Article pubs.acs.org/JPCA
Quantum Chemical Parametrization and Spectroscopic Characterization of the Frenkel Exciton Hamiltonian for a J‑Aggregate Forming Perylene Bisimide Dye D. Ambrosek,† A. Köhn,‡ J. Schulze,† and O. Kühn*,† †
Institut für Physik, Universität Rostock, D-18051 Rostock, Germany Institut für Physikalische Chemie, Universität Mainz, Duesbergweg 10-14, D-55099 Mainz, Germany
‡
ABSTRACT: Quantum chemical and quantum dynamical calculations are performed for a bay-substituted perylene bisimide dye up to its hexameric aggregate. The aggregate structure is determined by employing the self-consistent charge density functional tight-binding (SCC-DFTB) approach including dispersion corrections. It is characterized by a stabilization via two chains of hydrogen bonds facilitated by amide functionalities. Focusing on the central embedded dimer, the Coulomb coupling for this J-aggregate is determined by means of the time-dependent density functional theory (TDDFT) to be −514 cm−1. Exciton vibrational coupling is treated within the shifted oscillator model from which five strongly coupled modes per monomer are selected for inclusion into a minimal dynamic model. Performing wave packet propagations for a model employing up to 7 electronic states and 30 vibrational modes using the multiconfiguration time-dependent Hartree method, aggregate absorption spectra are obtained and compared to experiment.
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bisimides like PBI-1, no first-principle calculations of the spectroscopic properties have been performed so far. It should be noted, however, that there has been considerable effort directed toward the study of excitonic couplings in Haggregate-forming perylene bisimides but using a supermolecule approach for a dimer to deduce electronic states and couplings depending on intermolecular coordinates.12−14 While some authors claim the importance of charge-transfer excitons in π-stacked PBI systems,15,16 more accurate treatments of the electronic structure of PBI dimers indicate only minor charge-transfer contributions to the lowest pair of electronic excited states.14 This justifies a pure Frenkel-type effective Hamiltonian, as will be used in the present work. The coupling of the molecular electronic transitions to vibrational degrees of freedom is an important mechanism influencing excitation energy transfer.10 According to the strength of exciton−vibrational coupling (EVC), excitation energy-transfer regimes range from the Förster limit (strong EVC) to the (quasi-)coherent limit (negligible or weak EVC).7 Until recently, the discussion of EVC has been focused on the
INTRODUCTION Perylene bisimide derivatives are attractive candidates for building blocks of molecular aggregates to be used for artificial light harvesting applications.1 Of particular interest for applications requiring a high fluorescence are those derivatives that form J-aggregates (see, e.g., refs 2−4). Recently, we have started the investigation of the electronic properties of PBI-1 shown in Figure 1,5 which exhibits a number of outstanding properties upon aggregation such as a remarkably high fluorescence quantum yield4 as well as directed one-dimensional exciton motion along extended aggregates.6 While in ref 5 we focused on the properties of the PBI-1 monomers, it is the aim of the present work to extend this study to a description of aggregates. A straightforward way to build an essentially arbitrary long aggregate out of monomeric building blocks is the Frenkel exciton concept,7,8 which also provides the frame for simulation of linear and nonlinear spectroscopies (for reviews, see refs 9 and 10). Here, the monomers retain their electronic properties but are coupled via Coulomb interactions. Traditionally, the latter have been treated within the dipole approximation. For tightly packed aggregates such as the ones considered here, this approximation has been recognized to fail,11 and either full Coulomb coupling calculations are performed or the interaction is chosen empirically using experimental data. For J-aggregates consisting of perylene © 2012 American Chemical Society
Special Issue: Jörn Manz Festschrift Received: July 13, 2012 Revised: August 27, 2012 Published: September 4, 2012 11451
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Figure 1. A stick diagram of the investigated PBI-1 molecule (N,N′-di(N-(2-aminoethyl)-benzamide)-1,6,7,12-tetra(4-tert-butylphenoxy)-3,4:9,10perylenebiscarboximide). Notice that in the experiment, the 3, 4, and 5 positions at the phenyl ring in R are OC12H25, which are replaced by H in the present model. Further, R′ contains tert-butyl groups, which will not be considered here.
