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Article pubs.acs.org/JPCC

A Benchmark Study of H2 Activation by Au3 and Ag3 Clusters Salvador Moncho,† Edward N. Brothers,*,† and Benjamin G. Janesko‡ †

Chemistry Department, Texas A&M University at Qatar, Texas A&M Engineering Building, Education City, Doha, Qatar Department of Chemistry, Texas Christian University, Fort Worth, Texas 77005-1892, United States

‡

S Supporting Information *

ABSTRACT: We present a high-level computational study of the activation and disassociation of H2 on triatomic gold and silver clusters as well as benchmarks of various density functional theory (DFT) approximations. The reaction was modeled using complete basis set (CBS) extrapolated CCSD(T) energies at MP2/def2-QZVPP geometries. Our calculations considered several isomers of dissociated H2 on the metal trimer as well as transition states between them. High-level results were then used to benchmark 30 different semilocal, hybrid, double hybrid, and Rung 3.5 DFT functionals as well as Hartree−Fock and MP2 theory. The effect of optimizing the geometries using DFT was also studied with a smaller set of functionals. The results indicate that double-hybrid functionals, especially mPW2PLYP, accurately model this class of reactions, albeit at computational cost higher than standard DFT. The range-separated (screened) hybrids HSE06 and HISSb are also successful and provide a reasonable balance of computational cost and accuracy. These methods are particularly promising for treatments of coinage metal clusters and surfaces.

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INTRODUCTION Heterogeneous catalysts are critical in industrial chemistry.1 Computational simulations of heterogeneous catalysis have provided useful insights into technologically relevant catalysts.2 Density functional theory (DFT) with approximate exchangecorrelation (XC) density functionals has been widely used to model heterogeneous catalysis, providing reasonable accuracy at relatively modest computational cost. Typical approximate XC functionals are not systematically improvable, and there is no a priori way to know which functionals will work best for a given problem; this renders benchmarking essential.3 Gold and silver clusters catalyze several hydrogenation reactions.4−6 This has led to extensive interest in simulating H2 dissociation on coinage metals. Gordon and co-workers performed CCSD(T) calculations on H2 on neutral and anionic Au2 and Au3 and used the results to benchmark a few functionals.7 Chen and coworkers studied H2 dissociation on Aunx clusters with PW91.8 Roncero and co-workers studied larger clusters (2−10 atoms) and found that that the edges of planar gold clusters are particularly reactive.9 Similar results were found by Corma and co-workers.10 Fierro and co-workers, comparing between the results for a bulk surface and medium Au clusters (14−29 atoms), found using DFT that these two types of gold structures have qualitatively different reactivity.11 Wang and co-workers studied H atom adsorption on a wide range of gold and silver−gold clusters.12 Other benchmarks for homogeneous gold catalysis show good results for double-hybrid functionals.13,14 Finally, comparisons between a few DFT methods and different wave function approaches on the coordination of monatomic gold and silver systems have been studied.15 © XXXX American Chemical Society

We are interested in the efficacy of DFT, especially the most modern constructions, for modeling heterogeneous catalysis. This has to be ascertained through benchmark studies. Here we treat H2 dissociation reaction on Au3 and Ag3 clusters. These small clusters provide simple models for larger, technologically relevant coinage metal surfaces and supported clusters. While the tested clusters are too small to quantitatively model chemistry at realistic Au and Ag catalysts, DFT methods that perform poorly for these simple benchmarks are of questionable utility for larger and more realistic systems. The systems are treated at a very high level of theory, namely complete basis set (CBS) extrapolated CCSD(T) using geometries fully optimized at the MP2 level with a large basis (def2QZVPP). We particularly consider transition states for H2 activation and interconversion of bound isomers. We then use this reference data to benchmark several DFT methods. We also quantify the effect of DFT geometry optimization by recalculating the CCSD(T)/CBS energies with DFT geometries; this is a critical component as practical computational investigations often cannot afford MP2/ def2QZVPP geometries.

