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. 2022 Sep 1;9(9):3025–3034. doi: 10.1021/acsphotonics.2c00761

Do We Really Need Quantum Mechanics to Describe Plasmonic Properties of Metal Nanostructures?

Tommaso Giovannini †,*, Luca Bonatti †, Piero Lafiosca †, Luca Nicoli †, Matteo Castagnola †, Pablo Grobas Illobre †, Stefano Corni ‡,§, Chiara Cappelli †,*
PMCID: PMC9502030  PMID: 36164484

Abstract

graphic file with name ph2c00761_0007.jpg

Optical properties of metal nanostructures are the basis of several scientific and technological applications. When the nanostructure characteristic size is of the order of few nm or less, it is generally accepted that only a description that explicitly describes electrons by quantum mechanics can reproduce faithfully its optical response. For example, the plasmon resonance shift upon shrinking the nanostructure size (red-shift for simple metals, blue-shift for d-metals such as gold and silver) is universally accepted to originate from the quantum nature of the system. Here we show instead that an atomistic approach based on classical physics, ωFQFμ (frequency dependent fluctuating charges and fluctuating dipoles), is able to reproduce all the typical “quantum” size effects, such as the sign and the magnitude of the plasmon shift, the progressive loss of the plasmon resonance for gold, the atomistically detailed features in the induced electron density, and the non local effects in the nanoparticle response. To support our findings, we compare the ωFQFμ results for Ag and Au with literature time-dependent DFT simulations, showing the capability of fully classical physics to reproduce these TDDFT results. Only electron tunneling between nanostructures emerges as a genuine quantum mechanical effect, that we had to include in the model by an ad hoc term.

Keywords: atomistic, interband, gold, silver, tunneling, field enhancement

1. Introduction

The recent progress in nanoscience has allowed to experimentally reach the atomistic detail in the geometrical arrangement of metal nanoaggregates.1−5 This has paved the way for many technological applications, including the creation of local hot-spots featuring enormously enhanced electric field, that has allowed single molecule detection, and even submolecular resolutions, when coupled to surface enhanced spectral techniques.6−13 A deep understanding of the peculiarities of these structures may benefit of an unavoidable interplaying between theory and experiments. At subnanometric scales such as for atomistically defined needles/tips, quantum effects play an important role in the plasmonic response and therefore need to be considered.14−23 As a result, a unified theoretical approach to describe the plasmonic properties of metal nanoaggregates under different regimes needs to consistently take into account the physical phenomena underlying the quantum and classical response.19,24,25

Large-size nanoparticles are typically described by means of classical electrodynamics, such as the Mie theory,26 the Discrete Dipole Approximation (DDA),27 the electromagnetic Finite Difference Time Domain (FDTD),28 or the Boundary Element Method (BEM).29−34

Nonlocal corrections can be considered by exploiting spatially dependent dielectric function based models35 or hydrodynamic models,36−38 which are also able to account for the electron spill-out effect that determines the near-field generated in plasmonic hot-spots of d-metals.39−41 However, these models substantially discard atomistic details, which could become relevant when studying surface-assisted spectroscopic properties.42 It is also worth noting that electron spill-out and nonlocal effects can effectively be treated by means of surface response functions, which would need to be specified for different surface planes.43,44 Limitations of these approaches for nanoparticles have recently been discussed.45 In this context, ab initio modeling, at the Time-Dependent Density Functional Theory (TDDFT) level, is still considered the most accurate approach to deal with these effects;46−50 however, it can only treat relatively small metal nanoparticles (NP; with diameter < 5 nm); therefore, real-size systems cannot be afforded due to the prohibitively large computational cost. Such a situation naturally leads to the conclusion that an explicit quantum mechanical treatment of electrons, such as DFT and TDDFT, is mandatory to provide a realistic picture of plasmonic phenomena in this size regime. Such a conclusion is not really challenged by the existing classical atomistic approaches to nanoplasmonics,51−57 that, while delivering accurate results, are based on fitting very general classical response expressions on TDDFT calculations, retaining therefore the physical basis of the latter.

In this paper, we explore whether a classical atomistic method based on essentially classical ingredients (Drude conduction mechanism and classical polarizabilities to reproduce interband polarization) can reproduce the optical response of complex plasmonic nanostructures. To this goal, we propose a physically robust approach to describe the plasmonic properties of sizable metal nanoaggregates characterized by the presence of interband transitions. Together with collective electronic excitations, they determine the plasmonic response of noble metal nanoparticles. The model is based on the recently developed ωFQ method,32,58−61 in which each metal atom is endowed with a net charge, which varies as a response of the external electric field. The charge-exchange between atoms is governed by the Drude mechanism. As an element of mere quantum mechanical origin that we found essential to add to our classical atomistic picture, the Drude charge exchange is modulated by quantum tunneling, which guarantees a correct description of the optical response for subnanometric junctions.15,16,18,20,22,58,62,63 Although the method has been successfully applied to sodium nanostructures and graphene-based materials,58,59 the basic formulation of ωFQ overlooks interband contributions, thus, it is unsuitable to describe the plasmonic properties of nanostructures based on d-metals.64−68 Here, we substantially extend ωFQ so to assign each metal atom with an atomic complex-valued polarizability (i.e., a complex-valued dipole moment), appropriately tuned to model interband effects. The resulting approach is called ωFQFμ (frequency-dependent fluctuating charges and fluctuating dipoles) by analogy with a parent polarizable approach, which has been proposed by some of the present authors to treat completely different chemical systems.69−71 The theoretical basis of the approach stems from the evidence that d-states can efficiently be treated as polarizable shells placed at lattice positions.67

Notice that ωFQFμ was developed from a different perspective compared to other classical atomistic approaches.51−54 Indeed, ωFQFμ is built from textbook bulk metal physics rather than fitting of generic polarizability and capacity frequency-dependent expressions. This provides practical benefits as well. In fact, Drude-tunneling and interband regimes are perfectly decoupled in the two terms that depend on ωFQs58 and ωFμs, thus allowing for a fine, physically guided, tuning of the plasmonic response. In the following, we show that ωFQFμ is able to correctly reproduce the plasmonic properties of Ag and Au nanostructures as a function of size and shape, and also their plasmonic response when forming subnanometer junctions. Remarkably, the favorable scaling of the method permits to afford large systems (more than 104 atoms), which cannot be treated at the quantum-mechanical level. Also, the ability of ωFQFμ to fully retain the atomistic detail is crucial to reproduce not only the plasmonic response but also near-field enhancements, which play a key-role in near-field enhanced spectroscopies.72,73

2. Theoretical Model

ωFQFμ is a fully atomistic, classical approach which substantially extends ωFQ, which assigns to each metal atom a time dependent charge. Under the action of a time dependent external electric field, metal atoms exchange charge via the Drude conduction mechanism, which is further assisted by quantum tunneling, which limits the charge transfer among nearest neighboring atoms and makes the interaction decrease with the typical exponential decay.58 In particular, ωFQ charge equation of motion in the frequency domain (ω) reads:58

