INVESTIGATION OF CATALYTIC PROPERTIES AND ELECTRONICSTRUCTURE OF CORRELATED MATERIAL CeO2 WITH AB-INITIO

COMPUTATIONAL METHODS

A THESIS SUBMITTED TOTHE GRADUATE SCHOOL OF NATURAL AND APPLIED SCIENCES

OFMIDDLE EAST TECHNICAL UNIVERSITY

BURAK ÖZDEMR

IN PARTIAL FULFILLMENT OF THE REQUIREMENTSFOR

THE DEGREE OF MASTER OF SCIENCEIN

PHYSICS

AUGUST 2013

Approval of the thesis:

INVESTIGATION OF CATALYTIC PROPERTIES AND ELECTRONICSTRUCTURE OF CORRELATED MATERIAL CeO2 WITH AB-INITIO

COMPUTATIONAL METHODS

submitted by BURAK ÖZDEMR in partial fulllment of the requirements for thedegree of Master of Science in Physics Department, Middle East TechnicalUniversity by,

Prof. Dr. Canan ÖzgenDean, Graduate School of Natural and Applied Sciences

Prof. Dr. Mehmet ZeyrekHead of Department, Physics

Assoc. Prof. Dr. Hande TooliSupervisor, Physics Department, METU

Examining Committee Members:

Prof. Dr. O§uz GülserenPhysics Department, Bilkent University

Assoc. Prof. Dr. Hande TooliPhysics Department, METU

Prof. Dr. enay Katrco§luPhysics Department, METU

Assoc. Prof. Dr. Hüseyin OymakPhysics Department, Atlm University

Dr. Yavuz DedeChemistry Department, Gazi University

Date:

I hereby declare that all information in this document has been obtainedand presented in accordance with academic rules and ethical conduct. Ialso declare that, as required by these rules and conduct, I have fully citedand referenced all material and results that are not original to this work.

Name, Last Name: BURAK ÖZDEMR

Signature :

iv

ABSTRACT

INVESTIGATION OF CATALYTIC PROPERTIES AND ELECTRONICSTRUCTURE OF CORRELATED MATERIAL CeO2 WITH AB-INITIO

COMPUTATIONAL METHODS

Özdemir, Burak

M.S., Department of Physics

Supervisor : Assoc. Prof. Dr. Hande Tooli

AUGUST 2013, 51 pages

Density functional theory employed with Hubbard model addressing strong electronic

local correlation source of which considered to be the occupation of d or f type atomic

orbitals, since these are localized in space, is used to study ceria (CeO2) material. Since

ceria exhibits occupied atomic f orbitals on Ce sites and these sites are not equivalent in

correlation aspect when surface and defective structures considered, there has been no

account of single Hubbard eective U value that captures or improves all the properties.

Therefore, locally resolved Hubbard U, by this we mean dierent U values for every

distinct Ce site, is the rst approach which should be taken in order to nd a method

that improves the theoretical accounts of this material and this approach is in the focus

of this thesis. Ceria (111) and (110) surfaces, CO and Au adsorptions are investigated

with this approach.

Keywords: DFT, DFT+U, f-orbital, Ceria, Hubbard U, Catalysis, CO, Au

v

ÖZ

GÜÇLÜ KORELASYONLU CeO2 MALZEMESNN KATALTK ÖZELLKLERVE ELEKTRONK YAPISININ LK LKELER BLGSAYAR YÖNTEMLER LE

NCELENMES

Özdemir, Burak

Yüksek Lisans, Fizik Bölümü

Tez Yöneticisi : Doç. Dr. Hande Tooli

A§ustos 2013 , 51 sayfa

Ceria (CeO2) malzemesi, kayna§ d veya f atomsal yörüngesine ait elektronlar olan

güçlü elektronik yerel korelasyonun Hubbard modeli kullanlarak yo§unluk fonksiyoneli

teorisi ile incelendi. Ceria atomsal f yörüngesine ait elektronlar içerebilen Ce atomlarn-

dan olu³tu§u ve bu atomlarn korelasyon açsndan yüzey ve kusurlu yaplarda farkl

olu³undan, özelliklerinin tümünün do§ru bulundu§u ya da geli³tirildi§i tek bir etkili

Hubbard U de§eri literatürde rapor edilememi³tir. Bu malzemenin teorik incelemesinin

geli³tirilmesi için sahip olunmas gereken ilk yakla³m her farkl Ce atomuna farkl bir

Hubbard U de§eri uygulanmas yani yerel olarak çözümlenmi³ Hubbard U olmaldr ve

bu yakla³m tezin oda§ndadr. Ceria (111) ve (110) yüzeyleri, CO ve Au ba§lanmalar

bu yakla³mla incelenmi³tir.

Anahtar Kelimeler: DFT, DFT+U, f-orbital, Ceria, Hubbard U, Kataliz, CO, Au

vi

To my family

vii

ACKNOWLEDGMENTS

I want to thank to my advisor Assoc. Prof. Dr. Hande Tooli for her kindness and

patience. I am grateful for everything that I learned from her in a broad spectrum.

I am grateful to my family for their boundless supports that made possible this priceless

intellectual journey for me.

This work is nancially supported by TÜBTAK, The Scientic and Technological

Research Council of Turkey (Grant No. 112T542). The computatinal resources were

provided by UHeM, National Center for High Performance Computing (Grant No.

1001942012 and 10922010). XCRYSDEN computer program has been used to generate

the graphics [1].

viii

TABLE OF CONTENTS

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

ÖZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

TABLE OF CONTENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

CHAPTERS

1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 THEORY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1 Theorems Underlying DFT . . . . . . . . . . . . . . . . . . . . 7

2.2 Kohn-Sham Equations . . . . . . . . . . . . . . . . . . . . . . . 7

2.3 Planewave Expansion . . . . . . . . . . . . . . . . . . . . . . . 8

2.4 Pseudopotentials . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.5 Brillouin Zone Integration . . . . . . . . . . . . . . . . . . . . . 15

2.6 Mixing Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.7 Diagonalization . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.8 Exchange-Correlation . . . . . . . . . . . . . . . . . . . . . . . 16

2.9 Eective Hubbard Model . . . . . . . . . . . . . . . . . . . . . 20

ix

2.10 Hellman-Feynman Theorem . . . . . . . . . . . . . . . . . . . . 23

3 METHOD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.1 Pseudopotential Generation . . . . . . . . . . . . . . . . . . . . 25

3.1.1 LDA . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.1.2 GGA . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.2 DFT Calculations . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.3 Local Hubbard U . . . . . . . . . . . . . . . . . . . . . . . . . 31

4 LOCAL HUBBARD U & CERIA . . . . . . . . . . . . . . . . . . . . . 35

4.1 Literature Review of Ceria . . . . . . . . . . . . . . . . . . . . 35

4.2 Results & Discussion . . . . . . . . . . . . . . . . . . . . . . . 37

4.2.1 Bulk Ceria . . . . . . . . . . . . . . . . . . . . . . . . 37

4.2.2 (111) & (110) Surfaces of Ceria . . . . . . . . . . . . 38

4.2.3 CO Adsorption on (111) & (110) Surfaces of Ceria . . 41

4.2.4 Au Adsorption on (111) Surface of Ceria . . . . . . . 43

5 CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

x

LIST OF TABLES

TABLES

Table 3.1 All-electron and LDA pseudo energy dierences for various ionic states

calculated with pseudo potential acquired from the neutral reference state.

