Theoretical Analysis of Optical Absorption and Emission in Mixed

UES, Inc. Dayton Ohio 45432 , United States. J. Phys. Chem. A , Article ASAP. DOI: 10.1021/acs.jpca.8b01882. Publication Date (Web): April 11, 2018. C...
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Theoretical Analysis of Optical Absorption and Emission in Mixed Noble Metal Nanoclusters Paul N. Day,*,†,‡ Ruth Pachter,*,† and Kiet A. Nguyen†,‡ †

Materials and Manufacturing Directorate, Air Force Research Laboratory, Wright-Patterson Air Force Base, Ohio 45433, United States ‡ UES, Inc. Dayton Ohio 45432, United States S Supporting Information *

ABSTRACT: In this work, we studied theoretically two hybrid gold−silver clusters, which were reported to have dual-band emission, using density functional theory (DFT) and linear and quadratic response time-dependent DFT (TDDFT). Hybrid functionals were found to successfully predict absorption and emission, although explanation of the NIR emission from the larger cluster (cluster 1) requires significant vibrational excitation in the final state. For the smaller cluster (cluster 2), the ΔH(0−0) value calculated for the T1 → S0 transition, using the PBE0 functional, is in good agreement with the measured NIR emission, and the calculated T2 → S0 value is in fair agreement with the measured visible emission. The calculated T1 → S0 phosphorescence ΔH(0−0) for cluster 1 is close to the measured visible emission energy. In order for the calculated phosphorescence for cluster 1 to agree with the intense NIR emission reported experimentally, the vibrational energy of the final state (S0) is required to be about 0.7 eV greater than the zero-point vibrational energy.

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INTRODUCTION Properties of solution-phase noble metal clusters in the size regime of 1−3 nm have drawn significant interest in the last few year because of unique properties dependent on the structure,1 where more than 100 such clusters have been synthesized and characterized so far (ref 1 and references therein). Clusters of noble metal atoms can have light absorption and emission properties useful for applications,2 including photodynamic therapy,3−5 photovoltaics,6−9 optical memory,10−13 imaging,14−16 sensors,17−20 and biolabels.21−24 Noble metal clusters demonstrate strong luminescence,25−29 tunability in the absorption wavelength, and also a large two-photon absorption (TPA) cross-section.4,30,31 Materials that exhibit TPA have advantages in tighter spatial confinement, as well as in accessing electronic excitation of twice the incident photon energy, and can be particularly useful in photodynamic therapy,32−34 imaging,35−38 and optical memory,39,40 as well as in confocal microscopy41,42 and microfabrication.43−46 The possible applications of noble metal clusters are being expanded by using them as building blocks to form superclusters.9,29 We have investigated one-photon absorption (OPA) and TPA spectra of [Au25(SR)18]−1 by applying density functional theory (DFT) and time-dependent DFT (TDDFT). The large TPA cross-section was explained in terms of resonance enhancement.31 In addition, we predicted OPA and TPA for the [Ag44(SR)30]4− and Ag14(SR)12(PR′3)8 silver clusters and, for comparison, also for Ag31/32(SG)19 and Ag15(SG)11,47 where a larger TPA cross-section for the less symmetrical Ag31/Ag32 and © XXXX American Chemical Society

Ag15 was shown. In particular, the rugby-ball shaped Ag31(SR)19 cluster was predicted to have a TPA cross-section of over 700 GM, while for the more spherically symmetrical Ag44(SR)304− cluster, we predicted a peak TPA of about 14000 GM. Although linear and nonlinear optical absorption properties were addressed for noble metal clusters, predicting luminescence, which could be useful in emission applications, has been less characterized. For example, dual fluorescence was demonstrated for a silver sulfide cluster, Ag12(SR)6(CF3CO2)6(CH3CN)6 with R = methylnaphthalene when in crystal form,48 while Kasha’s rule,49 which states that fluorescence is observed from the first excited state, generally holds because nonradiative internal conversion from higherenergy excited states to the first excited state is much faster than fluorescence from these higher excited states. The dual emission was attributed to excimer stabilization through πbonding of the naphthalene groups in adjacent clusters in the solid, partially confirmed by our calculations.50 The heavy gold atoms facilitate intersystem crossing (ISC) between the singlet-spin states and the triplet-spin states, and the resulting long-lived emission is likely to be due to phosphorescence. Because of Kasha’s Rule,49,51,52 phosphorescence is expected only from the lowest lying triplet state, T1, back to the ground-state, S0 (T1 → S0). However, Received: February 23, 2018 Revised: April 3, 2018 Published: April 11, 2018 A

