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Time-Dependent Density Functional Theory Studies of Optical Properties of Ag Nanoparticles: Octahedra, Truncated Octahedra, and Icosahedra Gyun-Tack Bae and Christine M. Aikens* Department of Chemistry, Kansas State University, 213 CBC Building, Manhattan, Kansas 66506, United States S Supporting Information *

ABSTRACT: The effect of the size and shape of silver nanoparticles on their optical absorption properties is theoretically investigated to understand the plasmonic properties of these systems. Time-dependent density functional theory (TDDFT) calculations are employed to calculate the optical absorption spectra for a series of silver clusters (Agn, n = 6−85) in various charge states whose structures are octahedral, truncated octahedral, and icosahedral. Octahedral Agn clusters with n = 6, 19, 44, 85, truncated octahedral Agn clusters with n = 13, 38, 55, 79, and icosahedral Agn clusters with n = 13, 43, 55 are calculated. Charged systems are considered to obtain closed shell electronic structures. These calculations are performed with the ADF code with the BP86/DZ level of theory in the optimizations and the SAOP functional and LB94 functional in the excitation calculations. A sharp excitation peak originates from a mixture of orbital transitions, and a broad excitation arises from multiple excited states in octahedral, truncated octahedral, and icosahedral Agn clusters. We predict that the absorption peak maximum red shifts as the cluster becomes larger and blue shifts as the shape of clusters changed from octahedral to icosahedral.

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INTRODUCTION Recently, physicists, chemists, material scientists, and biologists have been increasingly interested in the plasmonic optical properties of metal nanoparticles.1−4 The conduction electrons in the metal undergo a collective resonance called a surface plasmon resonance (SPR). When a photon is absorbed by a metal nanoparticle or surface, plasmon excitation occurs and tranfers energy into the collective oscillations of the conduction electrons. The number of such modes as well as their frequency and width are determined by the electron density, effective mass, shape, size, dielectric function, and environment of the nanoparticle. Metals (silver, gold, titanium, cobalt, etc.) have a sharp and distinct optical response because of plasmonic properties. The plasmon resonance of noble metal nanoparticles such as silver and gold appears in the visible region. Therefore, the optical properties of noble metal nanoparticles lead to many uses in catalysis,5 biological labeling,6 nanooptics,7,8 and electronics.9,10 Metal nanoparticles have been researched for medical diagnostics such as DNA detection11 or drug delivery.12 The principal parameters of nanoparticles are their size, shape, and surface characteristics. Nanoparticles containing sharp tips such as triangular plates13 and icosahedra14 can exhibit multipolar plasmon modes. Therefore, size and shape dependences of optical properties have been studied experimentally and in many groups.15−20 Various silver and gold nanoparticles such as rods, cubes, plates, octahedra, truncated octahedra, and icosahedra have been reported.13,14,19,21−26 Theoretical methods have been developed to interpret the extinction spectra of nanoparticles such as Mie theory,27 the discrete dipole approximation (DDA),28 electromagnetic finite difference time domain (FDTD),29 and time-dependent density © 2012 American Chemical Society

functional theory (TDDFT). In classical physics, Mie studied the extinction spectra of spherical particles of arbitrary size using Maxwell’s equations in 1908. Even though Mie theory is quite simple, computationally cheap, and qualitatively correct, the model does not consider the discrete atomic structure of the real nanoparticle, and it can be applied only to spherical, ellipsoidal, and core−shell structures. Advanced classical electrodynamics methods for solving Maxwell’s equations for spherical nanoparticles and more complex structures include DDA and FDTD. The quantum mechanical TDDFT method is also good for describing nanosystems and for understanding in detail the electron excitation process; it is limited to systems smaller than DDA and FDTD. These techniques have been developed to examine the intense extinction spectrum, which is affiliated with localized surface plasmon excitation. Recently, there have been many theoretical studies that employ TDDFT for silver and gold nanoparticles with various sizes and shapes.22,30−39 Tetrahedral Agn (n ≤ 120) particles,31 silver and gold nanorod clusters (Agn, Aun, n = 12−120),32,37 and larger silver and gold nanoclusters (truncated octahedral, octahedral, cubic, and icosahedral shapes) have been investigated using TDDFT calculations.38,39 In general, a relationship between the surface plasmon resonance and the morphology (number of facets and vertices) of nanoparticles is of interest. Much research has been performed on icosahedral and decahedral nanoparticles as well as fcc related morphologies like cubes and truncated cubes with Received: January 24, 2012 Revised: March 7, 2012 Published: May 1, 2012 10356

dx.doi.org/10.1021/jp300789x | J. Phys. Chem. C 2012, 116, 10356−10367

The Journal of Physical Chemistry C

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their facets and vertices.14,40−46 As the number of facets of silver nanoparticles increases, there are fewer surface plasmon resonances, the main resonance is blue-shifted, and the fwhm of the spectra decreases with cubic, truncated cubic, truncated octahedral, icosahedral, and spherical nanoparticles using DDA calculations.33 In this Article, we use TDDFT to study the evolution of cluster spectra with size and shape for the octahedral, truncated octahedral, and icosahedral shapes. We show absorption spectra of charged clusters that have filled or unfilled orbitals for each size and shape. We examine the size and shape dependences of octahedral, truncated octahedral, and icosahedral clusters. We predict that the peak location maximum red shifts as the cluster becomes larger. In addition, the correlation between experimental and theoretical absorption spectra is analyzed.

