Molecular Plasmon−Phonon CouplingYao Cui,†,# Adam Lauchner,‡,# Alejandro Manjavacas,*,§ F. Javier Garcıa de Abajo,∥,⊥Naomi J. Halas,*,†,‡,∇,# and Peter Nordlander*,‡,∇,#

†Department of Chemistry, Rice University, Houston, Texas 77005, United States‡Department of Electrical and Computer Engineering, Rice University, Houston, Texas 77005, United States§Department of Physics and Astronomy, University of New Mexico, Albuquerque, New Mexico 87131, United States∥ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain⊥ICREA-Institucio Catalana de Recerca i Estudis Avancats, Passeig Lluís Companys, 23, 08010 Barcelona, Spain∇Department of Physics and Astronomy, Rice University, Houston, Texas 77005, United States#Laboratory for Nanophotonics, Rice University, Houston, Texas 77005, United States

*S Supporting Information

ABSTRACT: Charged polycyclic aromatic hydrocarbons(PAHs), ultrasmall analogs of hydrogen-terminated grapheneconsisting of only a few fused aromatic carbon rings, have beenshown to possess molecular plasmon resonances in the visibleregion of the spectrum. Unlike larger nanostructures, the PAHabsorption spectra reveal rich, highly structured spectralfeatures due to the coupling of the molecular plasmons withthe vibrations of the molecule. Here, we examine thismolecular plasmon−phonon interaction using a quantummechanical approach based on the Franck−Condon approx-imation. We show that an independent boson model can beused to describe the complex features of the PAH absorptionspectra, yielding an analytical and semiquantitative descriptionof their spectral features. This investigation provides an initial insight into the coupling of fundamental excitationsplasmonsand phononsin molecules.KEYWORDS: plasmonics, plasmon−phonon coupling, graphene, polyacenes, PAHs

Plasmons, collective oscillations of electrons,1,2 provide amechanism for subwavelength light confinement and

manipulation. Plasmon resonances in nanoparticles can betuned by changing the morphology or composition of thenanostructure.3−8 Recent studies have shown that chargedpolycyclic aromatic hydrocarbons (PAHs) support a set ofcollective resonances that are strongly dependent upon theelectron−electron interaction strength9,10 and are derived froma superposition of multiple in-phase electron excitations.11 Inmolecules composed of less than ∼50 carbon atoms, thesemolecular plasmon resonances lie in the visible region of thespectrum and have energies that depend on the charge state ofthe molecules in a manner analogous to the dependence ofgraphene plasmons on doping. As the smallest examples ofgraphene and as readily available chemical species, PAHsprovide an ideal platform for molecular plasmonics.Experimental observations of molecular plasmons in charged

aromatic PAHs have revealed rich spectral features resultingfrom the coupling of collective electronic modes and atomicvibrations.12 In our previous work, we used time-dependentdensity functional theory (TDDFT) to demonstrate thatcoupling of the molecular plasmon resonance to vibrational

modes of the molecule produces an energy level splittingobserved in the experimental measurements. The calculatedplasmonic−vibronic energy levels agreed qualitatively with theobserved extinction spectra and with previous TDDFTinvestigations of PAH phonon modes.13,14 This approachrelied on the Franck−Condon principle and is referred to hereas TDDFT-FC.15

Substrate lattice phonons are well known to affect theelectro-optical properties of graphene,16 whereas its intrinsicphonons, as measured using Raman spectroscopy, serve as keyindicators of structural integrity and chemical purity.17

However, limited effort has been invested to examine theeffect of intrinsic phonon coupling with electronic andplasmonic resonances in carbon allotropes, such as gra-phene18−21 and nanotubes.22 Here, we present a comprehen-sive experimental and theoretical investigation of the molecularplasmon−phonon coupling for different PAH molecules. Wealso provide a microscopic analysis of this coupling based on

Received: July 6, 2016Revised: September 25, 2016Published: September 26, 2016

Letter

pubs.acs.org/NanoLett

© 2016 American Chemical Society 6390 DOI: 10.1021/acs.nanolett.6b02800Nano Lett. 2016, 16, 6390−6395

the independent boson model (IBM),23 originally introducedto describe the coupling between electronic excitations andbosonic phonon modes in solids. This exactly solvableanalytical model provides a simple, intuitive expression formolecular plasmon−phonon coupling and enables semi-quantitative agreement with experimental data.The impact of molecular plasmon−phonon coupling on the

absorption spectra is illustrated schematically in Figure 1.

