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Orbital Renormalization Effects on the Coupling Between Molecular Excitations and Plasmons Justin E. Moore, and Lasse Jensen J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.5b10479 • Publication Date (Web): 24 Feb 2016 Downloaded from http://pubs.acs.org on March 1, 2016

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The Journal of Physical Chemistry

Orbital Renormalization Effects on the Coupling between Molecular Excitations and Plasmons Justin E Moore and Lasse Jensen∗ The Pennsylvania State University, Department of Chemistry, 104 Chemistry Building, University Park, PA 16802, USA E-mail: [email protected]

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Abstract Accurately describing the electronic structure of molecules on metal nanostructures is key to modeling their surface-enhanced properties. Particularly difficult is the modeling of the coupling between molecular excited states and plasmons. Here we present a computational efficient approach to study the renormalization effects on the molecular electronic structure and its optical properties due to the interactions with the metal surface. Accurate simulations of the renormalization effects are achieved by employing a hybrid atomistic electrodynamics and time-dependent density functional model. The coupling between the molecular absorption and the plasmon excitation depends strongly on the spectral overlap. Here we show that the renormalization effect for the benzene-tetracyanoethylene donor-acceptor complex interacting with a metal nanoparticle causes a 0.6 eV shift in the absorption band. Furthermore, we show that the coupling between the molecular absorption and the plasmon excitation is caused by interference between the molecular absorption, the image field of the metal nanoparticle, and the near field due to the plasmon excitation. The results presented here illustrate the importance of using first-principles simulations to understand in detail the coupling between molecular absorption and plasmon excitation.

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Introduction The interactions between molecules and plasmonic metal nanostructures give rise to many unique chemical processes such as plasmon mediated photocatalysis, 1–4 plasmon induced charge separation, 5–7 and surface-enhanced spectroscopy. 8,9 Furthermore, the strong coupling between molecular excited states and plasmons can lead to unique nanophotonic materials that are characterized by new hybridized states and can be controlled by modulating the spectral overlap between the plasmonic and molecular resonances. 10–13 This coupling is also exploited to enhance molecular fluorescence 14,15 and resonance effects in surface-enhanced resonance Raman scattering. 16 While an understanding of the mechanisms behind these events is emerging, the detailed microscopic description of the coupling between molecular excited states and plasmons remains sparse due to the complexity of modeling the plasmonic nanostructure-molecule interface from first-principles. The design and modeling of the plasmon-excited state coupling often assumes that the molecular absorption remains unchanged when bound to the surface of the metal nanoparticle. 17–19 While this often is a good approximation, the formation of dye aggregates, the introduction of new charge-transfer states, and shifts in the excited state energy levels all have a significant influence on the coupling strength. Thus, it becomes important to understand in detail the electronic structure of molecules adsorbed on metal nanoparticles to correctly interpret and design these hybrid optical materials. In particular, correctly describing the energy alignment between frontier electronic states in the molecule and the Fermi energy of the substrate becomes essential as this dictates the optical properties of the adsorbed molecule and its coupling strength with the plasmon. A variety of hybrid methods that combine a quantum mechanical description of the molecule and a classical description of the metal nanoparticle have been developed. 20,21 These models are promising, but mostly rely on describing the electronic structure of the molecule using density functional theory (DFT). However, DFT-based approaches

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miss non-local correlation effects and contain self-interaction errors when using traditional exchange-correlation (XC) functionals. 22 This leads to an incorrect alignment of the molecular energy levels relative to the metal’s Fermi level and a lack of renormalization of the fundamental energy (quasi-particle) gap due to image charge effects. Many-body perturbation theory, for example the GW approximation, can correctly describe this but remains computationally very expensive. 23,24 Finally, if renormalization of the optical transitions is of interest, as the case here, then the Bethe-Salpeter equation (BSE) needs to be solved, which adds significantly more computational expense. 23 Hence it is desirable to have cheap and efficient ways of describing the renormalization of the molecular energy levels near metal surfaces. Recent work has shown that the quasi-particle (QP) gap, defined as the difference between the ionization potential and electron affinity, of molecules can be calculated with high accuracy using long-range corrected functionals with optimum attenuation parameters. 25,26 The optimum attenuation parameters are obtained from first-principles by requiring that the energy of the highest-occupied molecular orbital is equal to the negative of the ionization potential. This is simply Koopmans’ theorem, which is known to be fulfilled for the exact functional. 27 Recently, Egger et. al. have presented a full DFT approach to describing molecules on metal surfaces using LC functionals and a self-energy correction model to achieve results comparable to G0 W0 at a significantly reduced computational cost. 28 However, this methods invokes periodic boundary conditions and thus cannot easily be used for studying the plasmonic nanostructures of relevance here. In this article, we demonstrate that accurate simulations of the renormalization effects on the coupling between molecular excited states and plasmons can be achieved by employing a hybrid atomistic electrodynamics model combined with a time-dependent DFT description of the molecule. We show that the discrete interaction model/quantum mechanical (DIM/QM) method 21,29 combined with a LC functional range tuning scheme can accurately describe the QP gap of molecules absorbed on metal surfaces. This method is not limited to planar surfaces and thus enables the simulation of molecules

