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A Discrete Interaction Model/Quantum Mechanical Method for Simulating Plasmon-Enhanced Two-Photon Absorption Zhongwei Hu, and Lasse Jensen J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.8b00893 • Publication Date (Web): 23 Oct 2018 Downloaded from http://pubs.acs.org on October 27, 2018

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Journal of Chemical Theory and Computation

A Discrete Interaction Model/Quantum Mechanical Method for Simulating Plasmon-Enhanced Two-Photon Absorption Zhongwei Hu and Lasse Jensen∗ Department of Chemistry, The Pennsylvania State University, 104 Chemistry Building, University Park, 16802, United States. E-mail: [email protected]

Abstract In this work, we extend the discrete interaction model/quantum mechanical (DIM/QM) model to simulate Plasmon-Enhanced Two-Photon Absorption (PETPA). The metal nanoparticle is treated atomistically by means of electrodynamics while the molecule is described using damped cubic response theory within a time-dependent density functional theory framework. Using DIM/QM, we study the PETPA of para-nitroaniline (p-NA) with a focus on the local and image field effects, the molecular orientation effects, and the molecule-nanoparticle distance effects. Our findings show that the enhancement is more complex than the simple |E|4 enhancement mechanism, where |E| is the local field at the position of the molecule. Due to specific interactions with the nanoparticle, we find a TPA dark state of p-NA can be significantly enhanced through a coupling with the plasmon excitation. The results presented in this work illustrate that the coupling between molecular excitations and plasmons can give rise to unusual and complex behavior in nonlinear spectroscopy that cannot simply be understood by

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considering the optical properties of the individual molecules and nanoparticles separately. The method presented here provides detailed insights into the enhancement of nonlinear optical properties of molecules coupled to plasmonic nanoparticles.

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Introduction

Two-photon absorption (TPA) is a nonlinear optical (NLO) process that involves simultaneous absorption of two photons and has applications in optical storage, 1,2 optical limiting, 3 biological imaging, 4–8 and photodynamic therapy. 9–12 To increase applications of TPA processes, significant efforts have been made to design chromophores with large TPA cross sections. This is typically done by synthesizing molecules with appropriate electron-donating and accepting functional groups such that the NLO properties are enhanced. 13–16 Alternatively, the TPA cross section can be enhanced by coupling the molecular response to the plasmonic response of metal nanoparticles. 7,17–22 Due to the strong dependence of NLO properties on the plasmonic near-field, this offers significant potential for enhancing the molecular TPA cross sections. However, so far the observed enhancements have been modest in part due to a lack of understanding of criteria needed for generating plasmon enhanced two-photon absorption (PETPA). The enhancement of linear and nonlinear optical properties of molecules is well known from surface-enhanced vibrational spectroscopy. 23–27 In surface-enhanced Raman scattering (SERS), the plasmonic near-field enhances the molecular signal such that single molecules can be detected. 28,29 Enhancement of nonlinear Raman processes has also been demonstrated. 30–32 In contrast to the very large enhancements found for vibrational spectroscopy, enhancements of the TPA cross section have been found to be around 102 -103 . 33–35 The TPA cross section is expected to scale with the fourth power of the plasmonic near field, which is the same scaling as SERS where enhancements of 106 − 1010 can be achieved. 22,36 Therefore, one could expect that it should be possible to achieve significantly larger enhancements of the TPA cross sections by careful design of the molecule-nanoparticle complex. Theoreti2


