Quantum Mechanical Approach to Solvent Effects on the Optical

J. Phys. Chem. C 2010, 114, 1553–1561

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Quantum Mechanical Approach to Solvent Effects on the Optical Properties of Metal Nanoparticles and Their Efficiency As Excitation Energy Transfer Acceptors ´ ngel Sańchez-Gonza´lez,† Aurora Munõz-Losa,† Sinisa Vukovic,†,§ Stefano Corni,‡ and A Benedetta Mennucci*,† Department of Chemistry and Industrial Chemistry, UniVersity of Pisa, Via Risorgimento 35, 56126 Pisa, Italy, INFM-CNR National Research Center on nanoStructures and bioSystems on Surfaces (S3), Via Campi 213/A, 41100 Modena, Italy, and Chemical Sciences DiVision, Lawrence Berkeley National Laboratory, 1 Cyclotrone Road, Berkeley, California 94720 ReceiVed: December 2, 2009

We present a time-dependent density functional theory (TDDFT) investigation of the solvent effect on the light absorption of metal nanoparticles and on their efficiency as excitation energy transfer (EET) acceptors from organic dyes. The calculations consider both the dye and the metal particle at quantum-mechanical (QM) level, thus including quantum size effects. The results are compared to those of a second method that exploits a continuous dielectric model for the metal nanoparticle while keeping the same QM level for the dye. Both methods use the polarizable continuum model (PCM) for the solvent. The comparison of these two approaches for gold and silver nanoparticles has clarified how their different electronic nature specifically couples with the solvent and leads to different optical properties and EET efficiency. Moreover, a critical comparison of the QM results with the popular Fo¨rster approach to EET has been performed, quantifying the inherent limitations of the latter for dye-nanoparticle EET in solution. 1. Introduction After photoexcitation, energy absorbed by a molecule (the donor) can be transferred efficiently over a distance of up to several tens of angstroms to another molecule (the acceptor) by the process of electronic energy transfer (EET).1 This photophysical process is ubiquitous in nature, playing a key role in the light-harvesting (LH) machinery of photosynthesis.2 Inspired by natural LH systems, researchers have designed and employed artificial antennae for the capture and energy conversion of light. Noble metal (e.g., gold and silver) nanoparticles (MNP) have been also used as acceptors in EET, and they have shown to be very effective acceptors.3 For example, EET from organic dyes to a metal nanoparticle has been shown to be a pivotal step in the working mechanism of the very recently discovered spacer-based nanolaser.4 EET from a chromophore to a MNP has also been extensively investigated theoretically.5 The common understanding of the origin of the exceptional optical properties of MNP is in terms of excitations of the MNP surface plasmons (SP). In fact, surface plasmons, collective excitations of the electron gas confined in the nanoparticle, have unusually large transition multipoles and, thus, are able to enhance many phenomena involving electromagnetic fields including EET. Recently, however, we have shown6 that the real picture is more complex than this and that nonplasmonic small metal clusters can be as intrinsically effective as plasmonic particles in EET. The aim of this article is to go farther in this detailed analysis of the optical phenomena involving MNPs and unravel the effects of the solvent on the MNP light absorption and the EET with an organic dye. The role of the solvent is in fact a * Corresponding author. E-mail: [email protected]. † University of Pisa. § Lawrence Berkeley National Laboratory. ‡ INFM-CNR.

fundamental aspect which is often overlooked in both its nature and consequences. It is well-known that a solvent (or a more general host medium) can strongly affect the absorption spectrum of metal nanoparticles.7 Experiments have in fact clearly shown that the presence of a solvent induces a red-shift in the extinction maximum of the MNP8 being such an effect even more important than the influence of the ligands attached on the surface of the metal.9 The polarizability of the solvent is involved with the red-shift, being the increase of the red-shift consistent with the polarity of the dielectric medium.10,11 A detailed analysis of the influence of the environment (ligands, solvent, and composition of both) in the surface plasmon resonance is also reported by Ghosh et al.12 Much less, however, is known on the effect that a solvent can have on the capacity of the nanoparticle to act as an effective EET acceptor. To better understand this aspect we have considered a realistic donor (the perylene diimide, PDI) and two different metal clusters as acceptors, Au20 and Ag20, and we have studied their optical properties and their EET efficiency in water using a quantum mechanical (QM) approach we have developed in our group.13 By using such an approach we can calculate the EET rate for a solvated D/A pair by including the effect of the solvent both on the properties of donor (D) and acceptor (A) moieties and on their interactions. In this study the solvent is introduced using a continuum description through the integral equation formalism version of the polarizable continuum model (IEFPCM).14 Within this framework, the system of interest (here the D/A pair) is described at QM level, and it is assumed to be embedded in a cavity of shape and dimension defined according to the geometrical structure of the pair. Such a cavity is surrounded by a continuous dielectric which represents the solvent. The two metal clusters have been selected as they are known to behave in a very different way. Photoabsorption studies of small silver clusters (2-21 atoms) have shown that the spectra