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adsorbed on plasmonic metal nanoparticles. To illustrate this, we examine the renormalization effect on the optical excitations of a donor-acceptor molecular complex adsorbed on a quasi-spherical nanoparticle. We find a large reduction in the optical excitation due to the renormalization effect, which is in good agreement with manybody results but at a fraction of the computational cost. Finally, we demonstrate that including this change in the optical excitation is essential when modeling the coupling between excited-states and plasmons.

Computational Details DIM/QM calculations were done using a local implementation within the NWChem 6.1.1 program package. 30 A full description of DIM/QM can be found in Refs. 21,29,31 In DIM/QM, the interactions between the molecules and the nanoparticle is described by an embedding operator given by Vˆ DIM (rj ) = −

X µind m,α rmj,α m

|rmj |3

(1)

where m denotes a DIM atom, j denotes a QM electron, and rmj is the vector between them. The induced dipoles are obtained self-consistently by solving a set of linear response equations based on assigning an atomic polarizability to each atom in the nanoparticle as α=

6 3 ǫ(ω) − ǫ0 R π ǫ(ω) + 2ǫ0

(2)

where R is the radius of the atom, and ǫ(ω) is the metal’s frequency-dependent dielectric function. We use a value of R = 1.4445 ˚ A for both silver and gold. For the dielectric function, we used a Drude model with the parameters for silver as ωp = 9.50 eV, γ = 0.0987 eV, and ε∞ = 5.0. These terms were chosen to match the visible region of the experimental dielectric function, giving a plasmon resonance at 337 nm. The parameters for gold were ωp = 9.000 eV, γ = 0.067 eV, and ε∞ = 11.84, taken from Ref. 32, but with ε∞ chosen to give a plasmon resonance at 500 nm. All DFT calculations

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were performed using the long-range corrected Perdew, Burke, and Ernzerhof hybrid (LC-ωPBEh) functional with a correlation consistent triple-ζ with polarization (ccpVTZ) basis set, both from NWChem’s standard library. Geometry for the benzeneTCNE complex was taken from Ref. 33; all other QM geometries were optimized with the aforementioned parameters in gas phase. Geometries were frozen when placed on the metal surfaces. This should be an reasonable approximation as the molecules is only weakly physisorbed on the surface and thus only slight geometric changes are expected. The optical absorption across section at a frequency ω was obtained as

σabs =

4πω Im{¯ αN P (ω)} c

(3)

where α ¯ N P (ω) is either the isotropic polarizability of the system or the specific polarizability component of interest

Results and Discussion Renormalization of the quasi-particle gap: Benzene As a molecule is brought near a metal surface, the energy levels of the molecule will renormalize due to the polarization response in the metal. This renormalization is a result of non-local correlation, a feature missing in traditional DFT functionals, but has been shown to follow a classical image-like potential. 34 Accounting for this renormalization is essential in theoretical calculations of conductance, as the incorrect energy level alignment otherwise leads to a large overestimation of the molecular conductance. 35,36 Recent work has also demonstrated the same energy alignment determines the off-resonance chemical effect in surface enhanced Raman scattering (SERS). 37–39 To study these effects using the DIM/QM model, we first consider benzene interacting with a Ag surface. The metal surface is described as a 6 layered slab model with 1600 atoms in each layer (40 x 40 square), and the benzene molecule is oriented such that its molecular plane is parallel to the surface. Since benzene only interacts

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Energy (eV)

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11.0

11.0

10.5

10.5

10.0

10.0

9.5

9.5

9.0

9.0

QP Gap DIM/QM HOMO-LUMO DIM/QM[γ opt ] HOMO-LUMO Free QP - G0 W0 Jellium QP - G0 W0

8.5 8.0

8.5 8.0

5

10

15

20 ∞

˚ Distance A Figure 1: Calculated HOMO-LUMO and QP gaps for benzene at varying distance from a silver surface. The G0 W0 data is taken from Ref. 24.

