Third-Order Nonlinear Optical Properties of Ag Nanoclusters

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Cite This: Chem. Mater. XXXX, XXX, XXX−XXX

Third-Order Nonlinear Optical Properties of Ag Nanoclusters: Connecting Molecule-like and Nanoparticle-like Behavior Rebecca L. M. Gieseking* Department of Chemistry, Brandeis University, 415 South Street, Waltham, Massachusetts 02453, United States

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S Supporting Information *

ABSTRACT: Molecules and materials with negative real parts of third-order nonlinear optical (NLO) polarizability Re(γ) are rare despite their advantages for applications such as all-optical switching. Although plasmonic metal nanoparticles can have a negative Re(γ) that increases in magnitude with a decrease in size, these properties are not yet understood from a quantum mechanical perspective. Here, we use quantum chemical approaches to model the NLO properties of prototypical silver nanoclusters. In linear Ag nanowires, the longitudinal excited-state properties are analogous to those of polyenes, leading to a positive Re(γ) that increases with nanowire length. In contrast, the transverse modes show plasmon-like mixing of excitations in the main one-photon excited state, leading to a negative Re(γ). The tetrahedral Ag20 cluster likewise has a negative Re(γ) due to plasmon-like mixing of excitations. On the basis of these results, we propose a new approach to obtain a large negative Re(γ) by identifying and tuning the structural features that contribute to plasmon-like mixing of excitations in the absorbing states.

1. INTRODUCTION Materials with large nonlinear optical (NLO) responses have been successfully used in a wide range of applications; for example, materials with a large second-order NLO response have been used for second-harmonic imaging1,2 and electrooptic modulation,3−5 and materials with large imaginary parts of third-order NLO response Im(χ(3)) have been used for twophoton imaging6,7 and optical power limiting.8,9 However, designing materials for applications such as all-optical switching that require large real parts Re(χ(3)) and small imaginary parts Im(χ(3)) of the material third-order NLO response has proven to be more challenging.10−12 In all-optical switching, one optical pulse is used to modulate a second pulse via the interaction of light with an NLO material. This modulation requires a material with a large nonlinear refractive index proportional to Re(χ(3)), enabling the speed of light through the material to change due to the added intensity of the modulating pulse. In a material for which Re(χ(3)) > 0, the effective linear polarizability increases with the intensity of light, so the nonlinear refractive index is positive; in a material with a negative Re(χ(3)), the converse is true. Although materials with both positive and negative Re(χ(3)) can in principle be used for all-optical switching, in practice a negative Re(χ(3)) provides significant advantages. A positive Re(χ(3)) induces self-focusing of the laser pulse13,14 and can lead to intensities large enough to cause dielectric breakdown;15 in contrast, a negative Re(χ(3)) leads to self-defocusing behavior, which allows higher intensities of light to be used with less concern about dielectric breakdown. Despite several decades of work on NLO materials, molecules and materials with negative third-order NLO © XXXX American Chemical Society

responses are relatively rare. No continuous inorganic materials have been shown to have a negative Re(χ(3)) in the longwavelength limit,16,17 leaving organic molecules18−22 and nanostructured inorganic or metallic materials23 as the only routes to negative Re(χ(3)). Because these are discrete systems, it is more convenient to discuss the microscopic (molecular) third-order polarizability γ than the material-scale third-order polarizability χ(3). The NLO properties can be qualitatively understood in terms of the excited-state properties using an essential-state model24−26 γzzzz ∝

+

μge 2 Δμge 2 Ege 3

∑ e′

−

μge 2 μee 2 ′ μge 2 μge ′

μge 4 Ege 3

(D term)

(T term)

(N term) (1)

where the z-axis is defined as the long axis of the molecule; this long-axis component typically dominates in linear conjugated molecules. This expression consists of three terms. The dipolar term D depends on the transition dipole moment μge between Special Issue: Jean-Luc Bredas Festschrift Received: April 1, 2019 Revised: June 4, 2019 Published: June 5, 2019 A

