Frenkel to WannierâMott Exciton Transition: Calculation of FRET

Article pubs.acs.org/JPCB

Frenkel to Wannier−Mott Exciton Transition: Calculation of FRET Rates for a Tubular Dye Aggregate Coupled to a CdSe Nanocrystal Thomas Plehn,† Dirk Ziemann,† Jörg Megow,‡ and Volkhard May*,† †

Institut für Physik, Humboldt-Universität zu Berlin, Newtonstraße 15, D-12489 Berlin, Germany Institut für Chemie, Universität Potsdam, Karl-Liebknecht-Straße 24-25, D-14476 Potsdam, Germany

‡

S Supporting Information *

ABSTRACT: The coupling is investigated of Frenkel-like exciton states formed in a tubular dye aggregate (TDA) to Wannier−Mott-like excitations of a semiconductor nanocrystal (NC). A double well TDA of the cyanine dye C8S3 with a length of 63.4 nm and a diameter of 14.7 nm is considered. The TDA interacts with a spherical Cd819Te630 NC of 4.5 nm diameter. Electronic excitations of the latter are described in a tight-binding model of the electrons and holes combined with a configuration interaction scheme to consider their mutual Coulomb coupling. To achieve a proper description of TDA excitons, a recently determined structure has been used, the energy transfer coupling has been defined as a screened interaction of atomic centered transition charges, and the site energies of the dye molecules have been the subject of a polarization correction. Even if both nanoparticles are in direct contact, the energy transfer coupling between the exciton levels of the TDA and of the NC stays below 1 meV. It results in FRET-type energy transfer with rates somewhat larger than 109/s. They coincide rather well with recent preliminary experiments. molecular electronic Qy excitation of pheophorbide-a. Using differently shaped NCs (spherical, hemispherical, and pyramidal) with roughly the same number of atoms, EET rates around 108/s were obtained if the molecule is attached to the NC surface. To treat large NCs with some thousands of atoms, we utilized the methodology of a semiempirical tight-binding description.15−20 A configuration interaction (CI) approach allowed electron−hole Coulomb interaction to be considered and Wannier−Mott-like excitations of the NC to be described. In the following, we proof a possible control of the EET rate by considering the coupling of a similar CdTe NC not to a single pheophorbide-a molecule but to a huge cyanine dye complex. Choosing molecular chains where the length can be altered systems are available which number and shape of exciton levels can be varied. Of particular interest in this respect are tubular dye aggregates (TDAs) formed by the cyanine dye C8S3 (see Figure 1). These TDAs have been intensively studied in refs 21−25. Related computations on the exciton levels and on linear absorption spectra can be found in refs 26−28. The TDAs offer a broad Frenkel-exciton spectrum, and first experiments on EET between this type of TDAs and NCs are undertaken in ref 29. A further motivation for the subsequent studies is our novel structure model for these

I. INTRODUCTION If the classical description due to Förster is valid, rates of excitation energy transfer (EET) among single molecules display the famous 1/R 6 dependence. R denotes the intermolecular distance, and the specific dependence is due to the fact that molecular electronic excitations are dominated by transition dipole moments. This 1/R6 dependence of the rate undergoes drastic changes if the simple model of interacting transition point dipoles is not further valid. By combining nanoparticles of varying composition and shape, other dependencies on the interparticle distance are obtained (see the recent studies1,2 and references therein). The coupling of molecules to differently shaped semiconductor nanocrystals (NCs) was of particular interest in this respect (cf., for example, refs 3−9). Although experimentally investigated for more than a decade, theoretical studies describing molecule−NC EET with an atomistic resolution found less interest. A dipyridyl porphyrin interacting with a Cd33Te33 NC coated by a Zn78S78 shell has been investigated in ref 10 using a DFT approach. In ref 11, EET rates for a CdSe−NC chlorophyll complex were determined. TD-DFT techniques could be applied in ref 12 to a tiny Cd6Te6 cluster interacting with a rhodamin cation. In a similar way, EET among different NCs was investigated.13 Recently, we described EET between a Cd1159Se1450 NC and the tetrapyrrole-type pheophorbide-a molecule.14 By extending existing studies in the literature, EET was characterized by a coupling which relates Wannier−Mott-like excitons of the NC (and not uncorrelated electron−hole pairs) to the intra© XXXX American Chemical Society

