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Tuning Nonradiative Lifetimes via Molecular Aggregation Alan Celestino, and Alexander Eisfeld J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.7b06259 • Publication Date (Web): 21 Jul 2017 Downloaded from http://pubs.acs.org on July 26, 2017

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The Journal of Physical Chemistry

Tuning Nonradiative Lifetimes via Molecular Aggregation Alan Celestino and Alexander Eisfeld∗ Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Strasse 38, D-01187 Dresden, Germany E-mail: [email protected]

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Abstract We show that molecular aggregation can strongly influence the nonradiative decay (NRD) lifetime of an electronic excitation. As a demonstrative example, we consider a transition-dipole-dipole-interacting dimer whose monomers have harmonic potential energy surfaces (PESs). Depending on the position of the NRD channel (qnr ), we dim ) can exhibit a completely different dependence on the find that the NRD lifetime (τnr

intermolecular-interaction strength. We observe that (i) for qnr near the Franck-Condon dim increases with the interaction strength; (ii) for q region, τnr nr near the minimum of dim ; the monomer excited PES, the intermolecular interaction has little influence on τnr

(iii) for qnr near the classical turning point of the monomer nuclear dynamics, on the dim decreases with the interaction strength. Our findings other side of the minimum, τnr

suggest design principles for molecular systems where a specific fluorescence quantum yield is desired.

1

Introduction

The lifetime of a molecular system’s electronically excited state (EES) is determined by radiative and nonradiative transitions 1–4 . While radiative electronic transitions in molecular systems commonly occur within nanoseconds, nonradiative transitions can be much faster, with picosecond or even femtosecond timescales. Thus nonradiative decay (NRD) processes can determine the EES lifetime in molecular systems. Often these fast NRD processes are useful, e.g. in switching between retinal isomers 5–8 , transfer to long-lived triplet states 9 , nonphotochemical quenching in photosynthesis 10–12 , DNA photoprotection 13–15 , singlet fission 16 , molecular rotary motors 17 , and molecular switches 18–20 . However, sometimes these NRD processes are unwanted, e.g. in light harvesting applications 21 or where large quantum yields are desirable 22,23 . In this article we show that molecular aggregation can strongly influence the timescale of NRD processes (NRD lifetime) in molecular systems. Molecular aggregates consisting 2


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of transition-dipole-dipole-interacting molecules have attracted interest for decades (see e.g. Refs. 24–26 and references therein). One reason is the significant changes of the optical properties (e.g. radiative lifetime) upon aggregation, caused by the formation of exciton states coherently delocalized over several molecules. As we will show below, the excitonic delocalization can also strongly influence the NRD lifetime. Molecular aggregates can be formed from a variety of molecules in a variety of environments. Traditionally, aggregates are formed via self-assembly in aqueous solution 24 , where the geometrical arrangement of the molecules can be tuned e.g. by adding specific side groups 27 or using DNA as a template 28 . In recent years, aggregates have also been created in ultracold, suprafluid He-nanodroplets 29–31 and on surfaces, and the latter can be used as templates to obtain various molecular arrangements 32–34 . This great flexibility in the choice of the molecule, the environment, and the molecular arrangement also implies a great variety of system parameters (e.g. transition dipole-dipole interaction, vibrational relaxation, molecular potential energy surface (PES), location of the nonadiabatic coupling region, etc.). In the present work we will therefore not focus on a specific situation, but rather provide some basic insights in the parameter dependence.

2 2.1 2.1.1

Theoretical Description and Results The Monomer and its Decay Dynamics Basic Model

Before we discuss the dipole-dipole interacting molecules, let us briefly specify our model of the individual molecules (monomers). The basic features of a monomer are sketched in Fig. 1 (a) and (b), where the relevant PESs are shown as a function of a single nuclear “reaction” coordinate q. Initially the molecule is in its electronic ground state | g i, in thermal equilibrium with respect to the ground state PES. After a vertical Franck-Condon transition

