Ab Initio Study of Excitation Energy Transfer between Quantum Dots

7548

J. Phys. Chem. C 2009, 113, 7548–7552

Ab Initio Study of Excitation Energy Transfer between Quantum Dots and Dye Molecules Hiroyuki Tamura,*,† Jean-Maurice Mallet,‡ Martin Oheim,§,|,⊥ and Irene Burghardt‡ AdVanced Institute for Material Research, Tohoku UniVersity, 2-1-1 Katahira, Aoba-ku, Sendai, 980-8577, Japan, De´partement de Chimie, Ecole Normale Supe´rieure, 24 rue Lhomond F-75231 Paris cedex 05, France, INSERM, U603, 75006 Paris, France, CNRS, UMR 8154, 75006 Paris, France, and Laboratory of Neurophysiology and New Microscopies, UniVersity Paris Descartes, 45 rue des Saints Peres, 75006 Paris, France ReceiVed: December 15, 2008; ReVised Manuscript ReceiVed: February 27, 2009

We investigate excitation energy transfer between a semiconductor nanocrystal (NC) and an organic dye molecule, using ab initio calculations. The electronic coupling is evaluated based on the full Coulombic interaction between the NC and dye transition densities. We explore the effect of the NC-dye relative configurations on the electronic coupling, using a (CdSe)6 cluster and a rhodamine cation as a simplified model system. Our analysis demonstrates the limitations of the commonly used Fo¨rster theory for the NC-dye system and highlights the importance of considering the full Coulombic interactions of donor and acceptor. We find that energy transfer involves several electronic states and can occur even from optically dark states of the NC. The consequences for larger NC-dye systems are discussed. I. Introduction 1,2

Fluorescence resonance energy transfer (FRET) or exciton transfer3 from a photoexcited donor to a fluorescent or dark acceptor (quencher) provides the basis for a variety of sensors and molecular switches.4,5 Donor and acceptor may be dye molecules, fluorescent proteins, and semiconductor nanocrystals (NCs) like quantum dots or nanorods. NCs, i.e., nanometersized clusters of ∼200 atoms are promising materials for FRETbased nanosensors,6,7 owing to their efficient light-harvesting function due to the wide absorption and narrow emission spectra, as well as their size-dependent spectral tunability, high photostability, and long-lived photoluminescence lifetime. Hybrid NC-organic donor-acceptor assemblies are being utilized for toxin detection, ion sensing,8 monitoring enzyme activity, as well as medical diagnosis and curing, e.g., in photodynamic therapy for the selective destruction of cancer cells.9 The dynamics of excitation energy transfer is largely determined by the electronic coupling V (i.e., exciton-exciton coupling) and vibronic coupling (electron-phonon coupling). V determines the energy-level splitting of the coupled exciton states at resonance (∆E ) 2V). A strong electronic coupling induces a coherent exciton transfer and necessitates an explicit quantum dynamical treatment. By contrast, the FRET rate k in a weak coupling (perturbative) regime can be described by Fo¨rster theory with k ∝ |V2|.1,2,10,11 In this case, the FRET rate is also proportional to the overlap between donor emission and acceptor absorption spectra, which can be assumed to be unaffected by the intermolecular electronic interaction. FRET phenomena in NC-dye systems simultaneously involve various processes, including photoexcitation, radiationless decay in the NC, NC f dye excitation energy transfer, dye f * Corresponding author, [email protected]. † Advanced institute for Material Reserch, Tohoku University. ‡ De´partement de Chimie, Ecole Normale Supe´rieure. § INSERM, U603. | CNRS, UMR 8154. ⊥ Laboratory of Neurophysiology and New Microscopies, University Paris Descartes.