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COMPUTATIONAL METHODS Calculations are done using the development version of the Gaussian suite of programs.16 Reference geometries are Received: October 5, 2012 Revised: March 22, 2013

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dx.doi.org/10.1021/jp3098704 | J. Phys. Chem. C XXXX, XXX, XXX−XXX

The Journal of Physical Chemistry C

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Figure 1. Optimized geometries (MP2/def2QZVPP level) of the different dihydrogenated Au3 and Ag3 cluster isomers, with their conversion reactions. Energies are CCSD(T)/CBS-extrapolated in kcal/mol.

optimized using MP217 with the def2QZVPP basis set and relativistic effective core potentials (hereafter abbreviated “QZ”).18 Such relativistic treatment of core electrons is important in gold chemistry19 and specifically for Au3 clusters.20 The computed geometries were assigned to minima or transition states based on their number of imaginary frequencies; in some cases IRC calculations have been used to confirm the connection of ground and transition states. CCSD(T)/CBS energies were evaluated by combining CCSD(T)/def2TZVPP (hereafter “TZ”) with MP2 correlation energy basis set extrapolation and a HF energy basis set extrapolation: ECCSD(T)CBS = EHFCBS + ECCSD(T)TZ − EMP2TZ + EΔMP2CBS. Correlation EΔMP2CBS was extrapolated from MP2/TZ and MP2/QZ calculations via the equation Ecorrn = EcorrCBS + An−3, where n is the cardinal of the basis (3 for TZ, 4 for QZ).21 EHFCBS was extrapolated from MP2/ TZ and MP2/QZ calculations via the equation EHFn = EHFCBS + Be−nα, where n is the cardinal of the basis (3 for TZ, 4 for QZ),22 using an α parameter optimized for diatomic transition metal systems.23 If not stated otherwise, the energies discussed in sections 3a and 3b are this extrapolated ECCSD(T)CBS energy and are used throughout this work as the reference or target values. The quality of coupled cluster calculations was evaluated by the T124 and T2 diagnostics. Post-HF calculations (MP2 and CCSD(T)) included the Gaussian suite’s default number of frozen core orbitals. The quality of the basis set was tested by calculating basis set superposition error (BSSE) using the counterpoise method.25 This was done for different gold and silver complexes using MP2 and then repeated for different DFT methods (HSE06, BLYP, and B3LYP). The QZ basis set gave very low BSSE for DFT methods (around 0.1 kcal), while the BSSE values of MP2 calculations were high, up to a few kcal/mol in some cases. While CBS extrapolations will reduce the energy errors due to BSSE in the reference/target values, one concern is that geometrical errors could arise. However, the geometries predicted with DFT, which shows low BSSE, and those predicted by MP2, which shows high BSSE, are similar. This leads us to believe that BSSE will not contribute significantly to geometry. Most DFT calculations used the QZ basis set and an “UltraFine” numerical integration grid corresponding to a pruned grid of 99 radial shells of 590 points; “M06 family” calculations used a larger integration grid with 199 radial shells of 590 points.26,27 A wide range of DFT functionals have been

selected including pure functionals (BLYP,28 B97D,29 M06-L,30 PBE,31 PBEsol,32 PW91,33 revPBE,34 SVWN,35 tHCTH,36 and TPSS37), hybrid functionals with exact exchange (B3LYP,38 BHandHLYP, 39 BMK, 40 M06, 41 M06HF, 42 M062x, 41 mPW1PW91,43 PBE0,44 and TPSSh37), range-separated hybrid functionals (HISSb,45 HSE06,46 LC-ωPBE,47 ωB97,48 ωB97X,48 and ωB97XD49), double-hybrids including exact exchange and MP2-like correlation (B2PLYP, 5 0 B2LYPD, 5 1 and mPW2PLYP52), and “Rung 3.5” functionals53 (Π1PBE54 and PBE+Π(s) 55 ). Rung 3.5 calculations used the smaller def2TZVPP basis set with f functions removed; HSE06 calculations with this basis set give RMSD ∼ 1 kcal/mol larger than HSE06/QZ. Results are also reported for HF, MP2, and SCS-MP256 in the QZ basis. A brief word about the relative computational cost of these methods is in order here. While there will some variation based on system size and efficiency of the specific code/ implementation being used, there are certain absolutes in timings that arise from the nature of the theory. For example, hybrids should always be slower than pure DFT as the hybrids include nonlocal exact exchange as an additional step. Screened hybrids will generally be faster than global hybrids in large systems with small bandgaps, as the screening removes the need to evaluate expensive long-range exchange. Finally, double hybrids should be slower than standard DFT, as they involve the standard SCF calculations, plus an atomic orbital to molecular orbital transformation to form the MP2-like correlation energy. While the exact time ratio between pure DFT and hybrid DFT or hybrid DFT versus double hybrids is intrinsically system- and code-dependent, the general trends of increasing expense is intrinsic to the methods. For the current systems of interest with the current implementation, range separated hybrids are about twice as expensive (in CPU time) as pure functionals, and double hybrids are about 40 times more expensive than pure functionals. All energies in this work are electronic energies without zeropoint or thermal corrections. Such corrections could be included later for comparisons against experiments at finite temperature, but their inclusion should not change the overall conclusions of this study.

