Modulating the Electronic Structure of Chromophores by Chemical

Article pubs.acs.org/JPCA

Modulating the Electronic Structure of Chromophores by Chemical Substituents for Efficient Energy Transfer: Application to Fluorone Andrew M. Sand, Claire Liu, Andrew J. S. Valentine, and David A. Mazziotti* Department of Chemistry and The James Franck Institute, The University of Chicago, Chicago, Illinois 60637, United States ABSTRACT: Strong electron correlation within a quasi-spin model of chromophores was recently shown to enhance exciton energy transfer significantly. Here we investigate how the modulation of the electronic structure of the chromophores by chemical substitution can enhance energy-transfer efficiency. Unlike previous work that does not consider the direct effect of the electronic structure on exciton dynamics, we add chemical substituents to the fluorone dimer to study the effect of electron-donating and electron-withdrawing substituents on exciton energy transfer. The exciton dynamics are studied from the solution of a quantum Liouville equation for an open system whose model Hamiltonian is derived from excited-state electronic structure calculations. Both van der Waals energies and coupling energies, arising from the Hellmann−Feynman force generated upon transferring the dimers from infinity to a finite separation, are built into the model Hamiltonian. Though these two effects are implicitly treated in dipole-based models, their explicit and separate treatment as discussed here is critical to forging the correct connection with the electronic structure calculations. We find that the addition of electron-donating substituents to the fluorone system results in an increase in exciton-transfer rates by factors ranging from 1.3−1.9. The computed oscillator strength is consistent with the recent experimental results on a larger heterodimer system containing fluorone. The oscillator strength increases with the addition of electron-donating substituents. Our results indicate that the study of chromophore networks via electronic structure will help in the future design of efficient synthetic light-harvesting systems.

I. INTRODUCTION Light-harvesting occurs in nature with very high efficiency. Several experimental1,2 and theoretical3−8 studies have suggested that quantum effects play an important role in excitation energy transfer in photosynthetic processes. In addition to biological systems,9−12 excitation energy transfer is important in the description of organic photovoltaics and semiconductors.13−15 A recent study has shown long-lived quantum coherence in synthetic heterodimer systems consisting of rigidly linked halofluorescin chromophores.16 Such systems can be used as experimental and theoretical models to understand persistent electronic coherences in biological and synthetic materials. In the manufacture of synthetic chromophore systems, general design principles for tuning quantum effects for optimal efficiency are needed. Previous studies have highlighted such principles, such as the importance of geometric orientation,17 environmental effects,6,8 and entanglement.18,19 In many of these studies, in part because the excitations are modeled by simple two-level systems or dipoles, the effect of the electronic structure of the individual chromophores on energy transfer has not been considered. Recently, it has been suggested that electron correlation within the chromophore molecules plays a significant role in exciton dynamics;3 in a quasi-spin model of the chromophore system containing an adjustable parameter for controlling the degree of electron correlation, it was found that the presence of significant electron correlation within the chromophore can enhance the energy-transfer efficiency by as much as 100%. Such results indicate accurate electronic © 2014 American Chemical Society

structure information is required for a realistic description of excitation (exciton) dynamics. Not only does the electronic structure of a chromophore have a significant effect on the outcome of the dynamics, but also it can be modulated through the addition of chemical substituents. In this paper we evaluate the effect of chemical substituents on the chromophores on the exciton dynamics in dimer systems. The chromophore size is selected to facilitate a systematic exploration of the electronic structure and correlation. We incorporate information from electronic structure calculations into a model exciton Hamiltonian. Both van der Waals energies and coupling energies, arising from the Hellmann−Feynman force generated upon transferring the dimers from infinity to a finite separation, are built into the model Hamiltonian. Although these two effects are implicitly treated in dipole-based models, their explicit and separate treatment is critical to forging the correct connection with the electronic structure calculations. We focus on the heterodimer system used in the recent experimental study. 16 We remove components of the heterodimer to generate a dimer of fluorone molecules, as shown in Figure 1a,b. The lowest-lying exciton states correspond to π → π* transitions. To study the effect of the electronic structure on exciton dynamics, we alter the π-system through the inclusion of electron-donating and electronReceived: April 21, 2014 Revised: June 10, 2014 Published: July 25, 2014 6085

dx.doi.org/10.1021/jp503900m | J. Phys. Chem. A 2014, 118, 6085−6091

The Journal of Physical Chemistry A

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withdrawing substituents on the monomers in the dimer system. Through the addition or depletion of electron density in the π-system, correlation may be expected to increase or decrease, respectively. Previous works have studied substituentdependent π−π interactions for ground-state benzene monomers.20 In principle, the model Hamiltonian approach we follow in this work can be used in conjunction with any excitedstate electronic structure method, including wave functionbased methods, two-electron density matrix-based methods,21−24 and time-dependent density functional theory (TDDFT).25 In the present work we chose TDDFT to characterize the electronic excited states.