Unit of energy is Ryd. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

Table 3.2 Eigenenergies for ghost state check acquired from all-electron solution

and LDA pseudo potential solution with Bessel function basis set having

kinetic energy cut-o of 40 Ryd. Unit of energy is Ryd. . . . . . . . . . . . 27

Table 3.3 All-electron and GGA pseudo energy dierence for various ionic states

calculated with pseudo potential acquired from the neutral reference state.

Unit of energy is Ryd. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

Table 3.4 Eigenenergies for ghost state check acquired from all-electron solution

and GGA pseudo potential solution with Bessel function basis set having

kinetic energy cut-o of 40 Ryd. Unit of energy is Ryd. . . . . . . . . . . . 29

Table 4.1 Bulk ceria. Hubbard U dependence of lattice constant, a0; band gap,

Eg; peak energy of localized gap states of Ce-4f character, E4f . Starred

value (4*) indicates the use of Wannier projection. Distances are in Å and

energies are in eV units. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

Table 4.2 Energies of (111) and (110) surfaces of ceria with dierent U values

applied to outermost Ce sites. U value of 4 eV is applied to the Ce sites

residing in bulk. d is the distance between the outermost Ce atomic layer

and the previous one in Å unit. Surface energies are in J/m2. . . . . . . . 39

xi

Table 4.3 LDA+U energy of CO adsorption on Ce-top site at (111) surface.

Ub=4 eV. E′ and E′′ are the corrected energies with the Hubbard and total

energy dierences, respectively. d is the distance of C to nearest neighbor

Ce atom. Energies are in eV. . . . . . . . . . . . . . . . . . . . . . . . . . . 42

Table 4.4 LDA+U energy of CO adsorption on Ce-top side at (110) surface.

Ub = 4 eV. d is the distance of C to nearest neighbor Ce atom. Energies are

in eV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

Table 4.5 LDA+U energy of adsorption and charge state (qAu) of Au on O-top

site on (111) surface of ceria. d is the distance of Au to nearest neighbor O

atom. Ub=4 eV. Energies are in eV. . . . . . . . . . . . . . . . . . . . . . . 44

Table 4.6 LDA+U energy of adsorption and charge state (qAu) of Au on O bridge

site at (111) surface of ceria. d is the distance of Au to nearest neighbor Ce

atom. Ub=4 eV. Energies are in eV. . . . . . . . . . . . . . . . . . . . . . . 44

xii

LIST OF FIGURES

FIGURES

Figure 1.1 Unit cell of ceria (CeO2). Big gray and small red balls represent Ce

and O atoms, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Figure 1.2 Optimized structures of ceria surfaces (a) (111) (side view), (b) (111)

(top view), (c) (110) (side view), (d) (110) (top view). . . . . . . . . . . . . 2

Figure 3.1 (a) All-electron and pseudo wave functions for the ultrasoft LDA

pseudopotential. (b) Logarithmic derivatives of the all-electron and pseudo

wave functions for the ultrasoft LDA pseudopotential calculated at r=3.01

a0, L is the logarithmic derivative and E is the energy of the scattered wave. 27

Figure 3.2 (a) All-electron and pseudo wave functions for ultrasoft GGA pseu-

dopotential, (b) Logarithmic derivatives of all-electron and pseudo wave

functions for ultrasoft GGA pseudopotential calculated at r=3.01 a0. . . . 28

Figure 3.3 Potential along the surface normal of CO adsorbed on ceria (111)

surface. The potential is calculated by averaging over layers parallel to the

surface. C and O atoms are labeled on the abscissa. cell b. indicates the

cell boundary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Figure 3.4 Au adsorbed at O-top site on (111) surface of ceria. . . . . . . . . . 33

Figure 4.1 Band structure of bulk ceria for (a) U=0 eV, (b) U=4 eV employed

with atomic projection. (c) Total and projected density of states of bulk

ceria at U=0 eV. Fermi level energy is set to zero. . . . . . . . . . . . . . . 38

xiii

Figure 4.2 Comparison of (a) valence and (b) conduction density of states for

ceria (110) surface found with Us values of 3, 4, and 6 eV. Ub=4 eV. . . . . 40

Figure 4.3 Comparison of (a) valence and (b) conduction density of states for

ceria (111) surface found with Us values of 3, 4, and 6 eV. Ub=4 eV. . . . . 40

Figure 4.4 DOS of ceria bulk, (111), and (110) surfaces calculated at U = 4 eV

employed with Wannier projection. . . . . . . . . . . . . . . . . . . . . . . 40

Figure 4.5 Optimized structures of CO at (a) Ce-top site (labeled as Un) of ceria

(111) surface, (b) Ce-top site of ceria (110) surface. Big grey and small red

balls represent Ce and O atoms, respectively. Atoms reside at surface are

darker in color. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

Figure 4.6 Optimized structures of Au at (a) O-top site of ceria (111) surface,

(b) O-bridge site of ceria (110) surface. . . . . . . . . . . . . . . . . . . . . 43

xiv

CHAPTER 1

INTRODUCTION

Ceria, CeO2, has a high oxygen storage capacity, in other words it can easily adsorb

and release oxygen, and plays therefore an actively role in CO oxidation and water-gas

shift reaction when used as a support for metal particles [2, 3]. Ceria is an insulator

having a cubic uorite structure (Fig. 1.1) that exhibits highly debated mixed-valency.

The upper valance band consists of Ce-4f atomic states along with the main contri-

bution of O-2p atomic states as found using density functional theory (DFT), but

an experimental study by Wuilloid et al. [4] showed that localized Ce-4f states are

completely empty. This controversial situation concerning whether ceria is a charge

transfer or Mott-type insulator poses serious theoretically challenges. In the band gap,

there is a localized empty band that almost completely consists of Ce-4f atomic states

which are not considered in the band gap calculation [4]. The amount of band gap

and energy of Ce-4f gap states are underestimated in conventional DFT method, in

an amount greater than the usual DFT band gap error of 1-2 eV, with respect to the

Figure 1.1: Unit cell of ceria (CeO2). Big gray and small red balls represent Ce andO atoms, respectively.

1

experiments [4, 5, 6]. This error stems from the inhomogeneous distribution of f elec-

trons in space not correctly described by standard local approximations. In order to

study this kind of interactions, Hubbard model and further approximations to it are

introduced. An eective Hubbard model is introduced where the hopping potential J

is buried inside on-site interaction potential U [7]. In the Hubbard model, the occu-

pation of the localized atomic d or f orbitals has to be calculated which can be done

by projecting Bloch states to atomic orbitals or Wannier orbitals. Although Wannier

orbitals are more suitable for the eective Hubbard U application, contribution of

Hubbard U to interatomic forces vanishes with this projection. Band gap of bulk ceria

is found to be inversely proportional to the eective Hubbard U (Ueff ) employed with

atomic projection method and directly proportional to the Ueff value employed with

Wannier projection method. 4f gap state energy is found to be directly proportional to

Ueff employed with both atomic and Wannier projection methods. Since both band

gap and 4f gap state energies are underestimated within local and semi-local density

approximations, Wannier projection method serves better than atomic projection for

the eective Hubbard U. Reduction energy of CeO2 to Ce2O3 is inversely proportional

to Ueff and matching with the experimental value is found at a specic Ueff value

employed with atomic projection method. By projectiong on Wannier functions, the

reduction energy is found to be independent of the value of Ueff and slightly overes-

timates the experimental value.[8, 9]

(a) (b)

(c) (d)

Figure 1.2: Optimized structures of ceria surfaces (a) (111) (side view), (b) (111) (topview), (c) (110) (side view), (d) (110) (top view).

