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A Theoretical Investigation of the Structure and Optical Properties of a Silver Cluster in Solid Form and in Solution Published as part of The Journal of Physical Chemistry virtual special issue “Mark S. Gordon Festschrift”. Paul N. Day,*,†,‡ Ruth Pachter,*,† and Kiet A. Nguyen†,§ †

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

ABSTRACT: Using density functional theory (DFT) and linear and quadratic response time-dependent DFT, we investigated the structure and optical properties of a silver sulfide cluster with the interesting property of dual emission that was observed when in crystal form but not in solution. Since the dual fluorescence is observed only in the crystal, a supposition of stabilization of a higher-energy excited state by an excimer-like complex was analyzed by calculations for a cluster dimer, formed through π-stacking of aromatic groups bonded to the sulfur atoms. However, because of the complexity of the system, a simple one-dimensional method for dimer optimization, which works moderately well in predicting the red-shifted fluorescence compared to its absorption in a naphthalene dimer, predicts only partially the red shift for the emission energy. Interestingly, calculations of the two-photon absorption (TPA) cross-section on the optimized isolated cluster as well as the crystal structure geometry indicate significant off-resonance TPA. While some materials have significantly larger TPA cross-sections, such a TPA cross-section off-resonance could be useful. The high density of states in the dimer system results in a higher probability for significant resonance enhancement and thus much larger TPA cross-sections. naphthalene,10 as shown in Figure 1. The cluster can be roughly described as a distorted ring of six silver atoms sandwiched between two six-member distorted rings with alternating silver and sulfur atoms. In the ring of six silver atoms, each of the six trifluoro-acetate groups bridges two silver atoms, with each silver atom bonded to two of these groups. Acetonitrile groups stabilize the other six silver atoms. When the fluorescence of this system is measured in chloroform solution, a single peak is observed at ∼2.9 eV (428 nm), while the crystal fluoresces with two distinct peaks, one at 1.88 eV (660 nm) and the second at 2.18 eV (570 nm), for an excitation energy of 3.35 eV (370 nm) or higher. When the excitation is at 3.10 eV (400 nm), a single emission peak at 1.79 eV (694 nm) is observed. The dual emission may be the result of excimer stabilization through πbonding of the naphthalene groups11 in adjacent clusters in the solid, a mechanism that has been proposed in previous studies.12,13 In this work, we aim to gain an understanding of the origin of the dual emission by comparing density functional theory (DFT) calculations for a single cluster to its corresponding π-

1. INTRODUCTION Kasha’s rule,1 which states that fluorescence is observed only from the first excited state, seemed to be violated by a dual emission phenomenon in early work by Eber et al.2 on azulene derivatives. The S2→S0 fluorescence in these molecules is attributed to a large S2−S1 gap, which stabilizes S2 against nonradiative conversion. In some substituted azulenes, S1→S0 fluorescence is also observed at low temperature, resulting in dual emission, although at higher temperatures the rate of S1→ S0 nonradiative conversion increases, and S1→S0 fluorescence is difficult to observe. Kasha’s rule usually holds, mainly because nonradiative internal conversion from higher-energy excited states to the first excited state is much faster than fluorescence from these higher excited states. In later work, dual emission was, for instance, attributed to locally excited (LE) and intramolecular charge transfer (ICT) states due to conformational changes in pyrylium salts3 or other fluorescent dyes4−8 (see references therein). A recent study claims to be the first to have observed actual Kasha’s rule-breaking with useful dual fluorescence, attributed to a confluence of photophysical phenomena.9 Interestingly, recently a silver sulfide cluster, Ag12(SR)6(CF3CO2)6(CH3CN)6, has been reported to show dual fluorescence when in crystal form and R = methyl © XXXX American Chemical Society

Received: October 28, 2016 Revised: December 12, 2016 Published: December 13, 2016 A

