DFT-Based Method for More Accurate Adsorption Energies: An

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A DFT-Based Method for More Accurate Adsorption Energies: An Adaptive Sum of Energies from RPBE and vdW Density Functionals Alyssa J. R. Hensley, Kushal Ghale, Carolin Rieg, Thanh Dang, Emily S. Anderst, Felix Studt, Charles T. Campbell, Jean-Sabin McEwen, and Ye Xu J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.6b10187 • Publication Date (Web): 26 Jan 2017 Downloaded from http://pubs.acs.org on January 27, 2017

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The Journal of Physical Chemistry

A DFT-Based Method for More Accurate Adsorption Energies: An Adaptive Sum of Energies from RPBE and vdW Density Functionals Alyssa J. R. Hensley,† Kushal Ghale,§ Carolin Rieg,† Thanh Dang,§ Emily Anderst,† Felix Studt,‡ Charles T. Campbell,& Jean-Sabin McEwen*,†,#, and Ye Xu*,§ †

Gene & Linda Voiland School of Chemical Engineering and Bioengineering, Washington State University, Pullman, WA 99164, USA § Cain Department of Chemical Engineering, Louisiana State University, Baton Rouge, LA 70803, USA ‡ Institute for Chemical Technology and Polymer Chemistry, Karlsruhe Institute of Technology, Karlsruhe 76131, Germany & Department of Chemistry and Department of Chemical Engineering, University of Washington, Seattle, WA 98195, USA # Department of Physics and Astronomy and Department of Chemistry, Washington State University, Pullman, WA 99164, USA Supporting Information Placeholder

metals are crucial for developing new materials for energy and environmental technologies, such as catalysts, fuel cells, batteries, sorbents, membranes and sensors. With the advent some 20 years ago of fast density functional theory methods with periodic boundary conditions (henceforth referred to as “DFT” for brevity) that were able to treat transition metal surfaces with reasonable energy accuracy,1-3 surface and material chemistry research has been revolutionized. Nearly 1,000 groups worldwide now employ DFT in research. This huge growth has been driven by the deep fundamental insights provided by DFT and its potential to help improve materials where surface properties are crucial.4-10 Several groups have already demonstrated some successes in using DFT to guide discovery of better catalysts.11-17 It is clear that computational chemistry will play an essential role in the future of heterogeneous catalysis research. However, many of these applications of DFT in catalysis research rely on accurate estimates of the relative energies of adsorbed intermediates, and the best current DFT methods have less than desirable energy accuracy. Recently, Wellendorff et al. reported 39 accurate experimental adsorption energies of different common catalytic intermediates on single-crystal late transition metal surfaces (referred to hereafter as the CE39 data set) and calculated the same set of adsorption energies using six popular exchange-correlation functionals (i.e. LDA, PBEsol, PW91, PBE, RPBE, and BEEF-vdW).18 These adsorbates were specifically chosen as benchmarks to guide the development of improved computational methods relevant to surface chemistry and heterogeneous catalysis on late transition

ABSTRACT: In recent years the popularity of density functional theory with periodic boundary conditions (DFT) has surged for the design and optimization of functional materials. However, no single DFT exchangecorrelation functional currently available gives accurate adsorption energies on transition metals both when bonding to the surface is dominated by strong covalent or ionic bonding and when it has strong contributions from van der Waals interactions (i.e., dispersion forces). Here we present a new, simple method for accurately predicting adsorption energies on transition metal surfaces based on DFT calculations, using an adaptively weighted sum of energies from RPBE and optB86b-vdW (or optB88-vdW) density functionals. This method has been benchmarked against a set of 39 reliable experimental energies for adsorption reactions. Our results show that this method has a mean absolute error and root mean squared error relative to experiments of 13.5 and 19.6 kJ/mol, respectively, compared to 20.4 and 26.4 kJ/mol for the BEEF-vdW functional. For systems with large van der Waals contributions, this method decreases these errors to 12.0 and 18.3 kJ/mol. Thus, this method more accurately predicts adsorption energies both for processes dominated by strong covalent or ionic bonding and for those dominated by dispersion forces than any current standard DFT functional alone.

1. Introduction Fast computational methods that provide accurate energies of surface chemical reactions on transition

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metals. The best of these functionals, the Bayesian error estimation functional with van der Waals correlation (BEEF-vdW, shortened to BEEF below),19 has the smallest mean absolute error (MAE) and root mean squared error (RMSE), 20.4 and 26.4 kJ/mol (per fragment, same below), respectively.18 The other functionals perform worse than BEEF, especially for cases with large vdW contributions (larger errors by 9 to 30 kJ/mol) (Fig. 1).

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kJ/mol, respectively, and offers much better accuracy when vdW contributions are large (MAE ca. 15 kJ/mol smaller than BEEF) while maintaining essentially the same accuracy as BEEF for pure chemisorption systems. Given the trends in DFT-based surface chemistry research to study larger adsorbates and include solvent molecules, vdW contributions will be increasingly important in atomistic modeling, so this improvement may prove impactful.

