Properties of Real Metallic Surfaces: Effects of Density Functional Semilocality and Van Der Waals Nonlocality

Properties of real metallic surfaces: Effects of density functional semilocality and van der Waals nonlocality

Abhirup Patraa,1, Jefferson E. Batesa, Jianwei Sunb, and John P. Perdewa,c,1

aDepartment of Physics, Temple University, Philadelphia, PA 19122; bDepartment of Physics and Engineering Physics, Tulane University, New Orleans, LA 70118; and cDepartment of Chemistry, Temple University, Philadelphia, PA 19122

Contributed by John P. Perdew, September 6, 2017 (sent for review July 27, 2017; reviewed by John F. Dobson and J. M. Pitarke) We have computed the surface energies, work functions, and 30) that remain computationally efﬁcient, including the recent interlayer surface relaxations of clean (111), (100), and (110) sur- strongly constrained and appropriately normed (SCAN) meta- faces of Al, Cu, Ru, Rh, Pd, Ag, Pt, and Au. We interpret the sur- GGA (30, 31). There have also been surface studies based upon face energy from liquid metal measurements as the mean of the the random phase approximation (RPA) (32). solid-state surface energies over these three lowest-index crystal SCAN captures intermediate-range van der Waals (vdW) faces. We compare experimental (and random phase approxima- interaction [responsible for the equilibrium binding of two tion) reference values to those of a family of nonempirical semilo- closed-shell atoms (30, 31, 33)], but capturing longer-ranged cal density functionals, from the basic local density approximation vdW interaction requires the addition of a nonlocal vdW cor- (LDA) to our most advanced general purpose meta-generalized rection as from the revised Vydrov–Van Voorhis 2010 (rVV10) gradient approximation, strongly constrained and appropriately functional (34, 35). The intermediate-range vdW interaction is normed (SCAN). The closest agreement is achieved by the sim- crucial for SCAN’s correct description of liquid water (36). plest density functional LDA, and by the most sophisticated one, Ref. 35 suggests that the vdW interaction is semilocal at short SCAN+rVV10 (Vydrov–Van Voorhis 2010). The long-range van der and intermediate range, but displays pairwise full nonlocality at Waals interaction, incorporated through rVV10, increases the sur- longer ranges, and displays many-body full nonlocality (37, 38) face energies by about 10%, and increases the work functions by at the longest and often least energetically important distances. about 3%. LDA works for metal surfaces through two known error Accounting for intermediate and long-ranged vdW interactions cancellations. The Perdew–Burke–Ernzerhof generalized gradient is especially important for layered materials (35, 39, 40) and ionic approximation tends to underestimate both surface energies (by solids (41–43). The vdW interactions are also needed to cor- about 24%) and work functions (by about 4%), yielding the least- rect the errors of GGAs for bulk metallic systems (42). Ref. 44 accurate results. The amount by which a functional underesti- reports long-range vdW interaction between two jellium slabs. mates these surface properties correlates with the extent to which The importance of the vdW contribution for the surface energy it neglects van der Waals attraction at intermediate and long and the work function will be demonstrated here. By naturally range. Qualitative arguments are given for the signs of the van accounting for both intermediate- and long-range interactions, der Waals contributions to the surface energy and work function. SCAN+rVV10 (35) represents a major improvement over pre- A standard expression for the work function in Kohn–Sham (KS) vious functionals for many properties of diversely bonded sys- theory is shown to be valid in generalized KS theory. Interlayer tems (31). Its pairwise interactions at long range even match the relaxations from different functionals are in reasonable agree- RPA binding energy curve for graphene on a nickel surface (35). ment with one another, and usually with experiment.

metallic surfaces | density functional theory | van der Waals interaction Signiﬁcance

