Plasmons in Molecules - The Journal of Physical Chemistry C (ACS

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Plasmons in Molecules Stephan Bernadotte,† Ferdinand Evers,‡,§,* and Christoph R. Jacob*,† †

Center for Functional Nanostructures and Institute of Physical Chemistry, Karlsruhe Institute of Technology (KIT), Wolfgang-Gaede-Strasse 1a, 76131 Karlsruhe, Germany ‡ Institute of Nanotechnology, Karlsruhe Institute of Technology (KIT), Hermann-von-Helmholtz-Platz 1, 76344 Eggenstein-Leopoldshafen, Germany § Institut für Theorie der Kondensierten Materie, Karlsruhe Institute of Technology (KIT), Wolfgang-Gaede-Strasse 1, 76131 Karlsruhe, Germany ABSTRACT: The classification of electronic excitations in molecules and molecular nanostructures plays an important role when tailoring materials with desired properties. One example of such a class of excitations are plasmons. Plasmons appear in solid-state physics, where they are characterized as density oscillations in an electron gas that are driven by the restoring forces associated with the electromagnetic field induced by the density oscillations themselves. Here, we investigate how this concept can be transferred to molecular systems by performing a step-by-step analysis, starting from three-dimensional bulk systems and ending with molecules. On the basis of this analysis, we propose to scale the electron−electron interaction in quantum-chemical response calculations in order to identify plasmons in molecules. This approach is illustrated for molecular chains and clusters. Our results show that the concept of plasmons is still applicable for extended molecular systems and demonstrate that the proposed scaling approach provides an easy way of characterizing electronic excitations.

1. INTRODUCTION The optical properties of nanostructured materials and nanoclusters are determined by their electronically excited states. Tailoring these optical properties for applications ranging from chemical and biological sensing1−3 over solar energy conversion4,5 to the fabrication of metamaterials6−9 is the goal of “plasmonics”, which has emerged as an important research field at the interface of physics, chemistry, and material science in the past decades. The quantum-chemical description of the electronic excitations underlying such applications can be instructive both for the interpretation of experiments and for designing new materials.10 However, the accurate treatment of electronic excitations relevant in plasmonics presents a challenge. First, for excited states it is usually more difficult than for the ground state to calculate their properties with the accuracy required for a meaningful comparison with experiment.11−13 Second, the size of the nanoclusters of interest poses an additional difficulty.14 Not only does the computational demand increase with the size of the system, but also a larger number of excited states appear in the relevant energy range. Without a suitable classification of these electronic excitations this hampers the interpretation of the computational results and makes it difficult to identify the important excitations (for a discussion of this problem in the context of vibrational spectroscopy, see refs 15−17). Of course, a classification of different types of electronic excitations is useful both for the interpretation of quantumchemical calculations and for selectively targeting excitations © 2013 American Chemical Society

that are of interest for predicting optical or other spectroscopic properties. Typically, the classification of electronic excitations within quantum chemistry is based on a rather technical approach,11 and driven by the need to decide which quantumchemical methods are appropriate for describing the excitations of interest.12 For instance, one distinguishes valence and Rydberg excitations for determining whether asymptotically corrected exchange−correlation potentials are needed when using time-dependent density-functional theory (TDDFT)18,19 and distinguishes charge-transfer and local excitations for deciding whether TDDFT with standard nonhybrid exchange−correlation functionals can be employed.20,21 An alternative route is chosen in solid-state physics where the electronic excitations in metallic systems are classified as singleparticle excitations and plasmons.22−24 The distinction of plasmons and single-particle excitations can be motivated from the homogeneous electron gas model.23,25 Here, plasmons are collective charge density oscillations with a strong induced electromagnetic field, which provides a restoring force. These plasmonic excitations often determine the optical properties of metals, whereas the single-particle excitations play a role, e.g., in damping processes. Plasmons in extended systems are usually well-described by classical electrodynamics.24,26 These classical models can also be applied to nanosystems, such as metal clusters. For larger metal clusters, they usually Received: November 15, 2012 Revised: December 20, 2012 Published: January 22, 2013 1863