DOI: 10.1021/acs.chemmater.9b01290 Chem. Mater. XXXX, XXX, XXX−XXX

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Chemistry of Materials

time scales;38,39 the NLO responses of plasmonic nanoparticles are dramatically larger than those of bulk metals because of the contributions of electronic transitions related to the plasmon.40 The purely electronic NLO response can be challenging to separate from thermal effects related to absorption,40−44 and the thermal and electronic contributions to Re(γ) may have opposite signs.45 Because of the large size and number of atoms, the properties are generally studied using classical electrodynamics descriptions rather than quantum mechanical pictures. In a classical Drude-type model, the conduction electrons can move freely within the nanoparticle but experience a sharp increase in potential energy at the boundaries of the nanoparticle, corresponding to an effective decrease in polarizability with an increasing intensity of light and thus negative Re(γ). Due to quantum confinement effects, the magnitude of the NLO response is often largest for small nanoparticles. For Ag nanoparticles, Re(γ) in the vicinity of the plasmon is negative and becomes larger in magnitude when the particle size decreases from 16 to 3 nm.23 However, precise structural control is challenging, and variations in size, shape, surface sites, and ligand protection can lead to large heterogeneity between nanoparticles. In recent years, there has been extensive work toward the synthesis and characterization of atomically precise silver and gold nanoclusters.46,47 These structures range in size from a handful of atoms to several hundred metal atoms, with variation in the ligand protection, shape, and crystalline arrangement of the metal atoms. Even relatively small clusters may have excited states with some plasmon-like collectivity, defined as excited states composed of a linear combination of several to many single-particle excitations with additive contributions to the transition dipole moment.48−53 The atomic precision provides the opportunity to develop detailed structure−property relationships that are impossible in larger nanoparticles and tune the structures to maximize the NLO response. However, although there have been a number of experimental and theoretical studies of the NLO properties of individual nanoclusters,54−57 to date there has been no systematic study of the structural evolution of Re(γ). Here, we evaluate the nonlinear optical properties of prototypical silver nanoclusters from a quantum mechanical perspective, focusing primarily on Ag nanowires where the atoms are arranged linearly. We first examine the excited-state properties and absorption spectra and then turn to the nonlinear optical properties. The longitudinal excited states are analogous to polyene excited states and lead to a large positive Re(γ) that increases with nanowire length. In contrast, the transverse modes show plasmon-like mixing of excitations in the main one-photon excited state, leading to a negative Re(γ). Application of these methods to the tetrahedral Ag20 cluster show that this cluster likewise has a negative Re(γ) due to plasmon-like mixing of excitations. These excited-state properties are analogous to those of P(C6H5)4+, which has been confirmed experimentally to have a negative Re(γ). On the basis of these results, we propose a new approach to obtain a large negative Re(γ) by evaluating and tuning the structural features that contribute to strong coupling of excitations and plasmon-like mixing of excitations in the absorbing states. The development of plasmonic metal nanoclusters provides a promising novel route toward materials with large negative Re(γ) values.

ground state g and one-photon excited state e as well as the energetic difference Ege and difference in state dipole moments Δμge between states g and e; this term is only non-zero in noncentrosymmetric molecules. The two-photon term T additionally involves transitions from one-photon state e to a series of two-photon states e′. The negative term N depends on only the one-photon-state properties and is the only term that can make negative contributions to Re(γ). We note that this model assumes that only one excited state e is optically coupled to the ground state along the z-axis; a more complete version of the sum-over-states (SOS) expression24 must be used for systems with multiple one-photon states.22 Among organic molecules, cyanines are the best-studied example of structures with negative Re(γ) values.10,19,27,28 Cyanines are linear conjugated molecules with an odd number of methine (CH) units bridging two identical end groups; these molecules typically bear one positive or negative charge and are closed-shell molecules. In polyenes and most other πconjugated molecules, the first one-photon state e and twophoton state e′ are comparable in energy (Figure 1), and a

Figure 1. Schematic energy diagram of the first few excited states for (left) polyene-like and (right) cyanine-like molecules. State e is the first one-photon-absorbing (OPA) state, and state e′ is the first twophoton-absorbing (TPA) state.