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weakly with the substrate, we will ignore geometric changes that occur upon binding. In Figure 1, we plot the QP gap as a function of the distance from the surface calculated using DIM/QM. The QP gap can be calculated using the ∆SCF method as the difference in ground state energies between the N and N − 1 electron systems, and the N and N + 1 electron systems, respectively. 40 The QP gap of the free benzene molecule is found to be 10.9 eV, which is in agreement with experimental results 41,42 and results obtained using more computationally expensive many-body methods. 24 At 4 ˚ A above the surface, we find that the QP gap has been reduced by 2.1 eV, again in good agreement with results obtained using the G0 W0 method, where a reduction of the gap by 2.3 eV was found. 24 In Figure 1 we also plot the HOMO-LUMO gap of the benzene molecule obtained using the DIM/QM and the LC-ωPBEh functional 43 with a constant range-separation parameter of γ = 0.2. This provides a good description of the QP gap for the free benzene molecule, but fails to capture the gap renormalization due to the surface polarization. Within DIM/QM, this polarization is described in terms of a set of image dipoles induced in the metal surface by the molecule. To demonstrate the extent of the image dipole effect, we calculate the magnitude of the electric field at the center of the benzene molecule due to the DIM dipoles. This extent of the image dipoles is shown in Figure 2 for the neutral benzene and cationic benzene at 4 ˚ A from the metal surface. Using a cutoff value of |E| ≥ 10−6 a.u., the image dipole effect for neutral benzene on silver is limited to DIM atoms up to 16 ˚ A away, while the effect for the cationic system extends A. The different extents of the image dipoles for the two systems reflect the out to 30 ˚ differential stabilization of the cationic molecule relative to the neutral molecule due to the surface polarization. The QP gap obtained using the ∆-SCF method combined with DIM/QM naturally incorporates this differential surface polarization effect, which leads to the reduction of the gap as the molecule gets closer to the surface. However, simply relating the QP gap with the HOMO-LUMO gap of the neutral molecule lacks this differential polarization, and hence does not correctly capture the renormalization effect.

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(a) Neutral Benzene

(b) Cationic Benzene

Figure 2: Image dipoles calculated using DIM/QM for benzene on a silver surface. Color indicates the magnitude of the electric field (|E|) at the center of the benzene molecule due to that dipole, from high (red) to low (dark blue) with a cutoff of |E| ≥ 10−6 a.u.. The black circle has a radius equal to the distance to the furthest significant dipole.

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This renormalization effect can be accounted for in a DIM/QM calculation by combining it with range-tuning of the attenuation parameter, γ. While several different schemes exist for range-tuning, 33 we chose, for simplicity, to enforce Koopmans’ theorem, such that: OP T

−εγHOMO = IP = Egs (N − 1; γ opt ) − Egs (N ; γ opt ),

(4)

opt

where εγHOMO is the energy of the HOMO for the N electron system and Egs is the total ground-state energy of the N or N −1 electron system calculated using DIM/QM. The optimization is then performed independently for each separation distance of the benzene molecule from the surface. In the following, we will denote the DIM/QM calculations employing the optimized attenuation parameter by DIM/QM[γ opt ]. In Figure 1, we show that the HOMO-LUMO gap obtained using DIM/QM[γ opt ] provides an excellent description of the QP gap and shows the expected renormalization due to the surface polarization. We find that DIM/QM[γ opt ] predicts QP gaps that are within 0.1 eV of that obtained using the ∆SCF method (see supporting information for a quantitative comparison). The range-tuning within DIM/QM leads to a large reduction of the attenuation parameter for the free molecules from γ f ree = 0.20 to γ surf = 0.098 at the closest separation. Effectively, the range-running introduces a screening of the long-range HF exchange due to the surface polarization. This screening represents the missing nonlocal correlation in the XC-functional and thus provides the correct description of the renormalization of the QP gap. As illustrated in Figure 3(a), the screening reduces the orbital energy of the LUMO and increases the orbital energy of the HOMO which when combined, leads to the observed reduction in the HOMO-LUMO gap as a function of distance from the surface. Detailed simulations using the G0 W0 23,34 model have shown that the renormaliza-