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RESULTS AND DISCUSSION a. Mechanism of the Cleavage of an H2 Molecule by an Au3 Cluster. Figure 1 shows calculated local minima for H2 B



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Figure 2. Plot of the RMSD of the methods for BPA relative energies for Ag vs Au. The line indicates equal RMSD for both systems.

binding and activation on Au3. The first step of H2 activation is coordination to Au3 to form complex Au3H2. (Bare Au3 is predicted to be in the 2B2 electronic state, with a 2A1 state 0.5 kcal/mol higher in energy.) The long H−H distance in Au3H2 (0.945 Å) suggests that is a coordinated H2 molecule, not two coordinated H atoms. Au3H2 is 20.0 kcal/mol more stable than the isolated reactants. Coordinated H2 can rotate through transition state AuTSH2,H2 to re-form Au3H2 with exchanged H positions. Au3H2 can also react to form the five isomers in Figure 1. AuHH has two terminal (“T”) hydrogens bound to the same Au atom, AuTT and AuTT′ have two terminal hydrogens bonded to different Au atoms, and AuST and AuST′ each have a bridging57 hydrogen shared (“S”) by two different Au atoms. (Descriptors “T” and “S” are used for describing the H positions; we describe a hydrogen atom as “T” or terminal where it is only bonded to one Au atom and we name them “S” or shared where they are bonded to two adjacent Au atoms.) All five complexes are more stable than the isolated reactants but less stable than Au3H2. This result is consistent with those Gordon and co-workers7 for AuST and AuHH with energies of 4.2 and 9.4 kcal/mol, respectively; here we found them to be 3.1 and 7.0 kcal/mol. Formation of AuHH from Au3H2 via transition state “AuTSH2,HH” is predicted to barrierless at the MP2/QZ level of theory. Any perturbation to the Au3HH geometry followed by optimization leads to the Au3H2 structure. This is consistent with previous work done at the CCSD(T) level.7,58 Migration of a H atom in AuHH to a bridging (“S”) interaction forms AuST, which is 3.9 kcal/mol less stable than AuHH. This migration occurs through transition state AuTSHH,ST. This transition state is 8.6 kcal/mol higher in energy than AuHH and 11.7 kcal/mol higher in energy than Au3H2. Complex AuST can evolve by the migration of the shared H to the second Au atom to form the AuTT complex with two terminal hydrogen atoms. Energetically, the formation of AuTT is favorable, with a relative barrier of just 2.5 kcal/mol to form a compound which is 5.8 kcal/mol more stable than AuTT (and

only 1.2 kcal/mol higher in energy than the initial coordinated complex). Further movement of the hydrogen to the next free side of the Au3 triangular cluster leads to the AuST′ complex, 5.3 kcal/mol less stable than AuTT, with a relative barrier of 10.1 kcal/mol. An “AuSS” isomer with two bridging hydrogens was searched for extensively in triangular Au3 clusters. However, the triangular cluster with two “S” atoms was found to be a transition state, AuTSTT,TT, for the exchange of H positions in AuTT. Another isomer AuSS′ is a local minimum, with the triangular structure of the Au3 cluster broken to form a distorted linear Au3 complex. AuSS′ is 5.5 kcal/mol less stable than AuTT and can be formed directly from AuTT via a relatively high energy transition state AuTSTT,SS. b. Mechanism of the Cleavage of an H2 Molecule by an Ag3 Cluster. Figure 1 also shows calculated local minima for H2 binding and activation on Ag3. The initial coordination complex Ag3H2 is only 4.2 kcal/mol more stable than the isolated reactants, and the short H−H bond (0.766 Å) suggests that the initial complex is noncovalent. The barrier of rotation of the H2 coordinated molecule, through AgTSH2,H2, is only 0.7 kcal/mol. (As for Au3, bare Ag3 is predicted to be in the 2B2 electronic state, with a 2A1 state 0.4 kcal/mol higher in energy. This is consistent with previous work.59,60) For Ag3, only two minima with separated H atoms that do not significantly distort the cluster have been found. AgST has one H in a terminal position and the second forming a bridge adjacent to the terminal H. AgSS has two bridging H atoms. Efforts to optimize a complex similar to AuHH always yield the noncovalent complex Ag3H2, and efforts to optimize a complex similar to AuTT lead to AgSS. Most attempts to find a structure similar to AuST′ lead to AgSS, although an AgST′ complex has been found where the triangular cluster is very distorted. The high energy of the AgST complex, and the lack of a minimum with two terminal Hs, suggest that coordination by a single metal atom is not favored for Ag3. However, the opposite behavior was observed in the gold complexes, where structures with two terminal Hs were the most stable, and the double bridge C



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Figure 3. RMSD in BPB relative energies for Ag vs Au; details as in Figure 2.