series of higher-energy two-photon states closely follow the first two excited states. Because of these two-photon states, the T term is larger than the N term and Re(γ) is positive. In contrast, in cyanines, the energy of the first one-photon state is significantly lower than that of the first two-photon state. Because of this energetic spacing, the N term dominates over the T term and Re(γ) is negative.19,28 Structural tunability has enabled the design of cyanine-based materials with large Re(χ(3)) values;10,27 however, further increases in Re(χ(3)) are limited by the tendency of long cyanines to symmetrybreak29−31 and the loss of the favorable NLO properties upon aggregation.27,32,33 Negative Re(γ) values can also be obtained in plasmonic metal nanoparticles, typically composed of noble metals such as silver or gold;23,34−36 however, for most applications, plasmonic nanoparticles have unacceptably large optical losses.12 In these structures, collective excitation of the conduction-band electrons leads to a strong absorption peak that depends strongly on the nanoparticle size, shape, and environment.37 Bulk noble metals inherently have large nonlinear optical responses on the femtosecond to picosecond B


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character, which we label Σn, and doubly degenerate molecular orbitals with π-type bonding character, which we label Πn; each orbital has n − 1 nodes along the long axis of the nanowire. For a nanowire with 2n Ag atoms, the HOMO is Σn and the LUMO is Σn+1; all of the Π-type orbitals are unoccupied. Our orbital occupations and nomenclature are consistent with previous modeling of Ag nanowires.53 We first consider the excited states and linear absorption spectra of the Ag nanowires. As has been shown previously,53,74 these nanowires have two main absorption peaks (Figure 2); the energies and transition dipole moments

2. COMPUTATIONAL METHODS The geometries of the Ag clusters were optimized using a density functional theory (DFT) approach with the BP86 functional58,59 and a double-ζ (DZ) Slater-type basis set. A frozen-core approximation was used for all orbitals through the Ag 4p orbital (DZ.4p). Scalar relativistic effects were incorporated using the zero-order regular approximation (ZORA).60 Excited states at the time-dependent DFT (TD-DFT) level were computed using the SAOP functional61 and an all-electron triple-ζ Slater-type basis set with polarization functions (TZP); this functional has previously been shown to provide reliable absorption spectra for noble metal nanoclusters.50,51,62 The lowest 200−750 excited-state energies were computed. All DFT computations were performed using the Amsterdam Density Functional (ADF) software package.63,64 The excited states were also computed using the semiempirical Intermediate Neglect of Differential Overlap (INDO) Hamiltonian65,66 and a configuration interaction approach incorporating only single excitations (SCI) or single and double excitations (SDCI). These computations used our recently developed INDO parameters for Ag.49,67 All possible single excitations and all double excitations within the first 10 occupied and 10 unoccupied orbitals were generated, and the lowest 7000 excitations were included in the CI matrix. The CI matrix was diagonalized to obtain the lowest 3000 excited states. The third-order polarizabilities were computed using a sum-over-states (SOS) approach24 by summing over these 3000 excited states. All INDO computations were performed using a homebuilt code incorporating portions of MOPAC201668,69 and the INDO/CI code of J. Reimers.70 The static (zero-frequency) third-order polarizabilities were also computed using a finite-field approach using both the SAOP/TZP and INDO/SCI models. The linear polarizabilities were computed for fields ranging from 0.001 to 0.016 au in increments of 2 ,71 and the third-order polarizability at each field strength was computed as γaabc(Fa/2) =

2αbc(0) − 2αbc(Fa) Fa 2

(2)

The third-order polarizability at the zero-field limit was computed using a linear fit of the third-order polarizabilities at finite fields. The finite-field computations were performed both at the SAOP/TZP level using analytical linear polarizabilities computed by the RESPONSE module of ADF and at the INDO/SCI level using linear polarizabilities computed via a sum-over-states approach summing over the first 3000 excited states.