dx.doi.org/10.1021/jp3113073 | J. Phys. Chem. C 2013, 117, 1863−1878

The Journal of Physical Chemistry C

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F = −4πe 2ρ0 Δx

provide a rather accurate description of the optical spectrum27−29 whereas extended models including quantummechanical corrections accounting for the presence of discrete energy levels can be used for more complicated cases.30,31 For smaller clusters consisting of only tens to hundreds of atoms, classical models are no longer applicable and a full quantumchemical treatment is usually necessary. In this case, the optical spectrum does not show a single strong plasmon peak anymore, but instead a spectrum with distinct peaks due to transitions between discrete energy levels is obtained, which is typical of molecular systems. When increasing the size of the clusters, the optical spectrum obtained from full quantum-chemical calculations approaches the one predicted by classical models.32,33 In such calculations a large number of electronic excitations contribute to the plasmon peak, and the distinction between plasmons and single-particle excitations present for the homogeneous electron gas is not immediately apparent anymore in standard quantum-chemical calculations. Therefore, it is not straightforward to connect such calculations to the classical models of plasmons and to identify the important plasmonic excitations. Alternatively, the emergence of plasmons in molecular chains and clusters has been studied by using explicitly time-dependent methods.34−37 However, in this case the optical spectrum is obtained, but individual electronic excitations cannot be identified anymore. Thus, despite the wide use of the term “plasmons” in such molecular systems, it often remains unclear how plasmonic excitations differ from other types of electronic excitations. In fact, all excitations with large oscillator strengths calculated in metal clusters are often referred to as plasmons (see, e.g., refs 32, 33, and 38). Here, we close this gap. We present a detailed analysis showing how the concept of a plasmon can be transferred to molecular systems and propose an approach for identifying plasmonic excitations in quantum-chemical calculations for molecules. This work is organized as follows. In Section 2 the plasmonic resonances of the three-dimensional homogeneous electron gas are introduced both in a classical picture and in the framework of (quantum-mechanical) linear response theory. In addition, the distinction of plasmons and single-particle resonances is discussed. Subsequently, the downscaling from a threedimensional system to a molecular wire is carefully analyzed. This is followed by an analysis of the electronic excitations in a linear sodium chain in Section 3. On the basis of our analysis, we then propose an approach for distinguishing plasmons and single-particle excitations in Section 4. This is applied to molecular chains and clusters in Section 5 and Section 6, respectively. Finally, a summary and concluding remarks are given in Section 7.

(1)

where e is the electron charge and ρ0 is the electron (particle) density. Here and in the following, we are using Gaussian units. Classically, the movement of the electrons can be described by Newton’s equation, m

∂ 2Δx + 4πe 2ρ0 Δx = −e ,ext ∂t 2

(2)

where m is the (effective) electron mass. Thus, one finds that the system behaves like a driven harmonic oscillator. Such a driven harmonic oscillator has a resonance frequency at

ω2 =

4πe 2ρ0 (3)

m

This resonance corresponds to a collective electronic excitation of the classical electron gas, in which all electrons move as one entity. Comparing with eq 1, we notice that this resonance frequency is determined by the restoring force caused by the induced electric field. 2.2. Quantum-Mechanical Description. As starting point for a microscopic treatment, one considers the induced density δρ(r,ω) as a response to an external scalar potential Φext(r,ω) oscillating with angular frequency ω, δρ(r, ω) =

∫ χext (r, r′, ω)Φext(r′, ω) d3r′

(4)

where the external density response function χext(r,r′,ω) has been introduced. This external response function depends on the angular frequency ω of the perturbation and its poles correspond to electronic excitation energies. The induced density gives rise to an induced Hartree potential, Φind(r, ω) =

∫ fCoul (r − r′)δρ(r′, ω) d3r′

where f Coul is the Coulomb kernel, e fCoul (r − r′) = |r − r′|

(5)

(6)

The induced Hartree potential adds to the external potential, yielding the total potential Φtot(r, ω) = Φext(r, ω) + Φind(r, ω)

(7)

The response equation, eq 4, can then be recast in the form

∫ χext (r, r′, ω)Φext(r′, ω) d3r′ = ∫ χirr (r, r′, ω)Φtot(r′, ω) d3r′

δρ(r, ω) =

(8)

where we have introduced the irreducible response function χirr(r,r′,ω) describing the response of the electron density to the change in the total potential. After substituting eqs 7 and 5 into the right-hand side of the above expression and rearranging, we obtain an integral equation,

