Vibronic energy localization in weakly coupled small molecular aggregates

Chemical Physics Letters 541 (2012) 49–53

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Vibronic energy localization in weakly coupled small molecular aggregates Johannes Wehner, Alexander Schubert, Volker Engel ⇑ Institut für Physikalische und Theoretische Chemie, Emil-Fischer-Str. 42, and Röntgen Research Center for Complex Material Systems, Am Hubland, 97074 Würzburg, Germany

a r t i c l e

i n f o

Article history: Received 5 April 2012 In final form 16 May 2012 Available online 25 May 2012

a b s t r a c t Within a one-exciton picture, molecular dimers (M1–M2) possess excited states of even and odd parity which correspond to linear combinations of locally excited configurations (M1 —M2 ; M1 —M2 ). If this symmetry is broken, the excitation energy localizes in one or the other monomer. We perform time-dependent quantum calculations on dimer and trimer aggregates which are subject to time-dependent perturbations. The latter induce exciton localizations which are influenced by the monomer vibrational degrees-of-freedom. This influence is characterized by comparison with purely electronic models. Ó 2012 Elsevier B.V. All rights reserved.

1. Introduction Molecular aggregates are promising building blocks for organic photovoltaic devices [1,2]. The basic principle is that, after photoexcitation, the absorbed energy moves through the aggregate and eventually reaches a region where charge separation is possible. The crucial quantity is the diffusion length which should be large, so that charges can be produced, and a photovoltaic current is created. If the excitonic energy localizes in a sub-unit of the aggregate, diffusion is hindered and relaxation mechanisms can take place so that the energy is trapped at a local site. Therefore, it is important to understand the process of localization in more detail. Many studies on exciton dynamics and localization have been performed. Here, we mention explicit time-dependent investigations on exciton delocalization in light harvesting systems employing femtosecond spectroscopy [3–5] and quantum dynamics simulations on the energy transfer in molecular aggregates [6,7]. Recently, it has been shown theoretically [8], that femtosecond two-dimensional optical spectroscopy [9–12] can be used to probe excitonic localizations. The simplest aggregate, i.e., a molecular dimer (M1–M2), has motivated various early studies, both experimentally and theoretically [13–16]. In 1964, Fulton and Gouterman treated dimers including a single vibrational mode per monomer and thus provided the first numerical study of a dimer system with vibronic structure [17]. They showed that the mixing of excited state configurations corresponding to locally excited monomers (M1 —M2 ; M1 —M2 ) leads to vibronic band progressions of states with even and odd parity. These states are symmetric and antisymmetric linear combinations of wave functions for the local excitations. If a dimer is prepared in such a state, the probability for the excitation energy to reside on one or the other monomer is equal. Suppose now, that an external perturbation is applied to

⇑ Corresponding author. Fax: +49 931 31 85331. E-mail address: [email protected] (V. Engel). 0009-2614/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.cplett.2012.05.034

the system. This, e.g., could be the interaction with a solvent to introduce a disorder [18–20] or an external electromagnetic field. Then, in general, the symmetry is broken and localization of the excitonic energy occurs. The purpose of the present Letter is an analysis of such localizations in small aggregates, and in particular, the influence of internal vibrational degrees-of-freedom on this phenomenon. That vibrations play an important role in this context is evident since the early work of Frenkel [21]. For example, the ‘exciton–phonon coupling’ strongly influences the energy transfer dynamics and also absorption/emission lineshapes [22,23]. An interesting point is the relation between the localization and the topology of the potential energy surfaces determining the dynamics of the nuclei [24]. In the present Letter, we primarily concentrate on a dimer but extensions to trimers are also discussed. The model is outlined in Section 2, the results and a short summary are given in Section 3. 2. Theory and model We regard photo-excited molecular dimers and trimers where a single vibrational degree-of-freedom is included per monomer and inter-monomer modes are not taken into account. The excited state dynamics is governed by the Hamiltonian [17,23]

b ~ Hð xÞ ¼

N X n¼1

b n ð~ jni H xÞhnj þ

N 1 X ½jni J hn þ 1j þ jn þ 1i J hnj;

