Simple Model Explaining and Predicting Coverage-Dependent Atomic Adsorption Energies on Transition Metal Surfaces
Nilay İnoğlu and John R. Kitchin, “Simple Model Explaining and Predicting Coverage-Dependent Atomic Adsorption Energies on Transition Metal Surfaces”, Physical Review B 82(4), 045414 (2010). https://doi.org/10.1103/physrevb.82.045414
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Question
The d-band model explains why low-coverage adsorption energies differ from metal to metal, and correlations with the surface d-band center can save many DFT calculations, provided the d-band properties can be predicted cheaply. Existing simple models and databases of d-band centers did not account for coverage. Building on earlier evidence that coverage-dependent adsorption energies on Pd(111) track adsorbate-induced broadening of the surface d band (Kitchin 2009), this paper asks whether a tight-binding style formula can predict surface d-band widths of clean and adsorbate-covered surfaces, and whether that formula can then be used to estimate atomic adsorption energies as a function of coverage (İnoğlu 2010).
Approach
The authors computed C, N, O and S adsorption in fcc hollow sites on the (111), (100) and (110) facets of Cu, Rh, Pd, Ag, Ir, Pt and Au, using four-layer slabs with 2x2 surface cells, 10 Å of vacuum, adsorption on one side only and the top two layers relaxed (forces below 0.05 eV/Å). The calculations used DACAPO with Vanderbilt ultrasoft pseudopotentials, a 340 eV cutoff, the PW91 functional and an 8x8x1 k-point grid, converged to within 0.02 eV/adsorbate; Pd data came from earlier work. Coverages ran from 0.25 to 1 ML, with energies referenced to gas-phase atoms. The d-band width was defined as the square root of the second moment of the atom-projected d band about the Fermi level (İnoğlu 2010).
The model follows Harrison’s tight-binding picture, in which band width scales with interatomic matrix elements summed over nearest neighbors. For a clean surface atom, the width is written as an atomic offset term plus a neighbor sum of (product of characteristic d-orbital radii)(3/2) divided by interatomic distance to the fifth power, plus a facet-specific constant that corrects for surface relaxation and electron redistribution. Radii and offsets for each metal were obtained by fitting widths from bulk calculations at varied volumes. For adsorbate-covered surfaces, a second term is added that is proportional to coverage and to (metal radius times an effective adsorbate radius)(3/2) divided by the metal-adsorbate distance to the 5/2 power, plus a facet-specific adsorbate correction. The two exponents were fit and then fixed at the nearby half-integer values, and the proportionality factor was 1.189. The resulting parameters form a new PW91-based “solid-state table” for the seven metals and four adsorbates (İnoğlu 2010).
Key findings
On Pd(111) and Pt(111) all four adsorbates show adsorption energies that become weaker linearly as the d-band width increases, and the width itself rises linearly with coverage, so narrow bands go with strong binding at low coverage and broad bands with weak binding at high coverage. Surface d-band centers and widths stayed linearly correlated at all coverages. The clean-surface formula reproduced d-band widths for bulk and for all three facets of the seven metals. With adsorbates included, estimated widths agreed with DFT with an average error below 5 percent and a largest error of 13 percent (on Ag). The main outliers were reconstructed structures, such as a surface carbide formed by a monolayer of C on Pd(110) and monolayers of C or N on Ir(111), where interlayer spacings expanded relative to the ideal bulk distances used in the model (İnoğlu 2010).
To predict adsorption energies, one needs a reference energy at a single coverage (0.25 ML was used), the slope linking adsorption energy to d-band width (obtainable from, for example, two DFT calculations at 0.25 and 1 ML), and average metal-adsorbate bond lengths. Applied to the (111) facets of all seven metals, this gave average errors below 4.3 percent when S on the noble metals Cu, Ag and Au was excluded. Those S cases failed because their energy shifts do not follow the d-band changes; the authors attribute this to filled d bands, Pauli repulsion and s and p electron contributions outside the model. The model ranks the adsorbate-induced electronic modification as S > N > C > O: larger adsorbate orbitals sitting closer to the surface perturb the d band more, giving stronger coverage dependence, with S the largest and O showing the smallest electronic and largest geometric contribution. Preliminary (unshown) results suggested good agreement for impurities in Pd-based (111) alloys and for A-on-B bimetallic overlayers (İnoğlu 2010).
Significance
The paper turns the qualitative picture of coverage effects as adsorbate-induced d-band broadening into a quantitative, inexpensive predictor. Once a metal’s characteristic radius and an adsorbate’s effective radius are tabulated, coverage-dependent d-band widths and adsorption energies can be estimated without new DFT calculations, or used to decide where DFT is worth running. The authors caution that the facet-specific constants may depend on the exchange-correlation functional, although they expect the coverage trends not to (İnoğlu 2010).