DOI: 10.1021/acs.jpca.8b01882 J. Phys. Chem. A XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry A interestingly, Lei et al.53 have reported dual emission in solution for two hybrid gold−silver clusters, Au10Ag2(2-pyridylethynyl)3(1-pyridyldiphenylphosphine) 6(BF4)5 (cluster 1) and Au6Ag2(1-pyridyldiphenylphosphine)6(BF4)4 (cluster 2). Experimentally, the clusters were found to violate Kasha’s rule and have dual emission when in solution but not in the solid state, where only a single emission is observed for each cluster, in the visible spectrum for cluster 1 and in the NIR for cluster 2. Lei et al.53 suggested that while the isolated cluster has two excited states stable enough to produce emission, interaction between adjacent clusters in the solid state may disrupt one of the states. Dual-peak phosphorescence was reported by Paul, Chakrabarti, and Ruud54 for an organic excimer, but in this case the peaks were separated by only 0.2 eV, and it was shown that the two peaks were the result of different vibrational modes in the same electronic transition; while Brancato et al.55 claimed that a coumarin derivative exhibited the first actual, observable Kasha’s-rule breaking. Wang et al.56 describe another strategy for dual emission. A recent study57 of silver superclusters details impressive temperature-dependent dual-emission. In this study, DFT and TDDFT methods are applied to investigate the unique dual-band emission behavior of cluster 1 and cluster 2 due to phosphorescence, as well as the potential for larger TPA. Our results, mostly consistent with experiment, provide an approach for designing noble metal nanoclusters with long-lifetime emission.

Figure 1. Core structures of the two clusters when the full cluster is optimized with the PBE0 functional: (a) cluster 1; (b) cluster 2.

symmetrical, with the core having close to D3h symmetry, and this symmetry is maintained when optimized with either the PBE0 or the PBE functional, but when the B3LYP functional is used in the optimization, the central gold atom shifts to one side, destroying the symmetry. The inset in Figure 2 shows the central plane of symmetry for cluster 1, including the three 2-pyridylethynyl groups which stabilize the four gold atoms. Cluster 2 can be obtained by removing the part of cluster 1 shown in Figure 2, and optimizing the structure of the remaining atoms. The structure optimized with the PBE0 functional was also optimized in the solvent dichloromethane (DCM) using PCM. The DFT-optimized structures for cluster 1 tend to have longer bonds between the core atoms than the crystal structure, with B3LYP yielding longer average bond lengths than PBE0 and PBE (Table S1 in the Supporting Information). When the PBE0 functional is used without the inclusion of solvation effects, the average for the Ag−Au bond lengths is nearly the same as in the crystal structure (XS), but the Au−Au average is 0.06 Å longer. The PBE functional yields nearly the same average for the Ag−Au bond lengths as PBE0, and a slightly larger average for the Au−Au bonds. When B3LYP is used, the Ag−Au average is 0.06 Å larger than in the XS, and the Au−Au average is 0.43 Å longer than in the XS. The inclusion of solvent effects increases the mean bond length for the Ag−Au bonds by an additional 0.03 Å, but has a negligible effect on the mean bond length for the Au−Au bonds. The bond lengths listed in bold font are unusually long when the B3LYP functional is used due to the shift of the central Au atom to one side of the cluster. For cluster 2, the gas-phase calculated geometries have longer mean bond lengths than the XS, by 0.16, 0.06, and 0.06 for the Ag−Au bonds, using B3LYP, PBE0, and PBE respectively, and for the Au−Au bonds, by 0.13, 0.06, and 0.07 (Table S2). Inclusion of solvation (by PCM) causes a small decrease in the mean Ag−Au bond length with both hybrid functionals, while the change in the mean Au−Au length due to solvation is negligible. Experimental OPA spectra53 revealed features at 2.27, 2.67, and 3.51 eV for cluster 1, although only the maximum at 3.51 eV is readily apparent. Our calculated (as well as the measured) spectra for cluster 1 are shown in Figure 2. While the first excited state in our calculations is in fair agreement with the