2. PLASMONS: FROM PERIODIC TO MOLECULAR SYSTEMS 2.1. Phenomenological Description. In solid-state physics, plasmons are typically discussed in the context of metals as long wavelength charge-density oscillations induced by an external perturbation.22,24 For a classical description, we consider a homogeneous electron gas with a homogeneous positively charged background in a three-dimensional (3D) cube of volume V. If an external electric field ℰext is applied, the electron gas exhibits a density response, which is given by a displacement of the electronic density against the background by a shift Δx. This induced density in return gives rise to an induced electric field acting as a restoring force F,

∫ ε(r, r′′, ω)χext (r′′, r′, ω) d3r′′ = χirr (r, r′, ω)

(9)

where we introduced the dielectric function, ε(r, r′′, ω) = δ(r − r′′) −

∫ χirr (r, r′′′, ω)fCoul (r′′′ − r′′) d3r′′′ (10)

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Defining the inverse dielectric function ε−1 via

∫ ε−1(r, r′′, ω)ε(r′′, r′, ω) d3r′′ = δ(r − r′)

within the orbitals, e.g., within the Hartree or Kohn−Sham approximation) and can serve as an approximation for χirr. This approximation of using the response function of a noninteracting reference system for χirr is called Random-Phase Approximation (RPA). It is closely related to time-dependent density-functional theory (TDDFT),42−45 where χ(0) is also used for the irreducible response function, but where an exchange−correlation kernel f xc enters in addition to the Coulomb kernel in eq 5. The poles of χ(0) describe excitations from occupied to unoccupied states and correspond to orbital energy differences, i.e., ω = (εa − εi)/ℏ. 2.4. Three-Dimensional Electron Gas. We consider a noninteracting homogeneous electron gas with volume V. The single-particle states are given by plane waves,

(11)

one can solve for χext and obtains χext (r, r′, ω) =

∫ ε−1(r, r′′, ω)χirr (r′′, r′, ω) d3r′′

(12)

To find the electronic excitations of the system, we have to determine the poles of the external response function χext. Considering eq 12, one can naively identify two different ways in which poles can arise.22,39 First, poles of χext should appear for frequencies that correspond to poles of χirr. Second, poles of χext could also arise if the dielectric function ε has a zero mode (i.e., an eigenvalue equal to zero) for a certain frequency, i.e., if

∫ ε(r, r′, ωplas)π(r′, ωplas) d r′ = 0

φk(r ) =

3

(13)

ε(k) =

ℏ2k 2 2m

(16)

where k = |k|. The energy of the highest occupied state is the Fermi energy EF, with the corresponding absolute value of the wavevector kF. The separation of the electronic excitations into singleparticle excitations and plasmons introduced above is applicable here. Therefore, we treat single-particle excitations and plasmons separately by determining the poles of the irreducible response function χirr, approximated by χ(0) within the RPA, and the zeros of the dielectric function ε, respectively. Subsequently, we clarify why this separate treatment is indeed appropriate and when it breaks down. The possible single-particle excitation energies (i.e., the poles of χ(0)) are obtained by subtracting the energy of the state with wavevectors k below the Fermi energy from a state with wavevector k + q above the Fermi level, where the wavevector of the single-particle excitation is q. Thus, the single-particle excitation energy is Δεsp(k, q) = ε(k + q) − ε(k) =

ℏ2 ℏ2 2 k·q + q m 2m

(17)

In macroscopic systems (i.e., in the limit that the volume V goes to infinity), the wavevectors can be treated as continuous, and from the boundary conditions |k| ≤ kF and |k + q| ≥ kF one finds for the single-particle continuum Δεsp,25 Δεsp(q) ≤ ℏvFq +

χ (0) (r, r′, ω) = −2e ∑ ϕi*(r )ϕa(r)ϕi(r′)ϕa*(r′)