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Figure 1: Potential energy surfaces (PESs) and nonradiative decay (NRD) dynamics of a single molecule. The top row shows sketches of the electronic excitation and relaxation processes along the PESs. After a vertical Franck-Condon transition, the electronic relaxation can occur in two different ways: (a) direct relaxation from the electronically excited state e to the electronic ground state g; (b) relaxation from the optically bright state e to the electronic dark state d. (c) Ground and first optically excited state harmonic PESs (case considered in the numerics). qs is the shift between PESs. (d) Time-resolved population decay for different positions of the NRD channel. The qnr values are illustrated as arrows in (c) according to the colors and linestyles of the curves in (d). (e.g. through a short laser pulse) to an EES | e i, which leaves the nuclear wavefunction unchanged 3 , the nuclear dynamics and the NRD set in. We denote the PESs of the electronic ~ and Ve (Q), ~ respectively. Here, Q ~ is the set of all relevant ground and excited states by Vg (Q) nuclear coordinates. Vibrational relaxation due to coupling to environmental degrees of freedom accompanies the coherent motion on the PESs. Typically, nonradiative transitions between molecular electronic states involve nuclear degrees of freedom and occur at points where the respective PESs are close or cross 1,35 . We assume a localized region in nuclear space where the electronic excitation can efficiently leave the electronic state | e i, which we call “the NRD channel”. Since we do not focus on a particular molecule and we are mainly interested in qualitative results, we model the NRD channel as an imaginary potential added to the excited PES. This PES is then given ~ = Ve (Q) ~ − iΓ(Q), ~ where we denote Γ(Q) ~ as the “decay-function”. Note that by V˜e (Q) this imaginary potential implies that the Hamiltonian of the system is non-Hermitian. We emphasize that this NRD channel can occur in any region of the PES. For instance, in β4


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Apo-8’-carotenal a NRD channel is believed to occur at the vertical Franck-Condon region 36 . As a concrete example, we consider harmonic monomer PESs along a single normal mode coordinate 4 (reaction coordinate q), with identical frequencies ω but shifted with respect to each other by qs (see Fig. 1 (c)). The monomer PESs are Vg (q) = 12 ω 2 q 2 and Ve (q) = Ee + 21 ω 2 (q − qs )2 . Here, Ee is the energy shift between the minima of the PESs (electronic transition energy). For simplicity, we take the decay-function to be a delta function Γ(q) = λδ(q − qnr ) centered at the position qnr , where λ is the NRD strength. The results in this article do not change qualitatively for a different well-localized decayfunction (e.g. a Gaussian function). Vibrational relaxation is taken into account via a bilinear coupling between the system and a bath of harmonic oscillators. The system part of the coupling operator is proportional to (q − qs | e ih e |). We have used the Ohmic spectral density j (˜ ω ) = (~2 /π)θ (˜ ω) γ ω ˜ exp (−˜ ω /ω0 ) (θ(˜ ω ) is the Heaviside step function). We simulate the system’s dynamics solving a multilevel Redfield equation 4 , obtained after tracing out the bath degrees of freedom. For more details on the modeling of vibrational relaxation, we refer to the Supplemental Material. All numerical results shown in this article are obtained for q

the parameter values qs = 1.5 ~/ω, γ = 1, ω0 = 10ω/π, NRD strength λ = 0.1~3/2 ω 1/2 , and temperature T = 0 (since we are not interested in thermal effects). The values of γ, ω0 , and λ guarantee fast vibrational relaxation compared to the timescales of the NRD and to nuclear oscillations. This case applies to many molecules, as vibrational relaxation typically takes place within a picosecond 3 . 2.1.2

Exemplary Calculations

As a reference for the dimer case later on, we now consider the dependence of the NRD on qnr for a single molecule. We focus on different locations qnr of the NRD channel leading to qualitatively different behaviors. These locations are qnr = 0 (at the vertical FranckCondon region), qnr = qs (minimum of the excited-state PES of the monomer), and qnr = 2qs (classical turning point to the right of this minimum). Note that qnr = 0 and qnr = 2qs

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enclose the classically accessible region in the monomer nuclear space. The numerically calculated population in the monomer EES P (t) is shown in Fig. 1 (d). Different NRD channel positions qnr are indicated by arrows in Fig. 1 (c) according to the colors and linestyles in Fig. 1 (d). P (t) depends sensitively on qnr , and decays approximately as a monoexponential

(mon) P (t) ≈ exp −t/τnr because vibrational relaxation is fast compared to NRD dynamics.