k ,l=1

(8)

(9)

(10)

where ϵm is the isolated monomer excitation energy, EVDW(R) is the van der Waals energy, and V(R) is the coupling energy. Diagonalization of this matrix yields eigenfunctions consisting of symmetric and antisymmetric combinations of eqs 7 and 8: 1 (ψ ± ψ01) ψ± = (11) 2 10 with corresponding eigenvalues E± = ϵm + E VDW (R ) ± V (R )

(12)

EVDW(R) and V(R) can be calculated using components of the electronic Hamiltonian:

(2)

ϵm = ⟨ψ10|HÂ + ĤB|ψ10⟩ − ⟨ψ00|HÂ + ĤB|ψ00⟩

(13)

̂ |ψ ⟩ − ⟨ψ |HAB ̂ |ψ ⟩ E VDW = ⟨ψ10|HAB 10 00 00

(14)

̂ |ψ ⟩ V (R ) = ⟨ψ10|HAB 01

(15)

We note that this result can be equivalently obtained using the Hellmann−Feynman theorem. Treating the interaction between the monomers as a perturbation, we can write the electronic Hamiltonian as ̂ (R ) Ĥ (λ ,R ) = HÂ + ĤB + λHAB

where ϵs is the excitation energy for chromophore s, Vs,t is the coupling between chromophores s and t, and bŝ and b†ŝ are excitonic annihilation and creation operators of excitons on the s chromophore.26 The excitonic annihilation and creation operators are expressible in terms of the spin orbital creation and annihilation operators:

(16)

Using ψ+ and ψ− as reference functions, the Hellmann− Feynman theorem28 reveals that the first-order corrections to the energy are given as E1,± = EVDW(R) ± V(R). The terms ϵm, EVDW(R), and V(R) can be determined by (i) direct calculation of the matrix elements in eqs 13−15,29,30 (ii) the Förster approximation (also known as the ideal dipole approximation), where

(4)

†

† bŝ = a p,1 a p,0

|ψ01⟩ = |ψ0A ⟩|ψ1B⟩

⎡ ϵm + E VDW (R ) V (R ) ⎤ ⎥ H=⎢ ⎢⎣ V (R ) ϵm + E VDW (R )⎥⎦

where m is the total number of wave functions spanning the nelectron space. The Lindblad operator accounts for environmental effects including dephasing and dissipation. The electronic Hamiltonian and the many-electron density matrix can be approximated with the tightly bound (or Frenkel) exciton model Hamiltonian † † † 1 Ĥ = ∑ ϵs(bŝ bŝ − bŝ bŝ ) + ∑ Vs,tbŝ bt̂ 2 s (3) s≠t

† bŝ = a p,0 a p,1

(7)

where Ĥ A and Ĥ B are Hamiltonian operators for the isolated monomers and Ĥ AB(R) accounts for intermolecular interactions between the monomers. We can express our model Hamiltonian in matrix form as

m

ρlk |Ψk⟩⟨Ψ|l

|ψ10⟩ = |ψ1A ⟩|ψ0B⟩

̂ (R ) Ĥ (R ) = HÂ + ĤB + HAB

II. THEORY The time evolution of a quantum system can be calculated using the quantum Liouville equation:3,8 d i D = − [Ĥ ,D] + L̂[D] (1) dt ℏ ̂ where L[D] is a Lindblad operator and D is the many-electron density matrix given by

∑

(6)

where |ψA0 ⟩ and |ψB0 ⟩ are the ground-state wave functions for monomers A and B and |ψA1 ⟩ and |ψB1 ⟩ the excited-state wave functions for monomers A and B corresponding to the π → π* transition. The model Hamiltonian can be obtained by including energy corrections to account for van der Waals interactions.27 The full electronic Hamiltonian is expressible as

Figure 1. Structures used in this work: (a) fluorone and derivatives; (b) an example fluorone dimer system showing the sandwich (mirrored) dimer geometry with Cs symmetry.