ℏ2 2 q 2m

Δεsp(q) ≥ −ℏvFq +

ia

2(εa − εi) (εa − εi)2 − ℏ2ω 2

(15)

which can be classified by their wavevector k due to the translational invariance. Consequently, the single-particle (orbital) energies become a function of the wavevector, the so-called dispersion,

for a nonzero π(r′,ωplas) at the frequency ωplas. This distinction allows for an attempt to classify electronic excitations: The first type (i.e., those originating from poles of χirr) will be called single-particle excitations, whereas the second type (i.e., those originating from zero modes of ε) will be labeled plasmons. Note that the dielectric function ε contains the bare Coulomb kernel f Coul whereas the irreducible response function χirr can be expressed via a perturbation expansion in the screened Coulomb interaction and is independent of f Coul in zeroth order. Thus, single-particle excitations and plasmons are distinguished by their different dependence on the Coulomb kernel. Two remarks should be made here: First, we notice that because ε−1 depends on χirr, the frequencies of poles of χirr will not be exactly equal to those of poles of χext, but the presence of f Coul leads to a (usually small) shift. This can be seen by multiplying eq 9 by (ω − ω0) where ω0 is a resonance frequency and solving for χext in the same manner as in eq 12. Second, if a frequency for which ε has a zero-mode (i.e., a plasmonic excitation) is close to a pole of χirr (i.e., a singleparticle excitation), these two types of excitations will mix if they do not have different symmetries. Thus, in this case the separation of the electronic excitations (i.e., the poles of χext) into plasmons and single-particle excitations may break down. 2.3. Random-Phase Approximation. The linear-response formalism presented in the previous section can be employed for the (quantum-mechanical) calculation of the excitation energies. The simplest ansatz for the ground-state electronic wave function is a Slater determinant with single-particle orbitals {ϕp}. The corresponding orbital energies are denoted as εp. The electrons occupy the lowest states according to the aufbau principle. Using the common quantum-chemical notation, we label the occupied orbitals with an index i and the unoccupied ones with an index a. From time-dependent perturbation theory for a system of noninteracting electrons, one can calculate the linear response function,40,41

×

1 exp(ik·r ) V

ℏ2 2 q , if q > 2kF 2m

(18)

(19)

where we introduced the Fermi velocity vF = ℏkF/m. To determine the energies of the plasmonic excitations, we calculate the density response function within the RPA by taking the Fourier transform of eq 14,

(14)

This response function contains no interaction effects of the electrons (except for the interaction effects that are included 1865


The Journal of Physical Chemistry C 2e χ (q, ω) = − (2π )3 (0)

∫|k| 0.6. 1873



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Figure 6. Change of the squared excitation energies of a tetrahedral Na20 cluster upon scaling the electronic interaction (left) and isosurface plots of the identified plasmons (right). Excitation energies are plotted in units of the HOMO−LUMO gap (Δ = 0.92 eV) and the coloring scheme described in the text is used to follow the plasmonic excitations, using the excitations highlighted by a black circle as reference points. For dipole-allowed excitations, the linear absorption spectrum is also included.

Figure 7. Change of the squared excitation energies of a tetrahedral Ag20 cluster upon scaling the electronic interaction (left) and isosurface plots of the identified plasmons (right). Excitation energies are plotted in units of the HOMO−LUMO gap (Δ = 1.49 eV) and the coloring scheme described in the text is used to follow the plasmonic excitations, using the excitations highlighted by a black circle as reference points. For dipole-allowed excitations, the linear absorption spectrum is also included.

symmetry axis. The identified plasmonic excitation is mainly responsible for the strong peak in the optical spectrum of Ag20 calculated at 3.3 eV with BP86/TZP32 (see also the linear absorption spectrum included in the figure). However, the coloring according to the overlap with the reference excitation at λ = 0.1 shows that the plasmonic excitation also mixes with two single-particle excitations at lower energies. Therefore,

these single-particle excitations also gain dipole oscillation strengths and also contribute to the strong peak in the optical spectrum. Additional plasmons can also be identified for the A1 and E symmetries. These show induced densities similar to those 1874