Since the (111) surface (Fig. 1.2 a, b) is the energetically most favorable surface

2

as conrmed by experiments and the (110) surface (Fig. 1.2 c, d) is observed in

ceria nanoparticles as the second most stable surface together with the (111) surface

complementing the nanoparticles [10, 11], the most important surfaces of ceria are (111)

and (110). The most important property of these surfaces is their high oxygen storage

capacity. These surfaces can easily adsorb and release oxygen and therefore catalyze

the CO oxidation. Together with the water-gas shift reaction these ceria surfaces

have a potential use in hybrid automobiles which store H2 to produce electricity as an

additional power to the engine [12, 13].

CO adsorption on (111) and (110) surfaces of ceria and its oxidation have been pre-

viously studied using DFT with various exchange-correlation functionals [14, 15, 16,

17, 18]. The relatively strong adsorption energy of CO to oxygen bridge site at (110)

surface is best reproduced with B3LYP functional [17] as being closest to the experi-

mental value [19]. This energy is overestimated with GGA+U functional when U value

best describing the bulk properties and oxygen vacancy formation properties such as

formation energy and excess electron localization [15, 20] is used. The Hubbard U

values that best describe the properties of bulk, surface, and nanoparticle structures

of ceria are disributed in an energy interval ranging from 2 eV to 6 eV. These nd-

ings suggest that Hubbard U potential applied through atomic projection is structure

dependent, and therefore not all Ce sites are equivalent in this respect.

Adsorption of certain metal particles on ceria surfaces is found to increase both water-

gas shift (WGS) reaction and CO oxidation reaction rates up to 10 times where re-

actions occur at the metal-ceria interface [21]. DFT studies show that signicant

geometrical distortions occur upon metal adsorption. For example an outward relax-

ation of 0.5 Å has been observed for the outermost oxygen atoms at the (111) surface

[22]. There is one controversy and one open point on these catalytic reactions over

metal-ceria structure. The charge state of the active form of metal particle (i.e. cat-

alytically active) is highly debated in the literature. Both metallic nanoparticles and

charged nanoparticles, which are mostly cationic (positively charged), are proposed as

the catalytically active site [23, 24, 25, 11]. Although these cationic forms (with nom-

inal charges +1 and +3) over ceria surfaces are experimentally observed, DFT studies

with the GGA approximation predict a relatively low positive partial charge (+0.34)

while the LDA approximation completely fails [26, 27, 28]. The open point concerns

3

the sintering resistance drop of the metal particles while the catalytic reactions are in

progress; sintering of metal nanoparticle results in the irreversible drop of the catalytic

reaction rates for both the WGS and CO oxidation [11, 29, 13, 30].Neutral metal par-

ticles, which are formed in a reducing environment, tend to agglomerate, and a drop in

catalytic activity is observed as a consequence of the decrease of metal particle-surface

interfacial area [11]. Charge state of the gold particles are observed to be independent

of the CO oxidation process [31].

4

CHAPTER 2

THEORY

The most basic properties of a quantum mechanical system are its ground state en-

ergy and the corresponding eigenstate. For this purpose, two basic approaches have

been developed both of which involve a variational method except for highly ideal-

ized specic cases. The rst one of these methods is the Hartree-Fock (HF) approach

which is a wave function based method where each electron is assumed to occupy a

single particle orbital. The many-body wave function is then taken as a product of

these functions, which is symmetrized to satisfy the Pauli exclusion principle in the

form of a Slater determinant. The second is the density-functional theory, DFT, which

employs the electronic density instead of the wave function for a complete description

of the system. Its foundations lies in the fact that there is one-to-one correspondence

between the external potential and the density. Solution to both of the problems in-

volves a self-consistency cycle. In DFT, neither the density nor the potential is known

prior to the solution of the problem therefore an initial guess is made for the electron

density and this requires a self-consistency cycle for the problem to be solved.

The many-body Hamiltonian for a system with Ne electrons and Nn nuclei is given by,

H = −1

2

Ne∑i

∇i2 −Ne∑i

Nn∑I

ZI

|~ri − ~RI |+

1

2

Ne∑i

Ne∑j 6=i

1

|~ri − ~rj |+

1

2

Nn∑I

Nn∑J 6=I

ZIZJ

|~RI − ~RJ |(2.1)

where length and energy are rescaled by Bohr radius and Hartree energy respectively,

i.e. atomic units are used. The index i labels the electrons, and their position vectors

are ~ri and ~rj . ~RI and ~RJ are the position vectors of nuclei and Z is the atomic number.

The rst term is the kinetic energy operator, the second is the electron-ion potential,

5

and the third is the electron-electron interaction, while the last term is the ion-ion

interaction term. Due to the fact that nuclei have much higher masses relative to the

electrons, nuclei are assumed to be static with respect to the electrons according to

the Born-Oppenheimer approximation. The kinetic term for the ions therefore is not

included in the Hamiltonian.

The single-particle density operator is dened as:

n(~r) =

N∑i=1

δ(~r − ~ri) (2.2)

from which, the density can be written as

n(~r) = Ne

∫|ψ(~r, ~r2, ..., ~rNe)|2d~r2...d~rNe (2.3)

The main idea behind DFT is to write the energy as a functional of density. This may

not be achieved for the kinetic energy and thus the Kohn-Sham ansatz [32] must be

employed, whereby an auxiliary system of non-interacting electrons occupying single-

particle orbitals called Kohn-Sham orbitals φi(~ri) with density

nKS(~r) =∑i

|φi(~ri)|2 (2.4)

corresponding to the density of the real system is constructed. With the denition of

the Kohn-Sham orbitals, the total energy now becomes,

E = −1

2

Ne∑i

∫d~rφ∗i (~r)∇2φi(~r)d~r +

∫n(~r)Vne(~r)d~r +

1

2

∫ ∫d~rd~r′

n(r)n(r′)

|~r − ~r′|+ Exc

(2.5)

where, Vne is the external potential and

Exc = ∆T + ∆Eee (2.6)

is called the exchange-correlation energy. ∆T is the dierence between exact kinetic

6

energy and the KS kinetic energy and ∆Eee is an error that is associated with the

approximation of writing the two body density n(~r, ~r′) as the product of two one-

body densities n(~r)n(~r′). In the local density approximation, the exchange-correlation

energy takes the following form,

Exc =

∫d~rn(~r)εhomxc (n) (2.7)

where εhomxc (n) is the exchange-correlation energy density of the homogeneous electron

gas. This form is explained in detail later in the text.

2.1 Theorems Underlying DFT

Solution to the electronic structure of solid state systems using DFT begins with the

two simple yet ingenious theorems known as Hohenberg-Kohn theorems [33].

Theorem I: There is a one-to-one mapping between density and external potential, so

that the density is uniquely determined for a given external potential.

Theorem II: It is possible to dene a universal functional for the energy in terms of

density and the global minimum of this functional gives the exact ground state energy.

As a result of the rst theorem, the density uniquely determines the external potential

and potential uniquely determines the ground state wave function, therefore the proof

of the second theorem is possible with the use of the variational principle.