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COMPUTATIONAL METHODS Calculations were carried out with the Gaussian58 and Dalton59 programs, with additional calculations carried out with ADF60,61 and GAMESS.62 The Def2TZVP basis set63 and the corresponding effective-core potential (ECP) were used for the gold and silver atoms, and the Def2SVP all-electron basis was used for the other atoms. In the ADF calculations, the zeroorder-regular-approximation (ZORA) was used along with the corresponding TZP basis set.64 The B3LYP,65−69 PBE0,70,71 and PBE72 functionals have been applied to both clusters. Systems with significant charge transfer require hybrid functionals to properly model the electronic wave function in the asymptotic regions, and some require range-separation, where the fraction of exact exchange increases in the asymptotic region, such as the coulomb-attenuated model (CAM).73 The nonhybrid functional, PBE, was used previously 53 for calculating the OPA of these two clusters. The geometry and energy of the S0 state was calculated by the usual spin-restricted DFT geometry optimization, and the T1 geometry and energy were obtained using unrestricted-spin DFT. The effect of solvent was evaluated by using the polarizable continuum model (PCM).74 Minima for higher energy states were obtained using TDDFT. Phosphorescence was calculated using quadratic response TDDFT in the Dalton program,75,76 using the mean-field option.77 Quadratic response TDDFT in Dalton was used to calculate TPA cross sections.

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RESULTS AND DISCUSSION First, we discuss the geometry of the clusters. Figure 1 shows the core structure (Au, Ag, and P atoms) of each cluster when the full cluster had been optimized with the functional PBE0 (starting from the crystal structure geometry53), while Figure S1 in the Supporting Information compares the structures obtained from optimization with the B3LYP, PBE0, and PBE functionals. The crystal structure for cluster 1 is highly B

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Figure 2. Experimental (Exp.) and calculated (PBE0) OPA for cluster 1, plotted against the excitation energy in eV. PCM denotes the inclusion of the solvent model in the calculation. The left vertical axis indicates the extinction coefficient (E.C.), while the right vertical axis is the oscillator strength (O.S.). The experimental peaks were reported by Lei et al.53 The inset shows the central plane structure of cluster 1.

Figure 3. Experimental (Exp.) and calculated (B3LYP and PBE0) OPA of cluster 2. PCM denotes the inclusion of the solvent model in the calculation.

with this reported value if the dominant vibrational state in S1 is about 0.2 eV above the vibrational zero-point energy. The measured and calculated OPA spectra of cluster 2 are given in Figure 3. The experimentalists53 reported absorption features at 2.22, 2.71, and 3.25 eV. The first two are clearly peaks, while the third is a more subtle shoulder. As with the larger cluster, including solvation via PCM increases the calculated intensity. The shape of the calculated spectrum is partly dependent on the line width used for each calculated excited state. The calculation using the B3LYP functional, along with PCM to include the effects of the dichloromethane solvent, yields a spectrum in good agreement with experiment when the set of Gaussian line widths used are 0.3 eV for the first, second, and fourth excited states (at 2.15, 2.34, and 2.75 eV), and 0.6 eV for the third and fifth excited states (at 2.44

feature at 2.67 eV, the calculations do not indicate any state nearly as low in energy as the experimental value of 2.27 eV. When PCM is used with PBE0 in the TDDFT calculations, the spectrum rises nicely in agreement with the peak near 3.5 eV, largely due to the strong excitation to state S20 at 3.44 eV. The main effect of including solvation is to increase the absorption intensity. The calculated electronic adiabatic absorption energy to the excited-state S1 (ΔE) is 2.10 eV, and, after correction for zeropoint vibrational energy (ΔH0), it changes to 2.04 eV, significantly lower than the calculated vertical absorption energy of 2.55 eV. While carrying out a Franck−Condon analysis on this system to generate the vibronic spectrum is not feasible, the reported feature at 2.27 eV is between these two values. The PBE0 TDDFT results could be in good agreement C

