Nonlocal Density Functional Theory Studies of Hydrogen Activation by

Article pubs.acs.org/JPCC

Simulating Gold’s Structure-Dependent Reactivity: Nonlocal Density Functional Theory Studies of Hydrogen Activation by Gold Clusters, Nanowires, and Surfaces John J. Determan,† Salvador Moncho,‡ Edward N. Brothers,‡ and Benjamin G. Janesko*,† †

Department of Chemistry, Texas Christian University, Fort Worth, Texas 76129, United States Department of Chemistry, Texas A&M at Qatar, Education City, Doha Qatar

‡

S Supporting Information *

ABSTRACT: Gold’s structure-dependent catalytic activity motivates the study of reactions on a range of gold nanostructures. Electronic structure methods used to model gold catalysis should be capable of treating atoms, clusters, nanostructures, and surfaces on an equal theoretical footing. We extend our previous density functional theory (DFT) studies of a model reaction, H2 adsorption and dissociation on unsupported Au3 clusters [J. Phys. Chem. C 2013, 117, 7487], to larger clusters, quasi-one-dimensional nanowires and nanoribbons, and surfaces. We focus on trends in DFT predictions made using various approximate exchange-correlation functionals. Most functionals predict qualitatively reasonable trends, i.e., decreasing adsorption energies and increasing dissociation barriers with increasing Au coordination number. However, significant quantitative differences motivate continued exploration of methods beyond the generalized gradient approximation.

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Monte Carlo approximations can treat periodic surfaces30 but are also rather expensive. Kohn−Sham density functional theory (DFT) is widely applied to metal clusters, nanowires, and surfaces.31−33 DFT calculations’ accuracy and computational expense largely depend on the approximation used for EXC[ρ], the formally exact exchange-correlation (XC) functional of the electron density ρ(r). EXC[ρ] incorporates all many-body effects.34 However, selecting the appropriate level of approximation is nontrivial. Generalized Gradient Approximations in DFT. Currently, most DFT calculations on metal surfaces use generalized gradient approximations (GGAs) to the XC functional. GGAs model the XC energy density at point r as a function of ρ(r) and its gradient. References 31−33 review GGA applications to surface chemistry. References 35 and 36 provide recent benchmarks of GGAs for bulk, surface, and nanoparticle properties. GGAs have been extensively applied to gold catalysis. A few representative studies relevant to the present work are discussed here. Several investigators have explored the structures and catalytic properties of isolated gold nanoparticles, nanoribbons, and surfaces. Häberlen and co-workers used the local density approximation and various GGAs to model high-symmetry bare gold clusters. They found that bond lengths, cohesive energies, ionization potentials, and electron affinities converged rather

INTRODUCTION Gold Catalysis. Chemical reactions at surfaces and interfaces are central to a wide range of processes including heterogeneous catalysis.1 Reactions on gold nanoparticles are of particular recent interest.2,3 While low-index gold surfaces are rather unreactive,4 supported and unsupported5−8 gold nanoparticles catalyze a range of reactions. These include low-temperature oxidation of hydrogen,9 CO,10,11 and other organics,12−15 as well as hydrogenation.16 Small gold clusters are also susceptible to deactivation by sintering and/or poisoning.17 Gold nanowires occupy a size regime intermediate between small nanoparticles and low-index surfaces. Proposed gold nanowires include the nanotubes and nanoribbons considered in this work (Figure 1). Supported gold nanowires have been shown experimentally to catalyze the water-gas shift reaction18,19 and to have reduced susceptibility to sintering.20 Nanowires can also provide lower barriers to O2 dissociation, possibly through reduced occupancy of the gold dz2 orbitals.21−24 Gold bilayers also show CO oxidation activity.25,26 Density Functional Theory Simulations of Catalysis. Electronic structure simulations of reactions on surfaces provide insights into catalysis, from fundamental models of reactivity27 to predictions of new catalysts.28 Modeling gold’s structuredependent catalytic activity requires electronic structure approximations that can treat gold clusters, nanowires, and surfaces on an equal theoretical footing. This task is challenging for existing ab initio methods. Coupled cluster approximations can treat 1−10 gold atoms,29 but their steep computational scaling is problematic for realistic metal surfaces. Quantum © 2014 American Chemical Society

