MM Study of Photoinduced Reduction of a Tetrahedral Ag

Article pubs.acs.org/JPCC

QM/MM Study of Photoinduced Reduction of a Tetrahedral Ag20+ Cluster by a Ag Atom Hanning Chen,*,†,‡ Mark A. Ratner,‡ and George C. Schatz*,‡ ‡

Argonne-Northwestern Solar Energy Research Center, Department of Chemistry, Northwestern University, 2145 Sheridan Road, Evanston, Illinois 60208, United States ABSTRACT: A hybrid quantum mechanics/classical mechanics (QM/MM) approach was developed to investigate photoinduced electron transfer (PIET) from a neutral Ag atom to an ionized tetrahedral Ag20+ cluster, both of which are solvated in water. In this approach, PIET was modeled as a coherent quantum process involving both vertical excitation and electron injection by our recently developed constrained real-time time-dependent density functional theory (C-RTTDDFT) (J. Phys. Chem. C 2011, 115, 18810), whereas the aqueous solvation structure for the (Ag−Ag20)+ complex was determined by the empirical flexible simple point charge (SPC/Fw) force field (J. Chem. Phys. 2006, 124, 024503). An electrostatic embedding scheme was chosen to accurately represent the mutual polarization between the QM subsystem (the (Ag−Ag20)+ complex) and the MM subsystem (water molecules) in a self-adaptive manner that turns out to be critical to the relative stability of the electron transfer diabatic states in addition to their electronic coupling strengths in both ground and excited states. It was found that photoinduced electron transfer through an indirect coherent route, which is mediated by a shortlived virtual excited state, can be substantially faster than the sequential two-step process, which is typically limited by the light absorption efficiency. Moreover, the unusually wide plateau of near-unity quantum yields that we found near the plasmon-like resonance wavelength of the (Ag−Ag20)+ complex implies the possibility of designing exceptionally efficient plasmon-enhanced photocatalytic systems with an easily tunable range of activation wavelength by varying their plasmonic architectures.

1. INTRODUCTION The photoinduced reduction of metallic ions in electrolytic solution has become an environmentally friendly technique1 to rapidly fabricate nanostructured films on solid substrates at a drastically reduced cost compared to the traditional electrochemical or vapor deposition.2 A key process in the photoreduction is the formation of charge-neutralized atomic clusters3 that can be used as colloidal seeds4 to precisely control the shape, size, and composition of the growing metallic nanoaggregates. When the colloidal seeds are plasmonexcitable, it is found that the growth pattern of the nanoaggregates is largely determined by their plasmonic response.5 For example, disk-shaped silver nanoparticles with different diameter-to-height ratios were readily synthesized by varying the incident wavelength from 368 to 514 nm when irradiating silver nitrate solution in the presence of citrate.6 In another study, both the gold shell thickness and the silver core radius of DNA-templated bimetallic nanoparticles7 can be regulated by the molar ratio between AgNO3 solution and NaAuCl4 solution, when this mixture is directly exposed to sunlight to trigger in situ photoreduction. When a laser beam with a diameter of 1 μm propagates through immobilized gold nanoparticles that are in contact with aqueous silver nitrate, the diameter of the silver deposits is found to be much larger than the size of the illuminated spot,8 demonstrating the laterally propagating nature of surface plasmon waves. Moreover, © XXXX American Chemical Society

scanning microscopy images clearly delineate larger silver nanoparticles with increasing radial distance from the laser beam center8 because of increased availability of silver ions in spite of attenuation of the propagating plasmon polaritons. Very interestingly, the geometries of the resultant metallic nanoaggregates are nearly always effectively optimized to give maximum overlap between their plasmon resonance peaks and the incident intensity profile,6 making the photoinduced selfassembly of metallic ions controllable simply by using a tunable light source with a wide range of wavelengths.9 Among the very few chemical elements whose conduction band electrons are capable of resonating with the visible incident light,10 silver and gold have been most widely used in plasmonic subwavelength optical devices11 such as surface enhanced Raman spectroscopy,12 plasmon-enhanced dyesensitized solar cells13 and long-ranged all-optical switching.14 The wide applications of silver and gold substrates can be largely ascribed to their extraordinary chemical stability in addition to their pronounced surface plasmon resonance, with the latter property resulting from smaller electromagnetic Special Issue: Nanostructured-Enhanced Photoenergy Conversion Received: October 14, 2012 Revised: December 4, 2012