2.2 Kohn-Sham Equations

In order to nd the ground state energy, the total energy (Eq. 2.5) is minimized with

respect to the Kohn-Sham orbitals,

δEeδφ∗i (~r)

=δTsδφ∗i (~r)

+

[δEextδn(~r)

+δEHartreeδn(~r)

+δExcδn(~r)

]δn(~r)

δφ∗i (~r)= εiφi(~r) (2.8)

where Ts (kinetic energy of the Kohn-Sham system), Eext, and EHartree are the rst,

7

second, and third terms in Eq. 2.5, and results in the system of Kohn-Sham equations

− 1

2∇2φi(~r) +

[Vext(~r) +

∫n(~r ′)

|~r − ~r ′|d~r ′ + εxc[n(~r)] + n(r)

δεxc[n(~r)]

δn(~r)

]φi(~r) = εiφi(~r)

(2.9)

where Eq. 2.7 is used. Since the potential here depends on the density and the

density itself depends on the solution to this equation, the density depends in turn on

the potential. Therefore, the problem is nonlinear. A self-consistent solution has to

be sought where convergence is reaced when the dierence between the densities or

potentials of subsequent iterations fall below a predetermined threshold (Sec. 2.6).

2.3 Planewave Expansion

In order to avoid the solution of the Kohn-Sham system of equations in real space,

Kohn-Sham single-particle orbitals are expanded in some basis set, converting the

problem into a set of eigenvalue equations. The eort is now shifted to diagonalization

of the matrix equation. The basis functions chosen for the expansion in this thesis are

plane waves since this is the most suitable choice for a periodic system. A planewave

expansion of the Kohn-Sham orbitals can be written as:

φi(~r) =∑~q

ci,~q1√Ωei~q·~r (2.10)

where ci,~q are the expansion coecients, Ω is the volume of the system, and ei~q·~r is the

planewave with momentum ~q.

In this basis, eq. 2.9 becomes

HC = εiC (2.11)

where the elements of the matrix C are the ci,~q expansion coecients, and H is the

matrix representation of the Hamiltonian in the planewave basis H~q,~q ′ . Due to the

8

lattice-periodic nature of the systems under study in this thesis, Eq. 2.11 can be

further simplied to

∑Hm′,m(~k)ci,m(~k) = εi(~k)ci,m′(~k) (2.12)

with

Hm′,m(~k) =1

2|~k + ~Gm|2δm′,m + Veff (~Gm − ~Gm′), (2.13)

where ~k is the planewave vector and ~Gm are the reciprocal lattice vectors of the

system. Since the potential has the periodicity of the lattice, only terms with the

~Gm − ~Gm′ argument survives. Veff includes all the interaction terms; the external

potential, Hartree term, and exchange-correlation term,

Veff = Vext(~r) + VH(~r) + Vxc(~r)

= Vext(~r) +

∫n(~r ′)

|~r − ~r ′|d~r ′ + ε[n(~r)] + n(r)

δε[n(~r)]

δn(~r).

(2.14)

Fourier transform of these terms are calculated as follows. The external potential can

be written as:

Vext =

nsp∑κ

nκ∑j=1

∑~T

V κ(~r − ~τκ,j − ~T ) (2.15)

where κ counts the number of atomic species. There are nsp number of species in

the cell and j counts the number of atoms for a particular species. Vector ~τκ,j gives

the position of a specic atom in the cell and ~T is the lattice translation vector. The

Fourier components read

Vext(~G) =

nsp∑κ=1

Ωκ

ΩcellSκ(~G)V κ(~G) (2.16)

where Sκ(~G) and V κ(~G) are the structure and form factors, respectively. The form

factor is the atomic potential to be replaced with a softer pseudopotential. Pseudo

9

potentials are explained later in the text.

For the Hartree term, the Fourier components read

VH(~G) = 4πn(~G)

G2. (2.17)

Finally, for the exchange-correlation term, the Fourier components can be written as:

Vxc(~G) =∑~G ′

nxc(~G− ~G ′)dεxcdn

(~G ′) + εxc(~G). (2.18)

In a DFT calculation, one has to introduce a kinetic energy cut-o that restricts the

number of planewaves used which are innite in principle, but it is well known that the

contribution of waves with high frequency in Fourier series is less since only valence

electrons with smoother wave functions are treated and systems are periodic. Thus, a

cut-o for the kinetic energy is dened as

1

2|~k + ~G|2 < Ecut. (2.19)

The suitable value of Ecut for a particular calculation must be carefully determined

via convergence tests.

2.4 Pseudopotentials

In a DFT calculation, ionic potentials are replaced with eective potentials, which

includes the potential of core electrons that do not take part in crystal binding as well

as the ions. As a result, the number of electronic degrees of freedom decreases and

also the valence states become smoother. Otherwise very localized or oscillatory states

result that need higher cut-o energies for the planewave basis set.

Pseudopotentials are generated for an isolated atom and then ported to the real crystal

system. After solving the all-electron Kohn-Sham equations for the isolated atom

giving the corresponding wave functions and energies, a potential is derived for each

valence wave function to produce the all-electron energies.

10

To understand the idea behind pseudopotentials, we study a simplied model. We

imagine an all-electron wave function |ψ〉 and write as a sum of a smoother wave

function |φ〉 and an expansion over the core states.

|ψ〉 = |φ〉+core∑n

an|χn〉 (2.20)

where |χn〉 are the core wave functions. By use of orthonormality and requiring that

energies of all electron and pseudo wave functions should match, the following eective

equation for the valence pseudo wave function is obtained

H|φ〉+core∑n

(E − En)|χn〉〈χn|φ〉 = E|φ〉 (2.21)

This can be taken as an eigenvalue equation with an extra potential for the pseudo

wave function;

(H + V )|φ〉 = E|φ〉 (2.22)

Due to the term (E−En), the new potential is less repulsive than the original coulombic

potential. Also, V is expected to be dierent for each state. For a pseudopotential

there are a number of conditions to be met [34], which are as follows:

1. Real and pseudo valence eigenvalues agree for a chosen atomic conguration.

2. Radial part of the real and pseudo atomic wave functions agree beyond a chosen

"core radius" rc.

3. The real and pseudo charge densities agree for R > rc for each valence state (norm

conservation).

∫ R

0|φ|2r2d~r =

∫ R

0|ψ|2r2d~r (2.23)

4. The logarithmic derivatives of the real and pseudo wave function and their rst

derivatives agree for r > rc

11

For the cut-o radius rc, one has to determine the optimum value since it is not

unique in any situation. There are two considerations about a pseudopotential. One

is the transferability and the second is the smoothness. Since a pseudopotential is

generated for an isolated atom, it should be tested that the pseudopotential performs

correctly in molecules and crystals, and this quality of the pseudopotential is called

transferability. The smoothness of pseudo wave functions is instead important for

computational reasons, since a smoother wave function requires a smaller basis set

for its accurate description. While smaller rc values makes a pseudopotential more

transferable, larger rc makes the pseudo wave function smoother resulting in faster

calculations. A suitable rc is a result of such a compromise and pseudo potentials have

to be tested carefully before actual usage.

In 1990, D. Vanderbilt proposed a scheme for even more smoother pseudo wave func-

tions which signicantly decreases the number of planewaves in the basis expansion

[35]. This family of pseudopotentials, known as ultra-soft pseudo potentials (USPPs),

relaxes the norm-conservation condition for pseudo wave functions, allowing for the

construction of the pseudo wave function for each real wave function and thus larger

cut-o radius rc. This in turn makes the pseudo wave functions much smoother.

