Application of Screened Hybrid Density Functional Theory to Ammonia

Article pubs.acs.org/JPCC

Application of Screened Hybrid Density Functional Theory to Ammonia Decomposition on Silicon Richard Sniatynsky,† Benjamin G. Janesko,*,† Fedwa El-Mellouhi,‡ and Edward N. Brothers‡ †

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

‡

S Supporting Information *

ABSTRACT: Screened hybrid exchange-correlation (XC) density functionals incorporating short-range exact exchange aid application of hybrid density functional theory to solids and surfaces. We explore screened hybrid XC functionals for a prototypical surface reaction, namely, adsorption and dissociation of ammonia on silicon. Screened hybrids are found to improve upon standard semilocal functionals for the dissociation barrier on Si9H12, reproducing accurate complete basis set extrapolated CCSD(T) results. Similar trends are found for realistic periodic Si(100)-2 × 2 surfaces. Screened hybrids also better reproduce experimental results for the relative barriers to different dissociation pathways. While the tested hybrid functionals tend to overestimate molecular adsorption energies, their good performance for kinetics motivates further exploration of screened exchange in surface chemistry.

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limitations. Semilocal functionals overdelocalize electrons28,29 and tend to underestimate reaction barriers. This phenomenon is well characterized for gas-phase reactions, where DFT calculations can readily be compared to accurate experimental and/or ab initio benchmarks.30−37 Several studies suggest that this underestimation also occurs for reactions on surfaces. Filippi and co-workers found that the PW9138,39 GGA underestimates accurate quantum Monte Carlo (QMC) adsorption barriers and reaction energies for H2 dissociative adsorption on Si(100).40 Dürr and co-workers showed that PW91 underestimates experimental adsorption barriers for H2 on Si(100).41 Nachtigall and co-workers find that the LSDA as well as the BP42 and BLYP43 GGAs underestimate QCISD(T) barriers for H2 dissociation on gas-phase silanes and a cluster model for Si(100).44 Pozzo and Alfè find that the PW91, PBE,45 and RPBE46 GGAs all underestimate the QMC barrier for H2 dissociation on Mg(0001).47 Kanai and Takeuchi found that PBE and the TPSS meta-GGA48 underestimate QMC barriers for hydrogen abstraction by styrene radical on hydrogen-terminated Si(001).49 A series of studies by Bickelhaupt and co-workers show similar errors for small organic molecules reacting with atomic Pd.50−52 Thus, there are significant limitations in applications of semilocal DFT to reactions at surfaces. Previous Semilocal DFT Studies of NH3 on Si Surfaces. Several authors have used DFT with semilocal XC functionals

INTRODUCTION Chemical reactions at surfaces and interfaces are central to many problems in chemistry, including heterogeneous catalysis,1 surface-enhanced spectroscopies,2 and synthesis and properties of nanomaterials.3 Reactions on silicon surfaces are particularly relevant to the manufacture of integrated circuits.4 There have been many experimental5−8 and computational9−22 studies on ammonia dissociation on reconstructed Si(100), a reaction relevant to silicon nitride deposition.23 This work applies new electronic structure approximations to this wellstudied reaction. Semilocal DFT for Surface Chemistry. Chemical reactions at surfaces are challenging for electronic structure theory. Accurate ab initio methods are often too computationally expensive to routinely treat the large clusters or periodic slabs needed to model realistic surfaces. Such systems are typically treated using Kohn−Sham density functional theory (DFT) using simple “semilocal” approximations for the manybody exchange-correlation (XC) density functional.24 Semilocal functionals model the XC energy density at each point in space r as a function of the electronic structure in an infinitesimal region around r.25 The local spin-density approximation (LSDA) uses the electron density ρ(r), generalized gradient approximations (GGAs) also depend on |▽ρ(r)|, and metaGGAs incorporate the density Laplacian and/or noninteracting kinetic energy density.26 (Electron spin dependence is suppressed throughout for conciseness.) Semilocal XC functionals’ favorable trade-off of reasonable accuracy and modest computational cost has made them widely used in surface chemistry.27 However, they have important © 2012 American Chemical Society

