecular motion). The room temperature amplitude of the coupling fluctuations along these normal modes is more than one order of magnitude larger than the excitonic coupling at the equilibrium position of nuclei (SI Figure 2). Applying a PDA results in couplings that are about 30 % larger than the values obtained with the TrEsp method (SI Figure 3). These deviations reflect the fact that the extension of the optically active π-electron system of the chromophores is comparable to the center-to-center interchromophore distance. Similarly to the first-order coupling constants, the second–order coupling constants gξ,ξ′ are largest for low frequencies (Figure 2 B). The dominating contribution from low-frequency interchromophoric modes to the coupling fluctuations highlights the need for an inclusion of the intrachromomophoric spectral density, required to bridge the electronic energy gap between the donor and the acceptor chromophore. From the experimental absorption and fluorescence spectra of the isolated donor and acceptor chromophore, the respective lineshape functions are obtained and fitted in Figure 3 with the semi-classical lineshape theory (eq 19). The parameters of this fit, (d/a)

which are the 0-0 transition energies ~ω10

(converted to wavelength units), the frequencies

(d/a)

of the two high-frequency vibrational modes (ν = 1, 2) and the reorganization energy

(d/a)

of the low-frequency modes of the donor (d) and the acceptor (a), are given in Table

ων

Eλ

1. Please note that there is a slight asymmetry between the experimental lineshape functions of fluorescence and absorption, which is not covered by the present harmonic oscillator model. An inclusion of anharmonicity effects would be needed. 53 Since the overlap between the donor fluorescence and the acceptor absorption line shape functions is important for the energy transfer, we have focussed on the fit of those two spectra for the determination of the lineshape parameters. In this way the effect of anharmonicities on the calculated rate constant is negligible.

14


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The various contributions to the rate constant of energy transfer resulting from the coupling constants gξ and gξ,ξ′ (Figure 2), lineshape parameters (Table 1) and the expressions derived above (eqs 13-17, 22 and SI eq 11), are compiled in Table 2. The overall inverse rate −1 constant kd→a varies between 25 ps and 60 ps for the different QC methods (used to calculate

the transition charges of the chromophores). For all methods, the rate constant is clearly (1)

dominated by the non-Condon contribution kd→a that contains the linear coupling constants (0)

gξ . The standard Förster rate constant kd→a as well as the second–order non-Condon rate (2)

constants kd→a are 2-3 orders of magnitude smaller. This is the main result of the present application of our non-Condon theory. donor

lineshape / a.u.

3 2,5

absorption

fluorescence

2 1,5 1 0,5 0 400

450

500

550

600

650

wavelength / nm acceptor 4

lineshape / a.u.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


3

absorption

fluorescence

2 1 0 400

450

500

550

600

650

wavelength / nm Figure 3: Experimental (dashed lines) lineshape functions for absorption (blue lines) and fluorescence (black lines) of the isolated donor (upper part) and acceptor (lower part) measured in carbon disulfide are compared to calculations using a semi-classical description of the electron-vibrational coupling (eq 19). The parameters of this model, extracted from the fit of the experimental lineshape functions, are given in Table 1.

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Table 1: Lineshape parameters used in eq 19 to calculate the lineshape functions in Fig. 3. (d)

ω1 / cm−1 760 (a) ω1 / cm−1 636

(d)

S1 0.29 (a) S1 0.24

(d)

ω2 1425 (a) ω2 1440

(d)

S2 0.67 (a) S2 0.65

(d)

ω10 / nm 474.5 (a) ω10 / nm 537.1

(d)

Eλ / cm−1 215 (a) Eλ / cm−1 165

The experimental time constant (9.4 ps in chloroform, 13.4 ps in toluene) is still a factor of 2 to 2.5 smaller than the smallest non-Condon time constant (25 ps), calculated here with the transition charges based on the HF-CIS method. Including in the latter calculations Table 2: Inverse rate constant 1/kd→a and different contributions (eqs 13-17, SI eq 11), all in units of ps, obtained by using different QC methods for the cal(0) (1) culation of atomic transition charges. kd→a represents F¨ orster theory, kd→a and (2) kd→a are obtained taking into account non-Condon effects, linear and quadratic (1,cl) in the normal mode coordinates, respectively. kd→a represents the classical ap(1) proximation (eq 22) of the first-order non-Condon contribution kd→a . −1 −1 −1 −1 (0) (1) (2) (1,cl) −1 method kd→a kd→a kd→a kd→a kd→a HF-CIS TD-DFT/B65LYP TD-DFT/BHHLYP TD-DFT/B3LYP TD-DFT/CAM-B3LYP