Vanderbilt proposes this method in three stages. First, one obtains a fully non-local

pseudopotential (Kleinman-Bylander type) without dening a semi-local pseudopo-

tential by using the wave function. Here locality is intended to describe the space

dependence of the potential. A local potential means that the potential vanishes be-

yond a chosen cut-o radius rlocc . Second, this method is generalized to more than one

energy at which the scattering properties are captured to ensure the transferability of

the pseudopotential. And nally, norm-conservation is relaxed. Let us now consider

each stage separately.

Stage I:

Schrödinger equation with the all-electron screened potential is

(T + VAE − εi)|ψi〉 = 0. (2.24)

A local part of the potential is separated and used to obtain the local wave function

12

|χi〉 (local since it vanishes at and beyond R where VAE = Vloc and ψi = φi)

|χi〉 = (εi − T − Vloc)|φi〉. (2.25)

A non-local pseudopotential may be dened as;

VNL =|χi〉〈χi|〈χi|φi〉

(2.26)

It can be shown that

(T + Vloc + VNL − εi)|φi〉 = 0. (2.27)

With this nal equation one constructs a pseudo wave function |φi〉 with the constraint

that it joins smoothly to |ψi〉 at rcl and satises the norm-conservation at R

〈ψi|ψi〉R = 〈φi|φi〉R

Stage II:

Now, previous scheme is generalized to more than one energies εi at which scattering

properties will be correct. This time instead of |χi〉 we use linear combinations of the

so-called beta functions dened as

|βi〉 =∑j

(B−1)ji|χj〉. (2.28)

The non-local part of the potential this time is

VNL =∑i,j

Bij |βi〉〈βj | (2.29)

where Bij = 〈φi|χj〉. Again we have

(T + Vloc + VNL − εi)|φi〉 = 0. (2.30)

13

Stage III:

At this stage we relax the norm-conservation constraint. Although this stage is not

necessary, it is crucial to have smoother pseudo wave functions therefore a smaller

planewave basis set and in turn faster DFT calculations. In doing so, we need to

dene an overlap operator for a generalized eigenvalue equation

(H − εiS)|φi〉 = 0. (2.31)

The overlap operator, in this case, is also dened with respect to the β functions.

S = 1 +∑ij

Qij |βi〉〈βj | (2.32)

where Qij , which will also be used to compensate for the density decit below the

cut-o radius through the so called augmentation charges in a DFT calculation, is

Qij = 〈ψi|ψj〉R − 〈φi|φj〉R (2.33)

The non-local pseudo potential now becomes

VNL =∑ij

Dij |βi〉〈βj | (2.34)

where

Dij = Bij + εQij . (2.35)

As a result of relaxing the norm-conservation constraint, we are free to choose a larger

cut-o radius for pseudo wave functions. Now, one also needs to nd the overlap matrix

and augmentation charges in a DFT calculation but this extra step of calculation is

not computationally heavy, since the computationally demanding step is the matrix

diagonalization which is now much faster due to the smaller basis set size.

14

2.5 Brillouin Zone Integration

In order to nd the total electronic energy from the Fourier-transformed Kohn-Sham

equations

H(~k)φ~k(~r) = εi(~k)φ~k(~r) (2.36)

we need to integrate the Fourier-transformed kinetic, exchange-correlation and elec-

tronic interaction energies over the rst Brillouin zone and nally summing over the

state index i one nds the total electronic energy Ee where the energy due to the exter-

nal potential is excluded. For practical purposes the ~k space integration is discretized.

Since for each ~k-point, the Kohn-Sham equations have to be solved, one needs to use

as small a ~k-point set as possible without compromising the accuracy. The widely used

scheme of Monkhorst and Pack [36] denes a uniformly distributed set of k-points by

the formula

~kn1,n2,n3 =

3∑i

2ni −Ni − 1

2Ni

~bi (2.37)

where ~bi are the smallest reciprocal lattice vectors corresponding to real space lattice

vectors ~ai dened as

~bi = 2π~aj × ~ak

~ai · (~aj × ~ak)(2.38)

This scheme makes the Fourier components orthonormal by use of the point group

symmetries in the Brillouin zone and as such the ~k-points selected this way produces

the best results with minimum number of points.

2.6 Mixing Schemes

Since the electronic density is obtained from the solution of the Kohn-Sham equations

which through the eective potential depends on the density itself, one needs to solve

the problem self-consistently. An initial guess for the density is made and used to

15

construct the eective potential. The Kohn-Sham equations are then solved. The

solution gives a dierent density from the previous density and the objective is to

reduce this dierence down to a predetermined threshold in consecutive steps. It turns

out however that there is a signicant convergence problem. A remedy to this problem

is suggested by the so-called mixing schemes. Some of the mixing schemes proposed

for the convergence problem are simple, such as the one proposed by Broyden [37]

and Anderson [38]. A simple mixing scheme involves the mixing of old and updated

densities in suitable proportions through

nN+1(~r) = αF [nN ] + (1− α)nN (2.39)

Function F [nN ] connects the density at step N+1 to the density at step N . α is just a

parameter for mixing, having values between 0 and 1, and which has to be tuned for a

faster convergence. Although this tuning of α is going to slow down the scf convergence

since the density dierence between iterations decreases, the added stability justies

the added time.

2.7 Diagonalization

The planewave basis expansion typically results in matrices of very large dimensions

and a full diagonalization of such matrices is time consuming. Therefore, approximate

diagonalization techniques are used all of which are iterative as far as the calculations

in this thesis are concerned. Newton, Lanczos, Davidson, and conjugate gradients are

some methods to name [39].

2.8 Exchange-Correlation

The electron-electron interaction portion of the exact DFT energy functional is given

by

Eee =1

2

∫ ∫d~rd~r′

n(~r, ~r′)

|~r − ~r′|(2.40)

16

where the two-particle density n(~r, ~r′) can not be accessed exactly but must be ap-

proximated by a form with one coordinate (local density approximation). In this

approximation, Eee is replaced by

Eee =1

2

∫ ∫d~rd~r′

n(r)n(r′)

|~r − ~r′|+ Exc (2.41)

where Exc is called the exchange-correlation energy. A local approximation is possible

since the exact amount of this exchange-correlation error is known for the uniform

electron gas in the high and low density limits. To clarify the local density approxi-

mation, we can write the kinetic energy of particles in an innite potential well as an

example [40]. The system for which an exact solution to its kinetic energy exists is the

non-interacting uniform gas of electrons. The kinetic energy is given by:

Texact =π2

2L2

N∑j=1

j2 =π2

2L2

N(N + 1)(2N + 1)

6(2.42)

where the kinetic energy of the jth electron is given by Tj =k2j2 = (πj)2

2L2 in atomic units.

L is the length of the one-dimensional box, j is an integer, and N is the number of

electrons. The leading contribution comes from

Texact =π2

6

N3

L2= L

π2

6

(N

L

)3

. (2.43)

Since we know the exact form of the kinetic energy, we can make the following approx-

imation:

T loc =

∫dxπ2

6n3(x) (2.44)

where T loc is the kinetic energy in the local approximation. The Fermi wavevector can

be introduced which for one dimension reads:

kF =πN

L= πn (2.45)

17

so

T loc[n(x)] =

∫dx

1

6k2Fn(x) (2.46)

where k2F /6 is kinetic energy density. For a uniform density, n(x) = NL ,

T loc = Texact. (2.47)

So we have arrived at the density functional of kinetic energy, T loc[n(x)], for the uni-

form electron gas in the local density approximation. For the particle in a box problem,

the system assumes uniform density for large N and therefore the local approximation

gives the exact kinetic energy. Low density conned in a small space results in error

which is related to the exchange-correlation error in DFT. This approximation is made

around the uniform density and it is called the local density approximation.

