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Semiempirical Modeling of Ag Nanoclusters: New Parameters for Optical Property Studies Enable Determination of Double Excitation Contributions to Plasmonic Excitation Rebecca L. Gieseking, Mark A. Ratner, and George C. Schatz J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.6b04520 • Publication Date (Web): 03 Jun 2016 Downloaded from http://pubs.acs.org on June 9, 2016

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The Journal of Physical Chemistry

Semiempirical Modeling of Ag Nanoclusters: New Parameters for Optical Property Studies Enable Determination of Double Excitation Contributions to Plasmonic Excitation

Rebecca L. Gieseking, Mark A. Ratner, George C. Schatz*

Department of Chemistry, Northwestern University 2145 Sheridan Road, Evanston, Illinois 60208, United States

* Corresponding author: [email protected]; (847)491-5657

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Abstract Quantum mechanical studies of Ag nanoclusters have shown that plasmonic behavior can be modeled in terms of excited states where collectivity among single excitations leads to strong absorption. However, new computational approaches are needed to provide understanding of plasmonic excitations beyond the single-excitation level. We show that semiempirical INDO/CI approaches with appropriately selected parameters reproduce the TD-DFT optical spectra of various closed-shell Ag clusters. The plasmon-like states with strong optical absorption comprise linear combinations of many singly excited configurations that contribute additively to the transition dipole moment, whereas all other excited states show significant cancellation among the contributions to the transition dipole moment. The computational efficiency of this approach allows us to investigate the role of double excitations at the INDO/SDCI level. The Ag cluster ground states are stabilized by slight mixing with doubly excited configurations, but the plasmonic states generally retain largely singly excited character. The consideration of double excitations in all cases improves the agreement of the INDO/CI absorption spectra with TDDFT, suggesting that the SDCI calculation effectively capture some of the ground-state correlation implicit in DFT. These results provide the first evidence to support the commonly used assumption that single excitations are in many cases sufficient to describe the optical spectra of plasmonic excitations quantum mechanically.


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1. Introduction Noble metal nanoparticles have been widely studied in recent years due to their plasmonic optical properties.1-4 In noble metal nanoparticles such as silver or gold, the collective excitation of the conduction band electrons in a localized surface plasmon excitation leads to intense absorption that is strongly dependent on nanoparticle size, shape and environment,1 in which the local electric field is enhanced by several orders of magnitude.5,6 As a result, plasmonic nanoparticles been used for a wide variety of applications, such as refractive-index-based colorimetric sensing,7,8 surface-enhanced Raman spectroscopy,9-12 and plasmon-enhanced absorption.13,14

Although classical electrodynamic descriptions of the nanoparticle plasmonic states have been widely studied,1,6,15 a quantum-mechanical understanding of the nature of plasmonic states in finite systems is still being developed.16,17 Classically, a plasmon is defined as a collective oscillation of all conduction electrons in the plasmonic material; however, typical quantummechanical approaches provide stationary eigenstates and are limited in practical terms to excitations of one or a few electrons from the valence band into the conduction band. Recent calculations using time-dependent density functional theory (TD-DFT)16,18-23 and configuration interaction singles (CIS)24,25 approaches to study optical excitations in silver and gold clusters have revealed strongly absorbing plasmon-like excited states that involve linear combinations of several single excitations that contribute additively to the transition dipole moment; plasmon-like excitations with analogous collective character have also been found in molecular systems.26,27 Although these states do not involve excitations of all the conduction electrons, it is helpful to



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use this somewhat broader definition to express what distinguishes plasmon excitations in clusters from other excitations.

Quantum-chemical models of plasmonic clusters provide useful insight into plasmon/molecule interactions, but they are limited to relatively small clusters. In clusters with dozens of conduction band electrons, predicting the absorption spectrum requires computing hundreds of excited states,18,20 which is computationally expensive using TD-DFT and CIS approaches. Another limitation of the current calculations is the restriction to only singly excited configurations. The commonly-used adiabatic approximation in TD-DFT limits the excited-state character to singly excited configurations, although attempts have been made to extend the theory to multiple excitations.28 Although many wavefunction-based approaches that include multiply excited configurations have been developed, the number of excited states and the relatively large cluster sizes of interest in nanoscience applications make wavefunction methods beyond CIS in most cases prohibitively expensive. A method that provides access to multiply excited configurations at a low computational cost could provide chemical insight that is not available using current approaches.