aspect of phase and energy relaxation, mostly described within second-order perturbation theory, for example, by means of the multilevel Redfield equations.17 The observation of long-lived quantum coherences in photosynthetic antenna complexes using two-dimensional spectroscopy18 has given a new twist to the issue of EVC, actually pointing to a perhaps less destructive role. This, of course, calls for a theoretical treatment that goes beyond Markovian perturbation theory, see, for example, refs 19 and 20. Recently, it has been even demonstrated that the observed coherences could be due to a mixing of electronic and vibrational states.21 In the case of molecular aggregates and, in particular, for the perylene bisimides considered here, the effect of strong EVC shows up already at the level of the linear absorption spectrum. It is shaped by a pronounced sideband at 1000−1500 cm−1 to the blue of the main (0−0) peak that can be assigned to a vibrational progression.1 This situation can be treated by defining respective vibronic exciton states, which incorporate selected vibrational coordinates in harmonic approximation (Huang−Rhys model).22−28 Due to numerical limitations, a straightforward application of this approach is restricted to small systems like dimers or trimers. Aggregates consisting of up to nine monomers have been treated using the multiconfiguration time-dependent Hartree (MCTDH) wavepacket propagation method.29 In order to treat still larger aggregates, further approximations have to be introduced such as the one-21 or two-particle approximation; the latter has been used to develop a Frenkel exciton polaron theory.30 Still, another alternative is the coherent scattering approximation employed within a Green’s function approach.31 In the present paper, we will develop a Frenkel exciton Hamiltonian for PBI-1 aggregates and characterize it by means of linear absorption spectra. This involves the determination of monomeric EVC parameters within a Huang−Rhys model, the calculation of Coulomb couplings for a PBI-1 dimer taken out of an optimized hexameric structure, and the calculation of absorption spectra, including seven electronic aggregate states and 30 nuclear degrees of freedom within the MCTDH approach. In the following section, we briefly summarize the basics of the Frenkel exciton Hamiltonian and give some details on the computational approach. Subsequently, results of numerical simulations are discussed in light of experimental data.
Hagg =
∑ Hm,g|0⟩⟨0| m
+
∑ (Tnuc + ∑ Vk ,g + Um ,eg)|m⟩⟨m| m
+
k
∑ Jmn|m⟩⟨n|
(1)
m,n
Here, the site index m runs from 1 to Nagg, that is, the number of monomers comprising the aggregate. In eq 1, |0⟩ is the state where all monomers of the aggregate are in their electronic ground state (|0⟩ = |g1, ..., gNagg⟩), and |m⟩ denotes the single exciton states corresponding to the case where the mth monomer is electronically excited to state em and all other monomers of the aggregate are in their electronic ground state (|m⟩ = |g1, ..., em, ..., gNagg⟩). The first sum in eq 1 is called zeroexcitation Hamiltonian. The other terms give the single excitation Hamiltonian; Tnuc is the total kinetic energy of vibrations, and Vk,a is the Born−Oppenheimer potential energy surface for nuclear motion at site k and in electronic state a. The energy gap, Um,eg, is given by the difference between the potential energy surfaces at the site m, where the excitation is localized, plus some correction due to the Coulomb coupling, that is Um ,eg = Vm ,e − Vm ,g +
∑ [Jmk (emgk, gkem) k≠m
− Jmk (g mgk , gkg m)]
(2)
Here, we used the general Coulomb integral Jmn (am , bn , cn , dm) =
∫ dr dr′
5am , dm(r)5bn , cn(r′) |r − r′|
(3)
with the generalized molecular charge density 5am , bm(r) = ρa
m , bm
(r) + δam , bmnm(r)
(4)
and the nuclear charge density nm(r) = − ∑ ezμδ(r − R μ) μ∈m
(5)
With the help of this definition, the resonant coupling between the monomer excitations in eq 1 is given by Jmn = Jmn(emgn,engm). Assuming that the nuclear degrees of freedom can be treated within the harmonic approximation and introducing normal mode coordinates for the mth monomer, {Qm,ξ}, the monomer part of eq 1 contains the vibrational Hamiltonian for the electronic ground state, given by