Received: September 15, 2012 Revised: November 16, 2012 Published: December 11, 2012 26396

dx.doi.org/10.1021/jp309185h | J. Phys. Chem. C 2012, 116, 26396−26404

The Journal of Physical Chemistry C

Article

HSE06 has been extensively applied to solids and surfaces,63 accurately treating lattice parameters,73,74 many60,73 (but not all75) band gaps and defect states,76 and improving somewhat upon semilocal DFT for level alignment77 and adsorption sites78 at metal surfaces. HSE06 calculations on small-band-gap solids can be significantly faster than global hybrid calculations.60,61 HISS-b uses shorter exchange screening lengths than HSE06 and thus should be even faster in systems where exchange screening is important.36 Both HSE06 and HISS-b dramatically outperform semilocal functionals for gasphase reaction barriers. For example, HISS-b gives mean absolute errors 1.7, 1.8 kcal/mol for standard sets of 38 gasphase hydrogen-transfer and non-hydrogen-transfer reaction barriers.35,36,79 The corresponding errors for the PBE GGA are 9.7, 8.6 kcal/mol.36 To summarize, it is clear that screened and global hybrids give comparable results in small systems and that both improve upon semilocal functionals for many properties of interest. The long-range piece of exact exchange can be problematic in some infinite systems. However, such systems can still be readily treated with hybrid DFT by incorporating screened nonlocal exchange. The resulting screened hybrid calculations can differ substantially from semilocal DFT. Extrapolation from results on small systems suggests that hybrid calculations incorporating screened exchange are more accurate than semilocal DFT. Previous Global Hybrid DFT Studies of NH3 on Si Clusters. We are not aware of any screened hybrid DFT studies of NH3 dissociation on reconstructed Si(100). However, several previous studies have applied global hybrids to cluster models of the Si surface.11,12,14,19−21 These calculations support our contention that semilocal DFT tends to underestimate the overall reaction barrier and the relative barrier of interdimer vs on-dimer ammonia dissociation. Smedarchina and Zgierski14 used the B3LYP80,81 global hybrid XC functional to model NH3 adsorption and dissociation on small clusters of 15 Si atoms. The authors found that the interdimer dissociation barrier was much higher, relative to the on-dimer barrier, than in semilocal DFT calculations. Their B3LYP interdimer barrier was only 0.6 kcal/mol (0.03 eV) below the on-dimer barrier.14 Periodic slab calculations with the PW91 GGA gave the interdimer barrier 0.10 eV below the ondimer barrier.17 The B3LYP global hybrid results were thus more consistent with the dominance of the on-dimer configuration seen experimentally.8,16 We will show that screened hybrids predict similar trends for on-dimer vs interdimer dissociation on realistic Si slabs. Smedarchina and Zgierski also reported a “rather large” overall barrier to ondimer dissociation.14 Current Work: Screened Hybrid DFT Studies of NH3 on Si Clusters and Surfaces. Here we apply the HSE06 and HISS-b screened hybrid functionals to NH3 adsorption and dissociation on the reconstructed Si(100) surface. First, we perform benchmark CCSD(T)/CBS-extrapolated ab initio calculations for NH3 adsorption and on-dimer dissociation on a Si9H12 cluster. While this small cluster cannot quantitatively model realistic periodic surfaces,11 DFT methods that cannot reproduce the benchmarks for the small cluster are of questionable utility for larger systems. Second, we show that the differences between semilocal vs screened hybrid predictions persist on large Si clusters and periodic Si slabs. Finally, we compare semilocal and screened hybrid predictions on Si(100) slabs to experiment. We find that standard semilocal DFT functionals are not close to CCSD(T) for the reaction

to model NH3 on periodic slab models of the reconstructed Si(100) surface.13,16,17,22 These calculations predict that NH3 adsorbs molecularly via a dative bond to the “down” atom on a buckled Si−Si dimer.22 (The reconstructed Si(100) surface, modeled with a six-layer slab as described below, is shown in Figure 3.) This metastable state can dissociate to adsorbed products NH2(a) + H(a).17 “On-dimer” and “interdimer” dissociations, respectively, leave the dissociated H atom on the same Si−Si dimer as NH2 or on an adjacent dimer in the same row.16,17 (Inter-row dissociation has also been explored.17) Semilocal DFT calculations predict that the interdimer product is less thermally stable than the on-dimer product but that interdimer formation has a lower barrier.13,17 The relatively low barrier suggests that the interdimer product might dominate at low temperature. However, experiments find that the on-dimer configuration is more common at low coverage16 and the interdimer configuration is rare at 65 K.8 This discrepancy between theory and experiment has been suggested to arise from hydrogen tunneling14 or preferential motions of adsorbed NH 3 . 17 We will demonstrate in this work that the aforementioned limitations of semilocal XC functionals may also play a role in this discrepancy. Screened Hybrid DFT. Semilocal XC functionals’ systematic overdelocalization can be ameliorated in “hybrids” incorporating the nonlocal exact exchange energy29,53,54 E Xex = −