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11.0

0

10.5 HOMO-LUMO gap (eV)

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Energy (eV)

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−2 HOMO LUMO

−4 −6 −8

10.0 9.5 9.0 8.5

HOMO-LUMO z0 = 0.9 Free HOMO-LUMO

−10 4

6

8

10

12 14 Distance A˚

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18

8.0 4

20

6

8

(a)

10

12 14 Distance A˚

16

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(b)

Figure 3: a) HOMO and LUMO energies using DIM/QM[γ opt ] for benzene at varying distances from a silver surface The dashed lines are the gas-phase values. b) Classical image charge model. The value of z0 = 0.9 ˚ A was taken from Ref. 28. tion effect can be described using a simple classical image charge model as:

V =

−1 , 4(z − z0 )

(5)

where the charge distribution of the HOMO and LUMO is approximated by a point charge sitting at a distance z from the metal surface with an image plane located at z0 . In Figure 3(b), we compare the gap renormalization predicted by the image charge model against the HOMO-LUMO gap obtained using DIM/QM[γ opt ] for benzene on the Ag surface as a function of distance. The image charge model depends strongly on the actual value for the location of the image plane. Here, we chose an image plane of z0 = 0.9 ˚ A, similar to that used for gold in the recent work by Egger et. al.. 28 Overall, we see that there is qualitatively good agreement between the two models, although the DIM/QM[γ opt ] results decay slightly faster away from the surface than the image charge model. One advantage of DIM/QM[γ opt ] is that there is no ambiguity in deciding the location of the image plane. In DIM/QM[γ opt ] the image field interactions are determined by the radii of the DIM atoms, however, these are chosen to describe the polarizability of the metal nanoparticle and thus not free variables. The process of tuning γ is system specific since γ has been shown to depend on the

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electron density. 44 We observe that for the benzene-silver system, the magnitude of which the HOMO level shift after γ-tuning is similar to that of the polarization energy of the cationic molecule. This implies that γ-tuning is simply incorporating the DIM polarization energy into the HOMO energy such that the correct renormalization of the QP gap is found. As the polarization energy is not expected to depend strongly on the actual γ value, it should be possible to transfer the optimum γ value found for the gas phase to that on the surface once the polarization energy is known. To illustrate this we use the fact that the HOMO energy shows an exponential dependence on the value of γ εγHOM O = A + B ∗ e(−Cγ) ,

(6)

where A is the HOMO energy for γ → ∞ and (A + B) is the HOMO energy for γ → 0. These three parameters can easily be obtained by fitting to the range-tuning data obtained for the gas phase molecule. Once the polarization energy of the N and N-1 electron systems has been calculated, the new optimum value can simply be found by solving for γ opt using Eq. 6 and the εγHOM O minus the polarization energy. A detailed description of this can be found in the supporting information. For benzene on the surface, we find approximate γ values that are within ∼2% of the actual values for all distances. Similar accuracy was also obtained for a water molecule interacting with the Ag surface which shows a stronger polarization due to its dipole moment. This clearly illustrates that the main effect of range tuning within DIM/QM is to incorporate the differential polarization energy into the HOMO energy such that the correct renormalization of the QP gap is achieved.