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THEORETICAL METHODS Frenkel Exciton Hamiltonian. The Frenkel exciton Hamiltonian is obtained from the expansion of the aggregate Hamiltonian into the basis of n-excitation states. It can be written as7 11452
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The Journal of Physical Chemistry A Hm ,g({Q m , ξ}) =
∑ ξ
⎞ ℏωm , ξ ⎛ ∂ 2 2 ⎟ ⎜ + Q m , ξ⎟ 2 ⎜⎝ ∂Q m2 , ξ ⎠
Article
transition dipole moment dm) and τ is a parameter mimicking the finite line width (dephasing time) of the real system. Notice that we have obtained a better agreement with experiment by using the Gaussian-type broadening ∝exp(−(t/τ)2). The present Frenkel Hamiltonian suggests the introduction of diabatic electronic states, α = (0,{m}). The nuclear wave function for these states, Ψα(Q,t), will be expanded into multiconfiguration time-dependent Hartree (MCTDH) form29,33
(6)
Here, we have used dimensionless normal mode coordinates by employing the oscillator’s length scale (ℏ/μm,ξωm,ξ)1/2, where μm,ξ and ωm,ξ are the reduced mass and frequency, respectively, of the ξth coordinate at site m. Assuming a linear displacement of the excited-state potential energy surface Vm,e with respect to the ground state (shifted oscillator model), the energy gap Um,eg in eq 2 is given by Um ,eg({Q m , ξ}) = Em +
∑ ℏωm,ξgm(ξ)Q m,ξ ξ
n j ... n j 1
Ψα(Q, t ) =
j1 ... jM
(7)
1 ∂Um ,eg ℏωm , ξ ∂Q ξ
Q ξ=0
(9)
Alternatively, one can define the reorganization energy for mode ξ, which is required to relax the energy to the excitedstate minimum after vertical electronic transition. The reorganization energy is given by
Em(reorg) = ℏωm , ξSm , ξ ,ξ
(10)
Hence, in order to calculate the monomer contribution to the aggregate Hamiltonian, one starts with the determination of normal modes of vibration for the electronic ground state at a stationary point. Next, the vertical electronic excitation energy has to be calculated as well as the gradient on the excited-state potential. Finally, this gradient vector needs to be projected onto the normal mode directions defined for the ground state in order to obtain the Huang−Rhys factors. Below, we will use the Huang−Rhys factor to select the Nvib most important modes for the monomer. Notice that, in order to keep the model simple, we have neglected the possible effect of Duschinsky mixing in the excited state. In principle, this would render the dynamics to become more complex, as shown recently, for instance, in ref 32. Further, note that, in principle, Jmn will depend on the nuclear coordinates as well and, here in particular, on intermolecular coordinates such as the distance between the monomers defined, for example, using the centers-of-masses. This dependence will be neglected in the following. Absorption Spectra. EVC in aggregates can be characterized by means of the absorption spectrum. Here, a timedependent formulation will be used, that is, the absorption spectrum is expressed as7 I(ω) = I0ω Re
∫0
∞
M
1
M
Here, the are the time-dependent expansion coefficients weighting the contributions of the different Hartree products, which are composed of single-particle functions (SPFs), ϕ(α) jk (Qk;t), for the kth degree of freedom in state α. It is important to notice that because each monomer has its own set of coordinates, the vector Q = (Q1, ..., QM) is identical to Q = (Q1,ξ=1, ..., Q1,ξ=Nvib, ..., QNagg,ξ=1, ..., QNagg,ξ=Nvib). Thus, the dimension of the vibrational space is M = Nagg × Nvib. Computational Methods. The gas-phase ground-state equilibrium geometry of the PBI monomer has been obtained using density functional theory (DFT) with the B3LYP functional and a 6-311G* split-valence basis set as described in ref 5. Convergence criteria of 10−7 for the geometry optimization and a fine quadrature grid (m4) was used. The first electronic excited singlet state was calculated employing time-dependent DFT (TDDFT). For these calculations, the TURBOMOLE program package was utilized.34 Starting from the optimized monomer structures, aggregates of increasing size have been prepared, with starting geometries resembling those suggested empirically in ref 4. Within the Frenkel exciton framework, Coulomb couplings between neighboring monomers will be needed to build up an aggregate Hamiltonian. Focusing on a single pair of monomers taken to be representative for a situation in a long aggregate, one needs to