2026 Volume 12 Article ID: 2025-0008
The electronic structures of tetragonal palladium oxide (PdO, space group P42/mmc) were calculated using the first-principles DFT+U and DFT+U+V methods. The calculations gave the band gap that reproduced the experimental values. It was found that the conduction band comprised the anti-bonding interactions of Pd 4dzx–O 2p and Pd 4dzx–O 2s, whereas the valence band top mainly comprised the anti-bonding interactions of Pd 4dz2–O 2p and Pd 4dx2-y2–O 2p. The DFT+U+V method, compared with the PBE and HSE06 functionals, gave larger Pd–O interactions near the valence band top. The electronic structures of rock salt PdO in the type-II antiferromagnetic phase were also computed as a comparison with nickel oxide (NiO).
Palladium oxide (PdO), which has a tetragonal P42/mmc structure (Figure 1 (a)) under ambient conditions [1], is a semiconductor with a band gap of 0.8–2.2 eV [2,3,4,5]. PdO is employed as an automotive catalyst for the reduction of nitrogen oxides and the oxidation of carbon monoxide and hydrocarbons [6,7,8,9]. Understanding the electronic structure of PdO is important when considering the catalytic mechanism. It has been reported that the density functional theory (DFT) calculation using the conventional local density approximation (LDA) or generalized gradient approximation (GGA) incorrectly give the metallic character for PdO [10,11,12,13,14]. One solution to this problem is using the Heyd, Scuseria, and Ernzerhof hybrid functional, HSE06, that considers the Hartree–Fock exchange in the short-range potential [15]. However, the high computational cost makes calculating large systems such as automotive catalysts impractical.
(a) Primitive unit cell of tetragonal PdO (space group: P42/mmc) with lattice constants of a = b = 3.042 Å and c = 5.351 Å [1]. (b) Primitive unit cell of rock salt PdO (space group: Fm
The DFT+U [16] and DFT+U+V [17] methods, where U and V represent on-site and inter-site electron-electron interactions, respectively, can correct the electron correlation effects in the conventional DFT calculations at a reasonable computational cost. For example, although the DFT calculations using the Perdew, Burke, Ernzerhof (PBE) functional erroneously provides the metallic character of nickel oxide (NiO, the 3d counterpart of PdO), the U correction leads to a wide band gap to reproduce the experimental value [17]. Furthermore, adding the V correction improves the interactions between the Ni d and O p states in the valence band. The U value for each atom and V value for each atomic pair, which depend on the oxidation states of atoms, can be determined from first-principles using linear-response theory [18,19,20,21].
In this study, the first-principles DFT+U and DFT+U+V methods were applied to PdO to understand its basic electronic structures for future applications to theoretical research on environmental catalysts. To the best of our knowledge, the DFT+U and DFT+U+V methods combined with the linear-response approach have not been applied to tetragonal PdO. Bennett et al. applied the LDA+U method to PdO, but a large U value of 8.0 eV, determined so as to reproduce the experimental band gap [22]. We also investigated the electronic structures of the rock salt PdO in the type-II antiferromagnetic phase (Figure 1 (b)), as a comparison with the antiferromagnetic NiO. The rock salt PdO can be grown on substrates, such as carbon and alumina, by the vacuum deposition technique [23]. In addition, the rock salt PdO, epitaxially bonded to the Sr3Ti2O7 surface, exhibits good catalytic activity for the purification of automotive exhaust gases [9]. Although the electronic structures of the rock salt PdO have been calculated [24, 25], its antiferromagnetic properties have not been studied previously.
The energy in the DFT+U+V method is obtained by adding the correction term for the electron-electron interactions, EU+V, to the approximate DFT energy, EDFT [17],
where EU+V is given by
First-principles calculations were performed within the DFT framework using the PBE functional as implemented in the Quantum ESPRESSO package [29,30,31]. The PBE+U [16] and PBE+U+V [17] approaches were employed for the on-site Pd 4d and first-neighbor inter-site Pd 4d–O 2p. The U and V values were determined based on the linear-response approach using the density functional perturbation theory (DFPT) [19,20,21], where the Löwdin orthogonalized atomic orbitals [32] were used as the projector functions. The V value for the first-neighbor inter-site Pd 4d–Pd 4d, which was less than 0.5 eV, was ignored because its inclusion had little effect on the electronic structures. Projector-augmented wave (PAW) pseudopotentials with Pd (4s,4p,4d,5s) and O (2s,2p) valence states were used with electronic wavefunction and charge density cutoffs of 60 Ry and 600 Ry, respectively. The Brillouin zone for the DFT calculations (PBE and HSE06) was sampled with the Γ-centered Monkhorst–Pack scheme [33]. The 6×6×4 and 6×6×6 k-point meshes were used for the tetragonal and rock salt PdO, respectively. The DFPT calculations were performed with the same q-point meshes. The k-point meshes were increased by a factor of three for the density of states (DOS) calculations.
The electronic structures of the rock salt NiO were calculated using similar computational conditions to the rock salt PdO. The PAW pseudopotentials with Ni (3s,3p,3d,4s) were used with electronic wavefunction and charge density cutoffs of 80 Ry and 800 Ry, respectively.
