Letter pubs.acs.org/NanoLett
Molecular Plasmon−Phonon Coupling Yao Cui,†,# Adam Lauchner,‡,# Alejandro Manjavacas,*,§ F. Javier Garcıá 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-Institució Catalana de Recerca i Estudis Avançats, 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 graphene consisting of only a few fused aromatic carbon rings, have been shown to possess molecular plasmon resonances in the visible region of the spectrum. Unlike larger nanostructures, the PAH absorption spectra reveal rich, highly structured spectral features due to the coupling of the molecular plasmons with the vibrations of the molecule. Here, we examine this molecular plasmon−phonon interaction using a quantum mechanical approach based on the Franck−Condon approximation. We show that an independent boson model can be used to describe the complex features of the PAH absorption spectra, yielding an analytical and semiquantitative description of their spectral features. This investigation provides an initial insight into the coupling of fundamental excitationsplasmons and phononsin molecules. KEYWORDS: plasmonics, plasmon−phonon coupling, graphene, polyacenes, PAHs
P
lasmons, collective oscillations of electrons,1,2 provide a mechanism for subwavelength light confinement and manipulation. Plasmon resonances in nanoparticles can be tuned by changing the morphology or composition of the nanostructure.3−8 Recent studies have shown that charged polycyclic aromatic hydrocarbons (PAHs) support a set of collective resonances that are strongly dependent upon the electron−electron interaction strength9,10 and are derived from a superposition of multiple in-phase electron excitations.11 In molecules composed of less than ∼50 carbon atoms, these molecular plasmon resonances lie in the visible region of the spectrum and have energies that depend on the charge state of the molecules in a manner analogous to the dependence of graphene plasmons on doping. As the smallest examples of graphene and as readily available chemical species, PAHs provide an ideal platform for molecular plasmonics. Experimental observations of molecular plasmons in charged aromatic PAHs have revealed rich spectral features resulting from the coupling of collective electronic modes and atomic vibrations.12 In our previous work, we used time-dependent density functional theory (TDDFT) to demonstrate that coupling of the molecular plasmon resonance to vibrational © 2016 American Chemical Society
modes of the molecule produces an energy level splitting observed in the experimental measurements. The calculated plasmonic−vibronic energy levels agreed qualitatively with the observed extinction spectra and with previous TDDFT investigations of PAH phonon modes.13,14 This approach relied on the Franck−Condon principle and is referred to here as TDDFT-FC.15 Substrate lattice phonons are well known to affect the electro-optical properties of graphene,16 whereas its intrinsic phonons, as measured using Raman spectroscopy, serve as key indicators of structural integrity and chemical purity.17 However, limited effort has been invested to examine the effect of intrinsic phonon coupling with electronic and plasmonic resonances in carbon allotropes, such as graphene18−21 and nanotubes.22 Here, we present a comprehensive experimental and theoretical investigation of the molecular plasmon−phonon coupling for different PAH molecules. We also provide a microscopic analysis of this coupling based on Received: July 6, 2016 Revised: September 25, 2016 Published: September 26, 2016 6390
DOI: 10.1021/acs.nanolett.6b02800 Nano Lett. 2016, 16, 6390−6395
Letter
Nano Letters the independent boson model (IBM),23 originally introduced to describe the coupling between electronic excitations and bosonic phonon modes in solids. This exactly solvable analytical model provides a simple, intuitive expression for molecular plasmon−phonon coupling and enables semiquantitative agreement with experimental data. The impact of molecular plasmon−phonon coupling on the absorption spectra is illustrated schematically in Figure 1.
