Letter pubs.acs.org/JPCL
Superatomic Orbitals under Spin−Orbit Coupling De-en Jiang,*,† Michael Kühn,‡ Qing Tang,† and Florian Weigend‡,§ †
Department of Chemistry, University of California, Riverside, California 92521, United States Institut für Physikalische Chemie, Karlsruher Institut für Technologie, Kaiserstraße 12, 76131 Karlsruhe, Germany § Institut für Nanotechnologie, Karlsruher Institut für Technologie, Postfach 3640, 76021 Karlsruhe, Germany ‡
S Supporting Information *
ABSTRACT: The Au25(SR)18− cluster has been the poster child of success in applying the superatom complex concept and remains the most studied system of all of the monolayerprotected metal clusters. In this Letter, we try to solve a mystery about this cluster: the low-temperature UV−vis absorption spectrum shows double peaks below 2.0 eV while simulation by scalar relativistic time-dependent density functional theory (TDDFT) shows only one peak in this region. Using a recently implemented two-component TDDFT, we show that spin−orbit coupling (SOC) leads to those two peaks by splitting the 1P superatomic HOMO orbitals. This work highlights the importance of SOC in understanding the electronic structure and optical absorption of thiolated gold nanoclusters, which has not been realized previously.
SECTION: Physical Processes in Nanomaterials and Nanostructures
xploration of thiolate-protected gold nanoclusters exploded after the publication of the crystal structures of Au102(SR)44 and Au25(SR)18− (RS− being a thiolate group, such as PhCH2CH2S−).1−3 Accompanying the structural determination, the superatom concept is used to understand the underlying electronic structure of these clusters. 4 Au102(SR)44 and Au25(SR)18− will have 58 and 8 “free” electrons, respectively, according to the superatom concept.4,5 Au25(SR)18− consists of an inner 13-atomic icosahedral core plus 6 dimeric RS−Au−SR−Au−SR motifs (see Figure 1); the entire system is of lower symmetry. The eight free electrons fill
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up the superatomic orbitals 1S and 1P as (1S)2(1P)6. The fivefold-degenerate 1D orbitals split according to the nonicosahedral symmetry of the entire cluster, leading to a doubly degenerate LUMO and a triply degenerate LUMO+1. This superatom orbital picture of Au25(SR)18− is supposed to dictate its optical absorption. Optical absorption has been used often as a fingerprint to identify thiolated gold nanoclusters because they are easy to obtain and often very characteristic to distinguish small-sized clusters.6 Being the most studied member of the thiolated gold cluster family, one would expect that the optical absorption of Au25(SR)18− has been well understood. However, this is not the case. Even at room temperature, the optical absorption of Au25(SR)18− displays an obvious shoulder at ∼780 nm (or 1.59 eV) next to the main peak at 690 nm (or 1.80 eV).7 The 690 nm peak was assigned to the HOMO−LUMO transition, but the 780 nm shoulder was missing in the previously published simulated spectrum correlating the optical absorption to the structure of Au25(SR)18−.3 In their low-temperature experiment, Ramakrishna et al. showed beautifully how the shoulder evolves out as a distinct peak at 1.67 eV, additionally to the main peak that shifts to 1.90 eV with decreasing temperature.8 Given the symmetry of the icosahedral Au13 core and the whole cluster, one would not expect that the triply degenerate HOMO would split to such a large degree (∼0.2 eV) under the superatom picture.
Figure 1. Structure of the Au25(SR)18− cluster. Here, we use CH3− for R−. Au, orange; S, yellow; C, brown; H, white.