NC back transfer, and vibrational energy redistribution. Thus, it can be expected that the assumptions underlying Fo¨rster theory represent an (over-) simplified view of the true transfer process occurring in the NC-dye system. In this context, the theoretical prediction of the electronic coupling V is an important issue for the microscopic modeling of energy transfer. Ab initio calculations of donor-acceptor supermolecules provide the most exact approach by providing adiabatic potential energy surfaces, from which the corresponding diabatic potentials and the electronic coupling can be determined by an appropriate diabatization scheme.12 However, such a treatment necessitates excited-state calculations of a donor-acceptor supermolecule and might not be feasible for large systems like NC-dye assemblies. In practice, the exciton states are often written as a product of the wave functions of the isolated donor and acceptor,1,2,10,13 so that the electronic coupling V reads (in atomic units),

〈

V ) ΦDeΦAg

|

|

∑ |rd -1 ra| ΦDgΦAe d,a

〉

(1)

where rd and ra are the electron positions in the donor and acceptor, respectively, and the summation runs over all pairs of electrons between the donor and acceptor. ΦDe, ΦAg, ΦDg, and ΦAe are the wave functions of the excited-state donor, ground-state acceptor, ground-state donor, and excited-state acceptor, respectively. That is, ΦDeΦAg and ΦDgΦAe are considered as the initial and final states, respectively. Importantly, eq 1 does not account for donor-acceptor electron exchange and relaxation of the wave functions due to donor-acceptor interactions; these are negligible unless the donor and acceptor are close to each other so that the exchange repulsions are nontrivial. Equation 1 can be rewritten as the integral of the Coulomb interaction between transition densities of the donor (FD) and acceptor (FA)

V)

∫ ∫ FD(rd) |rd -1 ra| FA(ra) drd dra

where

10.1021/jp811042t CCC: $40.75  2009 American Chemical Society Published on Web 04/10/2009

(2)

Nanocrystal-Dye Excitation Energy Transfer

J. Phys. Chem. C, Vol. 113, No. 18, 2009 7549

FD(rd) ) ΦDg(rd)ΦDe(rd)

(3a)

FA(ra) ) ΦAg(ra)ΦAe(ra)

(3b)

and

At large donor-acceptor (DA) distances, the electronic coupling is approximated by dipole or multipole expansions of the transition densities. In the dipole approximation, the transition densities in eq 2, FD and FA, are regarded as dipoles centered at the donor and acceptor, respectively.13 Accordingly, the electronic coupling is given by the Coulomb interaction between the dipoles as follows (in a.u.)

Vdip )

dD · dA RDA

3

-

3(dD · RDA)(dA · RDA) RDA5

(4)

where dD and dA are the donor and acceptor transition dipole moments and RDA and RDA are the vector and distance between the centers of mass of the donor and acceptor, respectively. However, the dipole approximation may break down at short DA distances. Violations of the dipole approximation have indeed been reported for various systems for which consideration of the full Coulomb interaction is essential.10,14-17 The exciton states in relatively large NCs are often modeled based on a quantum-confinement picture,18,19 where the electronic structure is determined by several fundamental parameters such as the cluster size, exciton radius, and effective mass. Recent experimental and theoretical studies suggest, however, that the optical properties of small NCs not only depend on the cluster size but also depend sensitively on their precise atomistic structure.20-22 These findings indicate the limitations of the simple quantum-confinement picture to describe the exciton states of NCs. Recently, the electronic structure of NCs20-26 and the exciton transfer of NC-dye systems27,28 have been investigated explicitly considering the atomistic structures, by using semiempirical methods,27 density functional theory (DFT),20-25,28 and the symmetry adapted cluster configuration interaction (SAC-CI) method.26 Regarding the electronic coupling, Curutchet et al.27 have very recently investigated the applicability of the dipole approximation for NC-dye systems and found that the dipole approximation can well reproduce the electronic couplings obtained by the full Coulomb interaction between transition densities even at contact NC-dye separations. By contrast, Schrier et al. have demonstrated that the dipole approximation cannot properly describe the electronic couplings between nanorods at short distances.17 Similar observations of non-Fo¨rster behavior have been made for various other molecular systems.10,14-16 To what extent the Fo¨rster theory and the dipole (or multipole) approximation hold depends on the geometries of the donor and acceptor as well as their relative configurations. Since possible NC-dye configurations have not been extensively explored, the applicability of the dipole approximation for NC-dye systems remains an open question. In the present study, we address the theoretical prediction of the electronic coupling between a NC and a dye molecule based on configuration interaction (CI) electronic structure calculations and the full Coulomb interaction between transition densities. Here, we consider a small cluster model of NC so that highaccuracy ab initio methods are feasible; these results can also be utilized as a benchmark to calibrate approximate methods (e.g., DFT). Note that the typical quantum-confinement picture of NCs does not hold for small clusters. Jose et al.22 have investigated the nucleation of CdSe by experimental spectra and time-dependent density functional theory (TDDFT) and found