Without molecular vibrations, the electronic transition, shownin panel (a), gives rise to a single absorption peak. By includingthe coupling to vibrations, the ground state can be connected todifferent vibrational levels in the excited state, which results indifferent peaks in the spectrum, as shown in panel (b). As a

consequence of this, the dominant peak is red-shifted comparedto the analysis without vibrations.In recent work on molecular plasmonics, we measured the

absorption spectra of isomeric three- and four-unit planararomatic ring molecules dissolved in tetrahydrofuran (THF).We interpreted these spectra using TDDFT calculations. Forthe charged linear three- and four-ring molecules anthraceneand tetracene, the vibrationally resolved absorption spectrawere simulated by only considering the coupling between thestrongest molecular plasmon and the vibrations of the moleculewithin the spectral region under consideration (1.2−3.0 eV).Here, we go beyond that approximation and investigate bothmolecules in a more detailed manner, analyzing the vibrationalstructure of additional molecular plasmon peaks.We start our study by analyzing the electronic modes of

tetracene, anthracene, and coronene doped with one electron.In panels (a), (b), and (c) of Figure 2 we show the inducedcharge densities of tetracene, anthracene, and coronene,respectively. The induced charge densities of the strongestmolecular plasmon of tetracene and anthracene, polarized alongthe long axis of each molecule, are shown. We label this modeas the longitudinal molecular plasmon (LP). Tetracene andanthracene present an additional weaker molecular plasmonpeak, located at 1.96 eV for the anthracene anion and at 1.38 eVfor the tetracene anion. Examining the induced charge of theseexcitations, we observe that they correspond to oscillationsalong a direction perpendicular to the molecular axis, andtherefore, we label them as transverse molecular plasmons(TP). Panels (d) and (e) of Figure 2 show their correspondingmeasured absorption spectra in tetrahydrofuran (THF) (graycurves), together with the TDDFT-FC calculations for thelongitudinal (orange curves) and transversal (green curves)spectra. For both molecules, the main features of theexperimental spectra are well reproduced (within the expectederror for TDDFT calculations24) by the sum of the vibration-

Figure 1. Molecular plasmon−phonon coupling. Schematics illustrat-ing the absorption spectrum of a molecule in the absence (a) and inthe presence (b) of coupling to molecular vibrations.

Figure 2. Absorption spectra of singly charged PAHs. Induced charge density plots for molecular plasmonic resonances oriented along thelongitudinal and transverse axes of (a) tetracene, (b) anthracene, and (c) coronene. Experimental extinction spectra (gray) are compared withTDDFT-FC absorption spectra (black) obtained by coupling to two different vibrationally expanded transverse plasmon resonances (TP, green) andlongitudinal molecular plasmon resonances (LP, orange) for (d) tetracene, (e) anthracene, and (f) coronene.

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ally resolved TDDFT spectra associated with each of the twomolecular plasmon modes (black curves).As a model molecule for studies on graphene surfaces,

coronene belongs to the class of pericondensed benzenoidaromatic molecules C6s

2H6s, with s = 2 and has D6h symmetry.Because of its planar 6-fold symmetry, the molecular plasmonmodes for coronene anion cannot be classified as longitudinalor transverse. Instead, the induced charge densities for the twomodes within the region 1.2−2.4 eV oscillate along the x axisand y axis, respectively (Figure 2c). The strength of the twomodes does not display the large asymmetry exhibited by theLP and TP modes in the linear PAHs, although the magnitudesof their respective molecular plasmon−phonon couplings differsignificantly, with the higher-energy molecular plasmonresonance exhibiting a more pronounced coupling to vibra-tional modes. Figure 2f shows the absorbance spectrum ofcoronene (C24H12) anion measured in THF solvent and thevibronic spectra for all peaks within the relevant spectral region,as calculated with the radical in vacuum.Figure 3 shows the main vibrational modes that couple to the

LP and the TP resonances in the anthracene anion. The peakstructure is dominated by coupling to only a few vibrationalmodes. For the longitudinal molecular plasmon (∼1.75 eV),one of the most intense bands (at 630 cm−1) is assigned tomode 17−1 (here, we use the notation n − x, where n is theexcited normal vibrational mode in the Gaussian standardnomenclature25 and x is its population). The atomic displace-ments associated with this vibrational mode (Figure 3c) involvea CCC deformation of the aromatic rings, with outer ringdeformation being out-of-phase relative to the center ringdeformation. Doubly populated vibrational modes, such as the17−2, also contribute to the spectrum (see SupportingInformation for more details). The peak at 1272 cm−1

corresponds to the 42−1 mode, associated with the CHbending in which the center ring evolves in a breathing mode.Similarly, the mode at 1425 cm−1, labeled as 48−1, correspondsto the CC stretching of the aromatic group. A doubly excitedmode resulting from the combination of modes 42−1 and 48−1 contributes to the vibrationally resolved absorption peakaround 1.9 eV, which leads to a high amplitude compared tothe ∼1.83 eV peak. For the transverse molecular plasmon(∼1.96 eV), the vibronic spectrum displays similar features butwith much smaller amplitude. The most intense bands areassociated with the singly and doubly populated 8, 48, and 52vibrational modes, which correspond to CCC deformations(with all three rings deforming in phase), CC stretching, andC−H bending (with the center ring in a CC stretchingmode), respectively. A comprehensive description of allvibrational modes is provided in the Supporting Information.The strongly coupled modes correspond almost exclusively

to in-plane vibrations. Examining the most strongly coupledvibrational mode for each molecular plasmon, we observe thatthe longitudinal molecular plasmon interacts strongly withvibrational modes in which the atomic displacements arealigned primarily along the transverse axis. Similarly, for thetransverse molecular plasmon, the atomic displacements of thecoupled vibrational modes are aligned along the longitudinalaxis. To provide a more intuitive understanding of themolecular plasmon−phonon coupling, we use the IBM todescribe plasmonic excitations interacting with vibrationalmodes (for more details, see the Supporting Information).Briefly, in this approach, the system is described using a model

