Exciton–vibrational coupling in the dynamics and spectroscopy of Frenkel excitons in molecular aggregates (original) (raw)

Introduction

The importance of vibronic excitations in the theory of molecular excitons had been recognized already in the original paper of Davydov in 1948 [1]. Comprehensive studies of the effect of vibronic states within exciton theory emerged in the late 50s and early 60s. Two of the central questions concerned the construction of a proper exciton–vibrational (EV) wave function and the applicability of different approximations to Davydov’s theory. The activities can be roughly divided into those focusing on periodic systems (molecular crystals or polymers) and those approaching the problem from the molecular side, essentially using the model of a molecular dimer. In a seminal paper, Simpson and Petersen [2] introduced a classification in terms of the relative strength of intermonomer electronic Coulomb coupling (CC) and intramonomer exciton–vibrational coupling (EVC) to distinguish weak and strong couplings according to the ratio of excitonic and local vibronic bandwidths. These cases were considered from both stationary state and time-dependent points of view, with a special focus on the qualitative discussion of the intermediate coupling regime. Applications covered monomer and crystal absorption spectra, which were discussed in light of earlier studies of vibronic excitations in carotenoid pigments, cyanine dye dimers, and polymers. This classification scheme was further developed in Refs. [3] and [4].

A more quantitative theory was put forward by Witkowski and Moffitt [5] on the basis of an EV Hamiltonian for a homo-dimer. Later this approach was extended by Fulton and Gouterman [6], who payed special attention to symmetry properties. Contrasting the regime of strong and weak EVC, as compared to the CC, they established the connection to the Jahn–Teller effect and vibrational intensity borrowing, respectively. In Ref. [7] they presented numerical simulations of dimer absorption spectra for various regimes of coupling strengths. Further, the effect of vibrational excitation in the electronic ground state on the degeneracy of excited states and their splitting due to CC was correctly pointed out for the first time. A systematic investigation of monomer, dimer, and crystal spectra based on the ideas of Ref. [5] can be found in Ref. [8].

A series of papers on spectral changes introduced by EVC was presented by McRae. In Ref. [9] a general theory of EVC was developed, for a model consisting of an array of non-rigid molecules treated in the harmonic oscillator limit, ordered on a rigid lattice. Explicit expressions were given for the vibrational part of interaction integrals in terms of local Franck–Condon factors. An application to a dimer has been put forward in Ref. [10], where, for the first time, a systematic discussion of the eigenstates of the Hamiltonian was presented together with absorption and fluorescence spectra as a function of the CC strength. Further developments include the application to polymer-like systems [9], the study of perturbation theory with respect to the CC [11], and the treatment of cases with two monomers per unit cell [12]. Finally, McRae used a variational approach to what was called the zone of molecular distortion, which surrounds the instantaneously excited monomer in a molecular crystal alike an exciton–polaron [13]. An early comprehensive review is given in Ref. [14].

Vibrational motion in an exciton state itself was treated by Merrifield on the basis of an effective Schrödinger equation [15]. This inspired the use of a Born–Oppenheimer type approximation for nuclear motion in excitonic states in Ref. [16], which fails, however, in cases of near-degeneracy of EV states. Merrifield also developed an approach to EVC in terms of the second quantization formalism employing a variational approach to the exciton band structure [17]. It was also used in Refs. [18] and [19] and extended to include configuration interaction in Ref. [20].

The formal basis for later developments of numerical strategies for calculating eigenspectra of larger excitonic systems in the presence of EVC was laid by Philpott [21], [22], [23]. He put forward the idea of the so-called n-particle (or multi-particle) expansion of the total wave function. In essence it is based on a decomposition into a basis containing up to n simultaneous vibronic (electronic excited state) and vibrational (electronic ground state) excitations. This basis is used to represent the Hamiltonian, whose eigenstates are determined by numerical diagonalization (also called direct diagonalization method). Needless to say, that several critical assessments of the validity of different levels of the n-particle approximation were performed, in particular when it comes to the simplest one-particle level, which is valid only if the local EVC dominates the intermonomer CC; see, e.g., Refs. [24], [25].