We estimate the monomeric NRD lifetime assuming that the nuclear wavefunction is in the ground state of Ve (q) at all times (with a different, time-dependent norm). The obtained √ (mon) (mon) (mon) expression τnr (qnr ) ≈ τnr (qs ) exp[ω(qnr − qs )2 /~] (τnr (qs ) ≈ π~3/2 /2λω 1/2 ) fits well the numerical results from Fig. 1 (d). In particular, the closer the NRD channel is to the minimum of the excited-state PES, the faster the NRD takes place.

2.2

Interacting Molecules

Figure 2: Sketch of the geometry (a) and energy levels (b) (vibronic levels suppressed) of the dimer. (a) The dimer’s geometry is defined by the relative orientation of the transition ~ (b) The eigenenergies of the ground state dipoles ~µ1 and ~µ2 (θ), and the relative position R. | gg i, single excitation manifold {| eg i, | ge i} for vanishing J (Ee ), and dimer singly excited states |+i and |−i (respectively Ee + J and Ee − J) for J < 0. To furnish a clear example on how the transition dipole-dipole interaction influences the NRD lifetime, we treat the case of a molecular dimer 30,37–41 in detail (see Fig. 2). The two monomers are assumed to be sufficiently far apart to neglect overlap between electronic wavefunctions. However, they interact via long-range Coulomb interaction. The Coulomb interaction depends on the dimer’s geometry, which is considered fixed (see Fig. 2 (a)). In the point-dipole approximation, which is often appropriate, the interaction strength can be

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Figure 3: Nonradiative decay dynamics for different J and qnr values. (a), (b), and (c): Population in the single excitation manifold P (t) as a function of time (notice the different (dim) x-axis range in (c)). (d), (e), and (f): NRD lifetime τnr (J) as a function of J (continuous (sat) blue line). The green dashed horizontal line is the NRD lifetime saturation value τnr . For clarity, the values J = −1 and J = −2 for which we plotted P (t) in (a), (b), and (c) are indicated by vertical lines matching the colors and linestyles in (a), (b), and (c).

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written as J ∝

1 R3

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~ · ~µ1 )(R ~ · ~µ2 )/R2 . Here, R ~ is the distance vector between ~µ1 · ~µ2 − 3(R

the centers of the two monomers and we consider ~µ1 and ~µ2 to be the transition dipoles of monomer 1 and 2, respectively. We stress that the specific form of this interaction is not relevant in the following. The electronic subspace is spanned by the states | gg i, | eg i, | ge i, and | ee i (see Fig. 2 (b)). For both monomers in the electronic ground state, the corresponding nuclear Hamilto~ 1, Q ~ 2 ) = Hg (Q ~ 1 ) + Hg (Q ~ 2 ), where Hg (Q ~ j ) = Kj + Vg (Q ~ j ), and Kj is the nuclear nian is Hgg (Q kinetic energy for the monomer j. Consequently, the initial state (before the Franck-Condon vertical transition) is the same as the thermal equilibrium of two uncoupled monomers. Because of large detuning in energy, the doubly excited state | ee i is not populated and we will not discuss it further. In the single excitation manifold, i.e. in the subspace spanned by the degenerate electronic states | eg i and | ge i, the transition dipole-dipole interaction leads to a coupling of the form J (| eg ih ge | + | ge ih eg |). The nuclear Hamiltonian in the single excitation manifold is then given by ~2 ~ 1 + Vg Q ~ 1, Q ~ 2 = Knuc + V˜e Q Hex Q

| eg ih eg |

~ 2 + Vg Q ~1 + V˜e Q

| ge ih ge |

+ J (| eg ih ge | + | ge ih eg |) ,

(1)

where Knuc = K1 + K2 and we note that Hex is non-Hermitian. The monomers have the same PESs and NRD channels as the single molecule considered when discussing Figure 1 (d). This model directly relates to previous studies of dimers, where NRD has not been taken into account (see e.g. Refs. 30,37–41 ). Each monomer is bilinearly coupled to a distinct harmonic bath. The system part of the coupling operator is now proportional to (qj − qs | πj ih πj |), where | π1 i = | eg i and | π2 i = | ge i. The baths are at the same temperature and have the same spectral density. As in the monomer case, we derive a multilevel Redfield master equation for the dimer by tracing out the environmental degrees