ð1Þ

n¼1

where ~ x ¼ ðx1 ; . . . ; xN Þ is the set of vibrational coordinates of the monomers, and N is 2 and 3 for the dimer and trimer, respectively. The electronic coupling element is denoted as J, and only next neighbor interactions are included. The operators

" # " # X 1 @2 x2 1 @2 x2 2 2 b n ð~ ; H xÞ ¼  þ  x Þ þ D E þ  þ ðx x n 0 2 @x2n 2 @x2m 2 2 m m–n ð2Þ

J. Wehner et al. / Chemical Physics Letters 541 (2012) 49–53

correspond to the excitation of monomer Mn in the chain. The potentials are taken as harmonic with the same force constant in the ground and excited state, and (x0, DE) are the shifts of the excited state potential in the vibrational coordinate and in energy, b n are replaced by energies E, respectively. If the Hamiltonians H i.e., the vibrational degrees-of-freedom are neglected, the Hamiltob describes a coupled N-level electronic system. nian H The following parameters are used in our numerical example: x = 0.175 eV, x0 = 2.57 eV1/2 and DE = 2.35 eV. They are determined by comparison with spectroscopic experiments on aggregates of perylene bisimide dyes [25,26]. We thus employ scaled units. Below, the numerical values of time are given in multiples of the vibrational period Tvib = 2p/x which amounts to 23.4 fs, whereas in the analytical considerations, the unit of time is eV1. The coupling element is taken as a constant with the value of J = 0.0175 eV. This coupling strength can be classified as weak according to the value of the Simpson–Peterson parameter s = 2J/ DM [27], where in our case the bandwidth of the monomer absorption spectrum is about DM = 2x [26], so that s = 0.1. In order to simulate an external perturbation, we introduce a time-dependent interaction operator

c t ¼ j1iW t h1j ¼ j1igðtÞkh1j; W

ð3Þ

where g(t) is an envelope function and k an energy shift. Thus, the perturbation disturbs only the configuration where monomer M1 is in its excited state, if not stated differently. The perturbation is switched on/off with a shape function of the form: 2

ðt 6 ti Þ : gðtÞ ¼ ebðtti Þ ;

2

ðt P tf Þ : gðtÞ ¼ ebðttf Þ ;

ð4Þ

and is kept constant (g(t) = 1) in the time interval [ti, tf], with values of ti = 42 and tf = 70. The width (full width at half maximum) of the Gaussian is taken as cw = 21. We solve the time-dependent Schrödinger equation for the Ncomponent total wave function wð~ x; tÞ ¼ ðw1 ð~ x; tÞ; . . . ; wN ð~ x; tÞÞ on a grid [28], where the initial wave function in each component is the ground state of the uncoupled and unshifted harmonic oscillators. 3. Results In what follows, we discuss the population dynamics in the coupled excited states, i.e., regard the time-dependent quantities Pn ðtÞ ¼ hwn ðtÞjwn ðtÞi. The energies in the different configurations (hwn ðtÞjHn jwn ðtÞi) correlate directly with the populations [29]. Therefore, we do not distinguish between population- and energy dynamics in what follows. We first neglect vibrations. For the unperturbed system (i.e., k = 0), the problem can be solved analytically [30]. The eigenenergies are determined as E± = E ± J. The population in the excited state configuration M 1 —M 2 (denoted as j1i) is calculated as

