d-Band Model

A research topic across the group’s papers.

The d-band model relates the chemical reactivity of a transition metal surface to the electronic structure of its surface d states, most often summarized by the d-band center (the average energy of the d-projected density of states relative to the Fermi level). A higher d-band center generally means stronger adsorbate binding. The d-band center is a widely used descriptor because it can be computed with density functional theory and connects changes in surface composition and geometry to changes in adsorption energies. See also Strain and ligand effects, Bimetallic surfaces and Tight-binding models.

Hydrogen binding versus d-band center on Ni/Pt(111) (2003)

For Pt, Ni, Ni-Pt surface alloys, a Ni monolayer on Pt and a Pt-Ni-Pt sandwich, DFT hydrogen binding energies at 0.25 ML varied roughly linearly with the surface d-band center. Surface Ni raised the d-band center and strengthened H binding, while subsurface Ni lowered the surface Pt d-band center and weakened it below both parent metals; the ordering survived spin polarization, relaxation and a 2% lattice contraction. This was an early experimental test of the d-band picture against TPD trends on a bimetallic surface (Kitchin et al. 2003). Summary: kitchin2003.

Subsurface 3d metals in Pt(111) (2004)

Kitchin, Nørskov, Barteau and Chen placed a layer of Ti through Ni beneath an unstrained Pt(111) surface and found with DFT that the surface Pt d-band filling stays near 9.3 electrons per atom, so broadening of the band by the subsurface metal forces its center down (from -2.44 eV on Pt(111) to between -2.60 eV for Ni and about -3.2 eV for V). The square root of the mean squared width varied linearly with the center as a rectangular band model predicts. Dissociative H2 and O2 adsorption energies fell nearly linearly with the d-band center, becoming endothermic for the lowest centers, and the authors argued that the d-band center is a more reliable descriptor than the density of states at the Fermi level or the number of d holes (Kitchin et al. 2004). Summary: kitchin2004.

d-band width, center and filling in bimetallic surfaces (2004)

Kitchin, Nørskov, Barteau and Chen showed with DFT that forming supported Ni, Pd and Pt monolayers changes the surface d-band filling by less than about 2%, so a change in d-band width forces a compensating shift of the d-band center: the rms width and the center are linearly correlated with slope near -1. The width is proportional to a tight-binding interatomic d-d matrix element summed over nearest neighbors, and the dissociative adsorption energy of H2 on these surfaces correlates with the d-band center, with higher centers giving stronger binding (Kitchin et al. 2004). Summary: kitchin2004a.

Extending the d-band model to carbide surfaces (2005)

Kitchin and coworkers tested whether the d-band center correlation they had found for H adsorption on bimetallic surfaces also holds for early transition metal carbides. With an unrestricted projection of the density of states, the d-band center correlated poorly with hydrogen binding energy, because low-lying carbon-derived states hybridized with the metal d band lower the center without affecting H bonding. Projecting with a short 1 Å cutoff reduced these contributions, and the hydrogen binding energy then followed the d-band center trend, though with more scatter than for bimetallics, attributed to band-shape distortions and possible direct H-C interactions. The work highlights that d-band descriptors can depend on how the electron density is partitioned (Kitchin et al. 2005). Summary: kitchin2005.

Adsorbate-induced d-band changes and coverage dependence (2009)

Kitchin applied the same constant-filling picture to adsorbate-covered surfaces. For C, N and O in fcc sites of Pd(111) between 0.2 and 1 ML, hybridization with the adsorbates broadens the surface Pd d band while its fractional filling stays nearly constant, so the d-band center moves down roughly linearly with coverage; width and center are again strongly correlated as in a rectangular band model. The lower d-band center explains why adsorption weakens as coverage rises, and carbon, which sits closest to the surface, perturbs the d band most (Kitchin 2009). Summary: kitchin2009; see also Coverage-dependent adsorption.

Sulphur-induced d-band changes on Cu (2009)

For S on Cu(111), (100) and (110), İnoğlu and Kitchin found the surface d-band filling stays near 0.99 (about 9.9 d electrons per Cu) at all coverages while the band broadens. A rectangular band at this filling predicts a slope of -1.16 between the square root of the second moment and the d-band center; the DFT data give -1.09. Adsorption energies on hollow and bridge sites weaken nearly linearly as the d-band center falls, but atop sites (weak overlap with the d band) and dimer or trimer structures do not follow the correlation, underscoring that such correlations apply to geometrically similar configurations (İnoğlu 2009). Summary: inoglu2009a; see also Catalyst poisoning.