Table 1. RMSD (in kcal/mol) for the Different Methods, Ordered by Overall BPB Errora complete set

Au only

Ag only

minima

TS only

RMSD (kcal/mol)

type

BPB

BPA

BPB

BPA

BPB

BPA

BPA = BPB

BPB

BPA

mPW2PLYP B2PLYP B2PLYPD HSE06 HISSb mPW1PW91 PBE0 B3LYP M06L ωB97XD SCS-MP2 BHandHLYP M06-2x TPSSh M06 BMK ωB97X MP2 TPSS ωB97 M06HF BLYP PBE PW91 tHCTH revPBE PBE+Π(s)b LC-ωPBE B97D Π1PBEb PBEsol SVWN HF

DH DH DH RS RS H H H P RS WF H H H H H RS WF P RS H P P P P P 3.5 RS P 3.5 P P WF

1.54 1.68 1.97 2.52 2.57 2.63 2.73 2.75 2.76 2.92 2.97 3.09 3.18 3.26 3.32 3.61 3.62 3.67 4.06 4.32 4.45 4.57 4.61 4.61 4.66 5.11 5.14 5.24 5.35 6.71 6.80 6.82 8.54

1.64 1.82 2.13 3.09 2.65 3.15 3.31 3.10 3.56 3.12 3.18 2.78 3.42 4.18 2.73 3.90 3.56 4.27 5.13 4.24 4.43 5.08 5.68 5.66 5.50 5.43 5.27 5.65 6.06 7.24 7.97 8.89 7.65

0.78 0.98 1.09 1.55 2.19 1.48 1.51 1.97 1.70 2.10 3.68 3.15 2.53 2.06 3.96 3.84 3.01 4.52 2.96 3.62 3.69 3.72 2.99 3.04 3.28 4.19 5.10 3.41 4.02 6.49 5.09 3.51 8.84

0.75 0.93 1.25 1.81 2.01 1.79 1.81 2.43 2.16 2.16 3.84 2.76 3.01 2.88 3.05 4.10 2.71 5.25 3.82 3.21 3.68 4.39 3.84 3.88 4.02 4.00 5.43 3.07 5.08 6.37 5.05 4.63 7.31

2.29 2.42 2.87 3.57 3.09 3.83 3.98 3.67 3.90 3.89 1.07 3.00 4.02 4.58 1.88 3.21 4.44 1.46 5.40 5.26 5.47 5.70 6.41 6.38 6.29 6.33 5.22 7.29 6.98 7.05 8.90 10.10 8.05

2.47 2.70 3.05 4.44 3.45 4.57 4.85 3.95 5.07 4.25 1.57 2.82 4.00 5.69 2.10 3.56 4.61 1.72 6.72 5.51 5.44 6.04 7.79 7.72 7.28 7.16 5.00 8.27 7.37 8.46 11.20 13.13 8.18

1.23 1.39 1.76 2.93 2.92 3.04 3.19 2.86 3.18 2.98 2.29 2.84 2.95 3.92 3.16 4.29 3.80 2.86 4.80 4.73 5.10 4.87 5.28 5.25 5.17 4.53 6.39 6.29 5.69 7.12 7.06 8.21 8.10

1.73 1.87 2.11 2.15 2.27 2.28 2.32 2.66 2.39 2.87 3.39 3.26 3.34 2.65 3.44 3.00 3.49 4.17 3.41 3.98 3.90 4.34 4.02 4.05 4.24 5.51 3.96 4.28 5.07 6.38 6.59 5.55 8.86

1.88 2.09 2.36 3.20 2.43 3.23 3.40 3.27 3.83 3.23 3.70 2.73 3.74 4.36 2.35 3.59 3.37 5.07 5.36 3.83 3.86 5.23 5.96 5.94 5.73 6.02 4.24 5.11 6.32 7.32 8.59 9.36 7.30

a

QZ basis set, MP2/QZ geometries. Each column presents error statistics for one of the nine subsets discussed above. The lowest RMSD for each subset is italicized. XC functionals are sorted by RMSD in the complete BPB set, and are labeled as follows: DH = double hybrid, RS = range separated hybrid, H = hybrid, P = pure DFT, WF = wave function and 3.5 = Rung 3.5 functionals. bModified def2-TZ basis set. D