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B3LYP-D3 geometry with stacked naphthyl ligands (shown in Figure 2) having a nearest C−C distance of 3.433 Å and a centroid-to-centroid distance of 3.96 Å. The nearest C−C distance was then varied, and TDDFT calculations were performed to find the minimum-energy geometry for each state. Absorption spectra were calculated by performing linearresponse TDDFT calculations on both the crystal structure (XS) and the ground-state optimized geometry, while the emission peaks were obtained through excited-state optimization using Gaussian 09.24 The polarizable continuum model (PCM)35 was used to include solvent effects. Dalton25 was used to calculate TPA spectra with quadratic response TDDFT, as was previously described.36 We note that in our previous TDDFT study on silver clusters,36 the PBE, PBE0, B3LYP, CAMB3LYP, LB94, and SAOP XC functionals were tested. The B3LYP functional performed best in the prediction of the one-photon absorption (OPA) spectra for Ag44(SR)304−, CAMB3LYP for Ag31(SR)19 and Ag32(SR)19, and PBE0 for Ag15(SR)11 calculations, while the LB94 and SAOP functionals had the lowest error in predicting the Ag14(SR)12(PR′3)8 spectrum. Figure 1. Structure of Ag12(SR)6(CF3CO2)6(CH3CN)6 with R = methylnaphthalene.

3. RESULTS AND DISCUSSION Xu et al.10 reported the room-temperature solid-state absorption spectrum of Ag12(SR)6(CF3CO2)6(CH3CN)6, as well as the excitation spectrum at room temperature and 77 K, monitoring the emission at 1.88 and 1.80 eV, respectively. The absorption spectrum, which is plotted in Figure 3, shows a steep rise that begins at ∼2.6 eV, levels off near 3.2 eV, has a brief steep rise near 3.5 eV, and continues to rise gradually beyond 5 eV. The low-temperature excitation spectrum, also shown in Figure 3, shows a broad band centered at 3.04 eV with apparent maxima at 3.01 and 3.09 eV and a second band centered at 3.36 eV. Calculated TDDFT excited-state energies and corresponding oscillator strengths were fitted to either Gaussian or Lorentzian line width functions to simulate the absorption spectrum, but in this case, comparison of the calculated excited-state energies with the low-temperature excitation spectrum is also useful. Calculated excited-state energies and corresponding oscillator strengths, obtained using the B3LYP, CAMB3LYP, and PBE0 functionals, are plotted in Figure 3, based on the structure of the cluster shown in Figure 1. The B3LYP results are in good agreement with the features in the measured absorption and excitation spectra, while the

bonded dimer, shown in Figure 2, and in addition, assess the utility of the cluster in two-photon absorption (TPA) applications. TPA applications using organic systems have been studied extensively,14−17 and more recently metal cluster systems have shown large TPA.18,19 Silver clusters with large TPA have shown promise as materials in imaging applications.20−22

2. COMPUTATIONAL METHODS The crystal structure was obtained from Xu et al.,10 and timedependent (TD) DFT calculations were performed on the individual cluster using the geometry of this crystal structure and on geometries obtained from DFT optimization. Calculations were performed with GAMESS,23 Gaussian 09,24 and Dalton25 using the PBE0,26,27 B3LYP,28−32 PBE0-D3, and B3LYP-D3 exchange correlation (XC) functionals, where D3 refers to the dispersion correction functional of Grimme et al.33 The LANL2DZ basis set was used.34 Dimer geometries were constructed starting from both the crystal structure10 and the

Figure 2. Dimer structure of the cluster given in Figure 1. B

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Figure 4. Effect of solvent on the OPA calculated with B3LYP for the Ag12(SR)6(CF3CO2)6(CH3CN)6 monomer. The upper panel shows the extinction coefficients obtained from fitting to a Gaussian line shape with fwhm of 0.2 eV, where the dashed blue line is the gas-phase data, and the solid green line is the data in chloroform using PCM. The lower panel shows the calculated oscillator strengths, using blue squares to denote the gas phase and green triangles for the chloroform solvation. The experimental data given in Figure 3 are also repeated here.