First, the electronic structures of the tetragonal PdO crystal (Figure 1 (a)) were calculated. The UPd value in the PBE+U method was determined to be 6.32 eV. Also, the UPd and VPd–O values in the PBE+U+V method were determined to be 6.87 eV and 1.22 eV, respectively. Table 1 shows the optimized lattice constants for the tetragonal PdO crystal. The calculated lattice constants were similar to the experimental ones for all the computational methods, although HSE06 gave the closest values. Thus, the following calculations were performed using the experimental lattice constants.
| PBE | PBE+U | PBE+U+V | HSE06 | Exp [1]. | |
| a | 3.074 | 3.095 | 3.099 | 3.041 | 3.042 |
| c | 5.420 | 5.443 | 5.449 | 5.348 | 5.351 |
Figure 2 shows the computed band structure and DOS. The PBE calculations resulted in a zero band gap. In contrast, the HSE06, PBE+U, and PBE+U+V calculations gave the band gap of 0.95, 1.41, and 1.51 eV, respectively, at the M point, which agreed with the experimental results of 0.8–2.2 eV [2,3,4,5]. It is noted that the experimental band gap varies depending on the measurement methods, i.e., electrical conductivity (1.5 eV) [2], optical transmittance (0.8 eV) [3], optical absorption (2.2 eV) [4], and REELS (1.5–2.0 eV) [5]. The DOS in the conduction band is similar for all the computational methods. In contrast, compared with the PBE and HSE06 functionals, the PBE+U and PBE+U+V methods increased the degree of the Pd–O interactions in the valence band between −2 and −4 eV. This result may agree with the XPS, which suggests that the valence band top mainly comprises the O 2p states [5].
Band structure and projected DOS for the Pd and O states using (a) PBE, (b) HSE06, (c) PBE+U, and (d) PBE+U+V methods. The symmetry points in the band structure are as follows: Γ = (0,0,0), X = (1/2,0,0), M = (1/2,1/2,0), Z = (0,0,1/2), R = (1/2,0,1/2), and A = (1/2,1/2,1/2). The DOS was broadened using the Gaussian function with a linewidth of 0.01 Ry.
The orbital contribution to the interaction between the Pd and O states was investigated. Figure 3 (a) shows the projected DOS for Pd 4d and O 2p, 2s, and Crystal Orbital Hamilton Population (COHP) [34, 35] between them in the ca-plane, computed using the DFT+U+V method. The conduction band comprised the interactions of Pd 4dzx − O 2p and Pd 4dzx − O 2s with the antibonding character. In contrast, the valence band top comprised the other interactions between the Pd 4d, particularly for 4dz2 and 4dx2-y2, and O 2p also with the antibonding character. Figure 3 (b) shows the degenerate highest occupied crystal orbitals (HOCO) and lowest unoccupied crystal orbital (LUCO) at the M point. These orbitals were delocalized over the Pd and O atoms.
(a) Projected DOS for Pd 4d and O 2p, 2s as well as COHP between them in the ca-plane. The positive and negative -COHP represent the bonding and antibonding interactions, respectively. The DOS and COHP were broadened using the Gaussian function with a linewidth of 0.01 Ry. The DOS scale for O 2s is 0.1 times that for O 2p. The COHP scale for Pd 4d–O 2s is half that for Pd4d–O 2p. (b) Degenerate HOCO and LUCO at the M point (isosurface value: 5×10−4 a.u.) calculated using the DFT+U+V method. These are evaluated at the Γ point of the (2×2×1) unit cell.
PdO has a square-planar geometry with the d8 configuration. The site symmetry of Pd coordinated by the four O atoms is D2h. The irreducible representations (irreps) of the Pd dz2, dx2-y2, dxy, dyz, and dzx orbitals in the ca-plane are ag, ag, b1 g, b3 g, and b2 g, respectively. It is noted that, in the Oh site symmetry, the irreps of the dz2 and dx2-y2 orbitals are eg, and those of the dxy, dyz, and dzx orbitals are t2 g. The Pd dzx orbital forms the σ bond with the O s and p orbitals, while the other Pd d orbitals form the π bond with the O p orbitals. Thus, only the Pd dzx orbital is unoccupied.
Finally, the electronic structures of the rock salt NiO and PdO in the type II antiferromagnetic phase were computed within the PBE+U+V method. The UNi and VNi–O values in NiO were determined to be 7.47 eV and 1.07 eV, respectively. The calculated band gap of NiO between the T and Γ points was 3.23 eV (Figure 4 (a)), which well reproduced the experimental value of 3.1–4.3 eV [36,37,38,39] and the previous computed value [17]. For PdO, the determined UPd and VPd–O values were 5.14 eV and 0.78 eV, respectively. The total energy per primitive unit cell of PdO in the type-II antiferromagnetic phase was lower than that in the ferromagnetic phase by 0.10 eV. This energy difference was small compared with the case of NiO (0.23 eV). Also, the calculated band gap of PdO between the T and Γ points (0.62 eV, Figure 4 (b)) was smaller than NiO. The small energy gap of PdO is considered to induce the cooperative Jahn-Teller distortion from the rock salt to tetragonal structures [40].
Band structure and DOS of rock salt (a) NiO and (b) PdO in the type-II antiferromagnetic phase (see Ref [41]. for the high symmetry points of the rhombohedral Brillouin zone). The DOS was projected onto Ni/Pd major state (Ni1/Pd1), minor state (Ni2/Pd2), and O state. The DOS was broadened using the Gaussian function with a linewidth of 0.01 Ry.
In summary, the electronic structures of tetragonal and rock salt PdO were calculated using the first-principles PBE+U+V method. The calculations gave the band gap of 1.51 eV for tetragonal PdO, which reproduced the experimental value. The band gap of rock salt PdO in the type-II antiferromagnetic phase was calculated to be 0.62 eV, which was smaller than that of NiO with 3.23 eV. These results indicate that correcting the electron-electron interaction term computed by the PBE functional is important for PdO of 4d transition metal oxides.
This study was supported by JSPS KAKENHI Grant Number JP22K05253 in Scientific Research (C) and JP24K23082 in Research Activity Start-up, and by JST-ALCA-Next Program Grant Number JPMJAN23C5. Numerical calculations were partly performed using the supercomputer system at the information initiative center, Hokkaido University, Sapporo, Japan.