Oxygen-induced d-band broadening on Au(111) and Pt(111) (2009)

Increasing oxygen coverage broadens the surface-atom d band on both Au(111) and Pt(111) while the d filling stays nearly constant, so the d-band center falls linearly with the d-band width, as a rectangular band of fixed filling predicts (Miller 2009). The oxygen adsorption energy then weakens nearly linearly with the falling d-band center on both metals, consistent with the Hammer-Nørskov model, with a steeper slope and more scatter on Au attributed to repulsion from its filled d band. Summary: miller2009.

d-band center shift as a segregation descriptor (2009)

Han, Kitchin and Sholl used the d-band center as an input to predict surface segregation rather than adsorption. The shift in the surface d-band center when an impurity substitutes into the unrelaxed close-packed host surface, combined with a bond strength difference and multiplied by the coordination deficit, gave the best of the correlations they tested for DFT segregation energies on flat and stepped surfaces. Adding atomic size did not help, which they attribute to the d-band center already depending sensitively on atomic size (Han et al. 2009). Summary: han2009; see also Surface segregation.

Predicting surface d-band widths with adsorbates (2010)

Because surface d-band center and width remain linearly correlated at all coverages, a model for the width is enough to track reactivity changes (İnoğlu 2010). The paper builds a PW91-parametrized “solid-state table” of characteristic d-orbital radii and offsets for seven late transition metals, effective orbital radii for C, N, O and S, and facet-specific correction constants (different for noble and other metals), which together predict d-band widths of bulk, clean facets and adsorbate-covered surfaces. It also notes the limits of the d-band picture: S on filled-d-band noble metals does not follow the d-band trends, which the authors attribute to Pauli repulsion and s and p electron effects. Summary: inoglu2010.

A DFT-parameterized solid-state table for d-band widths (2010)

İnoğlu and Kitchin fitted, for 11 late transition metals, a characteristic d-orbital length r_d and a residual width W_o from the linear dependence of DFT bulk d-band widths on the inverse fifth power of the nearest-neighbor distance. The d-band width of any atom is then estimated as W_o plus a nearest-neighbor sum of (r_d,i r_d,j)^(3/2)/d_ij⁵, with a facet-specific correction for surface atoms. The model reproduced DFT widths of bulk atoms and low-index surfaces with about 2% average error (9% worst case, Cu(110)), and estimated widths for overlayer/host alloys correlated with DFT d-band centers with a slope near -1, as a rectangular band model predicts, so the table can be used to estimate d-band centers without new DFT calculations (İnoğlu 2010). Summary: inouglu2010.

Rectangular band behavior in bulk 3d perovskites (2011)

Akhade and Kitchin tested the metal-derived constant-filling picture on the B-site d band of bulk cubic LaBO3 and SrBO3 (B = Sc to Cu). After separating the localized d band from hybridized O s,p states, the d filling stayed practically constant under strain and the d-band width (here the square root of the second moment about the d-band center) and the d-band center were linearly correlated for each compound, with positive slopes for early and negative slopes for late transition metals as a rectangular band model predicts. The sign change appeared near a fractional filling of 0.8 instead of the ideal 0.5, which the authors tentatively attribute to electron counting from projected densities of states (Akhade 2011). Summary: akhade2011.

d-band mediated coverage dependence across late transition metals (2011)

For oxygen on Rh, Ir, Pd, Pt, Cu, Ag and Au(111), Miller, İnoğlu and Kitchin found that the d-band width and center are linearly related on every surface, as the rectangular band model at constant filling predicts, and that adsorption energy is roughly linear in the d-band center on Rh, Ir, Pd and Pt. Even on the filled-d metals, where Pauli repulsion might change the picture, the trend largely survives (R2 of 0.71 for Cu and Au, 0.30 for Ag). The slope of the cross-metal configurational correlation was modeled empirically from the characteristic d-orbital radius and the cube of the fractional d filling, with higher filling giving faster weakening of adsorption with coverage (Miller et al. 2011). Summary: miller2011.

d-Band width as a descriptor for sulfur adsorption on alloys (2011)

For S on Pd(111) and Pt(111) at 0.25 to 1 ML, adsorption energies correlated nearly linearly with d-band center, which was linear in d-band width. Across 110 bimetallic overlayers, shifts in S adsorption energy relative to the pure metals tracked shifts in d-band width along a single line of slope 1.59, except for Ag and Au systems where filled d bands and Pauli repulsion break the simple d-band picture (İnoğlu 2011). Summary: inoglu2011.

The d-band center on 3d perovskite surfaces (2012)

Akhade and Kitchin carried the d-band analysis to BO2-terminated (001) surfaces of cubic LaBO3 and SrBO3 (B = Sc to Cu) with GGA (PW91) DFT. After separating the localized B d band from the O s,p region containing hybridized d states, unrelaxed surfaces showed a roughly linear d-band width versus center correlation whose slope changed sign with filling, qualitatively like a rectangular band model, although the sign change came at fractional fillings near 0.8 (La) and 0.6 (Sr) rather than 0.5, and surface relaxation degraded the correlation. Both the O dissociative adsorption energy and the surface oxygen vacancy formation energy correlated with the B-site d-band center, except for the d0 compounds LaScO3 and SrTiO3 and the filled-band Cu compounds. The authors suggest that a reported failure of the d-band center as a perovskite descriptor may stem from not deconvoluting the d band or from DFT+U localizing the d states (Akhade 2012). Summary: akhade2012; see also Perovskite oxides.