2. 1

where qi(ω), a complex-valued quantity, is the Fourier component at the frequency ω of the oscillating atomic charge on atom i. n0 is the metal density, τ the friction time, Inline graphic is the effective area connecting ith and jth atoms, and lij is their distance. ϕel is the electrochemical potential acting on each metal atom, which takes into account the interactions between the different atoms and their interaction with the external electric field, which oscillates at frequency ω. f(lij) is a Fermi-like function mimicking quantum tunneling:58

2. 2

where lij0 is the equilibrium atom–atom distance, whereas d and s are dimensionless parameters that determine the sharpness and the center of the Fermi function f(lij), respectively. Their values can be determined by comparing computed results with reference ab initio data (see also Section S1 in the Supporting Information (SI)). In eq 1, Drude and tunneling terms are collected in the K matrix. The parameters entering ωFQ all have a clear microscopic physical meaning. Therefore, their values can be either chosen by an independent experiment or fitted to reproduce higher level results, and then the soundness of their values can be judged. We took the latter perspective, as discussed in the SI.

Due to its physical foundations, ωFQ cannot describe the specificity of metals featuring d-electrons, which contribute to interband transitions and that substantially affect the plasmon response.64−68 Therefore, we here extend ωFQ into a novel method, ωFQFμ, in which each atom is assigned a charge and an additional source of polarization, that is, an atomic polarizability (to which an induced dipole moment is associated). The presence of the dipole moments is included in eq 1 by taking into account the interaction between charges and dipoles in the electrochemical potential. The induced dipole moments μi are instead obtained by solving the following set of linear equations:

2. 3

where, Eext, Eμ, and Eq are the external electric field and those generated by the other dipole moments and charges, respectively. αω is the atomic complex polarizability, which is introduced to describe interband transitions. Remarkably, αω can easily be obtained by extracting interband contributions from the experimental permittivity function (see Section S1 in the SI), with no need to introduce a posteriori adjustable parameters.

To effectively couple charges and dipoles, that is, to simultaneously account for Drude and interband transitions, eqs 1 and 3 need to be solved simultaneously. By explicitly indicating all terms, the problem can be recast as the following set of linear equations:

2. 4
2. 5

where Tqq, Tqμ, and Tμμ define charge–charge, charge–dipole, dipole–dipole interactions, respectively. By imposing Tμμ diagonal elements to correspond to 1/αω, eqs 4 and 5 can be written in a compact matrix formulation as

2. 6

where the complex A matrices include interaction kernels and the Kij terms (see Section S1 in the SI for more details). The right-hand side of eq 6 accounts for external polarization sources, that is, the electric potential and field calculated at atomic positions:

2.

Notably, any kind of plasmonic materials can be modeled by ωFQFμ because it integrates all relevant physical ingredients (i.e., Drude conduction, electrostatics, quantum tunneling and interband transition) via eq 1 and eq 3. Also, once complex-valued charges and dipoles are computed, the absorption cross section σabs and the induced electric field can easily be calculated (see Section S1 in the SI for more details).

To conclude this section, it is worth noting that other atomistic, classical models, belonging to the Discrete Interaction Model (DIM) class, have been proposed to describe the plasmonic response of noble-metal nanoparticles.54,74 Among them, the coordination-dependent–Discrete Interaction Model (cd-DIM)74 is able to properly describe the size dependency of the Plasmon Resonance Frequency (PRF) for Ag NPs and the plasmonic response of Ag dimers, which is governed by quantum tunneling. Within this model, both effects arise from a modification of the coordination of surface atoms and the consequent modification of the atomic polarizability that is considered to be a parametrized function of the coordination number. In our approach the first effect genuinely originates from the screening effect of d-electrons, which is physically modeled by the presence of the αω term. A correct description of the plasmonic response of dimers arises from the phenomenological introduction of the quantum tunneling via the Fermi function in eq 2.

3. Results and Discussion

3.1. Optical Response of Metal Nanoparticles: ωFQFμ versus TDDFT Results

Here the capability of ωFQFμ at describing typical plasmonic response properties of single metal nanoparticles is discussed.75 Although the model is completely general, it is here applied to Ag and Au nanoparticles (NP), for which αω values are obtained from the permittivity functions reported by Etchegoin et al. in ref (76) and fitted in ref (77) (see also Section S1 in the SI). For both metals, we first consider NPs with three different geometrical arrangements, namely, truncated cuboctahedron (cTO), icosahedral (Ih), and ino-decahedron (i-Dh), which are all characterized by atomistically defined edges.54 Their plasmonic properties are studied as a function of the size (from a minimum of 85 atoms, ∼5 Å radius, to a maximum of 43287 atoms, ∼65 Å radius).78−80 Note that geometry relaxation is not considered, because it only slightly affects optical responses.23,81−86

Although sizable NPs cannot be afforded by ab initio methods, they can indeed be treated by ωFQFμ,60 at a reasonable computational time (on average, 59 min on Intel(R) Xeon(R) Gold 5120 CPU @ 2.20 GHz, 28 processors, for each frequency given in input for the structure composed of 43287 atoms). As an example of the performance of ωFQFμ, Kuisma et al.87 have reported that the calculation of the optical spectrum of Ag561 (∼1.4 nm radius) in Ih geometry with the time-propagation (TP) approach to TDDFT requires a wall time of 42.0 h with 512 cores.87 For the same system, ωFQFμ only requires 25 s on the aforementioned platform. Remarkably, ωFQFμ and reference TDDFT data are very similar (see Figure 1a, bottom panel). Slight discrepancies in the PRF and band broadening among the two models can be justified by considering that TDDFT results substantially vary as a function of the basis set; this is demonstrated by the data shown in Figure 1a (top panel), which are taken from ref (87). Clearly, reference ab initio data display large variability of almost 1 eV when moving from linear combinations of atomic orbitals (LCAO) with different basis sets to real space grid calculation (GRID) results. We also remark that the width of peaks is chosen arbitrarily in TDDFT calculations.

Figure 1.

Figure 1

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(a) Computed ωFQFμ and TDDFT σabs for Ag147 (∼0.8 nm radius) and Ag561 (∼1.4 nm radius). TDDFT results are reproduced from ref (87) and obtained by exploiting different basis sets (LCAO optimized, def. DZ (double-ζ), def. DZP (double-ζ polarized)) and on a real-space grid (GRID). (b) ωFQFμ and TDDFT87 plasmon densities for Ag561. TDDFT plasmon densities adapted with permission from ref (87). Copyright 2015 APS Publications. (c) ωFQFμ and TDDFT88 σabs for Au147–Au1415 (∼1.9 nm radius). (d) ωFQFμ and TDDFT88 plasmon densities for Au1415. TDDFT plasmon densities adapted with permission from ref (88). Copyright 2014 ACS Publications. ωFQFμ isovalues are set to 0.002 and 0.0005 au for Ag and Au, respectively.