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We next consider the role of the metal, considering the BPA reference results for Ag3. This data set gives a rather different picture. The best overall results are obtained with the MP2 and SCS-MP2 wave function methods. The best DFT method is the M06 hybrid, with a moderate RMSD of 1.9 kcal/mol.61 Double hybrids and BHandHLYP also perform reasonably well. The PBE+Π(s) Rung 3.5 functional improves on most of the pure DFT methods. Figures 2 and 3 extend on this, directly comparing the RMSD for Au vs Ag. (Figure 2 uses the BPA choice of transition state energies, and Figure 3 uses BPB.) All methods perform better for Au, except for MP2, SCS-MP2, BMK, M06, and PBE+Π(s). (This is in part due to the larger absolute magnitude of the Ag PES; comparisons of relative RMSD for Au and Ag are discussed in the Supporting Information.) Moving out from the origin of Figure 2, the double hybrids provide the most accurate overall results, followed by range-separated and global hybrids. These show a wide range of performance. These are followed by the pure functionals which have similar trends but larger errors than hybrid methods. There however are a few exceptions, such as the large errors of SVWN and PBEsol and the excellent performance of M06L. Comparing Figures 2 and 3 shows that there is less separation (more parity) between pure and hybrid functionals for the BPB relative energies. This illustrates a fundamental challenge of benchmarking. For example, one question that benchmarking should be able to answer is the relative performance of functionals; i.e., is method X better than method Y? If we use BPA statistics, TPSS is not as good as wB97 for modeling gold, but if BPB statistics are used, the TPSS functional is better. It is thus important to consider multiple compilations of statistics for adequate benchmarking and analysis of functional efficacy. Considering more than one compilation allows discrimination between the effects of the selecting a particular set of relative energies and the more important general behavior that transcends selection; i.e., it allows the analysis of what is a selection artifact and what is a real limitation of a given method. We next consider the tested methods’ performance for local minima vs transition states (“minima” and “TS” in Table 1). Surprisingly, the semilocal functionals SVWN, PBE, and TPSS have lower errors for BPB transition states than for minima! One possible interpretation is that these transition states are not as delocalized as in typical gas-phase reactions, where semilocal DFT tends to perform poorly. This argument can be supported by geometrical data. For example, in AuTSHH,ST can be observed that the H rotating to form the bridge has almost the same AuH length as the one is not rotating (1.548 Å the moving versus 1.543 Å the other), effectively an early transition state, so the bond has not been significantly stretched. Similar distances can be found in Ag (with bonded H at 1.603 and 1.595 Å, respectively). SVWN, PBE, and TPSS still have rather large BPB transition state errors relative to global hybrids. As expected, hybrids optimized for kinetics, such as BMK, tend to give lower errors for BPB transition states than for local minima. Screened and double hybrids give rather consistent performance for transition states and local minima. Finally, we consider combined error statistics for all stationary points in the data set, in the columns of Table 1 labeled “total”. Both choices of transition state energy (BPA and BPB) give that double hybrids are the most accurate overall. Among the remaining functionals global and (especially) screened hybrids are reasonably accurate. The accurate screened hybrid results are particularly