Here, we benchmark the semiempirical INDO/SCI approach29,30 to reproduce the TD-DFT absorption spectrum of the tetrahedral Ag20 cluster. We show that this approach reproduces not only the absorption of the cluster but also the collectivity of the plasmon-like excited state observed in previous quantum-chemical calculations, in terms of additive contributions from many single excitations to the transition dipole moment. The same parameters give absorption spectra largely consistent with TD-DFT for ligand-free clusters of other shapes and sizes. We


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then extend this INDO/CI approach to investigate the role of double excitations in plasmonic spectra. These studies reveal that double excitations mix into the ground state to an extent that depends on cluster charge but have only a minor influence on the plasmonic features in optical spectra.

2. Computational methodology The geometric structures of the silver clusters were optimized using density functional theory (DFT), using the BP86 functional31,32 and a double-ζ (DZ) Slater-type basis set. To account for scalar relativistic effects, the zeroth-order regular approximation (ZORA) was used.33 Many of these clusters had been previously optimized at the same level,20 and our results are consistent. The excited states were computed using a TD-DFT approach with the SAOP functional34 and the DZ basis set, which has previously shown reasonable results for Ag clusters.20 Comparison to other functionals previously used for Ag clusters18,20 with a frozen-core DZ.4p basis set, including BP86, PBE,35 and LB94,36 is provided in the SI. All density functional theory calculations were performed using the Amsterdam Density Functional (ADF) 2014 program.37,38,39

Excited state energies were also computed using the semiempirical Intermediate Neglect of Differential Overlap (INDO) Hamiltonian29,30 and singles (SCI) or singles and doubles (SDCI) configuration interaction approaches. The main parameters used for the Ag atoms were those obtained by Anderson et al.,29 but with the one-electron, two-center resonance integrals βsp (for s and p orbitals; often referred to separately as βs and βp) and βd (for d orbitals) treated as freely tunable parameters and optimized as described below; unless otherwise specified, the values βsp



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= -3 eV and βd = -40 eV that arose from this optimization were used. The CI matrix was constructed by generating all possible single excitations among the valence orbitals and, in the SDCI calculations, all possible double excitations within the 25 highest occupied molecular orbitals and the 25 lowest unoccupied molecular orbitals. Because the number of excitations generated is substantially greater than the size of the CI matrix, this active space includes all double excitations of sufficiently low energy to be included in the CI matrix. Of the many excitations generated, the 2000-7000 excitations of lowest energy were included in the CI matrix; this matrix was then diagonalized to obtain the lowest 500-2000 excited states. These calculations were performed using a home-built code based on elements of Mopac 7.140 and the INDO/CI code written by Jeffrey Reimers.41 Our modifications allow for the use of a larger active space to generate single excitations and a larger CI matrix and improve the ground-state SCF convergence for the larger Ag clusters; the calculations are otherwise the same as those possible within the original Reimers code.

3. Results and Discussion 3.1. Tuning of INDO parameters and application to Ag clusters Since the semiempirical INDO Hamiltonian has been previously described in great detail,29,30,41 we only summarize the basics here. As is typical in semiempirical approaches, only the valence electrons are computed explicitly. Based on the Zero Differential Overlap approximation, all one-electron integrals involving three centers, all two-electron integrals involving three or four centers, two-center two-electron integrals except those of the Coulomb type are neglected; the remaining integrals are parameterized based on experimental or higher-level computational data. Most of the parameters are fixed at the values determined by Anderson et al.;29 however, the one-


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electron, two-center resonance integrals βsp (for s and p orbitals) and βd (for d orbitals) are fit to reproduce the results of interest. The off-diagonal terms of the Hamiltonian are computed by scaling the overlap integral Sµν by the average of the resonance integrals of the two orbitals

஺஻ ‫ܪ‬ఓఔ

൫ߚఓ஺ + ߚఔ஻ ൯ = ܵఓఔ 2

1

where µ and ν indicate atomic orbitals on atoms A and B, respectively, where A ≠ B. Since the original βsp and βd parameters for Ag were interpolated based on the first-row transition metals but were not refined,29 the previously determined values may not be suitable for reproducing the absorption spectra of Ag clusters.