3. RESULTS AND DISCUSSION As we are interested in developing a fundamental moleculartype understanding of the nonlinear optical properties of metal nanoclusters, we focus here on Ag nanowires consisting of an even number of Ag atoms arranged linearly. Although this specific model system cannot be directly studied experimentally, the structure is similar to nanowire-like structures that have been synthesized. Metal nanowires two atoms wide have been synthesized on a Si substrate,72 and crystal structures of a Ag8 cluster encapsulated by DNA show that six of the eight atoms are arranged approximately linearly.73 The fundamental understanding obtained from this simple model system will be possible to extend to structures like atomically precise metal nanoclusters, which have been widely studied experimentally.46,47 Our Ag nanowire model has the advantage of providing a simple set of systems that can be studied systematically as a function of length and has been used previously as a model system for the absorption53,74 and time evolution75 of plasmonic excited states. In these nanowires, the d band (valence band) is fully occupied and the sp band (conduction band) contains one electron per Ag atom. The sp band consists of molecular orbitals with σ-type bonding

Figure 2. Absorption spectra of linear Ag nanowires with lengths 2n = 2−20 (Ag2−Ag20) at the (a) SAOP/TZP, (b) INDO/SCI, and (c) INDO/SDCI levels.

are shown in Figure 3, and state decompositions are presented in the Supporting Information. At the SAOP/TZP level, the energy of the longitudinal absorption peak decreases from 3.29 to 0.77 eV as the length of the nanowire increases from two to 20 atoms (n = 1−10). For the shorter nanowires, this excited state consists almost solely of a Σn → Σn+1 transition; as the length increases, there is some mixing with the Σn−2 → Σn+1 transition and splitting of the absorption between two excited states within a 0.05 eV span. Although this state has been termed plasmon-like in several previous studies of Ag nanowires,53,74 this identification has been largely based on its strong absorption, and it has been previously noted53 (as C


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properties of molecular systems;78,79 although various TDDFT algorithms for including double excitations have been developed,80 typical linear-response TDDFT algorithms such as those used here include only single excitations. Thus, we turn to the semiempirical INDO approach, which has been shown to agree well with DFT absorption spectra of Ag nanoclusters49 and has contributed significantly to the development of organic NLO materials.19,28,81−84 We consider both INDO/SCI, which includes only single excitations, and INDO/SDCI, which also incorporates double excitations. At the INDO/SCI level, the longitudinal absorption peak is in excellent agreement with the SAOP/TZP spectrum in terms of both energy and oscillator strength; similar to the SAOP/TZP results, this excited state consists primarily of a Σn → Σn+1 transition. The energy of the first transverse peak is roughly 1 eV lower than that predicted at the SAOP/TZP level and is likewise dominated by a linear combination of Σ → Π-type transitions across the length range studied. The large increase in the transverse absorption cross section with an increase in length is related to a lack of mixing with d-band transitions. This is likely related to limitations of the INDO parameters used here; the d-band molecular orbital energies are underestimated at the INDO level, causing the energies of the associated transitions to be too high to mix effectively with the Σ → Π-type transitions. We note that the transverse absorption peak in Figure 2b is stronger than the longitudinal absorption peak, even though the transverse states have smaller transition dipole moments as shown in Figure 3b. This is in part because the transverse states are doubly degenerate, leading to a doubling of the absorption intensity. In addition, the intensity of the absorption peak is proportional to Egeμge2, leading to larger absorption intensities for the higher-energy states. Overall, the absorption spectra at the INDO/SDCI level are consistent with the INDO/SCI results, particularly in terms of the magnitudes of the transition dipole moments and the excited-state compositions; this is consistent with our previous INDO/SDCI results for other Ag clusters.49 The largest difference is in the excited-state energies. Because the inclusion of double excitations stabilizes the ground state, the energies of the absorption peaks are higher at the INDO/SDCI level than for the same length nanowire at the INDO/SCI level, a difference that becomes more prominent with an increase in nanowire length. Despite these minor differences, the overall picture of the linear optical properties is largely consistent across all three levels of theory. The consideration of double excitations unsurprisingly becomes much more critical upon examination of the twophoton-absorbing excited states. We focus first on the twophoton states coupled to the longitudinal absorbing state (Figure 4). Substantial differences appear between the INDO/ SCI and INDO/SDCI excited states, most notably due to the absence of the double Σn, Σn → Σn+1, Σn+1 excitation in the INDO/SCI model. At the INDO/SDCI level, the first several two-photon states are composed primarily of linear combinations of three excitations: Σn−1 → Σn+1, Σn → Σn+2, and the double Σn, Σn → Σn+1, Σn+1 excitation. The energy of the lowest of these excited states is lower than that of the first one-photon state for 2n ≥ 4. Interestingly, this composition and energetic spacing are quite similar to those of the prototypical example of linear polyenes shown in Figure 1.28,85−87 A higher-energy two-photon state has the largest μee′; for this state, μee′ is approximately 1.5μge because the three excitations in e′