1 2

2

∫ d3r ∫ d3r′ |γ|r(r−, rr′)′||

(1)

The one-particle density matrix γ(r,r′) = ∑iϕi (r)ϕ*i (r′) is constructed from the occupied orbitals {φi(r)} of the noninteracting Kohn−Sham reference system and obeys limr′→rγ(r,r′) = ρ(r). Global hybrids combining a semilocal XC functional with a constant fraction of exact exchange generally outperform semilocal functionals for gas-phase reaction barriers.30−37 Global hybrid functionals have also been applied to bulk insulators,55,56 semiconductor surfaces,57 and bulk conjugated polymers.58,59 However, the long-range component of exact exchange (large |r − r′| in eq 1) is computationally expensive60,61 in small-band-gap systems and formally problematic in bulk metals, where it is approximately canceled by long-range correlation.62−64 (References 58 and 59 reported numerical stability problems in calculating exact exchange in crystalline polyacetylene.) Screened hybrids incorporate some nonlocal information while eliminating the problematic long-range exchange. The HSE06 screened hybrid,65,66 designed as a screened version of the PBE0 global hybrid,67,68 includes 25% of the short-range exact exchange energy EXSRex = −

1 2

2

∫ d3r ∫ d3r′ |γ|r(r−, rr′)′||

erfc(ω|r − r′|)

(2)

Here “erfc” is the complementary error function and ω is an adjustable parameter. HSE06 also includes 75% short-range PBE exchange,69 100% long-range PBE exchange, and PBE correlation. The inverse screening length ω = 0.11 Bohr−1 is selected empirically. The “middle-range” screened hybrid HISSb incorporates two screening lengths and includes a larger fraction of exact exchange at intermediate |r − r′|.36,70 HISS-b incorporates a maximum fraction of 36% exact exchange at intermediate |r − r′| and zero exact exchange as |r − r′| approaches 0 or ∞ (cf. Figure 2 of ref 70.). Other screened hybrids have also been proposed.71,72 26397



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Figure 1. Calculated geometries and potential energy surfaces for NH3 adsorption and OD dissociation on a Si9H12 cluster model of the reconstructed Si(100) surface.

are hydrogen terminated. Slab geometries are optimized in QUANTUM ESPRESSO93 using PBE. PBE/planewave total energies are evaluated using a plane-wave basis of 50 Ry, a 6 × 6 × 1 k-point mesh for Brillouin zone integration, slabs separated by 3 lattice vectors, and ultrasoft pseudopotentials.94,95 Transition states for NH3 dissociation on Si slabs were evaluated with the climbing image nudged elastic band algorithm.96 Nudged elastic band geometry optimizations use 7 images, a plane-wave basis of 30 Ry, and a 4 × 4 × 1 k-point mesh for computational convenience. Geometries for NH3 on Si9H12 clusters reproduce ref 11, and geometries for Si slabs reproduce ref 97 (Supporting Information).

energy and reaction barrier of NH3 on Si9H12 while HSE06 and HISS-b are. These differences between semilocal vs screened hybrid functionals persist on realistic Si surfaces. Screened hybrids also better match available experimental results for reaction energies and barriers, though they appear somewhat problematic for the adsorption energy. The results motivate continued development and exploration of screened hybrids for surface chemistry.

