Article pubs.acs.org/JPCC
At What Size Do Neutral Gold Clusters Turn Three-Dimensional? Mikael P. Johansson,*,†,¶ Ingolf Warnke,‡,§ Alexander Le,‡ and Filipp Furche*,‡ †
Laboratory for Instruction in Swedish, Department of Chemistry, University of Helsinki, FI-00014 Helsinki, Finland Department of Chemistry, University of California, Irvine, 1102 Natural Sciences II, Irvine, California 92697-2025, United States
‡
S Supporting Information *
ABSTRACT: Recent experiments and calculations have established the transition from two-dimensional (2D) to three-dimensional (3D) structures at a cluster size of 8 and 12 atoms for gold cluster cations and anions. For neutral gold clusters, however, experimental data are scarce, and existing theoretical studies disagree on the 2D−3D crossover point. We present the results of global structure optimizations of neutral gold clusters Aun for n = 9−13 using a genetic algorithm and meta-generalized density functional theory. The relative energies of the lowest-lying isomers are computed using the revTPSS functional and the random phase approximation (RPA). Thermal, scalar relativistic, and spin−orbit effects are included, and basis set extrapolations are performed for the RPA calculations. For the 2D−3D transition of gold cluster cations and anions, this methodology yields near-quantitative agreement with cross section and electron diffraction measurements. For neutral gold clusters, the 2D and 3D structures are predicted to be almost isoenergetic at n = 11 gold atoms, while clusters with n > 11 are manifestly 3D. Thus, neutral gold clusters turn 3D at an unusually large size of 11 gold atoms.
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[CCSD(T)] method, Choi and co-workers38 as well as Götz et al.39 arrived at the same conclusion. Using second-order Møller− Plesset perturbation theory (MP2), David and co-workers,40 on the other hand, assigned a compact 3D structure as the minimum structure of Au10. Recently, Zanti and Peeters used hybrid DFT to study the bonding patterns in larger gold clusters based on Au2 units; in their work, Au10 was also suggested to be 3D.41 The hitherto most accurate calculations on neutral clusters may be CCSD(T) calculations showing Au8 to be still planar.42−45 The wide variation in previous theoretical results reflects the formidable challenge that small gold clusters present to electronic structure theory.22,23,46 Gold clusters are small-gap systems, and Hartree−Fock-based single-reference methods such as MP2 or, to a lesser extent, CCSD(T) are suspect, while multireference calculations including dynamic correlation effects are hardly affordable. Semilocal density functional results show a strong dependence of the 2D−3D crossover point on the functional.47 Mid- and long-range dispersion interactions may be important for a precise determination of the crossover point but are usually not accurately captured by semilocal functionals. Moreover, scalar-relativistic as well as spin−orbit (SO) effects must be included along with thermal corrections,22,48 and careful global searches for structures are necessary to identify all relevant isomers. Here, we attempt a definitive prediction of the 2D−3D crossover point of neutral gold clusters. We use the revTPSS
INTRODUCTION Despite the chemical inertness of bulk gold,1 smaller gold clusters and nanoparticles exhibit surprising activity as heterogeneous catalysts.2−16 Small gold cluster cations and anions are known to have planar two-dimensional (2D) structures up to a size of 817 and 1218−23 atoms, respectively, and dimensionality has been proposed as a crucial variable for the catalytic activity of these clusters.24 However, the experimental characterization of neutral gold clusters has been hampered by the difficulty of separating and probing clusters of different size without reliance on their mass/ charge ratio.25 For a long time, Au226 and Au327 remained the only experimentally characterized neutral gold structures. In 2008, Fielicke and co-workers28 reported the gas-phase structures of neutral Au7, Au19, and Au20 clusters tagged with krypton atoms. In a follow-up study,29 it was found that for clusters smaller than Au7, Kr binds to specific gold atoms, perturbing the vibrational spectra and possibly the structures. Recently, Au8 was assigned a planar structure.30 Thus, while electronic absorption spectra for neutral clusters Aun with n = 1− 5 and 7−9 coordinated with neon31 and for n = 7, 9, 11, and 13 coordinated with xenon32 have been measured, in the expected 2D−3D crossover region, only Au7 and Au8 have been distinctly characterized and shown to be planar.28,30 The exact 2D−3D crossover point remains unknown. Existing theoretical studies of neutral gold clusters are in stark disagreement, with predictions of the smallest three-dimensional ground-state structure ranging from 733,34 to 1435,36 atoms. Employing semilocal density functional theory (DFT), BonaĉićKoutecký and co-workers37 studied gold clusters of up to ten atoms and found the ground states of all to be planar. Employing the coupled cluster singles doubles and perturbative triples © XXXX American Chemical Society
Special Issue: John C. Hemminger Festschrift Received: June 10, 2014 Revised: July 24, 2014
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dx.doi.org/10.1021/jp505776d | J. Phys. Chem. C XXXX, XXX, XXX−XXX
The Journal of Physical Chemistry C
Article