Special Issue: John R. Miller and Marshall D. Newton Festschrift Received: November 7, 2014 Revised: December 22, 2014

A

DOI: 10.1021/jp5111696 J. Phys. Chem. B XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry B

Singly excited states of the TDA are deduced from the standard Frenkel-exciton Hamiltonian HFX =

∑ Em|ϕm⟩⟨ϕm| + ∑ Jmn|ϕm⟩⟨ϕn| m

m≠n

(1)

In its diagonal part, it covers the so-called site energies Em (excitation energy of molecule m). The off-diagonal part is formed by the EET coupling (excitonic coupling) Jmn which resonantly transfers excitation energy from one dye molecule to the other (cf. eq S3 of the Supporting Information). The ϕm are molecular product states with molecule m in the first excited state and all other molecules in the ground state. Single exciton states as eigenstates of HFX with energy E α̅ are of the form

|α ̅ ⟩ =

∑ Cα̅(m)|ϕm⟩

(2)

m

Figure 1. View on the TDA−NC complex displaying EET from the TDA to the NC (solvent molecules are not shown). The TDA is formed by five fragments of 12.68 nm (each with 828 C8S3 molecules as used in the MD simulations to fill the simulation box). The total length of the TDA is 63.4 nm (diameter of 14.7 nm) which is used to define exciton levels. One of the six ribbons which are turned around the TDA axis to form the inner and outer walls was highlighted. The spherical Cd819Te630 NC of 4.5 nm diameter is placed in the axial position close to the TDA surface.

The minimal version of the exciton Hamiltonian introduced so far has to be improved by a consideration of further couplings among the molecules, which may change the site energies as well as the matrix elements of the excitonic coupling.30,32 Electrostatic couplings due to permanent charge distributions in the ground and excited states of the dye molecules result in the shift ΔEm(el) =

∑ [Jmk (eg , ge) − Jmk (gg , gg )] k

(3)

of site energy Em (if necessary contributions due to solvent molecules can be incorporated into the k sum33,34). In contrast to the subsequent considerations, these couplings are part of the two-level model for HFX.35 In section 1 of the Supporting Information, it is demonstrated that corrections of the Frenkel-exciton state can be related to the need to go beyond the minimal version of the exciton Hamiltonian which only refers to two states per molecule. When studying a single molecule in a CI scheme, different configurations with multiple excitations have to be considered. This observation is also valid here. The resulting corrections can be explained by van der Waals (dispersive) interactions of the ground and first excited electronic states of a considered molecule with higher excited states of all environmental molecules. As determined in ref 37, dispersive shifts dominate the shifts due to electrostatic and excitonic couplings. We introduce the change of the molecular energy referring to state φma as ΔE(dis) ma (a = g, e labels the electronic ground and first excited states, respectively). Then, it follows32

TDAs as described in ref 30. Thus, the subsequent computations on EET from a TDA to a NC merge earlier work of refs 14 and 30. Therefore, the used theory is only briefly explained (for some details, see the Supporting Information). The paper is organized as follows. In the next section, we explain the novel aspects used to define exciton levels in the TDA. Section II.B gives a brief introduction to the NC exciton description. The EET rates are described in section III. The manuscript ends with some concluding remarks in section IV.