To tune the resonance integrals, we focus on the absorption spectrum of the tetrahedral Ag20 cluster; this cluster has been widely used as a prototypical example in computational studies.18,22,23,42,43 βsp was varied from -2 to -5 eV and βd from -10 to -50 eV; these ranges are consistent with those previously optimized for small Au clusters (βsp = -4 eV, βd = -50 eV)41 and the original parameters selected for Ag (βsp = -1 eV, βd = -27.94 eV).29 Since the plasmonic absorption peak involves sp → sp intraband transitions,18 the absorption spectrum is quite sensitive to changes in βsp (Figure 1, top). In general, as the magnitude of βsp increases, the first absorption peak shifts to higher energy, although a small absorption shoulder at ~2.2 eV remains regardless of βsp. For most values of βsp, there are multiple absorption peaks with significant intensity; however, when βsp = -3 eV, there is a single strong absorption peak corresponding to a triply degenerate excited state at 3.50 eV with oscillator strength of 3.12. This is in good qualitative agreement with TD-DFT results at the SAOP/DZ level, which have a single strong 7 ACS Paragon Plus Environment


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absorption peak at 3.72 eV with an oscillator strength of 2.43 (Figure 2). Comparison to several other functionals previously used for Ag clusters is provided in the SI; BP86 and PBE yield slightly smaller absorption energies of 3.58 and 3.55 eV, respectively, whereas LB94 overestimates the absorption energy of 4.05 eV.

Figure 1. INDO/SCI absorption spectra of the tetrahedral Ag20 cluster as a function of (top) βsp and (bottom) βd.


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Figure 2. Absorption spectra of tetrahedral Ag clusters at the INDO/SCI (solid lines) and SAOP/DZ (dotted lines) levels of theory.

The value of βd has a smaller effect on the shape of the absorption spectrum but affects the energy and splitting of the main absorption peak (Figure 1, bottom). Although the plasmonic state retains largely sp → sp intraband character throughout the range of parameters studied, the contribution of d → sp interband character increases from 4% to 11% as βd increases in magnitude from -10 to -30 eV but remains nearly constant for larger values of βd. For small values of βd, several excited states of similar energy have significant oscillator strengths leading to some broadening of the absorption peak as is shown more clearly in the stick spectra in the SI; this splitting decreases as βd increases. Although the best agreement between the plasmonic state energies at the INDO/SCI and SAOP/DZ levels is obtained when βd = -30 eV, a slightly larger βd value is necessary to minimize the broadening of the plasmonic absorption peak. However, when βd becomes too large, secondary absorption peaks around 5 eV increase in intensity relative to the main absorption peak. To balance these two limits, we select the coupling parameters of βsp = -3 eV and βd = -40 eV for further studies.



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The INDO/SCI approach with the same βsp and βd parameters likewise provides absorption spectra similar to TD-DFT for Ag clusters of other shapes and sizes. We first look at size effects for tetrahedral clusters, as shown in Figure 2. For Ag102+, INDO/SCI is in reasonably good agreement with TD-DFT results, with the absorption peak about 0.1 eV lower in energy than the SAOP/DZ results. INDO/SCI has a small shoulder on the low-energy side of the main absorption peak that is not present using TD-DFT, but one absorption peak still dominates. For the larger Ag35+ cluster,44 both INDO/SCI and SAOP/DZ have several excited states with large oscillator strengths within a narrow energetic range, resulting in relatively sharp absorption peaks; however, the INDO/SCI absorption peak is shifted to higher energy. This suggests that while INDO/SCI captures the main absorption features, there is some variability in the agreement of the exact absorption energies.

Similar trends are seen in the INDO/SCI absorption spectra for clusters of other shapes (Figure 3), including octahedra, cuboctahedra (truncated octahedra), and icosahedra. For all of the clusters studied, INDO/SCI is in reasonable agreement with SAOP/DZ, although INDO/SCI tends to have slightly more broadening of the absorption peaks. Although the absorption energies are generally comparable, there is some variation in the relative energies: INDO/SCI in most cases underestimates the absorption energies by several tenths of an eV, but overestimates the absorption energy of clusters such as Ag384+. INDO/SCI also typically gives larger total oscillator strengths than SAOP/DZ; however, since in some cases the absorption is split between several peaks of similar energy, the peak absorption cross-section is in some cases quite similar to SAOP/DZ. Despite these relatively minor differences, the INDO/SCI approach is generally able


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to reproduce the major features of the TD-DFT absorption spectra at a greatly reduced computational cost.