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intensity absorption of cluster 2, on the other hand, seems less structured, with some of the transitions having a small transfer of charge from the gold atoms to the silver atoms, phosphorus atoms, or pyridyl rings, while others, such as the strongest calculated transition at 3.43 eV ( f = 0.1247, see Figure S5) involves charge transfer from a pyridyl ring to the gold core. In addition, we compared TPA spectra to those we recently calculated for an Ag12 silver cluster,49 for which we calculated a significant off-resonance TPA cross-section. The calculated TPA spectrum for cluster 1 is shown in Figure S6, for which 10 TPA states were included, spanning the range 2.5−3.2 eV, while several states near 3 eV contribute small TPA cross sections in the range 0.4−0.6 GM when a Gaussian line width function with fwhm =0.5 eV is used, resulting in a peak TPA of about 2.4 GM. Figure S7 shows the calculated TPA spectrum for cluster 2. This calculation includes 30 excited states and predicts a modest TPA cross-section of 67 GM in the visible red spectrum having a transition energy of about 3.8 eV, corresponding to two photons of energy 1.9 eV (665 nm). The resonance enhancement for TPA at this energy is small, which means the interference from OPA is small. Next, we note that in solution, the two clusters demonstrate dual emission at photon energies of about 1.3 and 1.9 eV when excited by photons with energy 2.6 eV or higher, although the near-infrared (NIR) peak at 1.3 eV is much weaker in cluster 2.53 The emission lifetimes are in the range 0.5−5 μs, which is much longer than expected for fluorescence, and assumed to be due to phosphorescence. Phosphorescence is due to spin−orbit coupling that enables crossing from the excited singlet-spin states to triplet-spin states, and the long-lived emission is due to the decay of the triplet states back to the ground-state singlet (S0). As this is likely to occur only from the lowest energy triplet state (T1), the dual emission is an anomaly. Also of note is that in the solid-state, these clusters do not have dual emission, as cluster 1 has a single emission peak in the visible at 1.78 eV, and cluster 2 has a single emission peak in the NIR at 1.15 eV. The calculated results related to emission are summarized in Tables 1 and 2 for clusters 1 and 2, respectively. The adiabatic phosphorescence energy; i.e., the energy difference between the S0 and T1 states at their respective optimized geometries, is denoted ΔE. The S0 energy at the T1 geometry has also been calculated, and the energy difference between the two states at the T1 geometry is labeled T1 phos(v), for vertical phosphorescence. The vertical phosphorescence was also estimated, using a linear response calculation from the ground-state at the T1 geometry and solving for triplet excited states. This result is labeled T1 phos(v)rsp. A similar calculation was carried out in the Dalton program using quadratic response TDDFT with the spin−orbit mean-field approach.75−77 This method uses the single residue quadratic response formalism with one operator set to the dipole operator and one set to the spin−orbit operator, but simplifies the computation by creating an effective one-electron spin− orbit operator by averaging over the two-electron spin coupling.77 It yields phosphorescence lifetimes as well as the vertical phosphorescence energies. The vertical phosphorescence energies calculated with this method are labeled with the Dalton keyword MNFPHO and are nearly identical to the values calculated with linear response in the Gaussian program, with the exception of the B3LYP result on cluster 2. The calculated lifetimes are several orders of magnitude larger than the experimental values.