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COMPUTATIONAL METHODS

Calculations compare HSE06 and HISS-b to the PBE and revPBE82 GGAs, the TPSS meta-GGA, and the empirical M06L meta-GGA.83 Cluster calculations also treat the B3LYP80,81 and PBE067,68 global hybrids. Atomic orbital (AO) basis calculations on clusters and slabs use the GAUSSIAN 09 suite of programs.84 HISS-b and revPBE calculations use the development version of GAUSSIAN.85 LSDA calculations use Vosko−Wilk−Nusair correlation functional III.86 CCSD(T)/ CBS-extrapolated benchmark calculations use two-point extrapolations of the Hartree−Fock energy87 and MP2 correlation energy88 in the aug-cc-pVTZ and aug-cc-pVQZ basis set89 plus a cc-pVTZ ΔCCSD(T) correction. DFT calculations on finite clusters use B3LYP/6-311++G(3df,2pd) geometries and the 6-311++G(3df,2pd) basis set unless noted otherwise. AO basis slab calculations use the m6-311G* basis73 for Si and N and the 6-311G** basis for H. Si21H20 calculations in the m6-311G* basis set use PBE geometries evaluated with m6-311G* on Si and N and 6-31G on H to test the effect of PBE vs B3LYP geometries. AO basis calculations on periodic slabs use at least 300 k points and evaluate the screened exchange and nonlocal one-electron operators using at least 300 replica cells.90 Coulomb terms are evaluated using infinite lattice sums via the fast multipole method algorithm.91 AO basis NH3 adsorption energies include a counterpoise correction,92 with adsorption to Si slabs corrected by counterpoise corrections for NH3 on Si21H20. Slab calculations use a six-layer Si(100)-2 × 2 slab with lattice parameter 14.61353 Bohr, taken from the geometry of bulk Si optimized at this level of theory. The bottom three layers of Si are frozen to the bulk Si values, and dangling bonds on the bottom face

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RESULTS Accurate Benchmarks on a Small Cluster. Figure 1 shows the calculated geometries and selected potential energy surfaces for NH3 adsorption to a one-dimer Si9H12 cluster, a system treated in refs 9, 11, 12, 15, and 21. Table 1 reports the calculated potential energy surface. ΔEads is the ammonia adsorption energy, ΔErxn(OD) is the energy of the on-dimer dissociation product relative to adsorbed ammonia NH3(a), and ΔE‡(OD) is the corresponding transition state energy. All Table 1. Comparison of Several DFT Functionals to Reference CCSD(T)/CBS-Extrapolated Values (“Ref”) for the NH3 Adsorption Energy ΔEads, NH3(a) → NH2(a) + H(a) Barrier ΔE‡(OD), and Reaction Energy ΔErxn(OD) (eV) on Si9H12

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method

ΔEads

ΔE‡(OD)

ΔErxn(OD)