A similar comparison can be performed for Au Ih NPs. TDDFT absorption cross sections reproduced from ref (88) and calculated at the ωFQFμ level are reported in Figure 1c,d. Also, in this case, ωFQFμ can correctly reproduce PRF trends as a function of the NP size and the relative intensities of the bands for the different NPs.

One of the most peculiar features of noble metal NPs and, in general, of d-metals is that the PRF blue shifts as the size of the system decreases, in contrast to what happens for simple metals (and correctly reproduced by ωFQ58). The physical origin of this blue-shift has been studied in the past.67 The screening of the Coulomb interaction among conduction electrons (that determines the plasmon frequency) by the localized d-electron core interplays with conduction electrons spill-out at the cluster surface. In short, the d-electron core screens electron–electron repulsion, thus decreasing the d-metal plasmon frequency compared to what is expected on the basis of the free-electron density of the metal. At the surface, conduction electrons spill out of the structure; in ωFQ and ωFQFμ, the finite size of each atomic distribution accounts for this effect. In parallel, the d-electron core, which is localized on the metal atom (described by a point dipole in ωFQFμ) cannot effectively screen the electron–electron repulsion. As a result, the plasmon frequency moves back to the nonscreened, free-electron value. Remarkably, ωFQFμ can indeed describe this mechanism because it provides the right result for the right reason. In fact, if the d-electron core response is artificially switched-off (i.e., αω → 0 in eq 3), the plasmon frequency (which is overall increased), red shifts for metal nanoparticles, as it is expected based on spill-out effects only.58,82 This is demonstrated by the plots reported in Figure S3 in the SI.

The dependence of computed PRFs on the number of atoms is reported in Figure 2a for different geometries; the plots clearly demonstrate that ωFQFμ can indeed correctly reproduce the previously reported trends. Also, the linear fit of Ag Ih PRFs as a function of the inverse of the NP diameter permits to linearly extrapolate PRF = 3.47 eV for an infinite diameter. This value is in almost perfect agreement with the mesoscopic limit of 3.43 eV, as obtained at the quasi-static FDTD (QSFDTD) level,87 and in excellent agreement with the extrapolated ab initio value of 3.35 eV (see also Figure S4 in the SI).87

Figure 2.

Figure 2

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(a) Ag and Au PRF for cTO, Ih, and i-Dh NPs as a function of the number of atoms. (b) Ag densities calculated at the PRF for cTO (top), Ih (middle), and i-Dh (bottom) geometries. Charge and dipole contributions are plotted together with ωFQFμ plasmon densities. Isovalues are 0.002 and 0.0005 au for Ag and Au, respectively.

The investigation of plasmon densities (i.e., the imaginary part of the charge density induced by a monochromatic electromagnetic field oscillating at the PRF) is of fundamental importance for correctly characterizing the plasmon resonance. Computed densities at the PRF for the largest structures in each geometrical arrangement are depicted in Figure 2b. In all cases, they represent a dipolar plasmon. Our result can be compared with reference ab initio data for Ag561 (∼1.4 nm radius, see Figure 1b) and Au1415 (∼1.9 nm radius, see Figure 1d). In both cases, the agreement is almost perfect.

For the purpose of a deeper theoretical analysis, ωFQFμ can also be used to decouple the contributions of Drude and interband transitions to the total plasmon density. Charge and dipole contributions are graphically depicted in Figure 2b for selected Ag NPs (see Figure S5 in the SI for Au NPS). Clearly, the two plasmon densities are associated with dipole moments in opposite directions. This finding remarks the screening role of the d-electrons response.41,67 As explained above, this results in the typical blue shift observed for d-metals.

To further demonstrate the ability of ωFQFμ to correctly take into account screening effects in d-metal NPs, we can compare density distributions in inner regions. Computed ωFQFμ and TDDFT densities for Au1415 Ih NP (∼1.9 nm radius), in the central region of the cluster (defined for −2 < z < 2 Å), are depicted in Figure 3. Notice that densities are computed at the corresponding PRFs, which only differ by 0.01 eV. Coherently with the results reported in Figure 1d, positive and negative charges are located at the top and bottom surface regions, respectively. However, ωFQFμ and TDDFT densities are mainly characterized by a local charge distribution around each Au atom, which is polarized along an opposite direction with respect to the polarization of the surface density. Such a behavior is related to screening effects, which are correctly taken into account by our atomistic, yet classical model.

Figure 3.

Figure 3

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ωFQFμ and TDDFT88 Au1415 (∼1.9 nm radius) plasmon densities in the central region of the cluster (−2 < z < 2 Å). TDDFT plasmon densities adapted with permission from ref (88). Copyright 2014 ACS Publications.

In Figure 4a,b the total enhanced electric field (|E|/|E0|, where E0 is the external electric field intensity) at the PRF is reported. Such quantity (elevated to the fourth power) is related to field enhancement factors that are measured in SERS experiments.10 The dependence of |E|/|E0| factors as a function of the number of atoms and the NPs radius is reported in Figure 4a and b, respectively. Notice that enhancement factors are computed at a distance of 3 Å from the tip of each structure; according to many previous reports, it corresponds to the typical adsorption distance of molecular systems.89 |E|/|E0| color maps at the PRF for each geometrical arrangement are graphically displayed in Figures 4c. As expected, |E|/|E0| maximum values correspond to tips, and are reported for cTO geometries (for both gold and silver), where edges are the sharpest. Interestingly, for all arrangements, |E|/|E0| follows a Inline graphic trend (N being the number of atoms) and, thus, a linear trend with respect to NP radius, because the difference in the electric potential linearly increases with the NP intrinsic size. Remarkably, ωFQFμ is also able to quantify the differences between Ag and Au NPs, being the latter associated with much lower enhancement factors as compared to the former, as expected in this frequency range.

Figure 4.

Figure 4

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Ag and Au enhanced electric field |E|/|E0| calculated at 3 Å from the tips for cTO, Ih, and i-Dh NPs as a function of the number of atoms (a) and NP radius (b). (c) Ag and Au |E|/|E0| color maps for cTO (top), Ih (middle), and i-Dh (bottom) geometries.

As a final comment, it is remarkable that ωFQFμ is able to describe nonlocal effects. To demonstrate that, we artificially changed the total electric potential and field acting on a specific atom placed at the surface or at the centroid of Ag147 (∼0.8 nm radius) and Ag3871 (∼2.7 nm radius) structures in the Ih configuration. As a result, independently of the atom position, not only charge and the dipole of the perturbed atom are modified, but also those of other atoms (see Figure S6 in the SI).