2. Methods Periodic, self-consistent DFT calculations were performed using the Vienna Ab Initio Simulation Package (VASP)1, 20 with the optB88-vdW and optB86bvdW functionals.21-22 The two optB functionals, as well as the RPBE and BEEF-vdW functionals, are all conveniently implemented in the latest versions of VASP. The core electrons were treated using the projector augmented waves method23 and the valence electrons were expanded in plane wave basis sets. The adaptive summing method introduced here should work regardless of the basis set, whether plane waves or atomic orbitals, and regardless of how the reciprocal space is treated. Results generated using other exchange-correlation functionals discussed in this work (e.g. RPBE and PBE) are reproduced from Ref. 18. Final optimized adsorption structures from Ref. 18 were used as the initial structures that were further optimized using each optB functional. Gas-phase molecules used in the calculation of the adsorption energies were optimized in a simulation cell with at least 8 Å on a side. All surfaces were modeled as slabs containing four atomic layers with the bottom two layers kept fixed in their bulk positions and at least 10 Å vacuum between neighboring slabs. The MethfesselPaxton smearing method,24 with a smearing width of 0.05 eV, was used for the slab calculations. The convergence criteria for the electronic self-consistent energy and the ionic relaxation were set to 10-4 eV and 0.025 eV/Å, respectively. For each adsorption system, three levels of optimization were performed, consistent with that done in Ref. 18. First, the system was optimized using a cutoff energy of 400 eV and a Γ-centered k-point mesh of either (4×4×1) for p(2×2) supercells or (2×2×1) for p(3×3) supercells. Second, the system was optimized using a cutoff energy of 500 eV and a Γ-centered k-point mesh of either (6×6×1) for p(2×2) supercells or (4×4×1) for p(3×3) supercells. Finally, a static calculation was performed on the optimized geometry from the second step with a cutoff energy of 750 eV and a Γ-centered kpoint mesh of either (8×8×1) or (6×6×1) for p(2×2) or p(3×3) supercells, respectively. The adsorption energy values for the last step are reported herein. As shown in Table S5, increasing the energy cutoff from 750 eV to 820 eV has a negligible change on the adsorption energy

Figure 1. Comparison of the accuracy of different density functionals relative to the CE39 experimental data set for: (A) Chemisorption systems (Reactions 1-25 from Ref. 18); (B) systems where vdW interactions make large contributions to adsorption (Reactions 26-39 from Ref. 18); and (C) the full CE39 data set. For each data set, the bars illustrate the mean signed error (MSE, red bars), the mean absolute error (MAE, white bars), and the root mean squared error (RMSE, grey bars), per molecular fragment produced upon adsorption. The meaning of the SW results is explained in the main text.

We introduce here a DFT-based adaptive method that decreases the MAE and RMSE to 13.5 and 19.6


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for all 39 reactions studied for this effect. Therefore, our choice of 750 eV for energy cutoff was deemed sufficient. The lattice constant for each metal investigated in this work was calculated with a cutoff energy of 400 eV and a Γ-centered (12×12×12) k-point mesh (optB88) and with a cutoff energy of 750 eV and a Γ-centered (16×16×16) k-point mesh (optB86b). All of the lattice constants are listed in Table S5.

bonded adsorption systems by significantly underbinding such adsorbates.18 However, this has not prevented a lot of excellent research to be done with RPBE when adsorbate reaction energies were of greater concern,4 since RPBE is more accurate for chemisorption energies (Fig. 1A). The accurate description of adsorption energies both for processes dominated by charge transfer and for those dominated by dispersion forces thus remains a challenging task. Here we present a simple, accessible method that combines the best features of the RPBE and optB86b (or optB88) functionals but avoids much of their deficit. This approach is parameterized and tested against the CE39 experimental data set. It markedly improves the accuracy of the predicted adsorption energies of common adsorbates on late transition metal surfaces compared to both parent functionals (and to BEEF) for the full CE39 data set and its subsets (chemisorption and vdW adsorption). This improvement could have a major impact on catalysis and surface science research. The overall performance of the two optB functionals with respect to the ZPE-corrected experimental reaction energies is visualized in Fig. 1. These weighted errors are calculated in terms of the MAE and also the mean signed error (MSE) and the root mean squared error (RMSE) as defined in Ref. 18. Note that Fig. 1 shows the weighted average error quantities, where the weights (as defined in Ref. 18) have been chosen such that the DFT errors are per molecular fragment produced upon adsorption. The values of the different measures of errors are reported in Tables S1-S3 in the SI. Based on references cited in Ref. 18, the MAE of the experimental energies is certainly below 6 kJ/mol per molecular fragment. Fig. 1A shows that RPBE and BEEF perform quite well for the 25 predominantly chemisorption processes, but the optB functionals perform poorly for this subset of the CE39 data set and significantly over-bind the adsorbates. The BEEF functional performs the best for these 25 adsorbates among the six originally tested functionals, while RPBE is slightly less accurate (MAE = 17.7 vs. 16.0 kJ/mol).18 Fig. 1B shows the errors for adsorbates with large vdW contributions. Both RPBE and BEEF clearly under-bind this class of adsorbates. On the other hand, the optB functionals perform very well here with a MAE of 12 kJ/mol. Most importantly, none of these functionals perform satisfactorily for the overall CE39 data set, as shown in Fig. 1C.