Received: March 14, 2014 Revised: June 24, 2014 Published: July 10, 2014 15693

dx.doi.org/10.1021/jp5025922 | J. Phys. Chem. C 2014, 118, 15693−15704

The Journal of Physical Chemistry C

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supported gold clusters.11,56,57 Boronat and co-workers used PW91 calculations to suggest that defects stabilize two-layer clusters with several low-coordinate gold atoms.58 Rashkeev and co-workers used PBE to model TiO2-supported Au and Pt nanoparticles, highlighting the relation between CO or O2 binding and Au−Au bond weakening.59 Other investigators have also explored the role of defects and charging of supported clusters.60−62 Beyond the GGA. GGAs tend to over-delocalize electrons and overstabilize the stretched bonds of transition states. The resulting underestimates of reaction barriers are well-known in computational chemistry.63,64 Reference 65 reviews studies suggesting that these underestimations carry over to surfaces. In particular, benchmark studies of small gold clusters indicate that GGAs’ limitations persist for gold catalysts. GGAs overestimate accurate CCSD(T) benchmarks for O2 binding to neutral and anionic Au2 and Au3,66 underestimate the barrier to H2 dissociation on these clusters,67 potentially overstabilize planar Au8,68,69 and overestimate the rate of Au3-catalyzed H2 oxidation.70,71 In much of computational chemistry, GGAs have been superseded by hybrid functionals incorporating a fraction of nonlocal exact exchange Ex = −(1/2)∫ dr1 ∫ dr2 |γ(r1,r2)|2/|r1− r2|. Here γ(r1,r2) = Σi φi*(r1) φi(r2) is the nonlocal one-particle density matrix of the noninteracting Kohn−Sham reference system, and {φi(r)} are the occupied Kohn−Sham orbitals. (Both quantities are implicit density functionals.) Exact exchange admixture tunes GGAs’ over-delocalization, improving predictions of a broad range of properties. In particular, refs 67 and 72 showed that hybrids improve on GGAs for reactions of H2 on small Au clusters. Screened hybrid functionals73,74 remove the long-range piece of exact exchange, which can be formally75,76 and computationally77 problematic in metals. Empirical metaGGAs incorporating the noninteracting kinetic energy density78 also show promise for surface chemistry.79−83 Several recent studies suggest that screened hybrids and empirical meta-GGAs show promise for accurately treating metal clusters, nanowires, and surfaces on an equal theoretical footing.81−86 Our recent benchmarks indicated that the empirical M06L78 meta-GGA and the HSE0674,87,88 and HISS89,90 screened hybrids significantly improve GGAs’ systematic underestimate of reaction barriers for H2 on Au3.72 We also find that the mPW2PLYP “double hybrid” including a fraction of Görling−Levy second-order perturbative correlation reproduces CCSD(T) benchmarks at reduced cost. Our other studies of noncatalytic reactions, including ammonia dissociation on Si(001)65 and adatom diffusion on graphene,91 suggest that meta-GGAs and screened hybrids may systematically improve GGAs’ underestimated reaction barriers on surfaces. Present Work: GGAs and Beyond for Gold Clusters, Nanowires, and Surfaces. This work extends ref 72 by exploring H2 adsorption and dissociation on a suite of representative unsupported neutral gold clusters, nanowires, and surfaces. We report new CCSD(T) and mPW2PLYP benchmarks for larger Au clusters and report trends in the predictions of representative GGAs, meta-GGAs, and screened hybrids for H2 on Au clusters, nanowires (nanotubes and nanoribbons), and pristine and defective Au(111) surfaces. The results provide insights into gold’s structure-dependent reactivity and further illustrate the potential of beyond-GGA methods for heterogeneous catalysis.

Figure 1. Calculated initial adsorption geometries of H2 on Au clusters, nanoribbons, nanotubes, and defective and pristine Au(111) surfaces.

smoothly with cluster size.37 Hammer and Nørskov used GGA calculations to rationalize unfavorable H2 dissociation on pristine gold surfaces.23 Lopez and Nørskov used the RPBE38 GGA to model CO oxidation by an isolated Au10 nanoparticle, finding that O2 dissociation and subsequent oxidation were facile.39 Barrio and co-workers used the RPBE, Perdew−Wang 1991 (PW91)40, and Becke 88 (BPW91)41 GGAs to model H2 dissociation over Au(100) and Au(111) surfaces and finite Au clusters. They concluded that H2 dissociation required cooperative effects of several Au atoms.42 A subsequent PW91 study by Corma and co-workers found instead that a single lowcoordinate Au atom suffices to dissociate H2.43 Roldán and coworkers later used PW91 to model O2 dissociation over Au clusters up to Au225.. The authors found that O2 dissociation was dominated by low-coordinate Au atoms and that charging effects were reduced with increasing cluster size.44−46 Kang and coworkers performed a PW91 study of H2 dissociation over several small Au clusters.47 Ghebriel and Kshirsagar performed PW91 studies comparing H2 and H2S on small Au clusters.48 Hernández and co-workers used RPBE and PW91 to model the high catalytic activity of Au bilayers.49 An and co-workers used PBE to model CO oxidation over Au nanotubes and pointed to the role of undercoordinated surface atoms in catalysis.50 Li and co-workers used GGA calculations to explore the relative stability of gold nanowires and nanoparticles.51 Flores and co-workers used the PW91 GGA to find low barriers to H2 dissociation on linear Au chains, and higher barriers on planar Au nanostructures.52,53 Chen and co-workers considered the O2 dissociation activity of several gold nanowires.24 Moskaleva and co-workers used the kinked Au(321) surface as a model for catalytic activity of unsupported nanoporous gold.54 Several investigators have also explored catalysis by supported gold clusters. Landman and co-workers used the Perdew− Burke−Ernzerhof GGA (PBE, ref 55) to conclude that surface charging due to defects is important to catalytic activity of 15694