Extending d-band trends to oxides with outer electrons (2013)

Calle-Vallejo and co-workers noted that the d-band model had no established counterpart for transition metal oxides. They found that the number of outer electrons (valence electrons remaining on the metal after oxidation) captures *O, *OH and *OOH adsorption trends on 3d metals, monoxides and perovskites in one framework: for metals it reproduces the d-band expectation of strong binding on early and weak binding on late metals, and for octahedral oxides it reflects filling of nonbonding or antibonding eg and t2g levels (Calle-Vallejo et al. 2013). Summary: callevallejo2013.

Adsorbate p states as a descriptor for metals and monoxides (2014)

Xu and Kitchin showed that on fcc 3d metals the average energy of the adsorbed oxygen p states tracks the surface d-band center, and that a single correlation between this p-state energy and the dissociative O adsorption energy holds for metals, isotropically expanded metals and rock-salt monoxides, apart from strongly expanded Mn and Co. They used this to interpret how lattice expansion (d-band narrowing, or splitting into bonding and antibonding parts for mid-row metals) and oxidation (formation of low-lying metal-oxygen hybridized states) change oxide reactivity (Xu 2014).

Summary: xu2014.

The t_2g band as a descriptor for rutile oxides (2015)

In octahedrally coordinated oxides the d states split into e_g and t_2g sets, so the whole d band is not necessarily the right descriptor. For O adsorption on doped late 4d and 5d rutile dioxides, Xu and Kitchin found that the t_2g-band center of the surface cation, whose states lie near the Fermi level, correlated with adsorption energies better than the d, e_g or oxygen p-band centers, with separate trends for 4d and 5d cations. Under the ligand effect a higher t_2g center gave stronger binding, as in metals, but the t_2g filling was not conserved: the center moved because charge was transferred between cations. Under strain the slope relating t_2g-center shifts to adsorption scaled inversely with the idealized d filling: cations with more than half-filled d bands (Pd, Pt) followed the metal-like trend of weaker binding for a lower t_2g center, while those at or below half filling showed the opposite (Xu 2015). Summary: xu2015b.

Experimental valence band center versus H2 adsorption barrier on CuPd (2015)

Gumuslu and co-workers tested the d-band picture experimentally across a CuxPd1-x composition spread film (x = 0.30 to 0.97). Spatially resolved UPS gave a filled valence band center that shifted from about -3.9 to -3.3 eV toward the Fermi level as Cu decreased, with DFT reproducing the slope (offset about 0.8 eV) and showing the total filled-state center tracks the d-band center. Barriers for dissociative H2 adsorption from microkinetic fits of H2-D2 exchange fell at about 1.5 eV per eV of valence band shift on the Cu-rich side but became constant for Pd-rich alloys, a nonlinearity the authors attribute to hydrogen-induced Pd segregation decoupling the surface from the near-surface electronic structure (Gumuslu et al. 2015). Summary: gumuslu2015.

Limits of the d-band center for single atom alloys (2018)

For single atom alloys of Cu, Ag and Au with Pt, Pd, Ir, Rh or Ni dopants, the d-band width and center of the dopant atom remained roughly linearly correlated, consistent with the rectangular band model, but the dissociative adsorption energy of H2 scattered strongly against the d-band center. The authors concluded that the d-band center alone does not capture reactivity of these surfaces, pointing instead to the sharp atom-like d states of the isolated dopant or to Pauli repulsion from the noble metal host (Thirumalai 2018). Summary: thirumalai2018.

Learned corrections and the band-shape resonance effect (2024)

Huang, Wang, Kamanuru, Achenie, Kitchin and Xin built the rectangular-band, constant-filling relation between d-band width and center into a graph neural network that learns corrections to it, and trained it on 12,165 fcc surface sites (mean absolute error 0.032 eV for the d-band center, versus about 0.27 eV for the uncorrected physical model). Attribution analysis shows charge transfer contributes almost nothing, consistent with d-charge conservation, strain and ligand effects about 0.15 eV each, and a band-shape term about 0.20 eV. They attribute this last term to an orbital resonance effect: energy misalignment between interacting d levels skews the band (for example, skewness +0.48 for a Pt monolayer on Re(111)) and shifts its center without changing its second moment (Huang et al. 2024). Summary: huang2024; see also Interpretable machine learning.

d-band center and oxygen affinity of Mo2N versus Pt/Mo2N (2024)

Projected densities of states placed the d-band center of surface Mo on Mo2N(111) at −1.12 eV and that of a Pt overlayer on Mo2N(111) at −2.65 eV, close to Pt(111). The high-lying Mo states correlate with strong oxygen binding (−3.79 eV versus −1.31 eV on Pt/Mo2N), which favors alkoxide formation and C−O scission to propylene, while the Pt overlayer prefers α C−H scission and acetone formation (Porter et al. 2024). Summary: porter2024.