Hamiltonian consisting of the bare plasmon and the relevantvibrational modes q

∑ ∑ω ω= ℏ + + + ℏ+ + +H c c M a a a a[ ( )]q

q q qq

q q q0

where c+ and c and are the field operators for the plasmonmode of energy ℏω0, whereas aq

+ and aq are the field operatorsfor the bosonic vibrational modes of energies ℏωq calculatedusing the TDDFT approach. The coupling coefficients Mqbetween the plasmon and vibrations are treated as adjustableparameters. The Hamiltonian can be solved analytically,providing absorption spectra that can be compared directlywith the TDDFT-FC results and experimental data. To simplify

the equations we introduce the two parameters = ωℏ⎜ ⎟⎛⎝

⎞⎠gq

M 2q

q

Figure 3. Vibrational mode coupling of the anthracene molecularplasmon. Assignment of the main vibrational bands of (a) thelongitudinal molecular plasmon mode and (b) the transversalmolecular plasmon mode. The vibronic modes are denoted n − x,where n is the excited normal vibrational mode and x is its population.The horizontal axis shows the energy relative to the 0−0 transition. (c,d) Induced charge density plots for the two plasmon resonances. (e, f)Atomic displacement vectors for the three most prominent vibrationalmodes of (a) and (b).

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and Δ = ∑q ωqgq. For finite temperature β = 1/kBT, the opticalabsorption spectrum is obtained from the Fourier transform ofthe Green function, which can be written as the product of thetraces over the plasmonic and vibrational degrees of freedom as

∫ω ∝ − − Πω∞ −ΦA i te t e( ) Im{ d ( ) }i t t

0

( )

where the vibrational trace takes the form e−Φ(t), with

∑Φ = − + + −ω ω−t g N e N( ) [ (1 ) ( 1)(1 e )]q

q qi t

qi tq q

and

∑Π = + β ω ω

=

∞− ℏ −Δ − −Δ +t N n e( ) ( 1) ep

n

n n i n t

0

( ) [ (2 1)]0 0

(1)

In these expressions, Nq−1 = eβℏωq − 1 and Np

−1 = ∑n = 0∞

e−βℏ(ω0−Δn)n. Equation 1 represents a generalization of theconventional IBM to account for the bosonic statistics of theplasmon, which is useful in many-electrons systems. For thepresent case of small molecules, for which the molecularplasmon has a strongly anharmonic character, as it is supportedby a small number of electrons, we must restrict the summationto the n = 0 term.Because of its analytical structure, this model provides a

direct and intuitive account for how the temperature, thecouplings, and the population of the vibrational modesinfluence the absorption spectra (see Figures S1−S3 in theSupporting Information).We further measured the absorbance spectra of singly

charged linear PAHs with increasing lengths (two- to five-unitplanar aromatic rings) in THF solution (Figure 4). Vibration-ally resolved absorption spectra calculated by TDDFT-FC takeinto consideration only the strongest transition peak in theelectronic spectrum for each molecule within the spectralregion under examination. By considering the most prominent

phonon energies from the TDDFT-FC output, the IBM iscapable of modeling the fine structure in both measured andTDDFT-FC spectra (see the Supporting Information for moredetails). The addition of an extra benzene ring (n = 3, 4, 5)causes a red shift of the longitudinal plasmon in the linearPAHs, which agrees well with previous results showing that thelow-energy peak appears at a wavelength that scales almostlinearly with the number of benzene rings.10 This is similar tothe behavior observed in gold nanorods for which there exist anapproximate proportionality between the nanorod length andthe dipolar plasmon frequency.26

In this study, we have adjusted the coupling coefficients Mqto the TDDFT-FC-derived absorption spectra (see Methods).However, it would be equally valid to fit these parameters toexperimental data, providing a quick method for character-ization of the coupled modes observed in the absorptionspectra. Such characterization based on the IBM provides UV−vis spectroscopy with a degree of molecular specificity typicallyobserved with infrared vibrational spectroscopy.We also examined molecular plasmon−phonon coupling in

PAH molecules for the case of pyrene and its benzene-fusedderivative benzo[a]pyrene (Figure 5). The experimental spectraof the two molecules as singly charged anions are very similar,with the primary difference being that benzo[a]pyrene exhibitsa red shift relative to pyrene, similar to the red shift observedwith increasing aspect ratio for linear PAHs. From TDDFT-FCsimulations, we find that not only are the vibronic spectra

Figure 4. Absorption spectra of charged linear PAHs. Experimentalextinction spectra (left), calculated TDDFT-FC absorption spectra(center), and calculated IBM absorption spectra (right) for PAHmolecules with 2−5 benzene rings (top-to-bottom: naphthalene,anthracene, tetracene, and pentacene).