minimize boundary effects by choosing the computational model to be sufficiently large. Here, we will consider a hexameric aggregate and take the central dimer for evaluating the Coulomb coupling. These couplings have been determined for the dimer in the gas phase according to eq 3 using a prerelease version of the “intact” module of TURBOMOLE. The monomeric subunit of the PBI aggregate is composed of 124 atoms and 548 electrons, greatly limiting the computational methods capable of doing geometry optimizations of aggregate structures. Therefore, the hexameric aggregate structure has been optimized using the dispersion-corrected self-consistent charge density tight-binding DFT method (SCC-DFTB) as implemented in the DFTB+ package,35 and the parametrized two-center integrals are provided in the mio-0-1 set.36 MCTDH calculations of absorption spectra have been performed using the Heidelberg program package.37 Three systems have been considered, that is, the monomer, dimer, and hexamer having 2, 3, and 7 diabatic states, respectively. For each monomer, five nuclear degrees of freedom, Qm,1−Qm,5, are included in the model. For the primitive basis, we have chosen a 15-point harmonic oscillator discrete variable representation in the interval [−5:3] for the mode below 100 cm−1 (Qm,1) and [−4:3] for all other modes. For the SPF basis, the multiset method was used. Because there is no mode coupling in the
(8)
1 2 g (ξ ) 2 m
1
Cj1(α) ,...,j M (t)
The coupling of a particular mode to the electronic transition can be characterized by the (dimensionless) Huang−Rhys factor
Sm , ξ =
C (j α,...,) j (t )ϕj(α)(Q 1; t )... ϕj(α)(Q M ; t ) (12)
Here, Em is the electronic excitation energy including the contribution from the Coulomb coupling and the dimensionless coupling constant is given by g m (ξ ) =
M
∑
2
dt e iωt − (t / τ) ⟨Ψ0|d e−iHaggt / ℏd|Ψ0⟩ (11)
where d is the dipole operator taken in the form d = ∑m(dm | m⟩⟨0| + h.c.) (Condon approximation with a constant S0−S1 11453
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monomer case, the time-dependent Hartree ansatz is sufficient for the wave function propagation. In case of the dimer, converged spectra could be obtained by using two and three SPFs per excited state for weakly (Qm,1,Qm,2,Qm,5) and strongly (Qm,3,Qm,4) coupled modes, respectively. For the hexamer, this setup has been computationally too demanding, and we restricted the SPF basis to one and two functions for the weakly and strongly coupled modes, respectively.
Table 1. Harmonic Vibrational Frequencies, Which Are Appreciably Coupled to the S0−S1 Electronic Transition of the PBI Monomera mode 2 4 7 8 11 12 15 (Q1) 18 20 23 25 28 31 40 (Q2) 45 (Q3) 46 53 71 85 96 98 114 133 250 (Q4) 266 278 (Q5) 299 305
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RESULTS Monomer Properties. The PBI monomer has already been characterized in ref 5 using the MRCI/DFT approach. The vertical electronic transition energy was determined as Em = 2.13 eV (582 nm). In the experiment, a strong maximum at 580 nm was observed, which, according to a Franck−Condon analysis, can be mainly attributed to the 0−0 transition. The broad shoulder observed between 575 and 500 nm has been assigned to a vibrational progression of the same electronic transition. This appearance of the spectrum is typical for perylenes (see, e.g., ref 38). In ref 5, the vibrational progression had been investigated using an effective vibrational mode, determined within the range of 800−2000 cm−1. This effective mode had a frequency of 1415 cm−1 and a Huang−Rhys factor of 0.44. Using this one-mode Franck−Condon model, the spectrum could be well-reproduced, including the effect of the reorganization energy. However, the experimental spectrum is rather broad (the main peak has a width of 1110 cm−1), suggesting the contribution of many modes. In Table 1, we have compiled the information on all relevant modes, giving their frequencies, Huang−Rhys factors, and reorganization energies (cf. Figure 2). Strongly coupled modes cluster are in the range from ∼1300 to ∼1700 cm−1. These are essentially breathing modes of the central perylene core (cf. refs 5 and 38). At around 200 cm−1, two strongly coupled modes are found, which correspond to a twisting motion of the perylene core. Finally, there are many low-frequency modes (