Recently, the n-particle approach was extensively used by Spano and co-workers paying particular attention to EVC effects on absorption and emission spectra [26], [27], [28], [29], [30], [31]. This resulted, e.g., in the discovery of the different behavior of H- and J-aggregates with respect to the ratio of 0–0 to 0–1 peak intensities for a model including a single coupled high-frequency mode (typically vinyl stretching mode at 1400 cm−1). Here, 0–n refers to transitions from the electronic and vibrational ground state to the state with n vibrational quanta in the electronically excited state. Note, however, that in contrast to the monomeric case in an aggregate this classification is not strict due to the mixing of excitonic and vibronic excitations. According to Spano et al. in absorption this ratio decreases/increases with CC strength for H/J-aggregates correspondingly, whereas the situation is reversed in emission. On the other hand, thermal effects, i.e. a Boltzmann population of initial states, lead to an increase/decrease of this ratio with temperature for H/J-aggregates. An interesting aspect concerns the coherent enhancement of the 0–0 transition in emission. This effect does not occur for the 0–1 transition such that the intensity ratio between the 0–0 and 0–1 transitions reflects the coherence domain size of the exciton, see Refs. [32], [33], [34], [35] for an expansion on this topic. In Ref. [36] an analysis of multi-particle states in terms of Fermi resonances between one- and two-particle bands was presented. A multi-mode extension of the n-particle expansion to include high- and low-frequency modes was given in Ref. [37]. Here, synergistic effects were, for instance, observed where the low-frequency mode modifies the intensity of the vibronic progression in a way similar to the effect of diagonal disorder [37]. Further, it was found that side band intensities of weakly coupled modes can be enhanced by strongly coupled modes [38], [39].

A schematic overview on the effect of EVC on the exciton dynamics can be obtained by inspecting the molecular dimer model shown in Fig. 1 (for a more detailed account on this model, see also Ref. [40]). Here, we have included one harmonic oscillator coordinate and two electronic states, S0 and S1, per monomer. For this situation there are three electronic configurations: First, both molecules are in their electronic ground state, (S0,S0), and the two-dimensional harmonic potential energy surface (PES) is centered at the equilibrium position (axes origin at (0, 0)). In the two excited state configurations, (S1,S0) and (S0,S1), the PES is displaced along the coordinate of monomer 1 and 2, respectively. This gives rise to two intersecting PES as shown in Fig. 1a. As a consequence of the CC between the two excited electronic configurations the degeneracy of the PES is lifted leading to two adiabatic PES. The nuclear motion on the adiabatic PES is conveniently described employing symmetric (Q+) and antisymmetric (Q−) combinations of the local coordinates (Fig. 1a). While along Q+ one has a shifted harmonic oscillator-like PES, the motion along Q− is governed by a double minimum PES in the lower adiabatic state, Fig. 1b. The nuclear dynamics is rather different if coupled and uncoupled cases are compared. For instance, suppose that there has been a vertical laser pulse excitation of a wave packet from the electronic ground state. As a consequence of symmetry the initial motion of the wave packet’s center of gravity will be along Q+ in the coupled case. In other words, not only the excitons will be delocalized on the aggregate, the coupled vibrations will respond collectively to the excitation as well. In passing we note that the scenario shown in Fig. 1 is not unique to the case of molecular aggregates. For instance, gas phase hydrogen-bonded dimers have been discussed in the context of the present approach. Here, the special topology of the PES appears as a vibrational quenching of the electronic line splitting [41].

The generalization of the single mode to a multi-mode dimer model had been provided by Myers-Kelley in 2003 [42]. Applications to absorption spectra revealed that if the CC is large compared to the vibronic width of the monomer spectrum, the dimer absorption spectra exhibit simple Franck–Condon progressions, having reduced vibronic intensities compared with those of the monomer. When the CC is comparable to the vibronic bandwidth, the H-dimer absorption spectra were found to show irregular vibronic frequency spacings and intensity patterns. In Ref. [42] the theory for calculating hyper-Rayleigh and resonance Raman scattering signals of excitonic dimers was formulated (for earlier work, see Refs. [43], [44]). This approach was used by Leng et al. to calculate resonance Raman spectra for merocyanine dyes, as monomers in dichloromethane solution and as H-dimers in dioxane solution [45]. Furthermore, a time-dependent approach to the calculation of spectra was employed to take into account up to 30 vibrational modes. For a comparison of diagonalization and time-dependent methods, see Ref. [46]. Both approaches yielded spectra, which in the strong CC limit, resemble those of a single electronic transition with appropriately rescaled vibrational displacements, transition dipoles, and electronic 0–0 transition frequencies.