LSDA PBE revPBE TPSS M06-L HSE06 HISS-b B3LYP PBE0 ref

−1.43 −1.03 −0.85 −0.99 −0.96 −1.15 −1.24 −0.93 −1.16 −1.08

0.51 0.66 0.70 0.70 0.74 0.79 0.86 0.78 0.78 0.86

−0.97 −1.01 −1.04 −1.18 −1.31 −1.14 −1.27 −1.32 −1.13 −1.19



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calculations use identical geometries and large basis sets, permitting “apples to apples” comparison of the different XC functionals. The results in Table 1 are consistent with the literature. CCSD(T)/CBS-extrapolated reference values qualitatively agree with the QCISD(T)/6-311++G(2d,p) values in ref 12 (ΔEads = −1.11 eV, ΔE‡(OD) = 0.88 eV, ΔErxn(OD) = −1.35 eV). The reference ΔE‡(OD) is somewhat higher than the 15 kcal/mol (0.65 eV) MRSDCI value in ref 9, and the reference ΔErxn(OD) is less negative than the −42 kcal/mol (−1.82 eV) value in that reference, possibly due in part to differences in cluster geometry. B3LYP values for the Si9H12 cluster qualitatively agree with the two-dimer B3LYP/6-31G* calculations of Smedarchina and Zgierski14 (ΔEads = −1.55 eV, ΔE‡(OD) = 0.90 eV, ΔErxn(OD) = −1.02 eV), the onedimer B3LYP calculations of Widjaja, Mysinger, and Musgrave12 (ΔEads = −1.00 eV, ΔE‡(OD) = 0.82 eV, ΔErxn(OD) = −1.34 eV), and the one-dimer B3LYP/6311+G* values of Loh and Kang15 (ΔEads = −1.19 eV, ΔE‡(OD) = 0.86 eV, ΔErxn(OD) = −1.28 eV). The most important result in Table 1 is that, as in the aforementioned gas-phase studies,30−36 screened hybrids dramatically improve upon all tested semilocal functionals for the dissociation barrier relative to NH3(a). HISS-b is particularly accurate, consistent with its good performance for gas-phase kinetics.36 HSE06, designed as a screened version of the PBE0 global hybrid, is nearly identical to PBE0 in this small system.98 (Of course, PBE0 calculations are much more computationally expensive than HSE06 calculations in many bulk systems.60,61) HSE06 and HISS-b also give accurate reaction energies, outperformed only by the TPSS meta-GGA and the empirical M06-L meta-GGA. While all of the semilocal functionals underestimate the benchmark ΔE‡(OD), semilocal functionals can either over- or underestimate ΔE ads . This is consistent with previous demonstrations that semilocal functionals can be tuned to accurately reproduce adsorption energies.46,99,100 The PBE0 global hybrid and both PBE-based screened hybrids overestimate ΔEads, while the B3LYP global hybrid underestimates it. It is worth noting that van der Waals corrections tend to increase adsorption energies,57,101 such that vdW-corrected PBE-based functionals should further overestimate ΔEads while vdW-corrected B3LYP may be rather accurate.57 It is also worth noting that the PBE GGA for exchange overbinds some noncovalent interactions, such as the noble gas dimer Ne2.102 In contrast, PBE0 underestimates ΔErxn(OD), while B3LYP overestimates it. Figure 2 explores this effect, showing ΔEads, ΔE‡(OD), ΔErxn(OD) calculated for global hybrids of the PBE and BLYP GGAs, plotted as a function of the fraction of exact exchange. Calculations are performed nonself-consistently with PBE0/6311++G(3df,2p) orbitals and include a counterpoise correction for ΔEads. (Note that this “B1LYP” global hybrid is different from the three-parameter B3LYP.) For both functionals, the magnitudes of ΔEads, ΔE‡(OD), ΔErxn(OD) all increase with the fraction of exact exchange. The barrier ΔE‡(OD) is largely determined by the fraction of exact exchange, with both hybrids giving comparable barriers. However, the adsorption and reaction energies differ significantly: most of the PBE hybrids overestimate the benchmark ΔEads, while all of the BLYP hybrids overestimate the benchmark ΔErxn(OD). This suggests that screened hybrids of other GGAs and meta-GGAs might prove promising for treating surface chemistry.

Figure 2. Calculated ΔEads (top), ΔE‡(OD) (middle), and ΔErxn(OD) (bottom) for NH3 on Si9H12. Results are evaluated for PBE and BLYP global hybrids and plotted as a function of the fraction of exact exchange. Horizontal lines are accurate CCSD(T)/CBS-extrapolated values.

From Small Clusters to Periodic Slabs. The Si9H12 cluster discussed above is too small to quantitatively model a periodic Si(100) surface.11 Table 2 reports PBE and HISS-b ΔEads, ΔE‡(OD), and ΔErxn(OD) for a larger three-dimer Si21H20 cluster and a periodic, 6-layer Si(100)-2 × 2 slab. Each row of Table 2 compares PBE vs HISS-b for a given choice of model surface, basis set, and so on. The most important result in Table 2 is that the differences between PBE and HISS-b seen on the small cluster persist for larger clusters and periodic surfaces. This is consistent with ref 40, which showed that differences between semilocal DFT and QMC for H2 adsorption and dissociation on large Si clusters are approximately equal to the differences seen for Si9H12 . (The absolute values do change: for example, ΔE‡(OD) on Si21H20 is ∼0.3 eV higher than on Si9H12. Similar but smaller changes are seen for small-basis B3LYP calculations on onedimer vs three-dimer clusters in ref 11.) Calculations on Si21H20 show that the smaller m6-311G* atomic orbital (AO) basis set, useful for AO-basis calculations on Si slabs,60 changes the PBE and HISS-b results by only around 0.05 eV. The last three rows in Table 2 show that the PBE results do not change appreciably between Si21H20 and periodic Si slabs or between AO and plane-wave basis sets. This helps validate our computational approach and indicates that we are adequately representing reactions on full surfaces. Screened Hybrids on Periodic Slabs. Figure 3 shows calculated geometries and representative potential energy surfaces for NH3 adsorption and on-dimer and interdimer dissociation on a periodic, six-layer Si(100)-2 × 2 slab. Global hybrid results are not shown, due to their computational expense.60,61 All calculations use a unit cell containing two Si dimers, giving 1/2 NH3 per Si dimer. This is a higher surface coverage than some previous computational studies17 and will not provide a perfect treatment of experiments at low coverages. However, it suffices for comparing trends in the 26399