3. Results and Discussion 3.1 Evaluation of optB vdW functionals We began by systematically evaluating the accuracy of a recently developed vdW method against the CE39 data set. We tested two self-consistent vdW functionals, optB86b-vdW and optB88-vdW21-22 (shortened to optB86b and optB88 below). These and several other vdW functionals, which are based on the approach of Langreth and Lundqvist that combines vdW correlation with an existing exchange,25 are the only type of selfconsistent vdW functionals that have been developed so far. The early versions often do not provide accurate adsorption energies for vdW adsorbate systems. Such inaccuracies are remedied to a large extent with the optB86b and optB88 functionals. The reaction energies for the adsorption reactions in the CE39 data set calculated using these functionals (and several other DFT functionals including PW91, PBE, RPBE, and BEEF reproduced from Ref. 18) are shown in Table 1, and the comparison of these DFT energies to the CE39 experimental energies are presented in Tables S1-S3. Although these functionals were originally optimized for bulk and molecular systems respectively, Table S2 shows that they provide much improved energetic descriptions for adsorption systems with large vdW contributions. However, Table S1 shows that they consistently over-bind adsorbates with strong chemisorption. Systematic benchmarking results encompassing both vdW and chemisorbed adsorbates are scarce for other vdW methods. What is available in the literature points to similar deficiencies in those methods in describing the adsorption of small adsorbates on metal surfaces. For instance, when benchmarked against a set of 13 adsorbates on Pt(111), the PBE-dDsC functional26 performed well, yielding an MAE of 23 kJ/mol (optPBE-vdW yielded an MAE of 20 kJ/mol).27 However they over-estimated the CO adsorption energy on Pt(111) by at least 39 kJ/mol. In contrast, the RPBE functional28 was optimized for predicting the adsorption energies of small adsorbates on late transition metal surfaces, and does well for those strong chemisorption systems. RPBE does not do as well as some DFT functionals, such as PBE from which it was derived, for bulk metal properties and surface energies29, and it has very large errors for vdW-



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Table 1. Theoretical adsorption reaction energies, (in kJ/mol, rounded to whole numbers) for 39 adsorption reactions on transition metal surfaces as written1), using the BEEF-vdW, RPBE, optB86b-vdW, optB88-vdW, and the SW-R86b and SW-R88 “functionals” proposed in this work. Also shown are the x and f(x) values associated with each adsorption reaction for both SW-R86b and SW-R88. #

Surface reaction

Adsorbate SWSWcoverage BEEF5) RPBE5) optB86b5) optB885) R86b5) R885) as written2)

x

x

f(x)

f(x)

86b

88

86b

88

1

CO + Ni(111) → CO/Ni(111)

1/4

-146

-140

-202

-186

-140

-140

0.308 0.249

0.005 0.000

2

CO + Pt(111) → CO/Pt(111)

1/4

-133

-136

-175

-156

-136

-136

0.222 0.126

0.000 0.000

3

CO + Pd(111) → CO/Pd(111)

1/4

-156

-153

-210

-187

-153

-153

0.270 0.180

0.000 0.000

4

CO + Pd(100) → CO/Pd(100)

1/4

-150

-154

-201

-186

-154

-154

0.235 0.170

0.000 0.000

5

CO + Rh(111) → CO/Rh(111)

1/4

-162

-163

-201

-188

-163

-163

0.190 0.133

0.000 0.000

6

CO + Ir(111) → CO/Ir(111)

1/4

-170

-169

-205

-198

-169

-169

0.174 0.144

0.000 0.000

7

CO + Cu(111)→ CO/Cu(111)

1/4

-55

-56

-93

-82

-93

-56

0.397 0.316

1.000 0.000

8

CO + Ru(001) → CO/Ru(001)

1/4

-162

-162

-197

-185

-162

-162

0.178 0.123

0.000 0.000

9

CO + Co(001) → CO/Co(001)

1/4

-138

-137

-179

-167

-137

-137

0.237 0.180

0.000 0.000

10

NO + Ni(100) → N/Ni(100) + 1/4 -370 O/Ni(100) (separated)3)