3.2. Subnanometer Junctions

In this section we will show that ωFQFμ has the potential to describe hot-spots in subnanometer junctions in a physically consistent manner. This is possible due to the account for quantum tunneling (eq 1), which dominates the plasmon response in these systems. To showcase ωFQFμ performances, we study Ag2869/Au2869 (∼2.6 nm radius) cTO dimers. In particular, we select two different morphologies obtained by approaching two NPs so to obtain surface–tip or surface–surface geometrical arrangements (see Figures 5 and 6).90 Note that for both structures the structural relaxation effects are not taken into account, similarly to previous studies.81 However, to study the effects of structural relaxation we have modeled surface roughness as derived from an atomic adjustment on the surface of one of the two cTO NPs in the surface–surface arrangement (see Figure S7 in the SI).

Figure 5.

Figure 5

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Ag (a) and Au (b) cTO2869 dimers in surface–tip geometrical arrangement. Color maps of σabs and |E|/|E0| as a function of the PRF and the distance between the two NPs are reported in the middle and bottom panels, respectively.

Figure 6.

Figure 6

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Ag (a) and Au (b) cTO2869 dimers in surface–surface geometrical arrangement. Color maps of σabs and |E|/|E0| as a function of the PRF and the distance between the two NPs are reported in middle and bottom panels, respectively.

For the ideal cTO dimers, we investigate both σabs and |E|/|E0| calculated at the gap’s center as a function of the distance d between the two monomers, in the range 1–24 Å. Computed spectra (Figure 5a,b) are characterized by a high energy peak (∼3.5 eV for Ag and ∼2.2 eV for Au), which red-shifts as d decreases. However, a clear discontinuity occurs at around 4 Å, where quantum tunneling plays a relevant role; in fact, such a distance is close to the Ag–Ag and Au–Au equilibrium distances.

For Au dimers, a second peak at lower energies is visible (d < 2 Å, PRF < 1.5 eV), which blue-shifts as d decreases (see Figure 5b). This band is not present for Ag, because its PRF falls below the investigated frequency range. Highest energy PRF corresponds to the typical Boundary Dipolar Plasmon, BDP, whereas the low-energy peak is associated with a Charge Transfer Plasmon, CTP, where a dipole moment arises in the whole structure (see Figures S8 and S9 in the SI). Note that for distances at which the CT mechanism takes place (for d < 4 Å), the high energy peak is associated with a high-order charge transfer plasmon, usually called CTP′. The clear discontinuity highlighted in σabs plots is also evident in the case of the computed |E|/|E0| values, which are reported as a color map as a function of the distance in Figure 5, bottom panels. For Ag, the maximum |E|/|E0| is depicted before the two NPs enter in the CT regime, whereas the opposite holds for the Au dimer, for which it corresponds to the CTP peak.

The surface–surface arrangements show similar plasmonic features, independently of the metal nature (see Figure 6a,b). In fact, both spectra are characterized by the presence of two bands when d < 4 Å: a CTP and a CTP′ peak (very low in intensity for Ag and fused with CTP for Au), which blue-shift or red-shift as d decreases, respectively. For d > 4 Å spectra are instead dominated by the BDP mode. The associated |E|/|E0| factors are plotted as a function of the distance in Figure 6, bottom panels. The computed values are of the same order of magnitude as for the previous case. Also, the maximum enhancement for Ag is shown at about 3.3 eV, for a distance of about 4 Å, that is, when CT effects start to dominate the plasmonic response. For Au, the maximum |E|/|E0| occurs at lower distances (d ∼ 3 Å), however it is associated with the CTP′ plasmonic mode, differently from the tip–substate arrangement. Interestingly, in case of Ag, a clear additional region of enhancement is displayed at energies larger than 3.5 eV, for d > 4 Å. This region is associated with a high-order plasmon, which is indeed dark in the σabs.

Finally, note that, for both studied geometries, the maximum |E|/|E0| is 3–5 times larger than the corresponding factor obtained for the Ag monomer and even 10 times for Au.

4. Summary and Conclusions

We have discussed the prediction of a classical physics based model for nanoplasmonics, ωFQFμ. ωFQFμ is a general atomistic approach to describe the plasmonic features of complex metal nanostructures characterized by interband transitions. In the model, which is formulated in the frequency domain, each atom of the structure is endowed with both a complex-valued charge and dipole, which are determined by solving response equations to an external monochromatic electric field. The two polarization sources conceptually describe the two fundamental mechanisms occurring in d-metals, that is, Drude conduction (charges) and d-electrons polarization (dipoles). Remarkably, charges and dipoles mutually interact; therefore, all physical features of metal nanostructures are considered. ωFQFμ is fully classical, therefore nanostructures of realistic size can be computed with accuracy comparable to full ab initio calculations, but with enormously lower computational cost. Note that the interband polarizability in eq 3 arises from the quantum nature of the system; however, in our approach it is modeled without explicitly considering the quantum nature of the constituting atoms.

From a conceptual point of view, demonstrating that ωFQFμ is able to reproduce the findings of a fully quantum description of the system is clearly questioning the notion that an explicit quantum mechanical treatment is needed to describe the change in plasmonic properties upon shrinking of the nanostructure size. From the results presented in this work, it turns out that only quantum tunneling (relevant for nanoaggregates and nanojunction) is inaccessible by such classical physics, and we had in fact to phenomenologically include the tunneling in ωFQFμ. It is finally worth noting that another relevant quantum effect, Landau damping, is not included in this classical modeling and would in principle require adding a phenomenological correction to ωFQFμ.

From a computational perspective, the developed method paves the way for an investigation of the plasmonic properties of realistic metal nanoparticles, characterized by complex shapes that require an atomistic detail.

Acknowledgments

We gratefully acknowledge the Center for High Performance Computing (CHPC) at SNS for providing the computational infrastructure.

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acsphotonics.2c00761.

  • Details on ωFQFμ equations and parametrization; Computational details; Ag and Au single NPs absorption cross sections and comparison with reference data; Demonstration of ωFQFμ nonlocality; Plasmon densities for Ag/Au cTO2869 dimers (PDF)

This work has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 Research and Innovation Program (Grant Agreement No. 818064). S.C. gratefully acknowledge the European Union’s Horizon 2020 FET Project ProID (No. 964363) for funding.

The authors declare no competing financial interest.