clusters. In noble metal clusters, the electronic shell effect plays an important role in small clusters. Electronic shells in spherical systems are filled as 1S21P61D102S21F142P61G18, where S−P− D−F−G, etc., denote the angular momentum quantum number and 1, 2, etc., denote the principle quantum number. The S, P, etc., orbitals are delocalized throughout the cluster and correspond to linear combinations of the valence 5s electrons. The magic numbers corresponding to the closure of electronic shells are known to be 2, 8, 18, 20, 34, 40, 58, 92, 138, etc., for spherical (and often for approximately spherical) systems.54,55 Therefore, the Ag6−2 cluster (8 electrons) and the Ag19−1 cluster (20 electrons) are magic number clusters for octahedral shapes. The orbitals and excitation spectra of Ag6−2 and Ag19−1 clusters calculated at LB94/DZ.4p levels of theory are shown in Figure 1. The octahedral irreducible representation and the

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COMPUTATIONAL METHOD The Amsterdam Density Functional (ADF) program47 is used for all density functional theory calculations. The molecular structures of all neutral and charged clusters are optimized using the gradient-corrected Becke-Perdew (BP86) exchangecorrelation functional.48,49 The basis set employed in the optimizations is a double-ζ (DZ) Slater type basis set. The zeroth-order regular approximation (ZORA) is employed in the calculations to account for scalar relativistic effects.50 The SCF convergence is tightened to 10−8. A gradient convergence criterion of 10−3 or tighter and an energy convergence criterion of 10−4 or tighter are used to obtain well-converged geometries. Time-dependent density functional theory (TDDFT) is employed to calculate excited states. We use the LB9451 and SAOP52 functionals because these functionals are asymptotically correct models and are effective for calculating excitation spectra.53 Even though the SAOP functional is more recent, this functional does not allow us to employ a frozen core basis set. The LB94 functional is chosen because of its computational economy with frozen core basis sets (denoted DZ.4p for the frozen core double-ζ basis). The tolerance was set to 10−8, and the orthonormality was set to 10−10. The numerical integration accuracy of integrals for the Fock matrix elements and energy terms was increased to 10−6 (10−8 for small clusters). The numbering of the orbitals omits the core orbitals. Depending on the size of the cluster, the first 50−600 dipole-allowed transitions were evaluated for the optical absorption spectrum. The smoothed spectra shown in the figures are convoluted with a Lorentzian with a full width at half-maximum (fwhm) of 0.2 eV. All calculations (geometry optimizations and excitation calculations) have been performed employing the Oh point group symmetry for octahedral and truncated octahedral clusters, while the icosahedral clusters have been calculated using D5d or C5v point group symmetry as the complete Ih group is not supported by ADF.

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Figure 1. Orbitals and optical absorption spectra for Ag6−2 and Ag19−1 clusters using the LB94 functional. The orbitals are plotted with a contour of 0.002. Optical absorption spectra for Ag6+4 and Ag19+1 clusters using the LB94 functional are also presented.

closest spherical symmetry representation are denoted for each orbital. Peak positions are located at higher energies when the LB94 functional is used instead of the SAOP functional. For the octahedral Ag6−2 cluster, peak locations with the LB94 functional are 2.64, 3.53, and 5.36 eV, whereas the corresponding peak locations with the SAOP functional are 2.33, 3.09, and 4.85 eV (Figure 1S). Spectra using the SAOP functional are presented in the Supporting Information. The triply degenerate HOMO of Ag6−2 corresponds to a set of P-like orbitals. The unoccupied D-like orbitals are split into two sets (eg + t2g) because of the octahedral symmetry of this system. The 2S orbital lies between the two D sets. An examination of the orbital energies shows the HOMO (3t1u) is predicted at −0.57 eV, while the LUMO (3eg) lies at 1.32 eV,

RESULTS AND DISCUSSION

Octahedral Clusters. We first consider theoretical absorption spectra of Ag6, Ag19, Ag44, and Ag85 clusters with octahedral shapes. Charged silver clusters such as Ag6−2, Ag6+4, Ag19−1, Ag19+1, Ag44−2, and Ag85−1 clusters are considered, which remove or add electrons in the HOMO to yield a closed shell. For example, Ag6 clusters can add two electrons in the 3t1u orbital (HOMO) so that this orbital is completely filled. We also consider magic number clusters when treating the charged 10357



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the LUMO+1 (3a1g) lies at 2.58 eV, and the LUMO+2 (3t2g) lies at 3.56 eV. Thus, the HOMO−LUMO gap for Ag6−2 is predicted to be 1.89 eV using the LB94 functional. The LB94 HOMO−LUMO gaps for each cluster are presented in Table 1. Notably, the HOMO−LUMO gaps of silver nanoparticles decrease as the cluster size increases.

Table 2. Strong Optical Absorption Peaks in the Spectrum of Octahedral Clusters Using the LB94 Functional: The Clusters, Peak Energy, Oscillator Strengths, Orbitals, and Primary Transitions Responsible for These Peaks

Table 1. Calculated HOMO−LUMO Gaps of Silver Nanoparticles Using the LB94 Functional shape octahedra

truncated octahedra

icosahedra

cluster −2

clusters

peak energy

oscillator strength

A6−2

2.64

0.06

3.53

0.32

5.36 4.49 4.59

1.32 0.81 1.21

A19+1

4.65 4.72

0.85 1.17

A44−2

4.46

3.55

A85−1

4.36

5.08

ΔE (eV)