Figure 5. Absorption spectra of geometrically similar molecules. (a)Experimental (dashed) and TDDFT-FC (solid) absorption spectra forbenzo[a]pyrene (red) and pyrene (green). (b) Simulated vibronicresonances for benzo[a]pyrene (red) and pyrene (green). Vibrationvectors for the two strongest vibrational states, (c) in-phase CCCdeformation and (d) CC stretching, for pyrene and benzo[a]pyrene(top and bottom, respectively).

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similar qualitatively, but so are the atomic displacements in thevibrational modes and their relative couplings. This is shown inFigure 5, where the primarily vibronic components come froma longitudinal stretching mode (9−1 in pyrene, 10−1 inbenzo[a]pyrene) and a CC stretching mode (modes 62−1and 78−1 in pyrene and benzo[a]pyrene, respectively). Theadditional benzene ring has significantly shifted the molecularplasmon resonance with almost no modification of themolecular plasmon−phonon coupling.In conclusion, we have examined and analyzed the complex

vibrational structure observed in molecular plasmon absorptionspectra for a variety of charged PAHs. We find that molecularplasmons tend to couple more strongly to phonon modeswhose vibrational motion is perpendicular with respect to themain direction of polarization associated with the molecularplasmon. By using spectral information from TDDFT-FCsimulations, we have determined a few parameters that bringthe analytical IBM into close agreement with the experimentalspectra, thus resulting in a simple and intuitive expression thatallows us to conclusively elucidate the effect of molecularplasmon-vibrational coupling in these systems.Methods. Experimental Section. For spectroelectrochem-

ical characterization, PAH molecules were dried and dissolvedin dry solvent at 5 mM concentration with 500 mM supportingelectrolyte (tetrabutylammonium perchlorate, TBAP) pur-chased from Sigma-Aldrich and dried by recrystallization fromdry diethyl ether. Room temperature tetrahydrofuran was usedas the solvent for all molecules except pentacene, which wasdissolved in o-dichlorobenzene (at the same concentrations ofPAH and TBAP), heated to 160 °C, and subsequentlymeasured before cooling. Charge transfer was accomplishedby three-electrode cyclic voltammetry (Pt mesh electrodes withAg wire pseudo reference) performed concurrently withabsorbance spectroscopy under white-light illumination. Theexperimental absorption spectra presented in this paper wereobtained at voltages just below the reduction potential of eachmolecule. For additional details see Lauchner et al.12

Quantum Mechanical Calculations. All quantum mechan-ical calculations were performed using Gaussian 09.27 Fullgeometry optimizations and harmonic frequency calculationswere conducted for all molecules in vacuum and in both groundand excited states using DFT and TDDFT methods,respectively. The absorption spectra and transition densitieswere calculated using TDDFT. The vibrationally resolvedabsorption spectra were obtained within the framework of theFranck−Condon (FC) principle15,28 upon TDDFT-basedoverlap integrals between the vibrational wave functions ofthe ground and the excited states, as proposed by Barone etal.15 The rendered spectra were calculated by convoluting theresonant energy intensity with a Gaussian with a half width athalf-maximum (HWHM) of 135 cm−1, which still allows us toassign individual vibronic contributions to the total spectrum.The hybrid B3LYP functional and the 6-31+G(d) basis set wereused throughout all calculations following previous works.29,30

Independent Boson Model. The independent bosonmodel23 was extended from the standard fermion−bosoncoupling to also describe boson−boson interactions, asappropriate for plasmon excitations in the many electronlimit. The phonon energies were obtained from TDDFT-FCcalculations, whereas the coupling parameters were determinedby fitting the calculated vibrationally resolved absorptionspectra using a genetic algorithm.

■ ASSOCIATED CONTENT*S Supporting InformationThe Supporting Information is available free of charge on theACS Publications website at DOI: 10.1021/acs.nano-lett.6b02800.

Derivation of the independent boson model; analysis ofthe effect in the spectra of the different molecularplasmon−phonon couplings; information of the vibra-tional modes. (PDF)

■ AUTHOR INFORMATIONCorresponding Authors*E-mail: nordlander@rice.edu.*E-mail: manjavacas@unm.edu.*E-mail: halas@rice.edu.Author Contributions(Y.C. and A.L.) These authors contributed equally.FundingThis work was supported by the Robert A. Welch Foundationunder grants C-1220 (N.J.H.) and C-1222 (P.N.). A.M.acknowledges financial support from the Department of Physicsand Astronomy and the College of Arts and Sciences of theUniversity of New Mexico. F.J.G.A. acknowledges support fromthe Spanish MINECO (MAT2014-59096-P and SEV-2015-0522).NotesThe authors declare no competing financial interest.