and 3.05 eV), as well as for all higher excited states. Other choices for the line widths may also give reasonable agreement, but this set is fairly simple and works well. The PBE0 functional yields results similar to B3LYP for the first three excited states, but the calculated energies for S4 and higher energy states are more blue-shifted than the B3LYP results, and thus the PBE0 agreement with the measured spectrum above 2.5 eV is not as good. While the vertical absorption to S1, calculated using PBE0, is 2.07 eV, the adiabatic absorption energies (ΔE) and (ΔH0) are 1.524 and 1.523 eV, respectively. For this calculation to be in agreement with the experimental value of 2.22 eV, the vibrational energy of the final S1 state must be 0.7 eV above the zero-point vibrational energy. A more likely explanation is that the measured peak at 2.22 eV is the result of a combined contribution from the S1 state and the more intense S2 state. We analyze frontier molecular orbitals that give rise, in part, to the observed transitions, first for cluster 1, shown in Figure S2. The primary orbital transitions involved for the low-lying excited singlet states are denoted in Figure S3. The S1 state at 2.55 eV with an oscillator strength of 0.081, calculated at the PBE0:PCM level of theory, originates primarily from a HOMO to LUMO transition. The electron density in the HOMO is concentrated primarily in the gold core and in the ethynyl bonds of the central plane (see Figure 2), with only small contributions on the ligands. Comparatively, for the LUMO, there is transfer of charge to the pyridyl groups between the ethynyl bonds, as well as to the two silver atoms, which are located on opposite ends of the gold core. The largest contribution to the absorption spectrum, as calculated at this level of theory, is from the transition to state S20 at 3.44 eV, with an oscillator strength equal to 0.542. The primary orbital transition for this excited state is HOMO−2 → LUMO+3. The HOMO−2 orbital has a large density in the gold core and ethynyl bonds, as well as significant density on one of the pyridyl rings of the central plane, and also smaller amounts in the pyridyl and phenyl groups attached to phosphorus atoms. The LUMO+3 orbital has a large electron density on some of the pyridyl groups attached to phosphorus atoms. For cluster 2, molecular orbitals are depicted in Figure S4, and the dominant orbital transition for each excited state is labeled in Figure S5. For this cluster, the first excited state, S1, is also dominated by the HOMO to LUMO transition, but it has a lower transition energy (2.14 eV) and a much lower oscillator strength (0.013). Also important in the absorption spectrum are the more intense OPA states S2, S3, and S4 at 2.36 eV (O.S. = 0.112), 2.50 eV (O.S. = 0.071), and 2.87 eV (O.S. = 0.097), which are dominated by HOMO → LUMO+1, HOMO → LUMO+2, and HOMO → LUMO+3, respectively. The HOMO shows high electron density in the gold core, as do LUMO, LUMO+1, LUMO+2, and LUMO+3, but with varying amounts of the electron density transferred outward to the silver, phosphorus, and carbon atoms. The LUMO+1 orbital, in particular, has significant electron density mostly on the silver and phosphorus atoms with little density on the carbon atoms, while the LUMO+3 orbital has significant charge density on the pyridyl phosphine substituents. By comparing this cluster, which does not have the central Au4(2-pyridylethynyl)3 structure, with cluster 1, we see that the central structure stabilizes the HOMO, thus blue-shifting the absorption for cluster 1 relative to cluster 2, and also significantly increases the absorption cross-section. The ethynyl bonds of the central core are a large source for charge transfer to the pyridyl rings, resulting in the much larger absorption by cluster 1. The lower D

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for cluster 2 is over 45 eV, it is nearly the same for the S0 and T1 states of each cluster, and the corrected energy difference between the two minima, listed as ΔH0, is only slightly smaller than the ΔE value. The ΔH0 value calculated for cluster 1 is 1.94 eV using the PBE0 functional, overestimated as compared to the measured NIR emission peak, observed near 1.3 eV with a high quantum yield. Our calculated value is closer to the visible emission observed near 1.9 eV. The emission for cluster 1 calculated using PBE0 is illustrated in Figure 4. The electronic transition energies in Figure 4a, of 2.01 eV for adiabatic phosphorescence, and 1.66 eV for vertical phosphorescence, are significantly higher than the measured emission near 1.3 eV. Figure 4b illustrates that agreement with experiment for the NIR emission will require T1 → S0 emission to a vibrational state that is about 0.6 eV

Table 1. Emission Data for Cluster 1 from the Triplet-Spin States T1, T2, and T3a T1 phos(ΔE) T1 phos(ΔH0) T1 phos(v) T1 phos(v)rsp T1MNPHOS T1 lifetime T2 phos(v) T2MNFPHO T2 lifetime T3 phos(v) T3MNFPHO T3 lifetime

PBE

PBE0

B3LYP

exp.

