Article pubs.acs.org/JPCA
Quantum Mechanical Identification of Quadrupolar Plasmonic Excited States in Silver Nanorods Rebecca L. Gieseking, Mark A. Ratner, and George C. Schatz* Department of Chemistry, Northwestern University 2145 Sheridan Road, Evanston, Illinois 60208, United States S Supporting Information *
ABSTRACT: Quadrupolar plasmonic modes in noble metal nanoparticles have gained interest in recent years for various sensing applications. Although quantum mechanical studies have shown that dipolar plasmons can be modeled in terms of excited states where several to many excitations contribute coherently to the transition dipole moment, new approaches are needed to identify the quadrupolar plasmonic states. We show that quadrupolar states in Ag nanorods can be identified using the semiempirical INDO/SCI approach by examining the quadrupole moment of the transition density. The main longitudinal quadrupolar states occur at higher energies than the longitudinal dipolar states, in agreement with previous classical electrodynamics results, and have collective plasmonic character when the nanorods are sufficiently long. The ability to identify these states will make it possible to evaluate the differences between dipolar and quadrupolar plasmons that are relevant for sensing applications.
1. INTRODUCTION Plasmonic noble metal nanostructures have been of broad interest in recent years, since their unique optical properties can enhance important properties of nearby molecules such as absorption,1,2 fluorescence,3,4 and Raman spectroscopy.5−8 In a classical electrodynamics picture, the plasmonic states involve coherent oscillation of all of the conduction band electrons within the nanostructure.9−11 Although the majority of interest has been in the dipolar plasmon resonances where the full electron cloud responds in phase with the electric field of light (Figure 1), higher-order plasmon resonances such as quadrupoles have long been known to occur12 and in the past several years have gained attention for various applications.13−16 The quadrupolar modes are typically optically dark in small nanostructures, but can be excited via near-field radiation17−20 or, in larger nanostructures where the retardation of light
across the nanostructure is significant, by far-field radiation.9,21−23 Electrodynamics simulations9,12,23 and experimental studies22,24,25 consistently show that the quadrupolar modes are higher in energy and have much narrower absorption peaks than the corresponding dipolar modes. In rod-shaped nanostructures, electron energy loss spectroscopy (EELS) experiments have revealed the spatial distribution of the longitudinal plasmon modes with various multipolar character. In particular, the lowest-energy mode has two longitudinal lobes corresponding to a dipolar resonance, the second mode has three lobes along the length of the rod corresponding to a quadrupolar resonance, and the higher-energy modes have more lobes corresponding to higher-order multipolar modes.26,27 Perhaps counterintuitively, the quadrupolar modes can in some cases provide more enhancement of the spectroscopic or chemical properties of nearby molecules or semiconductor particles than the dipolar modes. In particular, the quadrupolar modes of various nanostructures have been shown to yield larger figures of merit for refractive index sensing,13,14,28,29 stronger enhancement of surface-enhanced Raman spectroscopy,15,30,31 and may have the dominant contribution to the emission enhancement of nearby semiconductor particles.16,32 Thus, an improved understanding of the chemical nature and structural dependence of the quadrupolar plasmon modes may contribute to the design of nanostructures for many applications. In recent years, there has been significant interest in studying plasmonic metal nanoclusters using quantum mechanical methods, including time-dependent density functional theory (TDDFT),33−36 configuration interaction singles (CIS),37,38
Figure 1. General schemes of instantaneous electron cloud displacements for (a) transverse dipolar, (b) longitudinal dipolar, and (c) longitudinal quadrupolar plasmon modes in nanorods. The red shaded area corresponds to the electron cloud. (d) Geometric structure of the Ag25+ pentagonal nanorod. © 2016 American Chemical Society
Received: September 23, 2016 Revised: October 26, 2016 Published: October 27, 2016 9324
DOI: 10.1021/acs.jpca.6b09649 J. Phys. Chem. A 2016, 120, 9324−9329
Article
The Journal of Physical Chemistry A and semiempirical INDO/CI39 approaches. Although these techniques are limited to relatively small clusters, extrapolation of the absorption energies to larger sizes suggests good agreement with classical electrodynamics results.33 Although the preliminary quantum mechanical identification of plasmonic states was based primarily on large oscillator strengths,33 more recent approaches have involved characterizing the collective nature of the plasmonic states in terms of the number of excitations contributing to the plasmonic state,40 the additivity of contributions of several or many excitations to the oscillator strength,39,41 and the influence of scaling the electron−electron interactions on the state energies.36 However, to our knowledge, the only quantum mechanical calculations to identify quadrupolar plasmons have been time-domain calculations involving the response of the system to a quadrupolar electric field,42,43 and these analyses have not yet been extended to the more widely used frequency-domain approach. Further work is needed to identify and understand the quadrupolar plasmonic states. Here, we propose an approach to identify quadrupolar plasmonic states by analyzing the transition densities associated with electronic excitations in Ag clusters. We show that in rodshaped clusters (hereafter denoted nanorods), there are longaxis excitations with quadrupolar character at energies higher than the long-axis dipolar plasmon states, and we show that in some cases these excitations are plasmonic. We use INDO/SCI in this study due to the low computational cost to compute the hundreds to thousands of excited states needed to obtain the quadrupolar modes and the ease of analysis because of the minimal basis set,39 but we show that similar results are found using a TDDFT approach used previously for dipolar modes.