consequence of this, the dominant peak is red-shifted compared to the analysis without vibrations. In recent work on molecular plasmonics, we measured the absorption spectra of isomeric three- and four-unit planar aromatic ring molecules dissolved in tetrahydrofuran (THF). We interpreted these spectra using TDDFT calculations. For the charged linear three- and four-ring molecules anthracene and tetracene, the vibrationally resolved absorption spectra were simulated by only considering the coupling between the strongest molecular plasmon and the vibrations of the molecule within the spectral region under consideration (1.2−3.0 eV). Here, we go beyond that approximation and investigate both molecules in a more detailed manner, analyzing the vibrational structure 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 induced charge densities of tetracene, anthracene, and coronene, respectively. The induced charge densities of the strongest molecular plasmon of tetracene and anthracene, polarized along the long axis of each molecule, are shown. We label this mode as the longitudinal molecular plasmon (LP). Tetracene and anthracene present an additional weaker molecular plasmon peak, located at 1.96 eV for the anthracene anion and at 1.38 eV for the tetracene anion. Examining the induced charge of these excitations, we observe that they correspond to oscillations along a direction perpendicular to the molecular axis, and therefore, we label them as transverse molecular plasmons (TP). Panels (d) and (e) of Figure 2 show their corresponding measured absorption spectra in tetrahydrofuran (THF) (gray curves), together with the TDDFT-FC calculations for the longitudinal (orange curves) and transversal (green curves) spectra. For both molecules, the main features of the experimental spectra are well reproduced (within the expected error for TDDFT calculations24) by the sum of the vibration-
Figure 1. Molecular plasmon−phonon coupling. Schematics illustrating the absorption spectrum of a molecule in the absence (a) and in the presence (b) of coupling to molecular vibrations.
Without molecular vibrations, the electronic transition, shown in panel (a), gives rise to a single absorption peak. By including the coupling to vibrations, the ground state can be connected to different vibrational levels in the excited state, which results in different peaks in the spectrum, as shown in panel (b). As a
Figure 2. Absorption spectra of singly charged PAHs. Induced charge density plots for molecular plasmonic resonances oriented along the longitudinal and transverse axes of (a) tetracene, (b) anthracene, and (c) coronene. Experimental extinction spectra (gray) are compared with TDDFT-FC absorption spectra (black) obtained by coupling to two different vibrationally expanded transverse plasmon resonances (TP, green) and longitudinal molecular plasmon resonances (LP, orange) for (d) tetracene, (e) anthracene, and (f) coronene. 6391
DOI: 10.1021/acs.nanolett.6b02800 Nano Lett. 2016, 16, 6390−6395
Letter
Nano Letters ally resolved TDDFT spectra associated with each of the two molecular plasmon modes (black curves). As a model molecule for studies on graphene surfaces, coronene belongs to the class of pericondensed benzenoid aromatic molecules C6s2H6s, with s = 2 and has D6h symmetry. Because of its planar 6-fold symmetry, the molecular plasmon modes for coronene anion cannot be classified as longitudinal or transverse. Instead, the induced charge densities for the two modes within the region 1.2−2.4 eV oscillate along the x axis and y axis, respectively (Figure 2c). The strength of the two modes does not display the large asymmetry exhibited by the LP and TP modes in the linear PAHs, although the magnitudes of their respective molecular plasmon−phonon couplings differ significantly, with the higher-energy molecular plasmon resonance exhibiting a more pronounced coupling to vibrational modes. Figure 2f shows the absorbance spectrum of coronene (C24H12) anion measured in THF solvent and the vibronic 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 peak structure is dominated by coupling to only a few vibrational modes. For the longitudinal molecular plasmon (∼1.75 eV), one of the most intense bands (at 630 cm−1) is assigned to mode 17−1 (here, we use the notation n − x, where n is the excited normal vibrational mode in the Gaussian standard nomenclature25 and x is its population). The atomic displacements associated with this vibrational mode (Figure 3c) involve a CCC deformation of the aromatic rings, with outer ring deformation being out-of-phase relative to the center ring deformation. Doubly populated vibrational modes, such as the 17−2, also contribute to the spectrum (see Supporting Information for more details). The peak at 1272 cm−1 corresponds to the 42−1 mode, associated with the CH bending in which the center ring evolves in a breathing mode. Similarly, the mode at 1425 cm−1, labeled as 48−1, corresponds to the CC stretching of the aromatic group. A doubly excited mode resulting from the combination of modes 42−1 and 48− 1 contributes to the vibrationally resolved absorption peak around 1.9 eV, which leads to a high amplitude compared to the ∼1.83 eV peak. For the transverse molecular plasmon (∼1.96 eV), the vibronic spectrum displays similar features but with much smaller amplitude. The most intense bands are associated with the singly and doubly populated 8, 48, and 52 vibrational modes, which correspond to CCC deformations (with all three rings deforming in phase), CC stretching, and C−H bending (with the center ring in a CC stretching mode), respectively. A comprehensive description of all vibrational modes is provided in the Supporting Information. The strongly coupled modes correspond almost exclusively to in-plane vibrations. Examining the most strongly coupled vibrational mode for each molecular plasmon, we observe that the longitudinal molecular plasmon interacts strongly with vibrational modes in which the atomic displacements are aligned primarily along the transverse axis. Similarly, for the transverse molecular plasmon, the atomic displacements of the coupled vibrational modes are aligned along the longitudinal axis. To provide a more intuitive understanding of the molecular plasmon−phonon coupling, we use the IBM to describe plasmonic excitations interacting with vibrational modes (for more details, see the Supporting Information). Briefly, in this approach, the system is described using a model
Figure 3. Vibrational mode coupling of the anthracene molecular plasmon. Assignment of the main vibrational bands of (a) the longitudinal molecular plasmon mode and (b) the transversal molecular 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 vibrational modes of (a) and (b).