-375

-498

-437

-375

-375

0.247 0.142

0.000 0.000

11 NO + Pt(111) → NO/Pt(111)

1/4

-149

-148

-215

-201

-149

-148

0.310 0.263

0.008 0.000

12 NO + Pd(111) → NO/Pd(111)

1/4

-186

-188

-250

-233

-188

-188

0.249 0.193

0.000 0.000

13 NO + Pd(100) → NO/Pd(100)

1/4

-174

-172

-244

-230

-172

-172

0.294 0.252

0.001 0.000

14 O2 + Ni(111) → 2O/Ni(111)

1/8

-418

-393

-501

-489

-393

-393

0.216 0.197

0.000 0.000

15 O2 + Ni(100) → 2O/Ni(100)

1/8

-471

-444

-561

-521

-444

-444

0.208 0.147

0.000 0.000

16 O2 + Pt(111) → 2O/Pt(111)

1/18

-190

-159

-275

-276

-275

-208

0.421 0.423

1.000 0.420

17 O2 + Rh(100) → 2O/Rh(100)

1/8

-387

-355

-490

-473

-355

-355

0.276 0.249

0.000 0.000

18 H2 + Pt(111) → 2H/Pt(111)

1/8

-48

-66

-87

-63

-66

-66

0.239 -0.052 0.000 0.000

19 H2 + Ni(111) → 2H/Ni(111)

1/8

-64

-79

-109

-99

-79

-79

0.274 0.200

0.000 0.000

20 H2 + Ni(100) → 2H/Ni(100)

1/8

-56

-68

-103

-92

-84

-68

0.341 0.261

0.442 0.000

21 H2 + Rh(111) → 2H/Rh(111)

1/8

-67

-84

-110

-97

-84

-84

0.235 0.133

0.000 0.000

22 H2 + Pd(111) → 2H/Pd(111)

1/8

-67

-88

-111

-88

-88

-88

0.209 -0.004 0.000 0.000

23 I2 + Pt(111) → 2I/Pt(111)4)

1/8

-269

-229

-351

-319

-313

-229

0.348 0.283

0.687 0.000

24

CH2I2 + Pt(111) → CH/Pt(111) + 1/4 -340 H/Pt(111) + 2I/Pt(111) (separated)3)

-286

-464

-403

-463

-286

0.383 0.291

0.997 0.000

25

CH3I + Pt(111) → CH3/Pt(111) + 1/4 -164 I/Pt(111) (separated)3)

-114

-226

-208

-226

-208

0.495 0.452

1.000 0.999

26 NH3 + Cu(100) → NH3/Cu(100)

1/4

-41

-36

-61

-60

-61

-36

0.412 0.396

1.000 0.001

27 CH3I + Pt(111) → CH3I/Pt(111)

1/4

-36

-16

-70

-67

-70

-67

0.772 0.761

1.000 1.000

1/4

-31

-16

-56

-31

-56

-31

0.713 0.477

1.000 1.000

28

CH3OH + Pt(111) → CH3OH/Pt(111)


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29 CH4 + Pt(111) → CH4/Pt(111)

1/4

-15

-1

-21

-22

-21

-22

0.952 0.955

1.000 1.000

30 C2H6 + Pt(111) → C2H6/Pt(111)

1/9

-21

-3

-31

-35

-31

-35

0.904 0.913

1.000 1.000

31 C3H8 + Pt(111) → C3H8/Pt(111)

1/9

-29

-4

-47

-51

-47

-51

0.915 0.922

1.000 1.000

32 C4H10 + Pt(111) → C4H10/Pt(111)

1/9

-38

-6

-60

-64

-60

-64

0.901 0.906

1.000 1.000

33 C6H6 + Pt(111) → C6H6/Pt(111)

1/9

-76

-10

-204

-161

-209

-161

0.952 0.938

1.000 1.000

34 C6H6 + Cu(111) → C6H6/Cu(111)

1/9

-82

-20

-69

-70

-69

-70

0.709 0.716

1.000 1.000

35 C6H6 + Ag(111) → C6H6/Ag(111)

1/9

-21

10

-63

-64

-63

-64

1.159 1.157

1.000 1.000

36 C6H6 + Au(111) → C6H6/Au(111)

1/9

-18

22

-76

-76

-76

-76

1.290 1.289

1.000 1.000

37 C6H10 + Pt(111) → C6H10/Pt(111)

1/9

-85

-23

-164

-149

-164

-149

0.860 0.846

1.000 1.000

-21

-5

-41

-39

-41

-39

0.879 0.872

1.000 1.000

-52

-30

-58

-91

-58

-91

0.483 0.669

1.000 1.000

38 H2O + Pt(111) → H2O/Pt(111) H2O + ⅓O/Pt(111) →

39

⅔(H2O···OH)/Pt(111) 1)

1/4 1/2

6)

See Ref. 18 for side and top view of calculated structures.

2)

Coverages here are defined as the number of reacted gas molecules in the reaction as written per metal surface atom, except as otherwise noted. 3)

In these reactions where two or more different adsorbates are produced, each adsorbate was calculated separately at the stated coverage of that adsorbate alone, and then the energies of these separated adlayers were properly summed. Thus, the local coverages here are different from those determined experimentally and reported in Ref. 18, but were chosen to get the computations done within reasonable timeframes. 4)

Calculated values for this reaction are referenced to molecular I2 gas, instead of an iodine radical as used experimentally in Ref. 18. Using I2(gas) as the reactant together with the experimental zero-Kelvin dissociation energy of I2 gas (148.5 kJ/mol) changes the experimental value from Ref. 18 to ∆Eexperimental = 2(– 230 kJ/mol) + (148.5 kJ/mol) = –312 kJ/mol, for direct comparison to these numbers. 5)

DFT calculated reaction energies, not including ZPE corrections. The BEEF and RPBE values are taken from Ref. 18. 6)

The coverage of Reaction 38 is different from that reported in the experiment of Ref. 18, but the same as used in the BEEF and RPBE calculations of Ref. 18 cited above.