It is primarily at their surfaces that solids interact with their he rapid development of electronic structure theory has environments. What is the physics behind the measurable Tmade it easier to analyze and describe complex metallic properties of clean metallic surfaces? To answer this ques- surfaces (1), but understanding the underlying physics behind tion, we calculate surface energies, work functions, and sur- surface energies, work functions, and interlayer relaxations has face interlayer relaxations for aluminum and seven d-electron remained a long-standing challenge (2). Metallic surfaces are of metals, using a sequence of exchange-correlation density particular importance because of their wide range of applica- functionals of increasing sophistication. While the simplest tions, including catalysis (3–8). A detailed knowledge of the elec- one, the local density approximation, works well through tronic structure is required for accurate theoretical investigations error cancellation, the usually more realistic Perdew–Burke– of metallic surfaces (9, 10). Ernzerhof functional underestimates both surface energies Consequently, metal surfaces have played a key role in the and work functions. The more advanced functionals, includ- development and application of Kohn–Sham density functional ing the new strongly constrained and appropriately normed theory (KS DFT) (11). The work of Lang and Kohn (12–14) in (SCAN) and SCAN+rVV10, demonstrate the unexpected impor- the early 1970s demonstrated the ability of the simple local den- tance of intermediate and long-range van der Waals attraction sity approximation (LDA) (11, 15) for the exchange-correlation (seamlessly included in the random phase approximation). (xc) energy to capture the surface energies and work functions of real metals. Their work stimulated the effort to understand Author contributions: A.P., J.S., and J.P.P. designed research; A.P. performed research; why simple approximate functionals work and how they can J.P.P. contributed new reagents/analytic tools; A.P., J.E.B., and J.P.P. analyzed data; and be improved (16, 17). Later, correlated wave function calcula- A.P., J.E.B., and J.P.P. wrote the paper. tions (18, 19) gave much higher surface energies for jellium, Reviewers: J.F.D., Grifﬁth University; and J.M.P., CIC nanoGUNE. but were not supported by further studies (20, 21) and were The authors declare no conﬂict of interest. eventually corrected by a painstaking Quantum Monte Carlo Published under the PNAS license. calculation (22). The too-low surface energies from the Perdew– 1To whom correspondence may be addressed. Email: [email protected] or Burke–Ernzerhof (PBE) (23) generalized gradient approxima- [email protected]. tion (GGA) led, in part, to the AM05 (24) and PBEsol (25) (PBE This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. for solids) GGAs, and to general purpose meta-GGAs (26– 1073/pnas.1713320114/-/DCSupplemental.

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(13, potential Fermi vacuum function the and determining work accurately should energy by the DFT one using practice, calculated In be function can an values. work remains theoretical still with experimental it compare which However, energy. question surface open the is than (48–51) crystals. uncertain bulk but of faces useful low-energy provide the for They are estimates (1970–1980). values old experimental rather as Available also considered energy. be surface can actual and “average” the metals an from solid-state different surface the generally The of is energy extrapolating (45–47). surface phase then method liquid phenomenological the and of a metal tension using liquid K the 0 measuring of to by tension determined surface Experimentally, been the have purity. energies and surface morphology however, over have we control since calculations, the- absolute surface to Accurate and bulk straightforward (12). accurate are from surface obtain energies new surface a face-dependent of create oretical and effects crystal the infinite untangle an to able functionals. be dispersion. long-range nonempirical also and intermediate- other systematic will the over we demonstrate Furthermore, SCAN and of accurate, LDA improvement accidentally why this understand be with better surfaces ener- can can we metallic surface functional, studying metallic purpose By real general functions. for work tested and been gies not had it However, nti ok eivsiaetesraeeege,wr func- work energies, surface the investigate we work, this In descrip- reasonable very gave 63) 62, (16, works Previous explain- and modeling of challenges theoretical the Despite relax- geometric is surfaces of property fundamental Another measure to easier is hand, other the on function, work The cleave to area unit per required work the is energy surface The ne rsa ae,i eraigodro akn est nthe (110). in and density (100), packing (111), lowest- of are the order layer, fcc, for decreasing face For and, in fcc. close-packed, faces, as crystal are Ru index fcc treated also and have hcp we Both convenience, hcp. really is which Structure. 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APPLIED PHYSICAL PNAS PLUS SCIENCES Table 1. Mean surface energy (σ¯ ) of (111), (100), and (110) surfaces of different metals, in joules per square meter Metals LDA PBE PBEsol SCAN SCAN+rVV10 RPA* Experimental (45, 46)

Al 1.08 0.89 1.06 1.03 1.16 1.07 1.14 ± 0.2 Cu 1.98 1.48 1.74 1.68 1.89 2.03 1.79 ± 0.19 Ru 3.19 2.48 2.89 2.77 2.99 3.45 3.04 ± 0.33 Rh 2.86 2.47 2.71 2.6 2.81 3.17 2.66 ± 0.29 Pd 2.19 1.59 1.90 1.8 2.04 2.25 2.00 ± 0.22 Ag 1.2 0.84 1.08 1.03 1.22 1.40 1.25 ± 0.13 Pt 2.26 1.79 2.12 1.92 2.15 2.84 2.49 ± 0.26 Au 1.41 0.87 1.16 1.06 1.29 1.55 1.51 ± 0.16 MAE 0.18 0.64 0.35 0.46 0.24 0.23

The mean absolute error (MAE) of each functional is also shown. *GPAW.