(AuSS) species was not an energy minimum. Calculations on model systems with one H atom (Au3H and Ag3H) confirm this behavior. The single hydrogen as a bridge on the silver trimer is 1.4 kcal/mol more stable than if the hydrogen is terminal (per ECCSD(T)CBS calculations). However, the gold trimer with a bridging hydrogen was found not to be a minimum, but rather a transition state which is 18.6 kcal/mol less stable than the gold trimer with a terminal H. Formation of the activated complex AgST has a high energy barrier of 32.4 kcal/mol. AgST can readily convert to AgSS through transition state AgTSST,SS. This is a valid MP2/QZ TS geometry, with a relative barrier of 2.9 kcal/mol (ΔEMP2QZ). However, CCSD(T)/TZ energies suggest that the energy of this transition state is equal or slightly lower than the AgST energy of the MP2 geometry, so the process may be even barrierless. Migration of one of the H atoms in AgSS occurs through a nonplanar transition state AgTSSS,SS. In addition, a transition state (AgTSSS,ST′) leading to complex AgST′ has been found, with a relative barrier of 23.4 kcal/mol. Complex AgST′ has a terminal H and a bridging H between two Ag atoms. This bridging H is not coordinated in the same way as in the other Ag complexes but is located between the Ag atoms, breaking the Ag−Ag bond. c. Creation of Benchmark Databases. The energetic data in Figure 1 is used to create two separate data sets for benchmarking DFT energies, each of which is further partitioned to give nine separate subsets. These are classified as follows. The “BPA” set contains the energies of all stationary points for X3H2 (X = Au, Ag), evaluated relative to the most stable geometry Au3H2 or Ag3H2. The “BPB” set instead evaluates the transition state energies relative to the adjacent reactant, rather than relative to the minimum energy geometry. This avoids an excessive dependence on the Au3H2 and Ag3H2 reference structures. To illustrate, the BPA reference energy of AuTSHH,ST is 11.7 kcal/mol, while the BPB energy of AuTSHH,ST is (11.7 − 3.1) = 8.6 kcal/mol different from reactant AuHH. We present error statistics for the combined BPA and BPB sets (2 subsets), the BPA and BPB sets for H2Au3 (2 subsets), the BPA and BPB sets for H2Ag3 (2 subsets), a subset containing all local minima energies, and the BPA and BPB energies of all transition states (2 subsets). Differences between these subsets are used to rationalize different aspects of each functional’s performance. d. Benchmark of DFT Energies for H2 Cleavage by Au3 and Ag3. Table 1 presents root-mean-square deviations (RMSD) in DFT predictions of the relative energies of the complexes in Figure 1, evaluated at MP2/QZ geometries. Results are presented for all nine of the subsets discussed above. Error statistics are not correlated with the T1 diagnostic,24 indicating that differences between DFT and CCSD(T) energies do not arise from problems with the CCSD(T) calculation (Supporting Information). We begin by considering error statistics for the Au3 clusters, using the BPA reference for transition state energies. Based on this criterion, the most accurate functionals are the double hybrids, specially mPW2BPLYP, which had an RMSD of 0.78 kcal/mol. mPW1PW91 and PBE0 and its screened version HSE06 give the best results among functionals without perturbative correlation. The Rung 3.5 functionals perform rather poorly, due in part to the aforementioned restrictions on basis set. Hartree−Fock and MP2 theory also perform poorly relative to the CCSD(T) reference. (Limitations of MP2 for modeling gold have been discussed previously.13−15) E



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Figure 4. Plot of the rate-determining step barriers depending of the metal for all methods.

Figure 5. Plot of the ΔE between the different orbital symmetries of Au3 (X-axis) and Ag3 (Y-axis).

use MP2/QZ geometries. All tested methods correctly predict that Ag32B2 is more stable than Ag32A1. Otherwise, the trends among different DFT methods are rather difficult to explain. Hartree−Fock theory, B2PLYPD (but not B2PLYP), and M06 and M06HF (but not M06-2x) predict a qualitatively wrong state for Au3. SVWN is surprisingly accurate. g. Comparison of DFT and MP2 Geometries. We have used two different approaches to quantify the effect of geometry optimization. First, we reoptimized a subset of stationary points with several DFT functionals and compared error statistics for DFT energies at DFT vs MP2/QZ geometries. Second, we reevaluated representative CCSD(T) energies at HSE06/QZ geometries, rather than MP2/QZ geometries, and determined how this second set of reference data changed the relative errors of the DFT methods. The general idea is to show how sensitive the results are to geometric differences and to see the relevance of our benchmarking to chemical studies where one functional is used throughout. We conclude that the geometries have relatively little effect on these benchmarks. Table 2 presents the statistical errors in DFT calculations, at both self-consistent and MP2/QZ geometries, relative to the reference CCSD(T)/CBS//MP2/QZ results. Results are presented for the energies of AuHH, AuTT, and AuTSHH,ST

encouraging for applications to periodic metallic systems, where perturbative correlation and long-range exact exchange can introduce formal and computational difficulties. e. Benchmark of Relative Rates. Practical theoretical treatments of catalysis often require not accurate absolute rates, but accurate predictions of trends among different catalysts.2 Accordingly, Figure 4 shows how the different DFT methods perform for the relative rate-limiting barrier heights for H2 dissociation on Au3 vs Ag3, i.e., the AuTSHH,ST vs the AgTSH2,ST both related to coordinated hydrogen molecule X3H2. CCSD(T)/ CBS calculations predict that the Ag barrier is 2.78 times the Au barrier; this target ratio is plotted as the green line in Figure 4. All tested methods are qualitatively correct, predicting a larger dissociation barrier on Ag3. The tested semilocal functionals give particularly accurate relative barriers, despite their systematic underestimate of the absolute barrier heights. Interestingly, the highly parametrized M06HF and M06-2X functionals significantly overestimate the relative ratio, while MP2, SCS-MP2, and several long-range-corrected functionals tend to underestimate it. Further studies of this effect are ongoing. f. Benchmark of Bare Au3 and Ag3. Figure 5 shows how the different DFT methods treat the energy difference between 2 B2 and 2A1 electronic states of Au3 and Ag3. Calculations again F