P1 ðtÞ ¼ j~c1 ðtÞj2 ¼

1 1 ½P1 ð0Þ þ P2 ð0Þ þ ½P1 ð0Þ  P 2 ð0Þ cosð2JtÞ; 2 2

ð5Þ

with P 2 ðtÞ ¼ j~c2 ðtÞj2 ¼ 1  P 1 ðtÞ, and ~cn ðtÞðn ¼ 1; 2Þ are the timedependent coefficients in the two states. From this equation it is clear that if, initially, both excited state configurations are populated equally, the populations (and thus the energy) in these states remain constant. It is only if, at some point, the symmetry is broken, oscillations appear with a characteristic period T = p/J which is determined by the energy separation jEþ  E j ¼ 2jJj. Next, the effect of a perturbation is included. In order to understand the numerical results, we perform a unitary transformation as

ð~c1 ðtÞ; ~c2 ðtÞÞ ¼ ðeiW t t c1 ðtÞ; c2 ðtÞÞ;

ð6Þ

so that the two-level Hamiltonian appearing in the time-dependent Schrödinger equation for the transformed coefficients ðc1 ðtÞ; c2 ðtÞÞ reads:

b HðtÞ ¼

2 h i X jniEhnj þ j1i JeiW t t h2j þ j2i JeiW t t h1j :

ð7Þ

n¼1

Here, we assumed the shape function g(t) to be slowly varying in time so that its time-derivative can be neglected (slowly varying envelope approximation [31]). The coupling terms now carry phase factors eiW t t and resemble those terms appearing in the description of stimulated absorption and emission processes. To see the effect of the perturbation, we use first-order time-dependent perturbation theory to calculate the coefficient in the perturbed state j1i:

Z t 0 ð0Þ 0 0 0 ð1Þ ð0Þ c1 ðtÞ ¼ eiEt c1 ð0Þ  i dt eiEðtt Þ J eikt eiEt c2 ð0Þ 0   J ð0Þ ð0Þ iEt c1 ð0Þ  c2 ð0Þðeikt  1Þ ; ¼e k

ð8Þ

where we set the pulse envelope to a constant (g(t) = 1), and the ð0Þ coefficients cn ð0Þ are real valued. The population in the first state then is (to first order in J):

P1 ðtÞ ¼ P1 ð0Þ 

2J ð0Þ ð0Þ c ð0Þc2 ð0Þ½cosðktÞ  1: k 1

ð9Þ

This shows that, for short times and positive values of the coupling J and the shift k, the population in the perturbed state initially increases as a function of time and we have, within the present approximation P1 ðtÞ þ P 2 ðtÞ ¼ 1. This is indeed found in the numerical calculation, see Figure 1, left panels. There, the populations for the two-level case are displayed for different energy shifts k, as indicated. Also shown are the respective perturbations (Wt). Obviously, the increase of the population in the perturbed state which is predicted for short time, continues until the perturbation reaches a constant value. Then, the populations stay constant, in the average. The reason is that, for longer times t, the time-integral in Eq. (8) is highly oscillatory so that the first-order correction does not contribute significantly to the population. Referring to the analogy with a photon-interaction this is the case of a non-resonant excitation. When the perturbation is switched off, the populations show the opposite trend as in the beginning and settle to an

population

50

1 0.8 0.6 0.4 0.2 0 0.8 0.6 0.4 0.2 0 0.8 0.6 0.4 0.2 0

P1(t)

0.0875 eV

Wt

P2(t)

0.160 eV

0.175 eV

0

40

80 time

0

40

80

120

time

Figure 1. Populations corresponding to the two excited state configurations M1 —M2 ðP 1 ðtÞÞ and M1 —M2 ðP 2 ðtÞÞ. Curves are shown for different perturbations Wt, which are included in the left hand panels. The numbers refer to the maximal energy shift k. Results for an electronic two-level system (left) are compared to calculations incorporating the monomer vibrational degrees-of-freedom. Three times tn are marked in the lower right panel, for details see text. Here and in all figures, the time is given in multiples of the vibrational period Tvib = 23.4 fs.