II. THE TDA−NC MODEL Before characterizing details of the EET from the TDA to the NC, we determine in a first step the exciton states of both constituents of the complete system. Then, we judge the strength of EET coupling among the TDA and the NC. A. Frenkel-Like Excitons of the TDA. A recently repeated structure analysis of the cyanine dye C8S3 TDAs (see the cryoTEM data of ref 24) combined with extensive MD simulations resulted in a redefined TDA spatial structure (it differs considerably from the one suggested in ref 23 and is described in detail in ref 30). Accordingly, the TDA consists of six ribbons which are wrapped around a common axis to let the TDA be formed by an inner and an outer cylinder (see Figure 1). Both cylinders have the same periodicity (resulting in differently dense inner and outer walls). The diameter of the whole TDA is 13 nm, and the width of a single ribbon amounts to 2 nm (the mean distance of neighboring molecules is 5 Å for both the inner and outer walls). The unit cell consists of two molecules which are shifted perpendicular to the ribbon by 9 Å. The stability of this structure has been confirmed by extensive molecular dynamics (MD) simulations (consideration of 828 C8S3 molecules, 81 672 water molecules, and 10 746 methanol molecules; simulation time up to 3 ns31). Since MD simulations generate thermal induced deviations from the ideal structure, they also offer a direct access to consider structural disorder.

(dis) ΔEma = −∑ ∑ k

f ,f ′

|Jmk (ff ′, ga)|2 Emfa + Ekf ′ g

(4)

f and f ′ count all higher excited energy levels ( f, f ′ > e). Emfa and Ekf′g denote the transition energies of molecule m and k, respectively. The two-molecule Coulomb matrix elements Jmk(f f ′, ga) have been defined in eq S3 of the Supporting Information. They relate transitions in the considered molecule m to transitions in all other molecules labeled by k. According to the actual position of molecule m in the TDA, the k summation notices the concrete surrounding. The overall site energy shift follows as (dis) (dis) ΔEm(dis) = ΔEme − ΔEmg

(5)

Considering the structure of the TDA, Figure 1, it is obvious that such a dispersion shift is different for molecules in the B


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Figure 2. TDA absortion spectrum (red curve) confronted with the NC absorption (blue curve). (The stick spectrum corresponding to the TDA has been broadened by 5 meV and that of the NC by 0.5 meV.)

inner wall and for those in the outer wall. In order to determine the different energy shifts, the expressions, eqs 4 and 5, have to be transformed into a tractable form to compute its dependence on the mutual position of the coupled molecules. This has been recently demonstrated in refs 30 and 32 and is briefly indicated in section 1.3 of the Supporting Information. In similarity to the site-energy shift, one also obtains an alternation of the excitonic coupling, i.e., screening (see, e.g., ref 36). It reduces the coupling Jmn between two molecules to f mnJmn.36,37 The prefactor f mn can be approximated by a constant 1/n2, where n is the optical refractive index.37 Using n = 1.78, we get f mn = 0.32, i.e., a considerable reduction of the excitonic coupling. Noting this small excitonic coupling, the splitting of the TDA absorption could be explained in ref 30 mainly by the site-dependent polarization. Nevertheless, the Frenkel-like excitons of the TDA form a band with a width larger than 100 meV (see Figures 2 and 3). It has been calculated by using five TDA fragments (with a single fragment defined by the MD simulation box), resulting in a total length of 63.4 nm. According to the total number of 4140 C8S3 molecules, the same number of exciton levels is considered (cf. also Figure 1). The two main peaks in the absorption spectrum are originated by the inner and outer walls, whose molecules undergo a somewhat different polarization shift.30 To evaluate the various exciton levels, we display in Figure 4 the exciton expansion coefficients |C α̅ (m)|2, eq 2. To present the data, the TDA has been cut along the main (longitudinal) axis and rolled out in such a way to have the view on the inner wall. Expansion coefficients for different exciton levels at the energies E α̅ of 2.058, 2.09229, 2.09234, and 2.18508 eV have been taken. The participation numbers Sα̅ = 1/∑m |C α̅(m)|4 are 79, 177, 376, and 53, respectively. Moving from the low to the high energy part of the chosen energies, the exciton