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already identified in Na20 that correspond to a breathing mode and to a displacement of the density along the C3 axis with two nodal planes. Even though these plasmons have a large induced density, their dipole oscillator strengths vanish because of their symmetry, and the lowest nonzero contributions to the oscillator strengths are due to the induced octupole and quadrupole moments, respectively. 6.3. Fullerene C60. Finally, we investigate C60 as an example of a molecule-like cluster. The first analysis of the plasmons of C60 was performed in ref 56. These calculations predicted strong peaks in the optical spectrum at ca. 6 eV and at ca. 20 eV, which were assigned to “π-plasmons” and “σ-plasmons”, respectively. However, this early study did not distinguish plasmons and single-particle excitations. Subsequently, plasmons in fullerenes were investigated theoretically in more detail by Ju et al. using model calculations assuming a spherically symmetric system.57 In this case, the plasmonic excitations may be labeled according to their angular momentum by quantum numbers (J,mJ). Only the plasmons corresponding to J = 1 have large dipole oscillator strengths and can be assigned to strong peaks in the optical spectrum of fullerenes. The point group of C60 is Ih and the excitations with J = 1 in a spherically symmetric model belong to the irreducible representation T1u, whereas those with J = 2 belong to the Hg representation. As icosahedral symmetry is not supported in our quantum-chemical calculations, all calculations have been performed for the D5d point group. The previously discussed excitations then do not transform like single irreducible representations anymore, but the three degenerate excitations corresponding to J = 1 belong to A2u and E1u, whereas the five degenerate excitations corresponding to J = 2 belong to A1g, E1g, and E2g. For simplicity, only the excitations transforming like the one-dimensional irreducible representations A2u and A1g are analyzed in the following. The dependence of the squared excitation energies as calculated with TDDFT on the scaling parameter λ is depicted in Figure 8. For the excitations with A2u symmetry, two plasmons with one nodal plane can be identified. These correspond to J = 1 and can be identified with the excitations assigned previously to the intense peaks at 5.9 eV and at 6.4 eV in the optical spectrum58 (see also the linear absorption spectrum included in the figure). Even though these plasmonic excitations, in particular the second one, are close to singleparticle excitations, virtually no mixing is observed here. In addition, a plasmon with three nodal planes (i.e., corresponding to J = 3) can be identified at higher excitation energy. Note, however, that also two of the lower-lying single-particle excitations have significant oscillator strengths and transition densities similar to those of the identified plasmons with J = 1. However, as their excitations energies do not increase with λ above ca. 0.5, we did not classify these excitations as plasmons. For the excitations with A1g symmetry, three plasmonic excitations can be found. These each have two nodal planes and correspond to J = 2. The lowest-energy plasmon starts to mix with single-particle excitations for higher values of λ, as can be seen from the change in the slope. Finally, we note that all plasmons identified here are due to transitions between π orbitals and can, therefore, be labeled π-plasmons, whereas transitions between σ-orbitals appear at excitation energies that are higher than those considered here.

Figure 8. Change of the squared excitation energies of a fullerene C60 cluster upon scaling the electronic interaction (left) and isosurface plots of the identified plasmons (right). Excitation energies are plotted in units of the HOMO−LUMO gap (Δ = 1.66 eV) and the coloring scheme described in the text is used to follow the plasmonic excitations, using the excitations highlighted by a black circle as reference points. For dipole-allowed excitations, the linear absorption spectrum is also included.

7. CONCLUSIONS In this work, we have investigated how plasmonic excitations can be identified in quantum-chemical calculations for molecular systems. To this end, we revisited the threedimensional electron gas model, where a distinction of singleparticle excitations, which correspond to poles of the irreducible response function χirr, and plasmons, which are due to zeros of the dielectric function ε, emerges naturally. By a step-by-step analysis going from the 3D electron gas over a 1D electron gas to an electron gas in a finite wire, we demonstrated that this separation is preserved for excitations with small wavevectors q. As a final step, we could show that for a molecular sodium chain, the results of a quantum-chemical calculation can be mapped onto the electron gas model. This demonstrates that in molecular systems a classification of electronic excitations as plasmons or single-particle excitations can still be possible. 1875



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chemistry. Another important area is the coupling of molecules to small nanoclusters, both in optical67,68 and in Raman spectroscopy.53,69,70 Here, indentifying the relevant plasmonic excitations is a prerequisite for a quantum-chemical treatment with subsystem approaches.14,71−73 Moreover, with a method for identifying plasmonic excitations available, it will also become possible to calculate these excitations selectively.74−76 Finally, the present development will also allow for the parametrization of empirical models for treating plasmons in nanoclusters against full quantum-chemical calculations.77,78