Among other applications of methods based on direct diagonalization of the EV dimer Hamiltonian in an n-particle basis we mention the work of Guthmuller et al. In Ref. [47] absorption spectra of streptocyanine and its covalently bonded dimer were studied using time-dependent density functional theory. In particular an excitonic dimer in one-particle approximation has been compared to a supermolecule calculation of the Franck–Condon progression. In the supermolecule approach the system is treated as a whole and not as Coulomb-coupled monomers. For the considered example, which does not have a pronounced vibronic structure in the absorption spectrum though, this comparison yielded a rather good agreement. Subsequently, these authors presented successful simulations for the more challenging case of a PDI dimer (3,4,9,10-perylenetetracarboxylic diimide) using fourteen modes within a two-particle approximation, but restricting the excitation space to four and two simultaneously excited vibronic and vibrational modes, respectively [48].

There are more alternatives to the direct diagonalization of the exciton Hamiltonian when it comes to the calculation of, e.g., absorption spectra under the influence of strong EVC. Here, we mention the response function formalism in combination with a cumulant approximation [49], [50], [51], the quantum state diffusion approach [52], [53], and the coherent exciton scattering (CES) approximation [54], [55]. The latter is a Green’s function approach, which expresses the spectrum in terms of monomeric and aggregate line shape functions. This becomes possible by invoking a mean-field approximation that involves an average of the monomer Green’s function with respect to electronic ground state vibrations. The breakdown of the mean-field treatment in the case of strong CC was investigated in Ref. [56]. The CES approach has been extensively used by Briggs and Eisfeld and their co-workers to systematically analyze J- and H-bands of organic dye aggregates [57], [58], [59]; for an application to exciton transfer, see also Ref. [60].

The discussion so far has been essentially restricted to linear spectroscopies. Here, the effects of EVC appear in the respective line shapes, e.g., as a broadening and/or as a vibrational progression. Since the information content of broad spectra is rather limited, nonlinear spectroscopies have been developed to unravel the underlying dynamics on their actual time scales. Early experiments were mostly performed using the pump–probe technique. Here, an ultrafast pump pulse resonantly excites exciton–vibrational dynamics, which is interrogated by a time-delayed probe pulse. In 1994 Sundström and co-workers had been the first to demonstrate coherent nuclear wave packet dynamics upon ultrafast excitation of the pigments in a light-harvesting complex (LH1) [61], [62]. Also exciton dynamics in the same complex was studied using the pump–probe technique [63]. Wave packet dynamics in a simpler system, the B820 dimer, was observed in Ref. [64]. Besides a typical low-frequency mode (<200 cm−1) also modes up to about 700 cm−1 were reported and assigned to a photoproduct formed in the excited state. In an earlier investigation, the signatures of coherent vibrational motion in this system with a frequency around 170 cm−1 were attributed to an intermolecular mode [65]. When commercial amplified solid state femtosecond laser systems appeared, it became possible to use a spectrally broad “white light” pulse as a probe, enabling analyses of transient absorption spectra in a broad wavelength range. Such studies were used, among other techniques, to investigate the size of the exciton coherence domain [66], [67], [68] and exciton self-trapping (polaron formation) [69] in light-harvesting systems (for a review, see also Ref. [70]).

Various other femtosecond nonlinear spectroscopic methods, once being exotic proof-of-the-principle experiments, turned out to be very valuable in the studies of exciton–vibrational dynamics and couplings [49]. Particularly, photon echo experiments were implemented by a number of groups [71], [72], [73]. It was shown that a version of the three-pulse photon echo spectroscopy, the so-called echo peak shift (3PEPS) measurement [74], [75], [76], can be used to extract the spectral density of the EVC and can give information about the exciton transfer in light-harvesting complexes [77]. Three-pulse transient grating was also used to demonstrate vibrational wave packets around 150 cm−1 in the LH2 complex [78], whereas LH2 oscillations around 100 cm−1 were found in a fluorescence upconversion study [79].

In 2005 Fleming and co-workers were the first to use the newly developed electronic two-dimensional (2D) spectroscopy [80] for studies of exciton dynamics in a light-harvesting complex [81]. In a number of subsequent 2D spectroscopy experiments with light-harvesting complexes, long-lasting beating signals were observed [82], [83], [84]. Since EVC in chlorophyll-like chromophores within pigment–protein complexes is typically quite weak [85], the beatings were originally assigned to electronic coherences. However, distinguishing between electronic and vibrational origin of the beating signals in 2D spectroscopy turned out to be not trivial [86], [87], [88], [89], [90], [91]. Due to excitonic effects in light-harvesting complexes, strong mixing between vibrational and electronic transitions can occur, leading to significant redistribution of the transition strengths [92]. As a result vibrational coherences can be excited. It was concluded that the observed quantum beats in 2D spectroscopy of light-harvesting complexes most likely originate from coherent vibrations [93], [94], [95]. Recently, it was further argued that such non-classical vibrations can assist exciton transfer in photosynthesis [96].