1.54 1.46 1.28 1.18

2.01 1.94 1.66 1.64 1.65 0.56 2.09 2.09 1.25 2.26 2.26 0.26

1.86 1.82 0.73 0.59 0.59 8.38 1.41 1.41 1.67 1.60 1.60 0.68

1.28 1.28 1.28 1.28 1.28 7.E-07 1.90 1.90 4.E-06

1.40

1.41

The “adiabatic phosphorescence”, i.e., the electronic transition energy between the triplet state and the ground-state singlet each evaluated at its equilibrium geometry, is labeled “phos(ΔE)” or after correction for zero-point vibrational energy, “phos(ΔH0)”. The “vertical” phosphorescence energy, i.e., the electronic transition energy between the triplet state and the ground-state singlet at the equilibrium geometry of the triple state, is labeled phos(v) if calculated directly or “phos(v)rsp” if calculated by a TDDFT response calculation. The vertical response phosphorescence calculation using the mean-field option in Dalton is denoted “MNFPHO”. Energies are in eV and lifetimes are in seconds. The T2 and T3 energies were evaluated at the T1 geometry.

a

Table 2. Emission Data for Cluster 2a T1 phos(ΔE) T1 phos(ΔH0) T1 phos(v) T1 phos(v)rsp T1MNFPHO T1 lifetime T2 phos(v)rsp T2MNFPHO T2 lifetime T3 phos(v)rsp T3MNFPHO T3 lifetime with PCM T1 phos(ΔE) T1 phos(ΔH0) T1 phos(v) T1 phos(v)rsp T2 phos(v)rsp T3 phos(v)rsp

PBE

PBE0

B3LYP

1.11 1.12 0.46 0.41 0.54 9.78 1.22 1.48 0.73 1.45 2.00 0.22

1.31 1.31 0.65 0.62 0.62 11.35 1.58 1.58 0.21 2.01 2.01 0.18

1.15 1.13 0.03 −0.12 1.33 0.14 1.33 1.83 0.10 1.83 1.85 0.07

1.30 1.30 1.30 1.30

exp.

1.25 1.25 0.57 0.57 1.63 1.95

1.10 1.09 0.00 −0.13

1.30 1.30 1.30 1.30 1.90

1.90 4.E-06

a

Abbreviations are as defined in Table 1. Energies are in eV and lifetimes are in seconds. The PBE functional was also used with the zero-order regular approximation (ZORA) and the corresponding TZP basis set in the ADF program, yielding ΔE = 1.03 and ΔH0 = 1.04.

Figure 4. Absorption and emission energies for cluster 1 calculated using PBE0. (a) Absorption is calculated as a vertical transition at the ground-state geometry. The dashed gray arrow indicates the relaxation of the S1 state to its minimum energy geometry. The purple dashed arrow denotes the intersystem crossing (ISC) to the T1 state. The phosphorescence is calculated two ways: as an adiabatic transition between the equilibrium geometries of the two states, and as a vertical transition evaluated at the equilibrium geometry of the T1 state. (b) Zero-point vibrational energy has been added to the electronic energy for the S0, S1 and T1 states. This yields an adiabatic absorption energy of 2.04 eV. The adiabatic phosphorescence energy is reduced from 2.01 to 1.94 eV when zero-point vibrational energy is included. Additional arrows indicate vibrational final states that would put the calculated results in good agreement with experiment.

However, these energy differences include only the electronic energy, while vibrational effects have also to be considered. Each state will have some vibrational energy even in the lowest vibrational quantum state, i.e. the zero-point vibrational energy. This can be estimated at each state’s minimum energy geometry by carrying out a Hessian calculation to obtain the vibrational frequencies in the harmonic approximation. While the zero-point vibrational energy for cluster 1 is over 50 eV and E