Figure 1. Ground-state equilibrium geometries of (a) the (CdSe)6 wurtzite cluster and (b) the Rh+ cation (C13H9O+). The symmetry axis is along the z axis.

that the wurtzite (CdSe)6 cluster is one of the minimal structures. In the present study, we assess the validity of the dipole approximation for a simple NC-dye model system comprised of the (CdSe)6 cluster and a simplified Rhodamine cation (Rh+) (Figure 1). We demonstrate illustrative examples where the dipole approximation breaks down and discuss the consequences for designing FRET-based nano(bio-) sensors. II. Methods II.A. Computational Details. We calculated the excitation energies of the (CdSe)6 cluster and the Rh+ ion using multiconfigurational quasi-degenerate second-order perturbation theory (MCQDPT).29 We consider the ground-state equilibrium geometries of the donor and acceptor and do not account for geometry relaxation in the excited states. The Hey/Wadt valence basis set30 is used for (CdSe)6 with a polarization d-function for Se (HW(d)), and the 6-31G(d) basis set is used for the Rh+. The GAMESS code31 is employed for all ab initio calculations in this study. For the (CdSe)6 cluster, the exciton states are generally characterized by single-electron excitations from the valence orbitals that are mainly comprised of the sp orbitals of Se (the contributions of the d orbitals of Cd are small).32 Thus, the active space for (CdSe)6 includes these Se valence orbitals as well as several important virtual orbitals contributing to the exciton states. Single and double excitations in the active space (i.e., limited CISD) are taken into account to construct the reference wave functions for the subsequent MCQDPT calculations. The ground-state equilibrium geometry of the (CdSe)6 cluster is obtained by the second-order Møller-Plesset perturbation theory (MP2) under C2h symmetry (Figure 1a), and the transition dipole moments are calculated using the MCQDPT method. For comparison, the excitation energies and the transition dipole moments of (CdSe)6 are also calculated using TDDFT with the Perdew-Wang 1991 (PW91) exchange-correlation functional,33 where the HW(d) basis set is used as well. For the Rh+ ion, all π orbitals are included in the active space, and complete active-space self-consistent field (CASSCF) wave functions are considered as the references for the MCQDPT calculations. The ground-state equilibrium geometry and the transition dipole moments of the Rh+ are calculated by the CASSCF method under C2V symmetry. II.B. Electronic Coupling. We evaluate V according to eqs 2 and 3, considering the full Coulomb interaction between the donor and acceptor. The present scheme is formally equivalent

7550

J. Phys. Chem. C, Vol. 113, No. 18, 2009

Tamura et al.

to the transition density cube method,10,15,16,27 which is based on a numerical grid representation of the transition densities. The grid size determines the accuracy and computational cost of the transition density cube method, where coarse grids may cause serious numerical errors.10 Differently from the transition density cube method, our scheme utilizes analytical Coulomb integrals in the atomic orbital (AO) basis

V)

∑ ∑ FRβFγδVRβγδ

(5)

R,β γ,δ

where VRβγδ are the two-electron AO integrals between the donor and acceptor

VRβγδ )

∫ ∫ χR(rd)χβ(rd) |rd -1 ra| χγ(ra)χδ(ra) drd dra (6)

where χR, χβ, and χγ, χδ are the AO basis functions expanding the molecular orbitals (MO) of the donor and acceptor, respectively. FRβ and Fγδ are the transition density matrices for the donor and acceptor in the AO basis. The electronic coupling (eq 2) is comprised of two-electron integrals of the relevant MOs. Thus, the transition density matrices read