D=

|ψ00⟩ = |ψ0A ⟩|ψ0B⟩

V (R ) =

(5)

μA⃗ ·μB⃗ r

3

−

3(μA⃗ ·R⃗)(μB⃗ ·R⃗) r5

(17) 27,31

and explicit determination of EVDW(R) is neglected, or (iii) the use of energies from excited-state electronic structure methods on the monomer and dimer systems.32,33 Although the term EVDW(R) affects only the spectrum of the Hamiltonian and not the excitonic dynamics, the determination of the dynamics-dependent V(R) from the electronic structure results

where 0 and 1 correspond to, for example, the highest occupied spin orbital of chromophore p and the lowest unoccupied spin orbital of chromophore p3. For a system of two identical chromophores the model Hamiltonian and many-electron density matrix can be written in the basis of monomer-wave function products: 6086



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requires the consideration of EVDW(R). Given the monomer excitation energy and the energies of the two exciton states of the dimer, the van der Waals correction EVDW(R) at each R can be obtained from the difference between the average of the two dimer excitation energies, denoted as ϵ1 and ϵ2, and the monomer excitation energy: E VDW (R ) = [ϵ1(R ) + ϵ2(R )]/2 − ϵm

proportional to the electronic coupling in the ideal dipole approximation. The fluorone molecules with electron-donating substituents give larger oscillator strengths than the fluorone molecules with electron-withdrawing substituents or without substituents. Interestingly, the electron-withdrawing substituents, with the exception of the nitro group, also show higher oscillator strengths than the unsubstituted fluorone. For the dimer of unsubstituted fluorone molecules, Figure 2 shows the energy splitting of the lowest-lying exciton states as a

(18)

and the coupling correction at each R is computed from the difference between the lowest dimer excitation energy and the van der Waals correction: V (R ) = ϵ1(R ) − (E VDW + ϵm)

(19)

Diagonalization of the model Hamiltonian reproduces the exciton states from the quantum mechanical calculation. The model excitonic Hamiltonian can be built from any quantum mechanical method, which resolves the exciton splitting of the excited states. In our study we use the monomer and dimer excitation energies calculated from TDDFT.

III. APPLICATIONS A. Methodology. All electronic structure calculations were performed using the GAMESS package.34 The ground-state geometries of the monomers were optimized at the B3LYP/631G* level of theory35,36 while Cs symmetry was preserved. Excited-state energies for each monomer were calculated using TDDFT within the adiabatic approximation37 at the B3LYP/631G* level of theory.38 The energies of the dimer systems, consisting of two monomers in a sandwich orientation with mirror-plane symmetry, were calculated with TDDFT. The time-dependent Liouville equation was solved a using a Fehlberg fourth-fifth-order Runge−Kutta method implemented in Maple 18.39 B. Results. Excitation energies, transition dipole moments, and oscillator strengths of the fluorone monomer are given in Table 1. Excitation energies correspond to the lowest π → π*

Figure 2. Energy splitting of the lowest-lying exciton states of the fluorone dimer as a function of distance as determined by TDDFT. The monomer excitation energy and the average energy of the two exciton states, also shown, are used to define the energy corrections in the model Hamiltonian.

function of distance. Similar curves, albeit not shown, have also been calculated for each substituted dimer system. We observe asymmetric splitting about the monomer excitation energy. The average of the energies from the exciton states, which are needed to define the model Hamiltonian, is also shown. The van der Waals correction at each distance is computed from the difference between the average exciton energy and the monomer excitation energy. The electronic coupling correction is computed from one-half of the difference between the energies of exciton states 1 and 2. Figure 3 shows the van der Waals energy correction for dimers of fluorones with different substituents. We do not observe any trend for electron-withdrawing and electrondonating groups among these energy corrections. Figure 4 shows the coupling energy corrections, which account for the splitting of the exciton states. We observe that the electrondonating substituents loosely agree with the trends of electrondonating ability given by Hammett parameters,40 which measure electron-donating character by considering the acidities of benzoic acid derivatives; both the amino and hydroxy substituents, which are electron donors, result in greater splitting. Similar to the trend observed with the oscillator strengths, the electron-withdrawing groups show an increase in energy splitting relative to the unsubstituted fluorone dimer system. To generate time dynamics of the excitation energy transfer, we examine the energy splittings of the exciton states in the dimer systems at a distance of 13 Å, a separation distance that is similar to the distance found in natural photosynthetic light harvesting in green-sulfur bacteria. This distance is sufficiently large enough to minimize any charge-transfer effects from