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dx.doi.org/10.1021/jp310149y | J. Phys. Chem. C XXXX, XXX, XXX−XXX

The Journal of Physical Chemistry C

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different components and to show how they work for the (Ag20-Ag)+ model system. Recently, the so-called constrained real-time time-dependent density functional theory (C-RT-TDDFT) has been developed to evaluate the PIET rate as a function of incident wavelength and radiation intensity for systems similar to what we are considering.24 In the C-RT-TDDFT approach, PIET is treated as a coherent process that is initialized by vertical photoexcitation of the donor to its excited state followed by parallel electron injection to the acceptor. Under this coherence assumption, the transition dipole moment via the excited states can be conveniently computed by tracking the time evolution of the diabatic electronic wave function overlap between photoexcited (initial) and ground (final) states. Once the free energy profile along the ET path is determined, one can also obtain the two key parameters, the ET driving force, ΔG and the reorganization energy, λ, that govern the thermally averaged nuclear wavepacket overlap according to the semiclassical Marcus theory.25 The overall PIET rate is simply then given by the product of the squared magnitude of the excited state transition dipole moment, μ, the radiation intensity, I, and the density-weighted Franck−Condon factor (DWFC) as shown below24

energy dissipation for these elements compared to other metals.10 From the perspective of electronic structure studies, relativistic effects are so uniquely strong in gold even compared to lighter noble metals15,16 that scalar relativistic effects and spin−orbit coupling must be considered when investigating its optical properties.17 Therefore, silver clusters are much better choices as model systems for studying plasmonic effects. As shown in a density functional theory (DFT) study on Ag55− clusters, the calculated density of electronic states using nonrelativistic pseudopotentials agrees nearly perfectly with experimental high-resolution photoelectron spectra.18 In contrast, a scalar relativistic correction must be added to the exchange−correlation functional to reproduce the several closelying low-symmetry isomers for Ag55− clusters.18 Similar findings concerning the large difference due to relativity between gold and silver were also reported by Huang et al.19 and Aikens et al.20 Jensen, Aikens, and Schatz20 have demonstrated that small tetrahedral silver clusters, such as Ag20, Ag84, and Ag120, provide useful models of plasmonic excitation, as the optical spectra of these clusters are dominated by a single intense conduction electron transition, sometimes broadened by interaction with dark background states. By combining these models with broadening of the excited states based on the bulk metal plasmon response,21 it is possible to mimic the optical properties of larger silver particles, while at the same time retaining a purely quantum mechanical description of the ground and excited states that can be used to study a variety of spectroscopic and photophysical properties.22,23 In this paper we will use one of these clusters, Ag20, as a model system for understanding photoinduced electron transfer (PIET) processes involving Ag+ ions in solution (Figure 1), which is one of

kPIET(ω) =

μ 2 (ω) I(ω) 2π DWFC(ω) PIET ℏ cε0

(1)

where ω is the incident angular frequency, ℏ is the reduced Planck constant, c is the speed of light, and ε0 is the vacuum permittivity. If geometry relaxation upon photoexcitation is assumed to cause much smaller motion along the ET reaction path than the motion induced by the radiationless electron injection, the frequency-dependent DWFC(ω) for PIET is given by24 DWFC(ω) =

2 1 e−(−ΔG0 −ℏω + λ1) /4λ1kBT 4πλ1kBT

(2)

where ΔG0 is the free energy difference between the two ET diabatic ground states, λ1 is the reorganization free energy for the final ET diabatic state, kB is the Boltzmann constant, and T is the absolute temperature. In this expression, the maximum DWFC(ω) is reached when the incident photon carries an energy that can be exploited to barely overcome the energy barrier collectively imposed by the ET driving force and reorganization energy, i.e., ℏω = −ΔG0 + λ1. Furthermore, if PIET does not result in drastic temperature fluctuations, the width of the Gaussian-shaped DWFC (ω) is mainly governed by λ1, usually leading to a narrow peak less than 1 eV wide at room temperature, as the values of λ1 are typically within a few electronvolts even in highly polar solvents.26 Thus, an accurate assessment of ΔG0 and λ1 is critical to a reliable evaluation of kPIET(ω) because its variation over ω is significant when ℏω is far from the optimal incident energy of −ΔG0 + λ1, which maximizes nuclear wavepacket overlap. The theory described in eq 1 was previously used to describe PIET based on a continuum solvation model of the solvent.24 Though computationally convenient, such continuum models are subject to potentially significant errors. Here we show how the theory can be refined using an explicit solvation model based on a QM/MM partitioning between system and solvent. Since its introduction by Warshel and Levitt in 1976 to study the stability of the intermediate carbonium ion in an enzymatic reaction,27 QM/MM has become a powerful tool to investigate