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most stable isomers for gold and silver, are included. Most density functionals have better results when used in the optimization (with the exception of wB97XD and M06). These differences affect only slightly the trends of functionals’ relative accuracies, and the two best methods as well as the worst method remain the same (B2PLYP, HSE06, and M06HF, respectively). However there are some differences in the middle of the list where the functional are very similar in accuracy. For example BMK and PBE perform worse than B3LYP using MP2 geometries, but they are better using DFT optimization. The main trend is that most accurate methods are less affected by the geometry optimization, and in general both overall rank and total accuracy are very similar to MP2 or DFT geometries. As a further test, one density functional was used for obtaining a full mechanistic pathway (including points that were not found at MP2 level) and computing also the energies for these geometries with the reference energy level, i.e., CCSD(T)/CBS. The HSE06 functional was chosen due to accurate performance; another obvious candidate was B2PLYP, but HSE06 was chosen due to lower computational cost. As observed in Figures 6 and 7, energies for both methods HSE06 and CCSD(T) are almost identical in the geometries that are obtained with both methods (a third profile including other geometries on the gold system is included in the Supporting Information). However, there are some differences in the obtained reactivity. For example, a new transition state has been found for the activation of the H−H bond with a low barrier in HSE06. However, the CCSD(T)/CBS barrier for this TS is very low (just 0.1 kcal/mol in the reverse direction). Furthermore, as could be expected from single-point calculations giving negative relative barriers for AgTSST,SS in most of the methods, neither AgTSST,SS nor AgST is obtained as a stationary point using HSE06. There is also a small difference in the AgST′ structure, that is not symmetrical in HSE06 optimization, due to a deviation of the terminal H. However, the difference of the new minima and of the symmetrical structure corresponding to TS of the movement of this H is less than a 0.1 kcal/mol. HSE06-optimized geometries with CCSD(T)/CBS geometries could have been used as a reference for the benchmark

Table 2. Summary of the RMSD (in kcal/mol) for the Different Methods with MP2 and DFT Geometries B2PLYP HSE06 BMK PBE B3LYP wB97XD BLYP M06 SVWN HF M06HF

RMSDDFT‑OPT

RMSDMP2‑OPT

1.0 1.8 2.0 2.0 2.0 2.4 2.8 3.2 3.5 4.4 4.5

1.2 1.9 2.3 2.4 2.3 2.3 3.3 2.8 4.7 3.3 5.6

Figure 6. Energy profile of the activation of H2 by a Au3 cluster, leading to the AuTT complex. Geometries included are those calculated at MP2/QZ level except for AuTSH2,HH, which was not found at this level.

relative to AuHH and AgSS relative to Ag3H2. It must be noticed that despite the existence of differences between the relative energies available in DFT and MP2 optimizations, the most important of them, i.e. both rate-determining barriers and the

Figure 7. Energy profile of the activation of H2 by a Ag3 cluster. Geometries included are those calculated at the MP2/QZ level. G



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Table 3. Summary of the RMSD (in kcal/mol) for the Different Methods with MP2 and HSE06 Geometriesa mPW2PLYP B2PLYPD B2PLYP B3LYP ωB97XD HISSb BHandHLYP HSE06 M06L PBE0 ωB97X TPSSh M06 ωB97 MP2 BLYP TPSS tHCTH LC-ωPBE PW91 PBE B97D HF SVWN

HSE06full

HSE06shared

MP2full

MP2shared

Difffull

Diffshared

1.0 1.2 1.6 2.6 2.7 2.7 2.9 3.0 3.0 3.3 3.3 4.0 4.0 4.0 4.3 4.9 5.0 5.3 5.5 5.8 5.9 5.9 8.0 9.2

1.0 1.2 1.7 2.7 2.8 2.8 2.7 3.1 3.1 3.4 3.3 4.2 3.8 4.0 3.9 5.2 5.3 5.6 5.6 6.1 6.2 6.3 7.7 9.5

1.6 1.8 2.1 3.1 3.1 2.7 2.8 3.1 3.6 3.3 3.6 4.2 2.7 4.2 4.3 5.1 5.1 5.5 5.6 5.7 5.7 6.1 7.7 8.9

1.7 1.8 2.1 3.1 2.9 2.6 2.8 2.8 2.9 3.0 3.3 3.7 2.8 3.9 4.0 4.9 4.5 5.0 5.3 5.0 5.0 5.9 6.7 7.6