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COMPUTATIONAL METHODS Overview of Tested Systems and Approximations. We treat finite Au clusters, one-dimensionally periodic Au nanowires, and pristine and defective Au(111) surfaces modeled as twodimensional periodic slabs. CCSD(T)/CBS-extrapolated benchmarks for Au clusters are compared to generalized Kohn−Sham DFT using a broad range of approximate XC functionals. We focus on six representative functionals: the PBE55 and revPBE92 GGAs, the highly parametrized M06-L78 meta-GGA, the HISS,89 and HSE0674,93−95 screened hybrids and the mPW2PLYP double hybrid.96 Some systems are also tested with the local density approximation (LDA) with Vosko−Wilk−Nusair correlation;97 the BLYP,41,98B97D,99 PBE, PW91,40 and revPBE GGAs; the nonempirical meta-GGA TPSS;100 and the global and long-range-corrected hybrids B3LYP,101 BHandHLYP,102 M06,103 M062X,103 mPW1PW91,104 PBE0,105,106 TSSPh,100 LC-ωPBE,107 ωB97,108 ωB97X,108 and ωB97XD.109 Basis Sets and Core Potentials. Unless noted otherwise, all calculations in this work use atom-centered Gaussian basis sets. Some test PBE calculations, and all geometry optimizations of periodic systems use plane-wave basis sets. All calculations combine a nonrelativistic Hamiltonian with relativistic effective potentials for the core electrons. Atom-Centered Basis Set Calculations. Calculations with atom-centered basis sets use the development version of the GAUSSIAN suite of programs.110 We use several choices of atom-centered basis set. Coupled cluster benchmarks and some DFT calculations use the def2NZVPP (N = D, T, Q; denoted “defNZ”) basis sets and effective core potentials.111 Other calculations use the LANL08 basis set and core potential on Au112,113 and the recommended 6-311G** basis set on H. Both core potentials were recently shown to give results reasonably close to all-electron scalar relativistic calculations.114 Calculations using atom-centered basis sets on periodic systems use a modified basis “modLANL08”. In this basis set, all basis function exponents are set to ≥0.11 au for Au nanotubes and nanoribbons and ≥0.12 au for pristine and defective Au(111).94 Atomcentered basis sets are taken from the EMSL basis set exchange.115,116 Technical details of the GAUSSIAN calculations are as follows. Open-shell systems are treated spin unrestricted. No effort is made to break spin symmetry in closed-shell systems. Calculations use “tight” self-consistent field (SCF) convergence and an “UltraFine” pruned numerical integration grid with 99 radial shells per atom and 590 angular points per shell. Some calculations use Fermi temperature broadening in early SCF iterations.117 GAUSSIAN calculations on periodic systems evaluate one-electron and exact exchange operators using at least 100 replica cells, and evaluate Coulomb terms using a fast multipole moment algorithm.118 Periodic calculations also use Marzari−Vanderbilt smearing for Brilloun-zone integration, with initial temperature 2000 K.119 Calculations on nanotubes and nanoribbons use one-dimensional (1D) periodicity and 200 kpoints. Calculations on slabs use 2D periodicity and 512 k-points. Global and long-range-corrected hybrid calculations on periodic systems are performed non-self-consistently from PBE orbitals. Test calculations indicate that self-consistent and post-PBE reaction energies and barriers are generally within ∼0.04 eV of each other. Plane-Wave Basis Set Calculations. Calculations with plane-wave basis sets use the Quantum ESPRESSO program.120 Calculations use kinetic energy cutoffs of 40 Ry for wave