Figure 1. Initial (a) and final (b) states of the (Ag20−Ag)+ complex for the photoreduction of the Ag20+ cluster. The distance between the Ag atom and the closest apex atom of the tetrahedral Ag20 cluster is fixed at 6.0 Å, i.e., R = 6 Å. The complex is fully solvated in water, and the density of the transferring electron in each state is illustrated by the blue shaded region. For visual clarity, only a small number of water molecules are explicitly represented in CPK style and others are intentionally blurred.

the key processes involved in the plasmon-driven growth of nanoparticles mentioned earlier.5 Such processes are difficult to describe theoretically, as photoexcitation and electron transfer compete with rapid damping (a few tens of femtoseconds), making it impossible to use conventional theories of electron transfer for these processes. At the same time, many aspects of electron transfer theory, such as the incorporation of solvent effects on the donor and acceptor states, need to be included in the theoretical description. Our goal in this paper is to develop new theoretical methods that enable us to combine these B



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2. COMPUTATIONAL METHODOLOGY AND SIMULATION DETAILS 2.1. (Ag20−Ag)+ Model for Photoreduction of the Ag20+ Cluster. As shown in Figure 1, our model system consists of a tetrahedral Ag20 cluster and an isolated Ag atom with a total charge of +1. The vector connecting the Ag atom and its closest apex atom in the Ag20 cluster is normal to the farthest tetrahedral surface, and the length of the vector, R, is fixed at 6 Å to ensure reasonable ET efficiency (based on the earlier study24). The entire (Ag20−Ag)+ complex that is treated by quantum mechanics is fully solvated by water molecules, which are modeled at the molecular mechanics level using the flexible simple point charge (SPC/Fw) force field.38 Earlier work has demonstrated excellent agreement with experiment using this force field for a variety of water properties, including the static dielectric function.38 Calculations, which we present later, will show that in the lowest energy state, the Ag20 cluster is ionized and the Ag atom is neutral (Figure 1a). In the photoreduction, an electron is transferred from the Ag atom to the Ag20 cluster, resulting in a drastically localized hole as shown in Figure 1b. In our simulations, the geometry of the (Ag20−Ag)+ complex is kept fixed under the assumption that PIET is fully driven by low-frequency solvent reorientation (i.e., neglecting high-frequency intracluster vibrations). The side length of our cubic simulation box is 30.0 Å and the number of water molecules is 872, yielding a mass density of 1.10 g/cm3. 2.2. QM/MM Simulations for Evaluating ET Driving Force and Reorganization Energy. The thermodynamic integration method39 was employed to evaluate the energetics of ET. To this end, a linearly interpolated Hamiltonian is defined