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Table 2. PBE and HISS-b Calculations for NH3 Adsorption Energy ΔEads, NH3(a) → NH2(a) + H(a) Barrier ΔE‡(OD), and Reaction Energy ΔErxn(OD) (eV), Evaluated for Different Basis Sets and Model Si Surfaces (Details in Text) ΔE‡(OD)

ΔEads

ΔErxn(OD)

surface

basis

PBE

HISS-b

PBE

HISS-b

PBE

HISS-b

Si9H12 Si21H20 Si21H20 Si(100)-2 × 2 slab Si(100)-2 × 2 slab

6-311++G(3df,2pd) 6-311++G(3df,2pd) m6-311G* m6-311G* plane-wave

−1.03 −1.31 −1.27 −1.26 −1.22

−1.24 −1.59 −1.56 −1.57

0.66 0.93 0.89 0.83 0.86

0.86 1.18 1.13 1.03

−1.01 −0.60 −0.63 −0.69 −0.67

−1.27 −0.82 −0.84 −0.93

Figure 3. Calculated geometries and potential energy surfaces for NH3 adsorption and OD dissociation on a six-layer Si(100)-2 × 2 slab.

Table 3. Comparison of Several DFT Functionals for NH3 → NH3(a) Adsorption Energy ΔEads and on-Dimer and Interdimer NH3(a) → NH2(a) + H(a) Reaction Energy and Barrier (eV)a

a

method

ΔEads

ΔE‡(OD)

ΔErxn(OD)

ΔE‡(ID)

ΔErxn(ID)

ΔE‡(ID−OD)

PBE revPBE TPSS M06-L HSE06 HISS-b

−1.26 −1.07 −1.26 −1.24 −1.44 −1.57

0.83 0.88 0.87 0.88 0.96 1.03

−0.69 −0.72 −0.84 −1.03 −0.81 −0.93

0.68 0.73 0.72 0.83 0.86 0.96

−0.52 −0.54 −0.66 −0.78 −0.56 −0.65

−0.15 −0.15 −0.15 −0.06 −0.10 −0.07

Calculations treat a six-layer Si(100)-2 × 2 slab.

cell: doubling our unit cell in the interdimer direction changes our PBE/plane-wave ΔErxn(ID) from −0.58 to −0.49 eV. The most important result in Table 3 is that the screened hybrid DFT methods predict higher reaction barriers and deeper reaction energies than any of the semilocal functionals, consistent with the results for Si9H12. Given the accuracy of screened hybrids for Si9H12 reaction energies and barriers (Table 1) and given that Table 2 and previous work40 show that differences between semilocal DFT and ab initio results are similar for small clusters vs slabs, we suggest that the most accurate reaction energies and barriers in Table 3 are from the screened hybrid calculations. This is consistent with previous studies showing that semilocal functionals systematically underestimate reaction barriers on surfaces.40,41,44,47,49 How-

predictions of different approximate XC functionals. Table 3 reports the adsorption energy ΔEads and on-dimer and interdimer reaction barriers and reaction energies evaluated with various approximate XC functionals. As above, all calculations use the same geometries, surface coverage, and basis sets. The semilocal DFT results in Table 3 are consistent with the literature. PBE slab calculations with a p(2 × 2) reconstructed surface by Lee and Kang13 give ΔEads = −1.21 eV, ΔE‡(OD) ≈ 0.7 eV, and ΔErxn(OD) = −0.78 eV. PW91 slab calculations with four Si dimers per unit cell by Bowler and Owen17 give ΔEads = −1.36 eV, ΔE‡(OD) = 0.87 eV, ΔErxn(OD) = −0.68 eV, ΔE‡(ID) = 0.77 eV, and ΔErxn(ID) = −0.33 eV. The difference in ΔErxn(ID) appears to arise from the different unit 26400



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the experimental ΔE‡(OD). This is consistent with their good performance for reaction barriers on Si9H12 (Table 1). HSE06 also provides accuracy comparable to the semilocal functionals for the “experimental” ΔErxn(OD). While these comparisons are not definitive, due to limitations in the choice of basis set, surface coverage, slab geometry, and so on, the trends provide additional support for the utility of screened exchange.