functions and 250 Ry for charge density and potential. Vanderbilt ultrasoft pseudopotentials121 are taken from the Quantum ESPRESSO page. Calculations use Marzari−Vanderbilt cold smearing with a Gaussian spreading of 0.02 Ry for the Brillouinzone integration,119 an 8 × 1 × 1 k-point mesh for nanowires, and an 8 × 8 × 1 k-point mesh for slabs. Periodic replicas are separated by >10 Å. CCSD(T) Benchmarks. Complete basis set extrapolated CCSD(T) benchmarks, evaluated in atom-centered basis sets, are obtained as in previous work.72 The total energy is approximated as ECCSD(T)CBS = EHFCBS + EΔMP2CBS + ECCSD(T)defTZ − EMP2defTZ. The complete basis Hartree−Fock energy EHFCBS is extrapolated from HF/defTZ and HF/defQZ calculations via EHFn = EHFCBS + Ae−Bn, where EHFCBS and A are fitted to the calculations, n is the cardinal of the basis (3 for TZ, 4 for QZ),122 and parameter B = 1.63 is optimized for diatomic transition metal systems.123 The complete basis MP2 correlation energy EΔMP2CBS is extrapolated from MP2/def2TZVPP and MP2/ def2QZVPP correlation energies via the equation Ecorrn = EcorrCBS + An−3, where EcorrCBS and A are fitted to the calculations and n is the cardinal of the basis (3 for TZ, 4 for QZ).122,124 The same extrapolation scheme is used in test calculations extrapolating the CCSD(T) correlation energy itself. Geometry Optimizations. All calculations use geometries obtained with the PBE GGA, permitting “apples to apples” comparisons of energies evaluated at given geometries. Cluster geometries are optimized with the LANL08 atom-centered basis set described previously. Transition states are located with the quadratic synchronous transit approach125,126 and confirmed to have one imaginary vibrational frequency. Transition states are tested by displacing toward reactants and products along the vibrational mode with imaginary frequency and reoptimizing. This occasionally yielded a different weakly bound adsorption geometry, with the H−H bond axis tilted with respect to the cluster edge. Differences among these physisorbed geometries are beyond the scope of the present work. Nanotube, nanowire, and slab geometries are optimized with plane-wave basis sets, except for a few test cases. Transition states are located with the climbing image nudged elastic band algorithm with seven images for the transition-state location.127 Au(111) is treated as a four-layer slab with 12 Au atoms per unit cell, and with the in-plane lattice vector constrained to the 4.1787 Å computed for bulk gold. This is modestly larger than the experimental value 4.08 Å,128 consistent with PBE’s known overestimate of lattice parameters. The defective Au(111) is obtained by removing alternating rows of atoms from the top layer of Au(111), similar to ref 43. Calculations on defective Au(111) optimize the top two layers of Au atoms. Calculations on pristine Au(111) freeze all Au atoms at the bulk position and constrain adsorbed H2 to 3-fold sites. Definitions of Energy Differences. Most results are described in terms of energy differences. The cohesive energy per atom of isolated gold systems is ΔEcoh = (nE(Au) − E(Aun))/ n, where E(Aun) is the total energy of the bare gold system. The adsorption energy of molecular H2 is ΔEads = E(Aun) + E(H2) − E(H2−Aun), where E(H2−Aun) is the total energy of the initial adsorption geometry. The reaction barrier and reaction energy for H2 dissociation relative to the adsorbed state are ΔE⧧ = E(TS) − E(H2−Aun) and ΔEdis = E(H−Aun−H) − E(H2−Aun). The barrier and energy for H2 desorption are ΔEdes⧧ = E(TS) − E(H−Aun−H) and ΔEdes = E(Aun) + E(H2) − E(H−Aun−H). ΔEcoh > 0 for stable structures. ΔEads > 0 denotes energetically favorable adsorption. ΔEdis > 0 denotes energetically unfavorable 15695



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Figure 2. Calculated ΔEcoh (a), ΔEads (b), ΔE⧧ (c), ΔEdis (d), and ΔEdes⧧ (e), for the systems of Figure 1

illustrates the calculated initial H2 adsorption geometry on clusters Au3−Au16, low- and high-coverage gold nanoribbons RL and RH, low- and high-coverage gold nanotubes T1L and T1H, gold nanotube T2, and the Au(111) surfaces. Calculated transition-state and product geometries are shown in Supporting Information Figures SI1 and SI2. The Au3 cluster geometry is based on refs 67 and 72. Au4 and Au5 structures and

dissociation. Calculated geometries and energies of all species are reported as Supporting Information. Figures use color coding: Au, yellow; H, white. Bond orders are drawn as a guide to the eye.

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RESULTS AND DISCUSSION Geometries. Figure 1 shows the calculated geometries of all tested gold clusters, nanowires, and surfaces. The figure 15696



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Table 1. Calculated Adsorption Energy ΔEads, Dissociation Energy ΔEdis, and Dissociation Barrier ΔE⧧ for H2 Dissociation on Au3 and Au4 Clusters (eV) Au3 functional type GGA

meta-GGA screened hybrid global hybrid

double hybrid long-range-corrected

ab intio

Au4

XC functional

ΔEads

ΔE⧧

ΔEdis

ΔEads

ΔE⧧

ΔEdis

LDA PW91 B97D PBE revPBE BLYP M06-L TPSS HSE06 HISS M06 PBE0 M06-2X mPW1PW91 TPSSh BHandHLYP B3LYP mPW2PLYP ωB97 ωB97X ωB97X-D LC-ωPBE CCSD(T)/TZ CCSD(T)/QZ CCSD(T)/(TQ)Z CCSD(T)/CBS

1.478 0.779 0.396 0.763 0.521 0.431 0.520 0.600 0.663 0.662 0.352 0.679 0.195 0.622 0.584 0.314 0.435 0.619 0.677 0.622 0.559 0.779 0.774 0.860 0.908 0.899

0.211 0.305 0.310 0.304 0.325 0.323 0.458 0.324 0.460 0.529 0.648 0.456 0.523 0.462 0.383 0.602 0.448 0.515 0.600 0.575 0.529 0.539 0.494 0.488 0.489 0.522

−0.014 0.065 0.003 0.067 0.083 0.064 0.277 0.117 0.292 0.401 0.504 0.293 0.337 0.295 0.204 0.495 0.248 0.330 0.559 0.505 0.398 0.510 0.298 0.323 0.341 0.389

1.516 0.789 0.455 0.773 0.528 0.403 0.562 0.625 0.745 0.779 0.421 0.771 0.290 0.714 0.643 0.430 0.482 0.705 0.789 0.728 0.677 0.918 0.845 0.912 0.955 0.955

−0.022 0.112 0.194 0.082 0.104 0.142 0.465 0.114 0.139 0.177 0.457 0.139 0.348 0.167 0.134 0.269 0.183 0.165 0.414 0.639 0.974 0.113 0.118 0.110 0.108 0.109