Results and Discussion The errors and mean absolute percentage errors (MAPEs) of Surface Energy. Surface energies from measured liquid metal the computed mean surface energies are shown in Fig. 1. The surface tensions (45–47, 72) are usually average surface ener- errors are computed with respect to the experimental values. Our gies over crystal faces. Hence, the experimentally measured sur- results are in agreement, within an acceptable margin, with those face energy can be compared with the mean of the surface ener- previously reported in the literature (65, 66, 73, 74). For Al, Fig. gies for (111), (110), and (100) surfaces (73). As we will see 1 (Left) illustrates the accuracy of all methods for simple metals later in this section, the calculated face dependencies are not that are close to the jellium limit. strong, which helps to justify this choice. Here we also use the Table 1 demonstrates that there is an overall systematic mean surface energies to compare with the experimentally mea- improvement from PBE to PBEsol or SCAN and then to sured values, but from a different perspective: LDA is known SCAN+rVV10 in the Al surface energy, due to the sequen- to yield accurate surface energies for jellium, within the uncer- tial incorporation of intermediate-range dispersion in PBEsol tainty of the latest quantum Monte Carlo (QMC) values (22), or SCAN and long-range dispersion in rVV10. We expect that and an equally weighted average over the three lowest-index PBEsol+rVV10 could be comparably accurate for solids and faces from LDA reproduces the experimental surface energies surfaces (but not for molecules, unlike the general purpose to within their uncertainties. LDA displays a remarkable error SCAN+rVV10). The long-range contribution from rVV10 in Al cancellation between its exchange and correlation contributions accounts for 12% of the total surface energy, and foreshadows (16, 17). Usually, the LDA exchange energy contribution to the the importance of including this contribution for the d metals. surface energy is an overestimate, while the correlation contribu- Transition and noble metal surfaces are more challenging due tion is a signiﬁcant underestimate, and their combination results to their localized d orbitals, which cause strong inhomogeneities in an accurate prediction. PBE improves both the exchange and in the valence electron density. These inhomogeneities lead to a the correlation contributions, but loses the remarkable error can- wider spread in the results from the different functionals. PBE cellation of LDA. yields the largest errors for the transition metal surface ener- In Table 1, we report the mean surface energies calculated gies, because it neglects most of the vdW interaction, i.e., it using different density functionals, including results from the radically underestimates the equilibrium binding energy of two RPA. Fig. 1 (Left) shows the error (in joules per square meter) of closed-shell atoms or molecules. PBEsol and SCAN incorporate the computed values of the mean surface energies compared with intermediate-range vdW, and so improve the surface energy. With the best available experimental results for each metal. The con- the addition of the long-range vdW from rVV10, SCAN+rVV10 sistent performance of SCAN+rVV10 can be seen in all cases, surpasses the accuracy of SCAN, indicating that the long-range whereas PBE and SCAN both perform less well. The RPA results vdW contribution to the surface energy is more important than are overall in good agreement with the experimental results; previously recognized. The intermediate- and long-range vdW however, the computational cost is higher. One can argue that attraction between separating half spaces must increase the work SCAN+rVV10 is the “best” candidate for predicting metallic needed to pull them apart, and thus the surface energy. surface energies, with its moderate computational cost and high In LDA, the vdW attraction is overestimated at intermediate accuracy. range but neglected at long range, leading to another remark-

Fig. 1. The errors in mean surface energy (σ¯)(Left) of the (111), (100), and (110) surfaces compared with experiment (45, 46). The MAPE of the surface energies (Right) for each functional. Note that the mean experimental uncertainty from Table 1 is 0.22 J/m2 or 12%.

