^{1}, Marek Sierka

^{1,a)}, Jens Döbler

^{1}and Joachim Sauer

^{1}

### Abstract

A periodic electrostatic embedding scheme is presented that uses the periodic fast multipole method. The convergence of properties with increasing cluster size is examined for cluster models of calcium fluoride. Properties investigated are the electron density, the density of states, the electronic excitation of color centers, and energies of defect formation. The embedded cluster method is applied to and oxygen vacancies in bulk as well as on its (111) surface. Employing the PBE0 functional, vacancy formation energies of 3.0 and 3.3 eV have been obtained for the bulk and the (111) surface, respectively. Formation of subsurface defects requires 3.33 eV (singlet open shell). The localization of the electrons left behind on defect formation in Ce states is discussed. Occupied Ce states are well localized on nearest Ce atoms for surface and subsurface vacancies. Localization apart from the vacancy was obtained for bulk. The total CPU time spent on the embedding part did not exceed 30 s on a single CPU even if 8000 basis functions of the cluster are involved.

This work was supported by the Deutsche Forschungsgemeinschaft (Center of Excellence UNICAT and Sonderforschungsbereich 546) and by the Stiftung Stipendien-Fonds des Verbandes der Chemischen Industrie. Thanks go to M. V. Ganduglia-Pirovano for discussions including access to results in Ref. 40 prior to publication.

I. INTRODUCTION

II. METHODS

A. The PEECM

B. Computational details

1. Periodic electrostatic embedded cluster calculations

2. Periodic DFT calculations

III. MODELS

IV. BULK AND (111) SURFACE RESULTS

A.

1. Defect-free bulk and surface structures

2. Defects

B.

1. Defect-free bulk and surface structures

2. Bulk oxygen vacancy

3. Oxygen vacancies at the (111) surface

V. CONCLUSIONS

### Key Topics

- Vacancies
- 49.0
- Crystal defects
- 26.0
- Density functional theory
- 24.0
- Surface structure
- 17.0
- Electrostatics
- 11.0

## Figures

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.

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.

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.

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.

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).

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).

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.

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.

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.

## Tables

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 .

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.

Oxygen defect formation energies and gaps between virtual and occupied Ce states (eV) for triplet and singlet spin states.

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