−0.190 −0.058 −0.067 −0.082 −0.059 −0.050 0.331 −0.084 0.010 0.060 0.401 0.012 0.258 0.034 −0.048 0.150 0.023 0.019 0.358 0.567 0.871 0.026 −0.035 −0.021 −0.007 0.016

basis sets, effective core potentials, and electronic structure codes give comparable results. Reference 130 highlights some challenges in obtaining such agreement between Gaussianorbital and plane-wave calculations. Figure 2 illustrates that most of the trends of refs 67 and 72 carry over to larger and more realistic systems. PBE adsorption energies are quite close to the available mPW2PLYP values and somewhat below CCSD(T) benchmarks. The HISS screened hybrid approximately reproduces the PBE adsorption energies. HISS predicts dissociation energies close to available CCSD(T) and mPW2PLYP values. HISS and M06L also better reproduce available CCSD(T) and mPW2PLYP dissociation barriers and energies and generally tend to increase ΔE⧧ and ΔEdis relative to PBE. This is consistent with the aforementioned GGA overdelocalization. Interestingly, while M06L gives a nearly constant increase in dissocation energies and barriers over revPBE, some HISS values for larger systems (e.g., T1H and Au(111) ΔE⧧, T1H and pristine and defective Au(111) ΔEdis) are below revPBE. This result, and the corresponding increase in HISS desorption barriers, is explored in detail in the subsequent discussion. Comparison with Previous Work. Figure 2 shows that all tested methods predict chemically reasonable trends consistent with previous work. All systems but Au3−Au4 have weak H2 adsorption ΔEads. This is consistent with the weak binding to high-coordinate Au atoms seen in refs 42 and 43 and the trends for ΔEads on neutral Au3−Au5 seen in refs 47 and 48. The low dissociation barrier ΔE⧧ on Au4 and increase in ΔE⧧ for Au4− Au5 are consistent with ref 47. (We find a higher dissociation barrier on Au3.) Dissociation on the long edges of the nanoribbon-like clusters Au8−Au16 generally becomes more favorable as the cluster size increases, with an odd−even effect

chemisorption sites follow refs 47 and 48. Nanoribbons RL and RH are based on the most stable gold nanoribbon treated in ref 129. Nanotubes T1L and T1H are the filled W6-1 nanowire from ref 24. Nanotube T2 is the hollow T(6,0) nanotube from ref 24. The defective Au surface is based on the extended defects of ref 43. Planar Au10, Au13, and Au16 clusters with H2 on the long edge are chosen to be similar to an Au nanoribbon. We emphasize that we consider a single H2 adsorption geometry on representative systems, in order to explore systematic trends in reactivity across a broad range of structures. While we believe that the chosen geometries are chemically reasonable, they are not guaranteed to be the global minima or the lowest-barrier dissociation pathways. One noteworthy result in Figure 1 is that our PBE/planewave calculations predict the isolated T1 nanowire to have a “zigzag” arrangement of the central Au atoms, with Au−Au−Au angles of 119.4°. Reference 24 instead found that the central Au atoms were linear. Test PBE/modLANL08 geometry optimizations with atom-centered basis functions predict the linear structure to be a stable local minimum, but find a zigzag geometry with Au−Au−Au angles of 134.8° to be more stable. Summary of Energetic Trends. Figure 2 summarizes the results of this work, showing cohesive energies ΔEcoh, H2 adsorption energies ΔEads, dissociation barrier ΔE⧧, dissociation energies ΔEdis, and desorption barriers ΔEdes⧧ for all tested structures. Individual values are tabulated in Supporting Information. The figure reports results for DFT with four representative XC approximations: the standard PBE GGA, the revPBE GGA widely used to model surface chemistry, the M06L meta-GGA, and the HISS screened hybrid.72 CCSD(T) and mPW2PLYP results are included where available. The end of this section reports several “sanity tests” showing that the different 15697



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Figure 2b and Supporting Information Table SI2 present calculated H2 adsorption energies ΔEads. As discussed previously, these are quite small for systems larger than Au4, consistent with weak binding to high-coordinate Au atoms.42,43 PBE, M06-L, and HISS give similar results, close to available mPW2PLYP calculations. revPBE gives systematically less favorable adsorption, with energetically unfavorable adsorption ΔEads < 0 for most species. (Recall that all geometries are optimized with PBE.) This is consistent with revPBE’s reduction of PBE chemisorption energies in other systems.92 We note as an aside that the low-coordinate Au atoms present on our clusters can give other adsorption geometries with more favorable ΔEads, consistent with previous work.42,43 For example, H2 adsorbs to the low-coordinated “end” Au atoms on Au10 with PBE ΔEads 0.176 eV, significantly larger than the 0.055 eV in Supporting Information Table SI2. Parts c−e of Figure 2 present perhaps the most important results in this work. These show that the M06-L meta-GGA and HISS screened hybrid generally predict dissociation barriers ΔE⧧ and desorption barriers ΔEdes⧧ that are higher than the GGAs, and dissociation energies ΔEdis that are less negative than the GGAs. For example, HISS gives a dissociation barrier ΔE⧧ 0.2 eV higher than revPBE on Au3; ∼0.1 eV higher than revPBE on RH, RL, T1L, and T2; and comparable to revPBE on Au10−Au16, T1H, and defective Au(111). HISS gives particularly large desorption barriers ΔEdes⧧ on RH, T1H, T2, and pristine and defective Au(111). This demonstrates that the improvements to GGAs seen for smaller clusters generally transfer to larger and more realistic systems. This result is also consistent with our previous studies of noncatalytic reactions on surfaces.65,91 One notable result in Figure 2c−2e is that screened hybrids’ tendency to destabilize transition states and dissociated products is moderated by their tendency to increase binding to metallic systems. This is especially clear for the Au(111) surface: HISS predicts ΔEdis for the defective surface, and ΔE⧧ and ΔEdis for the pristine surface, that are close to PBE. It is also visible in the ΔEdis for T1H nanowire. Figure 3 illustrates this in more detail, showing the calculated potential energy surface for H 2 dissociation on defective Au(111). The HISS screened hybrids