E9190 | www.pnas.org/cgi/doi/10.1073/pnas.1713320114 Patra et al. Downloaded by guest on September 23, 2021 Downloaded by guest on September 23, 2021 ar tal. et Patra face. crystallographic each for energy surface calculated the play properties. of set broader a for sys- LDA more than perform tematically to interactions expected vdW surroundings. be can long-range their SCAN+rVV10 and accurately, the and intermediate with treats surfaces dealing it metallic Incor- Because for second. clean important close of is a interactions interactions is vdW LDA of although poration functional, density cal slabs jellium for results the energy 76). upon surface (75, based expected the is overestimate This to GGAs, slightly. tends LDA, RPA our by meta-GGA. vdW described of and properly includes is analysis RPA short- that of (32). correlation vector magnitude the range jellium overestimates wave but of ranges, the all energy at the in attraction surface language, xc seen RPA vdW be the the can using cancellation Without same cancellation. error able i.2. Fig. al wt oeipt rmrf.7–0 n i.2dis- 2 Fig. and 77–80) refs. from inputs some (with 2 Table semilo- best the is SCAN+rVV10 MAPE. shows (Right) 1 Fig. ufc energies Surface t1119 .618 .418 .3() .5(4 .9(6,22 7) .0(65) 2.00 (74), 2.29 (66), 1.49 (65) 1.15 (54) (74), 2.35 1.17 (2), 2.23 1.89 (77) 1.12 (54), (65) 1.12 1.90 (74), 1.92 (66), 1.64 1.31 1.16 (77) 1.64 (54), 1.85 1.88 0.97 1.56 1.77 1.98 1.00 1.54 0.78 111 1.63 1.13 1.36 111 Pt 1.88 111 Ag Pd h1126 .924 .326 .8(4,25 7)24 7) .1(65) 2.61 (74), 2.47 (77) 2.53 (54), 2.78 (77) 2.99 2.61 2.33 2.55 2.40 2.39 2.09 2.49 2.67 2.14 111 2.81 111 Rh Ru u1112 .51109 .716 5) .4(8 .4(6,12 7) .4(65) 1.14 (74), 1.28 (66), 0.74 (78) 1.04 (54), 1.61 1.17 0.93 1.1 0.75 1.24 111 Au u1118 .315 .917 .6(54) 1.96 1.74 1.49 1.59 1.33 1.81 111 Cu l1109 .709 .111 .1() .7(4 .7(6,11 (74) 1.19 (66), 0.67 (54) 1.27 (2), 0.91 1.11 0.91 0.99 0.77 0.99 111 Al energies Surface 2. Table eetdmetals selected easSraeLAPEPEo CN+V1 ohrworks) (other +rVV10 SCAN PBEsol PBE LDA Surface Metals σ 0 .518 .120 .524 5)18 6) .3(4,24 (65) 2.47 (74), 2.73 (66), 1.81 (65) 1.35 (74), (54) 1.24 2.48 (65) 1.27 (65) (74), 2.23 1.20 (74), 2.33 (66), 1.55 2.32 (77) 1.26 (54), 2.25 1.29 (77) 1.21 (54), (65) 1.20 (77) 2.08 2.15 1.97 (74), 2.33 (66), 1.33 1.49 2.04 1.18 2.31 (77) 1.86 (54), 1.90 1.12 2.21 1.94 2.05 1.00 1.88 2.29 1.19 2.46 1.83 1.04 2.35 0.93 2.03 110 0.81 1.93 1.32 100 2.15 1.16 1.61 110 1.79 2.25 100 2.43 110 100 0 .427 .727 .029 5) .1(7 .9(4,30 (65) 3.01 (74), 2.79 (77) 2.88 (77) 2.81 (54), 2.90 (77) 3.45 2.82 3.00 2.76 3.12 2.71 3.3 2.77 2.81 2.97 2.55 3.11 2.77 2.94 2.86 3.25 3.04 2.27 110 3.02 3.42 100 3.34 110 100 1 .109 .61214 .9(4,15 8)09(6,17(4,14 (65) 1.41 (74), 1.7 (66), 0.9 (65) 1.36 (74), 1.63 (66), (80) 0.85 1.55 (54), 1.79 (79) 1.39 (54), 1.71 1.47 1.24 1.2 1.05 1.26 1.13 0.99 0.86 1.61 1.39 110 100 1 .316 .818 .223 (54) 2.31 (54) 2.09 2.02 1.91 1.84 1.71 1.88 1.76 1.63 1.48 2.13 1.99 110 100 1 .909 .110 1.19 1.18 1.09 1.08 1.11 1.08 0.96 0.95 1.09 1.15 110 100 111 (Left ), σ 100 ,and (Middle), σ i olsprsur ee)o h 11,(0) n 10 ufcsfrthe for surfaces (110) and (100), (111), the of meter) square per joules (in σ 110 (Right o h eetdmtl nti ok h hmclted r iia o l functionals. all for similar are trends chemical The work. this in metals selected the for ) SCAN u oR secue rmteMP n enabsolute mean polycrystalline and for MAPE is the from it excluded exper- for the is and have Ru hcp, we is so Ru function Ru, face-dependent 3. the Fig. work of in imental MAPE plotted the are and functions work faces) crystal over aged Function. Pd. and Work Rh, Ru, for broken oth- be of to with trend seems general trend agreement this The good 73). LDA in 66, our σ are (65, that reported and trends our (81), recently of and al. ers agreement values et excellent Sun PBE ﬁnd of We and those see. with to results easy ener- PBEsol by is surface PBE solids the used by of be elemental underestimation gies systematic could the the with 2 while overlaps of Table LDA, frequently shapes SCAN+rVV10 in construction. values equilibrium numbers Wulff the experimental the the metals, corresponding predict our to ﬁnd of not any for could we While 100 σ < 110 .2(77) 3.52 a ese rmFg o otmtl.However, metals. most for 2 Fig. from seen be can LDA h roso h enwr ucin (aver- functions work mean the of errors The PNAS | .5(6,28 7) .9(65) 2.49 (74), 2.82 (66), 1.85 ulse nieOtbr1,2017 17, October online Published .9(4,30 (65) 3.08 (74), 2.89 .4(4,21 (65) 2.19 (74), 2.24 (65) 2.15 (74), 2.17 .3(6,12 (74) 1.27 (66), 0.93 (74) 1.35 (66), 0.86 .5(4,19 (65) 1.94 (74), 1.95 ohrworks) (other GGA | σ 111 E9191 <