Table 1. Excitation Energies, Transition Dipoles, and Oscillator Strengths for the Fluorone Monomer (Labeled “Unsubstituted”) and the Substituted Fluorone Monomer Structuresa transition dipole (au) molecule

excitation energy (eV)

x

y

z

oscillator strength

unsubstituted carbonyl amino cyano fluoro hydroxy nitro

2.991 2.733 3.057 2.837 3.059 3.068 2.642

−1.731 1.833 2.457 −1.885 −1.874 2.105 −1.818

−0.0713 −0.2783 −0.0118 −0.0171 −0.1043 0.0486 0.0247

0.000 0.000 0.000 0.000 0.000 0.000 0.000

0.220 0.230 0.452 0.247 0.264 0.333 0.214

Excitation energies correspond to the lowest π → π* transition. Calculations are performed to the TDDFT/6-31G* level of theory using the B3LYP functional.

a

transition. We see a trend in the excitation energies, suggesting that electron-donating substituents, hydroxy and amino functional groups, raise the excitation energies and electronwithdrawing substituents, carbonyl, cyano, fluoro, and nitro functional groups, usually lower the excitation energies. For the dynamics of the system, however, the most important result is the observed trend in the oscillator strengths, as they are 6087



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Table 2. Components of the Model Hamiltonian for Each Dimer Systema Model Hamiltonians IDA

eq 10

molecule

ϵm

V

ϵm

EVDW

V

unsubstituted carbonyl amino cyano fluoro hydroxy nitro

109.839 100.352 112.309 104.183 112.343 112.694 97.033

−0.202 −0.232 −0.407 −0.240 −0.238 −0.299 −0.223

109.839 100.352 112.309 104.183 112.343 112.694 97.033

−0.085 −0.084 −0.048 −0.087 −0.086 −0.056 −0.046

−0.164 −0.176 −0.312 −0.183 −0.191 −0.214 −0.168

a

The monomer separation for each dimer system is 13 Å. Results calculated from the ideal dipole approximation (IDA) are also presented. Values are given in millihartrees (mhartrees).

Figure 3. Van der Waals energy correction (EVDW(R)), as a function of distance, is presented for each substituted fluorone dimer. The corrections are determined by TDDFT(B3LYP)/6-31G* calculations on the dimer system.

Figure 5. Exciton population in chromophore 1 is shown as a function of time for (a) the unsubstituted fluorone dimer, (b) the hydroxysubstituted fluorone dimer, and (c) the amino-substituted fluorone dimer, all in the sandwich orientation. The oscillator frequency of the unsubstituted fluorone dimer is 2.11 THz. The electron-donating hydroxy and amino substituents increase the frequency of oscillation to 3.07 and 4.08 THz, respectively.

Figure 4. Coupling energy (V(R)) as a function of distance R is presented for each substituted fluorone dimer. Similar to the trend observed with the oscillator strengths, the electron-donating groups show an increase in splitting relative to the unsubstituted fluorone dimer system.

exciton transfer than the unsubstituted and electron-withdrawing-substituted dimer systems. The addition of electrondonating groups to the conjugated system increases the electron density in the π-system, which increases electron correlation. In the case of the electron-withdrawing dimer systems, we might expect a depletion of electron density in the π-system, which would decrease exciton-transfer rates relative to the unsubstituted dimer system. It appears that the substitution of a hydrogen atom with a substituent (containing many more electrons than a hydrogen atom) serves to increase electron density, effectively offsetting the reduction in excitontransfer rates caused by the electron-withdrawing character. In an effort to reproduce the oscillation frequencies experimentally observed for synthetic chromophores,16 we studied dimer geometries using monomer orientations similar to those in the rigidly linked chromophores (Figure 6a,b). We use the same optimized monomer geometries that were used in the sandwich-oriented dimers. The orientation and distance between the two monomers is replicated from the linked dimer structure.

TDDFT. Table 2 shows the components of the model Hamiltonian in each of the dimer systems at this distance. For comparison purposes, the exciton splittings from the ideal dipole approximation (IDA) are calculated using eq 17 and the dipole data from Table 1. For molecules with electron-donating substituents, the IDA largely predicts the trend in exciton splittings with a few deviations from the TDDFT results. Quantitatively, however, the IDA predicts larger electronic coupling elements V than TDDFT for each of the dimer systems. Using the components given in Table 2, we can construct the model excitonic Hamiltonian for the 13 Å separated dimer systems. Results of the time dynamics with the model Hamiltonian are shown for several molecules in Figure 5a−c. The system is prepared with an exciton localized to state 1. All of the electron-withdrawing-substituted fluorones show exciton-transfer rates that are quite similar to those from the unsubstituted fluorone. However, we also observe that the dimer systems with the electron-donating groups display faster 6088