∑ CI(De)CJ(Dg)∑ cRicβj

(7a)

∑ CK(Ag)CL(Ae)∑ cγkcδl

(7b)

FRβ )

I,J

Fγδ )

K,L

∑ cRiχR(rd), R

k,l

φAk )

∑ cγkχγ(ra)

(8)

γ

and the CI wave functions read

ΦDe )

∑ CI(De)ΨI(D), I

ΦAg )

∑ CK(Ag)ΨK(A), K

ΦDg )

∑ CJ(Dg)ΨJ(D)

(9a)

J

ΦAe )

∑ CL(Ae)ΨL(A) L

(9b) (D)

(A)

TABLE 1: Excitation Energies (eV) and Transition Dipole Moments (Debye) (in Parentheses) of the Excited States of the (CdSe)6 Cluster Computed by the MCQDPT and TDDFT (PW91) Methods

i,j

Here, cRi denotes the Rth MO coefficient of the ith MO of donor, cβj denotes the βth MO coefficient of the jth MO of acceptor, C(De) and C(Dg) denote the CI coefficients for the configurations from the initial and final donor wave functions, and C(Ag) and C(Ae) denote those for the acceptor wave functions. These coefficients determine the wave functions, i.e., the donor and acceptor MOs read

φDi )

Figure 2. Excitation energies and transition dipole moments from the ground state for (CdSe)6 (blue) and Rh+ (red) calculated by the MCQDPT method. x, y, and z denote the direction of the transition dipole moment (see Figure 1). The 5Bu and 3Au states are degenerate. The blue circle indicates the dark state (3Ag) of (CdSe)6.

where Ψ and Ψ are the configurations (i.e., the determinants consisting of the MOs) taken into account in the CI calculations for the donor and acceptor, respectively. For simplicity, we use a state-independent MO set (c matrix) to describe the ground and excited states, where the MOs are optimized for the ground state (multiconfiguration self-consistent field calculations are not carried out for the excited states). When a pair of configurations from the initial and final donor (or acceptor) wave functions includes more than one difference in the electron occupations, the integral vanishes. Thus, only a limited number of configuration pairs and the corresponding MO pairs contribute to V. For example, the Hartree-Fock ground state only couples with the singly excited configurations; in this case, each configuration pair gives only one two-electron MO integral that consists of the excited electron and hole. The MO and CI coefficients in eq 7 can be obtained by ab initio calculations of the isolated donor and acceptor. The present scheme can be applied for any kind of CI wave functions. A calculation of the supermolecule is only required to obtain the

1Bu 2Bu, 1Au 3Bu, 2Au 4Bu 5Bu, 3Au

MCQDPT

TDDFT - PW91

2.94 (2.1) 3.07 (0.3) 3.30 (1.0) 3.47 (3.7) 3.59 (3.1)

2.43 (2.0) 2.71 (0.3) 3.00 (1.9) 3.41 (1.4) 2.91 (1.4)

two-electron AO integrals (eq 6). In this study, the wave functions of (CdSe)6 (namely, ΦDe and ΦDg in eq 3a) and those of Rh+ (ΦAe and ΦAg in eq 3b) are calculated using the CIS and CASSCF methods, respectively. III. Results III.A. Optical Properties. Figure 2 shows the excitation energies and transition dipole moments of the (CdSe)6 cluster and the Rh+ dye obtained by MCQDPT calculations. Here, we only consider singlet electronic states and omit the contribution of triplet states. The transition dipole moments are utilized for the estimation of Vdip according to eq 4. The first and the second lowest bright states of the Rh+ are considered as acceptor states. These states are assumed to be resonant with the excited states of the (CdSe)6 cluster. We focus here on the electronic coupling, while the line broadening and spectral overlap are not investigated. TDDFT is less time-consuming than CI methods and is supposed to be a practical tool for describing the singly excited states of NCs, although it may fail to describe some excited states such as the Rydberg and charge transfer states.34 Here, we assess the validity of TDDFT for the exciton states of the (CdSe)6 cluster by comparing with the accurate MCQDPT results. As can be inferred from Table 1, TDDFT (PW91) generally provides reasonable results, even though qualitatively different excitation energies are obtained for some of the higher excited states (namely, 5Bu and 3Au). Here, the energetic sequence of the electronic states is not correctly reproduced. Other standard exchange-correlation functionals (e.g., B3LYP) are found to predict the same trend as PW91. Following photoexcitation to the bright states of the NC, the excitation energy can transfer from the donor to acceptor. We also consider excitation energy transfer from the dark states of the NC, which can occur subsequent to radiationless decay from the bright to dark states of the NC. Here, only the singlet dark