Figure 3. Energies and transition dipole moments of the longitudinal and transverse excited states for linear Ag nanowires with lengths 2n = 2−20 (Ag2−Ag20) at the (a) SAOP/TZP, (b) INDO/SCI, and (c) INDO/SDCI levels.

our SAOP/TZP results likewise show) that this state does not have the collectivity among excitations commonly used as an indicator of plasmon-like character. For this reason, we refer to this state as single-particle-like and not plasmon-like. The transverse absorption peak remains between 5.96 and 6.45 eV across the length range studied and splits among two or more absorbing states for 2n ≥ 12. Unlike the longitudinal mode, the transverse state has nearly equal contributions from several excitations involving transitions from Σa to Πa (a = 1, 2, . . ., n) for nanowires as short as 2n = 4. This collectivity among many excitations that contribute to the main absorbing state is analogous to the collectivity in classical plasmons and has been widely used as an indicator that these excited states have plasmon-like character,48,49,76,77 and this state has been previously identified as plasmon-like due to this collectivity.53,75 For nanowires with 2n ≥ 12, there is significant mixing of transitions involving the d band into the transverse peak, damping the absorption. Although DFT is widely used to compute absorption spectra, its computational cost is prohibitively large for the thousands of excited states that must be computed to obtain converged SOS third-order polarizabilities. In addition, doubleexcitation contributions are critical to understanding the NLO D


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Table 1. Transverse One-Photon and Two-Photon ExcitedState Properties for Linear Ag Nanowires at the INDO/ SDCI Levela one-photon state 2n

energy (eV)

μge (D)

2

4.82

7.14

4

5.05

5.39

6.94

6.42

Figure 4. Energies (Ege′) and transition dipole moments (μee′) of the longitudinal two-photon excited states for linear Ag nanowires with lengths 2n = 2−20 (Ag2−Ag20) at the (a) INDO/SCI and (b) INDO/SDCI levels. 6

contribute additively to μee′. In contrast, at the INDO/SCI level, the absence of double excitations leads to a smaller μee′ and weaker mixing of the Σn−1 → Σn+1 and Σn → Σn+2 excitations, particularly for 2n < 10. Given the critical role of double excitations in the twophoton states, for the transverse states we focus solely on the INDO/SDCI results. Memory limitations in the code currently restrict the number of excited states that can be computed to 3000. Because of the high density of states, for nanowires of length 2n ≥ 8 this number of excited states is not sufficient to produce all of the relevant two-photon states. From the data that are available (Table 1), we see that the only states with significant couplings to the transverse modes involve a linear combination of double excitations where two electrons are excited from Σ-type orbitals to Π-type orbitals. Similar to the case for the one-photon transverse states, as the nanowire length increases, the character of the two-photon transverse states becomes more mixed among contributions from many different excitations. Unlike for the longitudinal states, this mixing does not redistribute oscillator strength to a single state with a large μee′. Instead, a handful of transverse excited states all have relatively small μee′ values. This is due to both the onephoton and two-photon states having significantly mixed character. Because μee′ is a sum of contributions from all possible pairings of one excitation in state e with one excitation in state e′, the mixing in character means that it is difficult to gain the additivity of terms required to obtain a large μee′. Interestingly, this mixing of excited-state characters leading to

composition Σ1 → Π1 (82%)

Σ2 → Π2 (58%) Σ2, Σ2 → Σ3, Π1 (17%)

Σ1 → Π1 (49%) Σ2, Σ2 → Σ3, Π1 (21%)

two-photon state energy (eV)

μee′ (D)

9.16

5.00

9.17

4.99

9.26

4.16

9.62

4.05

10.95

4.64

10.96

6.73

10.96

6.55

11.09

4.88

5.76

5.4

Σ2 → Π2 (22%)