10.1021/jp911426f  2010 American Chemical Society Published on Web 01/04/2010

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Sańchez-Gonza´lez et al.

of the smaller clusters have several discrete peaks whereas the larger clusters are dominated by a single broad peak.15 Although this broad absorption feature cannot be considered as a true collective excitation due to the small size of the cluster, it can be considered as a microscopic analog to the plasmon excitation observed in nanoparticles, with strong dipolar excitation that will produce a field close to the nanoparticle which is capable of enhancing optical phenomena. It has been shown that the optical properties of small gold clusters like Au20 are completely different from Ag20. The calculated spectrum for Au20 in fact displays many discrete peaks that are reduced by over a factor of 10 from the single strong peak that is seen for Ag20.16 This behavior can be understood as arising from relativistic effects, which cause the intraband (sp r sp) and interband (sp r d) transitions to overlap for gold. Thus the single strong intraband absorption feature typical of silver is mixed with many interband states in gold clusters, resulting in multiple absorption features that are spread out over a wide range of the vis/UV spectrum and with smaller oscillator strength per transition. The comparison between the two metals will allow for a detailed analysis of the effects of the solvent on the optical properties of the metal particles as well as on their efficiency as EET acceptors. This full quantum-mechanical analysis (indicated by QM/QM in the following, to remark the QM treatment of both the organic chromophore and the MNP) will be accompanied by a complementary study based on a combined QM/Continuum Metal (QM/CM) model17,18 in which the chromophore is still treated quantum-mechanically while the metal particle looses its atomistic nature to be substituted by a continuous body, characterized by its frequency dependent local dielectric permittivity. The results following from this combined QM/CM model in which the metal particle is, by definition, plasmonic (being described by experimental dielectric properties of the bulk metal, corrected for the finite electron mean free path) and the QM/QM description will give further information on the nature of the interactions exerted by the solvent. In both approaches the QM part of the system will be described using a time-dependent density functional theory (TDDFT).19 2. Methods and Computational Details 2.1. QM/QM Model. The EET rate between the donor (PDI) and the acceptor (Au20 or Ag20) is obtained in the weak coupling limit, by applying the Fermi golden rule, namely

kEET )

2π 2 VJ p

(1)

where J is the spectral overlap defined with respect to area normalized donor emission and acceptor absorption spectra and V is the electronic coupling between the donor and acceptor. In the present study, J and V are obtained using a TDDFT description. In particular, the overlap J is calculated assuming a Gaussian approximated shape for the emission and absorption bands centered at the transition energies obtained by the TDDFT description. The coupling V is calculated using the TDDFT transition densities of donor and acceptor (FDT and FAT, respectively), namely12

V )

1 r AT*(b′) r [ r + gxc]FDT(b) ∫ db′r ∫ dbF |b r - b′| r

(2)

where the first term represents the Coulomb contribution and the second one includes electron exchange and correlation contributions (through the kernel gxc which is specific of the functional used). In the case of metal clusters, however, it is necessary to take into account many different electronic states of the metal, which can collect the energy emitted from the PDI. Due to the tetrahedral symmetry of the pyramid, the metal cluster presents degenerated states with the same energy (same overlap) but different transition dipole orientations (different electronic coupling). With this consideration, the expression for the Fermi golden rule given in eq 1 has to be modified as

kEET )

2π p

degenerate

∑[ ∑ M

Vi2]JM

(3)

i(M)