Ag6 Ag6+4 Ag19−1 Ag19+1 Ag44−2 Ag85−1

1.89 3.41 0.68 0.13 0.49 0.04

Ag13−1 Ag13−5 Ag13+5 Ag38−2 Ag38+4 Ag55−1 Ag55−3 Ag55+5 Ag79−1 Ag79+5

0.75 1.08 2.43 0.32 0.53 0.28 0.16 0.49 0.11 0.06

Ag13−1 Ag13−5 Ag13+5 Ag43−1 Ag43+3 Ag55−3 Ag55+1

0.05 0.32 2.93 0.10 0.53 0.79 0.17

A19−1

Following the selection rule ΔL = ±1, the Ag6−2 cluster with eight electrons should have allowed P → D or P → S transitions. In Ag6−2 cluster (Figure 1 and Table 2) calculated with LB94 functional, the first peak (2.64 eV and oscillator strength f = 0.06) can be assigned to a linear combination of HOMO → LUMO (P → D) transition and HOMO → LUMO+1 (P → S) transitions, and the second peak (3.53 eV and f = 0.32) can be assigned to a constructive mixture of the HOMO → LUMO (P → D), HOMO → LUMO+1 (P → S), and HOMO → LUMO+2 (P → D) transitions. The third strong peak at 5.36 eV (f = 1.32) is calculated to be the HOMO → LUMO+2 (P → D) transition. In the Ag6+4 cluster (Figure 1), there are also three peaks as in the Ag6−2 cluster. These peaks with the LB94 functional lie at 4.34, 5.54, and 5.84 eV, whereas these peaks with the SAOP functional lie at 4.26, 5.36, and 5.66 eV (Figure 1S). The next larger octahedral cluster is the Ag19 cluster, which has one electron in the HOMO (5a1g orbital). Adding or removing one electron yields the Ag19−1 and Ag19+1 clusters, respectively. The two strong peaks in Ag19−1 lie at 4.49 eV (f = 0.81) and 4.59 eV ( f = 1.21) using the LB94 functional (Figure 1). One of two strong peaks located at 4.49 eV arises from HOMO−2 to LUMO+2. The primary transiton is D → P. Another peak located at 4.59 eV arises from a mixture of HOMO−HOMO−2 to LUMO−LUMO+2 and LUMO+5. The primary transitions are D → P, D → F, and S → P (Table 2). Loss of one electron gives the Ag19+1 cluster that has two strong

orbital

transition from occupied orbital

transition to unoccupied orbital

P→D P→S P→D P→D P→S P→D D→P S→P D→P D→F D→F D→F D→F D→P D→P D→F D→F D→F D→F P→S P→D F→G F→G G→H G→H G→H D→P H→I H→I H→I H→I D→F D→F D→F G→H P→S

HOMO HOMO HOMO HOMO HOMO HOMO HOMO−2 HOMO HOMO−1 HOMO−1 HOMO−1 HOMO−2 HOMO−2 HOMO−1 HOMO HOMO HOMO−1 HOMO−1 HOMO HOMO−2 HOMO HOMO−4 HOMO−5 HOMO−3 HOMO−1 HOMO−1 HOMO HOMO−1 HOMO−3 HOMO−2 HOMO−3 HOMO−4 HOMO HOMO−4 HOMO−5 HOMO−7

LUMO LUMO+1 LUMO LUMO+2 LUMO+1 LUMO+2 LUMO+2 LUMO+2 LUMO+2 LUMO+5 LUMO+1 LUMO+1 LUMO LUMO+4 LUMO+4 LUMO+6 LUMO+2 LUMO+1 LUMO+2 LUMO LUMO+6 LUMO+1 LUMO LUMO+3 LUMO+3 LUMO+4 LUMO+11 LUMO+8 LUMO+1 LUMO+5 LUMO+2 LUMO+4 LUMO+12 LUMO+6 LUMO+3 LUMO

peaks at 4.65 eV (f = 0.85) and 4.72 eV (f = 1.17) using the LB94 functional (Figure 1). The peak located at 4.65 eV arises from HOMO−1 to LUMO+4. The primary transition is D → P. The other peak located at 4.72 eV arises from HOMO−HOMO−2 to LUMO−LUMO+2, LUMO+4, and LUMO+6. The primary transitions are D → P, D → F, and P → S (Table 2). Using the SAOP functional, the strong peak in Ag19−1 cluster is located at 4.04 eV with an oscillator strength of 1.22. Loss of one electron gives the Ag19+1 cluster, which has a sharp peak at 4.21 eV (Figure 1S). The next larger octahedral cluster is the Ag44 cluster that would have four electrons in the triply degenerate 18t1u orbital (HOMO). Adding two electrons yields the Ag44−2 cluster, which exhibits a sharp peak at 4.46 eV (f = 3.55) with the LB94 functional calculation (Figure 2). The strong peak arises from mixed transitions that are P → D, F → G, and G → H (Table 2). It should be noted that the G-like orbitals are not completely filled, so transitions both into and out of these orbitals contribute to the strong peak. As compared to one 10358