Figure 3. Absorption spectra of octahedral Ag19 and Ag44 clusters, cuboctahedral Ag38 clusters, and icosahedral Ag43 clusters of various charge states at the INDO/SCI (solid lines) and SAOP/DZ (dotted lines) levels of theory

Previous quantum-chemical studies of Ag clusters have shown that the collectivity of the plasmonic state can be described in terms of a linear combination of many single excitations that contribute additively to the total transition dipole moment µge from the ground state g to an excited state e.16,20 Detailed analysis of the INDO/SCI results confirm that this method provides a comparable description of the collectivity of the plasmon-like state; we compare here the plasmonic state of Ag20 at 3.50 eV with a non-plasmonic excited state at 3.25 eV to highlight the differences between the natures of these two states (Table 1).45 The plasmon-like state comprises a linear combination of many singly excited configurations, nearly all of which contribute 11 ACS Paragon Plus Environment


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additively to the transition dipole moment. The largest contribution of opposite sign along the z axis is an order of magnitude smaller than the largest positive contribution. The transition density for this state is large along two edges of the tetrahedron as has been previously observed at the LB94/DZ level.16 The non-plasmonic state is likewise a linear combination of many singly excited configurations, but there is significant cancellation among the major contributions to µge leading to much weaker absorption into this state. This cancellation is also apparent in the total transition density: there are few regions of large transition density, and the transition densities in the central layers of the cluster partially cancel the transition densities on the edges of the tetrahedron.

Table 1. CI configurations, contributions to the transition dipole moment, and transition densities for representative plasmonic and non-plasmonic states of Ag20. All configurations with dipole moment contributions larger than 0.3 Debye are listed.

Energy Configuration (eV)

3.50

3.25

H-1 → L+6 H-5 → L+2 H-4 → L+8 H-3 → L+7 H → L+9 H-3 → L+1 H-4 → L H-4 → L+7 H-3 → L+8 H-2 → L+3 H-8 → L+3 Total H-5 → L+2 H → L+9 H-1 → L+6 H-3 → L+5 H-4 → L+4 H-5 → L+6

Contribution to µge (Debye) x y z 0.00 0.00 3.47 0.00 0.00 2.55 0.00 0.00 1.68 0.00 0.00 1.68 0.00 0.00 1.61 0.00 0.00 1.36 0.00 0.00 1.36 0.00 0.00 1.14 0.00 0.00 1.14 0.00 0.00 0.41 0.00 0.00 -0.32 0.00 0.00 14.97 0.00 0.00 3.29 0.00 0.00 2.79 0.00 0.00 0.68 0.00 0.00 0.43 0.00 0.00 0.43 0.00 0.00 -0.30 12 ACS Paragon Plus Environment

Transition density

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H-3 → L+1 H-4 → L H-4 → L+7 H-3 → L+8 H-4 → L+8 H-3 → L+7 Total

0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.00 0.00 0.00 0.00 0.00 0.00 0.00

-0.33 -0.33 -0.98 -0.98 -1.45 -1.45 2.13

The decomposition of µge for each excited state into components from each configuration reveals strong differences between the plasmonic state and all other excited states of the Ag20 cluster. The magnitudes of the configuration contributions and the degree of cancellation can be summarized by comparing µge for each state to the hypothetical maximum possible µge, computed by adding the vector magnitudes of each configuration's contribution to the transition dipole moment (Figure 4). We note that due to the treatment of symmetry in these calculations, in triply degenerate states the maximum possible µge values are larger for the pair of states with µge in the xy plane than for the state with µge along the z axis. This analysis reveals that the large µge for the triply degenerate plasmonic state results both from large configuration contributions to µge and from a large degree of additivity among those contributions. This is consistent with previous studies of plasmonic nanoclusters that identified plasmonic states on the basis of large transition dipole moments.16,18,20 In contrast, all other excited states have much smaller µge values because of much stronger cancellation among the configuration contributions when the individual terms are reasonably large. The consistency of the key characteristics of the plasmonic state with previous TD-DFT calculations demonstrates that INDO/SCI is sufficiently reliable to describe the main chemical features of the plasmonic absorption.