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of freedom 4 and performing Born, Markov, and secular approximations. This equation is i ∂ρ † = − Hex ρ − ρHex + L [ρ] , ∂t ~

(2)

where t is the time, ρ is the density matrix of the dimer, and L is a dissipator. For more details on the modeling of the vibrational relaxation, we refer to the Supplemental Material. As our initial state, we consider the result of a Franck-Condon transition to the first excited adiabatic electronic state. For J < 0, the case we show here, this corresponds to |+i = √ √ (| eg i + | ge i)/ 2 (for J > 0 it corresponds to |−i = (| eg i − | ge i)/ 2). These two electronic states are the eigenstates of the system if nuclear degrees of freedom are neglected (see Fig. 2 (b)). The results we discuss here are not fundamentally changed by choosing a different initial condition within the single excitation manifold, nor by considering J > 0. We use the same values of qs , γ, ω0 , λ, and T , as in the monomer case discussed above. To get a feeling for the values of the transition dipole-dipole interaction strength used below, let us take the vibrational relaxation to occur within some hundreds of femtoseconds in our model. Then, the largest value of the interaction strength that we consider in the following, J = −10, corresponds to roughly −1000 cm−1 , which is in the order of magnitude achievable in experiments. 2.2.1

Decay Dynamics of a Dimer

The numerical results for the dimer are shown in Figure 3. From the top to the bottom row, qnr = 0, qnr = qs , and qnr = 2qs are shown, respectively. In the left column, the population in the single excitation manifold P (t) is shown for different values of J. As in the monomer case, P (t) approximately follows a monoexponential decay (see Figure 3 (a)-(c))

(dim) and can therefore be written as P (t) ≈ exp −t/τnr . The numerically fitted NRD lifetime (dim) τnr is plotted as a function of J as continuous blue lines in the right column (Figure 3

(d)-(f)). As one can see from Figure 3 (d)-(f), P (t) depends on J, and this dependence is

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(dim) different for different values of qnr . For all qnr , the NRD lifetime τnr varies monotonically (sat) with J and it saturates for small J (J < 0) at the value τnr , which depends on the

specific choice of qnr , qs and λ. This saturation value can be analytically determined to be (sat) (sat) τnr (qnr ) ≈ τnr (qs /2) exp[ω(qnr −qs /2)2 /~] (see discussion about the adiabatic limit below), (sat) with τnr (qs /2) ≈ π 1/2 ~3/2 /2λω 1/2 , and is plotted as a dashed green line in Figure 3 (d)-(f). (dim) The value of qnr determines whether τnr increases or decreases with J. For qnr > 3qs /4 (dim) (qnr < 3qs /4), τnr increases (decreases) with |J|.

The transition dipole-dipole interaction can suppress (trigger) fluorescence of (non-)fluorescent molecules when they form dimers or larger aggregates. Although in the examples shown in Figure 3 (d)-(f) the NRD lifetime maximally varied over approximately one order of magnitude (Figure 3 (f)), this is not limited on the range of NRD lifetime variation. The range of (dim) (mon) (sat) variation of τnr with J is at least the ratio τnr /τnr ≈ exp [ω (3qs2 /4 − qnr qs ) /~]. Since

it depends exponentially on the shift between the monomer PESs qs , the range of tunability becomes exponentially larger for larger qs . We stress that the full quantum nonadiabatic dynamics is considered in our numerical simulations, i.e. the correct dimer PESs, nonadiabatic couplings and excitonic relaxation (sat) is naturally included in the numerics. The analytic expression of τnr we used in the

discussion of our numerical results was derived in the adiabatic limit. In this limit, namely √ when |J| ≫ ~1/2 ω 3/2 qs / 2, we can consider the nuclear dynamics to be confined within the adiabatic PES associated with the electronic state |+i (|−i) for J < 0 (J > 0). The corresponding (complex) PESs are given by ~ 1, Q ~ 2) = V˜± (Q