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LB94 functional lies at 4.36 eV with an oscillator strength of 5.08. The primary transitions of the sharp peak are D → P, H → I, D → F, G → H, and P → S (Table 2), which again correspond to the expected selection rule of ΔL= ±1. The peak energy maximum shifts when the cluster becomes larger. We consider the relationship of the peak shift to both the radius (L) of nanoparticles and the number of electrons for size dependence of silver nanoparticles using both the LB94 and the SAOP functionals. The radius L is defined as the distance between the center and one of the vertices. The linear fits between the peak location maximum and radius of nanoparticles are y = 0.35127x + 3.84239 with an R2 value of 0.82 and y = 0.41056x + 3.20188 with an R2 value of 0.63 for the LB94 and SAOP functionals, respectively, in Figure 3. Because the Ag6−2 and Ag6+4 clusters have a similar radius but greatly different optical absorption, the fit is not ideal. We have also considered the relation between N−1/3 (N = number of electrons) and peak location. The fits between peak location and number of electrons are y = 2.78683x + 3.69793 with an R2 value of 0.92 and y = 3.44801x + 2.95377 with an R2 value of 0.95 for the LB94 functional and SAOP functional, respectively, in Figure 3. The fits between peak location and number of electrons of nanoparticles are more accurately predicted as a cluster becomes larger than ones between peak location and radius of nanoparticles because the R 2 values of the linear fit between peak location and number of electrons are much closer to 1. Thus, as the octahedral silver clusters become larger, the absorption peak maximum is predicted to shift toward 3.70 or 2.95 eV using the LB94 functional or SAOP functional, respectively. It is known31,37 that the LB94 functional often overestimates the energy, so the SAOP functional prediction of 2.95 eV is most likely the closest to the experimental system. The experimental extinction spectra of octahedral particles show significant spread from 3.10 to 2.07 eV.40 Previous DDA simulations agree remarkably well with the experimental data.40 Truncated Octahedral Clusters. We have optimized truncated octahedral clusters for Ag13, Ag38, Ag55, and Ag79. These structures are formed by removing low-coordinated atoms from octahedral clusters. For instance, the six vertex atoms of the octahedral Ag19 cluster are removed to create the truncated octahedral Ag13 cluster. As for octahedral nanoparticles, charged clusters of truncated octahedral silver

Figure 2. Orbitals and optical absorption spectra for Ag44−2 and Ag85−1 using the LB94 functional. The orbitals are plotted with a contour of 0.002.

strong peak predicted with the LB94 functional, there are two strong peaks with the SAOP functional that lie very close in energy at 3.93 eV (f = 1.21) and 4.04 eV (f = 1.76) (Figure 1S). The largest cluster of octahedral shape in our research is the Ag85 cluster. Ag85−1 cluster is formed by adding one electron in the triply degenerate 31t2g orbital (HOMO). In Figure 2, the single sharp peak of the Ag85−1 cluster with the

Figure 3. Plots of absorption peak energy (eV) versus radius (L) of octahedral clusters and peak energy versus N−1/3 (N = number of electrons). 10359



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nanoparticles are examined with completely filled and unfilled orbitals. The Ag13 cluster with cuboctahedral symmetry is considered with three charge states of −1, −5, and +5. The neutral Ag13 cluster has five electrons in the triply degenerate 6t2g orbital (HOMO), which is a D-like orbital. Addition of one electron to fill the HOMO yields the Ag13−1 cluster; adding one electron in the 6t2g orbital and four electrons in the 5eg orbital (the second D-like orbital) forms the Ag13−5 cluster. In addition, we remove five electrons in the HOMO to make the charge state of +5, which corresponds to a filled P shell and completely unfilled D shell. The orbitals and excitation spectra of Ag13+5 cluster calculated at LB94/DZ.4p levels of theory are shown in Figure 4.

In Ag13 clusters, the charge states of −1, −5, and +5 exhibit single sharp peaks at 4.86 eV (f = 1.65), 4.98 eV (f = 2.18), and 4.59 eV (f = 0.98), respectively, using the LB94 functional (Figure 4). In the Ag13+5 cluster, the sharp peak at 4.59 eV arises from a linear combination of HOMO to LUMO through LUMO+2. Thus, the primary transitions are P → D and P → S (Table 3). In the Ag13−1 cluster, the primary transitions are D → P, D → F, P → D, and P → S (Table 3). In the Ag13−5 cluster, the sharp peak arises from HOMO through HOMO−2 to LUMO through LUMO+4, and the primary transitions are D → F, D → P, and P → S (Table 3). Using the SAOP functional, the sharp peaks fall at 4.36 eV ( f = 1.36), 4.46 eV (f = 1.88), and 4.40 eV (f = 0.71) for charge states of −1, −5, and +5, respectively (Figure 2S). The next truncated octahedral cluster Ag38 has four electrons in the triply degenerate 16t1u orbital (HOMO), which corresponds to the 2P shell. The charge states of −2 and +4 of Ag38 cluster are made by adding two electrons and removing four electrons so that the 16t1u orbital is completely filled or unfilled. The charged Ag38 clusters have sharp peaks at 4.52 eV (−2 charge; f = 5.33) and 4.62 eV (+4 charge; f = 3.93) with the LB94 functional (Figures 4) and at 4.01 eV (−2 charge; f = 3.74) and 4.20 eV (+4 charge; f = 2.80) with the SAOP functional (Figure 2S). The single sharp peak of Ag38−2 cluster with the LB94 functional is shown in Figure 4. The peak is comprised of a mixture (linear combination) of P → D, F → G, P → S, and F → D transitions (Table 3). In the Ag38+4 cluster using the LB94 functional, the mixed transitions are F → G, S → P, D → P, and F → D (Table 3). The next truncated octahedral silver cluster is the Ag55 cluster. The neutral Ag55 cluster has five electrons in the triply degenerate 15t1g orbital (HOMO), which is a G-like orbital. Because the next higher orbital also has G-like symmetry, we add one and three electrons to fill in the 15t1g orbital and 11a1g orbital for charge states of −1 and −3, respectively. The Ag55−3 cluster has 58 electrons, a spherical magic number. In addition, we remove five electrons in the 15t1g orbital to yield the +5 charge of the Ag55 cluster. Using the LB94 functional, the Ag55−3 cluster has a single sharp peak (Figure 5), while other charges exhibit two or three peaks. The single sharp peak of Ag55−3 cluster at 4.54 eV ( f = 3.54) has mixed transitions of G → H, P → D, F → D, and F → S (Table 3). F → S does not follow the spherical selection rule of ΔL = ±1; it should be noted that these orbitals have t1u and a1g representations in the Oh point group, and therefore transitions between these orbitals are allowed because of the symmetry-lowering in this cluster. In the Ag55−1 cluster, the two sharp peaks lie at 4.39 eV (f = 2.04) and 4.56 eV (f = 3.04) in Figure 5. The primary transition of the peak at 4.39 eV is F → I, and the primary transitions of the peak at 4.56 eV are G → H, F → D, F → I, F → S, and P → G (Table 3). The three sharp peaks of the Ag55+5 cluster lie at 4.39 eV (f = 1.01), 4.46 eV (f = 2.26), and 4.61 eV (f = 1.31) in Figure 5. The strongest peak of three peaks has the mixed transitions of G → H, P → G, F → D, P → D, F → S, F → G, and F → I (Table 3). Again, the selection rules are somewhat relaxed from the higher symmetry spherical case. The absorption spectra of Ag13−1, Ag13−3, and Ag13+5 clusters using the SAOP functional are presented in Figure 2S. The largest truncated octahedral silver cluster examined here is the Ag79 cluster. The charge states of −1 and +5 of the Ag79