consistent with changing cluster multiplicity. The energetically unfavorable chemisorption to Au10 and the increased chemisorption with increasing cluster size are consistent with the Au7 and Au10 clusters of ref 53. The larger Au16 cluster gives results similar to the RL and RH nanoribbons. Facile H2 activation by defective Au(111) is consistent with ref 43, which gave a GGA dissociation energy of −12.6 kcal mol−1 (−0.55 eV) comparable to our PBE ΔEdis. Our calculations on the defective Au(111) surface use a different adsorption geometry than ref 43, and give a weaker ΔEads. Au atoms at the surfaces of nanotubes T1L, T1H, and Au(111) give unfavorable H2 activation, consistent with previous work.23 CCSD(T) Benchmarks. While correlated single-reference methods are problematic for bulk metals, they can give useful insights into small metal clusters.66−68,71,72 Table 1 presents the calculated potential energy surfaces for H2 on Au3 and Au4, comparing DFT calculations to CCSD(T)/CBS-extrapolated benchmarks. Unlike refs 47, 67, and 72, we consider here only a single geometry of adsorbed and dissociated H2. The benchmarks are largely consistent with refs 67 and 72. The CCSD(T) chemisorption energy on Au3 is 0.90 eV, consistent with the increasing CCSD(T) ΔEads 0.41 eV (small basis) and 0.71 eV (large basis) of ref 67. The CCSD(T) dissociation barrier ΔE⧧ and dissociation energy ΔEdis on Au3 are 0.52 and 0.39 eV, consistent with the 0.59 and 0.41 eV large-basis CCSD(T) values of ref 67. It is worth noting the possible multireference character of these systems. CCSD(T)/defTZ calculations on these systems give T1 diagnostics131 as large as 0.03. We test whether secondorder many-body perturbation theory is appropriate for these systems by performing CCSD(T)/defQZ calculations for H2 on Au3. CBS extrapolation of the CCSD(T)/defTZ and CCSD(T)/ defQZ correlation energies yields similar results (Table 1), validating our use of MP2 in CBS extrapolations. The DFT results in Table 1 are also largely consistent with refs 67 and 72. The LDA overbinds as usual. All other tested DFT methods underestimate the CCSD(T) adsorption energy. This is qualitatively consistent with ref 67, which showed that B3LYP, RPBE, and PW91 gave ΔEads on neutral Au3 smaller than or approaching large-basis CCSD(T). All tested GGAs underestminate the CCSD(T) dissociation energy and barrier, while M06L and HISS provide more reasonable values, consistent with refs 67 and 72. As in ref 72, the mPW2PLYP double hybrid is quite close to CCSD(T). Functionals that reproduce mPW2PLYP for larger Au clusters are thus likely to be reasonably accurate. Trends among Gold Clusters, Nanowires, and Surfaces. We next discuss the chemical trends illustrated in Figure 2. Figure 2a and Supporting Information Table SI1 present calculated cohesive energies per atom ΔEcoh. All tested functionals predict that the cohesive energies increase with system size and average coordination number, consistent with ref 51. As expected, calculated ΔEcoh are below the 3.83 eV atomization energy reported experimentally for bulk gold.132 The revPBE ΔEcoh are generally smaller than PBE, consistent with revPBE’s increased stabilization of strongly varying atom-like densities. The HISS screened hybrid and mPW2PLYP double hybrid give ΔEcoh approximately between PBE and revPBE. This decrease in ΔEcoh relative to PBE is generally consistent with previous screened hybrid calculations on bulk metals.77 The HISS ΔEcoh are close to revPBE for small clusters but somewhat larger than revPBE for larger systems.