APPLIED PHYSICAL PNAS PLUS SCIENCES Fig. 3. The errors in mean work functions (φ¯) (Left) for the (111), (100), and (110) surfaces predicted by each functional. MAPE of the face-dependent work functions (Right) for the same systems. Note that the mean experimental uncertainty from Table 3 is 0.09 eV or 2.0%.

error (MAE). Calculated values of the work function for each improves upon PBE through its incorporation of vdW contri- face can be found in Table 3 (with some inputs from refs. 82– butions to the surface potentials. Although PBEsol and SCAN 101), and are plotted in Fig. 4. Our results for LDA and PBE differ in many ways, both incorporate intermediate-range vdW are generally within ∼0.15 eV of those reported in the litera- interactions. Their overall performance for work functions is ture (66, 102). For Al, LDA overestimates the work function quite similar, and, typically, the errors from these functionals for the (111) surface by 0.1 eV, but is dead on experiment for are within the experimental uncertainties. They also outperform the other two faces. PBE and SCAN perform similarly for Al, LDA for the work functions, which was not the case for the sur- but show larger deviations from one another for the d-block face energies above. metals. PBEsol and SCAN+rVV10 yield the smallest errors The inclusion of intermediate-range vdW interactions is not for Al. enough, however, as the long-range contributions can still raise Fig. 3 also shows the errors in the calculated values of the the work function by an appreciable amount. The (110) sur- work function for the transition and noble metals. These systems face of Rh is one such case, where the addition of rVV10 to have entirely or partly ﬁlled d orbitals which are localized on the SCAN increases the work function by nearly 0.2 eV, signiﬁ- atoms. Hybridization between the d and s orbitals varies with the cantly reducing the error compared with experiment. Incorpo- crystallographic orientation, resulting in changes in the surface rating the long-range dispersion amounts to between 3% and dipole and, consequently, the work function. The redistribution 6% of the total work function, underscoring the importance of of the d electrons in noble metals also impacts the work function, its inclusion. Although LDA and SCAN+rVV10 were of similar and these changes vary from one face to another (103). quality for the surface energies, SCAN+rVV10 clearly takes the From Fig. 3, it is clear that PBE systematically underestimates top spot for computing accurate work functions. We note that the work function, and its accuracy is erratic. In general, SCAN the trend φ110 < φ100 < φ111 predicted by Smoluchowski (104) is

Table 3. Work functions φ (eV) for the (111), (100), and (100) surfaces of different metals SCAN LDA GGA Metals Surface LDA PBE PBEsol SCAN +rVV10 (other work) (other work) Experiment