The local density approximation to the exchange interaction

Elocx [n] =

∫d3rn(r)εx(n) (2.48)

is found by evaluating the Fock integral for the Slater determinant of plane-wave

orbitals which gives the exact exchange energy per electron for the uniform electron

gas:

εx =3kF4π

(2.49)

therefore the local approximation assumes the following form,

Elocx [n] = Ax

∫d3rn4/3(~r) (2.50)

where the Fermi wavevector for three dimensional case, kF = (3π2n)1/3, is used and

so Ax = −(3/4)(3π)1/3.

18

In order to make a local density approximation to the exchange-correlation energy of

the fully interacting system of electrons Ceperley and Alder [41] used Monte-Carlo

calculations where Hartree-Fock wave functions are used as the initial guess and found

the exchange-correlation energies for various densities together with reproducing exact

analytical result in low density limit (Wigner crystal) found by Wigner [42] and the

high density limit which purely consists of exchange interactions. Interpolating the

results of Ceperley and Alder's stochastic calculations in the intermediate densities

and extrema construct the local density approximation.

Later, Perdew and Zunger [43] corrected the self-interaction error resulting from this

stochastic process where a single-particle density is used. For example in Hartree-Fock

theory, where the exchange interaction is treated exactly, this self-interaction is exactly

cancelled by the self-exchange term

Uασ + Exc[nασ, ~r = 0] = 0 (2.51)

where Uασ is the interaction energy without the exchange part (Eq. 2.41) and α,

σ are spin indices. ~r is the distance vector from the particle which inuences the

potential, therefore ~r = 0 means that the particle is interacting with itself. Since it

is known that the exact exchange is the term that cancels self-interaction energy by

self-exchange energy, the following equation can be written.

Uασ + Ex[nασ, 0] = 0 (2.52)

therefore the correlation term satises

Ec[nασ, 0] = 0 (2.53)

Finally the self-interaction corrected exchange-correlation energy is

ESICxc = Eapproxxc −∑ασ

δασ (2.54)

19

where the superscript approx is for the approximation made (e.g LSD, GGA) not

related to SIC and δασ is

δασ = U [nασ] + Eapproxxc [nασ, 0] (2.55)

Together with this self-interaction correction (SIC), the local density approximation

(LDA) to the exchange-correlation energy is known as PZ81 and is used widely and

also in this thesis.

2.9 Eective Hubbard Model

Local density approximation (LDA) and generalized gradient approximation (GGA)

fail to predict the correct band gap of transition metal oxides or more generally Mott

insulators. The band gap of these insulator systems are a result of the interaction of

the same type of electrons. Due to the low occupation of the atomic orbitals combined

with the inhomogeneous space dependence of the density of the d and f orbitals, the

correlational interaction in these orbitals are considerable and fall out of the region

where LDA and GGA are adequate since these approximations work for nearly ho-

mogeneous density. Transition metal oxides with occupied d orbital are predicted by

local approximations to be metallic instead they are observed to be insulator (e.g. FeO,

NiO, CeO2 [7, 8]). To remedy this shortcoming, Anisimov et al.[44] added the on-site

(applied only to specied orbitals; atomic d and f orbitals) Hubbard-like interaction

term EHub to the LDA functional,

ELDA+U [n(~r)] = ELDA[n(~r)] + EHub[nIσmm′]− Edc[nIσ] (2.56)

where ELDA+U is the Hubbard potential U corrected energy, ELDA is the LDA energy,

EHub is the Hubbard energy, nIσm is the occupation of the atomic orbital of the Ith

atom having spin σ and angular momentum quantum number m. Double counting

term Edc accounts for the on-site correlation that corresponds to the part already

present in LDA, therefore it is subtracted from the Hubbard energy functional. Then

the rotationally invariant form with respect to the atomic orbital basis set used to

20

dene occupancies nI,σm is introduced [45] where atomic Hartree-Fock Slater integrals

are used for the orbital dependent part of EHub:

EHub[nImm′ ] =

1

2

∑m,σ,I

〈m,m′′|Vee|m′,m′′′〉nI,σmm′nI,−σm′′m′′′

+

(〈m,m′′|Vee|m′,m′′′〉

− 〈m,m′′|Vee|m′′′,m′〉)nI,σmm′n

I,σm′′m′′′

(2.57)

with the matrix elements dened through expansion of the Coulomb potential in spher-

ical harmonics as

〈m,m′′|Vee|m′,m′′′〉 =

2l∑k=0

ak(m,m′,m′′,m′′′)F k (2.58)

where the ak(m,m′,m′′,m′′′) are given by

ak(m,m′,m′′,m′′′) =

4π

2k + 1

k∑q=−k

〈lm|Ykq|lm′〉〈lm′′|Y ∗kq|lm′′′〉 (2.59)

and F k are the radial integrals dening electron-electron interaction in HF theory.

Here F k are treated as parameters corresponding to

U = F 0 =1

(2l + 1)2

∑m,m′

〈m,m′|Vee|m,m′〉 (2.60)

and

J =F 2 + F 4

14=

1

2l(2l + 1)

∑m6=m′,m′

〈m,m′|Vee|m′,m〉. (2.61)

Finally, the screening term can be written as

Edc[nI ] =

∑I

U

2nI(nI − 1)−

∑I

J

2[nI↑(nI↑ − 1) + nI↓(nI↓ − 1)]. (2.62)

21

The basis-set independent simplied scheme of M. Cococcioni [7] assumes zero J or

Ueff = U − J that mimics the eects of the non-spherical interactions resulting in

basis-set independent atomic orbital occupations and Hubbard U value. This scheme

reads as follows:

EU [nImm′] = EHub[nImm′]− Edc[nI]

=U

2

∑m,σ,I

(nI,σmm −∑m′

nσmm′nσm′m)

=U

2

∑I,σ

Tr[nI,σ(1− nI,σ)]

(2.63)

where nI,σmm′ is the occupation of the atomic site I with spin σ and magnetic quantum

numbers m and m′;

nIσmm′ =∑kv

fσkv〈ψσkv|ϕIm〉〈ϕIm′ |ψσkv〉. (2.64)

In Eq. 2.64, ψσkv is the valence electronic wave function, having ~k and spin σ, fσkv is the

occupation number of the state (~kv), and ϕIm is some localized orbital (e.g. atomic,

Gaussian, Wannier). Projecting the Bloch states directly on to atomic orbitals intro-

duces a serious problem. Atomic orbitals of the two neighboring atoms have an overlap,

and normally in Hubbard U these electrons are treated with the hopping potential J.