Figure 3. Calculated reaction path for H2 dissociation on the defective Au(111) surface. Results are shown for seven images from the nudged elastic band calculation. Calculated geometries are illustrated for steps 1, 4, and 6. “PBE/pw” results use a plane-wave basis set. 15698



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are close to revPBE near the transition state and closer to PBE near the product. Screened hybrids’ tendency to stabilize metallic systems is also visible in the difference between HISS and revPBE ΔEcoh in Figure 2a. It is consistent with previous reports that screened hybrids increase metal bandwidths,77 predict metal surface energies only comparable to PBE, and predict chemisorption energies that are larger (more stable) than PBE for CO on Rh(111) and Pt(111).84 It is also consistent with the large HISS desorption barriers. Overall, the results in Figure 2c− e strongly motivate continued exploration of meta-GGA and screened hybrid density functionals in studies of heterogeneous catalysis. Parts c−e of Figure 2 also give insights into the chemical reactivity of gold particles. Our results show an approximate Brønsted−Evans−Polanyi (BEP) relation,133−135 with large dissociation barriers approximately corresponding to unfavorable dissociation energies. However, Supporting Information Figure SI3 shows that this relation is to some extent obscured by the details of the individual systems. H2 dissociation tends to become more thermodynamically favorable on the “edges” of clusters and nanowires Au7−RH as the system size increases. As in previous work, the highly coordinated gold atoms on nanotubes T1L, T2L, and the Au(111) surface give relatively high dissociation barriers and relatively unfavorable dissociation energies. Systematic Errors of GGAs. Our study of graphene adatoms showed that different GGAs give a nearly linear relation between predicted adsorption energies and reaction barriers,91 such that no one GGA could simultaneously reproduce accurate adsorption energies and accurate reaction barriers. This “Procrustean dilemma” is familiar to developers of new GGAs.136 Screened exchange admixture generally increases the predicted reaction barriers of adsorbed species, illustrating how hybrids’ demonstrated utility for gas-phase reactions can carry over to surface chemistry. Figure 4 illustrates this effect for H2 on gold, showing a bivariate analysis of predicted reaction barriers versus adsorption or reaction energies. (Similar analyses may be found in refs 91 and 137.) Figure 4a shows that the LDA and all tested GGAs predict a nearly linear relation between ΔEads and ΔE⧧ for H2 on Au3. GGAs that predict stable chemisorption and a negative ΔEads predict low dissociation barriers. GGAs that predict weak chemisorption predict somewhat higher dissociation barriers. Critically, no tested GGA can simultaneously reproduce both the accurate adsorption energy and the accurate dissociation barrier. In contrast, many of the tested beyond-GGA functionals approach the CCSD(T) reference. While the figure does not indicate a single “optimal” functional, the overall improvements available beyond the GGA are clear. Parts b and c of Figure 4 illustrate that these systematic improvements carry over to larger and more realistic systems. The panels illustrate that the tested GGAs give a nearly linear relation between ΔEads and ΔE⧧ for H2 dissociation on RL nanoribbon, and between ΔEdis and ΔE⧧ on defective Au(111). (The ΔEads on Au(111) are too small to illustrate these trends.) GGAs that give more stable adsorption, i.e., a more positive ΔEads and a more negative ΔEdis, tend to give relatively low dissociation barriers ΔE⧧. The tested beyond-GGA functionals tend to give dissociation barriers above the trend for GGAs. This trend is reduced for the pristine Au(111) surface, presumably due to the increased binding to metallic systems discussed above. Detailed Study of Gold Nanowires. The gold nanotubes and nanoribbons tested here represent an interesting inter-

Figure 4. Calculated dissociation barrier ΔE⧧ plotted vs adsorption or dissociation energy. GGAs are lines + points; beyond-GGA calculations are points. Results are shown for (a) Au3 cluster, (b) RL nanoribbon, and (c) defective Au(111). The beyond-GGA methods give barriers above the trend for GGAs, and closer to the available reference values.

mediate case between small gold clusters and gold surfaces. We close with a more detailed discussion of their predicted chemistry. Table 2 reports detailed potential energy surfaces for the low-coverage RL nanoribbon. H2 adsorbs weakly to the RL and RH nanoribbon edges in a distorted side on geometry. We found no stable geometries with H2 adsorbed to the nanoribbon face. H2 dissociates through a 15699



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Table 2. Exploration of Other XC Functionals for H2 on Low-Coverage Nanoribbon RLa functional type GGA

meta-GGA

Screened hybrid

Global hybrid

a

XC functional

ΔEads

ΔE⧧

ΔEdis

ΔEmig⧧

ΔEmig

LDA PW91 PBE/planewave PBE/modLANL08 RPBE BLYP M06-L MN12-L TPSS HSE06 HISS MN12-SX M06 PBE0 M06-2X MPW1PW91 TPSSh BMK M06-HF BHandHLYP B3LYP

0.645 0.109 0.134 0.093 −0.139 (−0.183) (−0.003) 0.080 0.032 0.068 0.081 (−0.113) (−0.081) (0.077) (−0.208) (0.028) (−0.016) (−0.021) (−0.069) (−0.162) (−0.138)

0.018 0.256 0.213 0.244 0.317 (0.387) (0.447) 0.142 0.342 0.356 0.390 (0.335) (0.611) (0.337) (0.455) (0.359) (0.367) (0.356) (0.242) (0.568) (0.445)

−0.739 −0.430 −0.506 −0.437 −0.334 (−0.271) (−0.258) −0.600 −0.401 −0.240 −0.201 (−0.232) (0.127) (−0.220) (0.167) (−0.197) (−0.314) (−0.181) (0.221) (0.161) (−0.116)

0.227 0.159 0.184 0.161 0.133 (0.098) (0.184) 0.114 0.177 0.188 0.216 (0.109) (0.160) (0.172) (0.093) (0.166) (0.179) (0.134) (0.058) (0.123) (0.115)

−0.254 −0.285 −0.263 −0.284 −0.298 (−0.294) (−0.325) −0.314 −0.308 −0.452 −0.529 (−0.497) (−0.507) (−0.505) (−0.782) (−0.509) (−0.393) (−0.641) (−1.140) (−0.733) (−0.470)

Results in parentheses are post-PBE.