electrostatic couplings and van der Waals interactions between a chemically active region and its surrounding medium in solvent-driven chemical and biomolecular reactions,28 many of which involve interspecies ET.29 The elegance of QM/MM primarily lies in its deliberate balance between physical accuracy and numerical efficiency attained by treating the reacting species with delocalized wave functions while modeling the solvent molecules by localized point charges,30 thus allowing the occurrence of ET in the QM subsystem under the influence of nuclear reorientation and electronic polarization in the MM region. The QM/MM methodology not only notably extends the time scale obtainable using ab initio molecular dynamics simulations by several orders of magnitude but also can greatly reduce the statistical sampling error down to a level comparable with the one induced by thermal fluctuations, i.e., kBT, if advanced free energy sampling techniques such as umbrella sampling, metadynamics or thermodynamic integration are applied.31 For example, in an earlier study on the direct decarboxylation of N-methylorotic acid, the error bar of the activation free energy was reduced to 1.6 kcal/mol after a short 20 ps QM/MM metadynamics simulation using the interatomic distance between the two dissociating carbon atoms as the collective variable.32 Recently, a remarkably small statistical uncertainty of 0.6 kcal/mol was reported for the binding free energy change when the hydroxyl group of the Cox-2 inhibitor was substituted by a methyl group, owing to the significantly enhanced sampling efficiency associated with incorporating the replica exchange technique into the free energy perturbation method.33 Inspired by these encouraging results, we carried out simulations combining thermodynamic integration and QM/ MM in the present study (see section 3) to determine free energy profiles for ET between metallic entities, allowing for solvation in aqueous solutions upon ionization. Note, however, that the electronic coupling strength inside a solute complex is also very sensitive to instantaneous solvation structures noted for two adenine base pairs in stilbene-linked DNA hairpins.34 Recent evidence also includes the intramonomer and intermonomer couplings of the light-harvesting Fenna− Matthews−Olson (FMO) trimer,35 which undergoes a rather large conformational change at room temperature away from its crystal structure. Thus, nonequilibrium solvation effects should be taken into account in the calculation of the electronic coupling strength. An intuitive solution within the framework of constrained DFT (C-DFT)36 is the C-DFT/MM approach,37 wherein the solute in a quantum diabatic state is solvated by the classically represented solvent molecules at the empirical force field level. As will be discussed later, the CDFT/MM approach is also employed to evaluate the electronic coupling strengths in both ground and excited states for our model systems. The remainder of the paper is organized as follows. In section 2, the computational methodology and simulation details are introduced, in addition to the description of our model system. In section 3, the ground state and excited state ET within the (Ag−Ag20)+ complex are investigated by using the Marcus formula and C-RT-TDDFT method. Finally, some possible applications of C-RT-TDDFT to plasmon-enhanced ET, in combination with other theoretical methods, are discussed in section 4.

H(η) = (1 − η)HI + ηHF

(3)

where HI and HF are the Hamiltonians of the initial and final ET states, respectively, and η is a switching parameter. Besides the standard DFT Hamiltonian that includes kinetic energy, Coulomb interaction and exchange−correlation terms, an additional position-dependent Hartree potential was added to HI (or HF) to achieve the desired charge distributions through a two-cycle optimization on the system’s wave function and its imposed constraining potential as introduced in a previous CDFT study.40 In this sense, HI and HF only differ in the constraining Hartree potential to reflect the exchange of holecarrier entities upon photoreduction of the Ag20+ cation. A total of 11 thermodynamic integration windows were sampled along the reaction path corresponding to 11 equally spaced values of η, which range from 0 to 1 with a spacing of 0.1. In each window, a hybrid C-DFT/MM molecular dynamics calculation was performed to equilibrate the system at 300 K with a Nose−Hoover thermostat41 for 1 ps followed by a 3 ps production run, during which the values of ∂H(η)/∂η were collected for every time step of 1.0 fs to yield an ensemble average of ⟨∂H(η)/∂η⟩. Finally, ΔG0 is given by a discrete summation 1

ΔG0 =

∑ η=0

∂H(η) ∂η

dη η

(4)

and the final state’s reorganization energy, λ1, is obtained from λ1 = ⟨HF − HI⟩η= 0 + ΔG0 C

(5)



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Similarly, the reorganization energy for the initial state, λ0, is expressed as λ 0 = ⟨HI − HF⟩η= 1 − ΔG0

Note that an empirical damping factor, Γ, of 0.1 eV is introduced in eq 11 to reflect the effects of vibronic coupling and quantum dephasing. A procedure for evaluating ΔHIF(t) using the diabatic wave function overlap was described by Chen et al.24 In our present study, a short electric pulse with a duration of 0.0121 fs and a field strength of 1.47 × 106 V/m was chosen for Eext, which is polarized along the vector connecting the isolated Ag atom and its nearest apex atom in the Ag20 cluster. A RTTDDFT trajectory was propagated for a total of 4000 steps with a step size of 0.0121 fs. The trajectory is long enough to converge the optical and ET properties within the wavelength range of interest. With ΔG0, λ1, and μPIET(ω) determined, we are able to calculate kPIET according to eq 1. In our C-RTTDDFT simulations, all atoms of the MM subsystem were frozen during this propagation due to the short time (less than 50 fs) needed to converge the evaluation of eq 11.