■ ACKNOWLEDGMENTSWe thank Julien Bloino for helpful discussions.

■ REFERENCES(1) Maier, S. A. Plasmonics: Fundamentals and Applications; SpringerScience & Business Media: New York, 2007.(2) Novotny, L.; Hecht, B. Principles of Nano-Optics; CambridgeUniversity Press: New York, 2006.(3) Halas, N. J.; Lal, S.; Chang, W.-S.; Link, S.; Nordlander, P.Plasmons in strongly coupled metallic nanostructures. Chem. Rev.2011, 111, 3913−61.(4) Alvarez-Puebla, R.; Liz-Marzan, L. M.; García de Abajo, F. J. LightConcentration at the Nanometer Scale. J. Phys. Chem. Lett. 2010, 1,2428−2434.(5) Akselrod, G. M.; et al. Probing the mechanisms of large Purcellenhancement in plasmonic nanoantennas. Nat. Photonics 2014, 8,835−840.(6) Bryant, G. W. Approaching the quantum limit for plasmonics:linear atomic chains. J. Opt. 2016, 18, 074001.(7) Christensen, T.; Wang, W.; Jauho, A.; Wubs, M.; Mortensen, N.A. Classical and quantum plasmonics in graphene nanodisks: Role ofedge states. Phys. Rev. B: Condens. Matter Mater. Phys. 2014, 90,241414.(8) Pustovit, V. N.; Urbas, A. M.; Shahbazyan, T. V. Energy transferin plasmonic systems. J. Opt. 2014, 16, 114015.(9) Guidez, E. B.; Aikens, C. M. Quantum mechanical origin of theplasmon: from molecular systems to nanoparticles. Nanoscale 2014, 6,11512−27.(10) Manjavacas, A.; et al. Tunable molecular plasmons in polycyclicaromatic hydrocarbons. ACS Nano 2013, 7, 3635−3643.(11) Krauter, C. M.; Bernadotte, S.; Jacob, C. R.; Pernpointner, M.;Dreuw, A. Identification of Plasmons in Molecules with Scaled AbInitio Approaches. J. Phys. Chem. C 2015, 119, 24564−24573.(12) Lauchner, A.; et al. Molecular Plasmonics. Nano Lett. 2015, 15,6208−6214.

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(13) Yamakita, Y.; Kimura, J.; Ohno, K. Molecular vibrations of[n]oligoacenes (n = 2−5 and 10) and phonon dispersion relations ofpolyacene. J. Chem. Phys. 2007, 126, 064904.(14) Dessent, C. E. H. A density functional theory study of theanthracene anion. Chem. Phys. Lett. 2000, 330, 180−187.(15) Barone, V.; Bloino, J.; Biczysko, M.; Santoro, F. Fully IntegratedApproach to Compute Vibrationally Resolved Optical Spectra: FromSmall Molecules to Macrosystems. J. Chem. Theory Comput. 2009, 5,540−554.(16) Freitag, M.; Low, T.; Zhu, W.; Yan, H.; Xia, F.; Avouris, P.Photocurrent in graphene harnessed by tunable intrinsic plasmons.Nat. Commun. 2013, 4, 1951.(17) Ferrari, A. C.; Basko, D. M. Raman spectroscopy as a versatiletool for studying the properties of graphene. Nat. Nanotechnol. 2013, 8,235−46.(18) Ando, T. Anomaly of Optical Phonon in Monolayer Graphene.J. Phys. Soc. Jpn. 2006, 75, 124701.(19) Hwang, E. H.; Sensarma, R.; Das Sarma, S. Das. Plasmon−phonon coupling in graphene. Phys. Rev. B: Condens. Matter Mater.Phys. 2010, 82, 195406.(20) Saito, R.; Furukawa, M.; Dresselhaus, G.; Dresselhaus, M. S.Raman spectra of graphene ribbons. J. Phys.: Condens. Matter 2010, 22,334203.(21) Buljan, H.; Jablan, M.; Soljacic, M. Graphene plasmonics:Damping of plasmons in graphene. Nat. Photonics 2013, 7, 346−348.(22) Charlier, J.-C.; Eklund, P. C.; Zhu, J.; Ferrari, A. C. In CarbonNanotubes SE - 21; Jorio, A., Dresselhaus, G., Dresselhaus, M., Eds.;Springer: Berlin Heidelberg, 2008; Vol. 111, pp 673−709.(23) Mahan, G. D. Many-Particle Physics; Springer Science &Business Media: New York, 1990.(24) Bloino, J.; Biczysko, M.; Crescenzi, O.; Barone, V. Integratedcomputational approach to vibrationally resolved electronic spectra:anisole as a test case. J. Chem. Phys. 2008, 128, 244105.(25) Barone, V.; Bloino, J.; Biczysko, M. Vibrationally-resolvedelectronic spectra in Gaussian 09. 2009; http://dreamslab.sns.it/pdf/vibronic_spectra_G09-A02.pdf (accessed September 1, 2016).(26) Novotny, L. Effective Wavelength Scaling for Optical Antennas.Phys. Rev. Lett. 2007, 98, 266802.(27) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.;Robb, M. A.; Cheeseman, J. R.; Montgomery, J. A., Jr.; Vreven, T.;Kudin, K. N.; Burant, J. C.; Millam, J. M.; Iyengar, S. S.; et al. Gaussian09, Revision A.02; Gaussian, Inc.: Wallingford, CT, 2009.(28) Bloino, J.; Biczysko, M.; Santoro, F.; Barone, V. GeneralApproach to Compute Vibrationally Resolved One-Photon ElectronicSpectra. J. Chem. Theory Comput. 2010, 6, 1256−1274.(29) Malloci, G.; Mulas, G.; Cappellini, G.; Joblin, C. Time-dependent density functional study of the electronic spectra ofoligoacenes in the charge states − 1, 0, + 1, and + 2. Chem. Phys. 2007,340, 43−58.(30) Malloci, G.; Cappellini, G.; Mulas, G.; Mattoni, A. Electronicand optical properties of families of polycyclic aromatic hydrocarbons:A systematic (time-dependent) density functional theory study. Chem.Phys. 2011, 384, 19−27.