Supplementary Material

ph2c00761_si_001.pdf (8.6MB, pdf)

References

  1. Junno T.; Deppert K.; Montelius L.; Samuelson L. Controlled manipulation of nanoparticles with an atomic force microscope. Appl. Phys. Lett. 1995, 66, 3627–3629. 10.1063/1.113809. [DOI] [Google Scholar]
  2. Ishida T.; Murayama T.; Taketoshi A.; Haruta M. Importance of size and contact structure of gold nanoparticles for the genesis of unique catalytic processes. Chem. Rev. 2020, 120, 464–525. 10.1021/acs.chemrev.9b00551. [DOI] [PubMed] [Google Scholar]
  3. Sau T. K.; Rogach A. L. Nonspherical noble metal nanoparticles: colloid-chemical synthesis and morphology control. Adv. Mater. 2010, 22, 1781–1804. 10.1002/adma.200901271. [DOI] [PubMed] [Google Scholar]
  4. Liz-Marzán L. M. Tailoring surface plasmons through the morphology and assembly of metal nanoparticles. Langmuir 2006, 22, 32–41. 10.1021/la0513353. [DOI] [PubMed] [Google Scholar]
  5. Grzelczak M.; Pérez-Juste J.; Mulvaney P.; Liz-Marzán L. M. Shape control in gold nanoparticle synthesis. Chem. Soc. Rev. 2008, 37, 1783–1791. 10.1039/b711490g. [DOI] [PubMed] [Google Scholar]
  6. Willets K. A.; Van Duyne R. P. Localized surface plasmon resonance spectroscopy and sensing. Annu. Rev. Phys. Chem. 2007, 58, 267–297. 10.1146/annurev.physchem.58.032806.104607. [DOI] [PubMed] [Google Scholar]
  7. Zhang R.; Zhang Y.; Dong Z.; Jiang S.; Zhang C.; Chen L.; Zhang L.; Liao Y.; Aizpurua J.; Luo Y.; Yang J. L.; Hou J. G. Chemical mapping of a single molecule by plasmon-enhanced Raman scattering. Nature 2013, 498, 82–86. 10.1038/nature12151. [DOI] [PubMed] [Google Scholar]
  8. Jiang S.; Zhang Y.; Zhang R.; Hu C.; Liao M.; Luo Y.; Yang J.; Dong Z.; Hou J. Distinguishing adjacent molecules on a surface using plasmon-enhanced Raman scattering. Nat. Nanotechnol. 2015, 10, 865–869. 10.1038/nnano.2015.170. [DOI] [PubMed] [Google Scholar]
  9. Chiang N.; Chen X.; Goubert G.; Chulhai D. V.; Chen X.; Pozzi E. A.; Jiang N.; Hersam M. C.; Seideman T.; Jensen L.; Van Duyne R. P. Conformational contrast of surface-mediated molecular switches yields Ångstrom-scale spatial resolution in ultrahigh vacuum tip-enhanced Raman spectroscopy. Nano Lett. 2016, 16, 7774–7778. 10.1021/acs.nanolett.6b03958. [DOI] [PubMed] [Google Scholar]
  10. Langer J.; et al. Present and Future of Surface-Enhanced Raman Scattering. ACS Nano 2020, 14, 28–117. 10.1021/acsnano.9b04224. [DOI] [PMC free article] [PubMed] [Google Scholar]
  11. Benz F.; Schmidt M. K.; Dreismann A.; Chikkaraddy R.; Zhang Y.; Demetriadou A.; Carnegie C.; Ohadi H.; De Nijs B.; Esteban R.; et al. Single-molecule optomechanics in “picocavities. Science 2016, 354, 726–729. 10.1126/science.aah5243. [DOI] [PubMed] [Google Scholar]
  12. Yang B.; Chen G.; Ghafoor A.; Zhang Y.; Zhang Y.; Zhang Y.; Luo Y.; Yang J.; Sandoghdar V.; Aizpurua J.; et al. Sub-nanometre resolution in single-molecule photoluminescence imaging. Nat. Photonics 2020, 14, 693–699. 10.1038/s41566-020-0677-y. [DOI] [Google Scholar]
  13. Liu P.; Chulhai D. V.; Jensen L. Single-Molecule Imaging Using Atomistic Near-Field Tip-Enhanced Raman Spectroscopy. ACS Nano 2017, 11, 5094–5102. 10.1021/acsnano.7b02058. [DOI] [PubMed] [Google Scholar]
  14. Teperik T. V.; Nordlander P.; Aizpurua J.; Borisov A. G. Quantum effects and nonlocality in strongly coupled plasmonic nanowire dimers. Opt. Express 2013, 21, 27306–27325. 10.1364/OE.21.027306. [DOI] [PubMed] [Google Scholar]
  15. Zhu W.; Esteban R.; Borisov A. G.; Baumberg J. J.; Nordlander P.; Lezec H. J.; Aizpurua J.; Crozier K. B. Quantum mechanical effects in plasmonic structures with subnanometre gaps. Nat. Commun. 2016, 7, 11495. 10.1038/ncomms11495. [DOI] [PMC free article] [PubMed] [Google Scholar]
  16. Urbieta M.; Barbry M.; Zhang Y.; Koval P.; Sánchez-Portal D.; Zabala N.; Aizpurua J. Atomic-Scale Lightning Rod Effect in Plasmonic Picocavities: A Classical View to a Quantum Effect. ACS Nano 2018, 12, 585–595. 10.1021/acsnano.7b07401. [DOI] [PubMed] [Google Scholar]
  17. Savage K. J.; Hawkeye M. M.; Esteban R.; Borisov A. G.; Aizpurua J.; Baumberg J. J. Revealing the quantum regime in tunnelling plasmonics. Nature 2012, 491, 574–577. 10.1038/nature11653. [DOI] [PubMed] [Google Scholar]
  18. Marinica D.; Kazansky A.; Nordlander P.; Aizpurua J.; Borisov A. G. Quantum plasmonics: nonlinear effects in the field enhancement of a plasmonic nanoparticle dimer. Nano Lett. 2012, 12, 1333–1339. 10.1021/nl300269c. [DOI] [PubMed] [Google Scholar]
  19. Esteban R.; Borisov A. G.; Nordlander P.; Aizpurua J. Bridging quantum and classical plasmonics with a quantum-corrected model. Nat. Commun. 2012, 3, 825. 10.1038/ncomms1806. [DOI] [PubMed] [Google Scholar]
  20. Esteban R.; Zugarramurdi A.; Zhang P.; Nordlander P.; García-Vidal F. J.; Borisov A. G.; Aizpurua J. A classical treatment of optical tunneling in plasmonic gaps: extending the quantum corrected model to practical situations. Faraday Discuss. 2015, 178, 151–183. 10.1039/C4FD00196F. [DOI] [PubMed] [Google Scholar]
  21. Campos A.; Troc N.; Cottancin E.; Pellarin M.; Weissker H.-C.; Lermé J.; Kociak M.; Hillenkamp M. Plasmonic quantum size effects in silver nanoparticles are dominated by interfaces and local environments. Nat. Phys. 2019, 15, 275–280. 10.1038/s41567-018-0345-z. [DOI] [Google Scholar]
  22. Scholl J. A.; García-Etxarri A.; Koh A. L.; Dionne J. A. Observation of quantum tunneling between two plasmonic nanoparticles. Nano Lett. 2013, 13, 564–569. 10.1021/nl304078v. [DOI] [PubMed] [Google Scholar]