Received: August 18, 2014 Accepted: September 10, 2014 Published: September 10, 2014
© 2014 American Chemical Society
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One alternative hypothesis is that spin−orbit coupling (SOC) is responsible for this peak structure. It has been shown that SOC effects greatly affect the electronic structure, electron affinity, and magnetic properties of center-doped M@ Au12 icosahedral superatoms.9 In addition, as demonstrated previously for a ligand-free gold cluster, considering SOC in the excited-state calculation is important.10 While one-component, scalar relativistic effects are routinely taken into account in quantum chemical calculations, the consideration of SOC requires a two-component theory, which is computationally much more demanding. In the present work, we investigate the role of SOC on the optical absorption of Au25(SCH3)18− using the implementation of a two-component time-dependent density functional theory (2c-TDDFT) in the TURBOMOLE program suite10,11 that accounts for SOC effects on excitations of (Kramers-restricted) closed-shell systems. For convenience, the formalism behind 2c-TDDFT, which is based on the work of Wang et al.,12 is briefly summarized in the Supporting Information (SI). We computed the lowest 192 transitions (up to 2.5 eV) of Au25(SCH3)18− using the Perdew−Burke−Ernzerhof (PBE) functional of the generalized gradient approximation for electron exchange−correlation,13 which has been shown to be very successful in predicting structures and energetics of thiolated gold nanoclusters.14 Relativistic effects including SOC were accounted for by Dirac−Fock effective core potentials (dhf-ECPs).15 Basis sets optimized for these ECPs were of polarized double-ζ valence quality [dhf-SV(P)-2c].16 For reasons of computational economy R = CH3 was chosen as it has been shown that the optical absorption feature of Au25(SR)18− in the visible range does not change with −SR group.7 Figure 2 displays our simulated 2c-TDDFT spectrum in comparison with the experimental low-temperature UV−vis absorption spectrum of Au25(SR)18−. The overall agreement of the key peaks is excellent. Most importantly, the first peak at 1.67 eV, which appears as a shoulder at room temperature and becomes distinct at 78 K, is reproduced in our 2c-TDDFT simulation, indicating that SOC gives rise to this peak. Of course, the peak positions are systematically underestimated by about 0.4 eV from the simulation due to the PBE functional used. This underestimate is very common in predicted optical absorption of thiolated gold nanoclusters whenever a pure DFT functional is used,3,5,17,18 but it usually does not prevent one from understanding the optical transitions qualitatively. We believe that the overall excellent agreement between experiment and simulation in terms of relative peak positions and strengths shown in Figure 2 indicates that the 2c-TDDFT simulation qualitatively correctly captures the optical transitions of Au25(SR)18−. To further confirm our point, we simulated the optical absorption without SOC by using the one-component (that is, scalar relativistic) TDDFT (Figure 3), as commonly practiced when simulating these clusters’ optical absorption spectra. Without SOC, there is only one peak (α) below 1.8 at 1.38 eV. This peak has been attributed to the HOMO−LUMO transition of the Au25(SR)18− cluster. The frontier orbitals are in line with the superatom concept, as shown in Figure 4. The HOMO is of P character, while the twofold-degenerate LUMO and threefold-degenerate LUMO+1 form a D orbital. After taking into account SOC, the α peak splits into two, α1 and α2. The energy difference is about 0.20 eV, readily resolvable by
Figure 2. Experimental (top)8 and simulated (bottom) optical absorption spectra of Au25(SR)18−. The experimental spectrum is at 78 K for R = C6H13; the simulated spectrum is for R = CH3 with a 0.05 eV broadening. 2c-TDDFT/PBE is used for the simulation.
Figure 3. Simulated optical absorption spectra of Au25(SCH3)18− with and without SOC.