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Supporting information for:

Molecular Plasmon-Phonon Coupling

Yao Cui,†,‡ Adam Lauchner,¶,‡ Alejandro Manjavacas,⇤,§ F. Javier Garcıa de

Abajo,k,? Naomi J. Halas,⇤,†,¶,#,‡ and Peter Nordlander⇤,#,‡

†Department of Chemistry, Rice University, Houston, TX, USA

‡Laboratory for Nanophotonics, Rice University Houston, TX, USA

¶Department of Electrical and Computer Engineering, Rice University, Houston, TX, USA

§Department of Physics and Astronomy, University of New Mexico, Albuquerque, New

Mexico 87131, United States

kICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology,

08860 Castelldefels (Barcelona), Spain

?ICREA-Institucio Catalana de Recerca i Estudis Avancats, Passeig Lluıs Companys, 23,

08010 Barcelona, Spain

#Department of Physics and Astronomy, Rice University, Houston, TX, USA

E-mail: manjavacas@unm.edu; halas@rice.edu; nordland@rice.edu

S1

Independent Boson Model (IBM)

The independent boson model derived by Mahan (S1) for a fermionic excitation interacting

with bosonic modes can be extended to deal with the situation in which a bosonic plasmon

in large systems interacts with bosonic vibrational modes. The Hamiltonian for a plasmonic

mode of energy h!0 interacting with vibrational modes of energies h!q with a coupling

strength Mq is

H = c+c[h!0 +X

q

Mq(aq + a+q )] +X

q

h!qa+q aq,

where c, c+, aq, and a+q are the bosonic annihilation and creation field operators for the

plasmonic and vibrational modes, respectively. Using the canonical transformation H =

esHe�s with

s = c+cX

q

Mq

h!q

�

a+q � aq�

,

we obtain

H = c+c(h!0 � h�c+c) +X

q

h!qa+q aq, (1)

where we have defined � =P

q M2q /(h

2!q). We can separate this Hamiltonian as H(t) =

H1+H2, withH1 = c+c(h!0�h�c+c) andH2 =P

q h!qa+q aq. Using the same transformation

for the operators c and c+, we obtain c = cX and c+ = c+X+, being

X = exp

"

�

X

q

Mq

h!q

�

a+q � aq�

#

. (2)

In order to obtain the absorption spectrum of the system we first need to calculate the

following Green function

G(t) = �i hT c(t)c+(0)i ,

where T is the time-ordering operator. Therefore, for t > 0, we have

G(t) = �iNTr�

e��HeiHt/hce�iHt/hc+

.

S2

Here, N�1 = Tr�

e��H

and � = 1/kBT , being kB the Boltzmann constant and T the

temperature of the system. This expression can be rewritten using Eqs. (1) and (2) as

G(t) = �iNTrn

e��HeiHt/hcXe�iHt/hc+X+o

.

Now, taking into account that [c,H2] = 0, [c,X] = 0, and [X,H1] = 0, we can separate the

expression above into two parts: one involving the trace over the plasmon mode, ⇧(t), and

the other involving the trace over the vibrational modes, e��(t),

G(t) = �iNpTrp�

e��H1eiH1t/hce�iH1t/hc+

NvTrv�

e��H2eiH2t/hXe�iH2t/hX+

= �i⇧(t)e��(t),

where we have also separated the normalization constant as N = NpNv, with N�1p =

Trp�

e��H1

and N�1v = Trv

�

e��H2

. The trace over the vibrational modes can be eval-

uated using Feynman’s method for disentangling of operators, as shown by Mahan (S1).