Figure 4. Orbitals and optical absorption spectra for Ag13+5 and Ag38−2 using the LB94 functional. The orbitals are plotted with contours of 0.01 (Ag13+5 cluster) and 0.002 (Ag38−2 cluster). Optical spectra for Ag13−1, Ag13−5, and Ag38+4 clusters using the LB94 functional are also shown. 10360



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Table 3. Strong Optical Absorption Peaks in the Spectrum of Truncated Octahedral Clusters Using the LB94 Functional: The Clusters, Peak Energy, Oscillator Strengths, Orbitals, and Primary Transitions Responsible for These Peaks oscillator strength orbital

clusters

peak energy

A13+5

4.59

0.98

A13−1

4.86

1.65

A13−5

4.98

2.18

A38−2

4.52

5.33

A38+4

4.62

3.93

P→D P→S P→D D→P D→F D→F D→P P→D P→S D→F D→F D→F D→F D→F D→P P→S P→D P→D F→G F→G F→G F→G F→G F→G F→G P→S F→D F→D F→G F→G F→G F→G F→G F→G F→G S→P D→P F→D F→D



HOMO HOMO HOMO HOMO HOMO HOMO HOMO HOMO−1 HOMO−1 HOMO HOMO−1 HOMO−1 HOMO−1 HOMO HOMO−1 HOMO−2 HOMO HOMO HOMO−3 HOMO−2 HOMO−3 HOMO−3 HOMO−2 HOMO−1 HOMO−2 HOMO HOMO−1 HOMO−2 HOMO−2 HOMO−1 HOMO−2 HOMO−2 HOMO−1 HOMO HOMO−1 HOMO−3 HOMO−4 HOMO HOMO−1

LUMO+1 LUMO+2 LUMO LUMO+4 LUMO+2 LUMO+3 LUMO+5 LUMO LUMO+1 LUMO+4 LUMO+4 LUMO+3 LUMO+1 LUMO+1 LUMO+2 LUMO LUMO+7 LUMO+4 LUMO+2 LUMO LUMO+1 LUMO LUMO+3 LUMO+2 LUMO+1 LUMO+6 LUMO+7 LUMO+4 LUMO+3 LUMO+1 LUMO+2 LUMO+1 LUMO+4 LUMO+3 LUMO+2 LUMO LUMO LUMO+8 LUMO+5

oscillator strength orbital

clusters

peak energy

A55−3

4.54

3.54

A55−1

4.39 4.56

2.04 3.04

A55+5

4.46

2.26

A79+5

4.41

2.89

G→H G→H G→H G→H G→H G→H P→D F→D F→S F→I G→H G→H G→H G→H G→H F→D F→I F→S P→G G→H G→H G→H G→H P→G P→G F→D P→D F→S F→G F→I P→D H→I H→I H→I D→F G→H G→H P→S



HOMO−2 HOMO−1 HOMO−3 HOMO−2 HOMO−3 HOMO HOMO−5 HOMO−4 HOMO−4 HOMO−3 HOMO−1 HOMO HOMO−2 HOMO−1 HOMO−2 HOMO−3 HOMO−3 HOMO−3 HOMO−4 HOMO HOMO−1 HOMO HOMO−1 HOMO−3 HOMO−3 HOMO−2 HOMO−3 HOMO−2 HOMO−5 HOMO−2 HOMO−7 HOMO HOMO−1 HOMO HOMO−2 HOMO−3 HOMO−6 HOMO−7

LUMO+1 LUMO+6 LUMO+5 LUMO+3 LUMO+1 LUMO+6 LUMO LUMO+4 LUMO+2 LUMO+14 LUMO+2 LUMO+7 LUMO+6 LUMO+3 LUMO+2 LUMO+5 LUMO+14 LUMO+4 LUMO LUMO+3 LUMO+6 LUMO+4 LUMO+3 LUMO LUMO+1 LUMO+7 LUMO+2 LUMO+5 LUMO LUMO+15 LUMO LUMO+8 LUMO+6 LUMO+7 LUMO+4 LUMO+3 LUMO+2 LUMO+1

location versus N−1/3) than the fits with the LB94 functional because the data of Ag13 clusters with the LB94 functional scatter widely. We predict that the absorption peak maximum will shift toward around 3.54−3.64 eV for larger truncated octahedral clusters using SAOP functional calculations. The R2 values found for the truncated systems are not quite as good as for the standard octahedral clusters, which may be due in part to the variety of edge lengths, etc., in the truncated systems. Icosahedral Clusters. The shapes of icosahedral clusters are closer to spherical clusters than the shapes of octahedral and truncated octahedral clusters. In addition, icosahedral clusters have a larger number of nanoparticle facets than do octahedral and truncated octahedral clusters. Therefore, a relationship between the excitation energy and the more spherical shape is expected. The first icosahedral silver cluster is Ag13, which was calculated using D5d symmetry. The Ag13−5 and Ag13+5 clusters are