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2.3. C-DFT/MM Calculations for Ground State ET. In our C-DFT/MM simulations, performed by the CP2K molecular simulation package,42 the electrostatic coupling scheme was selected to describe interactions between the QM and MM subsystems.43 Within the nonperiodic QM region, the DFT calculations were carried out with the Goedecker−Teter−Hutter (GTH) dual-space Gaussian pseudopotential,44 the Perdew−Burke−Ernzerhof (PBE) exchange−correlation functional45 and the polarized valencedouble (PVDZ) basis set.46 In the MM calculations, the smooth particle mesh Ewald (SPME) method47 was used to treat the long-range electrostatic interactions with a relative cutoff of 1.0 × 10−6, and the VDW potentials were truncated at a radial distance of 11.4 Å. A charge distribution function, Qc, is defined to facilitate the definition of the initial and final diabatic states: Q c = Q Ag − Q Ag 20

3. RESULTS 3.1. Free Energy Profile of the Ground State Electron Transfer Reaction. To assess the energetics of thermally activated electron transfer between the two diabatic states, namely (Ag20+−Ag) and (Ag20−Ag+), eleven intermediate states were constructed by linearly superimposing the two corresponding diabatic Hamiltonians with different relative weights. A plot of the averaged Hamiltonian derivative with respect to the reduced reaction coordinate, ⟨∂H(η)/∂η⟩, is shown in Figure 2. This shows an essentially monotonically

(7)

where QAg20 and QAg are the atomic charges of the Ag20 cluster and Ag atom, respectively. The atomic charges were determined by the real-space-based Becke weighted population analysis,48 which has yielded satisfactory results in many previous C-DFT studies.36,49 After the system’s energies in the two ET states are minimized under the constraint of the defined charge distributions by C-DFT, the diabatic coupling strength, HIF, is given by the off-diagonal term of the following matrix: ⎛ E I E IF ⎞ ⎟C C+⎜ ⎝ E FI E F ⎠

(8)

where C is the diabatic transformation matrix to impose strict orthogonality, † denotes the Hermitian transpose, EI and EF are the energies of the ET states, and EIF and EFI are their coupling energies. For more detailed definitions of C, EI, EF, EIF, and EFI, please refer to refs 40 and 24. In combination with ΔG0 and λ0, the ground state ET rate, kg, for the thermally activated oxidation of the Ag20 cluster by the Ag+ cation can be readily calculated by the Marcus formula:25 2 2π 1 kg = e−(ΔG0 + λ0) /4λ0kBT |HIF|2 ℏ 4πλ 0kBT (9)

Figure 2. Profile of the derivative of the mixed Hamiltonian, H(η), with respect to the coupling parameter, η.

decreasing function that goes through zero close to η = 0.5. As a result, the free energy profile, i.e., ΔG as a function of η, which is plotted in Figure 3 (note that ΔG(η = 0) has been

2.4. C-RT-TDDFT/MM Simulations for PIET. The evaluation of the excited state electronic coupling strength, HIF(ω), follows our recently developed C-RT-TDDFT procedure:24 HIF(ω) =

μPIET 2 (ω) I(ω)

(10)

where the PIET transition dipole moment, μPIET(ω), can be obtained by tracking the time evolution of the diabatic coupling strength perturbation, ΔHIF(t) = HIF(t) − HIF(t=0), followed by a Fourier transform and a normalization with respect to the time-varying driving electric field, Eext(t): μPIET (ω) =

∫ dt ΔHIF(t )e(iω−Γ)t ∫ dt Eext(t )

Figure 3. Free energy profile, ΔG, along the reduced reaction coordinate, η.

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transitions,20 the weak intensity of which is nearly invariant over the whole ultraviolet wavelength range. Given the similarity of σabs(ω) for the Ag20 cluster and the (Ag20+−Ag) complex, it is apparent that as expected the tetrahedral silver cluster rather than the isolated silver atom is the excitable entity. 3.3. Coherent Photoinduced Electron Transfer CrossSection. Under light irradiation, the two diabatic ground states can couple indirectly with the aid of an intermediate excited state, and their coupling strength is a function of the incident wavelength, which determines the choice of the intermediate state. Because the coupling strength is also linearly proportional to the radiation intensity, it is more convenient to use the PIET dipole moment, μPIET(ω), for quantification. This is given by