Al 111 4.36 4.2 4.24 4.19 4.23 4.25 (82) 4.02 (66) 4.26 ± 0.03 (85), 4.32 ± 0.06 (87) 100 4.41 4.27 4.32 4.35 4.42 4.38 (82) 4.09 (66) 4.41 ± 0.03 (87), 4.32 ± 0.06 (87) 110 4.08 3.96 3.98 3.99 4.00 4.3 (82) 4.3 (66) 4.06 ± 0.03 (89), 4.23 ± 0.13 (87) Cu 111 5.20 4.88 4.98 4.98 5.09 4.94 (91), 4.9 ± 0.02 (87) 100 4.79 4.42 4.43 4.47 4.54 4.59 ± 0.03 (94), 4.73 ± 0.1 (87) 110 4.68 4.38 4.48 4.47 4.53 4.59 (93), 4.56 ± 0.1 (87) Ru 111 4.78 4.37 4.51 4.38 4.65 5.33 (77) 4.71 (51) 100 5.1 4.78 4.86 4.9 4.97 5.03 (77) 110 4.68 4.42 4.55 4.52 4.72 4.65 (77) Rh 111 5.23 5.00 5.12 5.16 5.20 5.3 (100), 5.46 ± 0.09 (87) 100 5.44 5.12 5.38 5.34 5.37 5.25 (77) 5.11 (101), 5.3±, 0.15 (87) 110 4.9 4.53 4.66 4.65 4.83 4.98 (77) 4.8 ± 0.05 (99), 4.86 ± 0.21 (87) Pd 111 5.66 5.32 5.52 5.39 5.47 5.64 (2) 5.25 (66) 5.44 ± 0.03 (83), 5.67 ± 0.12 (87) 100 5.54 5.12 5.25 5.19 5.26 5.11 (66) 5.3 (86), 5.48 ± 0.23 (87) 110 5.32 4.95 5.07 5.04 5.09 4.87 (66) 5.2 (84), 5.07 ± 0.2 (87) Ag 111 4.97 4.49 4.66 4.57 4.63 4.75 ± 0.01 (88), 4.53 ± 0.07 (87) 100 4.64 4.26 4.35 4.3 4.37 4.42 ± 0.02 (90), 4.36 ± 0.05 (87) 110 4.61 4.16 4.28 4.21 4.26 4.25 ± 0.03 (92), 4.1 ± 0.15 (87) Pt 111 6.08 5.72 5.85 5.90 5.97 6.06 (2) 5.69 (66) 6.08 ± 0.15 (95), 5.91 ± 0.08 (87) 100 6.06 5.69 5.82 5.94 6.01 5.66 (66) 5.9 (99), 5.75 ± 0.13 (87) 110 5.6 5.18 5.31 5.27 5.36 5.52 (80) 5.26 (66) 5.4 (96), 5.53 ± 0.13 (87) Au 111 5.49 5.12 5.19 5.32 5.41 5.63 (103) 5.15 (66) 5.3 to 5.6 (100), 5.33 ± 0.06 (87) 100 5.49 5.07 5.17 5.26 5.28 5.53 (103) 5.1 (66) 5.22 ± 0.04 (104), 5.22 ± 0.31 (87) 110 5.36 4.94 5.02 5.17 5.3 5.41 (103) 5.04 (66) 5.2 (105), 5.16 ± 0.22 (87) MAE 0.16 0.21 0.11 0.11 0.08

