Side view of anion-terminated fluorite-type (111) surface. Layers of ions are displayed. Light circles denote anions and dark circles cations. A single defect site in the L1 and L3 layers is called surface and subsurface defect, respectively.
Changes in the HOMO-LUMO gap with increasing radius of the QM cluster for various models: bulk, (111) surface, -center (fluorine vacancy), and isomorphic F/Cl substitution. The arrow marks the periodic limit for defect-free bulk structure. The PEECM calculations are performed with PBE/TZVP for experimental (exp) or PBE optimized (PBE) lattice constants of 5.46 and 5.547 Å, respectively.
TDOS and local DOS of models calculated with PBE/TZVP method. On top: TDOS of bulk (periodic DFT). Below: TDOS of bulk (PEECM, cluster) and the contributions from local Ca , , and F , functions calculated with Mulliken population analysis. At the bottom: TDOS of the corresponding -center (PEECM). Arrows show the positions of HOMO and LUMO.
Spin density of the -center calculated with PEECM using the QM cluster. Black spheres denote cations and white ones anions. x marks the position of the missing F atom.
TDOS (also A, B, C, D) and local DOS of cerium oxides calculated with PEECM (PBE0/DZVP-46). Symbols s, p, d, and f denote local contributions (obtained by Mulliken population analysis) from local Ce , , , and O , functions, respectively. Arrows display occupied Ce states. Bottom: the QM cluster is used for bulk . Top: (A) bulk. and (dashed line) clusters; (B) (111) surface. (dashed line) and clusters; (C) surface defect. , (D) subsurface defect. (singlet).
HOMO-LUMO gap of bulk (PEECM) as a function of QM cluster radius . Lattice constant is (Ref. 39), except (exp) with (Refs. 81–83). The numbers represent the charges of the QM clusters. Arrows display periodic limits calculated with GTO basis set for PBE (Ref. 38) and with plane waves for PBE0 (Ref. 39).
Fragments of the QM cluster showing spin density distribution of O vacancy in bulk ceria calculated with PEECM (PBE0/DZVP-46) under restriction of symmetry point group ; white: O atoms; black: Ce atoms; red: nearest O atoms; and blue: nearest Ce atoms.
Spin density distribution of O vacancies at (111) surface (top view) for PBE0/DZVP-46. A: L1 ; B and C: L3 singlet and triplet, respectively . Orange: oxygen atoms of first layer (L1); red: oxygen atoms of L3; dark red: oxygen atoms of L4 beneath each triangular Ce atom; and blue: cerium atoms of L2 (triangular) and L5. Dotted line: mirror plane.
Basis and auxiliary basis sets as well as ECPs used in the calculations. DZVP and TZVP denote double zeta valence split plus polarization functions and triple zeta basis sets, respectively. The number of electrons replaced by ECP is given by . For basis sets including ECPs we add the suffix .
Oxygen defect formation energies and gaps between virtual and occupied Ce states (eV) for triplet and singlet spin states.
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