Figure 3. Absolute values of NC−TDA energy transfer coupling Vαα̅ specified for the lowest bright NC exciton α = 1 and versus the energy Eα̅ of TDA excitons (red curve). The NC is placed in the axial position to the TDA with a minimal NC−TDA atom−atom distance of 3.8 Å. Uniform level broadening of Γ = 5 meV has been assumed. Blue curve: room temperature thermal distribution fα . ̅

localization turns from the inner wall to the outer wall. In the present study, we did not carry out any configuration averaging, which is somewhat justified by the consideration of more than 4000 molecules (effect of self-averaging). Due to the favorite positioning of the C8S3 molecules between neighboring ribbons, the expansion coefficients for the lowest energy levels are larger perpendicular to the ribbons than along them. Thus, excitons are localized across different ribbons (either of the inner or outer wall). We see the same behavior in Figure 4 for the highest exciton level. Those in between (E α̅ = 2.09229, 2.09234 eV) show a rather diffusive distribution. Interestingly, their participation numbers are somewhat larger than those of the two levels at the border of the band. B. Wannier−Mott-Like Excitons of the NC. In order to describe singly excited electron−hole pair states for a NC, we C


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Be aware of the fact that the present approach does not account for two or more electron−hole pair excitations, since electron− electron and hole−hole Coulomb interaction have been neglected. The low energy part of the NC exciton spectrum is shown in Figure 2. There the NC absorption is confronted with that of the TDA, indicating that the NC levels coincide well with those of the TDA.

III. RATES OF EET To judge the type of EET coupling between the TDA and the NC, we have to determine respective coupling matrix elements Vαα̅ . They couple the de-excitation of a Frenkel-like exciton |α̅ ⟩ in the TDA to an excitation of a Wannier−Mott-like exciton |α̅ ⟩ in the NC. As in the case of simple EET among molecules, Vαα̅ is determined by transition densities. However, densities have to be introduced which describe transitions into the respective exciton states (details are given in section 3 of the Supporting Information). Concrete computations for the present system are shown in Figure 3. There, Vα = 1α̅ is drawn versus the TDA energies E α̅ (α = 1 labels the lowest bright NC exciton level). The general view of Vαα̅ does not change if we consider the coupling to higher lying NC exciton levels. Although the NC is placed rather close to the TDA surface, all coupling matrix elements take values below 1 meV. This value is smaller than the broadening due to electron−vibrational coupling. We cannot expect the formation of TDA−NC hybrid exciton levels, which would require contributions due to higher orders of Vαα̅ . Instead, we stay at the lowest order contribution. Accordingly, EET has to be described by the following rate of TDA−NC EET 2π k TDA → NC = ∑ f |Vαα̅|2 δ(Eα̅ − Eα) ℏ α , α α̅ (9)

Figure 4. Expansion coefficients |C α̅ (m)|2 of TDA Frenkel-like exciton states, eq 2. For presentation, the TDA has been cut along the longitudinal axis and rolled out (view on the inner wall; radian measure for a positioning perpendicular to the longitudinal axis). Upper left panel, exciton level with energy Eα̅ = 2.058; upper right panel, Eα̅ = 2.09229 eV; lower left panel, Eα̅ = 2.09234 eV; lower right panel, Eα̅ = 2.18508 eV (color code: blue, zero; light gray, small values; ruby-colored, large values).

and by the rate for the reverse process kNC → TDA =

∑ Eaea+ea − ∑ Ea̅ ha+̅ ha̅ + ∑ ∑ V aa(e̅−, bh)̅b ea+ha+̅ hb ̅ eb a

a

a,b a ,b

(6)

Electrons are represented by the energies Ea and the second quantization operators e+a and eb. For the holes, we introduced E a ̅ and h a+̅ as well as hb ̅ . The electron−hole pair interaction (e − h) (e − h) (e − h) V aa = −W aa + Jaa , b̅b , b̅b , b̅b ̅

̅

(7)

̅

is constituted by the Coulomb matrix element W and the exchange matrix element J(e−h). Some details are given in section 2 of the Supporting Information. The single exciton states with energy Eα can be introduced as (e−h)

|α ⟩ =

∑ Cα(aa ̅ )ea+ha+̅ |ψ0⟩ a,a

∑ fα |Vαα̅|2 δ(Eα − Eα̅ ) α ,α

(10)