To identify plasmons in general molecules, we propose to analyze the excitations by scaling the electron−electron interaction in the response calculation. While single-particle excitations are only slightly affected by this scaling, plasmonic excitations are sensitive to variations in the scaling parameter λ. More specifically, their squared excitation energies should increase linearly with the scaling parameter λ. This approach has been implemented for TDDFT and was applied for molecular chains and clusters. It can be easily incorporated into any quantum-chemical program package. Depending on the system, a number of different cases can be observed. First, in many cases plasmons and single-particle excitations can be clearly distinguished with our analysis. While for some excitations, identified as plasmons, the squared excitation energy increases linearly with the scaling parameter λ, the energies of most excitations, identified as single-particle excitations, are largely independent of λ. Moreover, the composition of the plasmonic excitations is independent of λ and plasmonic and single-particle excitations do not mix even if they become close in energy. Such a behavior is found for the Na20 chain, but also for some of the plasmonic excitations in other systems. Second, for some molecular systems, such as the considered alkene chain, the identified plasmonic excitations mix with certain single-particle excitations if they become close in energy for a certain value of λ. This shows up in our plots as avoided crossings. In these cases, the coloring scheme applied here turns out to be useful for following the plasmonic excitations. Finally, in some cases a strong mixing of plasmonic excitations and single-particle excitations is observed. This is the case for the plasmon with three nodes in the considered Au20 chain. Here, the s−s plasmonic excitation mixes with d−s interband singleparticle excitations with the same (approximate) wavevector. Thus, the distinction of plasmons and single-particle excitations breaks down in this case. With the analysis tool developed here for identifying plasmonic excitations, it becomes possible to investigate the dependence of plasmons in molecular clusters as a function of the cluster size. In particular, it will be interesting to study the minimal size of a molecular system that is required to clearly distinguish plasmons from single-particle excitations and to investigate the transition to the regime in which classical models of plasmons become appropriate.29 The possibility of identifying specific excitations as plasmons opens the way to directly connecting quantum-chemical calculations to such classical models. We plan to explore these questions in our future work. The scaling approach introduced here for identifying plasmons is not limited to TDDFT, but can also be applied in combination with wave function based quantum-chemical response methods,14 such as linear-response coupled cluster theory59,60 or the algebraic diagrammatic construction (ADC) scheme.61−63 By doing so, one can overcome the shortcomings of common TDDFT exchange−correlation kernels in the description of charge-transfer excitations.50,51 If such chargetransfer excitations, referred to as excitons in solid-state physics, are described correctly, our scaling analysis should reveal these as an additional, third class of excitations for which the excitation energy decreases with the scaling parameter λ. Already in combination with TDDFT, the possibility to identify plasmons will allow for dedicated quantum-chemical studies for applications in plasmonics. One example is the design of molecular metamaterials64−66 guided by quantum

8. COMPUTATIONAL DETAILS All calculations in this work were performed with a locally modified version the ADF program package,79,80 using the BP86 exchange−correlation functional81,82 and a triple-ζ basis set with one set of polarization functions (TZP). All molecular structures were optimized with the same computational methodology. For the calculations on the gold chain, scalarrelativistic corrections were included via the zeroth-order regular approximation (ZORA).83−86 For the preparation of the figures using our coloring scheme, we employed the scripting framework PYADF87 in combination with the MATPLOTLIB package.88 The linear absorption spectra included in the figures are plotted by using a Lorentzian line shape for each excitation, with an area proportional to the calculated dipole oscillator strength and a full width at half-maximum of 0.2ℏω/Δ.

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AUTHOR INFORMATION

Corresponding Author

*Tel.: +49 721 608 26444 (F.E.) and +49 721 608 48032 (C.R.J.). Fax: +49 721 608 26468 (F.E.) and +49 721 608 48496 (C.R.J.). E-mail: [email protected] (F.E.) and [email protected] (C.R.J.). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank the DFG-Center for Functional Nanostructures for financial support. REFERENCES

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Plasmons in Molecules - The Journal of Physical Chemistry C (ACS

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