ever, we note that the rather inaccurate screened hybrid adsorption energies in Table 1 likely also carry over to the periodic slab calculations. Comparison with Experiment. Comparisons with experiment tend to support our assertion that screened hybrids accurately predict reaction energies and barriers for NH3 dissociation on reconstructed Si(100). Table 3 shows that the HSE06 and HISS-b screened hybrids and the empirical M06-L meta-GGA predict that ΔE‡(ID) is higher, relative to ΔE‡(OD), than predicted by standard semilocal functionals. While the effect is not large, the screened hybrids are more consistent with the experimental result that the on-dimer configuration is more common at low coverage16 and the interdimer configuration is rare at 65 K.8 This is compatible with previous suggestions that the discrepancy between semilocal DFT and experiment also arises in part from hydrogen tunneling14 or preferential motions of adsorbed NH3.17 Table 3 also shows that all functionals predict that the on-dimer dissociation transition state is lower energy than the than the free reactants (i.e., ΔE‡(OD) + ΔEads < 0), consistent with the experimentally observed tendency of ammonia to undergo dissociative chemisorption on reconstructed Si(100) even at low temperatures.15 Table 4 shows additional quantitative comparisons to experiment. These follow the comparisons to experiment in

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CONCLUSIONS Our results illustrate the promise of screened hybrid XC functionals for modeling NH3 on Si(100) and motivate further exploration of screened hybrids for surface chemistry. “Apples to apples” comparison with accurate ab initio calculations on Si9H12 show that the HSE06 and HISS-b screened hybrid functionals accurately model the barrier and reaction energy for adsorbed NH3 decomposition. They are less accurate for the initial adsorption energy, a phenomenon suggested to arise from the interplay of admixture of exact exchange and the form of the underlying GGA. Differences between semilocal vs screened hybrid values persist for larger clusters and periodic surfaces. Comparison to experiment provide additional evidence that the screened hybrids give reasonable treatments of the chemistry of adsorbed NH3 but overestimate the initial adsorption energy. Efforts to design new approximate XC functionals for surface chemistry99,100 could thus likely benefit from incorporating screened exact exchange.

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Table 4. Comparison of Calculated Si Slab Values (kcal/ mol) to Experiment method

ΔEads

ΔE‡(OD)

ΔErxn(OD)

PBE revPBE TPSS M06-L HSE06 HISS-b Expt.

−26 −22 −26 −26 −30 −33 −266,15

15 16 16 16 18 19 227

−19 −20 −23 −27 −22 −25 −215,6

ASSOCIATED CONTENT

S Supporting Information *

Calculated geometries and energies of all species. 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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ref 11. Temperature-programmed desorption (TPD) experiments6 have been correlated with an ΔEads for NH3(a) of −26 kcal/mol.15 Other TPD experiments were used to estimate a total desorption energy of NH2(a) + H(a) of 47 kcal/mol.5,11 Combining these desorption energies, and assuming following ref 11 that experiments measure the on-dimer pathway gives an experimental ΔErxn(OD) of −47 − (−26) = −21 kcal/mol. Molecular beam experiments give an activation energy for adsorbed NH3 dissociation [NH2(a) − H(a)]‡ = 4.0 ± 0.4 kcal/mol below the vacuum level.7 Combining this with the experimental ΔEads gives an experimental ΔE‡(OD) of −4.0 − (−26) = 22 kcal/mol. (In other words, ΔEads is the “NH3(a)” energy in Table 4 of ref 11, ΔE‡(OD) is the difference between the “NH3(a)” and the “[NH2(a) + H(a)]‡” values, and ΔErxn(OD) is the difference between the “NH3(a)” and the “NH2(a) + H(a)” values.) Following ref 12, all calculated energies in this section include the zero-point energy (ZPE) calculated from the one-dimer cluster model. The values in Table 4 differ from those in Table 3 due to these ZPE corrections. The results in Table 4 are qualitatively consistent with previous work. The PBE values are consistent with those of Lee and Kang,13 as shown in Table 4 of ref 11. The HSE06 and HISS-b screened hybrids tend to overestimate the experimental ΔEads but are more accurate than the semilocal functionals for

ACKNOWLEDGMENTS This work was supported by the Qatar National Research Fund through the National Priorities Research Program (NPRP Grant No. 09-143-1-022). The authors thank Myung Ho Kang for providing Cartesian coordinates of Si slabs from ref 97.

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REFERENCES

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Application of Screened Hybrid Density Functional Theory to Ammonia

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