We remark that the theory presented here is valid in the weak coupling limit, i.e., when the intrinsic decay rate of the acceptor excited states is much greater than the EET rate. Assuming a decay rate of ∼1013-1014 s-1 for the metal excited state,18 kEET larger than ∼1012-1013 s-1 (as some of those reported in the article) indicates a possible breakdown of the weak-coupling approximation and a reformulation of the rate constant.20 2.2. Modeling Solvent Effects on the Electronic Coupling in the QM/QM Approach. In this study the solvent is introduced using a IEF-PCM description.14 Within this framework, the quantum-mechanical PDI/M20 pair is assumed to be embedded in a cavity of shape and dimension defined according to the geometrical structure of the molecules; in our study the cavity is the sum of two separated cavities, one for the donor and the other for the acceptor moiety. The solvent is described as a polarizable continuum characterized by its dielectric properties (the dielectric constant and the refractive index), which responds to the presence of the QM system through a set of induced (or apparent) charges placed on the surface of the molecular cavities. In turn, such charges act back on the QM system from which they are generated: this mutual polarization effect is solved through a modified self-consistent field scheme. In addition, within the PCM framework it is possible to introduce nonequilibrium effects which appear anytime a fast process in the QM system originates delays in the response of the solvent.21 This is exactly what happens during an electronic excitation and a following EET process. In both cases the response of the solvent will be incomplete in the sense that only its fast degrees of freedom (of electronic nature) will be in equilibrium with the final state of the QM system while the rest will remain frozen in the initial configuration corresponding to the QM system before the change. Within the PCM framework, the screening effect on the electronic coupling is not introduced as a scaling factor as in the Fo¨rster model (see section 3) but it enters in the definition of the coupling through an additional term which sums to the definition given in eq 2. Namely, if we mimic the solvent polarization induced by the donor transition density in terms of PCM charges, such an additional term becomes13

Vexplicit )

(

1 r AT*(b) r ∑ ∫ dbF |b r -b s k| k

)

q(Sk ;εopt, FDT)

(4)

As described above, a nonequilibrium response for the environment (indicated by the dynamic or optical part of the permittivity, εopt, i.e. the square of the refractive index) is used.

Optical Properties of Metal Nanoparticles

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Now, if we want to recover a PCM screening factor s to be compared with the Fo¨rster estimate we have to reformulate it as

TABLE 1: Values Used to Define the Dielectric Constants for Metal Particlesa met

Vs + Vexplicit Vtotal s) ) Vs Vs

(5)

where Vs is the sum of the Coulombic and exchange-correlation terms defined as in vacuum (see eq 2) but with transition densities modified by the environment. 2.3. QM/CM Model. The QM/QM calculations of the PDIM20 pair will be compared with calculations obtained for PDI interacting with hypothetical bulk-like metal particles described in terms of their dielectric response. These calculations are obtained using a method that we have developed to study molecules close to MNPs17 by generalizing the PCM originally developed for solvents. In this generalization, the donor is still described at QM level using a TDDFT description (exactly as in the QM/QM approach described above) but now the metal is considered as a polarizable dielectric particle with the proper shape and dimension. Thus the TDDFT equations for the donor include now a new PCM-like term to take into account the interaction of its transition density with the oscillating metal polarization induced by the transition density. This polarization is expressed as that of the solvent, e.g. by means of apparent charges placed on the delimited surface which follows the shape of the metal nanoparticle. These charges are determined by the frequency dependent dielectric permittivity for the corresponding metal, which is in general complex. As a result, the excitation energies (ω) of the chromophore become complex and their imaginary part determines the EET rate to the metal, namely18

kEET ) -2Im{ω}

(6)

This expression is the counterpart of the EET rate defined in eq 3 for the QM description of the metal cluster. To describe the response of the CM, a frequency-dependent permittivity taken from experimental data (corrected for the limited mean free-path according to the Drude model22) is used

{(

2 εmet(ω, R) ) εmet exp (ω) + Ω

1

i ωω+ τ

-

)

1 (R + VFτ) ωω+i τR

[

]}

(7)

In this expression R is an effective radius of the particle, Ω is the plasma frequency, νF is Fermi velocity, and τ is the bulk relaxation time (the values used are from Palik23 and they are reported in Table 1). As the emission spectrum for the PDI remains almost unchanged passing from gas-phase to water, all of the values collected in the table have been used for both environments, vacuum and water. 2.4. Computational Details. For PDI, the TDDFT calculations have been performed using Becke three parameters hybrid functional with Lee-Yang-Parr correlation function (B3LYP)24 and the basis set chosen was 6-31+G(d). Regarding to DFT calculations for the Au20 and Ag20, including the geometry optimization, the same DFT level was used in combination with the LANLDZ pseudopotentials25 to include scalar relativistic

Re{ε } Im{εmet} ω τ Ω νF

Ag

Au

-9.906 -4.329 0.08678 1320 0.332 0.64

-4.329 2.140 0.08678 306 0.293 0.64

a The experimental emission wavelength of PDI (λ ) 525 nm) has been used. All of the values (in atomic units) have been extracted from Palik.23