Hamiltonian consisting of the bare plasmon and the relevant vibrational modes q H = c +c[ℏω0 +
∑ Mq(aq + aq+)] + ∑ ℏωqaq+aq q
q
where c and c and are the field operators for the plasmon mode of energy ℏω0, whereas a+q and aq are the field operators for the bosonic vibrational modes of energies ℏωq calculated using the TDDFT approach. The coupling coefficients Mq between the plasmon and vibrations are treated as adjustable parameters. The Hamiltonian can be solved analytically, providing absorption spectra that can be compared directly with the TDDFT-FC results and experimental data. To simplify ⎛ Mq ⎞2 the equations we introduce the two parameters gq = ⎜ ℏω ⎟ ⎝ q⎠ +
6392
DOI: 10.1021/acs.nanolett.6b02800 Nano Lett. 2016, 16, 6390−6395
Letter
Nano Letters and Δ = ∑q ωqgq. For finite temperature β = 1/kBT, the optical absorption spectrum is obtained from the Fourier transform of the Green function, which can be written as the product of the traces over the plasmonic and vibrational degrees of freedom as
∫0
A(ω) ∝ −Im{−i
∞
phonon energies from the TDDFT-FC output, the IBM is capable of modeling the fine structure in both measured and TDDFT-FC spectra (see the Supporting Information for more details). The addition of an extra benzene ring (n = 3, 4, 5) causes a red shift of the longitudinal plasmon in the linear PAHs, which agrees well with previous results showing that the low-energy peak appears at a wavelength that scales almost linearly with the number of benzene rings.10 This is similar to the behavior observed in gold nanorods for which there exist an approximate proportionality between the nanorod length and the dipolar plasmon frequency.26 In this study, we have adjusted the coupling coefficients Mq to the TDDFT-FC-derived absorption spectra (see Methods). However, it would be equally valid to fit these parameters to experimental data, providing a quick method for characterization of the coupled modes observed in the absorption spectra. Such characterization based on the IBM provides UV− vis spectroscopy with a degree of molecular specificity typically observed with infrared vibrational spectroscopy. We also examined molecular plasmon−phonon coupling in PAH molecules for the case of pyrene and its benzene-fused derivative benzo[a]pyrene (Figure 5). The experimental spectra of the two molecules as singly charged anions are very similar, with the primary difference being that benzo[a]pyrene exhibits a red shift relative to pyrene, similar to the red shift observed with increasing aspect ratio for linear PAHs. From TDDFT-FC simulations, we find that not only are the vibronic spectra
dteiωt Π(t )e−Φ(t )}
where the vibrational trace takes the form e−Φ(t), with Φ(t ) =
∑ gq[Nq(1 − eiω t ) + (Nq + 1)(1 − e−iω t )] q
q
q
and ∞
Π(t ) = Np ∑ (n + 1)e−βℏ(ω0 −Δn)ne−i[ω0 −Δ(2n + 1)]t n=0
N−1 q
βℏωq
N−1 p
(1)
∑n ∞ = 0
In these expressions, = e − 1 and = e−βℏ(ω0−Δn)n. Equation 1 represents a generalization of the conventional IBM to account for the bosonic statistics of the plasmon, which is useful in many-electrons systems. For the present case of small molecules, for which the molecular plasmon has a strongly anharmonic character, as it is supported by a small number of electrons, we must restrict the summation to the n = 0 term. Because of its analytical structure, this model provides a direct and intuitive account for how the temperature, the couplings, and the population of the vibrational modes influence the absorption spectra (see Figures S1−S3 in the Supporting Information). We further measured the absorbance spectra of singly charged linear PAHs with increasing lengths (two- to five-unit planar aromatic rings) in THF solution (Figure 4). Vibrationally resolved absorption spectra calculated by TDDFT-FC take into consideration only the strongest transition peak in the electronic spectrum for each molecule within the spectral region under examination. By considering the most prominent
Figure 5. Absorption spectra of geometrically similar molecules. (a) Experimental (dashed) and TDDFT-FC (solid) absorption spectra for benzo[a]pyrene (red) and pyrene (green). (b) Simulated vibronic resonances for benzo[a]pyrene (red) and pyrene (green). Vibration vectors for the two strongest vibrational states, (c) in-phase CCC deformation and (d) CC stretching, for pyrene and benzo[a]pyrene (top and bottom, respectively).