This leads to

NvTrv�

e��H2eiH2t/hXe�iH2t/hX+

= e��(t),

where �(t) =P

q gq [Nq (1� ei!qt) + (Nq + 1) (1� e�i!qt)], with gq =⇣

Mq

h!q

⌘2

, and N�1q =

e�h!q� 1. Similarly, the trace over the plasmon mode can be evaluated directly as

⇧(t) = NpTrp�

e��H1eiH1t/hce�iH1t/hc+

= Np

X

n

(n+ 1) e�i[!0��(2n+1)]te��h(!0��n)n.

Using these expressions, the absorption spectrum is obtained from the imaginary part of the

Fourier transform of G(t)

A(!) / Im

(

iNp

1X

n=0

(n+ 1)e��h(!0��n)ne�P

q gq(2Nq+1)

Z 1

0

dt ei!te�i[!0��(2n+1)]t

⇥

1X

l=0

1

l!

"

X

q

gqNqei!qt

#l 1X

m=0

1

m!

"

X

q

gq (Nq + 1) e�i!qt

#m9

=

;

.

S3

In realistic situations the molecular plasmon energy is much larger than the thermal

energy (i.e. h!0 � kBT ). Furthermore, for the cases that we analyze in the manuscript

it is su�cient to consider l,m 3 in the summations, which corresponds to a maximum

population of 3 in the vibrational modes. Under these conditions, the absorption is given by

A(!) / e�P

q gq(2Nq+1)Im

(

i

Z 1

0

dt ei(!�!0+�)t

"

1 +X

q

gqNqei!qt

+1

2

X

q,q0

gqgq0NqNq0ei(!q+!q0 )t +

1

6

X

q,q0,q00

gqgq0gq00NqNq0Nq00ei(!q+!q0+!q00 )t

#

"

1 +X

q

gq (Nq + 1) e�i!qt +1

2

X

q,q0

gqgq0(Nq + 1)(Nq0 + 1)e�i(!q+!q0 )t

+1

6

X

q,q0,q00

gqgq0gq00(Nq + 1)(Nq0 + 1)(Nq00 + 1)e�i(!q+!q0+!q00 )t

#)

.

This expression can be simplified even more if we neglect the e↵ect of the temperature (i.e.

if we take T = 0), which, as shown in Figure S3, has a small impact in the systems under

consideration. Therefore, within that limit, and making use ofR10 dtei⌦t = lim�!0

i⌦+i�

, the

absorption reduces to

A(!) / � e�P

q gqIm

(

lim�!0

"

1

! � !0 +�+ i�+X

q

gq! � !0 +�� !q + i�

+1

2

X

q,q0

gqgq0

! � !0 +�� !q � !q0 + i�+

1

6

X

q,q0,q00

gqgq0gq00

! � !0 +�� !q � !q0 � !q00 + i�

#)

.

This is the equation we use to generate the IBM spectra shown in the right column of Figure 4

of the main paper. The di↵erent parameters appearing in this expression are obtained as fol-

lows: the phonon energies are taken from the corresponding TDDFT-FC calculations, while

the coupling parameters are determined by fitting the TDDFT-FC vibrationally-resolved ab-

sorption spectra using a genetic algorithm. The maximum value of l used in the model varies

for the di↵erent molecules. For anthracene and pentacene, we use a maximum l = 2, which

is enough to reproduce the TDDFT-FC spectrum; while for naphthalene and tetracene, we

S4

sum up to l = 3 in order to get all the peaks appearing in the spectra. In all cases we fix

h� = 135 cm�1.

References

(S1) Mahan, G. Many-Particle Physics; Springer: Berlin, 2000.

TDDFT-FCl = 2l = 1l = 0

2.01.81.6 2.2 2.4Energy (eV)

Abs

orpt

ion

(arb

.u.)

TDDFT-FCl = 3l = 2l = 1l = 0

1.81.61.4 2.0 2.2Energy (eV)

Abs

orpt

ion

(arb

.u.)

a b

Figure S1: E↵ect of the population of the vibrational modes on the spectrum. (a) Longitu-dinal molecular plasmon of anthracene: the black curve shows the TDDFT-FC calculation,while the blue, yellow, and green curves represent the IBM fitted spectrum with maxi-mum population equal to 2, 1, and 0, respectively. (b) Longitudinal molecular plasmon oftetracene: the black curve shows the TDDFT-FC calculation, while the red, blue, yellow,and green curves represent the IBM fitted spectrum with maximum population equal to 3,2, 1, and 0, respectively.

S5

All ModesM5 = 0M4 = 0M3 = 0M2 = 0M1 = 0

2.01.81.6 2.2 2.4Energy (eV)

Abs

orpt

ion

(arb

.u.)