Figure 3. Experimental absorption spectrum (dashed red line), the low-temperature excitation spectrum (solid black line), and the calculated excitation energies and oscillator strengths for the cluster shown in Figure 1. Results from B3LYP are shown as green triangles, from CAMB3LYP as red diamonds, and from PBE0 as blue circles. Open symbols denote the optimized geometry, while solid symbols are the results when using the crystal-structure geometry. The intensity scale for the measured results is arbitrary. The excitation spectrum was obtained at 77 K by monitoring emission at 1.80 eV.

and usually the unoccupied orbitals are lowered more than the occupied orbitals, leading to a solvent-induced red shift. The first excited state is an exception, since the highest occupied molecular orbital (HOMO) is lowered more than the lowest unoccupied molecular orbital (LUMO) in the PCM calculation. Thus, the first excited state has a blue shift of 0.06 eV, from 3.01 eV in the gas phase to 3.07 eV in chloroform. While in the gas phase this first excited state appears as a small shoulder-like peak in the spectrum, due to the blue shift of this state and the red shift of a higher-energy excited state, it cannot be distinguished in the fitted spectrum from the more intense peak at 3.26 eV. The energies for absorption and emission peaks are listed in Table 1. While the gas-phase emission for this cluster was calculated with the B3LYP functional to be 2.69 eV, the fluorescence in chloroform was calculated to be 2.78 eV, in fair agreement with the measured value of 2.90 eV. At the optimized geometry of the first excited state, the PCM again lowers the HOMO more than the LUMO, and thus the calculated fluorescence energy is 0.09 eV higher in solution than in the gas phase. Next, we consider whether excimer formation in the cluster dimer can be responsible for the observed dual fluorescence and dramatic red shift in the emission energy. In general, the lowest-energy excited state (S1) can decay back to the ground state (S0) either radiatively (fluorescence) or nonradiatively. Higher-energy excited states (Sn) have the additional possibility of decaying nonradiatively to a lower-energy excited state followed by fluorescence. Usually, nonradiative decay from the higher excited states to S1 is rapid, so fluorescence from these higher-excited states back to S0 is not observed; hence, Kasha’s rule. To break Kasha’s rule and have dual fluorescence, a higher

PBE0 results are too blue-shifted by 0.2−0.5 eV, and the CAMB3LYP results are too blue-shifted by a full electronvolt or more, consistent with the amount of exact exchange included in these functionals. The excited states calculated from the optimized geometries are red-shifted by 0.1−0.2 eV relative to those calculated using the XS geometry. The spectra for B3LYP and PBE0 when fitted to a Gaussian line function (fwhm of 0.2 eV) are shown in Figure S1 in the Supporting Information. To calculate the energy of fluorescence from the first excited state, S1, an excited-state geometry optimization was performed. At the minimum-energy geometry on the potential energy surface (PES) of S1, the S1 energy is lower than it was at the minimum on the S0 surface, while the S0 energy is higher than it was at its ground-state minimum, and the difference between these two energies is the estimate of the emission energy. To compare the measured emission spectrum10 for the Ag12(SR)6 cluster in solution, for which fluorescence is observed at a single wavelength, following Kasha’s rule, we repeat the process, but using PCM to include the effects of the chloroform solvent, first calculating the OPA spectrum for the cluster at the solvated ground-state geometry, followed by the optimization of the energy of the first excited-state. The oscillator strengths have been fitted to Gaussian lineshape functions with a full width at half-maximum (fwhm) of 0.2 eV, and the results are shown in Figure 4 (the experimental spectra are also included). The two intense peaks calculated in the gas-phase at 3.34 and 3.67 eV are slightly red-shifted by the solvation model to 3.26 and 3.60 eV and become more intense. The solvation model tends to lower the energy of each orbital, C