11.46

4.06

6.34

7.83

Σ3 → Π5 (24%) Σ2 → Π4 (16%) Σ2 → Π2 (12%)

11.46

4.21

11.46

4.59

11.55

4.29

composition Σ1, Σ1 → Π1, Π1 (80%) Σ1, Σ1 → Π2, Π2 (19%) Σ1, Σ1 → Π1, Π1 (79%) Σ1, Σ1 → Π2, Π2 (19%) Σ1, Σ1 → Π1, Π1 (52%) Σ1, Σ1 → Π2, Π2 (19%) Σ1, Σ2 → Π1, Π2 (22%) Σ2, Σ2 → Π2, Π2 (18%) Σ1, Σ2 → Π1, Π2 (34%) Σ2, Σ2 → Π1, Π3 (10%) Σ2, Σ2 → Π2, Π2 (10%) Σ1, Σ2 → Π1, Π2 (64%) Σ1, Σ2 → Π1, Π2 (64%) Σ1, Σ2 → Π1, Π2 (26%) Σ1, Σ1 → Π1, Π1 (13%) Σ1, Σ2 → Π1, Π2 (12%) Σ2, Σ3 → Π2, Π3 (21%) Σ1, Σ3 → Π1, Π3 (13%) Σ2, Σ3 → Π2, Π3 (17%) Σ1, Σ3 → Π1, Π3 (10%) Σ2, Σ3 → Π2, Π3 (21%) Σ1, Σ3 → Π1, Π3 (13%) Σ2, Σ3 → Π2, Π3 (9%)

All one-photon states with a μge of >4 D and all two-photon states with a μee′ of >4 D (where e is a one-photon state) are listed. All excitations that contribute ≥10% to the state of interest are listed; for states where no excitation contributes >10%, the largest contribution is listed. a

suppression of two-photon absorption is similar to what was seen computationally and confirmed experimentally for P(C6H5)4+ and analogous molecules.22 To benchmark the NLO properties of the Ag nanowires, we compare the sum-over-states (SOS) polarizabilities at the INDO/SCI and INDO/SDCI levels to finite-field (FF) computations at the SAOP/TZP and INDO/SCI levels. Although the FF calculations are straightforward to perform, they have a major disadvantage in that it is challenging to extract structure−property relationships that relate the electronic structure to the NLO properties. The SOS approach E


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Hamiltonian involving coupling between orbitals on the same atom to zero. Because the transverse electric field on the linear nanowire would show up only in the terms of the Hamiltonian that are set to zero, the INDO/SCI-based FF calculations are unable to model the transverse linear polarization. Thus, these calculations are not reported in Figure 5.