For several of the nanorods, the excited states were also computed using the PBE functional and a triple-ζ Slater-type basis set with polarization functions (TZP) and a frozen-core approximation within the Amsterdam density functional (ADF) program.47 The lowest 200−500 excited states of A1 symmetry (C5v-symmetric nanorods) or A1′ symmetry (D5h-symmetric nanorods) were computed, and the transition quadrupole moments were computed using the full transition densities of each excited state.
3. RESULTS AND DISCUSSION We focus in this paper on small pentagonal Ag nanorods, which consist of alternating layers of individual Ag atoms and layers of five Ag atoms arranged in a pentagon (Figure 1). All of
2. COMPUTATIONAL METHODOLOGY The geometries of the Ag nanorods were optimized using a density functional theory approach with the BP86 functional44,45 and a double-ζ (DZ) Slater-type basis set with a frozen-core approximation, and scalar relativistic effects were included using the zeroth-order regular approximation (ZORA).46 All of these calculations were performed using the Amsterdam density functional (ADF) program.47 These structures have been previously optimized,48,49 and our results are consistent. The excited states were computed using the INDO/SCI approach. All possible single excitations among the valence molecular orbitals were generated, and the lowest 4000−7000 excitations were included in the CI matrix; this matrix was then diagonalized to obtain the lowest 1000−1500 excited states. These calculations were performed using our previously benchmarked INDO parameters for Ag clusters39,50 using a home-built code incorporating elements of Mopac 7.151 and Jeffrey Reimers’s INDO/CI code.52 Since most of the nanorods have geometries with C5v symmetry and because our modified INDO/CI code does not account for symmetry, the transition densities do not have perfectly symmetric or antisymmetric character and the excited states with dominantly quadrupolar character may have small nonzero transition dipole moments; in addition, states with significant dipolar character may also have quite large transition quadrupole moments. When the transition quadrupole moments were computed, only the states with very small transition dipole moments were considered; the cutoff for consideration of the quadrupole moments ranged from 0.5 D for Ag19+ to 2 D for Ag55+. Because all the clusters are much smaller than the wavelength of light, optical absorption due to the retardation of light across the cluster is negligibly small and not considered here.
Figure 2. INDO/SCI absorption spectra of pentagonal Ag nanorods.
the clusters considered are closed-shell with a single positive charge, and the lengths range from Ag19+ (3 pentagonal layers; aspect ratio = 1.62) to Ag55+ (9 pentagonal layers; aspect ratio = 5.16). We focus on cationic pentagonal nanorods as these have narrower absorption peaks and therefore more easily identified plasmonic states.48,49 The absorption spectra of these nanorods depend strongly on the length (Figure 2). For the shortest nanorods, INDO/SCI yields one broad absorption peak that includes both longitudinal and transverse plasmonic modes. For the nanorods Ag31+ and longer, there are two distinct absorption peaks corresponding to the lower-energy longitudinal plasmonic modes and the higher-energy transverse modes. These spectra are qualitatively similar to the absorption spectra previously computed using TDDFT48,49 but show somewhat more broadening of the two peaks, which is consistent with previous INDO/SCI results for Ag clusters of other shapes.39 As has been previously observed, the dipolar plasmonic states are highly collective: across the series no individual excitation contributes more than 30% to the CI composition of the plasmonic states, and the degree of collectivity does not follow any noticeable trends with cluster size. The main contributions to the transition dipole moments of the plasmonic states are strongly additive; for example, in Ag55+, the transition dipole 9325
DOI: 10.1021/acs.jpca.6b09649 J. Phys. Chem. A 2016, 120, 9324−9329
Article
The Journal of Physical Chemistry A moment μge from the ground state to the main plasmonic state is 27.2 D but no individual excitation contributes more than 5.8 D to μge. As previously seen in the INDO/SCI39 and TDDFT33 spectra of various Ag clusters, most of the nanorods have a grouping of several excited states with some dipolar plasmonic character within a narrow energetic window. This is most prominent in Ag43+, where the first absorption peak corresponds to six excited states with μge > 10 D within an energetic window of 0.08 eV. The plasmonic character can also be observed by examining the transition densities of the states with large transition dipole moments (Figure 3), which show significant dipolar character resulting from additive contributions of the component excitations.
systems with nonzero dipole moments, only the transition quadrupole moments of states with negligibly small transition dipole moments were considered. Comparison of several transition quadrupole moments using the atomic transition densities vs the full transition density suggest the error using the atomic approximation is