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The Journal of Physical Chemistry A Table 1. Peak Energies for Absorption and Emission (eV)a absorption measured A.S.(298 K) measured E.S(77K) measured (298 K) measured (298 K) PBE0//PBE0 PBE0//XS B3LYP//B3LYP B3LYP//XS CAMB3LYP//B3LYP CAMB3LYP//XS B3LYP//B3LYP B3LYP-D3//B3LYP-D3 B3LYP-D3//XS

solid solid CHCl3 THF cluster cluster cluster cluster cluster cluster CHCl3 dimer dimer

emission

peak1

peak2

peak3

3.21 3.01

3.55 3.09

3.36

3.26 3.34 3.02 3.14 3.92 4.04 3.26 3.03 3.15

3.82 3.88 3.34 3.60 4.13 4.65 3.60 3.11 3.20

4.10 4.21 3.67 4.00 4.46

3.16 3.56

S01

S02

1.88 1.88 2.90 2.72

2.18 2.18

S03

2.69

2.78 3.00 3.05

3.10 3.11

3.15 3.56

a

Measured data include the absorption spectrum (A.S.) and excitation spectrum (E.S.). Calculated absorption peaks obtained using a Gaussian lineshape with fwhm = 0.2 eV. Measured results are from Xu et al.10.

function of the separation distance and compare to previous calculations11,38 and experiment.40,41 In the PBE0-D3 results, the La and Lb curves are in fair agreement with the DMRGCASPT2 results,38 and each has a minimum deep enough to support a bound state at a separation geometry of ∼3.23 Å, with the Lb minimum ∼0.4 eV higher in energy. The energy curves for the Lc and Ld states have shallow minima at separations of 3.63 and 3.79 Å, respectively, at energies of 0.8 and 1.0 eV higher than that of La. The calculated absorption energy (corresponding to either the naphthalene monomer or else the dimer at large separation) of 4.42 eV using PBE0-D3 is in good agreement with the measured absorption energy of 4.45 eV. The emission energy calculated using the onedimensional excited-state optimization described above is 3.15 eV, in surprisingly good agreement with the measured value of 3.13 eV. However, when a full excited-state optimization is performed on the S1 (La) state of the naphthalene dimer, the calculated emission energy is reduced to 2.65 eV. The three excited states Lb, Lc, and Ld are significantly higher in energy than La and readily undergo nonradiative conversion to La, and thus the naphthalene system has just the single fluorescence from La, following Kasha’s rule. The one-dimensional optimization described above was then used on the larger π-bonded dimer of interest in this paper because of the difficulty in performing excited-state geometry optimizations on such large systems. To mimic the crystal structure, a dimer was first constructed from the optimized monomer, and linear-response TDDFT calculations were performed with the separation distance between the π-bonded carbon atoms varied from 2.3 to 4.2 Å. Figure 6 shows calculated state energies for the Ag12(SR)6(CF3CO2)6(CH3CN)6 dimer as a function of separation distances. The ground state has a very shallow minimum at 3.4 Å. The two states that are dominated by H→L and H→L+1 transitions appear to be degenerate and have an absorption energy of ∼3.03 eV and an emission energy of 3.00 eV at r = 3.25 Å. Two states dominated by H-1→L and H-1→L +1 also appear to be degenerate and have an absorption energy of ∼3.11 eV and an emission energy of 3.10 eV at r = 3.35 Å. Similarly, two states dominated by H-3→L and H-3→L+1 also appear to be degenerate and have an absorption energy of ∼3.16 eV and an emission energy of 3.15 eV at r = 3.35 Å. The calculated energies for these states are very flat along this

excited state, say S2, must be sufficiently stable against rearrangement to S1. Unsubstituted naphthalene does not show dual fluorescence, but its interesting absorption and emission properties have been extensively studied. Naphthalene in the ground state has a weakly bound dimer, but the excimer is more strongly bound. This results in fluorescence from the excimer at a significantly lower transition energy than for absorption, which is generally attributed to the monomer. While the absorption and fluorescence of this system have been studied theoretically at various levels of theory, including TDDFT,37,38 RI-CC2,37 CASPT2,11,38 MCQDPT,39 EOM-CCSD,38 and DMRG-CASPT2,38 only the latter described the two excited states La and Lb and the energy inversion that occurs for these two excited states. In Figure 5, we show our TDPBE0-D3 calculated energies for the ground and first five excited states of the naphthalene dimer as a