Nanocrystal-Dye Excitation Energy Transfer

J. Phys. Chem. C, Vol. 113, No. 18, 2009 7551

Figure 4. Simplified, schematic illustrations of the transition densities of (a) the 2A1 state of Rh+ and (b) the 3Ag state of the NC and 1B2 state of Rh+, where the dotted circle indicates the NC-dye interface.

Figure 3. Calculated V (eV) (eq 2) between the donor and acceptor states with corresponding NC-dye arrangements, where Vdip (eV) (eq 5) is given in parentheses for comparison (absolute values are shown). For (a) and (b), the transition dipole moments of the 4Bu (NC) and 1B2 (dye) states take parallel and collinear orientations, respectively, where the arrows indicate the directions of the transition dipole moments. For (a), V between 3Ag (NC dark state) and 1B2 (dye) is also shown. V (Vdip) of 1Bu (NC) f 2A1 (dye) and 4Bu (NC) f 2A1 (dye) are shown for (c) and (d), where the relative configurations of the transition dipole moments are identical. V between 3Ag (NC dark state) and 2A1 (dye) are also shown.

states are considered, although the triplet dark states can be involved in the exciton dynamics via spin-orbit coupling.19 The rate of radiationless decay in NCs is an open question, even though some experimental35-38 and theoretical26,28,39 investigations have been carried out. Vibronic coupling induces radiationless decays, which may compete with the donor-acceptor energy transfers. Multiple exciton generation26,39 and charge transfer may also occur as possible processes. The detailed analysis of these dynamics is beyond the scope of this paper. III.B. Illustrative Calculations of Electronic Coupling. We now analyze several typical cases where the dipole approximation breaks down. Several combinations of donor/acceptor electronic states including the NC dark states and several representative relative orientations are considered. We assume fairly large donor-acceptor separations so that the use of eq 1 is appropriate owing to the small exchange repulsion; in the following calculations, the overlap integrals between the donor and acceptor MOs are at most ∼0.05. Parts a and b of Figure 3 show the electronic couplings between the strongest bright state of the NC (the 4Bu state) and that of the dye (the 1B2 state), for different NC-dye orientations. For the purpose of illustration, we arrange the NC and dye in such a way that the transition dipoles of the donor and acceptor take parallel (Figure 3a) and collinear (Figure 3b) orientations. The ab initio full Coulomb coupling is slightly stronger for the parallel orientation (0.017 eV) than for the collinear orientation (0.015 eV), for the same RDA (8 Å). This is in contrast to the electronic coupling estimated from the dipole approximation (eq 4), which predicts a larger electronic coupling for the collinear orientation (0.037 eV) as compared with the parallel orientation (0.0184 eV). Here, Vdip of eq 4 was calculated based on the ab initio transition dipole moments at the NC and dye centers.