Figure 4. Absorption spectra of charged linear PAHs. Experimental extinction spectra (left), calculated TDDFT-FC absorption spectra (center), and calculated IBM absorption spectra (right) for PAH molecules with 2−5 benzene rings (top-to-bottom: naphthalene, anthracene, tetracene, and pentacene). 6393
DOI: 10.1021/acs.nanolett.6b02800 Nano Lett. 2016, 16, 6390−6395
Nano Letters
■
similar qualitatively, but so are the atomic displacements in the vibrational modes and their relative couplings. This is shown in Figure 5, where the primarily vibronic components come from a longitudinal stretching mode (9−1 in pyrene, 10−1 in benzo[a]pyrene) and a CC stretching mode (modes 62−1 and 78−1 in pyrene and benzo[a]pyrene, respectively). The additional benzene ring has significantly shifted the molecular plasmon resonance with almost no modification of the molecular plasmon−phonon coupling. In conclusion, we have examined and analyzed the complex vibrational structure observed in molecular plasmon absorption spectra for a variety of charged PAHs. We find that molecular plasmons tend to couple more strongly to phonon modes whose vibrational motion is perpendicular with respect to the main direction of polarization associated with the molecular plasmon. By using spectral information from TDDFT-FC simulations, we have determined a few parameters that bring the analytical IBM into close agreement with the experimental spectra, thus resulting in a simple and intuitive expression that allows us to conclusively elucidate the effect of molecular plasmon-vibrational coupling in these systems. Methods. Experimental Section. For spectroelectrochemical characterization, PAH molecules were dried and dissolved in dry solvent at 5 mM concentration with 500 mM supporting electrolyte (tetrabutylammonium perchlorate, TBAP) purchased from Sigma-Aldrich and dried by recrystallization from dry diethyl ether. Room temperature tetrahydrofuran was used as the solvent for all molecules except pentacene, which was dissolved in o-dichlorobenzene (at the same concentrations of PAH and TBAP), heated to 160 °C, and subsequently measured before cooling. Charge transfer was accomplished by three-electrode cyclic voltammetry (Pt mesh electrodes with Ag wire pseudo reference) performed concurrently with absorbance spectroscopy under white-light illumination. The experimental absorption spectra presented in this paper were obtained at voltages just below the reduction potential of each molecule. For additional details see Lauchner et al.12 Quantum Mechanical Calculations. All quantum mechanical calculations were performed using Gaussian 09.27 Full geometry optimizations and harmonic frequency calculations were conducted for all molecules in vacuum and in both ground and excited states using DFT and TDDFT methods, respectively. The absorption spectra and transition densities were calculated using TDDFT. The vibrationally resolved absorption spectra were obtained within the framework of the Franck−Condon (FC) principle15,28 upon TDDFT-based overlap integrals between the vibrational wave functions of the ground and the excited states, as proposed by Barone et al.15 The rendered spectra were calculated by convoluting the resonant energy intensity with a Gaussian with a half width at half-maximum (HWHM) of 135 cm−1, which still allows us to assign individual vibronic contributions to the total spectrum. The hybrid B3LYP functional and the 6-31+G(d) basis set were used throughout all calculations following previous works.29,30 Independent Boson Model. The independent boson model23 was extended from the standard fermion−boson coupling to also describe boson−boson interactions, as appropriate for plasmon excitations in the many electron limit. The phonon energies were obtained from TDDFT-FC calculations, whereas the coupling parameters were determined by fitting the calculated vibrationally resolved absorption spectra using a genetic algorithm.