Figure 5. State energies as a function of separation distance for naphthalene dimer. Black lines and symbols denote the ground-state S0, while purple denotes S0 or La, green denotes Lb, orange denotes Lc, red denotes Ld, and blue denotes Le. Solid lines indicate the PBE0D3 energies obtained by varying the separation distance, while circles denote PBE0-D3 energies at optimized geometries. Dashed lines show DMRG-CASPT239 results, and triangles denote CASPT211 results. The experimental40,41 data are denoted by squares. D

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Figure 6. State energies as a function of separation for dimer of the optimized cluster Ag12(SR)6(CF3CO2)6(CH3CN)6. Excited states are identified by the molecular orbital transition with the largest contribution. Solid black: S0, solid red: H-1→L, dashed red: H-1→L +1, solid purple: H-3→L, dashed purple: H-3→L+1, solid green: H→ L, dashed green: H→L+1, solid blue:H→L+2.

Figure 7. State energies as a function of separation for dimer of Ag12(SR)6(CF3CO2)6(CH3CN)6 using the crystal-structure geometry. Excited states are identified by the molecular orbital transition with the largest contribution. Solid black: S0, solid red: H-1→L, dashed red: H1→L+1, solid purple: H-2→L, solid green: H→L, dashed green: H→L +1, dashed blue: H-9→L.

coordinate, and thus the emission energies are barely lower than the absorption energies. Another state, characterized by the H→L+2 transition, had an absorption energy of 3.57 eV and emission at 3.27 eV with r = 3.10 Å. However, the higher energy and low oscillator strength of this state near the groundstate minimum make it unlikely that this state would contribute to fluorescence. These calculations do not predict the much lower emission energies (1.9 and 2.2 eV) that were measured. Indeed, the DFT-optimized geometry of the Ag12(SR)6(CF3CO2)6(CH3CN)6 cluster has a less compressed silver sulfide core compared to the crystal structure, where the central ring of six silver atoms has bond lengths of 2.98 Å, while in the DFT-optimized structure, these bonds average ∼9% longer (3.26 Å). We do not expect that such an optimization will be appropriate to mimic the crystal structure, but it was performed for comparison with calculations using the crystal structure model, as described in the following. A dimer was thus constructed from the crystal-structure geometry directly, and TDDFT calculations were performed to determine the state energies and oscillator strengths at various separation distances, as plotted in Figure 7. For consistency with the calculations for the cluster in solution, solid-state calculations were not performed. We note that in the case of a relatively flat excited-state potential energy surface, such a model may not be sufficient to fully reproduce the experimental emission shift, but the goal is to obtain a qualitative trend. The ground-state minimum-energy geometry found using this method was not at a separation distance near 3.4 Å, as expected, but closer to 3.18 Å. From this ground-state minimum, the first four excited states can be denoted by the most prominent orbital transition: H-1→L+1 (3.13 eV, f = 0.05), H-2→L (3.14 eV, f = 0.02), H→L (3.15 eV, f = 0.02), and H→L+1 (3.20 eV, f = 0.01). Together, these peaks compose the first peak in the absorption spectrum near 3.14 eV. As can be seen in Figure 7, the excited state with the lowestenergy minimum is H→L, which has a minimum energy at a separation of 3.03 Å, and the predicted emission energy is 3.06 eV. The first excited state, H-1→L+1, relaxes to a separation of 3.13 Å, where it can emit at 3.12 eV. The H-2→L state has a minimum near a separation of 3.23 Å, so its predicted emission