Next, we consider energy transfer to the lowest bright state (2A1) of the Rh+ dye (Figure 3c,d). Even though the two geometries under study correspond to an equivalent relative configuration of the transition dipole moments, the electronic couplings based on the full Coulomb interaction exhibits considerably different values. These differences again indicate the limitation of the dipole approximation for molecules having an anisotropic geometry. Figure 4a illustrates the transition density associated with the Rh+ 2A1 state, which is poorly captured by the primitive dipole approximation. In this case, the local distributions of the transition densities play a crucial role in the donor-acceptor Coulomb interaction. The dipole approximation overestimates the electronic coupling for the geometry of Figure 3b and underestimates the coupling for the geometry of Figure 3c. A similar trend was found for nanorod-nanorod electronic couplings at short distances.17 Finally, we investigate the electronic couplings between the NC dark state (3Ag) and the Rh+ acceptor states (2A1 and 1B2). Considerable electronic couplings are obtained even though the NC 3Ag state is optically dark (Figure 3). To explain this effect, we consider in Figure 4b a (simplified) schematic illustration of the transition densities of the NC 3Ag and Rh+ 1B2 states. The full Coulomb integral of such a relative configuration does not vanish, where the moiety of the QD’s transition density that faces the dye (dotted circle in Figure 4b) largely contributes to the Coulomb integral. In summary, all of the examples shown here illustrate that the proximity of the donor-acceptor species and their nonspherical shape precludes Fo¨rster type predictions of the electronic couplings. IV. Discussion and Conclusions Although we focus on a simplified NC-dye model, the present investigation considering the full Coulomb interaction provides some fundamental insight into the characteristics of the NC-dye excitation energy transfer, which could help the theoretical modeling and interpretations of the FRET phenomena in more realistic and larger systems. Further ab initio studies considering larger NCs are needed to extensively assess the validity of multipole expansions and of the quantum-confinement picture in various cases. The present scheme to estimate the electronic coupling V can be applied for any donor-acceptor system, provided that the wave functions of the separated donor and acceptor are available. In order to analyze the exciton transfer dynamics, the vibronic couplings and resonance conditions of the exciton states can also be investigated based on ab initio calculations, but these aspects are beyond the scope of this paper.

7552

J. Phys. Chem. C, Vol. 113, No. 18, 2009

Reasonable estimations of the excitation energies are necessary for the modeling of the FRET mechanism. The comparison between TDDFT vs MCQDPT calculations suggests that TDDFT may predict a qualitatively wrong sequence of the excitation energies of NCs, even though TDDFT is less computationally expensive than the CI methods and might be a practical tool to treat exciton states of large NCs. The electronic coupling is sensitive to the NC-dye relative orientations, which would be important for the material design to control the FRET phenomena. In some cases, the dipole approximation might be applicable for the electronic couplings of NC-dye systems, e.g., for a spherical quantum dot and chlorophyll.27 Our calculations, however, suggest that in certain cases the consideration of the full Coulomb interactions is essential to properly estimate the electronic coupling of NC-dye systems. Several exciton states including the dark states of NC can contribute to the FRET process. The nontrivial contributions of the optically dark states indicate the limitations of modeling the FRET mechanism based on experimental spectra. One may be able to apply some reduction schemes for the transition densities beyond the dipole approximation.16,18 For example, the 2A1 transition density of the Rh+ can be approximated by a collection of three dipoles that represent the respective local transition densities of the six-membered rings (Figure 4a). Equally, the description of the 3Ag transition density of the (CdSe)6 cluster (Figure 4b) can be improved by considering higher-order multipoles. For a related example, Baer et al.18 have proposed that the description of the electronic coupling between spherical quantum dots can be improved by accounting for a quadrupole-dipole interaction. However, such reduction schemes invoke intuition, and their validity should in general be verified by comparing with the electronic coupling based on the full Coulomb interaction. When the NC-dye distance is short enough, the electronic coupling is sensitive to the local distributions of the transition densities around the NC-dye interface. Such features are beyond the limits of multipole approximations. Thus, the full Coulomb interaction would equally be important for large NC-dye systems, in addition to the realistic modeling of the atomistic structures such as the effect of surface morphologies. Also, in some NC-dye systems, the dye molecules might be directly adsorbed on the NC surface. In such cases, the exchange repulsion may not be negligible; thus CI or TDDFT calculations of the whole NC-dye supermolecule may be required to appropriately estimate the electronic coupling through the energy-level splitting of the exciton states. Acknowledgment. This study was carried out at Ecole Normale Supe´rieure Paris and was supported by the Groupement d’Inte´reˆt Publique - Agence Nationale de la Recherche (GIPANR) project ANR-05-NANO-051 (NanoFRET). We thank Anne Feltz for helpful discussions. References and Notes (1) Fo¨rster, T. Discuss. Faraday Soc. 1959, 27, 7. (2) Lakowicz, J. R., Principles of Fluorescence Spectroscopy, 2nd ed.; Kluwer Academic: New York, 1999. (3) Scholes, G. D.; Rumbles, G. Nat. Mater. 2006, 5, 683. (4) Stephens, D. J.; Allan, V. J. Science 2003, 300, 82. Weijer, C. J. Science 2003, 300, 96. Miyawaki, A.; Sawano, A.; Kogure, T. Nat. Cell Biol. 2003, 5, S1. Jares-Erijman, E. A.; Jovin, T. M. Nat. Biotechnol. 2003, 21, 1387.