Figure 1. Graphical representation of the QM/QM vertex and face arrangements studied. Both longitudinal and perpendicular PDI orientations are shown together with the definition of the effective distance.

effects. We note that potentials are not able to account for spin-orbit effects which may have an influence on electronic spectra, in particular for gold clusters. In the literature there are studies which have analyzed the effects of including also spin-orbit effect for gold. According to these studies, the spin-orbit coupling can have an impact in the absorption spectra (resonances and oscillator strengths) but this is especially true for nanowires, in contrast to more compact gold clusters where this spin-orbit effect tends to be quenched.26 The calculations have been limited to the singlet excited states with excitation energies below 4.5 eV (this has meant to consider 100 excited states for gold and 84 for silver). All of the calculations have been performed using a locally modified version of the Gaussian package27 in which we have implemented both the QM/QM and QM/CM approaches described above. 3. Results In this section the numerical results will be presented as follows: first we shall summarize the solvent effects on the optical absorption of the MNPs then we shall move to analyze EET rates both in gas-phase and in water. Both in the analysis of absorption and in that of EET, the QM/QM approach will be compared with the QM/CM approach. The full QM analysis on different relative arrangements of PDI and metal clusters (see Figure 1) made of gold and silver will in fact be used to rationalize the solvent effects in terms of changes induced in the QM nature of the metal particles and in their QM interactions with the chromophores. By contrast, the QM/CM analysis will shed light on the effect of the solvent on the electrodynamic interactions between

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Figure 3. Plasmonic spectra for the tetrahedral CM nanoparticles of gold and silver in vacuum (blue) and in water (red). Energies are in eV and intensities are in arbitrary units.

Figure 2. Au20 (yellow) and Ag20 (black) absorption spectra in vacuum and in water. Energies are in eV and intensities are in arbitrary units. Both spectra have been obtained using a Gaussian band shape with 0.2 eV width.

chromophore and metal particle as well as the solvent induced changes on the plasmonic properties of the metal particles. As the QM/QM analysis is based on clusters made of 20 metal atoms, in the QM/CM analysis we have considered tetrahedral continuous particles obtained in terms of 20 interlocked spheres centered on the same position of the atoms in the real cluster and having radius)3 Å. 3.1. Solvent Effects on Metal Absorption Spectra. QM/ QM Approach. In Figure 2, we report Au20 and Ag20 absorption spectra obtained at TDDFT level in gas-phase and in water. As it can be seen from the plots, our calculations confirm the expected different behavior of the two metals. The absorption spectrum for the silver is mainly constituted by a single excitation while the spectrum for Au20 is formed by a convolution resulting from distinct excitations, although in practice it is dominated by a few dipole allowed states. In addition, gold and silver spectra significantly differ also for the relative height, with the silver maximum being much higher than gold one. While we are not aware of experimental data concerning the absorption band of Au20, for Ag20 in an argon matrix a value of 3.70 eV has been reported.16 The effect of the Ar matrix has been estimated to be a red-shift of 0.14 eV,28 and the best estimate for Ag20 band maximum is thus 3.84 eV, which is close to our computed gas-phase value of 3.59 eV. However, it should be noted that the precise value of this absorption band is quite sensitive to the used DFT xc functional and the basis set.29 The two metal clusters significantly differ also with respect to the sensitivity to the solvent. When the clusters are in water, the position of the gold spectrum remains almost unchanged. On the contrary, the Ag20 spectrum suffers a significant shift toward the red. In addition, the presence of the solvent changes the substructure of the gold spectrum but not that of Ag20. In fact, while in gas-phase the gold spectrum is dominated by two equally strong excitations (with oscillatory strengths 0.25 and 0.27) in water, the most intense excitations are three with

different oscillatory strengths (namely ca. 0.31, 0.56 and 0.17). By contrary, for Ag20, the spectrum is dominated by only one, very strong, excitation both in gas-phase and in solution; the effect of the solvent is only a further increase of the oscillatory strength from 2.44 to 3.14. Comparison with the QM/CM Approach. By using the CM description, we can apply the PCM approach also for the metal and calculate the dispersion of imaginary part of the metal polarizability for each of the metal particles studied. The imaginary part of the polarizability R is related to the absorption cross section σ by the formula

σ ) 4π(ω/c) Im(Rxx + Ryy + Rzz)

(8)