However in eective Hubbard U, J is zero and the eects of J are considered to be

minor and claimed to be mimicked with the potential U. It is evident that applying the

Hubbard potential U to the atomic orbitals will introduce an error since these include

also the electrons of the neighboring atom. It is shown that Wannier projection per-

forms far better in electronic structure calculation than the atomic projection (e.g in

CeO2 system [8]). Wannier functions for a band are dened from the Bloch functions

of the same band through a unitary transformation [46]. A Wannier function about a

lattice point n, constructed from Bloch functions ψk(r) is dened as

w(r− rn) = N−1/2∑k

e(−ik·rn)ψk(r) (2.65)

22

Wannier functions are orthogonal about dierent lattice points n, m

∫dV w∗(r− rn)w(r− rm) = 0, n 6= m (2.66)

However, Wannier functions for a Bloch state are arbitrary with a phase factor eiφn(k),

which preserves Wannier center rn, even for an isolated band. A Bloch band is isolated

if it is not degenerate with any other band anywhere in the Brillouin zone. And

in case of composite bands (e.g. CeO2) the indeterminacy is stronger. In order to

overcome this shortcoming, maximally localized Wannier functions are constructed

[47]. Although the problem with atomic projection is solved with this approach, this

time the electronic contribution to interatomic forces related to Hubbard U will vanish,

due to the orthogonality of the Wannier functions. This issue will be addressed in the

results of this thesis.

In eect, DFT+U introduces a penalty amount of U for fractional occupations there-

fore increasing the potential dierence between occupied and unoccupied atomic states.

In LDA/GGA approximations there is a problem of continuous derivatives of the en-

ergy with respect to occupations at integer occupations and this leads to atoms with

fractional charge states upon dissociation of bound systems, for example molecules

and crystals. Eective Hubbard U value can be calculated to solve this problem of

continuous energy derivative instead of empirical determination [7].

2.10 Hellman-Feynman Theorem

After having obtained the ground state energy through the self-consistent solution of

the Kohn-Sham equations, forces on the atoms can be calculated, for example, to nd

the surface structure from the initial guess for the surface structure that is constructed

from the bulk system. Forces are calculated with the use of the Hellman-Feynman

theorem and the relevant equations are as follows.

Forces acting on the ions arising from the electrons described by the ground-state wave

function |ψ〉 can be written as

23

Fi,e = − ∂E

∂Ri= − ∂

∂Ri〈ψ|H|ψ〉 = −〈ψ| ∂H

∂Ri|ψ〉 (2.67)

where Fi,e is the force acting on ith atom with the position vector Ri, and H is the

DFT Hamiltonian. With the addition of forces arising from ion-ion interaction one

obtains the total force on an atom i

Fi = −∫dn(r)n(r)

∂Vext∂Ri

− ∂EII∂Ri

(2.68)

where EII is the ion-ion interaction energy. Therefore, by nding the structure that

has the minimum total force one obtains the so-called optimized structure in the Born-

Oppenheimer approximation.

24

CHAPTER 3

METHOD

3.1 Pseudopotential Generation

D. Vanderbilt's ultrasoft pseudopotential (USPP) generation code[35] is used to gen-

erate Ce pseudopotentials in both LDA and GGA approximations. For both LDA and

GGA pseudopotentials the 5s and 5p states are included in the valence states therefore

the valence electrons are 5s25p64f15d16s2. The parameters reported below are chosen

so as to produce the most accurate results regarding all-electron and pseudo energies

for reference and dierent ionic congurations and logarithmic energy derivatives of

the pseudopotential. In an all-electron calculation conducted as a benchmark both

valence and core electrons are included in the DFT solution. In order to check the

existence of ghost states, which are extra energy states not present in the result of

all-electron calculation, DFT calculations with a Bessel function basis set are carried

out and checked for any extra energy levels that fall below the all-electron energy

levels. These low-lying energies are a characteristic signature of ghost states. The

neutral and ionic congurations of charge +1, +2, +3, and +4 refer to the electronic

congurations 5s25p64f15d16s2, 5s25p64f15d16s1, 5s25p64f15d16s0, 5s05p64f15d06s0,

5s25p64f05d06s0, respectively (Table. 3.1, 3.3). These congurations and the logarith-

mic derivatives are used as transferability tests. Transferability is the pseudopotential's

ability of giving accurate results when used in dierent environments. Singular point

of the logarithmic derivative of the radial part of the DFT solution with elastically

scattered wave is related to the phase shift of the scattered wave. Therefore, matching

of the singular points in the logarithmic derivatives of the pseudo and real potentials

ensures the transferability of the pseudopotential.

25

3.1.1 LDA

For the PZ81 (LDA) pseudo potential the cut-o radius of the local potential is chosen

to be 2.4 a0. For each angular momentum channel the cut-o radii are as follows: 2.2

a0 for l = 0, 2.1 a0 for l = 1, 2.0 a0 for l = 2, and 1.5 a0 for l = 3. The logarithmic

derivatives for each angular momentum channel are calculated at r = 3.01 a0 (greater

than the local potential cut-o). Seven beta functions in the form of projectors χ are

used for the 5 wave functions (4f, 5s, 5p, 5d, 6s) where two projectors are used for the

4f wave function and one projector is used for each of the remaining wave functions.

Energy dierences up to the +4 conguration are below 1 mRyd (≈0.01 eV), however

for the +4 state, the 4f energy dierence is about 24 mRyd (0.33 eV, Table 3.1).

Considering that charge states of Ce atoms in CeO2 are +4 and +3 in some cases of

atom adsorption at the surfaces, also taking into account that there is a pseudo energy

error dierence between +3 and +4 congurations about 0.3 eV, an error bar of 0.3

eV is expected for atoms and molecule adsorptions. But this calculated error is for

the 4f states of Ce and to these states Hubbard U potential will be applied, therefore

this error can be compensated by slightly increased Hubbard U potential values. The

second largest error is made for the 5d state and the calculated error is about 0.1 eV.

There is a one to one map of the eigenvalues with the all-electron energies and therefore

there is no ghost state (Table. 3.2). Singularities (phase shifts) of the all-electron wave

functions and pseudo wave functions match for each angular momentum channel as

seen in the logarithmic derivative graph (Fig. 3.1).

Table3.1: All-electron and LDA pseudo energy dierences for various ionic states cal-culated with pseudo potential acquired from the neutral reference state. Unit of energyis Ryd.

nlm ∆E (+0) ∆E (+1) ∆E (+2) ∆E (+3) ∆E (+4)430 0.000000 -0.000832 -0.001913 0.004168 0.024468500 -0.000001 0.000137 0.000441 0.001042 0.003516510 0.000002 -0.000044 -0.000041 0.000997 0.009321520 -0.000001 -0.000074 -0.000172 0.000556 0.010361600 -0.000002 0.000035 0.000163 0.000124 0.000694

26

(a)-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0 2 4 6 8 10 12

ψ

r (radius)

full - 4fpseudo - 4f

full - 5spseudo - 5s

full - 5ppseudo - 5p

full - 5dpseudo - 5d

full - 6spseudo - 6s

(b)-4

-3

-2

-1

0

1

2

3

4

-6 -5 -4 -3 -2 -1 0 1 2

L [

Ryd

]

E [Ryd]

full - spseudo - s

full - ppseudo - p

full - dpseudo - d

full - fpseudo - f

Figure 3.1: (a) All-electron and pseudo wave functions for the ultrasoft LDA pseu-dopotential. (b) Logarithmic derivatives of the all-electron and pseudo wave functionsfor the ultrasoft LDA pseudopotential calculated at r=3.01 a0, L is the logarithmicderivative and E is the energy of the scattered wave.