Renormalization of the optical gap: Image field effects Recent work has demonstrated that the non-local correlation also reduces the optical gap of a benzene-tetracyanoethylene(TCNE) donor-acceptor molecule (see Figure 4(a)) on a metal surface. 23 Here we will show that the DIM/QM[γ opt ] can describe both the renormalization of the QP gap and the optical gap. As mentioned above, the DIM/QM

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(a) benzene-TCNE complex

(b) benzene-TCNE on silver NP

Figure 4: Cartoon of the structure of (a) the benzene-TCNE complex, and (b) the benzeneTCNE complex on the Ag10179 nanoparticle. model does not rely on defining an image plane and thus can be used for complex surfaces relevant for plasmonic applications. To illustrate this, we will consider the interactions between the benzene-TCNE complex and an icosahedral Ag10179 nanoparticle. The molecular complex is oriented 4 ˚ A from the vertex of the silver icosahedron, see Figure 4(b). For this situation, the concept of an image plane is not well defined and thus the image charge model cannot easily be used to predict the renormalization effects, although, this could be overcome by mapping the image charge model onto a continuum electrodynamic representation. The QP gap of the free molecule is found to be 6.3 eV using LC-ωPBEh with γ = 0.20, in good agreement with the experimental results. 45,46 Range-tuning for the benzene-TCNE-nanoparticle system using DIM/QM leads to an optimum value of γ surf = 0.1234. Using this value, we find that the QP gap for the complex on the NP is reduced to 5.3 eV. A recent study of the same molecular complex on an Al metal surface using G0 W0 found a larger renormalization of around 2.8 eV. 23 There

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are two main reasons for this. The first is that in their simulations of the complex, they used a supercell without truncations of the Coulomb interactions which lead to the molecule effectively seeing two surfaces and thus a larger polarization effect. The second is that the polarization effect due to a planer metal surface is larger than a curved metal surface found on the the nanoparticle used in this work. To determine this effect, we also calculated the renormalization effect for benzene-TCNE on a flat silver surface and found a QP gap renormalization of 2.5 eV in better agreement with the G0 W0 result. Also, the optimized gamma parameter of γ f lat = 0.0912 is similar to that found for benzene on the Ag surface. Thus, it is important to incorporate a realistic description of the actual surface to capture the relevant polarization. 6

0.10

4 0.08 ˚ /molecule) σOPA (A

2 0 −2

0.06

2

˚ 2/molecule) σOPA (A

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−4 −6 −8

0.04

0.02

−10 −12 300

350

400

450 λ (nm)

500

(a) Full absorption spectra

550

0.00

600

400

450

500 λ (nm)

550

600

(b) Absorption spectra near the molecular resonance

Figure 5: Absorption cross section of the benzene-TCNE complex in gas phase (•) and on the vertex of a Ag10179 icosahedron calculated using DIM/QM (H) and DIM/QM[γ opt ] (N). The absorption spectra in (a) shows the full range from 300-600 nm, whereas the spectra in (b) focuses on the region with the molecular resonance. In both spectra the absorption contribution from the bare metal nanoparticle is not included. In the DIM/QM method, the interactions between the molecule and the metal surface are described in terms of two distinct mechanisms. 21 The first mechanism can be thought of as an image field effect, which leads to the renormalization of the energy levels as described above. The second mechanism results from the polarization induced in the metal nanoparticle due to the interactions with the external field. For excitations near the plasmon frequency of the metal nanoparticle, this mechanism describes the

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enhanced near field and is the major contribution to the coupling between the plasmon and the molecular excitation. However, as of yet, renormalization effect has not been considered when describing this coupling. To illustrate the influence of the image field, we study the renormalization of the lowest charge-transfer (CT) excitation in the benzene-TCNE complex at the vertex of a Ag10179 using DIM/QM. The lowest CT excitation from benzene to TCNE is found to be at 3.0 eV using the LC-ωPBEh functional with γ = 0.2. Experimental results find this transition at 3.6 eV in the gas phase, 47 and recent calculations using BSE find a value of 3.2 eV 23 in reasonable agreement with the result obtained here. 23 Using the range-tuned BNL functional 33 the transition is found to be at 3.8 eV, significantly higher in energy than the results found in this work and from the BSE calculations. For comparison, traditional functionals underestimate this transition by more than 1.5 eV. 23,33 In Figure 5(a), we compare the absorption cross section for the free molecule with that obtained using DIM/QM with γ = 0.2, and DIM/QM[γ opt ] with only the image field effect included. We see that the image field interactions with the plasmon induces a significant absorption band around 300-350 nm even though the free molecule do not show any significant molecular absorption in this region of the spectrum. In this region of the spectra we find only small differences between the DIM/QM with γ = 0.2, and DIM/QM[γ opt ] results as the the interactions are determined by the nanoparticle. Furthermore, we notice that the derivative lineshape found around 300-350 nm is a characteristic signature of the interference between the polarizability of the molecule and the nanoparticle. 29,48 We will discuss this feature in more detail below. To understand the image field contribution to the molecular absorption we plot in Figure 5(b) only the wavelength around the molecule resonances. We see that the image field effect introduces a small enhancement of the absorption cross section and a slight blue shift due to the interactions with the metal nanoparticle. To account for the renormalization effects on the optical gap, we plot the absorption cross section for the complex obtained using DIM/QM[γ opt ] in Figure 5. Once the non-local correla-