has an advantage in that it is straightforward to examine contributions of each excited state to the total NLO response; however, a very large number of excited states is required to fully converge the NLO properties, particularly in the transverse direction. As we will show, double excitations are critical for obtaining the correct sign of Re(γ) using an SOS approach but are not necessary using a FF approach. This is due to inherent differences in the formalisms. Using an SOS approach, only one excited-state calculation is performed, and the two-photon states, which often have significant doubleexcitation character, are explicitly included in the T term of the SOS expression (eq 1). Within the FF model, several singlepoint calculations are performed at varying electric field strengths, and the linear polarizability is computed either analytically or using an SOS approach. Within the SOS model, the linear polarizability depends on only states that are optically coupled directly to the ground state, so excited states with large double-excitation character (which are generally not optically coupled to the ground state) can be neglected; the linear polarizability can also be computed analytically. The independent SCF calculations at different electric field strengths capture the changes in the ground-state electronic structure that lead to small differences in polarizability. We first examine the longitudinal (z-axis) component of the static (zero-frequency) third-order polarizability, Re(γzzzz). FF calculations at the SAOP/TZP level suggest that Re(γzzzz) is positive for nanowires of length 2n ≥ 4 and increases with nanowire length. Because the longitudinal excited states are polyene-like, it is unsurprising that the sign of Re(γzzzz) is also the same as that of polyenes. Because of the strong anisotropy of the nanowires, Re(γzzzz) is the dominant contributor to the orientationally averaged Re(γ) for nanowires of length 2n ≥ 6. Using an SOS approach, the NLO properties depend strongly on the inclusion of double excitations. The INDO/ SDCI-based SOS results are in reasonable agreement with the SAOP/TZP-based FF results, predicting positive Re(γzzzz). INDO/SDCI-based SOS overestimates the magnitude of Re(γzzzz) by a factor of 4−6 relative to those of SAOP/TZP, which reflects the large sensitivity of NLO properties to small changes in the electronic structure. In contrast, INDO/SCIbased SOS incorrectly predicts a negative Re(γzzzz) for all nanowires and overestimates the magnitude of Re(γzzzz) by a factor of 30−50 relative to that from SAOP/TZP-based FF computations. The linear absorption gives rise to large N terms in the SOS expression at the INDO/SDCI and INDO/SCI levels; however, double excitations are required to accurately compute the two-photon excited states and the T term in the SOS expression and are critical in predicting the sign of Re(γzzzz). Interestingly, a FF approach using linear polarizabilities computed at the INDO/SCI level is in excellent agreement with the INDO/SDCI SOS results, with deviations on the order of 25%. In contrast, the transverse component Re(γxxxx) [=Re(γyyyy) by symmetry] is several orders of magnitude smaller than the longitudinal component, which is unsurprising given that the transverse absorption is weaker and at a much higher energy than the longitudinal absorption. At the SAOP/TZP-based FF level, Re(γxxxx) is negative for Ag2, Ag4, and Ag6 and positive for the longer nanowires, but |Re(γxxxx)| < 3 × 10−36 esu for all nanowires studied. The nanowires with positive Re(γxxxx) values exhibit strong mixing of the transverse modes with dband excitations, which is not observed using INDO-based models. We note that the INDO model sets all terms in the

Figure 5. (a) Longitudinal and (b) transverse components of Re(γ) for linear Ag nanowires with lengths 2n = 2−20 (Ag2−Ag20).

Using the INDO/SCI-based and INDO/SDCI-based SOS models, Re(γxxxx) < 0 for all nanowire lengths; Re(γxxxx) is a factor of 3−6 smaller at the INDO/SDCI level than at the INDO/SCI level due to the differences in the two-photon states. The negative Re(γxxxx) is consistent with the small transition dipole moments between excited states in the transverse direction, leading to a small T term. We note that the limited number of transverse excited states that can be computed may lead to an underestimation of the T term and an overly negative a value of Re(γxxxx) for the longer nanowires. The computed negative Re(γ) for Ag20 in the following section and the strong similarities to the excited states of P(C6H5)4+ where a negative Re(γ) has been experimentally confirmed22 suggest that the negative Re(γxxxx) in this case is a real effect and not an artifact of the limitations of our method. Although experimental results are not available for Ag nanowires, the large negative Re(γ) measured for 3 nm Ag nanoparticles23 also suggests that our attribution of a negative Re(γ) to plasmon-like systems is physically reasonable. As discussed above, the plasmon-like mixing of excitations in the main absorbing states leads to the small transition dipole F


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evidence that mixing of excitations in the one-photon excited states is a route to achieving negative Re(γ). Interestingly, a negative Re(γ) related to mixing of excitations in the one-photon states has been previously observed computationally and confirmed experimentally for P(C6H5)4+ and structurally related molecules.22 The electronic structure of these molecules is analogous to the plasmon-like excited states in the metal nanoclusters: several excited states within a narrow energy range have comparably strong onephoton absorption, and these states exhibit strong mixing of additive contributions from several excitations. The weak coupling of the one-photon states to higher-lying two-photon states leads to an overall negative Re(γ). This suggests that plasmon-like mixing of excitations may be a viable route to negative Re(γ) not only in metal clusters but also in organic molecules. The commonalities in the excited-state character between metal nanoclusters and P(C6H5)4+ suggest that P(C6H5)4+ and analogous molecules may be a useful model for molecular plasmons and serve as an alternative to the more commonly studied acenes.