energy is 3.14 eV. H→L+1 relaxes to 3.08 Å, where it could emit at 3.11 eV. Thus, the much lower emission energies are still not predicted, although lower than for the dimers constructed from the optimized monomer geometry. Compared to the optimized system of Figure 6, where the ground state and excited states had shallow minima of only ∼0.2 eV, the data in Figure 7 show a minimum of 0.5 eV in the ground state and greater than 0.8 eV for the H →L state. In the naphthalene dimer, the deep minimum for the excited state was at a significantly smaller separation geometry than for the ground-state minimum, and the combination of this deep minimum and the increased energy of the ground state at this geometry resulted in the emission energy being significantly smaller than the absorption energy. However, for the naphthalene dimer, a full geometry optimization was performed on both the ground state and the first excited state, so the emission energy calculated from the fully optimized system can be compared to that calculated from a simple one-dimensional optimization. The minimum energy for the first excited state, S1, was found in a full optimization at a separation of 3.14 Å, and the S1−S0 gap at this geometry predicts an emission energy of 2.65 eV, while in the one-dimensional optimization, the excited-state minimum is at a separation of 3.23 Å, and this gap is 3.15 eV. The decrease in the predicted emission energy from the geometry optimization is ∼0.5 eV, which comes from the combination of the excited-state minimum being 0.2 eV lower in energy and the ground-state energy at this geometry being 0.3 eV higher. Thus, we rationalize the inadequacy in predicting the low emission energy in the inability of full optimization of the dimer. Ab initio molecular dynamics simulations to account for small distortions in the structure will be performed in future work. The geometry optimization of excited states is often difficult, and the one-dimensional optimization used for the dimer system has the advantage that minima can be found for multiple excited states without the calculation of gradients. This method could be used for other dimer systems of molecules or metal clusters and is likely to work best when the cluster structure is quite stiff compared to the bond connecting the two clusters. The disadvantage of this method is that without a full E

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The Journal of Physical Chemistry A optimization of each state, the accuracy of the minimum obtained is not known. From the data plotted in Figures 6, the ground-state minimum occurs at a separation of 3.40 Å when the optimized geometry of the cluster is used, while as shown in Figure 7, the minimum occurs at a separation of 3.18 when the XS geometry is used. The excitation energies and oscillator strengths obtained in these dimer calculations have a correspondence to the results of the calculations on the individual clusters that were shown in Figure 3, with two excited states for each excited state found in the monomer calculations; that is, each cluster in the dimer contributes slightly perturbed versions of the excited states from the isolated cluster. As seen in Figure 4, while using a Gaussian line width of 0.2 eV for the excited states showed good agreement with the excitation spectrum, it is not likely to closely simulate the measured absorption spectrum. The measured absorption spectrum indicates that the higher-energy excited states have a larger line width, so we used line widths of 0.37, 0.43, and 0.53 eV for the first three excited states and a line width of 0.8 eV for all the other calculated excited states. The results are shown in Figure 8 for the B3LYP and PBE0 functionals with the two geometries. The B3LYP//XS calculation has the best agreement with experiment, while the PBE0 results are too blue-shifted.

δf 0(E1 + E2) =

8π 4 E1E2g (E1 + E2)|Sf 0(u1, u 2)|2 (ch)2

where E1 and E2 are the photon energies, c is the speed of light, h is Planck’s constant, and g is a linewidth function, for which we choose a Gaussian form. The TPA probability can be evaluated by a sum-over-states expression |Sf 0(u1, u 2)|2 ⎡ (u1·μ )(μ ·u 2) (u 2 ·μi0 )(μfi ·u1) ⎤ i0 fi ⎥ + = ∑⎢ Ei − E2 + i Γi ⎥⎦ ⎢ E − E1 + i Γi i ⎣ i N

2

where E1 and E2 are the energies of the two photons with unit polarization vectors u1 and u2, respectively. The transition dipole moments are given by μij, the state energies by Ei, and the state decay constants by Γi. The linewidth function g peaks when E1 + E2 = Ef, so when both photons have the same energy, the peak TPA cross-section is proportional to (1/(Ei /Ef − 1/2))2