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cal simulations of PETPA could provide the necessary insights into designing such systems but have so far been limited to modeling the expected enhancements based on electrodynamics simulations. Quantum mechanical methods have been developed to simulate TPA cross sections near metal nanoparticles but the near field effect is not considered. 37 Thus, it is important to develop first-principle methods that can combine a quantum mechanical description of the molecule with the plasmonic near field to understand PETPA. In this work, we extend the discrete interaction model/quantum mechanical (DIM/QM) method 38–45 to describe general third-order NLO properties of molecules interacting with plasmonic nanoparticles. In the DIM/QM method, the nanoparticle is treated using atomistic electrodynamics such that the local environment of the plasmonic nanoparticle can be incorporated into the simulations of the molecular optical properties. DIM/QM has previously been used to describe SERS, 46 surface-enhanced Raman optical activity (SEROA), 43 plasmonic circular dichroism, 47 and surface-enhanced hyper-Raman scattering. 45 Here we combine DIM/QM with damped cubic response theory to enable simulations of plasmon enhanced third-order NLO properties such as third-harmonic generation and degenerate TPA. 45 Although the implementation can describe general third-order NLO properties, we will focus on PETPA of para-nitroaniline (p-NA) placed in the junction of a dimer nanoparticle. The molecular TPA properties will be calculated using time-dependent density functional theory (TDDFT). While TDDFT offers a good compromise between accuracy and computational cost, the results are known to depend strongly on the chosen exchange-correlation (XC) functionals. Recent benchmarks against accurate coupled cluster results have shown that TDDFT tends to underestimate the TPA cross section using long-range corrected functionals. 48,49 Semi-local functionals tend to perform slightly better due to fortuitous error cancellation. 49 The emphasis in this work is on how the near-field changes the TPA properties where TDDFT is expected to be sufficiently accurate. Using this model system, we will examine the role of the near field and image field on PETPA. We will also investigate the molecular orientation and distance effects on PETPA.

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Theory

In the DIM/QM method we solve an effective time-dependent Kohn-Sham (TDKS) equation 38–41 i

∂ φi (r, t) = hKS [ρ(r, t)]φi (r, t), ∂t

(1)

with the effective KS operator being given as X ZI 1 hKS [ρ(r, t)] = − ∇2 − + 2 |r − RI | I

Z

ρ(r, t) 0 δE XC dr + + Vˆ DIM (r, t) + Vˆ pert (r, t). (2) |r − r 0 | δρ(r, t)

In the equation above, Vˆ DIM (r, t) is the DIM embedding operator and Vˆ pert (r, t) is the external perturbation. In the following we will work in the frequency domain, where the subscripts α and β denote Cartesian coordinates, i and j denote QM electrons, m and n denote DIM atoms, I and J denote QM nuclei, and the Einstein summation convention is employed for repeated Greek indices. These two operators can be further decomposed and written in terms of the polarization operator Vˆ pol (rj , ω), the perturbation operator Vˆ ext (rj , ω), and the local field operator Vˆ loc (rj , ω), as 38–41 Vˆ DIM (rj , ω) =

X

(1)

µind m,α (ω)Tjm,α

(3)

m

and Vˆ pert (r, ω) = Vˆ ext (rj , ω) + Vˆ loc (rj , ω) X (1) = Vˆ ext (rj , ω) + µext m,α (ω)Tjm,α .

(4)

m (1)

ind/ext

Tjm,α is the first-order interaction tensor, and µm,α

(ω) represents the frequency-dependent

dipoles of the DIM subsystem as induced by the QM system/external field. 38–41 The frequency-dependent second hyperpolarizability γ for the DIM/QM system can be

4


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expressed within a cubic response formalism by utilizing the 2n + 1 rule as 50 γαβγδ (−ωσ ; ω1 , ω2 , ω3 ) = h X n Tr n U α (−ωσ )Gβ (ω1 )U γδ (ω2 , ω3 ) + U β (ω1 )Gα (−ωσ )U γδ (ω2 , ω3 ) P

− U α (−ωσ )U β (ω1 )εγδ (ω2 , ω3 ) − U β (ω1 )U α (−ωσ )εγδ† (−ω2 , −ω3 ) + U α (−ωσ )Gγδ (ω2 , ω3 )U β (ω1 ) + U β (ω1 )Gγδ (ω2 , ω3 )U α (−ωσ ) − U α (−ωσ )U γδ (ω2 , ω3 )εβ (ω1 ) − U β (ω1 )U γδ (ω2 , ω3 )εα (−ωσ )

(5)