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We have found that the sigmoid function is an excellent form of f(x) for x between 0 and 1 (as is almost always the case):

3.2 Adaptively weighted summing for improved accuracy These findings suggest that a properly weighted mixture (Emix) of the optB adsorption energy (E(optB)) and RPBE adsorption energy (E(RPBE)) may have improved accuracy over both parent energies: ∙ 1 ∙

! ⁄"

(3)

This function makes a smooth and symmetric transition from 0 to 1 centered at a, with a width set by b. Therefore, this method features an adaptive weighting factor that varies from one adsorbate system to another based on the calculated adsorption energies according to the two parent functionals. We optimized the values of a and b by minimizing the MAE between Emix and the CE39 experimental data set, using a generalized reduced gradient algorithm. We used the MAE to represent the best estimate of true accuracy, since with the MAE negative errors do not cancel positive errors (as in the MSE), nor are data points given different weights (as in the RMSE). The best parameters are {a, b} = {0.343, 0.00672} for optB86b and {0.424, 0.00400} for optB88. The RPBE energies reported in Ref. 18 are used here. The functions f(x) are plotted in Fig. 2. The results from this “sigmoidweighted” (SW) sum of the RPBE and optB86b/88 energies henceforth will be referred to as SW-R86b and SW-R88, respectively. The values for x and f(x) and the resulting Emix thus calculated for each reaction are listed in Table 1. The different measures of errors for SWR86b and SW-R88 are shown in Fig. 1 for comparison to the individual functionals. We made no attempt to use other functional forms for f(x). The SW “functionals” have by far the smallest errors. We place “functionals” in quotes because these constructs are not true density functionals. For the pure chemisorption systems (Fig. 1A), the SW-R86b and SW-R88 paired functionals perform much better than the optB functionals and have a similar accuracy to RPBE and BEEF. A reverse trend is observed for the vdW-dominated subset of the CE39 data set (Fig. 1B), where SW-R86b and SW-R88 perform similarly to the optB functionals (MAE = 12.0 and 13.6 kJ/mol, respectively), with greatly improved performance over RPBE and BEEF (MAE = 56.6 and 28.4 kJ/mol, respectively). Finally, when compared to the overall CE39 data set, SW-R86b and SW-R88 have better performance than any of their parent functionals (optB86b, optB88, and RPBE), with a MAE of 13.5 and 14.4 kJ/mol respectively, which are ca. 7 kJ/mol smaller than BEEF. Further, the MSE and RMSE for SW-R86b and SW-R88 are |6| kJ/mol or less, and 19.6 kJ/mol, respectively, both clearly improved over BEEF (MSE and RMSE of 10.1 and 26.4 kJ/mol). Significantly, the overall improved accuracy of the SW functionals is achieved without developing a new actual vdW functional. This SW approach for adaptively weighting the sum of energies from RPBE and optB is far better than any

(1)

where f(x) is some smooth function that defines the weight to be assigned to each parent energy, with f equal to 0 for pure chemisorption and 1 for pure vdW interactions. We define x as:

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(2)

This definition is motivated by our previous study on tbutyl adsorption on Pt(111), which involves large contributions from both chemisorption and vdW interactions and suggests that the optB86b adsorption energy minus the PBE adsorption energy provides a good estimate of the vdW contribution to the adsorption energy (as estimated experimentally by comparing the adsorption energies of t-butyl and methyl).30 In a few cases the adsorbate is unbound according to RPBE (i.e. positive adsorption energy) and Eqn. (2) gives x > 1.0. We define f(x) as 1.0 for these systems. In two cases (i.e. Reactions 18 and 22) the RPBE adsorption energy is slightly more exothermic than the optB value, causing x to become negative. In such cases, we define f(x) as 0.

Figure 2. Best-fit sigmoid weighting functions, f(x) (Eqn. 3), found for SW-R86b and SW-R88, that minimize the MAE between the Emix (Eqn. 1) and 39 experimental energies in the CE39 data set. These functions are plotted versus x (Eqn. 2), which estimates the fraction of the adsorption energy due to vdW contributions. The triangles and circles correspond to the subsets of the CE39 data set: chemisorption (Reactions 1-25) and those with large vdW contributions (Reactions 26-39), respectively. Systems with intermediate values of f(x) are shown as black symbols highlighted in pink, orange, and yellow for Reactions 16, 20, and 23, respectively (see Table 1).