This resonance enhancement factor can be large if an intermediate state i has an energy close to half the energy of the final state f. Increasing the resonance enhancement factor is a method used in the design of TPA materials with large cross sections; otherwise, the TPA cross-section is increased through the increase of the product of the two relevant transition dipole moments, known as intrinsic enhancement. Most attempts to design TPA materials have relied on intrinsic enhancement through materials with a large polarizability, such as π-electron dense aromatic systems, or electron donor−acceptor systems. The design of TPA materials based on resonance enhancement is more challenging due to the difficulty in accurately predicting state energies. The TPA probability |Sf 0(μ1,μ2)|2 can be obtained most efficiently from a single residue quadratic response calculation, as we have described previously,42 avoiding a full sum-overstates evaluation. The TPA spectra, calculated for the silver sulfide at two different geometries of the monomer and for one geometry of the dimer, are shown in Figure 9. When the optimized geometry is used for the monomer, two peaks are predicted, the first at a transition energy of 3.37 eV with a peak TPA cross-section of 116 GM and the second at 4.04 eV with a peak TPA of 133 GM. This fitted spectrum was obtained from the TPA transitions to 30 excited states in the range of 3.00−4.13 eV. When the XS geometry is used, the calculated resonances are closer in energy, resulting in a single broad peak centered at 3.75 eV with a peak TPA cross-section of 68 GM. These data were obtained from 24 excited states in the range of 3.13−4.10 eV. While the calculated TPA peaks are rather modest, these transitions are far from any intermediate-state resonance enhancement. The TPA spectrum was also calculated for the dimer using the XS geometry. The fitted spectrum is similar to that from the corresponding monomer, although the excited-state energies and TPA probabilities are significantly different. This TPA spectrum has a peak at 3.59 eV of 58 GM, obtained from 20 excited states in the range of 3.12−3.63 eV. Accurate calculation of the TPA for this system may require the inclusion of additional states that extend past 4 eV, as with the monomer.

Figure 8. Calculated OPA for the Ag12(SR)6(CF3CO2)6(CH3CN)6 cluster’s dimer. The two upper panels use the XS geometry with the separation r = 3.18 Å, while the next two panels use the structure from the optimized geometry with r = 3.40 Å. The measured solid-state room-temperature absorption spectrum is given in the bottom panel.

Clusters of gold18,19,42 or silver36,43 have shown the potential for large TPA and for control of the absorption properties through varying the internal structure and the ligands. As we have mentioned previously,36,42 the residue of the TDDFT quadratic response is proportional to the TPA probability and thus provides for an efficient method to calculate the TPA cross-section F

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AUTHOR INFORMATION

Corresponding Authors

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

Paul N. Day: 0000-0002-6333-6359 Notes

The authors declare no competing financial interest.

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

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Figure 9. TPA calculated for the Ag12(SR)6(CF3CO2)6(CH3CN)6 monomer and dimer using B3LYP-D3. The data from the monomer using the XS geometry are given by the dashed blue line, while the data obtained from the B3LYP-D3 optimized structure are given by the dotted green line. The TPA calculated for the dimer using the XSderived geometry is given by the solid red line. A Gaussian line width function with fwhm = 0.5 eV was used.

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4. CONCLUSIONS Linear and quadratic response TDDFT have been used to study the optical properties of a silver cluster that exhibits dual fluorescence in the solid state but just single fluorescence in solution. The absorption and emission energies calculated for the cluster dissolved in chloroform using TD-B3LYP:PCM are in good agreement with the experiment. Since the crystal structure indicates that the naphthalene ligands of adjacent clusters are π-bonded, excimer formation was proposed as the mechanism leading to dual fluorescence, analogous to the excimer formation and red-shifted fluorescence in naphthalene. Attempts at full-geometry optimization of the cluster dimer did not lead to a reasonable structure, and one-dimensional optimizations of this system in the ground and excited states yield smaller emission red shifts than measured, but the TDDFT calculations on both the monomer and the dimer were shown to have reasonable agreement with the experimental absorption and excitation spectra. The first absorption peak in the solid-state excitation spectrum measured at 77 K is in good agreement with that calculated using TD-B3LYP on either the monomer or the dimer when a Gaussian line width of 0.2 eV is used. The experimental linear absorption spectrum indicated that the excitations to higher-energy states have a larger line width, and by making this adjustment, we obtained good agreement with the absorption spectrum as well. The TPA cross-section calculated for the isolated cluster is modest, but the lack of any nearby resonance enhancement could be advantageous in some applications. Because of a large density of states for the dimer system, the TPA spectrum for this system could be obtained only over a small frequency range.

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DOI: 10.1021/acs.jpca.6b10868 J. Phys. Chem. A XXXX, XXX, XXX−XXX

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