There is a number of books devoted to theory and/or experiments of exciton dynamics and spectroscopy (see, e.g., Refs. [97], [98], [99], [100], [101], [102]). This review sets its focus on effects of strong EVC in molecular aggregates. In terms of dynamics this implies a treatment beyond low-order perturbation theory. Apart from a few illustrative examples, the huge body of perturbative approaches such as the Bloch- or Redfield-theories is not covered in detail. Further, effects of slow environments, i.e. static disorder, leading to inhomogeneous broadening (see, e.g., Refs. [103], [104], [105]) will not be considered.

The remaining text is organized as follows: Section 2 starts with a brief summary of the Frenkel exciton Hamiltonian and introduces the Huang–Rhys model for incorporating EVC. On this basis, two different strategies for the design of a system–bath model are discussed. Given the importance of direct diagonalization methods, the n-particle approach is introduced in Section 2.3 and illustrated for the case of a dimer with one coupled vibrational mode per monomer. In Section 3 we will cover the exciton–vibrational quantum dynamics from the perspectives of two recent developments: First, the solution of the time-dependent Schrödinger equation using the ML-MCTDH method. This essentially facilitates a numerically exact treatment of many coupled Degrees Of Freedom (DOFs), up to approaching the continuum limit. Second, the HEOM method, which enables one to solve problems in dissipative quantum dynamics beyond perturbation theory and Markov approximation. Recently, this method has been used extensively in exciton research. Therefore, a quite detailed derivation will be given using a stochastic decoupling approach, which did not attract much attention so far. Section 4 presents a number of applications, starting with a systematic investigation of the spectrum of the molecular dimer Hamiltonian as a function of EVC and CC strengths. This is followed by illustrative examples for the application of the ML-MCTDH method. After discussing examples for the Quantum Master Equation approach in Section 4.4, the HEOM method is illustrated in Section 4.5. Section 4.6 is devoted to the recent discussion of the role of EVC in photosynthetic light-harvesting. Finally, we present results on the laser control of exciton dynamics in Section 4.7. The review concludes with an outlook in Section 5.

Section snippets

Exciton–vibrational coupling in Frenkel exciton theory

The Frenkel exciton Hamiltonian rests on the assumption that the electronic structure of a molecular aggregate can be obtained by considering its interacting monomers. Further, if charge transfer phenomena are neglected, electron and hole forming the exciton are localized on the same monomer. In principle, the properties of respective molecular electronic states depend on all nuclear DOFs of the considered system. In practice this dependency is available at most in the limit of classical nuclei

Quantum dynamics of Frenkel excitons

Relevant molecular exciton transfer systems usually work under system–bath conditions, i.e. rather large chromophores are embedded into a solvent, protein or similar environments. Therefore, at first glance the consideration of the coherent limit of exciton transfer seems to be of rather academic nature. However, one can argue that in the limit, where the CC considerably exceeds the system–bath coupling, the dynamics will be quasi-coherent and a description in terms of the Schrödinger equation

Monomeric building blocks of aggregates and photosynthetic antennae

The following applications do not attempt to be exhaustive as far as the discussion of different aggregates or other light-harvesting systems like photosynthetic antennae is concerned. We rather will focus on generic models and a few examples for specific systems. The state of the art concerning artificial aggregates made from organic dyes has been summarized in Ref. [246]. A review on the status in photosynthetic light-harvesting can be found in Ref. [247]. The specific examples have either

Conclusions and outlook

The last decade has witnessed an enormous progress in our understanding of exciton dynamics in natural light-harvesting complexes and artificial molecular aggregates. In particular the combination of novel experimental techniques like 2D spectroscopy with powerful simulation tools like the HEOM and the ML-MCTDH approaches has led to a revision of the role of EVC for exciton dynamics. Instead of merely providing a heat bath, molecular vibrations, ranging from high-frequency intra-molecular ones

Acknowledgments

We gratefully acknowledge financial support by the Deutsche Forschungsgemeinschaft through the Sfb 652. This work was partly (S.P.) performed within the state contract of the RF Ministry of Education and Science for Siberian Federal University for scientific research in 2014 (Reference number 1792). Further, funding from the Natural Science Foundation of China (Grant No. 21373064) is acknowledged (Y.Y.). M.S. thanks S. Karsten for performing benchmark calculations to test the HEOM code using