Letter
ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.nanolett.6b02800. Derivation of the independent boson model; analysis of the effect in the spectra of the different molecular plasmon−phonon couplings; information of the vibrational modes. (PDF)
■
AUTHOR INFORMATION
Corresponding Authors
*E-mail:
[email protected]. *E-mail:
[email protected]. *E-mail:
[email protected]. Author Contributions
(Y.C. and A.L.) These authors contributed equally. Funding
This work was supported by the Robert A. Welch Foundation under grants C-1220 (N.J.H.) and C-1222 (P.N.). A.M. acknowledges financial support from the Department of Physics and Astronomy and the College of Arts and Sciences of the University of New Mexico. F.J.G.A. acknowledges support from the Spanish MINECO (MAT2014-59096-P and SEV-20150522). Notes
The authors declare no competing financial interest.
■ ■
ACKNOWLEDGMENTS We thank Julien Bloino for helpful discussions. REFERENCES
(1) Maier, S. A. Plasmonics: Fundamentals and Applications; Springer Science & Business Media: New York, 2007. (2) Novotny, L.; Hecht, B. Principles of Nano-Optics; Cambridge University 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-Marzán, L. M.; García de Abajo, F. J. Light Concentration at the Nanometer Scale. J. Phys. Chem. Lett. 2010, 1, 2428−2434. (5) Akselrod, G. M.; et al. Probing the mechanisms of large Purcell enhancement 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 of edge states. Phys. Rev. B: Condens. Matter Mater. Phys. 2014, 90, 241414. (8) Pustovit, V. N.; Urbas, A. M.; Shahbazyan, T. V. Energy transfer in plasmonic systems. J. Opt. 2014, 16, 114015. (9) Guidez, E. B.; Aikens, C. M. Quantum mechanical origin of the plasmon: from molecular systems to nanoparticles. Nanoscale 2014, 6, 11512−27. (10) Manjavacas, A.; et al. Tunable molecular plasmons in polycyclic aromatic 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 Ab Initio Approaches. J. Phys. Chem. C 2015, 119, 24564−24573. (12) Lauchner, A.; et al. Molecular Plasmonics. Nano Lett. 2015, 15, 6208−6214. 6394
DOI: 10.1021/acs.nanolett.6b02800 Nano Lett. 2016, 16, 6390−6395
Letter
Nano Letters (13) Yamakita, Y.; Kimura, J.; Ohno, K. Molecular vibrations of [n]oligoacenes (n = 2−5 and 10) and phonon dispersion relations of polyacene. J. Chem. Phys. 2007, 126, 064904. (14) Dessent, C. E. H. A density functional theory study of the anthracene anion. Chem. Phys. Lett. 2000, 330, 180−187. (15) Barone, V.; Bloino, J.; Biczysko, M.; Santoro, F. Fully Integrated Approach to Compute Vibrationally Resolved Optical Spectra: From Small 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 versatile tool 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.; Soljačić, 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 Carbon Nanotubes 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. Integrated computational 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-resolved electronic 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. Gaussian 09, Revision A.02; Gaussian, Inc.: Wallingford, CT, 2009. (28) Bloino, J.; Biczysko, M.; Santoro, F.; Barone, V. General Approach to Compute Vibrationally Resolved One-Photon Electronic Spectra. J. Chem. Theory Comput. 2010, 6, 1256−1274. (29) Malloci, G.; Mulas, G.; Cappellini, G.; Joblin, C. Timedependent density functional study of the electronic spectra of oligoacenes in the charge states − 1, 0, + 1, and + 2. Chem. Phys. 2007, 340, 43−58. (30) Malloci, G.; Cappellini, G.; Mulas, G.; Mattoni, A. Electronic and optical properties of families of polycyclic aromatic hydrocarbons: A systematic (time-dependent) density functional theory study. Chem. Phys. 2011, 384, 19−27.
6395
DOI: 10.1021/acs.nanolett.6b02800 Nano Lett. 2016, 16, 6390−6395