  23. Barbry M.; Koval P.; Marchesin F.; Esteban R.; Borisov A.; Aizpurua J.; Sánchez-Portal D. Atomistic near-field nanoplasmonics: reaching atomic-scale resolution in nanooptics. Nano Lett. 2015, 15, 3410–3419. 10.1021/acs.nanolett.5b00759. [DOI] [PubMed] [Google Scholar]
  24. Baumberg J. J.; Aizpurua J.; Mikkelsen M. H.; Smith D. R. Extreme nanophotonics from ultrathin metallic gaps. Nat. Mater. 2019, 18, 668–678. 10.1038/s41563-019-0290-y. [DOI] [PubMed] [Google Scholar]
  25. Chen X.; Liu P.; Jensen L. Atomistic electrodynamics simulations of plasmonic nanoparticles. J. Phys. D Appl. Phys. 2019, 52, 363002. 10.1088/1361-6463/ab249d. [DOI] [Google Scholar]
  26. Mie G. Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen. Ann. Phys.-Berlin 1908, 330, 377–445. 10.1002/andp.19083300302. [DOI] [Google Scholar]
  27. Draine B. T.; Flatau P. J. Discrete-dipole approximation for scattering calculations. J. Opt. Soc. Am. A 1994, 11, 1491–1499. 10.1364/JOSAA.11.001491. [DOI] [Google Scholar]
  28. Taflove A.; Hagness S. C.; Piket-May M.. Computational Electromagnetics: The Finite-Difference Time-Domain Method; Elsevier: Amsterdam, The Netherlands, 2005. [Google Scholar]
  29. Myroshnychenko V.; Carbó-Argibay E.; Pastoriza-Santos I.; Pérez-Juste J.; Liz-Marzán L. M.; García de Abajo F. J. Modeling the optical response of highly faceted metal nanoparticles with a fully 3D boundary element method. Adv. Mater. 2008, 20, 4288–4293. 10.1002/adma.200703214. [DOI] [Google Scholar]
  30. García de Abajo F. J.; Howie A. Retarded field calculation of electron energy loss in inhomogeneous dielectrics. Phys. Rev. B 2002, 65, 115418. 10.1103/PhysRevB.65.115418. [DOI] [Google Scholar]
  31. Mennucci B.; Corni S. Multiscale modelling of photoinduced processes in composite systems. Nat. Rev. Chem. 2019, 3, 315–330. 10.1038/s41570-019-0092-4. [DOI] [Google Scholar]
  32. Bonatti L.; Gil G.; Giovannini T.; Corni S.; Cappelli C. Plasmonic Resonances of Metal Nanoparticles: Atomistic vs Continuum Approaches. Front. Chem. 2020, 8, 340. 10.3389/fchem.2020.00340. [DOI] [PMC free article] [PubMed] [Google Scholar]
  33. Marcheselli J.; Chateau D.; Lerouge F.; Baldeck P.; Andraud C.; Parola S.; Baroni S.; Corni S.; Garavelli M.; Rivalta I. Simulating Plasmon Resonances of Gold Nanoparticles with Bipyramidal Shapes by Boundary Element Methods. J. Chem. Theory Comput. 2020, 16, 3807–3815. 10.1021/acs.jctc.0c00269. [DOI] [PMC free article] [PubMed] [Google Scholar]
  34. Coccia E.; Fregoni J.; Guido C.; Marsili M.; Pipolo S.; Corni S. Hybrid theoretical models for molecular nanoplasmonics. J. Chem. Phys. 2020, 153, 200901. 10.1063/5.0027935. [DOI] [PubMed] [Google Scholar]
  35. Luo Y.; Fernandez-Dominguez A.; Wiener A.; Maier S. A.; Pendry J. Surface plasmons and nonlocality: a simple model. Phys. Rev. Lett. 2013, 111, 093901. 10.1103/PhysRevLett.111.093901. [DOI] [PubMed] [Google Scholar]
  36. Ciraci C.; Pendry J. B.; Smith D. R. Hydrodynamic model for plasmonics: a macroscopic approach to a microscopic problem. ChemPhysChem 2013, 14, 1109–1116. 10.1002/cphc.201200992. [DOI] [PubMed] [Google Scholar]
  37. Raza S.; Bozhevolnyi S. I.; Wubs M.; Mortensen N. A. Nonlocal optical response in metallic nanostructures. J. Phys.: Condens. Matter 2015, 27, 183204. [DOI] [PubMed] [Google Scholar]
  38. Ciraci C.; Della Sala F. Quantum hydrodynamic theory for plasmonics: Impact of the electron density tail. Phys. Rev. B 2016, 93, 205405. 10.1103/PhysRevB.93.205405. [DOI] [Google Scholar]
  39. Toscano G.; Raza S.; Jauho A.-P.; Mortensen N. A.; Wubs M. Modified field enhancement and extinction by plasmonic nanowire dimers due to nonlocal response. Opt. Express 2012, 20, 4176–4188. 10.1364/OE.20.004176. [DOI] [PubMed] [Google Scholar]
  40. David C.; García de Abajo F. J. Surface plasmon dependence on the electron density profile at metal surfaces. ACS Nano 2014, 8, 9558–9566. 10.1021/nn5038527. [DOI] [PubMed] [Google Scholar]
  41. Toscano G.; Straubel J.; Kwiatkowski A.; Rockstuhl C.; Evers F.; Xu H.; Asger Mortensen N.; Wubs M. Resonance shifts and spill-out effects in self-consistent hydrodynamic nanoplasmonics. Nat. Commun. 2015, 6, 1–11. 10.1038/ncomms8132. [DOI] [PubMed] [Google Scholar]
  42. Bonatti L.; Nicoli L.; Giovannini T.; Cappelli C. In silico design of graphene plasmonic hot-spots. Nanoscale Adv. 2022, 4, 2294–2302. 10.1039/D2NA00088A. [DOI] [PMC free article] [PubMed] [Google Scholar]
  43. Gonçalves P.; Christensen T.; Rivera N.; Jauho A.-P.; Mortensen N. A.; Soljačić M. Plasmon–emitter interactions at the nanoscale. Nat. Commun. 2020, 11, 1–13. 10.1038/s41467-019-13820-z. [DOI] [PMC free article] [PubMed] [Google Scholar]
  44. Echarri A. R.; Gonçalves P.; Tserkezis C.; de Abajo F. J. G.; Mortensen N. A.; Cox J. D. Optical response of noble metal nanostructures: quantum surface effects in crystallographic facets. Optica 2021, 8, 710–721. 10.1364/OPTICA.412122. [DOI] [Google Scholar]
  45. Babaze A.; Ogando E.; Stamatopoulou P. E.; Tserkezis C.; Mortensen N. A.; Aizpurua J.; Borisov A. G.; Esteban R. Quantum surface effects in the electromagnetic coupling between a quantum emitter and a plasmonic nanoantenna: time-dependent density functional theory vs. semiclassical Feibelman approach. Opt. Express 2022, 30, 21159–21183. 10.1364/OE.456338. [DOI] [PubMed] [Google Scholar]