cluster are made by adding and removing electrons in the triply degenerate 29t2g orbital, which has five electrons. Using the LB94 functional, the Ag79+5 cluster has three sharp peaks at 4.39 eV (f = 1.44), 4.41 eV ( f = 2.89), and 4.45 eV (f = 0.83) (Figure 5); the Ag79−1 cluster has a broad spectrum (Figure 5). The primary transitions of the Ag79+5 cluster at 4.41 eV are P → D, H → I, D → F, G → H, and P → S in Figure 5 and Table 3. Ag79+5 has one strong sharp peak at 3.97 eV (f = 2.49), while broad peaks are shown in the Ag79−1 cluster with the SAOP functional (Figure 2S). As the truncated octahedral clusters become larger, we predict the peak maximum shift using radius and number of electrons (LB94 functional and SAOP functional) in Figure 6. The radius L is defined as the distance between the center and the vertices. The better fits with the SAOP functional show y = 0.22048x + 3.63739 with an R2 = 0.88 (peak location versus radius) and y = 1.94589x + 3.53693 with an R2 = 0.78 (peak 10361



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collection at peaks is observed in the Ag13−1 cluster, which is not a magic number cluster. In Figure 7, we show the orbitals and spectra of icosahedral clusters Ag13+5, Ag43+3, and Ag55−3 with the LB94/DZ.4p level of theory. In Tables 4−6, we show the symmetries, peak energy, oscillator strength, orbitals, and primary transitions of each icosahedral cluster. A sharp peak at 4.66 eV ( f = 1.08) for the Ag13+5 cluster arises using the LB94 functional (Figure 7). Dipole allowed irreducible representations of D5d symmetry are A2u and E1u. The peak appears as two peaks in the spectrum due to splitting because D5d has lower symmetry than Ih. The primary transitions are P → D and P → S. The next larger icosahedral silver cluster is Ag 43, which was calculated using C5v symmetry. The charge states of −1 and +3 are examined for excitation energies. The magic number cluster Ag43+3 with 40 electrons has single sharp peaks of 4.46 eV ( f = 4.77) and 4.04 eV ( f = 2.84) with the LB94 functional (Figure 7) and the SAOP functional (Figure 3S), respectively. Using the LB94 functional, the sharp peak of Ag 43+3 cluster appears as two peaks (A1 and E1 symmetries) in the spectrum due to splitting peaks because C5v has lower symmetry than Ih. The primary transitions are F → G, P → D, P → S, and F → D in Figure 7. The Ag 43−1 cluster with 44 electrons, which is not a magic number, has broad peaks with both the LB94 functional (Figure 7) and the SAOP functional (Figure 3S). The next larger icosahedral cluster is Ag 5 5 (C 5 v symmetry). Charge states of −3 and +1 are calculated for excitation energies. The magic number cluster, Ag 55−3 (58 electrons), exhibits much clearer peaks with one or two peaks using the LB94 functional (Figure 7) and the SAOP functional (Figure 3S) than the absorption spectra of Ag 55+1 cluster (54 electrons), which is not a magic number. In the Ag55−3 cluster, two strong peaks at 4.41 eV ( f = 2.41) and 4.55 eV ( f = 3.49) are exhibited with the LB94 functional. In Figure 7, the primary transitions of the icosahedral Ag55−3 cluster are S → H, G → H, P → D, G → F, P → I, and P → S. Comparison of the orbitals and orbital energies for the icosahedral clusters shows that the D, G, H, etc., shells are not split as much as in the octahedral systems; this is likely due to the higher symmetry of the icosahedral clusters. The HOMO−LUMO gaps of magic number clusters in icosahedral clusters are larger than ones of other clusters because icosahedral clusters are close to a spherical model (Table 1). The energy gaps of the magic clusters Ag 13+5 (2.93 eV) and Ag13−5 (0.32 eV) with 8 electrons and 18 electrons, respectively, are much larger than the gap for the Ag13−1 cluster (0.05 eV). The magic clusters Ag 43+3 (40 electrons) and Ag55−3 (58 electrons) have significantly larger energy gaps than non magic clusters (Ag43−1 and Ag55+1) in Table 1. We also predict the peak location maxima as the cluster becomes larger using radius and N−1/3 relationships with both the SAOP and the LB94 functionals. We found fits of y = 0.19579x + 4.1205 with R2 = 0.64 (peak location versus radius with the LB94 functional) and y = 1.55057x + 3.72411 with R2 = 0.55 (peak location versus number of electrons with the SAOP functional). Even though the values of R2 are low due to the scattered data of Ag 13 clusters and small quantity of data, we predict that the absorption peak maximum will shift toward around 4.12 eV

Figure 5. Orbitals and optical absorption spectra for Ag55−3 and Ag79+5 using the LB94 functional. The orbitals are plotted with a contour of 0.005. Optical spectra for Ag55+5, Ag55−1, and Ag79−1 clusters using the LB94 functional are shown below the line.

magic number clusters because they have 18 electrons and 8 electrons, respectively. Ag13−5 and Ag13+5 clusters have single sharp peaks at 5.01 eV (f = 2.60) and 4.66 eV ( f = 1.08) (LB94 functional) in Figure 7 and at 4.48 eV (f = 2.26) and 4.48 eV (f = 0.78) (SAOP functional) in Figure 3S, while a broad 10362



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Figure 6. Plots of absorption peak energy versus radius (L) of truncated octahedral shape and peak energy versus N−1/3 (N = number of electrons).

