Accuracy of first-principles lateral interactions: Oxygen at Pd(100)

We employ a first-principles lattice-gas Hamiltonian (LGH) approach to determine the lateral interactions between O atoms adsorbed on the Pd(100) surface. With these interactions, we obtain an ordering behavior at low coverage that is in quantitative agreement with experimental data. Uncertainties in the approach arise from the finite LGH expansion and from the approximate exchange-correlation (xc) functional underlying the employed density-functional theory energetics. We carefully scrutinize these uncertainties and conclude that they primarily affect the on-site energy, which rationalizes the agreement with the experimental critical temperatures for the order-disorder transition. We also investigate the validity of the frequently applied assumption that the ordering energies can be represented by a sum of pair terms. Restricting our LGH expansion to just pairwise lateral interactions, we find that this results in effective interactions which contain spurious contributions that are of equal size, if not larger than any of the uncertainties, e.g., due to the approximate xc functional.

Figure 1

(Color online) Top view of the Pd(100) surface, illustrating the considered pool of 17 lateral interactions between O atoms in on-surface hollow sites. $V_m^p$ $(m=1,2,3,4,5)$ are the two-body (or pair) interactions at first-, second-, third-, fourth-, and fifth-nearest-neighbor distances. $V_n^t$, $V_n^q$, $V_n^{\mathit{qu}}$ are considered compact trio, quattro, and quinto interactions, respectively. Light shaded spheres represent Pd atoms, and small dark shaded spheres O atoms.

Figure 2

(Color online) Top view of five ordered adlayers with O in on-surface hollow sites. The coverage of each configuration from left to right panel is $1∕9$, $1∕4$, $1∕2$, $3∕4$, and 1 ML, respectively. Light shaded spheres represent Pd atoms, small dark shaded spheres O atoms, and the lines indicate the surface unit cells.

Figure 3

Coverage $\left(\Theta \right)$ dependence of the calculated DFT binding energies of 25 ordered configurations with O atoms in on-surface hollow sites.

Figure 4

(Color online) Formation energies $\Delta E_f$ as computed with DFT for the 25 ordered configurations shown in Fig. 2 (big circles), as well as for all configurations obtained by direct enumeration and using the LGH expansion (small circles), see text. The line is the convex hull for the DFT data, identifying three ordered ground states shown as insets: $p(2\times 2)\text{−}\mathrm{O}$ at $\Theta =0.25$ ML (left inset), $c(2\times 2)\text{−}\mathrm{O}$ at $\Theta =0.5$ ML (top inset), and $(2\times 2)\text{−}3\mathrm{O}$ at $\Theta =0.75$ ML (right inset). $\mathrm{Pd}=\text{large}$, light shaded spheres, $\mathrm{O}=\text{small}$, dark shaded spheres.

Figure 5

$\Theta \text{−}T$ diagram of the critical temperatures $T_c$ for the order-disorder transition. Shown with asterisks are the experimental data from Ref. 19, and with solid circles the values obtained when using the optimum LGH expansion. Additionally shown by empty circles are the values obtained when using the LDA as exchange-correlation functional in the DFT calculations (see text).

Figure 6

$\Theta \text{−}T$ diagram of the critical temperatures $T_c$ for the order-disorder transition. Compared are the curves obtained when using the first-principles lateral interactions for different LGH expansions compiled in Table and each time using 24 DFT configurations in the parametrization: Solid line, full circles: $m=9$; dashed line, empty circles: $m=10$; dotted line, empty circles: $m=11$.

Figure 7

$\Theta \text{−}T$ diagram of the critical temperatures $T_c$ for the order-disorder transition. Shown with asterisks are the experimental data from Ref. 19, and with solid circles the values obtained when using the optimum GGA-based LGH expansion, cf. Table . These curves are compared to those obtained when considering only pair interactions in the LGH: Using the GGA values compiled in Table (empty circles) or the empirically adjusted values by Liu and Evans (empty squares), see text.