Therefore, the maximum of Im(R) with the frequency gives an estimate of the surface plasmon resonance. This estimate is here done for the two metals both in gas-phase and in water and is reported in Figure 3. Exactly as the TDDFT, also the CM description shows a significant solvent-induced red-shift for silver (from 3.1 eV in vacuo to 2.6 eV in water) and a much smaller shift for gold (from 2.3 eV in vacuo to 2.2 eV in water). As for the absolute value of the energy of the absorption maximum, we note that the TDDFT results are blue-shifted w.r.t the CM values. This was expected, because the CM does not include quantum size effects (beside the limited mean-free path correction), while the TDDFT does. 3.2. Solvent Effects on EET Rates and Electronic Couplings. In the previous section we have explained the differences observed in the absorption properties of metal particles when described at the QM or CM level due to quantum-size effects present in the former but not in the latter approach. These differences correspond to different spectral overlap with the selected donor emission spectrum. As a result, the comparison between QM/QM and QM/CM EET rates is necessarily altered. A possible way to avoid this inconsistency would be to find two different donor systems with very similar electronic characteristics (both in the ground and in the excited state) but with shifted emission spectra so to have the same relative position with respect to the TDDFT and the plasmonic spectra, respectively. A much simpler way to restore the consistency in


Figure 4. PDI emission (blue), Au20 (yellow), and Ag20 (black) absorption spectra in vacuum and in water after the shift to get coincidence with the CM plasmon resonances. Energies are in eV and intensities are in arbitrary units. Both PDI and M20 spectra have been obtained using a Gaussian band shape with 0.3 and 0.2 eV width, respectively.

the comparison, is however to introduce a shift of the TDDFT spectra of the metal clusters so to have their maxima coincident with the corresponding CM results. These shifted spectra are reported in Figure 4 together with the emission spectra of PDI. When the spectra are shifted, the different behaviors of the two metals and between gas-phase and water become more evident. For Au20 in gas-phase there is almost a perfect coincidence with PDI spectrum and all the few states with large transition dipoles can contribute to the rate with high overlap. On the contrary, for Ag20 in gas-phase the overlap with the PDI spectrum is much less effective (there is a difference of 0.7 eV between the two maxima). Moving to water, we observe a small offset between gold and PDI spectra, whereas in the case of silver, the overlap becomes much more effective. These changes in the relative positions, together with the ones in the shape of the band will be reflected in the EET rate as shown in the following section. QM/QM Approach. In Figure 5 we report the QM/QM results for the EET decay rates obtained for the different clusters (Ag20 and Au20) in gas-phase. In this analysis we have treated different arrangements for the QM/QM systems: PDI can be either placed at the vertex (V) of the metal tetrahedral cluster or at a face (F) with perpendicular or longitudinal orientations (see Figure 1). For these arrangements we have considered different intermolecular effective distances (from 5 to 60 Å for F/P and from 15 to 70 Å for V/L) defined as the distance between the geometrical centers of the donor and acceptor systems. From the analysis reported in the previous section, we expect that the Au20 gives larger EET rates due to the much better overlap of its spectrum with that of PDI. This is confirmed by the data reported in Figure 5: the lines corresponding to gold are always above those referring to silver for the same orientation (for large distances the differences in the rates between the two metals is almost 1 order of magnitude).


Figure 5. log(kEET) (s-1) for PDI/M20 in vertex and face arrangement with longitudinal and perpendicular orientation with respect to the effective distance (Å), for the shifted spectra in vacuum.

Figure 6. log(kEET) (s-1) for PDI/M20 in vertex and face arrangement with longitudinal and perpendicular orientation with respect to the effective distance (Å), for the shifted spectra in water.

In Figure 6 we show the results for the EET rates of the same systems in water as obtained using the PCM description of the solvent. As we have seen in the previous section, the effect of water on the absorption spectrum of silver is to shift it closer to the