The TDA−NC transfer takes place from a thermal distribution fα = 1/Z × exp(−E α̅ /kBT) of Frenkel-type excitons. For the ̅ reverse process, the transfer starts from a thermal distribution fα of Wannier−Mott-like excitons. The used golden-rule-type expressions cannot be translated into a more compact notation, since any further simplification of the Vαα̅ is not possible. (One may formally introduce a dipole−dipole approximation which however is completely inadequate for the present system of strongly delocalized and spatially rather closely positioned excitations.) For computations, the δ-function is broadened according to Γ/π × 1/([Eα − E α̅ ]2 + Γ2) with a state independent Γ. To discuss the EET rate, we note Figure 2, displaying the TDA and NC absorption spectrum and thus the main exciton levels. Figure 3 also shows the EET coupling. A comparison indicates that the relevant NC levels are all addressed by transitions from the TDA (the thermal distribution and the couplings Vαα̅ are large enough). Figure 5 displays kTDA→NC, eq 9, versus the NC−TDA distance. This distance increases due to a NC shift perpendicular to the TDA surface. Three different positions of the NC to the TDA are chosen (NC in axial position and shifted away along the TDA). We studied various positions and orientations of the NC along the TDA. However, only the direct TDA−NC distance has an effect on the rate.

follow our earlier work of ref 14 and the methodology of refs 17−20. These approaches combine a tight-binding description of the individual electron and hole states with a consideration of the Coulomb attraction in a CI methodology. The Hamiltonian valid for an interacting electron−hole pair and Wannier−Mott-like exciton formation has the standard form (see also section 2 of the Supporting Information). HWMX =

2π ℏ

(8) D



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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Financial support by the Deutsche Forschungsgemeinschaf t through Sfb 951 is gratefully acknowledged.

Figure 5. EET rate kTDA→NC versus NC−TDA distance (chosen is the minimal NC−TDA atom−atom distance at the given positioning). Blue curve, NC is placed in axial position to the TDA (cf. Figure 1) located at the TDA center; yellow curve, NC is moved 12.7 nm away from the central position along the longitudinal TDA axis; red curve, as yellow curve but for a shift to a terminal position at 25.4 nm (room temperature conditions; the TDA is formed by a 5-fold repetition of the basic segment of 12.7 nm, cf. Figure 1).

Moving the NC perpendicularly away from the TDA decreases the rate. For the closest NC−TDA distance, a lifetime of the TDA excitation of about 250 ps is realized.

IV. CONCLUSIONS A theoretical description has been given of the coupling between Frenkel-exciton-like states of a tubular dye aggregate (TDA) formed by the cyanine dye C8S3 and Wannier−Mottlike excitations of a CdTe nanocrystal (NC). The TDA excitons have been computed starting with a recently determined structure and defining the excitonic coupling as an interaction of atomic centered transition charges. Polarization corrections of the excitation energies due to the actual molecular position in the TDA explain the double peak structure of the absorption spectrum. Electronic excitations of the NC were described in a tight-binding model of the electrons and holes. It has been combined with a configuration interaction scheme to consider the Coulomb coupling between both. Even in van der Waals contact the excitation energy couplings between the exciton levels of the TDA and of the NC stay below 1 meV. This range was found already recently for a CdSe NC coupled to a single tetrapyrrole-type molecule.14 Here, we obtain a somewhat larger transfer rate of about 4 × 109/s. Recent preliminary experiments of ref 29 confirm this value.

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ASSOCIATED CONTENT

* Supporting Information S

In the first section, the generalized Frenkel-exciton model of the TDA is introduced; afterward, formulas for the EET coupling are quoted; the configuration interaction scheme for Frenkel excitons is used to derive explicit formulas for disperion corrections; the computation of the site energy shift due to environmental induced dispersion is explained; a second section deals with the description of NC electron−hole pairs; EET coupling matrix elements between the TDA and the NC are derived in the third section. This material is available free of charge via the Internet at http://pubs.acs.org. E