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weighted sum using a single weighting factor f of the type: ∙ 1 ∙ (4) We found the best single value for f for the entire CE39 data set by minimizing the MAE between Emix and the experimental energy, including all 39 reactions in this MAE. The lowest MAE value was found to be 23.9 kJ/mol when combining the RPBE and optB86b functionals (best f = 0.704). This MAE is 10.4 kJ/mol larger than the MAE of 13.5 kJ/mol for SW-R86b. Similarly, a MAE value of 21.2 kJ/mol was obtained when combining the RPBE and optB88 functionals (best f = 0.794), again much larger than the MAE value of 14.4 kJ/mol for SW-R88. We note that f(x) is nearly a step function with f(x) = 0 or 1 or nearly so for most of the 39 systems in the CE39 data set. This means that almost all of the adsorption reactions fall into two categories: predominantly chemisorption systems with f ≲ 0.1 (for small adsorbates including H, O, I, CO, NO) and adsorptions with large vdW contributions for which f ≳ 0.9 (i.e. H2O, alkanes). However, one cannot know in advance for a given system whether f(x) = 0 or 1. For example, the authors of Ref. 18 classified the CE39 reactions into “chemisorption only” or “large vdW contributions”, with full knowledge of the RPBE and BEEF-vdW adsorption energies, but our results here proved that those intuition-based assignments would give wrong values for f(x) in many cases. That is, if one chose f(x) to be 0 or 1 based on that original classification, f(x) would be very incorrect in many cases. For example, in SW-R86b, f(x) would have been 0 but is not for Reactions #7 (f(x)=1.000), #16 (1.000), #20 (0.442), #23 (0.687), #24 (0.997), #25 (1.000). Similarly, in SW-R88, f(x) would have been 0 but is not for Reactions #16 (f(x)=0.420) and #25 (0.999). Further, in SW-R88, f(x) would have been 1 but is not for Reaction #26 (f(x)=0.001). Thus, the method described here for calculating x and f(x) is absolutely essential in achieving proper mixing. Furthermore, the resulting values of f(x) reveal new physical insight into the nature of the bonding, and strengths and weaknesses of the individual functionals used in this adaptive sum. Note that several of the systems in the CE39 data set include large contributions to their adsorption energies from both normal chemisorption and vdW interactions. For example, benzene, cyclohexene and the H2O-OH complex on Pt(111) all have strong C-Pt or O-Pt chemisorption bonds but also have strong vdW attractions. For these three systems, SW-86 and SW-88 give MAEs (relative to experiments) of 32 and 17 kJ/mol, respectively, while the MAE for BEEF is 46 kJ/mol. Since f(x)=1 for these systems, this improvement is essentially due to the fact that optB-vdW functionals are more accurate than BEEF for such systems.

To test the predictive capabilities of our method beyond the CE39 data set, we used a cross validation approach whereby we divided the CE39 data set into 39 different training sets, with a different one of the 39 adsorption energy values left out of each to serve as the test set. The remaining 38 experimental adsorption energies were used to fit the sigmoid function, and its accuracy in predicting the remaining “test” value was then determined. This was repeated 39 times. This procedure was done for both the SW-R86b and the SWR88 functionals. As can be seen when examining Figure S1 in the SI, the values of a and b hardly change from one fitting set to another. This cross validation approach of repeatedly dividing the data into a fitting set and a test set is commonly used to validate the general applicability of fitted lateral interaction potentials between adsorbates in comparison to DFT energies, where it has been demonstrated that a good measure of the robustness of the fitted potential is to calculate the corresponding average prediction error, √Scv.31-33 Here, we arrive at a √Scv value of 19.8 kJ/mol for SW-R86b and 20.6 kJ/mol for SW-R88. These prediction error values are very close to the RMSE values for both functionals (19.6 kJ/mol), to which the √Scv would reduce if all the fits were identical. These results strongly indicate that SW-R86b and SW-R88 are robust and relatively insensitive to the number of experimental values used to construct them. Such robustness indicates that our method will be transferable to other adsorbates on transition metals. Its robustness derives from the fact that it uses a weighting factor in summing the RPBE and optB energies that is adaptively determined for each new adsorbate/surface combination based on the RPBE and optB energies for that particular system. To our knowledge, such an adaptive approach has never been done before in pairing DFT functionals. It may even suggest a whole new class of approaches to DFT calculations using adaptive sums of energies. The adaptive weighting used in the SW functionals introduced here distinguishes them markedly from previous energy mixing schemes,34-36 including the Specific Reaction Parameter (SRP) approach, which is the only example so far where energies from two different DFT functionals were used in a weighted sum to more accurately fit experimental results for adsorption energies.37-38 SRP functionals use a single weighting factor in the energy sum for all systems to which that specific SRP functional is applied, whereas the SW functionals introduced here use a weighting factor that is adaptively determined for each system based on the calculated energies from the two functionals being summed. Thus, it is much better designed to describe diverse adsorption systems beyond the data set used to train it, possibly even including adsorptions on materials other than transition metals.