−0.6 −0.6 −0.5 −0.5 −0.4 0.1 0.1 −0.1 −0.5 0.0 −0.2 −0.2 1.2 −0.2 0.1 −0.1 −0.1 −0.2 −0.2 0.2 0.2 −0.1 0.3 0.3

−0.7 −0.6 −0.4 −0.4 −0.2 0.2 −0.1 0.4 0.2 0.5 0.0 0.5 1.0 0.1 −0.1 0.3 0.7 0.6 0.3 1.2 1.2 0.4 1.0 2.0

BPA is used throughout. Sets labeled “full” are the complete sets of structures found by the method; sets labeled shared are only the structures that overlap between HSE and MP2.

a

study; here we evaluate that choice and hence another aspect of the effect of using DFT geometries. In Table 3 we compare the RMSD values obtained if we use the HSE06 geometries in place of the MP2 geometries. Most of methods have slightly lower mean deviations when the HSE06 geometries are used; this can be attributed to the removal of some geometries found at the DFT level. For a better comparison, RMSDs from the subset of shared geometries are included; these are relative energies of AuHH, AuST, AuTT, AuSS′, AuST′, AuTT′, AuTSHH,ST, AuTSTT,SS′, AuTSST,TT, AuTSTT,ST′, AuTSH2,H2, AuTSTT,TT′, AgSS, AgST′, AgTSH2,ST, AgTSSS,SS, and AgTSSS,ST′. In this case several methods that lead to slightly better results with HSE06 geometries have slightly worse results (as the own HSE06 and PBE0, TPSSh, BLYP, LC-wPBE, B97D).

relative barriers is consistent, so accordingly they can be used to determine the relative activity. Considering everything in the preceding paragraph, the authors endorse and encourage the use of either HSE06 or HISSb for the study of reactions at surfaces for two reasons. First, as this study has shown, it has good accuracy when compared to high levels of theory; i.e., these functionals are expected to adequately model the physics of a system. Second, these methods have computational cost and scaling such that they are applicable to large and/or periodic systems; i.e., they are expected to be applicable to a wide range of systems. This combination of breadth and accuracy makes them good candidates for future studies and makes them the authors’ preference. Absolute deviations from reference values are larger for Ag systems than for Au systems. However, this can be related with the larger value of several Ag relative energies and not a systematic error with the DFT calculations. In fact, the lack of clear trends were relative RMSD have been analyzed, and the fact that the ratio for rate-determining barriers are approximately constant for most methods suggests that there is not a systematic deviation in the methods. Optimization of the geometries using DFT does not change significantly the trends observed in the accuracy of the different methods; CCSD(T)/CBS energies obtained DFT geometries lead to a very similar set of relative energies. In addition, despite differences in the exact order of the functionals depending on the geometry optimization (HSE06, MP2 or the tested method), general trends for the quality of the methods are the same. Hence, the results obtained with single-point calculations in this kind of system can be extrapolated to studies done by DFT optimizations, a situation that is likely to arise in standard investigations of chemistry on metal surfaces.

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CONCLUSIONS This work presents high-level ab initio calculations of the adsorption and dissociation of H2 on Au3 and Ag3 clusters, with the calculations suggest that the process is faster and energetically more favorable on gold. One major goal of this work is to compare several different DFT methods for these reactions versus some wave function methods used as a reference. Doublehybrid functionals have very good results in all the approaches used in this work. However, these methods are computationally quite expensive when compared to standard DFT, and thus exchange only hybrids or pure functional are more convenient for application in medium or large systems. In terms of absolute accuracy, range separated functionals such as the three-range HISSb functional or the screened HSE06 are very promising and have lower average deviations. That said, it must be noted that although semilocal pure density functionals underestimate the barriers for both processes, the ratio between the gold and silver H



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ASSOCIATED CONTENT

S Supporting Information *

Cartesian coordinates of all geometries optimized at MP2/QZ level, with the potential energies with the different computed methods; imaginary frequency values of all transition states; complete tables with the relative energies of the intermediates and transition states obtained with all DFT methods from MP2 geometries; additional plots with the correlation between the RMSDs for minima and transition states, between T1 diagnostic and RMSD, and between the relative RMSD for gold and silver; potential energy profile including the Au3H2 isomers not included in Figure 6. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This publication was made possible by NPRP Grant No. 09-1431-022 from the Qatar National Research Fund (a member of Qatar Foundation) and a generous allocation of computer time from Texas A&M University at Qatar’s Research Computing Group.

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REFERENCES

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J


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