moments between excited states. This suggests that increasing the plasmon-like character of the absorbing states may be a route toward the rational design of metal nanostructures with negative Re(γ) values. To further confirm that plasmon-like excited states lead to negative Re(γ) values, we have also computed the optical and NLO properties of the tetrahedral Ag20 cluster, which has been used widely as a model system for plasmon-like properties.48,49,88,89 At the INDO/SDCI level, the tetrahedral Ag20 cluster has an absorption peak at 4.01 eV resulting from a triply degenerate excited state (Figure 6). As shown in Table 2, there

4. CONCLUSIONS Developing design principles for new structures to achieve negative Re(γ) is important for the development of chip-scale all-optical switching devices. Here, we have taken a molecular approach based on excited states to understand the NLO properties of silver nanoclusters. In linear nanowires, the longitudinal excited-state and NLO properties are analogous to those of polyenes, with a large positive Re(γ) that increases with nanowire length. In contrast, the transverse excited states show the strong mixing among many excitations that is characteristic of plasmon-like excited states. This mixing leads to weak optical coupling between the one-photon and twophoton states and negative Re(γ). A similar electronic structure and negative Re(γ) have been observed previously for P(C6H5)4+. Computations on the tetrahedral Ag20 cluster, known to have plasmon-like excited states, further suggest that plasmon-like mixing of excitations is a useful route toward achieving negative Re(γ). The sum-over-states approach used here has an advantage in that it is possible to analyze the contributions of specific excited states to the NLO response. However, the current computations are limited by the thousands of excited states required to fully converge the sum-over-states expressions and the high density of excited states at high energies. Although the code could be extended to compute more than the 3000 excited states used here, alternate approaches such as real-time models to compute the polarizabilities90 may also enable computation of the NLO response, including the contributions of higher-energy states. This novel plasmon-based approach to achieve a negative Re(γ) could lead to the development of a new class of materials with negative material-scale third-order polarizabilities. Overall, these results suggest that it is possible to achieve negative Re(γ) in atomically precise nanoclusters, if the cluster has plasmon-like excited-state properties. Further research is needed to understand the effects of structural features such as ligands and enable the design of metal nanostructures with negative third-order polarizabilities.

Figure 6. Absorption spectrum of the tetrahedral Ag20 cluster at the INDO/SDCI level.

Table 2. Excited-State Properties of the Tetrahedral Ag20 Cluster at the INDO/SDCI Levela one-photon states energy (eV)

μge (D)

4.009

12.76

composition H → L + 10 (12%) H−2→L+4 (12%) H→L+9 (10%) H−5→L (10%)

two-photon states energy (eV)

μee′ (D)

8.033

3.28

8.281

3.69

8.322

3.57

8.342

5.07

8.365

3.79

composition H − 6, H → L, L (11%) H − 6, H → L, L (3%) H − 6, H → L, L + 3 (7%) H → L + 50 (5%) H − 6, H − 3 → L, L + 3 (4%)

All one-photon states with a μge of >4 D and all two-photon states with a μee′ of >4 D (where e is a one-photon state) are listed. All excitations that contribute ≥10% to the state of interest are listed; for states where no excitation contributes >10%, the largest contribution is listed. a

are only a handful of two-photon excited states with significant transition dipole moments to those states, and the μee′ values are significantly smaller than μge. Both the one-photon and the two-photon states have considerably mixed character; the excitation with the largest contribution contributes ≤12% to the total excited-state character. Because of the weak coupling between excited states, the orientationally averaged Re(γ) is −6356 × 10−36 esu at the INDO/SDCI level using an SOS approach. We note that at the INDO/SCI level using a FF approach, the orientationally averaged Re(γ) is −966 × 10−36 esu; although the magnitude is smaller, this comparison provides evidence that the negative sign is not simply an artifact of truncation in the SOS model. This provides further G


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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.chemmater.9b01290. Longitudinal and transverse excited-state properties at the SAOP/TZP, INDO/SCI, and INDO/SDCI levels (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: 781-736-2511. ORCID

Rebecca L. M. Gieseking: 0000-0002-7343-1253 Notes

The author declares no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by start-up funding at Brandeis University.

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Third-Order Nonlinear Optical Properties of Ag Nanoclusters

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