Table 4. Strong Optical Absorption Peaks in the Spectrum of Icosahedral Ag13+5: The Symmetries, Peak Energy, Oscillator Strengths, Orbitals, and Primary Transitions Responsible for These Peaks peak oscillator symmetry energy strength orbital A2u

4.66

1.08

E1u

4.66

1.08

P→D P→D P→S P→D P→S P→D

transition from occupied orbital 7e1u 5a2u 5a2u 7e1u 7e1u 5a2u

(HOMO) (HOMO−1) (HOMO−1) (HOMO) (HOMO) (HOMO−1)

transition to unoccupied orbital 8e1g 7a1g 8a1g 8e2g 8a1g 8e1g

(LUMO) (LUMO+1) (LUMO+3) (LUMO+2) (LUMO+3) (LUMO)

Table 5. Strong Optical Absorption Peaks in the Spectrum of Icosahedral Ag43+3: The Symmetries, Peak Energy, Oscillator Strengths, Orbitals, and Primary Transitions Responsible for These Peaks peak oscillator symmetry energy strength orbital A1

4.46

4.77

E1

4.46

4.73


F→G 46e1 (HOMO−5) 46e1 (HOMO−5) 45e2 (HOMO−4) 31a1 (HOMO−3) 46e2 (HOMO−2) P→D 47e1 (HOMO) 32a1 (HOMO−1) P→S 32a1 (HOMO−1) F→D 31a1 (HOMO−3) P→D 47e1 (HOMO) 32a1 (HOMO−1) F→G 46e1 (HOMO−5) 45e2 (HOMO−4) 45e2 (HOMO−4) 46e1 (HOMO−5) 46e2 (HOMO−2) 46e2 (HOMO−2) P→S 47e1 (HOMO)

transition to unoccupied orbital 49e1 48e1 47e2 33a1 48e2 50e1 34a1 35a1 34a1 49e2 50e1 47e2 48e1 48e2 33a1 49e1 48e2 35a1

(LUMO+2) (LUMO) (LUMO+1) (LUMO+4) (LUMO+3) (LUMO+6) (LUMO+5) (LUMO+11) (LUMO+5) (LUMO+7) (LUMO+6) (LUMO+1) (LUMO) (LUMO+3) (LUMO+4) (LUMO+2) (LUMO+3) (LUMO+11)

using the LB94 functional calculation and around 3.72 eV using the SAOP functional calculation. Comparison of Nanoparticle Shapes. One of the most clear observations from Figures 3, 6, and 8 is that peak location

Figure 7. Orbitals and optical absorption spectra for Ag13+5, Ag43+3, and Ag55−3 using the LB94 functional. The orbitals are plotted with a contour of 0.005. Optical spectra for Ag13−1, Ag13−5, Ag43−1, and Ag55+1 clusters using the LB94 functional are shown below. 10363



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The absorption peak maxima calculated using the LB94 functional for larger clusters such as octahedral Ag146+4, cuboctahedral Ag147+5, and icosahedral Ag147−1 lie at 4.37, 4.54, and 4.50 eV, respectively.38 These values are in good agreement with our linear fit of octahedra (4.23 eV), truncated octahedra (4.44 eV), and icosahedra (4.44 eV) (peak location versus number of electrons with the LB94 functional). It should be noted that the geometries in the current work and in ref 38 were determined with different functionals, which may account for the systematic underestimation of this work as compared to the values in ref 38. To examine the shape dependence, we compare absorption spectra between truncated octahedra and icosahedra with the same charges using the LB94 functional and SAOP functionals. The icosahedral Ag13+5 cluster has peaks that are blue-shifted with respect to truncated octahedral ones; the icosahedral Ag13−1 cluster has red-shifted peaks, while the icosahedral Ag13−5 cluster has slightly blue-shifted ones with respect to the corresponding truncated octahedral ones using the LB94 and SAOP functionals (Figure 9). In the Ag55−3 cluster, the icosahedral cluster is blue-shifted using the SAOP functional, while the icosahedral cluster is not blue-shifted with respect to the truncated octahedral cluster using the LB94 functional. We also compare absorption spectra between tetrahedra31 and octahedra with similar size using the LB94 functional. The absorption peak maximum of tetrahedra Ag20 cluster lies at 4.14 eV, while the absorption peak maximum of octahedral Ag19−1 is 4.56 eV. In addition, the absorption peak maximum of octahedral Ag85−1 cluster (4.38 eV) lies at 0.76 eV higher energy than does the absorption peak maximum of tetrahedral Ag84 cluster (3.62 eV). The differences in energy are much greater for tetrahedra versus octahedra than for the shapes compared in the current work. Experimental extinction spectra using UV/vis absorption spectra are reported for cubic, truncated octahedral, and octahedral silver nanoparticles.40 The truncated octahedral silver nanoparticles are blue-shifted with respect to octahedral ones and appear at λ = 2.59 eV and λ = 2.89 eV (2.07−3.10 eV range for octahedral shape). Thus, the absorption spectra are strongly associated with the shape like the edges of the polyhedra.40 From the experimental data, we predict that the absorption peaks of icosahedral clusters that have more facets than octahedral clusters will be

Table 6. Strong Optical Absorption Peaks in the Spectrum of Icosahedral Ag55−3: The Symmetries, Peak Energy, Oscillator Strengths, Orbitals, and Primary Transitions Responsible for These Peaks peak oscillator symmetry energy strength orbital A1