Tamura et al. (5) Becker, K.; Lupton, J. M.; Mueller, J.; Rogach, A. L.; Talapin, D. V.; Weller, H.; Feldmann, J. Nat. Mater. 2006, 5, 777. (6) Clapp, A. R.; Medintz, I. L.; Mattoussi, H. ChemPhysChem 2006, 7, 47. Medintz, I. L.; Uyeda, H. T.; Goldman, E. R.; Mattoussi, H. Nat. Mater. 2005, 4, 435. Pons, T.; Medintz, I. L.; Sykora, M.; Mattoussi, H. Phys. ReV. B 2006, 73, 245302. (7) Parak, W. J.; Pellegrino, T.; Plank, C. Nanotechnology 2005, 16, R9. (8) Peng, H.; Zhang, L.; Kja¨llman, T. H. M.; Soeller, C.; Travas-Sejdic, J. J. Am. Chem. Soc. 2007, 129, 3048. Shi, L.; De Paoli, V.; Rosenzweig, N.; Rosenzweig, Z. J. Am. Chem. Soc. 2006, 128, 10378. Snee, P. T.; Somers, R. C.; Nair, G.; Zimmer, J. P.; Bawendi, M. G.; Nocera, D. G. J. Am. Chem. Soc. 2006, 128, 13320. Fernandez-Arguelles, M. T.; Yakovlev, A.; Sperling, R. A.; Luccardini, C.; Gaillard, S.; Medel, A. S.; Mallet, J.M.; Brochon, J.-C.; Feltz, A.; Oheim, M.; Parak, W. J. Nano Lett. 2007, 7, 2613. (9) Tsay, J. M.; Trzoss, M.; Shi, L.; Kong, X.; Selke, M.; Jung, M. E.; Weiss, S. J. Am. Chem. Soc. 2007, 129, 6865. Bagalkot, V.; Zhang, L.; Levy-Nissenbaum, E.; Jon, S.; Kantoff, P. W.; Langer, R.; Farokhzad, O. C. Nano Lett. 2007, 7, 3065. (10) Krueger, B. P.; Scholes, G. D.; Fleming, G. R. J. Phys. Chem. B 1998, 102, 5378. Scholes, G. D.; Jordanides, X. J.; Fleming, G. R. J. Phys. Chem. B 2001, 105, 1640. (11) Markvart, T.; Greef, R. J. Chem. Phys. 2004, 121, 6401. (12) Simah, D.; Hartke, B.; Werner, H. J. J. Chem. Phys. 1999, 111, 4523. (13) May, V.; Ku¨hn, O. Charge and Energy Transfer Dynamics in Molecular Systems; Wiley-VCH: Weinheim and Chichester, 2004. (14) Wong, K. F.; Bagchi, B.; Rossky, P. J. J. Phys. Chem. A 2004, 108, 5752. (15) Czader, A.; Bittner, E. R. J. Chem. Phys. 2008, 128, 035101. (16) Barford, W. J. Chem. Phys. 2007, 126, 134905. (17) Schrier, J.; Wang, L.-W. J. Phys. Chem. C 2008, 112, 11158. (18) Baer, R.; Rabani, E. J. Chem. Phys. 2008, 128, 184710. (19) Scholes, G. D. AdV. Funct. Mater. 