51

J. Wehner et al. / Chemical Physics Letters 541 (2012) 49–53

equal value after the perturbation vanishes. This can be understood employing similar arguments as given above to explain the rise of the population P1(t) upon the initial action of the perturbation. Comparing the curves for the three perturbation-strengths it is found that hardly a difference exists. Finer oscillations can be seen in the populations by closer inspection (not visible in Figure 1). They show the expected beating determined by the energy difference of the two eigenenergies found upon diagonalization of the two-level Hamiltonian for the coupled non-degenerate system, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i.e., jEþ  E j ¼ 2 ðk=2Þ2 þ J 2 . Let us next turn to the dynamics in the full vibronic problem. The respective populations are compared to those for the two-level case in Figure 1. For the lowest perturbation (k = 0.0875 eV) a similar behavior is found as long as the perturbation is active. Minor oscillations are seen in the populations in the region of constant perturbation exhibiting a period which is in accord with the beating period for the disturbed two-level system, which is calculated as 2p=jEþ  E j. Thus, the average populations in the two states remain constant at quite different values, so that indeed a localization of the excitonic energy on the perturbed monomer is maintained. Nevertheless, the localization is lost after the perturbation is turned off smoothly, and one finds (besides the fast oscillation corresponding to the vibrational period Tvib of the unperturbed oscillator), a slower oscillation period which results from the complicated vibronic structure of the perturbed system. The situation changes if the shift is increased to a value of k = 0.16 eV. Then, in the time interval between ti and tf, oscillations with larger amplitudes are seen. This already suggests a stronger interaction between the two localized dimer configurations. The interaction is further increased if the energy shift k assumes the value of the vibrational quantum x. Figure 1, lower right panel, shows that in this case, there is a fast and efficient population

transfer between the configurations which means that excitonic energy is transferred periodically. This is not unexpected because the vibrational structure corresponding to the Hamiltonians Hn (n = 1,2) is only slightly modified by the weak coupling J. Thus, a shift of one potential by one vibrational quantum brings the two configurations to resonance making an efficient energy exchange possible. Here, the importance of the vibrational motion is obvious. It is of interest to take a closer look at the wave-packet dynamics at times when the effective population transfer occurs in the case of resonance (k ¼ x). Therefore, we display in Figure 2 plots of the probability densities jwn ðx1 ; x2 ; tÞj2 in the two states jni for three selected times t n . These times of t 1 = 47.65, t2 = 51.56 and t3 = 55.46 are marked in Figure 1. At time t 1 , when the perturbed state j1i is populated by about 80% its wave function is nodeless (panel a). It is found that the probability density does not show temporal changes around this time. Thus we conclude, that the wave function is mainly composed out of the vibrational ground state. On the other hand, the wave packet in state j2i shows a clear node so that one vibrational quantum of excitation is found (panel b). After the population transfer has taken place and the population ratio is inverted at time t = t2, the component jw1 ðx1 ; x2 ; t2 Þj2 exhibits a node (panel c), whereas jw2 ðx1 ; x2 ; t 2 Þj2 has not changed its nodal structure but mainly gained amplitude (panel d). To understand this behavior we regard times around t = t1 and apply time-dependent perturbation theory. The first order wave function is:

  Z t 0 0 0 0 ð1Þ ð0Þ ð0Þ w2 ðtÞ ¼ eiH2 t w2 ð0Þ þ i dt eiH2 t J eikt eiH1 t w1 ð0Þ ;

ð10Þ

0

where, for simplicity of notation, the time t1 is denoted as t = 0. The ð0Þ interaction term in the first-order correction to w2 ðtÞ has the form of a perturbation describing the field-induced absorption initiated by an electric field with photon energy k and field-strength J. Inserting complete sets of eigenfunctions fum;n g of Hn (where ðmÞ denotes the set of vibrational quantum numbers) with energies Em;n yields:

" ð1Þ w2 ðtÞ

iH2 t

¼e

ð0Þ w2 ð0Þ

þi J

X

# ð0Þ k;1 jw1 ð0ÞiI mk ðtÞ

um;2 hum;2 juk;1 ihu

;

m;k

ð11Þ with the time-integrals:

Imk ðtÞ ¼

Z

t

0

0

dt eiðEm;2 Ek;1 kÞt :

ð12Þ

0

Figure 2. Probability densities in the two electronic states shown at times t1 = 47.65 (a and b), t2 = 51.56 (c and d), and t3 = 55.46 (e and f), as marked in Figure 1. The left hand panels contain the functions jw1 ðx1 ; x2 ; tÞj2 for the perturbed dimer configuration M1 —M2 ; the right hand panels those (jw2 ðx1 ; x2 ; tÞj2 ) for the configuration M1 —M2 .

These integrals are highly oscillatory functions of time except for those, where the resonance condition Em;2  Ek;1  k ¼ 0 is fulfilled. The latter are the only terms contributing substantially to the perturbed wave function. Furthermore, because the initial ð0Þ function w1 , at time t1, is mainly composed out of the ground state u0;1 , only terms with Em;2  E0;1  k ¼ 0 remain. This means that the prepared wave function carries one vibrational quantum more, i.e., shows a node. The resulting function at time t = t2 resembles the first excited state of an antisymmetric stretch eigenfunction as is seen in Figure 2, panel d. On the other hand, the effective ð0Þ depopulation of the ground state w1  u0;1 means that at time t2 it does no longer appear in an eigenfunction-expansion of the wave function in state j1i. What is mainly left then is the first excited vibrational state so that a node appears in the wave function (panel c). If we proceed and regard the excitation of state j1i, starting with the wave function of the populated state j2i at t2, the argument is similar to that given above. Because the coupling – in first order – now contains a perturbation with the oscillating phase factor eþikt , this resembles a stimulated emission process with the same resonance condition as above. Starting from the first excited vibrational state, this means that one vibrational quantum

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J. Wehner et al. / Chemical Physics Letters 541 (2012) 49–53

is annihilated leading to a state j1i wave function possessing mainly ground state character at time t = t3 (Figure 2, panel e). The depopulation of the first excited state in j2i (present at t3 ) obviously leaves the next higher vibrational wave function, i.e., another node appears (panel f). We note that the next half-cycle of the population transfer curve ends up with wave functions resembling the ones shown in Figure 2 at time t 1 , which can be rationalized following the reasoning given above.

1

∆ = 0.0875 eV

J = 0.0700 eV

0.8

0.6

0.4

0.2

0

J = 0.0175 eV 0.8

0.6

0.4

0.2

0

0

40

80

120

time Figure 3. Population dynamics in the dimer. In both cases, the coupling acting perturbation is of the strength k = 0.0875 eV. As compared to the curves in Figure 1, the upper panel contains curves for a stronger electronic coupling element J. The lower panel illustrates the case where the perturbation is switched on/off faster.

1

d

P2(t)

0.8

The calculations presented up to now are restricted to a fixed electronic coupling J = 0.0175 eV. To get some insight what happens for larger couplings, we show in Figure 3, upper panel, the population dynamics for a coupling of J = 0.07 eV. It is seen that the general behavior discussed above does not change, i.e., the localization on the perturbed monomer unit is also found. The difference is that the degree of localization decreases which is accompanied with an even distribution of population after the perturbation is turned off. Also, the simple scaling with the coupling strength as suggested by Eq. (9) does not hold. Here, the particular form of the time-dependent perturbation is of importance. This can be taken from the curves displayed in the lower panel of Figure 3. There, the interaction Wt is taken such, that the switch on/off process takes five vibrational periods. This leads to a reduced localization (as compared to the case displayed in Figure 1, upper right panel) and larger oscillations during the time where the constant perturbation is present. What we have found so far and for the dimer is, that a timedependent perturbation can lead to a fast energy localization in the monomer which experiences the perturbation. This localization is removed if the energy induced shift is comparable to the vibrational level spacing. Because the latter changes in larger aggregates it is not clear, if the trends established for the dimer are also present in larger aggregates. As an example we regard the trimer system. In Figure 4, we compare populations obtained for two different scenarios. The right hand panels show the case (Md1 ) where the perturbation acts on the first monomer in its excited state (state j1i), and the left hand panels (M d2 ) illustrate the situation where state j2i is perturbed. In each case, the initial population is equally distributed over the three states. We note that this does not reflect the situation which corresponds to the population distribution obtained from the eigenvectors in the three-level system. Let us first discuss the case with a perturbation of the configuration j1i (first monomer in excited state). For our choice of initial conditions, the intermediate state j2i is more populated at all times than states j1i and j3i if no perturbation is present. This is reasonable because the net population transfer into the intermediate state which couples to both of the others is the largest. If the perturbation is switched on for the value of k = 0.0875 eV, a very similar behavior as for the dimer is encountered. Namely, the population localizes in the perturbed configuration. The other two configurations are equally populated,