25.1 40.6 59.0 50.6 50.2

9130 12200 1265 3510 796

25.3 40.9 61.9 51.3 53.6

5530 9700 56100 9680 9660

29.9 46.7 67.7 53.7 57.6

only the three normal modes with the lowest frequencies (inset of Figure 2) gives an inverse (1)

(1,cl)

rate constant of 1/kd→a = 40 ps. The high-temperature (classical) approximation kd→a (last (1)

column in Table 2) of the dominating kd→a works well. Further evidence for the dominant contribution of the low-frequency interchromophoric vibrations to the rate constant is obtained from the excellent correlation between the quantum result for the non-Condon rate (1)

(f)

constant kd→a and the reorganization energy Eλ (eq 21), as shown in SI Figure 4. The classical rate constant in eq 22 above predicts a linear increase of the rate constant with increasing temperature. Such a linear temperature dependence has indeed been measured. 29 We investigated the temperature dependence of the rate constant obtained with the classical expression (eq 22) and the full quantum result (eqs 14, 16 and 17), using the 16


Page 17 of 25 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


HF-CIS transition charges. The quantum and classical expression show a very similar linear temperature dependence with the quantum rate constant being about 20 % larger than the classical result (SI, Figure 5). The experimental inverse rate constant in toluene decreases from 14.2 ps at 258 K to 13.2 ps at 328 K, 29 whereas the calculations predict a decrease of the quantum inverse rate constant from 29 ps to 24 ps and a decrease of the classical time constant from 36 ps to 27 ps. Hence, the qualitative trend is captured by the present theory, but some quantitative deviations remain. Probably, the largest uncertainty is due to the QC transition densities of the chromophores, which have, therefore, been calculated with various QC methods. Despite the rescaling of the transition charges to yield the experimental transition dipole moment, the overall inverse rate constant varies between 25 ps and 60 ps, obtained with HF-CIS and TD-DFT/BHHLYP, respectively. Obviously, not just the magnitude of the transition dipole, but the detailed shape of the transition density matters. These results suggest that the present system may serve as a suitable test case for future QC calculations. Besides the refinement of the QC method for the geometry optimization and the calculation of transition charges, two other open points for further refinement of the theory and the calculation scheme concern: (i) the inclusion of the dielectric environment in the NMA 54 and in the calculation of screening/local field correction effects of the excitonic coupling, 23–25 and (ii) the investigation of anharmonic effects in the conformational dynamics of the dyad, e.g., by molecular dynamics simulations. 55,56 In summary, we have presented a theoretical framework for the structure-based description of fluctuation-induced energy transfer between molecules and applied it to a perylene biscarboximide dyad with orthogonal transition dipole moments. In particular, we find that the linear contributions of a few low-frequency interchromophoric modes modulating the excitonic coupling in combination with a few high-frequency intrachromophoric modes bridging the energy gap between donor and acceptor are sufficient to explain the order of magnitude of the excitation energy transfer rate constant in this system. To pinpoint the

17



proposed mechanism experimentally, time-resolved optical spectroscopy in the visible (VIS)/ terahertz (THz) spectral region would be helpful. In these experiments, after optical excitation of the donor with a VIS pump pulse, one would investigate energy transfer to the acceptor by a VIS probe pulse in the presence and absence of a THz excitation 57 of the interchromophoric low-frequency modes modulating the excitonic coupling. Supporting Information. Derivation of coupling constants gξ and gξ,ξ′ of the spectral density of coupling fluctuations, Derivation of rate constant, Higher-order non-Condon contributions to the rate constant, Lineshape functions of chromophores, Determination of dipole strength of chromophores from experiment, Transition dipole moments and excitonic couplings obtained with different QC methods, Visualization of normal modes and fluctuating couplings, Excitonic couplings in PDA, Correlation between non-Condon rate constant (1)

(f)

kd→a and reorganization energy Eλ of coupling fluctuations, Temperature dependence of non-Condon rate constant. ˆ Acknowledgement. We would like to thank Vladislav Sláma and Frantiˆsek Sanda for discussions and for sharing their results prior to publication. 56

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