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PDI spectrum, while the gold spectrum remains almost unchanged with only a small offset. Due to the solvent effects, the gap between the maxima of PDI emission and Ag20 absorption spectra reduces from 0.7 to 0.4 eV. This shift is enough to allow a remarkable increase of the rate. This is reflected in the clear solvent-induced reduction of the differences between gold and silver rates, which are now almost coincident at all distances with silver rates being now slightly larger than gold ones. As in gas-phase also in water, the vertex arrangement shows a much larger sensitivity to the orientation of the PDI than the face arrangement: V/L rates are significantly larger than V/P for both metal clusters while F/L and F/P rates are very similar. This behavior is a clear indication of the multipolar nature of the transition densities. In fact, in the dipolar approximation, vertex and face arrangements should give exactly the same distance dependency with respect to longitudinal and perpendicular orientations, respectively: for each effective distance the relative orientation of the PDI and metal cluster transition dipoles is the same in V/L and F/L or in V/P and F/P systems. The specific behavior shown by the vertex arrangement is intrinsically related with the shape of the metal cluster and it indicates that EET rates are quite sensitive to the relative orientation of the metal and the dye. As it appears from the graphs reported above, a strong component of the solvent effect on the EET rate is related to the changes induced in the absorption spectra of the two metal clusters and, as a final result, on the overlap factor J; however, it is here interesting to analyze also the effect that water has on the electronic coupling V. The effects of the solvent on the coupling are in general of two different types: on the one hand the properties of donor and acceptor will be modified both in their ground state and in the excitation process, on the other hand there will be an explicit effect of the solvent on the interaction between them. This second effect, generally the most important one in EET processes, is commonly indicated as screening. Such a screening effect due to the solvent is commonly introduced in the definition of the rate constant as a factor that directly multiplies the electronic coupling VfsV. Fo¨rster30 originally suggested to use a factor (s ) 1/n2) which only depends on the optical permittivity (εopt ) n2) of the solvent whatever is the nature of the D/A pair, their relative orientation, and their distance. One of the main points of the PCM approach to EET is that the screening effect is a function of the donor and acceptor transition densities, their orientation, and their distance.31 To analyze the screening effect of water on the PDI/M20 coupling, we have selected the metal transition with the largest transition dipole moment. This selection is straightforward in the silver cluster where the spectrum is characterized by a single intense transition but not in the gold for which various equally intense transitions appear. We have thus limited the present analysis to Ag20 only (we have however verified that the results remain very similar for the various intense transitions of gold). The resulting screening effect (here represented in terms of the factor s) is reported in Figure 7 for both PCM and Fo¨rster model. As it can be seen from the graph, PCM calculations confirm that s is a function of the distance and the orientation. Only at large distances it becomes constant with a value around 0.63, which is well above the Fo¨rster factor (0.56). These differences in the s factor, if converted in EET rates, should lead to increases at least of 20% with respect to the Fo¨rster estimates. In addition to the screening effect, the presence of the solvent should also modify the coupling by affecting the transition


Figure 7. Evolution of the screening factor s with the effective distance (Å) in the V/L (red lines) and F/P (blue lines) arrangements for PDI/ Ag20. For comparison also the Fo¨rster estimate is reported.

Figure 8. Vs2(water)/Vs2(vacuum) for PDI/Ag20 in the V/L (red lines) and F/P (blue lines) arrangements with respect to the effective distance (Å).

properties of both donor and acceptor moieties. This effect, that from now on we shall indicate as “implicit effect”, is not straightforward to get in classical models, and in fact it is generally neglected. By contrast, the “implicit effect” is taken into account in the PCM framework: the transition properties of both PDI and metal cluster are in fact calculated in the presence of the PCM solvent. If we now analyze such an implicit term we find that the changes are quite significant. This is shown in Figure 8, where we report the distance dependence of Vs2(water)/Vs2 (Vacuum) where Vs(water) is the coupling calculated in water without the screening effect. Once again, to simplify the analysis, we have reported results only for the silver transition with the largest transition dipole moment. It is known that the solvent increases the Coulombic contribution through the modification of the corresponding transition densities; as a result, Vs2(water)/Vs2(vacuum) is larger than 1 for all intermolecular distances. From Figure 9 we can see that, in the F/P orientation, implicit solvent effects are larger than in V/L: this is possible only if solvent induces also a rotation of the transition densities which makes V/L orientation less favorable in terms of the coupling with the PDI transition density. This has been verified in terms of the corresponding transition dipoles. The larger distance-dependence of the V/L orientation is an indirect evidence of the multipolar nature of the transition densities which is more important when the PDI is placed in the V/L than in the F/P orientation (see next section for further details). At large distances V/L and F/P orientations converge to the same value, that is, the orientational factor influence is lost with the distance.



Figure 9. log(kEET) (s-1) for PDI/Ag20 in the V/L (red lines) and F/P (blue lines) arrangements with respect to the effective distance (Å). The dashed lines refer to calculations including only the implicit solvent effect in the coupling whereas full lines refer to calculations including both implicit and explicit effect in the coupling. Dotted lines refer to calculations in vacuum. All values have been obtained with overlap factor J ) 1.