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Why and and Energies Functions, Surface Work Raises Attraction vdW the Why of code the using calculated 114. jellium, ref. stabilized and jellium of ties rdc,det orgtdsraercntuto 12 not (112) reconstruction surface calculation. corrugated our exper- in a included the to surface, due (100) predict, Au experi- the the for imental with However, comparison results. in mental relaxations interlayer accurate with. compare to results experimental few a only r ihyacrt o h ukltiecntnso l fthe of all of constants lattice bulk the in for metals. are accurate work highly this are in calculated results 66. PBE ref. with and agreement LDA the that for that, note 108–113). to (52, important sur- interlayer data is three different experimental It the predict the with of may compared layers functionals relaxations four xc top Different the faces. for relaxation percentage one. latter rear- include the here layers not calculations top but Our effect the equilibrium. former within the new ions a the reach result. which to in a range 107), as forces. 106, and (105, relax Pt interionic for Au the always observed been in have layers reconstructions changes surface surface Moreover, to between due distances equilibrium ion the The given change a can of ions position neighboring fewer of Studies. presence Other and Relaxation Lattice Surface both properties. for 31) (30, performance correc- bulk well-balanced to and a itself surface achieving lends rVV10, systems by bonded diversely tion for behav- systematic SCAN The of results. LDA ior well, the as worsen refer- would errors it experimental their but the ence, reduce underestimates functional likely bare would the GGAs since the to Addition rVV10 uncertainties. of experimental pre- within accurately functions work and dicts systematically other SCAN+rVV10 the rVV10, for from observed is but Rh, and Ru, metals. Al, for observed not ar tal. et Patra i.4. Fig. ytewr–nryterm h ufc nryi h exter- the is energy surface the theorem, work–energy the By nms ae,SA+V1 n CNpeitreasonably predict SCAN and SCAN+rVV10 cases, most In al S4 Table Appendix, SI S1–S3 Tables Appendix, SI potential the to contribution long-range a incorporating By okfunctions Work alsS5–S10 Tables d 12 smc togrta n forfunctionals our of any than stronger much is % φ 111 nthe in (Left hw htSA n SCAN+rVV10 and SCAN that shows alsS1–S3 Tables Appendix, SI ), IAppendix SI hwtetbltdvle fthe of values tabulated the show φ d 100 23 ,and (Middle), and % d hwsraeproper- surface show 34 ttesrae the surface, the At ,w aefound have we %, φ 110 (Right loshow also o h ufcssuidi hswr.Teceia rnsaesmlrfralfunctionals. all for similar are trends chemical The work. this in studied surfaces the for ) lcrnot)t h B okfnto fte(1)surface (111) the Metals of function (in work correction PBE vdW the long-range to and electronvolts) Intermediate- 4. Table function, (PBE work GGAs energy, two surface without The LDA, and correction. with the vdW (SCAN) long-range using meta-GGA a recent Au) a and and Pt, PBEsol), and Ag, Cu, Pd, (Al, metals Rh, of Ru, properties surface important three studied We Summary cor- vdW attractive an function. work of the the addition but raises electron the rection more Thus, one density. has bulk it same since system, positive singly the function. work the deﬁnes Eq. that difference that gener- energy shows The total exchange.) (115) needed exact theorem with Janak GGA of alized hybrid integral an a and for meta-GGA, operator (It a the for operator. operator multiplication which differential a in a be theory, becomes to KS constrained not generalized is a potential calcu- in xc out functional carried hybrid are and change lations) meta-GGA energy most total (like this calculations Eq. equate one-electron-like function to the work the theorem to of Janak’s theory energy uses total (114) elec- The the removed. when so occurs that is large increase tron is energy length, total that screening least and distance the thus wavelength a and Fermi to bulk metal the the with from is compared electron surface an remove the to in correction. vdW stored attractive an energy of interactions addition the the by Thus, vdW raised separation. separated. negative fragments the the fully when are after disappear are which and fragments, there two before the separation, between crystal the bulk Before the of energies change not does energy process. bulk separation the the that in thus Note and energy. work surface this the to to contribution attrac- positive are a screening separa- necessitate fragments they these or a tive, between wavelength forces to Fermi vdW the fragments bulk Since the macroscopic length. than two greater the much carry tion and plane a Cu Rh Au Ag Pt h eta ytmwl aealwrvWttleeg than energy total vdW lower a have will system neutral The needed work least the is surface metal a of function work The total the comparing by reached be also can conclusion This hw o hswr stedfeec ewe CNrV0adPBE. and SCAN+rVV10 between difference the is work this for Shown PNAS | e.117 Ref. −0.04 −0.03 ulse nieOtbr1,2017 17, October online Published 3 0.06 0.29 0.30 ihnK F.Ormeta-GGA Our DFT. KS within 3 tl rvdsthe provides still hswork This | 0.21 0.20 0.14 0.29 0.25 E9193