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The Journal of Physical Chemistry A higher than the S0 vibrational zero-point. The dual emission could then be explained by emission also to the lowest vibrational level of S0. Coupling between adjacent clusters in the solid-state may change the Franck−Condon structure such that emission to the lower vibrational state dominates, and thus the NIR emission is not observed in the solid-state. With this explanation, cluster 1 would not be breaking Kasha’s rule. A second possible explanation is that either the T2 or T3 state is stable enough against nonradiative decay, and thus provides the second phosphorescence peak. As T2 and T3 are about 0.4 and 0.6 eV higher in energy than T1, respectively, such a transition would be in good agreement with experiment if the S0 vibrational state were 0.4−0.6 eV larger than the zero-point vibrational energy. The experiment53 also reported that treatment with nBu4NCl to remove the silver ions quenched the visible emission but left the NIR emission unchanged. Since the LUMO has significant density on the silver atoms this implies a link between the absorption to S1, dominated by the HOMO to LUMO transition, and the visible emission at 1.9 eV. The reported excitation spectra53 for the visible emission shown in their Figure S4, which peaks at 2.93 eV, also implies that the main path to the visible emission is S0 → S1 → T1 → S0. The excitation spectra for the NIR emission peaks at 3.53 eV, implying that the main path for the NIR emission is S0 → S20 → T20 → Tn → S0, v = m, where the T20 to Tn (Tn may be T2 or T3) is a radiationless transition and the S0 final vibrational state, v = m, would need to be 0.7 eV or more above the ground vibrational state. This explanation seems most consistent with all the experimental data, and is consistent with the authors’ claim of Kasha’s-rule breaking. Lei et al.25,53 analyzed the excited states for this system using the nonhybrid PBE functional, and their discussion seems to imply that the dual emission was from two distinct excited states, but that the NIR emission results from excitation to a HOMO−1 → LUMO dominated S1 state. However, hybrid functionals are usually needed to accurately model charge transfer, while the PBE functional predicts transition energies that are too low. Figure 5 shows the emission data for cluster 2. The adiabatic phosphorescence energy calculated for T1 → S0 using PBE0 is 1.31 eV, and in very good agreement with the experimental NIR emission energy. The effect of including vibrational zeropoint energy is negligible, so ΔH0 is also 1.31 eV. The adiabatic phosphorescence energy calculated from the T2 state, of 1.74 eV, is in fair agreement with the observed visible emission energy of 1.90 eV. The visible emission could also correspond to emission from the T3 state. So for cluster 2, the results support a Kasha’s-rule-breaking dual emission from two different triplet states.

Figure 5. Absorption and emission energies from cluster 2 calculated using PBE0. (a) Vertical absorption is shown, while both vertical and adiabatic phosphorescence is shown. (b) Zero-point vibrational energy is included in the energies for S0, S1, and T1. The adiabatic absorption is only 1.52 eV, so an additional 0.7 eV of vibrational energy above the zero-point vibration is required to yield good agreement with experiment. The adiabatic approximation to the phosphorescence from T1 remains at approximately 1.31 eV when zero-point vibrational energy is included.

agreement with the measured visible emission. The calculated T1 → S0 phosphorescence ΔH(0−0) for cluster 1 is close to the measured visible emission energy. The intense NIR emission reported experimentally for cluster 1 may be primarily due to emission that originated with absorption to a higher energy singlet-state, and after intersystem crossing then relaxed to the T2 or T3 state, which undergoes phosphorescence to vibrationally excited S0. In order for the calculated phosphorescence using PBE0 to agree with the intense NIR emission reported experimentally, the vibrational energy of the final state (S0) would be about 0.7 eV greater than the zero-point vibrational energy. Since dual emission in solution was observed for each of the two clusters at similar energies, the different mechanisms for the emission may seem surprising, but it is consistent with the different emission behavior observed for the two clusters in the solid-state. For cluster 2, a modest TPA cross-section is predicted when the photon energy is 1.9 eV (wavelength = 653 nm).

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CONCLUSIONS The OPA, TPA, and emission of two gold−silver clusters have been modeled using DFT. The OPA spectra calculated using the hybrid functionals B3LYP and PBE0 tend to be in good agreement with the reported measured spectrum. For cluster 1, the results using the PBE0 functional yield the best results, while for cluster 2, both the B3LYP and PBE0 results are in good agreement with experiment. The excitation energies calculated with the PBE functional are too low to be in good agreement with experiment. Emission is modeled as being phosphorescence from the triplet-spin states. For cluster 2, The ΔH(0−0) value calculated using the PBE0 functional for the T1 → S0 transition is in good agreement with the measured NIR emission, and the calculated T2 → S0 value is in fair F

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ASSOCIATED CONTENT

* Supporting Information S

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpca.8b01882. Bond lengths calculated for the cluster cores, molecular orbitals, TPA spectra, and additional OPA analysis (PDF)

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AUTHOR INFORMATION

Corresponding Authors

*(P.N.D.) E-mail: [email protected]. *(R.P.) E-mail: [email protected]. ORCID

Paul N. Day: 0000-0002-6333-6359 Ruth Pachter: 0000-0003-3790-4153 Kiet A. Nguyen: 0000-0003-0363-313X Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We gratefully acknowledge support from the Air Force Office of Scientific Research and the computational resources provided by the AFRL DSRC.

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REFERENCES

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