shifted to zero), is a roughly symmetric function with a broad peak near η = 0.5. The peak corresponds to a saddle point on the free energy surface where the transferring electron (or hole) is nearly equally shared by the Ag20 cluster and the Ag atom. Figure 3 indicates that an activation energy of 0.77 eV needs to be surmounted at η = 0.5 when an electron is adiabatically transferred from Ag20 to Ag+. The free energy difference between the two diabatic states, ΔG0, is only 0.23 eV, and the difference favors (Ag−Ag20+) compared to (Ag+−Ag20). Presumably this difference shows that delocalizing the hole over the larger cluster is energetically and entropically preferred and is not overcome by the presumably larger solvation energy associated with the smaller ion. In the context of diabatic processes, the solvent reorganization energy has to be taken into account for the evaluation of the effective activation energy as implied by the Marcus formula.25 It turns out that the reorganization free energies for (Ag−Ag20+) and (Ag+−Ag20) are λ0 = 3.62 eV and λ1 = 3.38 eV, respectively, as calculated by using eqs 5 and 6. To determine the ground state electron transfer rate, an instantaneous atomic configuration was randomly chosen from the 3 ps production trajectory of the C-DFT/MM simulations for each of the two diabatic states, i.e., η = 0 and η = 1. After the diabatic transformation,36 the ground state electronic coupling strength, ΔHIF, was determined to be 2.00 × 10−3 eV. According to eq 9, the electron transfer rate for the ground state, kg, is determined to be 1.46 × 10−3 s−1, which is quite slow due largely to the exceptionally high reorganization energy compared to the electron transfer driving force. 3.2. Absorption Cross-Section of the (Ag20+−Ag) Complex. Before discussing excited state electron transfer, we want to examine the complex’s light absorption efficiency, which is best expressed using the absorption cross-section, σabs(ω). The detailed procedure for evaluating σabs(ω) by realtime time-dependent density functional theory was described by Chen et al.22 As shown in Figure 4, the (Ag20+−Ag) complex

⟨Ψ|I μ ⃗ |ΨM(ω)⟩⟨ΨM(ω)|Ĥ 0|ΨF⟩ (12) Γ ̂ where μ⃗ is the dipole operator, H0 is the Hamiltonian operator, Γ is the empirical damping factor, and Ψ1, ΨM, and ΨF are the wave functions of the initial, intermediate, and final states, respectively. Note that the intermediate state in this expression is the superposition of states that is excited by the initial pulse. Numerically, μPIET(ω) can be evaluated by tracking the time evolution of the diabatic wave function overlap between the initial and final states without explicitly constructing the intermediate states.24 As shown in Figure 5, μPIET2(ω) exhibits a very sharp peak at 3.27 eV with a width of ∼0.5 eV and a maximum value of ∼1.5 μPIET (ω) =

Figure 5. Squared magnitude of the PIET dipole moment as a function of the incident photon energy.

× 104 Debye2, whereas the intermediate excited states activated at other wavelengths are much less efficient, as demonstrated by their near-zero μPIET2(ω). Because both the location and width of the μPIET2(ω) peak are in line with the plasmonic peak of σabs(ω), it is apparent that plasmonic excitation is the primary driving force for excited state electron transfer. Now let us consider the PIET cross-section, σPIET(ω), which is defined by

Figure 4. Absorption cross-section for the (Ag20+−Ag) complex.

exhibits a narrow plasmon-like absorption peak at 3.22 eV, which can be ascribed to sp → sp intraband transitions.20 Compared to the experimental data on the gas phase Ag20 cluster,50 our calculated plasmon peak is red-shifted by ∼0.5 eV primarily due to the well-known incorrect asymptotic behavior of the commonly used DFT functionals.51 Moreover, the explicit treatment of the highly polarizable water molecules in our simulation may also cause a reduction in excited state energies,52 leading to a further red shift in the plasmon peak. The broad continuous absorption band on the blue side of the plasmon peak in Figure 4 stems from the d → sp interband

σPIET(ω) =

ℏωke(ω) I(ω)

(13)

This provides a direct measure of PIET efficiency which may be compared with the absorption cross-section, and it also determines the PIET transition rate for a given intensity.24 As illustrated in Figure 6, the peak in σPIET(ω), which is centered at 3.28 eV, is slightly narrower and sharper than that in σabs(ω). More important, the maximum in σPIET(ω) reaches ∼1700 Å2, which is over 2 orders of magnitude higher than the peak value E



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As shown in Figure 8, the electron injection rate is found to be very rapid for wavelengths corresponding to the plasmon

Figure 6. Photoinduced electron transfer cross-section, σPIET(ω).

of σabs(ω). The abnormally large σPIET(ω) at the plasmon wavelength suggests an ultraefficient electron transfer route that is no longer limited by the absorption efficiency of the initial state. This value, which is close to the maximum cross-section possible for the Ag20 cluster, can only arise if there is strong coupling between the plasmonically excited state and the final diabatic ground state. 3.4. Quantum Yield and Electron Injection Rate. The efficiency of turning incident light energy into charge transfer can be quantified by the dimensionless quantum yield (QY): QY% =