We find that our PBE calculations with atom-centered basis sets agree quite well with plane-wave PBE calculations. Supporting Information Tables SI1−SI4 show that PBE/planewave calculations on nanoribbon RH and the pristine and defective Au(111) surface generally return energy differences within 0.01−0.05 eV of the PBE/modLANL08 values. Figure 3 illustrates this for the defective Au(111) surface. We also find that the three tested atom-centered basis sets give similar results. The tables also show that, for Au16 clusters, the defTZ, LANL08, and modLANL08 basis sets give ΔEads that agree to within ∼0.005 eV, and ΔE⧧ and ΔEdis that agree to within ∼0.05 eV. The post-PBE and self-consistent calculations in Supporting Information Tables SI1−SI4 are also generally comparable, validating our post-PBE calculations on Au(111). We also conclude that even a relatively large Au16 cluster is not a perfect representation of the low-coverage Au nanoribbon RL. For example, the PBE/modLANL08 ΔEdis in Supporting Information Table SI4 differ by ∼0.15 eV between Au16 and RL. Test calculations suggest that this arises from the somewhat different geometries, rather than from a limitation of the calculation. PBE/modLANL08 calculations on a finite system consisting of two RL unit cells give ΔE⧧ and ΔEdis of 0.22 and −0.55 eV, close to the 0.24 and −0.44 eV values shown for RL itself in Supporting Information Table SI4.

bent transition state with H−H bond length 1.125 Å and the H− H bond axis 77.7° to the Au surface, yielding two H atoms adsorbed between adjacent pairs of Au atoms on the nanoribbon edge. The reactivity is somewhat coverage-dependent, with the high-coverage RH nanoribbon giving a generally higher dissociation barrier and less negative dissociation energy. The low-coverage RL nanoribbon has a second and more stable adsorption geometry in which an H atom migrates along the nanoribbon edge. Table 2 includes the migration barrier and energy ΔE⧧mig and ΔEmig relative to the initial product. The trends are similar to those discussed previously, with the HSE06 and HISS screened hybrids predicting significantly higher migration barriers than any GGA. This is again consistent with our previous work on migration barriers for adatoms on graphene, and further motivates exploration of screened hybrid functionals for surface chemistry. All of the methods predict relatively low migration barriers, suggesting that transport of chemisorbed H atoms along Au nanoribbons should be quite facile. H2 also adsorbs weakly to the high- and low-coverage nanowire T1L and T1H, with the H−H bond at angle 152.4° with respect to the nanotube axis. Figure 2b shows that dissociation barriers ΔE⧧ are significantly higher than for the RL and RH nanoribbons or the planar Au clusters, and approach the very high barrier seen on pristine Au(111). This is consistent with increased reactivity of the low-coordinate Au atoms at the cluster and nanoribbon edges. The coverage dependence is smaller for the nanowires, though Figure 2c shows that most functionals predict a more favorable ΔEdis for the high-coverage T1H nanowires. Sanity Tests. We conclude by highlighting “sanity tests” comparing our different choices of basis set and self-consistency. Properly designed atom-centered and plane-wave basis sets should in principle converge to the same basis set limit. More broadly, DFT calculations with atom-centered and plane-wave codes should generally be expected to give similar results.

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CONCLUSION The calculations presented here give new insights into the reactivity of gold nanostructures in a regime intermediate between small nanoparticles and pristine surfaces. Molecular adsorption of H2 to Au nanotubes and nanoribbons is generally quite weak. H2 dissociation is generally rather unfavorable on most of the tested structures, though the results depend sensitively on the cluster size. Comparisons of different XC functionals point to the importance of approximations beyond the GGA for systematically improving calculated reaction kinetics. The results further motivate vigorous exploration of 15700



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beyond-GGA approximations in modeling heterogeoeous catalysis and surface chemistry.

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ASSOCIATED CONTENT

S Supporting Information *

Text giving full citations of refs 110 and 120 and URLs for pseudopotentials, figures showing transition state and product geometries and BEP relations between ΔEads and ΔE⧧, tables listing calculated ΔEcoh, ΔEads, ΔE⧧, ΔEdis of Figure 2, and calculated geometries and total energies of all systems. This material is available free of charge via the Internet at http://pubs. acs.org.

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AUTHOR INFORMATION

Corresponding Author

*Tel.: 817-257-6202. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the Qatar National Research Foundation through the National Priorities Research Program (NPRP Grant No. 09-143-1-022).

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REFERENCES

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