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Figure 4. Relationship between the actual and maximum possible transition dipole moments of Ag20 for the lowest 500 excited states. The triply-degenerate plasmonic excited state is indicated by red squares; the states with transition dipole moments along the x and y axes have larger maximum µge than the z-axis state due to the treatment of symmetry.

3.2. Effect of double excitations

Applications of the INDO Hamiltonian to Ag clusters also enable chemical understanding of the plasmonic excited states that is not accessible via the quantum-chemical approaches used to date. In particular, to our knowledge all quantum mechanical studies of plasmonic Ag clusters to date have used approaches that effectively include only singly excited configurations;18,20,24,46 we note that the plasmon-like states in polyenes have been shown to have little double-excitation character using ADC approaches.47 In principle, because the plasmonic state is inherently collective, it should involve not only single excitations but also double and higher-order excitations up to the total number of conduction-band electrons. However, in practice it has been widely assumed that singly excited configurations will dominate the absorbing state because only 14 ACS Paragon Plus Environment

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configuration pairs that differ by one orbital contribute to the transition dipole moment. Calculations incorporating higher-order excitations are required in order to evaluate the validity of this assumption. Since the INDO/CI approach is computationally inexpensive, the CI procedure can be extended to include both single and double excitations (SDCI) to evaluate the role of doubly excited configurations in the ground and excited states of plasmonic Ag clusters. We focus on the small tetrahedral clusters Ag102+ and Ag20 and investigate the effects of cluster shape and charge by studying cuboctahedral and icosahedral Ag13 clusters with several charge states.

Because the selection of which single and double excitations to include in the CI matrix is based solely on the excitation energies, the CI matrices include far more double excitations than single excitations (Figure 5). The participation of doubly excited configurations in the ground-state wavefunction depends strongly on the cluster size, shape, and charge. For clusters with a large positive charge, the ground state comprises less than 5% double excitations; as the cluster charge become more negatively charged, the contribution of double excitations increases to nearly 20%. The shape of the cluster can also have a significant effect: while the cuboctahedral and icosahedral Ag135+ clusters have quite similar doubles contributions, the cuboctahedral Ag135cluster (red) has a much larger contribution of double excitations to the ground state than the equivalent icosahedral cluster. The ground state is also stabilized by mixing with double excitations, by amounts ranging from 0.3 eV for the most positively charged clusters to 1.5 eV for the most negatively charged.



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Figure 5. (Top) CI active spaces and (bottom) ground-state stabilization of Ag clusters. Red labels indicate cuboctahedral clusters; blue, icosahedral; and green, tetrahedral.

The excited states involve linear combinations of single and double excitations. In most clusters, the lowest-energy excited states involve primarily single excitations; at energies greater than twice the first excited-state energy, the contributions of double excitations become significant. As the clusters become more negatively charged, the first excited-state energy and the onset of double excitations shift to lower energy (Figure 6). In the highly positively charged clusters, the plasmonic excitation is substantially lower in energy than any double excitations, so the plasmonic absorption peak is purely a linear combination of single excitations. In the neutral or negatively charged clusters, the plasmonic state is higher in energy than the onset of double 16 ACS Paragon Plus Environment

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excitations, so there are many doubly excited configurations at energies close to that of the plasmonic state. However, in most clusters, the plasmonic state retains at least 90% singleexcitation character. The only exceptions are the cuboctahedral Ag13- and Ag135- clusters, which have significant participation of doubly excited configurations in the ground-state wavefunctions and several excited states within relatively narrow energetic ranges that have comparably large oscillator strengths. In these two clusters, transitions between pairs of doubly excited configurations in the ground-state and excited-state wavefunctions also contribute significantly to the oscillator strengths of the absorbing states.