+ U γδ† (−ω2 , −ω3 )U α (−ωσ )εβ (ω1 ) + U γδ† (−ω2 , −ω3 )U β (ω1 )εα (−ωσ ) oi − U γδ† (−ω2 , −ω3 )Gα (−ωσ )U β (ω1 ) − U γδ† (−ω2 , −ω3 )Gβ (ω1 )U α (−ωσ ) h + Tr hxc (r, r0 , r00 , r000 , ω1 , ω2 , ω3 )Dα (−ωσ )Dβ (ω1 )Dγ (ω2 )Dδ (ω3 ) oi X n gxc (r, r0 , r00 , ω2 , ω3 )Dα (−ωσ )Dβ (ω1 )Dγδ (ω2 , ω3 ) , + P

where ω1 , ω2 , and ω3 denote the three incident frequencies with ωσ being the sum of them, P “Tr” stands for the trace, and P represents a summation over corresponding terms obtained by permuting (±ω1 , β) and (±ω2 , ±ω3 , γδ). Details about the permutation can be found in Reference 50. We note that the γ expression shares the same form as what used for an isolated QM system, 50 however, all matrices accounts for the influence of the metal nanoparticle through inclusion of both the local field operator and the DIM operator. As shown previously, 42,45 the first-order transformation matrix is given by

α Uia (±ω) =

Gαia (±ω) , ε0a − ε0i ∓ (ω + iΓ)

(6)

where only the elements from the occupied-virtual block of U are shown, Γ corresponds to a phenomenological energy broadening term for the excited state, and ε0a , ε0i represent the KS one-electron energies of the virtual and occupied orbitals, respectively. The DIM/QM

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first-order KS matrix in the molecular orbital (MO) basis reads

Gαia (ω) = hi| [Vˆαext (r, ω) + Vˆαloc (r, ω) + Vˆ Coul + Vˆxc (r) + VˆαDIM ] |ai ,

(7)

where Vˆαloc (r, ω) and VˆαDIM accounts for the interactions with the metal nanoparticle. Similarly, the second-order transformation matrix U βγ is given as

Uijβγ (±ω1 , ±ω2 ) =

 n o  β γ γ β 1  U (±ω )U (±ω ) + U (±ω )U (±ω ) diagonal  1 2 2 1 ij ij ij ij  2        T βγ (±ω1 , ±ω2 ) + Gβγ  ij (±ω1 , ±ω2 )   ij 0 εj − ε0i ∓ (ω1 + ω2 + iΓ)

(8)

off-diagonal

where the constant part of the second-order transformation matrix can be found from

Tijβγ (±ω1 , ±ω2 )

all h X γ β Gβik (±ω1 )Ukj (±ω2 ) + Gγik (±ω2 )Ukj (±ω1 ) =

(9)

k

i β γ − Uik (±ω1 )εγkj (±ω2 ) − Uik (±ω2 )εβkj (±ω1 ) . and thus only depends on the first-order matrices, with the first-order Lagrangian matrix given by

εα (±ω) = Gα (±ω) + ε0 U α (±ω) − U α (±ω)ε0 ± (ω + iΓ)U α (±ω).

(10)

The non-constant part of the second-order transformation matrix is given by

βγ Gβγ (±ω1 , ±ω2 ) = C 0† Dβγ (±ω1 , ±ω2 ) × (2J) + νxc (±ω1 , ±ω2 ) C 0

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(11)

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where Dβγ is given as Dβγ (±ω1 , ±ω2 ) = C βγ (±ω1 , ±ω2 )nC 0† + C 0 nC βγ† (∓ω1 , ∓ω2 ) (12) β

γ†

γ

β†

+ C (±ω1 )nC (∓ω2 ) + C (±ω2 )nC (∓ω1 ), in which the second-order perturbed MO coefficients C βγ (±ω1 , ±ω2 ) = C 0 U βγ (±ω1 , ±ω2 ) depend on the U βγ matrix mentioned above. For details about the iteration process such as the self-consistent solutions to U α and U βγ , as well as the adiabatic local density approximation to second- and third-order XC kernels (gxc and hxc ), see Reference 50. It is also important to point out that, due to the inclusion of multiple incident frequencies and their different combinations at the second order, care must be taken when dealing with the second-order couplings in the molecule-nanoparticle system due to the frequency-dependent local field operator.