The analysis of solvent induced effects on the coupling is summarized in Figure 9 where we plot the logarithm of the EET rate for Ag20 in water as obtained using two different sets of values for the coupling: (i) one obtained including only the implicit contribution and (ii) one obtained including both implicit and screening contributions. We recall that the plotted rate constants have been obtained by eliminating the effect of the overlap factor J (a fixed overlap of 1 is used for all points). To have a more direct comparison, also vacuum results are reported. From the graphs it is clear that, at all orientations, the two different solvent effects (the implicit and the screening) almost compensate and the final rate in water remains close to the vacuum one. Comparison with the Fo¨rster Approach. One of the most common approaches used to obtain the energy transfer rate is the classical dipole-dipole approach originally proposed by Fo¨rster. This model considers that the electronic coupling can be substituted by its Coulombic contribution that generally includes more than 95% of the total coupling. In turn, this term can be described by the classical ideal dipole approximation as T

V ) sVs ) sVdip-dip

T

1 µDµA ) 2 3 κ n R

(9)

where µDT and µAT are the transition dipole moments of donor and acceptor, R is their separation, and κ is the orientational factor defined by the mutual orientation of two molecules. In eq 9 the Fo¨rster approximation of the solvent screening has been introduced. This model is valid when the intermolecular separation is much larger than the molecular dimension. On the contrary, it has been clearly shown32 that at short distances and according to the orientation the error introduced with eq 9 can be larger than 50%. This disagreement is due to two different aspects. First, the Fo¨rster model neglects multipolar contributions. As we have shown in a previous paper,6 the metal clusters present several states with null (or very low) transition dipoles. In terms of the dipole-dipole theory, the energy transfer process is not possible for these states and, therefore, their contribution to the total EET rate is zero. However, when a QM method not

Figure 10. log(kEET) (s-1) for PDI/Ag20 in the V/L (red lines) and F/P (blue lines) arrangements with respect to the logarithm of the effective distance (Å) as obtained using the QM/QM (full lines) and the Fo¨rster (dashed lines) description of the coupling.

restricted to dipolar coupling is employed, a not negligible electronic coupling is obtained also for these states. This coupling is determined by the interaction of the higher-order multipoles of the metal transition density matrix and PDI. The second aspect is related to the solvent effect that in the Fo¨rster model reduces to a constant screening factor s (the inverse of the square of the solvent refractive index) independent of the D/A intermolecular distance and orientations. In the previous subsection, we have shown that the Fo¨rster estimate of s is smaller (i.e., the screening is larger) than the PCM function (see Figure 7), and it does not catch the distance dependence of such quantity. To have a complete picture of the reliability of the Fo¨rster approach, in Figure 10 we report the logarithm of the EET rate for the V/L and F/P arrangements of the PDI/Ag20 system in water with respect to the logarithm of the effective distance obtained using the full QM coupling and the Fo¨rster approximation reported in eq 9. For the F/P orientation, we see in Figure 10 a quite good behavior of the Fo¨rster approximation, in spite of the important approximations introduced. Only at the shorter distances (comparable to the nanoparticle size), discrepancies between the QM and the Fo¨rster lines appear as expected by a less reliable dipolar description of the transition densities at these distances. This good behavior, however is favored by a compensation of errors. In fact, the neglect of the multipolar terms leads to an increase of the coupling while the use of a smaller s factor (see Figure 7) leads to a decrease of the coupling. Such error cancellation is fortuitous, and in general one cannot rely on it. In fact, in the same figure it is shown that for the V/L orientations relevant discrepancies are visible at much larger distances. It is also to be remarked that the trend of the QM results with the distance is different from the 1/d6 expected from the Fo¨rster approximation, as we already noted in ref 6. Comparison with the QM/CM Approach. As the focus of our analysis is on the solvent effect, the comparison between QM/QM and QM/CM models is more effectively done on the ratio kEET(water)/kEET(vacuum) for PDI close to the CM particles with respect to their effective distance (for the correct definition of the rate constant within the CM description, we refer to Methods section). This comparison is summarized in Figure 11.

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Sańchez-Gonza´lez et al. 4. Conclusions

Figure 11. kEET(water)/kEET(vacuum) in the V/L (red) and F/P (blue) arrangements with respect to the effective distance (Å) as obtained using the QM/CM (upper graph) and QM/QM (lower graph). Continuos lines refer to gold and dashed lines to silver.