σPIET(ω) × 100% σPIET(ω) + σabs(ω)

Figure 8. Electron injection rate, ke(ω), as a function of incident photon energy.

maximum, reaching a maximum value of ∼180 fs−1 at an incident energy of 3.31 eV. This extremely rapid rate is consistent with the other properties we have found for the PIET process, and it indicates that coupling between the plasmonic intermediate and the final state is very strong. Indeed, the rate is so fast that the plasmonic intermediate is essentially a virtual orbital similar to what exists in many multiphoton phenomena, such as two-photon absorption26 and Raman spectroscopy.53

(14)

As shown in Figure 7, the quantum yield is very close to unity for wavelengths close to the plasmon maximum, and it decays

4. DISCUSSION Plasmonic nanoparticles have become widely used in solar cell applications as supporting substrates13,54,55 mainly because of their unique capability to efficiently concentrate incident power flux into photochemically active components, such as organic dye molecules56 or silicon-based semiconductors,57 whose stimulated electronic transitions initialize the key charge separation step to generate desired photocurrent. The prevailing mechanism of plasmonic enhancement is the locally amplified electromagnetic field through near-field resonance and far-field scattering as is found in nearly all plasmon-related optical properties.11 Another possible enhancement mechanism is via resonant energy transfer wherein plasmonic substrates couple with acceptor chromophores through nonradiative dipole−dipole interactions,58 which are significantly reinforced by the large donor−acceptor spectral overlap.59 Moreover, a very recent study60 used modeling to show that the hole− electron generation efficiency of a Cu2O shell is increased ∼10 000-fold by the presence of Au core even when a thick insulator layer of SiO2 is placed between them to exclude other electron transfer mechanisms. The present results are different from our earlier studies using the same PIET formalism but with a continuum model for the solvent.24 In particular, the earlier result involved smaller reorganization energies, and therefore a much larger ground state electron transfer rate. However, the excited state rate constant was much smaller than in the present work, reflecting much weaker electronic coupling. As a result, the electronic branching ratio was much smaller, with a maximum value on the order of a 5% when the Ag20−Ag distance is 6 Å. These results indicate the strong sensitivity of the results to choice of solvation model, presumably reflecting errors in the continuum approximation that we used previously. Thus the present work represents an important step forward in the

Figure 7. Dependence of PIET quantum yield on incident photon energy.

particularly rapidly on the red side. For example, when ℏω = 2.5 eV, the quantum yield is less than 10% and it becomes nearly negligible when ℏω < 2.0 eV. On the blue side, the decay is substantially slower, allowing for a quantum yield of ∼50% for ℏω = 4.3 eV corresponding to an incident light energy over 1.0 eV above the plasmonic peak. The broad plateau in the quantum yield with a width of ∼1.0 eV suggests that quantum yield can be further optimized for solar applications by varying the nanoarchitecture of the silver clusters to maximize overlap between the solar power spectrum and PIET cross-section profile. Another interesting property of PIET is the electron injection rate, ke(ω), for the transferred electron: ke(ω) =

2π σPIET(ω) Γ ℏ σabs(ω)

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development of methods for describing PIET in a plasmonic system. In addition to methodology development, the present study conceptually proposes a viable alternative path for electron transfer through direct electronic wave function overlap between excited initial and ground final diabatic states.24 In this mechanism, the plasmonic entities uniquely play the dual role of light harvester and electron donor/acceptor, whereas the plasmonic excited states serve as transient intermediates to coherently bridge the initial and final states. As shown by our results, the coherent indirect path can be more efficient than the conventional two-step sequential route because of extremely rapid electron injection upon photoexcitation. In further work, we will consider incorporating the C-DFT/MM method into our previously developed time-domain QM/ED approach22 to develop a three-tier QM/MM/ED theorem for multiscale and multiresolution modeling of photoinduced electron transfer in a more realistic and therefore a more complicated environment.

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. *Fax: 847-491-7713 Phone: 847-491-5657 Email: schatz@ chem.northwestern.edu. Present Address †

Department of Chemistry, The George Washington University, 725 21st Street, NW, Washington, DC 20052, Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The research was supported by U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Award Number DE-SC0004752. The computational resources utilized in this research were provided by Shanghai Supercomputer Center.

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