APPLIED PHYSICAL PNAS PLUS SCIENCES and interlayer relaxation were calculated and compared with the (118) in combination with projector augmented wave (PAW) best available experimental values. The choice of xc functional method (119, 120). We used the PAW pseudopotentials rec- has a noticeable effect on the surface properties of metals, espe- ommended in the VASP 5.3.5 manual for LDA and PBE. For cially on the surface energy (116). For surfaces, the performance example, the electrons treated as valence are 5d 10 6s1 for Au. of SCAN is comparable to that of PBEsol, but PBEsol is a special Relativistic effects are included in the construction of the pseu- purpose functional for solids, while SCAN is a general purpose dopotential, but not otherwise. Because the PBE pseudopo- functional. tential is transferable, we also used it for PBEsol and SCAN. The vdW forces are present at metallic surfaces. They non- For both bulk and surface computations, a maximum kinetic negligibly increase the surface energies and work functions, as energy cutoff of 700 eV was used for the plane wave expan- we have seen from numerical calculations and from qualitative sion. The Brillouin zone was sampled using Gamma-centered arguments. Ferri et al. (117) found that vdW corrections to PBE k-mesh grids of size 16 × 16 × 16 for the bulk and 16 × 16 × 1 can increase work functions significantly for some metals, while for the surfaces. The top few layers (up to four) of the slab slightly reducing them for others, whereas we find consistent were translated without reconstruction until the total energy increases for all of the metals we have studied. Our results are changes converged to less than 1.0 × 10−6 eV and the residual compared with theirs in Table 4. atomic forces converged to less than 0.01 eV/A.˚ Dipole correc- We have also justified the one-electron-like Eq. 3 for the work tions were used to cancel the errors of the electrostatic poten- function, even when the vacuum potential and Fermi energy are tial, atomic forces, and total energy, caused by periodic boundary calculated in a generalized KS scheme such as the one we used condition. for SCAN and SCAN+rVV10. For the slab geometry, 20 A˚ of vacuum was used to reduce the All tested functionals predict comparable interlayer relax- Coulombic interaction between the actual surface and its peri- ations. Unlike the surface energies and work functions, these odic image. These fcc surfaces are built using a cell containing relaxations show no interesting trend as vdW attraction is added one atom per layer. Theoretical lattice constants, obtained by fit- from PBE to SCAN to SCAN+rVV10. ting the Birch–Murnaghan equation of state for the bulk with LDA overestimates the intermediate-range vdW attraction each functional (see SI Appendix, Table S4), are used to build but has no long-range component. These two errors of LDA these cells. We used Pt (111) to test the convergence of the sur- may cancel almost perfectly for surface energies. PBE under- face properties with respect to different computational variables estimates the intermediate-range vdW and has no long-range such as k mesh, cutoff energy, layer, and vacuum thickness of the vdW, so it underestimates surface energies (by about 25%) and slab geometry. Four- to twelve-layer slabs were used in the linear work functions (by about 5%). PBEsol and SCAN have real- fit for the surface energy, and eight-layer slabs were used for the istic intermediate-range vdW and no long-range vdW, so they work function. All of the computed surface properties presented are more accurate than PBE but not as good as LDA for pre- in this work are well converged with respect to these computa- dicting surface energies. The asymptotic long-range vdW inter- tional variables. actions missing in semilocal functionals can make up to a 10% The RPA calculations were made with the GPAW software. difference in the surface energy or a 3% difference in the work The PAWs included the scalar relativistic effect on the core, and function. SCAN+rVV10 stands out in this regard, as it is a bal- the electrons treated as valence were 5d 10 6s1 for Au. Because anced combination of the most advanced nonempirical semilo- the RPA calculations are expensive, we have used only four-layer cal functional to date and the flexible nonlocal vdW correction slabs. Our calculations with SCAN+rVV10 suggest that using from rVV10. In addition to delivering superior performance for only a four-layer slab overestimates the face-averaged surface layered materials (35), SCAN delivers high-quality surface ener- energy by less than 0.05 J/m2, except in Pd, where the overes- gies, work functions, and surface relaxations for metallic sur- timation is by 0.1 J/m2. faces. SCAN+rVV10 includes realistic intermediate- and long- range vdW interactions, so it tends to yield more systematic and ACKNOWLEDGMENTS. A.P. thanks A. Ruzsinszky, H. Peng, Z. Yang, and C. accurate results than LDA, PBEsol, or SCAN (however, all func- Shahi for their help and suggestions. A.P., J.S., J.P.P., and the design of the tionals other than RPA underestimate the surface energies of Pt project were supported by the National Science Foundation (NSF) under and Au). More-accurate measurements for these properties are Grants DMR-1305135 and DMR-1607868. J.E.B.’s contributions to the design of the project were supported by NSF Division of Materials Research under needed to validate the performance of new and existing density Grant DMR-1553022, while his computational work was supported by the functionals. Overall, we find that SCAN is a systematic step up Department of Energy Basic Energy Sciences under Grant DESC0010499. in accuracy from PBE, and that adding rVV10 to SCAN yields a Most computational aspects of this work were supported by the Center highly accurate method for diversely bonded systems. for the Computational Design of Functional Layered Materials, an Energy Frontier Research Center sponsored by the US Department of Energy (DOE), Office of Sciences, Basic Energy Sciences under Award-DE-SC0012575. This Computational Details. We performed first-principles DFT calcu- research used resources of the National Energy Research Scientific Comput- lations using the Vienna Ab Initio Simulation Package (VASP) ing Center, a DOE Office of Science User Facility.

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