As it can be seen from the graphs, both descriptions indicate a larger solvent effect on silver particles/clusters than in gold ones. In the QM/QM description, we have shown that this larger solvent effect leading to a faster transfer is due to the shift of the Ag20 absorption spectrum which can thus better overlap with the PDI emission. As a matter of fact, QM/CM and QM/QM approaches give qualitatively similar but quantitatively different increases in kEET due to the solvent (in the QM/QM the increase is almost twice than in the QM/CM). This difference can be rationalized on the basis of the different widths of the plasmon bands obtained at the QM and CM levels (see Figures 2 and 3). In fact, the CM band is wider than the QM one, implying that the same frequency shift along the band corresponds to a smaller variation of the absorption cross section for the CM band, i.e., to a smaller variation of the quenching power with respect to the QM treatment. This result indicates that the bandwidth chosen to simulate the absorption spectrum of the QM metal cluster affects the estimate of the solvent effects, and, when a direct comparison with experiments is done, such bandwidth should be estimated quite precisely. Finally, in the QM/QM description of silver we observe a different behavior of the two orientations (V/L and F/P) at shorter distances that is by far less evident in the QM/CM description. Such dependence of the results on the orientation can be explained looking at the previous graphs on the screening (Figure 7) and implicit effects (Figure 8): in the F/P orientation in fact the screening is smaller than in the V/L and the implicit effect is larger. These two effects sum up to give a net larger solvent-induced increase of kEET in the F/P orientation.

In this article, we have presented a study of the aqueous solvent effect on the light absorption of small metal clusters, Au20 and Ag20, and on the rate of EET between an organic dye and these nanoparticles. The calculations have been performed by two methods previously developed in our group. The first method (QM/QM) considers both the dye and the metal particle at the QM (namely TDDFT) level, using the polarizable continuum model (PCM) for the solvent; the second method (QM/CM) assumes a continuum dielectric model for both the solvent and the metal nanoparticle. Our main conclusions from these calculations are: 1. The solvatochromic shift of the nanoparticle absorption is very much dependent on the investigated metal (small for Au, large for Ag). The QM and CM approaches give similar solvatochromic shifts; however, they differ in the absolute values of the absorption band maximum. In fact, the CM does not include any quantum-size induced blue-shift, which is naturally taken into account by the QM description. 2. Solvent affects the EET process via three different mechanisms: (i) the solvatochromic shift of the nanoparticle absorption relative to the dye emission affects kEET via the band overlap factor J. The extent of such shift, and the related variation of the EET rate is very much dependent on the investigated metal, as mentioned above; (ii) the solvent screens the Coulombic interactions between the dye and the nanoparticle (explicit effect on the coupling); (iii) the solvent gives rise to changes in the nanoparticle transition densities that affect the electronic coupling and, in turn kEET (implicit effect on the coupling). For the investigated systems, factors (ii) and (iii) tend to compensate, and the extent of the solvent effect is mostly determined by (i). 3. The explicit solvent effects are poorly reproduced by the empirical screening factor used to correct the Fo¨rster theory in solution, both for the absolute value and for the trend as a function of the distance, as already verified for organic molecules in solutions.32 When the overall kEET obtained by the Fo¨rster equation is compared to the QM/QM results for Ag20, the differences found for certain orientations may be small, due to error compensation. However, this is specific of such orientations, and the error compensation cannot generally be invoked to justify the use of the Fo¨rster approach without further checks. 4. For both Ag20 and Au20, in vacuo and in solution, the QM/ QM and the QM/CM give EET rates of the same order of magnitude, once the differences in the relative position of donor emission and nanoparticle absorption wavelength are removed. Therefore, they offer a similar description of the electronic coupling and of the solvent effects. However, the quantitative estimate of the solvent effects may be different, since for Ag20 the QM/QM model yields an increase of kEET two times larger than the QM/CM model. Such difference is related to the different band shapes of the nanoparticle in the two descriptions, which demonstrates that to obtain reliable results both the position of the nanoparticle absorption band and its shape should be correctly reproduced. For the prediction of band maximum, the QM/QM model has an edge on the QM/CM approach, since QM/CM is not including any quantum-size induced band shift. The shape of the band is instead difficult to be reproduced for both models, although the QM/CM can effectively incorporate complex band shapes that result from the empirical frequency dependence of the metal dielectric function. The present article represents a step forward for the microscopic understanding of the EET processes involving metal

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Quantum Mechanical Approach to Solvent Effects on the Optical

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