2 X 1

j=1

2

~ j ) + Vg (Q ~ j ) ± J, V˜e (Q

(3)

and non-adiabatic couplings between these PESs are negligible. Notice from Eq. (3) that ~ 1 and Q ~ 2 are not coupled. Thus, for each coordinate the NRD channel the coordinates Q ~ j )) is the same as for the uncoupled monomers. However, the (which appears via V˜e (Q

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potential on which the nuclear wavepacket moves has a different shape (and in particular a different minimum) from the monomer’s excited-state PES. Considering the PESs from our numerics, we obtain for V± (q1 , q2 ) (the Hermitian part of V˜± (q1 , q2 )) a well known result 41–43 : V± (q1 , q2 ) = ±J +

P

j

ω 2 (qj − qs /2)2 /2 + ω 2 qs2 /4. The adiabatic PESs are thus shifted by

qs /2 in each coordinate with respect to the ground-state PES of the monomer. Taking these PESs into account, and considering fast vibrational relaxation compared to the NRD and nuclear oscillations timescale, we obtained the analytic formula for the saturated NRD (sat) (mon) lifetime τnr . This was performed in the same way as for deriving τnr , but assuming the

nuclear wavefunction to be always in the ground state of V+ (q1 , q2 ). Comparing the analytic (sat) (mon) formulas for τnr with τnr , one observes that their formulas only differ by the shift in the

nuclear coordinate, qs for the monomer and qs /2 for the saturated dimer. This is because the minimum of V+ (q1 , q2 ) is at (q1 = qs /2, q2 = qs /2), while the minimum from the excited-state PES of the monomer j lies at qj = qs . 2.2.2

Extension to Larger aggregates

An extension of the NRD lifetime analysis to longer aggregates can also be performed 44,45 . The adiabatic PESs of an N -mer (when the electronic wavefunction is delocalized over N monomers) can be estimated in the adiabatic limit. For harmonic monomer PESs, the adiabatic PESs are harmonic and shifted in all reaction coordinates by qs /N (they are also shifted in energy). The ratio between monomeric and saturated N -meric NRD lifetimes (mon) (sat,N ) behave as τnr /τnr ∼ exp [−ωqs (qnr (2N − 2)/N − qs (N 2 − 1) /N 2 ) /~].

2.2.3

Consequences to the Fluorescence Quantum Yield

Let us finally discuss the consequences of our results for the fluorescence quantum yield Yfl of an aggregate, which we express as

Yfl =

τnr , τfl + τnr 11


(4)


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where τfl and τnr are the fluorescence and the nonradiative lifetime of the aggregate, respectively. As discussed in the present work, the nonradiative lifetime can change strongly upon aggregation, compared to the monomeric value. Also the fluorescence lifetime τfl changes upon aggregation 46 . There are two important cases: (i) J-aggregates: here bright states located at the bottom of the exciton band provide the dominant contribution to the fluorescence lifetime. These states possess an oscillator strength that can be considerably enhanced compared to that of a monomer. This enhancement depends essentially on the number of monomers over which the aggregate wavefunction is coherently delocalized and on the geometrical arrangement of the monomers 33,47–49 . In the extreme case the fluorescence life(mon)

time can become N times shorter than that of the monomer, τfl = τfl

/N , where N is

the number of monomers. This means that for constant τnr the quantum yield increases. The mechanism described in the present work can either increase Yfl further (as exemplified in Figure 3 (e) and (f)) or it can reduce the Yfl (as exemplified in Figure 3 (d)). As we have shown, our mechanism can change τnr by several orders of magnitude and therefore considerably alter the the fluorescence quantum yield. (ii) H-aggregates. Here the bright states are located at the top of the exciton band. At typical temperatures only dark states at the bottom of the band are occupied. Here, no emission occurs, and one does not expect that the change in the nonradiative lifetime will strongly alter this result. Beside the pure J- and H-aggregates, there exist many cases where the optical selection rules are not so strict as for ideal J- and H-aggregates (see e.g. 50 ). Here one expects that the mechanism discussed in the present work will play an important role.