Figure S2: E↵ect of the di↵erent plasmon-phonon coupling constants Mi on the spectrumof the longitudinal molecular plasmon of anthracene. The dark blue curve shows the IBMspectrum obtained with the five most significant phonon modes, while the other curves showthe IBM fitted spectrum obtained by setting the plasmon-phonon coupling constant Mi tozero, one at a time, as indicated in the legend.

S6

T = 1000 KT = 300 KT = 0 KTDDFT-FC

2.01.81.6 2.2 2.4Energy (eV)

Abs

orba

nce

(arb

.u.)

Figure S3: E↵ect of temperature on the spectrum of the longitudinal molecular plasmon ofanthracene. The dark blue curve shows the IBM spectrum obtained with the temperatureT = 0K while the yellow and green curves show the IBM fitted spectrum obtained by settingthe temperature T = 300K and T = 1000K, respectively.

S7

Table S1: Energies of the di↵erent phonon modes and the corresponding coupling constantsfor the IBM spectrum of naphthalene shown in Figure 4.

Phonon energy (eV) Coupling constant Mi (eV)0.062 0.03550.094 0.00130.179 0.12510.184 0.1012

Table S2: Energies of the di↵erent phonon modes and the corresponding coupling constantsfor the IBM spectrum of anthracene shown in Figure 4.

Phonon energy (eV) Coupling constant Mi (eV)0.078 0.03710.158 0.06770.177 0.09570.048 0.00020.094 0.0093

Table S3: Energies of the di↵erent phonon modes and the corresponding coupling constantsfor the IBM spectrum of tetracene shown in Figure 4.

Phonon energy (eV) Coupling constant Mi (eV)0.072 0.02680.051 0.03640.076 0.01240.107 0.02510.169 0.08390.176 0.04140.191 0.05340.023 0.00080.072 0.00490.094 0.00660.149 0.00270.195 0.0286

S8

Table S4: Energies of the di↵erent phonon modes and the corresponding coupling constantsfor the IBM spectrum of pentacene shown in Figure 4.

Energy (eV) Coupling constant Mi (eV)0.032 0.00010.075 0.00110.094 0.00020.098 0.02090.151 0.00090.161 0.00100.175 0.00270.177 0.08070.192 0.01150.194 0.0012

Table S5: Energies and intensities of the di↵erent phonon modes appearing in the spectrumof the longitudinal molecular plasmon of anthracene (⇡ 1.75 eV).

Mode-Harmonic Energy (cm�1) Energy (eV) Intensity (dm3 mol�1 cm�1)0-0 0 0 515207-1 387 0.048 134117-1 630 0.078 1507017-2 1261 0.156 224723-1 762 0.094 170242-1 1272 0.158 1019048-1 1425 0.177 10520

42-1;17-1 1902 0.236 313148-1;17-1 2055 0.255 297748-1;42-1 2697 0.334 1887

S9

Table S6: Energies and intensities of the di↵erent phonon modes appearing in the spectrumof the transversal molecular plasmon of anthracene (⇡ 1.96 eV).

Mode-Harmonic Energy (cm�1) Energy (eV) Intensity (dm3 mol�1 cm�1)0-0 0 0.000 30138-1 385 0.048 17058-2 770 0.095 46033-1 1019 0.126 21548-1 1428 0.177 97248-2 2855 0.354 17352-1 1509 0.187 106852-2 3017 0.374 18455-1 1614 0.200 584

48-1;8-1 1813 0.225 53352-1;8-1 1894 0.235 57455-1;8-1 1999 0.248 33152-1;48-1 2936 0.364 35555-1;48-1 3042 0.377 19755-1;52-1 3123 0.387 197

52-1;48-1;8-1 3321 0.412 185

Table S7: Energies and intensities of the di↵erent phonon modes appearing in the spectrumof the longitudinal molecular plasmon of pyrene (⇡ 2.54 eV).

Mode-Harmonic Energy (cm�1) Energy (eV) Intensity (dm3 mol�1 cm�1)0-0 0 0.000 1310009-1 418 0.052 1195041-1 1131 0.140 345845-1 1217 0.151 414151-1 1417 0.176 370662-1 1798 0.223 3049062-2 3597 0.446 5667

62-1;9-1 2216 0.275 3631

Table S8: Energies and intensities of the di↵erent phonon modes appearing in the spectrumof the longitudinal molecular plasmon of benzo(a)pyrene (⇡ 2.14 eV).

Mode-Harmonic Energy (cm�1) Energy (eV) Intensity (dm3 mol�1 cm�1)0-0 0 0.000 8180010-1 337 0.042 1392011-1 375 0.046 382913-1 454 0.056 332464-1 1382 0.171 648778-1 1867 0.232 1330078-2 3735 0.463 2877

78-1;10-1 2204 0.273 2792

S10