  46. Zhu M.; Aikens C. M.; Hollander F. J.; Schatz G. C.; Jin R. Correlating the crystal structure of a thiol-protected Au25 cluster and optical properties. J. Am. Chem. Soc. 2008, 130, 5883–5885. 10.1021/ja801173r. [DOI] [PubMed] [Google Scholar]
  47. Rossi T. P.; Zugarramurdi A.; Puska M. J.; Nieminen R. M. Quantized evolution of the plasmonic response in a stretched nanorod. Phys. Rev. Lett. 2015, 115, 236804. 10.1103/PhysRevLett.115.236804. [DOI] [PubMed] [Google Scholar]
  48. Weissker H.-C.; Mottet C. Optical properties of pure and core-shell noble-metal nanoclusters from TDDFT: The influence of the atomic structure. Phys. Rev. B 2011, 84, 165443. 10.1103/PhysRevB.84.165443. [DOI] [Google Scholar]
  49. Marchesin F.; Koval P.; Barbry M.; Aizpurua J.; Sánchez-Portal D. Plasmonic response of metallic nanojunctions driven by single atom motion: quantum transport revealed in optics. ACS Photonics 2016, 3, 269–277. 10.1021/acsphotonics.5b00609. [DOI] [Google Scholar]
  50. Sinha-Roy R.; Garcia-Gonzalez P.; Weissker H.-C.; Rabilloud F.; Fernandez-Dominguez A. I. Classical and ab Initio Plasmonics Meet at Sub-nanometric Noble Metal Rods. ACS Photonics 2017, 4, 1484–1493. 10.1021/acsphotonics.7b00254. [DOI] [Google Scholar]
  51. Morton S. M.; Jensen L. A discrete interaction model/quantum mechanical method for describing response properties of molecules adsorbed on metal nanoparticles. J. Chem. Phys. 2010, 133, 074103. 10.1063/1.3457365. [DOI] [PubMed] [Google Scholar]
  52. Morton S. M.; Jensen L. A discrete interaction model/quantum mechanical method to describe the interaction of metal nanoparticles and molecular absorption. J. Chem. Phys. 2011, 135, 134103. 10.1063/1.3643381. [DOI] [PubMed] [Google Scholar]
  53. Jensen L. L.; Jensen L. Electrostatic interaction model for the calculation of the polarizability of large noble metal nanoclusters. J. Phys. Chem. C 2008, 112, 15697–15703. 10.1021/jp804116z. [DOI] [Google Scholar]
  54. Jensen L. L.; Jensen L. Atomistic electrodynamics model for optical properties of silver nanoclusters. J. Phys. Chem. C 2009, 113, 15182–15190. 10.1021/jp904956f. [DOI] [Google Scholar]
  55. Zakomirnyi V. I.; Rinkevicius Z.; Baryshnikov G. V.; Sørensen L. K.; Ågren H. Extended discrete interaction model: plasmonic excitations of silver nanoparticles. J. Phys. Chem. C 2019, 123, 28867–28880. 10.1021/acs.jpcc.9b07410. [DOI] [Google Scholar]
  56. Rinkevicius Z.; Li X.; Sandberg J. A.; Mikkelsen K. V.; Ågren H. A hybrid density functional theory/molecular mechanics approach for linear response properties in heterogeneous environments. J. Chem. Theory Comput. 2014, 10, 989–1003. 10.1021/ct400897s. [DOI] [PubMed] [Google Scholar]
  57. Zakomirnyi V. I.; Rasskazov I. L.; Sørensen L. K.; Carney P. S.; Rinkevicius Z.; Ågren H. Plasmonic nano-shells: atomistic discrete interaction versus classic electrodynamics models. Phys. Chem. Chem. Phys. 2020, 22, 13467–13473. 10.1039/D0CP02248A. [DOI] [PubMed] [Google Scholar]
  58. Giovannini T.; Rosa M.; Corni S.; Cappelli C. A classical picture of subnanometer junctions: an atomistic Drude approach to nanoplasmonics. Nanoscale 2019, 11, 6004–6015. 10.1039/C8NR09134J. [DOI] [PubMed] [Google Scholar]
  59. Giovannini T.; Bonatti L.; Polini M.; Cappelli C. Graphene plasmonics: Fully atomistic approach for realistic structures. J. Phys. Chem. Lett. 2020, 11, 7595–7602. 10.1021/acs.jpclett.0c02051. [DOI] [PMC free article] [PubMed] [Google Scholar]
  60. Lafiosca P.; Giovannini T.; Benzi M.; Cappelli C. Going Beyond the Limits of Classical Atomistic Modeling of Plasmonic Nanostructures. J. Phys. Chem. C 2021, 125, 23848–23863. 10.1021/acs.jpcc.1c04716. [DOI] [PMC free article] [PubMed] [Google Scholar]
  61. Yamada A. Classical electronic and molecular dynamics simulation for optical response of metal system. J. Chem. Phys. 2021, 155, 174118. 10.1063/5.0067144. [DOI] [PubMed] [Google Scholar]
  62. Duan H.; Fernández-Domínguez A. I.; Bosman M.; Maier S. A.; Yang J. K. Nanoplasmonics: classical down to the nanometer scale. Nano Lett. 2012, 12, 1683–1689. 10.1021/nl3001309. [DOI] [PubMed] [Google Scholar]
  63. Scholl J. A.; Koh A. L.; Dionne J. A. Quantum plasmon resonances of individual metallic nanoparticles. Nature 2012, 483, 421–427. 10.1038/nature10904. [DOI] [PubMed] [Google Scholar]
  64. Pinchuk A.; Kreibig U.; Hilger A. Optical properties of metallic nanoparticles: influence of interface effects and interband transitions. Surf. Sci. 2004, 557, 269–280. 10.1016/j.susc.2004.03.056. [DOI] [Google Scholar]
  65. Pinchuk A.; Von Plessen G.; Kreibig U. Influence of interband electronic transitions on the optical absorption in metallic nanoparticles. J. Phys. D: Appl. Phys. 2004, 37, 3133. 10.1088/0022-3727/37/22/012. [DOI] [Google Scholar]
  66. Balamurugan B.; Maruyama T. Evidence of an enhanced interband absorption in Au nanoparticles: size-dependent electronic structure and optical properties. Appl. Phys. Lett. 2005, 87, 143105. 10.1063/1.2077834. [DOI] [Google Scholar]
  67. Liebsch A. Surface-plasmon dispersion and size dependence of Mie resonance: silver versus simple metals. Phys. Rev. B 1993, 48, 11317. 10.1103/PhysRevB.48.11317. [DOI] [PubMed] [Google Scholar]
  68. Santiago E. Y.; Besteiro L. V.; Kong X.-T.; Correa-Duarte M. A.; Wang Z.; Govorov A. O. Efficiency of hot-electron generation in plasmonic nanocrystals with complex shapes: surface-induced scattering, hot spots, and interband transitions. ACS Photonics 2020, 7, 2807–2824. 10.1021/acsphotonics.0c01065. [DOI] [Google Scholar]