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Conclusions

In conclusion, we have shown that molecular aggregation can modify the NRD lifetime. As a proof of concept, we have considered in detail the simplest molecular aggregate featuring this

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phenomenon: a transition-dipole-dipole-interacting dimer with harmonic monomer PESs. We have shown that the relationship between the NRD lifetime and the intermolecularinteraction strength depends sensitively on the NRD channel position. In particular the NRD lifetime can increase with, decrease with, or be practically insensitive to the intermolecularinteraction strength. This indicates that quantum yield measurements can, e.g., be exploited for the detection of molecular aggregation; pinpointing of NRD channel locations in molecules; or to infer the geometry of molecular aggregates. We have also performed simulations for different values of the shift between monomer q

(dim) can feature dips PESs and γ. If qs is larger than in Figure 3, e.g. qs = 3.5 ~/ω, τnr

(and peaks) at certain J values - apart from spanning many (e.g. 4 for qnr = 2qs ) orders of magnitude upon varying J. For γ much smaller than the one used in Figure 3, P (t) shows oscillations reflecting the underdamped nuclear dynamics, however, the main trends observed in Figure 3 are preserved. For arbitrary (nonharmonic) monomer PESs, the transition dipoledipole interaction can impose more severe modifications to the NRD dynamics. This is because the shape of the dimer PESs from eq. 3 can differ from the monomer excited-state PES’s shape. For instance, the dimer’s PESs may present a minimum even if the excitedstate PES of the monomer does not. This can lead to a fundamental change of the nuclear dynamics in the diabatic case, e.g. stabilizing a photodissociation. Monomer PESs of different shapes will thus give rise to different changes in the NRD dynamics upon aggregation. Finally, let us mention that besides the mechanism discussed in the present work, the nonradiative lifetime can also be affected by other means upon aggregation, e.g. steric hindrance of torsional motion in the monomer. However, we want to emphasize that the mechanism discussed here should anyway be considered, since it can strongly change the nonradiative lifetime (and thus the fluorescence quantum yield) despite other coexistent mechanisms.

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Acknowledgement We thank C. Bentley for useful discussions.

Supporting Information Available We provide details on the monomer and dimer models used to generate Figure 1 (d) and Figure 3 of the article.

References (1) Simons, J. Energetic Principles of Chemical Reactions; Jones & Bartlett Pub., 1983. (2) Medvedev, E. S.; Osherov, V. I. Radiationless Transitions in Polyatomic Molecules; Springer Series in Chemical Physics; Springer-Verlag, 1995; Vol. 57. (3) Lakowicz, J. R. Principles of Fluorescence Spectroscopy; Springer Verlag, 2006. (4) May, V.; K¨ uhn, O. Charge and Energy Transfer Dynamics in Molecular Systems, 3rd ed.; WILEY-VCH, 2011. (5) Polli, D.; Altoè, P.; Weingart, O.; Spillane, K. M.; Manzoni, C.; Brida, D.; Tomasello, G.; Orlandi, G.; Kukura, P.; Mathies, R. A. et al. Conical intersection dynamics of the primary photoisomerization event in vision. Nature 2010, 467, 440– 443. (6) Hahn, S.; Stock, G. Quantum-mechanical modeling of the femtosecond isomerization in rhodopsin. J. Phys. Chem. B 2000, 104, 1146–1149. (7) Hahn, S.; Stock, G. Femtosecond secondary emission arising from the nonadiabatic photoisomerization in rhodopsin. Chem. Phys. 2000, 259, 297 – 312.

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Tuning Nonradiative Lifetimes via Molecular Aggregation - The

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