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tion is taken into account we find that the CT transition is significantly red-shifted by 0.6 eV (∼ 100 nm). This is in good agreement with the recent BSE calculations, which finds a ∼ 0.8 eV red-shift, especially if we account for the stronger image charge effect in the BSE calculations. 23 In fact, we obtain a red-shift of 0.8 eV when benzeneTCNE is range-tuned on a flat silver surface due to the larger polarization effect. We also find that the absorption cross section is slightly broader by ∼ 0.008 eV (FWHM) than the results obtained using the global This is likely a result of a detuning of the CT resonance from the plasmon excitation (∼ 330 nm) of the nanoparticle due to the renormalization effects. Thus, we find that DIM/QM[γ opt ] can describe both the renormalization of the QP gap and the optical gap at significantly lower computational cost than methods based on many-body perturbation effects. Furthermore, DIM/QM[γ opt ] is not limited to only studying planar substrates but can tackle complex geometries like metal nanoparticles of relevance for studying plasmonic materials. 20

20

15

10

10 0

˚ /molecule) σOPA (A

−10

2

˚ 2/molecule) σOPA (A

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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−20 −30

5 0 −5 −10 −15

Plasmon resonance (x 3.2 ∗ 10−3 ) XX

−40

−20

2

Free Complex (x102)

Free Complex (x10 )

−50 300

350

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450 λ (nm)

500

Plasmon resonance (x 6.5 ∗ 10−3 ) XX

550

−25 350

600

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450

500

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λ (nm)

(a) Benzene-TCNE on silver with local fields

(b) Benzene-TCNE on gold with local fields

Figure 6: Absorption cross-section of the benzene-TCNE complex on the vertex of a Ag10179 icosahedron (a) and a Au10179 icosahedron (b) using DIM/QM[γ opt ] with local field effects. In (b) the free complex’s absorption is plotted using γ opt to show the shift in excitation due to the renormalization.

Renormalization of the optical gap: Local field effects The strongest coupling between the metal nanoparticle and the molecule comes from the enhanced near field arising from the plasmon excitation. The coupling between the

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plasmon near field and the molecular CT excitation of the benzene-TCNE complex is shown in Figure 6(a). We only plot the XX component of the polarizability tensor that is aligned with the molecular transition moment as this dominates the absorption. The absorption, along the other components, can be found in the supporting information. The absorption spectrum does not include the absorption of the bare plasmon, but otherwise includes all interactions between the molecule and the nanoparticle. We find that the molecular absorption is 40 times stronger around the CT transition and more than 200 times stronger on the low energy side of plasmon excitation. The strong enhancement of the molecular absorption is found even though the molecular excitation is detuned from the plasmon excitation by more than 1 eV. This large detuning is caused by the renormalization of the optical gap as described above. Also, a strong negative absorption is found near the plasmon maxima. The negative absorption indicates the energy-transfer between the molecule and the metal nanoparticle due to the plasmon excitation. Furthermore, we note that the absorption line shape is highly asymmetric which is typical of interference with a broad plasmon. 49 The free benzeneTCNE complex does not have any appreciable absorption around 350 nm, however, the coupling with the image field introduces a small absorption band as shown in Figure 5. This absorption band arises from the excitation of the plasmon due to the image field interaction with the molecule. Therefore, the interference shown in Figure 6(a) is the result of the coupling between the molecular absorption induced by the image field from the nanoparticle and the near field from the plasmon excitation. The coupling between the molecular resonance and the plasmon is expected to become stronger as the overlap between their absorption bands increases. To increase this overlap, we used a gold nanoparticle with a plasmon resonance at 500 nm. In Figure 6(b), we plot the absorption spectrum of the benzene-TCNE complex interacting with the gold nanoparticle. The shape of the absorption band is similar to that of the silver nanoparticle, but with the negative absorption band around the plasmon excitation reduced by a factor of two. This is easily explained, since the plasmon absorption band for the

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gold nanoparticle is about a factor of two smaller than that of the silver nanoparticle. However, the increased absorption band at the low energy side of the plasmon has about the same strength for both nanoparticles. This is caused by the stronger coupling between the molecular absorption and the plasmon for the gold nanoparticle.