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Overall, these results show that the adaptive method proposed here for pairing the RPBE and optB adsorption energies creates a more accurate prediction of experimental adsorption energies for both chemisorptions and vdW adsorptions than any of the parent functionals, and offers a readily available solution that unifies the treatment of both types of adsorptions with equal accuracy, a crucial ability that is lacked by any single density functional to date. The additional computational time costs for this method are modest, adding roughly 140% compared to the cost of deploying RPBE alone (see SI). Our approach is not limited to RPBE and optB86b/88. It can be based on another standard DFT functional instead of RPBE (and conceivably another vdW method instead of optB86b/88), to achieve better accuracy than the parent methods. Just as we have done above with RPBE, we have also tested an “SW-86b functional” where we combined PBE with optB86b. The best parameters were found to be {a, b} = {0.271, 0.00400}. The averaged errors for the full CE39 data set were found to be: MAE = 17.3 kJ/mol, RMSE = 23.3 kJ/mol, MSE = -12.7 kJ/mol. Similarly, we developed an “SW-88 functional” where we combined PBE with optB88. The best parameters were found to be {a, b} = {0.264, 0.00323}, giving average errors of MAE = 18.2 kJ/mol, RMSE = 23.3 kJ/mol, MSE = -9.2 kJ/mol. The combinations of optB with PBE are not quite as accurate in predicting adsorbate energies as their combinations with RPBE, but still better than the existing single functionals (cf. Fig. 1C). This highlights the power and flexibility of our method in significantly improving the accuracy of existing density functionals for adsorption energies. Other methods have been developed recently that are achieving high accuracy for systems with large vdW contributions, but these do not simultaneously provide the accuracy for pure chemisorption systems that we have demonstrated here for the SW functionals. Even the many body dispersion (MBD) method developed by Tkatchenko et al.,39 which is highly accurate for molecules with large vdW contribution to adsorption,4043 nonetheless does poorly for molecules in the chemisorption group. For instance, we calculated CO and NO adsorption on Pt(111) using the PBE+MBD@rsSCS method39 as implemented in VASP, on p(2×2) supercells at the 500 eV cutoff energy level as described above. The adsorption energies for CO and NO were found to be -152 and -176 kJ/mol, respectively, which are comparable to the -156 kJ/mol for CO (PBE and PW91) and -177 (PW91) and -166 (PBE) kJ/mol for NO, but are over-bound by 28 and 57 kJ/mol compared to experiment (-124 kJ/mol for CO and -119 kJ/mol for NO).18

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3.3 Treatment of coadsorbates Some surface reactions involve several adsorbates, for example when both reactants and product are adsorbates. For such reactions, one should apply the SW functionals by separately calculating the energies of each adsorbate relative to the gas-phase using the procedure outlined above, and then construct an appropriate thermodynamic cycle to get the reaction energy of interest (using accurate experimental gas-phase energies as needed to complete this cycle). This approach simply recognizes the fact that the parameters in the sigmoid function were determined using adsorbate energies relative to the gas phase. More specifically, the parameterization was done using stable gas-phase molecules (and not radicals), so radicals should be avoided as the gas-phase reference species. One may apply this method to study adsorbateadsorbate interactions where, for example, the distance between two different adsorbates (A* and B*) is varied by changing the unit cell size in which the two adsorbates are placed. To do this, the gas-phase reference for this coadsorbate system must include both the gas-phase species that serve as references for both A* and B* alone, and calculate the appropriate x value using Eq. (2) with the two adsorption energies (E(optB) and E(RPBE)) referring to the total adsorption energy from the gas-phase combination to the coadsorbed combination. One shortcoming of the SW approach is that the two separate adsorbates use two different x values in Eq. (2), but the coadsorbed system uses a single x value for their combination, so that there can be a difference between the combined energy to make the coadsorbed system and the sum of the energies for two separate isolated adsorbates, even in the limit of no interaction energy between the two different adsorbates. To prove that this difference is small, we calculated the SW-R86b and SW-R88 coadsorption energies for all 136 binary combinations of the 17 different adsorbates on Pt(111) in the CE39 data set, excluding pairs of identical adsorbates and assuming no interactions between the coadsorbates (i.e., simply summing the optB or RPBE adsorption energies for the individual adsorbates to get the corresponding total optB or RPBE energies, which are then used to calculate x for the coadsorbed pair as a single system). The absolute value of the difference (per adsorbed fragment) between this SW energy of the coadsorbed pair and the sum of the SW energies for the two separate adsorbates, averaged over all 136 pairs, was only 6.3 kJ/mol for SW-R86b and 8.3 kJ/mol for SW-R88. Lists of the pairwise coadsorption energies and these absolute differences are shown in Tables S6 to S8 for pairing one chemisorbed species and one largely vdW adsorbed species, two chemisorbed species, and two largely vdW adsorbed species, respectively. This mean absolute difference


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drops further to 5.6 kJ/mol (for SW-R86b) if we include in this average the 17 combinations of pairs of identical adsorbates, which have 0 kJ/mol difference. These differences are very small compared to the MAEs in Fig. 1, indicating that this is not a major concern, but nevertheless a weakness of the SW approach. The difference arises mostly from combinations where x is near 0 for one adsorbate and near 1 for the other. (When x is very near to either 0 or 1 for both adsorbates, there is essentially no difference since their combination has that same x value.) Because of this difference, the MAE per fragment in the SW-R86b energies of these coadsorbed pairs relative to summed experimental energies was 2.6 kJ/mol larger than the MAE in the SW-R86b energies for the single adsorbates (16.1 kJ/mol vs. 13.5 kJ/mol; for SW-R88 they are 17.0 kJ/mol vs. 14.4 kJ/mol). One can easily remove this additional 2.6 kJ/mol error by subtracting the above-mentioned SW energy difference for non-interacting coadsorbates from all coadsorbate calculations after each coadsorbate calculation (i.e., at whatever distance they are placed apart for any particular structure being calculated).

method for estimating entropies based on an SWweighted sum analogous to Eqn. (1) is proposed, as discussed in the Supporting Information. A serious limitation of this SW method is that it is not a true functional that can always provide energy versus geometry. It often only provides adsorbate energies, but it does it better than any single DFT functional. This limited information is very important, however, because it is the adsorbate energies that are most important in microkinetic modeling in catalysis. If one tried to use f(x) to create a true functional, one should beware that the derivative of f(x) has small discontinuities at x = 0 and x = 1. Since f(x) is very close to zero or unity, respectively, on both sides of these points, one could perhaps find a numerical method to bypass this problem, or simply use the sigmoid function in Eq. (3) as a very close approximation for f(x) at all x values.