2008, 18, 1157. (20) Kasuya, A.; Sivamohan, R.; Barnakov, Y. A.; Dmitruk, I. M.; Nirasawa, T.; Romanyuk, V. R.; Kumar, V.; Mamykin, S. V.; Tohji, K.; Jeyadevan, B.; Shinoda, K.; Kudo, T.; Terasaki, O.; Liu, Z.; Belosludov, R. V.; Sundararajan, V.; Kawazoe, Y. Nat. Mater. 2004, 3, 99. (21) Botti, S.; Marques, M. A. L. Phys. ReV. B 2007, 75, 035311. (22) Jose, R.; Zhanpeisov, N. U.; Fukumura, H.; Baba, Y.; Ishikawa, M. J. Am. Chem. Soc. 2006, 128, 629. (23) Schrier, J.; Wang, L.-W. Phys. ReV. B 2006, 73, 245332. (24) Troparevsky, M. C.; Kronik, L.; Chelikowsky, J. R. J. Chem. Phys. 2003, 119, 2284. (25) Deglmann, P.; Ahlrichs, R.; Tsereteli, K. J. Chem. Phys. 2002, 116, 1585. (26) Isborn, C. M.; Kilina, S. V.; Li, X.; Prezhdo, O. V. J. Phys. Chem. C 2008, 112, 18291. (27) Curutchet, C.; Franceschetti, A.; Zunger, A.; Scholes, G. D. J. Phys. Chem. C 2008, 112, 13336. (28) Kilin, D. S.; Tsemekhman, K.; Prezhdo, O. V.; Zenkevich, E. I.; von Borcyskowski, C. J. Photochem. Photobiol., A 2007, 190, 342. (29) Nakano, H. J. Chem. Phys. 1993, 99, 7983. Nakano, H. Chem. Phys. Lett. 1993, 207, 372. (30) Hay, P. J.; Wadt, W. R. J. Chem. Phys. 1985, 82, 270. Hay, P. J.; Wadt, W. R. J. Chem. Phys. 1985, 82, 284. Hay, P. J.; Wadt, W. R. J. Chem. Phys. 1985, 82, 299. (31) Schmidt, M. W.; Baldridge, K. K.; Boatz, J. A.; Elbert, S. T.; Gordon, M. S.; Jensen, J. H.; Koseki, S.; Matsunaga, N.; Nguyen, K. A.; Su, S.; Windus, T. L.; Dupuis, M.; Montgomery, J. A. J. Comput. Chem. 1993, 14, 1347. (32) We have found this trend in preliminary CASSCF calculations for some lower excited states. (33) Perdew, J. P.; Wang, Y. Phys. ReV. B 1992, 45, 13244. (34) Towada, Y.; Tsuneda, T.; Yanagisawa, S.; Yanai, Y.; Hirao, K. J. Chem. Phys. 2004, 120, 8425. (35) Harbold, J. M.; Du, H.; Krauss, T. D.; Cho, K.-S.; Murray, C. B.; Wise, F. W. Phys. ReV. B 2005, 72, 195312. (36) Huxter, V. M.; Kovalevskij, V.; Scholes, G. D. J. Phys. Chem. B 2005, 109, 20060. (37) Wang, H.; de Mello Donega, C.; Meijerink, A.; Glasbeek, M. J. Phys. Chem. B 2006, 110, 733. (38) Muljarov, E. A.; Zimmermann, R. Phys. ReV. Lett. 2007, 98, 187401. (39) Kilina, S. V.; Craig, C. F.; Kilin, D. S.; Prezhdo, O. V. J. Phys. Chem. C 2007, 111, 4871. Prezhdo, O. V. Chem. Phys. Lett. 2008, 460, 1.

JP811042T