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Computational Details

All calculations in this work were carried out using a locally modified version of the Amsterdam Density Functional (ADF) program package. 51–53 Geometry optimization was performed using the Becke-Perdew(BP86) 54,55 XC potential with a triple-ζ polarized slater type (TZP) basis set from the ADF library. Unless otherwise stated, response properties were calculated using the statistical average of orbital model exchange-correlation potentials (SAOP) 56 with the TZP basis set. The SAOP potential was chosen due to its correct Coulombic decay of the potential at long distances which is important for the description of response properties. 56,57 Although large basis sets are expected to be needed for accurate calculations of TPA cross section, we will focus on results obtained using the smaller TZP basis set to limit the computational requirements. Solvent effects are not included in the simulations. No symmetry was assumed in the simulations. The excited state lifetime is included phenomenologically using a damping parameter Γ = 0.1 eV, which was previously found to be acceptable. 58,59

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Nanoparticles in this work were simulated using the discrete interaction model (DIM), 60–62 which treats the nanoparticle atomistically using classical electrodynamics. Silver and gold (FCC) unit cells were used to build the icosahedral dimer structures, and the frequencydependent complex dielectric functions of silver and gold were obtained from Johnson and Christy. 63 The polarizability interaction model (PIM) version of DIM/QM was used for all PETPA simulations, which describes the system as a collection of interacting polarizabilities. For degenerate linearly polarized TPA the cross section can be calculated as 50 N π 3 αf2 ω 2 ~3 X h Im γααββ (−ω; ω, ω, −ω) σTPA (ω) = 15e4 αβ

(13)

i + γαββα (−ω; ω, ω, −ω) + γαβαβ (−ω; ω, ω, −ω) , where αf is the fine structure constant, and the σTPA unit is given as Göppert-Mayer (1 GM = 10−50 cm4 s photon−1 ). 64 The integer value N is related to the experimental setup and in this work N = 4 is used for all simulated TPA spectra. 48

4 4.1

Results and Discussion PETPA of Para-Nitroaniline

Para-nitroaniline (p-NA) is a prototypical push-pull molecule, which has been widely used for spectroscopic studies due to its rich optical properties. The large TPA properties of p-NA is a result of the electron donating -NH2 and electron accepting -NO2 groups giving a charge-transfer character of the lowest excited state. The excited state corresponds to a π−π ∗ transition from the HOMO to the LUMO of p-NA. Here we investigate the PETPA properties of p-NA situated in a junction between either a silver or gold nanoparticle dimer as depicted in Figure 1. The p-NA molecule is placed vertically between two icosahedral Ag2057 or Au2057 nanoparticles with its nitrogen atoms set 4.0 ˚ A away from the corresponding nearest silver/gold atoms. The resulting PETPA spectra based on the Ag2057 - and Au2057 -dimer 8


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Figure 1: (a) Simulated TPA spectrum of an isolated p-NA molecule. Simulated PETPA spectra for p-NA placed vertically in the junction of an (b) Ag2057 -dimer system and (c) an Au2057 -dimer system. Maximum TPA cross sections found are 16 GM, 2277 GM, and 3504 GM, respectively. substrates are shown in Figures 1(b) and 1(c), respectively. For comparison, in Figure 1(a) we also plot the regular TPA spectrum of an isolated p-NA molecule. The lowest TPA absorption band for p-NA is found at 1.7 eV with a cross section of 16 GM. This is in good agreement with experimental results for which the band peak has been estimated at 1.6 eV with a cross section of 12 GM. 65,66 Since solvent effects are neglected in the simulations, the good agreement is a little fortuitous. Comparing the PETPA spectra with the regular TPA spectrum, we see that the plasmonic effects significantly enhance the TPA cross sections as one would expect. For the silver substrate the TPA cross section is enhanced 142 times, where for the gold substrate the TPA cross section is enhanced by 219. The TPA cross section scales with the local field as |E|4 and thus one should expect a similar enhancement as that of SERS. This can be seen from Eq. 5 where the U α matrix (or Gα matrix) depends linearly on the local field operator and U αβ (or Gαβ ) depends quadratically on the local field operator. Simulations of the local field enhancements at different frequencies can be found in the Supporting Information. If we calculate the local field at the center of the molecule, we find an estimated enhancement of 77 for the Au2057 -dimer and 64 for the 9