0.0875 eV P1(t)

d

M2

M1

0.6 0.4

P2(t)

Wt

0.2

P1(t) P3(t)

0

0.160 eV

population

0.8 0.6 0.4 0.2 0

0.175 eV

0.8 0.6 0.4 0.2 0

0

40

80 time

0

40

80

120

time

Figure 4. Population dynamics in the trimer-system. Perturbations of different strengths act on the intermediate monomer (Md2 , left panels) and on the first monomer (Md1 , right panels) in their respective excited states. In the former case, the populations P1(t) and P3(t) are identical.

J. Wehner et al. / Chemical Physics Letters 541 (2012) 49–53

in the average. If the perturbation is switched off slowly, no appreciable localization is found at longer times. Rather, oscillations around different average values are found. The picture changes with increasing value of the perturbation. For the largest value of k which equals the vibrational quantum x = 0.175 eV, the same strong population oscillations as in the dimer case are seen. This again can be explained by the fact that the shift brings the slightly perturbed vibrational levels of the different configurations into resonance which is followed by a strong population transfer between all three excited states. After the interaction Wt settles to a value of zero, ongoing oscillations in the populations are seen but, in the average, no localization can be identified. If the weakest perturbation now acts on state j2i, the population localizes in this configuration (Figure 4, left upper panel). Here, because of symmetry, the populations of the other two configurations are identical. An increase of the parameter k leads (as in all other cases) to a more efficient population transfer. The latter is most pronounced in the case of resonance (k ¼ x). Thus, the tendencies found in the case of the dimer are transferable to the trimer system. To summarize, we investigate the localization of excitonic energy in small molecular aggregates which are subject to a perturbation. The perturbation consists of a time-dependent energy shift of the excited state of one monomer. For a coupled two-level system which corresponds to a dimer without nuclear degrees of freedom, it is seen that a localization of the population in the perturbed level occurs which goes in hand with the localization of the excitonic energy. Here, the value of the energy shift is unsubstantial. Employed a model which incorporates a single vibrational mode per monomer unit it is seen that for small shifts, the vibronic system behaves similar as the purely electronic system. This suggests that even a small perturbation can lead to a localization of the exciton. However, if the perturbation is increased and the energy shift becomes comparable to the vibrational quantum of the monomer vibration, a strong coupling between the configurations occurs which leads to large amplitude population oscillations. This means that the excitation energy no longer is localized but is transferred resonantly on an ultrashort time-scale. At times when this occurs, the wave functions in the two excited state configurations change their nodal character which, employing perturbation theory, can be interpreted in terms of stimulated absorption or emission processes. The general trends identified for the dimer are also seen in the trimer which is an example for a system with three vibrational modes and three coupled excited states.

53

In the present Letter we treated the case of a relatively weak electronic coupling. As an outlook, it will be worthwhile to extend the study to cases with stronger coupling and also to larger aggregates possessing more vibrational degrees of freedom. Acknowledgment We gratefully acknowledge financial support by DFG within the GRK 1221. References [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]

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