(aug-cc-pwCV5Z-PP)76 along with the Stuttgart−Köln 2005 scalar-relativistic ECP.77 The RPA correlation energy was extrapolated to the complete basis set (CBS) limit using the physically motivated two-point scheme by Halkier and coworkers.78 For example, a two-point extrapolation using quadruple and quintuple-ζ energies is denoted aug-cc-pwCV[Q,5]Z. The universal RI-J61 auxiliary basis set was used for the TPSS calculations, while the exact exchange energy calculations were performed without density fitting techniques. Auxiliary basis sets for the RI-RPA correlation energy calculations50 were taken from the Turbomole V6.5 auxiliary basis set library, and 100 frequency integration points were used. SO coupling was not computed at the RPA level; instead, the RPA results reported include the SO interactions computed at the revTPSS level. The single-point ZORA calculations were performed with the ADF program package;79,80 all other calculations were performed with TURBOMOLE.81 Default convergence parameters and thresholds were used, with the following, tighter exceptions: The initial geometry optimization was performed using the fine m4 numerical grid82 and a gradient convergence criterion of 10−4 au; geometries for the vibrational analysis were obtained with the ultrafine m5 grid and a gradient convergence criterion of 10−5 au, along with a self-consistent field (SCF) energy convergence criterion of 10−8 au; and RPA calculations were performed with the ultrafine m5 grid, with energies and density matrices converged to 10−7 au. The ADF calculations were performed with a SCF convergence criterion of 10−7, using an integration accuracy parameter of 7.5, with diffuse fit functions added with the AddDiffuseFit keyword; SCF convergence benefited greatly from the LISTi method.83 Figures were prepared with Molden,84 Avogadro,85,86 and PyMOL.87 Basis Set Convergence and Importance of Core− Valence Correlation. The basis set dependence of the relative energies between planar and compact gold structures was previously explored for standard semilocal DFT methods,21 with the conclusion that at least quadruple-ζ quality basis sets are required for reliable energetics, both for scalar-relativistic energies and the SO contribution. In this section, we explore the basis set convergence and importance of semicore correlation for the RPA correlation energy. The basis set convergence of RPA correlation energies is known to be slow,88 similar to correlated wave function methods, because high angular momentum functions are necessary for describing the electron−electron Coulomb cusp.78,89 Further, when studying species of different dimensionality and spatial extent, intramolecular basis set superposition error (BSSE) may be a problem,90−92 increasing the demands on basis set size. The favorable scaling of RPA compared to wave function methods of comparable accuracy allows us to explore the basis set behavior near to the CBS limit, at least for relative energies between isomers of gold clusters. With very similar dependence of basis set size on the ability to describe the correlation energy, the results of this study should be relevant to correlated wave function methods as well. The individual energies of the different clusters and basis sets considered are supplied as Supporting Information; here, we summarize the main findings. Triple-ζ energy differences exhibit large, unsystematic errors of up to 10 kJ/mol. The errors of the quadruple-ζ basis sets are notably smaller, and at the quintuple-ζ level, the relative energies can be considered practically converged. For the 5Z basis set, diffuse functions were found to have little effect on the relative correlation energies. Due to the exceptionally poor performance of the triple-ζ basis sets, [T,Q]Z extrapolation yields relative
meta-generalized gradient approximation functional, large basis sets, and structures generated by a genetic algorithm (GA), and we include relativistic and thermal corrections. We calibrate our methods by the known crossover points of cationic and anionic gold clusters and by comparison to random phase approximation (RPA) calculations. RPA methods49 have recently become available for larger molecules50 and enable theoretical calibration of semilocal DFT results for small-gap compounds. RPA is known to work for bulk metals51 and molecular transition-metal compounds,52 includes dispersion,53,54 is nonempirical, and is thus highly predictive.
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COMPUTATIONAL DETAILS General. A global search of the potential energy surfaces of clusters with 9−13 gold atoms was performed using a GA55,56 based on the formulation by Deaven and Ho57 as implemented in Turbomole.58 The potential energy surfaces were evaluated at the TPSS (meta-GGA) level59 with polarized double-ζ quality def2-SVP basis sets60,61 and the Stuttgart−Köln 1990 effective core potential (ECP) with 19 valence electrons.62 The resolution of the identity approximation for the Coulomb energy (RI-J method) was used throughout.63 The GA optimizations were assumed to be converged if no new minimum structures occurred among the lowest-energy isomers during a certain number of generations. Another indicator of convergence was the presence of both enantiomers for chiral structures. For each cluster size, on the order of 1000 geometry optimizations were necessary to converge the GA search. However, even for extensive GA searches, there is no guarantee that the global minimum is found in a finite number of steps. For completeness, simulated annealing ab initio molecular dynamics simulations64 were also performed but did not generate lower-energy structures and failed to find several of the low-energy structures obtained from the GA search. The thus-obtained low-energy structures were subsequently reoptimized at the TPSS level using the triple-ζ quality 7s5p3d1f basis set specifically devised for gold clusters.17 For Au11, this shortened bond lengths by 2−3 pm but did not qualitatively alter the relative isomer energies. All optimized structures were confirmed to be minima by analytical second derivative calculations.65,66 Zero-point vibrational energies and thermal corrections were evaluated using the rigid rotor−harmonic oscillator approximation, after adjusting any very low energy vibrations (