4.41 4.55

2.41 3.49

E1

4.41 4.55

2.39 3.50


S→H 38a1 (HOMO−11) G→H 61e1 (HOMO−1) 59e2 (HOMO−3) 60e1 (HOMO−4) 41a1 (HOMO−2) 41a1 (HOMO−2) P→D 59e1 (HOMO−6) 40a1 (HOMO−5) G→F 60e2 (HOMO) P→I 59e1 (HOMO−6) P→S 40a1 (HOMO−5) S→H 38a1 (HOMO−11) S→H 38a1 (HOMO−11) G→H 59e2 (HOMO−3) 60e1 (HOMO−4) 61e1 (HOMO−1) 60e1 (HOMO−4) 41a1 (HOMO−2) 60e2 (HOMO) P→D 59e1 (HOMO−6) 40a1 (HOMO−5) G→F 61e1 (HOMO−1) P→S 59e1 (HOMO−6) S→H 38a1 (HOMO−11) P→I 40a1 (HOMO−5) 59e1 (HOMO−6)

transition to unoccupied orbital 44a1 (LUMO+5) 64e1 (LUMO+7) 62e2 (LUMO+8) 63e1 (LUMO+4) 44a1 (LUMO+5) 45a1 (LUMO+9) 62e1 (LUMO+2) 42a1 (LUMO) 64e2 (LUMO+12) 68e1 (LUMO+22) 43a1 (LUMO+3) 44a1 (LUMO+5) 63e1 (LUMO+4) 63e1 (LUMO+4) 44a1 (LUMO+5) 22a2 (LUMO+6) 22a2 (LUMO+6) 64e1 (LUMO+7) 62e2 (LUMO+8) 61e2 (LUMO+1) 62e1 (LUMO+2) 46a1 (LUMO+11) 43a1 (LUMO+3) 63e1 (LUMO+4) 68e1 (LUMO+22) 23a2 (LUMO+21)

maxima of octahedral, truncated octahedral, and icosahedral clusters are red-shifted as the cluster becomes larger because the slopes of the fits are positive. Furthermore, the peak location maximum increases (to around 2.95 eV for octahedra, 3.54 eV for truncated octahedra, and 3.72 eV for icosahedra) as the number of facets increases using the SAOP functional. In other words, the peak locations are blue-shifted with shape dependences, and these results are in good agreement with a previous theoretical study using DDA calculations.33

Figure 8. Plots of peak energy versus radius (L) of icosahedral shape and peak energy versus N−1/3 (N = number of electrons). 10364



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Figure 9. Comparison of absorption spectra for truncated octahedral and icosahedral clusters including the +5, −1, and −5 charge states of Ag13 and the −3 charge state of Ag55.

The peak intensities of octahedral, truncated octahedral, and icosahedral nanoparticles are plotted in Figure 10. While the peak intensities increased geometrically in octahedral clusters, the peak intensities are not changed at around 40−50 after Ag13 clusters size in truncated octahedral and icosahedral shapes.

blue-shifted. However, it should be noted that these are relatively minor effects. In our results, the absorption spectra of Ag 13−5, Ag 13 +5 , and Ag 55−3 clusters with truncated octahedra and icosahedra have good agreement with the experimental trends. 10365



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Figure 10. Plots of peak intensity versus cluster size.

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CONCLUSIONS

Corresponding Author

In this study, the size and shape dependences of octahedral, truncated octahedral, and icosahedral silver nanoparticles Agn (n = 6−85) have been examined. These particles possess delocalized orbitals composed of valence 5s electrons. The shapes of these orbitals are similar to orbitals arising from a spherical potential; however, the orbitals are split due to symmetry-lowering to octahedral or icosahedral point groups. Silver nanoparticles exhibit an optical absorption spectrum with a remarkably sharp absorption maximum depending on orbital occupation. The filled or unfilled completely orbitals in the HOMO as well as magic number clusters are considered and lead to sharp strong peaks. The primary transitions are calculated for the peaks of each cluster. The strong peaks arise from a linear combination of single particle transitions that combine constructively. Generally, the single particle transitions follow the spherical selection rule ΔL = ±1, although this is relaxed in a few instances because of symmetry-lowering. As expected from the spherical jellium model, magic number clusters have a correlation with optical properties. In icosahedral clusters, the absorption spectra of magic number clusters are clearer than those for nonmagic clusters. However, the spherical magic numbers are not as important for octahedra and truncated octahedra. The LB94 functional calculations show much higher energies than do SAOP functional calculations. The peak location maxima have a linear dependence on both 1/L and the number of electrons as the cluster size changes. As the clusters get larger, the absorption peak locations are predicted to lie around 2.95, 3.54, and 3.72 eV for octahedra, truncated octahedra, and icosahedra, respectively, using SAOP functional calculations. We find that the peak locations are blue-shifted as the shape changes from octahedra to icosahedra. In addition, the peak locations are red-shifted as the cluster size increases. Therefore, we conclude that the peak locations are red-shifted with increased cluster size because the slopes of fits of peak location maxima for octahedra, truncated octahedra, and icosahedra are positive and blue-shifted with increased number of facets in silver nanoparticles. In addition, the results are consistent with other experimental and theoretical studies.

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AUTHOR INFORMATION

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This material is based upon work supported by the Air Force Office of Scientific Research under AFOSR Award No. FA9550-09-1-0451.

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REFERENCES

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ASSOCIATED CONTENT

S Supporting Information *

Figures of absorption spectra of octahedral, truncated octahedral, and icosahedral clusters using the SAOP functional. This material is available free of charge via the Internet at http://pubs.acs.org. 10366



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