4. Conclusions To conclude, the optB86b and optB88 vdW functionals have been evaluated for their ability to predict the adsorption energies in the CE39 experimental data set, which contain both pure chemisorption and vdWdominated reactions specifically chosen to be used to benchmark DFT functionals for modeling common heterogeneous catalytic reactions. The optB functionals are found to perform well for the smaller subset of CE39 that involves large vdW contributions, but poorly for pure chemisorption, for which RPBE does much better. We show that an adaptively weighted sum of the energies from RPBE and optB, i.e. SW-R86b and SWR88, perform far better than both RPBE and optB for the CE39 data set. Most importantly, it offers a marked improvement over the best single functional, BEEF, with a MAE of only 13.5 kJ/mol, vs. 20.4 kJ/mol for BEEF. For adsorbates with large vdW contributions to bonding, the improvement is even greater (MAE = 12.0 kJ/mol, vs. 28.4 kJ/mol for BEEF). Our method offers a simple and effective solution that unifies the treatment of both types of adsorptions with equally high accuracy, a crucial ability that is lacked by any single density functional to date. It is transparent and easily accessible, and does not require prior categorization of a system as chemisorption or vdW adsorption, and can accommodate adsorptions that have large contributions from both. Further, this method is readily extensible when accurate calorimetry measurements of additional adsorptions become available in the future.

3.4 Applicability beyond adsorption energies Finally, since this method is much more accurate than any current functional for the initial and final states of elementary steps on metal surfaces (i.e., the minima in the potential energy surface (PES)), it is expected to be better also at predicting the energies of transition states (saddle points on the PES). To do this, one would calculate the transition state energy (relative to suitable gas phase species) separately in RPBE and in optB-vdW, then calculate x for the transition state and sum the RPBE and optB-vdW energies with weighting factor f(x) determined by exactly the same weighting algorithm as used above for stable adsorbates. We have not yet tested this, since no database of accurate, experimentally measured activation energies for adsorbate reactions on metal surfaces is presently available.44 Similarly, although the SW functionals do not directly provide geometries or vibrational frequencies, when f(x) is close to 0 or 1, which is true in almost all cases tested, one can simply use the corresponding geometries that come from RPBE and optB, respectively. In those cases, one can perhaps better understand this SW method by thinking of f(x) as a variable that simply decides which functional to use (RPBE or optB-vdW) for any particular adsorbate, pair of adsorbates, or transition state. Given the nearly stepfunction behavior of f(x) (see Fig. 2), this is almost always an appropriate interpretation of this method. The modeling of geometries, vibrational frequencies, entropies and kinetic pre-factors for intermediate cases where f(x) is closer to 0.5 is problematic, since the two functionals can give very different geometries for the same adsorbate or transition state. For these situations, a

ASSOCIATED CONTENT

Supporting Information. The Supporting Information is available free of charge on the ACS Publications website as a PDF file, which contains:



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Detailed computational methods; comparison of the computational costs of the proposed functional pairing method; suggestions on modeling adsorbate geometries, vibrational frequencies, entropies and kinetic prefactors using SW-R86b and SW-R88; calculated adsorption energies for all 39 adsobate systems in the CE39 data set using BEEF, RPBE, optB86b, optB88, SW-R86b, and SW-R88 functionals and the corresponding x and f(x) values for SW-R86b and SW-R88; lattice constants for the bulk metals calculated using optB86b and optB88; and lists of the pairwise coadsorption energies and absolute differences between this SW energy of the coadsorbed pair and the sum of the SW energies of the two separate adsorbates. AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected], [email protected]. Funding Sources

No competing financial interests have been declared. ACKNOWLEDGMENT

Work at WSU was funded by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Chemical Sciences, Biosciences and Geosciences (award #DE-SC0014560). C.R. thanks the Research Internships in Science and Engineering program, which is supported by the Federal Republic of Germany through the Federal Ministry of Education and Research. K.G. and Y.X. thank the Donors of the American Chemical Society Petroleum Research Fund for support for this work at LSU. C.T.C. thanks the U.S. National Science Foundation for support for this work (grant #CHE-1361939), and Líney Árnadóttir for helpful discussions. The computational work was conducted using high performance computational resources provided by WSU, LSU, and NERSC, which is supported by the Office of Science of the U.S. Department of Energy under contract #DE-AC02-05CH11231.

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DFT-Based Method for More Accurate Adsorption Energies: An

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