Figure 6. Contribution of double excitations to the excited states of the icosahedral Ag135+ (top) and Ag135- (bottom) clusters. Red squares indicate the plasmonic states of each cluster. 17 ACS Paragon Plus Environment


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For the positively charged and neutral clusters, the consideration of double excitations has relatively minimal effects on the absorption spectra (Figure 7). The plasmonic absorption peaks shift slightly to higher energy due to stabilization of the ground state, and the oscillator strength slightly decreases. Both of these changes tend to improve the agreement between INDO/SDCI and SAOP/DZ. For Ag102+, the inclusion of double excitations also decreases the size of the lowenergy shoulder of the main absorption peak, likewise improving agreement with the TD-DFT results.


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Figure 7. Absorption spectra of several Ag clusters. Solid lines indicate INDO/SDCI, dashed lines indicate INDO/SCI, and dotted lines indicate SAOP.

In the negatively charged clusters, the double excitations play a much more significant role. The INDO/SCI absorption spectra are in fairly poor agreement with TD-DFT: for the cuboctahedral Ag13- and the cuboctahedral and icosahedral Ag135- clusters, the absorption is split between two large peaks, both of which are significantly lower in energy than the SAOP/DZ absorption peak. However, the double excitations have quite large effects on the absorption spectra and substantially improve the agreement with TD-DFT. When double excitations are considered, the stabilization of the ground state results in a shift of the absorption to significantly higher energy, and the absorption spectra are dominated by one main peak.

These results suggest that in many cases singly-excited approaches are adequate to model the plasmonic state, and the inclusion of higher-order excitations serves primarily to stabilize the 19 ACS Paragon Plus Environment


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ground state. For negatively charged clusters, double excitations have a more important effect within the semiempirical model: INDO/SDCI appears to capture some of the ground-state correlation that is implicit in the commonly-used DFT functionals, which significantly improves the agreement of INDO/SDCI with TD-DFT for these clusters.

4. Conclusions

Quantum mechanical calculations using INDO/CI approaches can accurately model the absorption spectra and plasmonic character of Ag nanoclusters when appropriate resonance parameters are selected. This method is in good agreement with TD-DFT in terms not only of the absorption spectrum, but also the transition densities and the collective behavior of the plasmonic state. In particular, the plasmonic state consists of a linear combination of many singly excited configurations that contribute additively to the transition dipole moment, whereas other excited states show significant cancellation among the configurations leading to weak absorption. Although the current calculations are limited to Ag nanoclusters, a similar parameterization approach should be applicable to other plasmonic metals such as gold.

The development of a computationally efficient approach to model plasmonic nanostructures can aid in developing further understanding of the quantum mechanical properties of plasmons and allows for rapid screening of structures of interest. In particular, this approach allows us to obtain new physical insight about the role of double excitations in quantum mechanical descriptions of the plasmonic states, which has previously been inaccessible. Although double excitations at the INDO/SDCI level may significantly stabilize the ground state, particularly in negatively charged


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clusters, the plasmonic states retain dominantly singly excited character. Significantly, in all of the clusters studied, the inclusion of double excitations improves the agreement of the INDO/CI absorption spectra with TD-DFT, with relatively minor shifts seen for neutral and cationic clusters but substantial improvement for anionic clusters where the ground-state correlation is more significant. This suggests that the INDO/SDCI model in effect captures some of the electron correlation present in typical DFT functionals. These results provide the first direct evidence to support the common assumption that singly excited approaches offer a reasonable model to represent the spectra of plasmonic excited states.

Supporting Information INDO/SCI absorption spectra of Ag20 with additional sets of resonance parameters, comparison of TD-DFT absorption spectra with several functionals, and contributions of double excitations to excited states.

Acknowledgements The authors thank Jeffrey Reimers for providing his INDO/CI code, Christine Aikens for providing optimized structure of many of the Ag clusters, and Lindsey Madison and Adam Ashwell for helpful discussion. This research was supported by DOE grant DE-FG0210ER16153.

References (1) Kelly, K. L.; Coronado, E.; Zhao, L. L.; Schatz, G. C. The Optical Properties of Metal Nanoparticles: The Influence of Size, Shape, and Dielectric Environment. J. Phys. Chem. B 2003, 107, 668-677. (2) Eustis, S.; El-Sayed, M. A. Why Gold Nanoparticles Are More Precious Than Pretty Gold: Noble Metal Surface Plasmon Resonance and Its Enhancement of the Radiative and 21 ACS Paragon Plus Environment


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