8000

XX - CDA (x10) XX

6000 4000 2000 0 α ¯I

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−2000 −4000 −6000 −8000 −10000 350

400

450

500 λ (nm)

550

600

Figure 7: Comparison between the XX component of the imaginary polarizability for the benzene-TCNE complex on Au10179 icosahedron calculated using DIM/QM and CDA. The center-to-center distance of 42.7 ˚ A is used in the CDA simulations. A simple explanation for the interference can be obtained by considering the nanoparticle and molecule as two interacting anisotropic point polarizable objects using the coupled dipole approximation (CDA). 50 The imaginary part of the polarizability for the combined system as obtained using the CDA is shown in Figure 7. Overall, we find qualitative agreement between the CDA results and the DIM/QM simulations, although, there are differences such as a smaller blue-shifted coupling found using CDA

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as compared to DIM/QM. This is likely a result of the simple point dipole approximation, which is not expected to be quantitative for molecules adsorbed on the surface. The atomistic description within DIM/QM more accurately represents the local environment felt by the adsorbed molecule. This includes large changes in the electric field strength across the dimension of the molecule. A visualization of these local fields is given in the supporting information. However, this qualitative agreement can be used to gain insights into the interference by considering an approximate analytical solution for two interaction isotropic point objects 29 αIXX ≈ αINP + αIM +

I I R 4(αR NP αM + αNP αM ) r3

(7)

where r is the separation. Thus, the interference comes from the coupling between the absorption of the molecule/nanopaticle (αIM /αINP ) and the dispersion in the reR fractive index of the nanoparticle/molecule (αR NP /αM ). While the CDA provides a

qualitative picture of the interference, it becomes important to go beyond the simple dipole approximation to fully understand the coupling between the plasmon and the molecular absorption. Finally, considering the molecule as an isotropic medium, as is often the case in models for plasmon-exciton coupling, is also suspect due to the highly anisotropic coupling between the two systems as evident in the DIM/QM results.

Conclusions In conclusion, we have presented a computational efficient approach to study the renormalization effects of the quasi-particle and optical gap of molecules adsorbed on metal nanoparticles. We have demonstrated that accurate simulations of the renormalization effects on the coupling between molecular excited states and plasmons is possible by employing a hybrid atomistic electrodynamics and time-dependent DFT model. We find, in general, very good agreement with many-body results, but at a fraction of the computational cost. Furthermore, the method is not limited to planar surfaces and thus enables the simulation of molecules adsorbed on plasmonic metal nanoparti-

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cles. This is illustrated by considering the coupling between the molecular absorption of a donor-acceptor charge-transfer complex with the plasmon excitation. We show that the interference between the molecular absorption, the image field of the metal nanoparticle, and the near field due to the plasmon excitation determines the overall coupling. We also show that the coupling between the molecular absorption and the plasmon is significant, even with a large energy separation. The results presented here show the importance of using first-principles simulations to understand, in detail, the coupling between molecular absorption and plasmon excitation.

Acknowledgement This work was supported by the NSF award CHE-1362825. We acknowledge support received from Research Computing and Cyberinfrastructure, a unit of Information Technology Services at Penn State. This work used the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by National Science Foundation grant number ACI-1053575.

Supporting Information Available The supporting information contains numerical results for the renormalization of the HOMO-LUMO gap, an illustrative example of the γ prediction model, plots of electric field enhancement, image field effects for benzene-TCNE on a Au10179 icosahedron, and additional component of the absorption for the benzene-TCNE complex on the metal nanoparticles.

This material is available free of charge via the Internet at

http://pubs.acs.org/.

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TOC Graphic.

LUMO LUMO

HOMO HOMO

Keywords: electronic structure, orbital renormalization, surface effects, DFT, long-range corrected functional

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Orbital Renormalization Effects on the Coupling ... - ACS Publications

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