Ag2057 -dimer assuming the |E|4 scaling. This is in reasonable agreement with the DIM/QM enhancements although a factor of 2-3 smaller. This could be due to that the local field was only evaluated in the center of the molecules and thus likely underestimates the actual field felt by the molecule in the junction. As the field varies fast over the dimension of the molecule, field gradient effects could be important. 45 The reason for the modest enhancement is that the TPA absorption of p-NA is far from the plasmon excitation of the two dimer systems. At their corresponding plasmonic excitation energies, i.e., 3.6 eV for the Ag2057 -dimer and 2.4 eV for the Au2057 -dimer, the |E|4 local field enhancements are found to be 3355 and 180, respectively. More interesting is the splitting of the TPA band into two peaks at 1.6 eV and 1.8 eV for the Ag2057 -dimer system. Such a splitting is not observed for the Au2057 -dimer where the TPA band only shifts slightly to lower energy. Since the local field enhancement is found to be similar as described above, the main difference between the two dimer systems is the position of the plasmon resonance. In the Supporting information we show that the splitting is distance dependent with mainly the lowest band being sensitive to the distance. At a distance of 6 ˚ A between the nitrogen atoms and the corresponding nearest Ag atoms, there is no splitting due to the weaker interactions between the molecule and the plasmon. Reducing the distance to 2.5 ˚ A leads to a larger splitting with the lowest band found at 1.1 eV and the higher band at 1.8 eV. Intriguingly, the lowest molecular resonance of p-NA at 3.4 eV is near the plasmon resonance of the Ag2057 -dimer system at 3.6 eV. Therefore, the splitting could arise due to a coupling between the molecular resonance and the plasmon excitation. In the following we will examine the origin of this peak spitting in more detail.

4.2

The Role of Image and Local Fields

Within the DIM/QM framework, the molecule-nanoparticle interaction can be split into two parts: the image field and the local field. The image field arises from the interactions between the charge distribution of the molecule and the nanoparticle, which can be treated 10


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Figure 2: (a) Simulated TPA spectrum of an isolated p-NA molecule. Simulated PETPA spectra for p-NA placed vertically in the junction of an (b) Ag2057 -dimer system and (c) an Au2057 -dimer system, where the local fields are not included. Maximum TPA cross sections for them are found as 16 GM, 12 GM, and 17 GM, respectively. as a mutual polarization process. The local field contributions are due to the interactions between the external field and the nanoparticle, i.e., the direct excitation of the plasmon. To better understand the roles of these two distinct molecule-nanoparticle interactions on TPA of molecules, we will examine them individually. In Figure 2, we plot the TPA spectra of the two systems without the inclusion of the local field effect. Without the local field, we see that the TPA cross sections for both molecule-substrate systems are significantly smaller. This is not surprising as the local field effect gives rise to the largest enhancement and that the image field effect is only a minor contribution. This is similar to SERS where the EM mechanism dominates the enhancement. For both the Au2057 - and Ag2057 -dimer systems, we find that the cross section is similar to that of the free molecule. However, for the Ag2057 -dimer there is a slight reduction of the cross section where a slight enhancement is found for the Au2057 -dimer. Furthermore, we see that the two TPA bands for the Ag2057 -dimer are still present without the local field effect and thus is the result of the image field effect. Only one band is seen for the Au2057 -dimer. As the symmetry of the molecule is reduced in the junction, a possible explanation is that a previously two11



photon dark state becomes allowed. In p-NA there is a two-photon dark state at 1.92 eV that could become TPA active through the coupling with the nanoparticle. However, it is not simply the reduction of the symmetry that leads to this effect as only one band is seen for the Au2057 -dimer. Therefore, the likely origin of the second band in the TPA spectrum is due to a coupling between the molecular resonances and the plasmon excitation. One possible explanation for the splitting is that the lowest TPA allowed transition couples to the nanoparticle which results in the shift to 1.57 eV whereas the two-photon dark state leads to the peak at 1.8 eV. Another possibility is that the peak splitting is due to a two-photon plasmon exciton coupling between the molecular excitation and the plasmon.