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IV. DISCUSSION AND CONCLUSIONS The modulation of the electronic structure of chromophores through the addition of chemical substituents, we have shown, affects the rate of exciton energy transfer between chromophores. We have studied the fluorone dimer and its chemical substitutions through a model Hamiltonian derived from electronic structure calculations. The addition of electrondonating groups to fluorone enhances the coupling between exciton states which increases the rate of exciton energy transfer. The addition of electron-withdrawing groups, however, produces a slight increase or decrease of exciton energy transfer. Previous studies, using a Lipkin−Meshkov−Glick N-electron model, have shown that by increasing the degree of electron correlation in chromophore systems, one can significantly increase exciton-transfer rates.3 In this work the electronic structure of the chromophores is changed through functional group substitutions. These substitutions affect the amount of electron correlation in the system by changing both the number of electrons in the system and the distribution of the electrons in the molecule. Adding any substituent, which replaces a hydrogen atom with a many-electron atom or functional group, increases the number of electrons in the system and is thus expected to increase the amount of electron correlation. Electron-donating substituents increase the electron density in the π-system, further increasing the degree of electron correlation in the chromophore system. On the other hand, electron-withdrawing substituents decrease the electron density in the π-system, resulting in a competing process which reduces the electron correlation. In our study the most accelerated exciton-transfer rates occur in systems substituted with electron-donating groups, whereas systems substituted with electron-withdrawing groups result in either slight increases or decreases in exciton-transfer rates. We have found oscillation frequencies in our dimer systems near to those that have been experimentally observed for similar dimer systems. It is well-known that TDDFT often struggles to provide a good description of excited-state energies and oscillator strengths,41 in particular those involving charge-transfer (CT) states.33,42,43 Works have also shown that spurious CT states can even affect the proper description of non-CT valence states.44,45 In the present calculations we have employed sufficiently large distances between the chromophores to minimize the importance of CT states. We have systematically examined electronic-structure effects with other effects excluded. Future work will also explore the effect of coupling of vibrational modes to excitons, known as vibronic coupling,10,46 and the influence of geometric parameters like chromophore separation and orientation.17 The present calculations on fluorone and its substituents highlight an important design principle for synthetic chromophoric systems: though the addition of a substituent can be used to tune a system’s excitation energy, the substituent also has a significant effect on the energy (exciton)-transfer rate between chromophores in the system. This principle, in conjunction with future considerations of environmental and orientation effects, can assist in the design of more energy efficient chromophores and light-harvesting materials.

Figure 6. (a) Rigidly linked synthetic dimer system studied in ref 16. (b) Fluorone dimer system used in our calculations. The geometry of each monomer is individually optimized, and the orientation and distance between the monomers is replicated from the synthetic dimer geometry.

Following the construction of the model Hamiltonian, we generate the time dynamics shown in Figure 7a−c. The system

Figure 7. Exciton population in chromophore 1 is shown as a function of time for (a) the unsubstituted fluorone dimer, (b) the hydroxysubstituted fluorone dimer, and (c) the amino-substituted fluorone dimer, all with the monomer orientation derived from the synthetic chromophore used in ref 16. The oscillator frequency of the unsubstituted fluorone dimer is 6.02 THz. The electron-donating hydroxy and amino substituents increase the frequency of oscillation to 7.87 and 10.31 THz, respectively.

is prepared with an exciton localized to state 1. We observe a significant enhancement in the exciton-transfer rate relative to the rates seen in the dimer with sandwich orientation. This result is expected as the smaller monomer separation distance results in stronger exciton coupling between the monomers. The synthetic chromophore shown in Figure 6a had an experimentally observed oscillation frequency of 5.8 ± 0.1 THz.16 We obtain similar oscillation frequencies: 6.02, 7.87, and 10.31 THz for the unsubstituted, hydroxy-substituted, and amino-substituted dimer systems, respectively. Though the synthetic chromophore has different substituents (including the linkers) that affect the electronic structure of the system, the similar fluorone backbones result in similar oscillation frequencies between the synthetic and computationally modeled dimer systems. 6089



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AUTHOR INFORMATION

Corresponding Author

*D. A. Mazziotti. E-mail: [email protected]. Phone: 1-773834-1762. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS D.A.M. gratefully acknowledges the ARO (Grant No. W91 INF-1 1-504 1-346 0085), the NSF CHE-1152425, and the Keck Foundation for their generous support.

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