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(33) Megow, J.; Kulesza, A.; Qu, Z.-w.; Ronneberg, T.; BonačićKoutecký, V.; May, V. A Harmonic Approximation of Intramolecular Vibrations in a Mixed Quantum-Classical Methodology: Linear Absorbance of a Dissolved Pheophorbid-a Molecule as an Example. Chem. Phys. 2010, 377, 10−14. (34) Jmk(ge, ge) is the static Coulomb coupling between molecule m in the first excited electronic state with molecule k in the electronic ground state, while Jmk(gg, gg) is the Coulomb interaction between both molecules in the electronic ground state. The partial charges have been computed as follows. The geometry of the monomeric cyanine dye has been optimized at the DFT level, and the absorption energy for the first excited state has been calculated in the frame of TDDFT employing the long-range corrected hybrid CAM-B3LYP functional42 and the TZVP atomic orbital basis set43 for all atoms. Atomically centered partial charges (electronic ground and first excited state) and transition charges (only for heavy atoms) have been fitted to the electrostatic potential of the respective charge/transition charge density by using the CHELPG procedure.44 The differential dipole moment Δd was fitted to the result of hole burning experiments for similar cyanine dyes (0.7D < |Δd| < 1.1D).45 We chose Δd = 1D, and we note that other values within the range will change the absorption spectrum only slightly. (35) Zhu, H.; May, V.; Röder, B. Mixed Quantum Classical Simulations of Electronic Excitation Energy Transfer: The Pheophorbide-a DAB Dendrimer P4 in Solution. Chem. Phys. 2008, 351, 117− 128. (36) Adolphs, J.; Müh, F.; Madjet, M. E.-A.; Renger, T. Calculation of Pigment Transition Energies in the FMO Protein. Photosynth. Res. 2008, 95, 197−209. (37) Megow, J.; Renger, T.; May, V. Mixed Quantum-Classical Description of Excitation Energy Transfer in Supramolecular Complexes: Screening of the Excitonic Coupling. ChemPhysChem 2014, 15, 478−485. (38) Phillips, J. C.; Braun, R.; Wang, W.; Gumbart, J.; Tajkhorshid, E.; Villa, E.; Chipot, C.; Skeel, R. D.; Kale, L.; Schulten, K. Scalable Molecular Dynamics with NAMD. J. Comput. Chem. 2005, 26, 1781− 1802. (39) Case, D. A.; Darden, T. A.; Cheatham, T. E.; Simmerling, C. L.; Wang, J.; Duke, R. E.; Luo, R.; Merz, K. M.; Wang, B.; Pearlman, D. A.; Crowley, M.; Brozell, S.; Tsui, V.; Gohlke, H.; Mongan, J.; Hornak, V.; Cui, G.; Beroza, P.; Schafmeister, C.; Caldwell, J. W.; Ross, W. S.; Kollman, P. A. AMBER 8; University of California: San Francisco, CA, 2004. (40) Wang, J.; Wolf, R. M.; Caldwell, J. W.; Kollman, P. A.; Case, D. A. Development and Testing of a General Amber Force Field. J. Comput. Chem. 2004, 25, 1157−1174. (41) Darden, T.; York, D.; Pedersen, L. Particle Mesh Ewald: An N· log(N) Method for Ewald Sums in Large Systems. J. Chem. Phys. 1993, 98, 10089−10092. (42) Yanai, T.; Tew, D. P.; Handy, N. C. A New Hybrid ExchangeCorrelation Functional Using the Coulomb-Attenuating Method (CAM-B3LYP). Chem. Phys. Lett. 2004, 393, 51−57. (43) Weigend, F.; Ahlrichs, R. Balanced Basis Sets of Split Valence, Triple Zeta Valence and Quadruple Zeta Valence Quality for H to Rn: Design and Assessment of Accuracy. Phys. Chem. Chem. Phys. 2005, 7, 3297−3305. (44) Breneman, M.; Wiberg, K. B. Determining Atom-Centered Monopoles from Molecular Electrostatic Potentials. The Need for High Sampling Density in Formamide Conformational Analysis. J. Comput. Chem. 1990, 11, 361−373. (45) Chowdhury, A.; Wachsmann-Hogiu, S.; Bangal, P. R.; Raheem, I.; Peteanu, L. A. Characterization of Chiral H and J Aggregates of Cyanine Dyes Formed by DNA Templating Using Stark and Fluorescence Spectroscopies. J. Phys. Chem. B 2001, 105, 12196− 12201.

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