  69. Giovannini T.; Puglisi A.; Ambrosetti M.; Cappelli C. Polarizable QM/MM approach with fluctuating charges and fluctuating dipoles: the QM/FQFμ model. J. Chem. Theory Comput. 2019, 15, 2233–2245. 10.1021/acs.jctc.8b01149. [DOI] [PubMed] [Google Scholar]
  70. Giovannini T.; Riso R. R.; Ambrosetti M.; Puglisi A.; Cappelli C. Electronic transitions for a fully polarizable qm/mm approach based on fluctuating charges and fluctuating dipoles: linear and corrected linear response regimes. J. Chem. Phys. 2019, 151, 174104. 10.1063/1.5121396. [DOI] [PubMed] [Google Scholar]
  71. Giovannini T.; Egidi F.; Cappelli C. Molecular spectroscopy of aqueous solutions: a theoretical perspective. Chem. Soc. Rev. 2020, 49, 5664–5677. 10.1039/C9CS00464E. [DOI] [PubMed] [Google Scholar]
  72. Verma P. Tip-enhanced Raman spectroscopy: technique and recent advances. Chem. Rev. 2017, 117, 6447–6466. 10.1021/acs.chemrev.6b00821. [DOI] [PubMed] [Google Scholar]
  73. Zhang W.; Yeo B. S.; Schmid T.; Zenobi R. Single molecule tip-enhanced Raman spectroscopy with silver tips. J. Phys. Chem. C 2007, 111, 1733–1738. 10.1021/jp064740r. [DOI] [Google Scholar]
  74. Chen X.; Moore J. E.; Zekarias M.; Jensen L. Atomistic electrodynamics simulations of bare and ligand-coated nanoparticles in the quantum size regime. Nat. Commun. 2015, 6, 8921. 10.1038/ncomms9921. [DOI] [PMC free article] [PubMed] [Google Scholar]
  75. Ringe E.; McMahon J. M.; Sohn K.; Cobley C.; Xia Y.; Huang J.; Schatz G. C.; Marks L. D.; Van Duyne R. P. Unraveling the effects of size, composition, and substrate on the localized surface plasmon resonance frequencies of gold and silver nanocubes: a systematic single-particle approach. J. Phys. Chem. C 2010, 114, 12511–12516. 10.1021/jp104366r. [DOI] [Google Scholar]
  76. Etchegoin P. G.; Le Ru E.; Meyer M. An analytic model for the optical properties of gold. J. Chem. Phys. 2006, 125, 164705. 10.1063/1.2360270. [DOI] [PubMed] [Google Scholar]
  77. Johnson P. B.; Christy R.-W. Optical constants of the noble metals. Phys. Rev. B 1972, 6, 4370. 10.1103/PhysRevB.6.4370. [DOI] [Google Scholar]
  78. Johnson H. E.; Aikens C. M. Electronic structure and TDDFT optical absorption spectra of silver nanorods. J. Phys. Chem. A 2009, 113, 4445–4450. 10.1021/jp811075u. [DOI] [PubMed] [Google Scholar]
  79. Bae G.-T.; Aikens C. M. Time-dependent density functional theory studies of optical properties of Ag nanoparticles: octahedra, truncated octahedra, and icosahedra. J. Phys. Chem. C 2012, 116, 10356–10367. 10.1021/jp300789x. [DOI] [Google Scholar]
  80. Cottancin E.; Celep G.; Lermé J.; Pellarin M.; Huntzinger J.; Vialle J.; Broyer M. Optical properties of noble metal clusters as a function of the size: comparison between experiments and a semi-quantal theory. Theor. Chem. Acc. 2006, 116, 514–523. 10.1007/s00214-006-0089-1. [DOI] [Google Scholar]
  81. Chen X.; Jensen L. Morphology dependent near-field response in atomistic plasmonic nanocavities. Nanoscale 2018, 10, 11410–11417. 10.1039/C8NR03029D. [DOI] [PubMed] [Google Scholar]
  82. Kreibig U.; Vollmer M.. Optical Properties of Metal Clusters; Springer Science & Business Media, 2013; Vol. 25. [Google Scholar]
  83. Novo C.; Gomez D.; Perez-Juste J.; Zhang Z.; Petrova H.; Reismann M.; Mulvaney P.; Hartland G. V. Contributions from radiation damping and surface scattering to the linewidth of the longitudinal plasmon band of gold nanorods: a single particle study. Phys. chem. Chem. Phys. 2006, 8, 3540–3546. 10.1039/b604856k. [DOI] [PubMed] [Google Scholar]
  84. Juvé V.; Cardinal M. F.; Lombardi A.; Crut A.; Maioli P.; Pérez-Juste J.; Liz-Marzán L. M.; Del Fatti N.; Vallée F. Size-dependent surface plasmon resonance broadening in nonspherical nanoparticles: single gold nanorods. Nano Lett. 2013, 13, 2234–2240. 10.1021/nl400777y. [DOI] [PubMed] [Google Scholar]
  85. Foerster B.; Joplin A.; Kaefer K.; Celiksoy S.; Link S.; Sönnichsen C. Chemical interface damping depends on electrons reaching the surface. ACS Nano 2017, 11, 2886–2893. 10.1021/acsnano.6b08010. [DOI] [PubMed] [Google Scholar]
  86. Douglas-Gallardo O. A.; Soldano G. J.; Mariscal M. M.; Sánchez C. G. Effects of oxidation on the plasmonic properties of aluminum nanoclusters. Nanoscale 2017, 9, 17471–17480. 10.1039/C7NR04904H. [DOI] [PubMed] [Google Scholar]
  87. Kuisma M.; Sakko A.; Rossi T. P.; Larsen A. H.; Enkovaara J.; Lehtovaara L.; Rantala T. T. Localized surface plasmon resonance in silver nanoparticles: Atomistic first-principles time-dependent density-functional theory calculations. Phys. Rev. B 2015, 91, 115431. 10.1103/PhysRevB.91.115431. [DOI] [Google Scholar]
  88. Iida K.; Noda M.; Ishimura K.; Nobusada K. First-principles computational visualization of localized surface plasmon resonance in gold nanoclusters. J. Phys. Chem. A 2014, 118, 11317–11322. 10.1021/jp5088042. [DOI] [PubMed] [Google Scholar]
  89. Payton J. L.; Morton S. M.; Moore J. E.; Jensen L. A hybrid atomistic electrodynamics–quantum mechanical approach for simulating surface-enhanced raman scattering. Acc. Chem. Res. 2014, 47, 88–99. 10.1021/ar400075r. [DOI] [PubMed] [Google Scholar]
  90. Kim M.; Kwon H.; Lee S.; Yoon S. Effect of nanogap morphology on plasmon coupling. ACS Nano 2019, 13, 12100–12108. 10.1021/acsnano.9b06492. [DOI] [PubMed] [Google Scholar]

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