Figure 3: DIM/QM simulated PETPA for p-NA placed vertically in the junction of an Ag2057 -dimer system, where the γ IDRI tensors are used for (a) and (b) while the γ TPA tensors are used for (c) and (d). The local fields are not included for simulating the PETPA spectra shown in (b) and (d) To differentiate between the two possible explanations of the two bands in the PETPA spectrum, it becomes necessary to consider other possible one-photon resonance enhancements of the TPA cross sections. To do this, we consider the imaginary part of the intensity dependent refractive index (γ IDRI ) which includes both saturated linear absorption and two-photon absorption processes. Previously 67 we have used this approach to investigate the importance of one-photon resonance enhancements for monolayer protected gold clus12


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ters, and thus can provide insights into the resonance coupling between the molecule and plasmon excitation in TPA. Moore and Jensen 44 have shown that such resonant moleculeplasmon coupling could lead to a derivative line shape for the plasmon-enhanced one-photon spectrum, attributed to the interference between the polarizability of the molecule and the nanoparticle. 68 Therefore, if the peak splitting is due to two-photon plasmon exciton coupling mechanism, we would expect that γ IDRI show a significant one-photon contribution in the vicinity of the split TPA peaks. The reason for this is that we do not consider directly two-photon processes in the metal nanoparticle as the local field is obtained using linear response theory. On the other hand, if the peak splitting is due to an enhancement of the two-photon dark state by coupling with the plasmon, we would expect that the one-photon contributions should occur at larger energies closer to the plasmon excitation. The simulated PETPA spectra using γIDRI (with and without local field) are plotted in Figure 3 and are compared with their γTPA counterparts. We see that the TPA cross sections simulated using γIDRI and γTPA are very similar in the vicinity of the two-photon resonances. The TPA cross sections simulated using γIDRI do become negative for energies beyond ∼2 eV. Therefore, the origin of the two bands is most likely due to a two-photon-dark molecular state that becomes allowed through the coupling with the plasmon excitation. The negative TPA cross section observed using γIDRI results from a one-photon process into either the lowest molecular resonance at 3.4 eV, the plasmon resonance at 3.6 eV, or a combination of both. Our results show that the coupling between molecular excitations and plasmons can give rise to unusual and complex behavior in nonlinear spectroscopy that cannot simply be understood by considering the optical properties of the individual molecules and nanoparticles separately.

4.3

Molecular Orientation Effects on PETPA

The molecular orientation of surface adsorbates influences both the signal intensity and the spectroscopic signatures, and thus can aid in analyte identification and contribute to the nanostructure design. 69,70 This is for example used in surface-enhanced vibrational spec13



Figure 4: (a) Simulated TPA spectrum of an isolated p-NA molecule. DIM/QM simulated PETPA spectra for p-NA sitting in the junction of an Ag2057 -dimer system with its molecular dipole lies along the (b) x-axis, (c) y-axis, and (d) z-axis directions. troscopies where the surface-selection rules can be used to determine the molecular orientation. 71–76 Here we will examine how the enhancement of the TPA cross section depends on the orientation of the molecule relative to the plasmonic dimer. We will consider three orientations for p-NA placed within Ag-dimer, where the long-axis of p-NA is aligned with the x-axis, y-axis, and z-axis, respectively. The Ag-dimer is aligned along the z-axis. In Figure 4 we plot the simulated TPA spectra for the three orientations and compare with the TPA spectrum of the free molecule. As expected, we see the largest enhancement when the molecule is oriented along the z-axis (aligns with the local field in the junction). When the molecule is aligned along the x-axis, the TPA cross section around 1.7 eV is reduced as compared with the free molecule due to the screening of the field in the junction perpendicular to the molecule. We note that when the molecule is aligned along the x-axis and the y-axis, a small shoulder is found around 1.85 eV whereas a two-band feature is are seen when p-NA is aligned with the z-axis. This is consistent with the interpretation of the second band as arising from a plasmon-enhanced TPA dark state.