optical absorption, and in excellent agreement with the observed experimental splitting (0.23 eV from Figure 2). The splitting of the HOMO−LUMO transition into α1 and α2 peaks can be traced back to the energetic splitting of the HOMO upon including SOC. The HOMO 1P orbital splits into an energetically higher-lying doubly degenerate and a energetically lower-lying nondegenerate component (Figure 5), separated by about 0.20 eV. This is similar to the well-known splitting of atomic p orbitals into a doubly degenerate p3/2 and a nondegenerate p1/2 component under the influence of SOC.19 (We use “1P3/2” and “1P1/2” to indicate this analogy in Figure 5.) We found that the splitting of the LUMO under SOC is 3287
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those of the experiment by about 0.4 eV (Figure 2). One way to improve the agreement is to implement the 2c-TDDFT for hybrid functionals such as B3LYP, which has been shown to yield much better agreement with the experiment for optical absorption.28,37 However, such implementation is very hard to do, and even if implemented, studies of systems that are of similar size to the one examined in this Letter are probably prohibitive. A workaround approach is to estimate the difference between PBE and B3LYP by scalar-relativistic TDDFT and then shift the 2c-TDDFT/PBE spectrum by this value. The experimental low-temperature UV−vis absorption spectrum of Au25(SR)18− also showed two distinct peaks in higher-energy regions (above 3.0 eV),8 for which onecomponent TDDFT simulation showed only one peak.3 Again, SOC might be responsible for this peak splitting. However, one has to compute up to ∼400 excitations in the 2cTDDFT to reach those high energies, which would be too demanding on the memory and beyond our current computing capability. Hopefully this difficulty can be overcome in the future by a large-memory computer. In summary, we have applied the recently implemented 2cTDDFT method to study the effect of the SOC on the optical absorption of Au25(SR)18−. We found excellent agreement between our simulation and the experimental low-temperature UV−vis spectrum. More importantly, we convincingly showed that the twin peaks at 1.67 and 1.90 eV are due to the SOC, which causes the triply degenerate HOMO orbitals to split with an energy difference of about 0.20 eV. This work suggests that one should take SOC into account when simulating and discussing the optical absorption of molecule-like, monolayerprotected gold clusters and their derivatives. The SOC effect should be also important in understanding their electrochemistry and valence photoelectron spectra.
Figure 4. Calculated (one-component) frontier orbitals of Au25(SCH3)18−. The three-fold near-degenerate HOMO orbitals show p character, while the five LUMO and LUMO+1 orbitals show d character. Note that the cluster has different orientations in each of the pictures in order to show the orbital character more clearly.
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ASSOCIATED CONTENT
S Supporting Information *
The formalism behind 2c-TDDFT and detailed analysis of spinor orbitals involved in the excitations giving rise to the α1 and α2 peaks. This material is available free of charge via the Internet at http://pubs.acs.org.
Figure 5. Energy diagram to show (a) splitting of the triply degenerate superatomic 1P orbitals due to SOC and (b) the effect on the optical absorption (see Figure 3 for α1 and α2 peaks).
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small (about 0.03 eV). Hence, α1 and α2 peaks correspond to the transitions from 1P3/2 and 1P1/2 to the LUMO, respectively. Detailed analysis of spinor orbitals involved in the excitations giving rise to the α1 and α2 peaks is shown in the SI, fully confirming the simple picture shown in Figure 5. The dramatic role of SOC in the electronic structure of Au 25 (SR) 18 − has many implications. This is because Au25(SR)18− is the most widely studied cluster of the thiolated gold family. Doping,20−31 tuning R groups,32−34 and changing S to Se35,36 have all been explored to obtain Au25 derivatives. The immediate implication of the present work is that one should take into account SOC when trying to correlate their optical absorption to the electronic structure. Further, SOC should be also invoked for discussing the electronic structure of other clusters of similar size. There are many examples, such as Au15(SR)13,6,37 Au18(SR)14,6 Au20(SR)16,38 Au22(SR)18,39 Au23(SR)16,40 Au36(SR)24,41 and Au38(SR)24,42 to list a few. We plan to examine them in the near future. Of course, we are fully aware that the absolute energies of the simulated 2c-TDDFT absorption peaks still underestimate
AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS D.J. and Q.T. are supported by the University of California, Riverside startup fund. We thank Prof. G. Ramakrishna for providing us with the data file of the experimental absorption spectrum. M.K. is funded by the Carl Zeiss Foundation and also thanks TURBOMOLE GmbH for financial support.
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