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Figure 5: PETPA enhancement factor (log scale) simulated using DIM/QM (solid curves) and the |E|4 estimation (dashed curves) for p-NA sitting vertically in the junction of an Ag2057 -dimer. Only the two peak frequencies, i.e., 1.57 eV (the 1st peak, green) and 1.76 eV (the 2nd peak, blue), are considered.

4.4

Distance Effects in PETPA

To gain further insights into the enhancement mechanism in PETPA, we will examine how the TPA cross section scales with the increased separation between the two nanoparticles in the junction. As mentioned above, we expect that the TPA cross section scales with the local field as |E|4 . The local field in the junction is known to depend strongly on the nanoparticle separation and thus should show a large effect on the TPA cross section. By focusing on the two strong bands, we calculate the enhancement factor relative to the free molecule’s cross section as a function of the distance between the two nanoparticles. The enhancement factors obtained using DIM/QM are compared to the |E|4 estimations in Figure 5. Here the distance refers to the separation between the nitrogen atoms of p-NA and the nearest silver atoms. The |E|4 enhancement is based on the local field in the center of mass (COM) of the junction. In the supporting information we also plot the comparisons with the |E|4 enhancement calculated at the nitrogen positions. The enhancement factor is shown to reach around 103.5 at the shortest separation but

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quickly drops off to around 102 for larger separations. The largest enhancement is seen for the higher energy band. However, for separations larger than 5 ˚ A, the two bands in the PETPA spectrum merge and thus the enhancement factors become identical. The |E|4 enhancement is found to be smaller by almost one order of magnitude at all distances. Simulations of the |E|4 enhancement at the nitrogen positions do lead to large enhancements, however, the estimates are still smaller than that obtained using DIM/QM. The larger enhancement predicted by DIM/QM results most likely from a combination of image field effects and that the field varies strongly over the dimensions of the molecule.

5

Conclusion

In this work we have presented an extension of DIM/QM to simulate third-order response properties of molecule interacting with nanoparticles. The method combines an atomistic description of the nanoparticle using an electrodynamics model and a TDDFT description of the molecule using damped cubic response theory. The implementation includes both image and local field effects and thus can provide insights into plasmon enhancements of molecular nonlinear-optical properties such as TPA. We have tested the model by simulating the PETPA of p-NA in the junction of a nanoparticle dimer. The importance of local and image field effects, the molecular orientation effects, and the molecule-nanoparticle distance effects in PETPA has been demonstrated. Our results show that PETPA depends sensitively on the orientation of the molecule relative to the junction and the separation of the nanoparticles in the dimer. A general enhancement mechanism |E|4 , where |E| is the local field at the position of the molecule, is expected. However, we find a more complex enhancement that cannot simply be described by a local field enhancement. This is partly due to image field effects and reduction of molecular symmetry in the junction. We find that the interactions with the metal nanoparticles can enhance an otherwise TPA dark state through a coupling with the plasmon excitation. In fact the plasmon enhancement of the TPA dark state makes

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it become the strongest band in spectrum. Our results highlight that the coupling between molecular excitations and plasmons can give rise to unusual and complex behavior in nonlinear spectroscopy that cannot simply be understood by considering the optical properties of the individual molecules and nanoparticles separately.

Acknowledgement L.J. acknowledges support from the NSF award CHE-1362825. Portions of this work were conducted with Advanced CyberInfrastructure computational resources provided by The Institute for CyberScience at The Pennsylvania State University (http://ics.psu.edu).

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Quantum Mechanical Method for

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