Table3.2: Eigenenergies for ghost state check acquired from all-electron solution andLDA pseudo potential solution with Bessel function basis set having kinetic energycut-o of 40 Ryd. Unit of energy is Ryd.

nlm occupation ae energy pseudo energy430 1 -0.41236 -0.375386500 2 -2.99288 -2.992878510 6 -1.71247 -1.712476520 1 -0.23400 -0.233373600 2 -0.28560 -0.282615

3.1.2 GGA

For the PBE (GGA) pseudo potential the cut-o radius of the local potential is chosen

to be 2.5 a0. For each angular momentum channel the cut-o radii are as follows: 2.5

a0 for l = 0, 2.1 a0 for l = 1, 2.5 a0 for l = 2, and 2.0 a0 for l = 3. The logarithmic

27

derivatives for each angular momentum channel are calculated at r = 2.51 a0 (greater

than the local potential cut-o). Eight beta functions in the form projectors χ are used

for the 5 wave functions (4f, 5s, 5p, 5d, 6s). Energy dierences between all-electron

and pseudo potentials are found to be better than LDA pseudo potential considered

previously. An energy error bar of 0.1 eV mostly caused by 5p and 5d states, making

in total an error bar of 0.2 eV, is calculated (Table 3.3). As in the case of LDA, all

the eigenvalues match with the all-electron energies therefore there is no ghost state

(Table. 3.4). Singularities of the all-electron wave functions and pseudo wave functions

match for each angular momentum channel as seen in the logarithmic derivative graph

(Fig. 3.2).

(a)-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10 12

ψ

r (radius)

full - 4fpseudo - 4f

full - 5spseudo - 5s

full - 5ppseudo 5p

full - 5dpseudo - 5d

full - 6spseudo - 6s

(b)-6

-4

-2

0

2

4

6

-6 -5 -4 -3 -2 -1 0 1 2

L [

Ryd

]

E [Ryd]

full - spseudo - s

full - ppseudo - p

full - dpseudo - d

full - fpseudo - f

Figure 3.2: (a) All-electron and pseudo wave functions for ultrasoft GGA pseudopoten-tial, (b) Logarithmic derivatives of all-electron and pseudo wave functions for ultrasoftGGA pseudopotential calculated at r=3.01 a0.

28

Table3.3: All-electron and GGA pseudo energy dierence for various ionic states cal-culated with pseudo potential acquired from the neutral reference state. Unit of energyis Ryd.

nlm ∆E (+0) ∆E (+1) ∆E (+2) ∆E (+3) ∆E (+4)430 0.000000 -0.000813 -0.001489 0.005530 0.000015500 -0.000001 0.000052 0.000449 0.001564 0.010578510 0.000001 -0.000078 0.000068 0.001323 0.015578520 0.000000 -0.000085 -0.000081 0.000628 0.012897600 0.000001 0.000012 0.000174 0.000695 0.004383

Table3.4: Eigenenergies for ghost state check acquired from all-electron solution andGGA pseudo potential solution with Bessel function basis set having kinetic energycut-o of 40 Ryd. Unit of energy is Ryd.

nlm occupation ae energy pseudo energy430 1 -0.39559 -0.395052500 2 -2.98314 -2.983050510 6 -1.70705 -1.707055520 1 -0.22486 -0.223859600 2 -0.27523 -0.271883

3.2 DFT Calculations

DFT calculations are carried out using the Quantum Espresso code [48]. For the

exchange-correlation part of the interactions, the local density approximation (LDA)

in the form of PZ81 is used. Further improvement to the exchange-correlation potential

for the Ce-4f electrons is made through the rotationally invariant simplied form of

the Hubbard U [7], where Hubbard U is an eective potential, Ueff = U − J . Atomic

potentials replaced with ultrasoft pseudopotentials are in accord with the exchange-

correlation functional used. The electrons treated as valence for the Ce atoms are

5s25p64f15d16s2, for the O atoms are 2s22p4, for the C atoms are 2s22p2, and for the

Au atoms are 5d106s1. The kinetic energy and charge density cut-os of the plane-

wave basis set are selected as 40 Ryd and 300 Ryd, respectively. Both cut-os have

been tested for convergence. For Brillouin zone integration, a k-point grid is generated

with the Monkhorst-Pack scheme [36]. The calculations are not spin polarized.

Calculations of bulk ceria suggested the value of Ueff equal to 4 eV in order to matches

the LDA lattice constant to the experimental value of 5.41 Å. Therefore, a Ueff of 4

29

eV and a lattice constant of 5.41 Å are used throughout this thesis.

Vacuum thicknesses of approximately 12 Å are used for the calculations of the (111)

and (110) CeO2 surfaces, as well as CO and Au adsorptions at these surfaces. A 1.22

x 0.707 rectangular cell with a 4 x 6 x 1 k-point grid and a 0.708 x 1 rectangular

cell with a 4 x 1 x 3 k-point grid are used to model the (111) and (110) surfaces of

ceria, respectively. Here unit of length is one lattice constant (5.41 Å). CO adsorptions

at (111) surface are modelled with a 1 x 2 supercell, which corresponds to 25% (1/4

of surface atoms) of CO coverage, with a 7 x 6 x 1 k-point grid where the surface

normal is in the z direction and adsorption at the (110) surface is modelled with a

2 x 2 supercell, which means the same amount of CO coverage as for adsorptions at

(111) surface. A 4 x 1 x 3 k-point grid has been used, and the surface normal is in

the y direction. Au adsorptions at (111) are modelled with a 2 x 3 cell, corresponding

to 8.3% (1/12 of surface atoms) of Au coverage, using a 4 x 3 x 1 k-point grid. CO

molecule and Au atom energies are calculated in supercells with dimensions of 10 Å in

each direction and at the gamma point. For CO adsorption, dipole eld interactions

between periodic cells are reported in the literature and corrected accordingly, however

these studies used GGA approximation. In our LDA calculations no dipole eld was

found at vacuum located midway between the CO molecule and the end of the cell,

as apparent from the zero derivative of the potential at around 30 Bohr (Fig. 3.3).

Therefore, an asymmetric slab model is used for the surface and related structures.

Optimized structures are obtained for Hellman-Feynman forces below 0.02 eV/Å where

the rst two and three atomic layers are xed to their bulk positions in one side of the

(111) and (110) surface structures, respectively. The surface energies are calculated

using the following formula:

Esurf = (Ef − Eb)/A− (Ei − Eb)/(2A) (3.1)

where Ef is the energy of the optimized structure, Ei is the energy of the structure

before optimization, Eb is the energy of a bulk cell having identical numbers of Ce and

O atoms to the surface slab, and A is the surface area. The term (Ei − Eb)/(2A) is

subtracted because, for the asymmetric slab, one side of the surface structure, ions of

which are xed to their bulk positions, also faces vacuum therefore there is an energy

30

-2

-1.5

-1

-0.5

0

0.5

1

0 C O 30 cell b.

po

ten

tia

l [V

]

position [Bohr]

Figure 3.3: Potential along the surface normal of CO adsorbed on ceria (111) surface.The potential is calculated by averaging over layers parallel to the surface. C and Oatoms are labeled on the abscissa. cell b. indicates the cell boundary.

dierence because of this pseudo surface. Division of this energy dierence with 2A

is necessary because this energy dierence is calculated by using the structure before

optimization theferore the two sides of the structure that faces vacuum are equivalent,

so the surface area is two times the surface area of one side.

3.3 Local Hubbard U

As briey mentioned earlier in the introduction chapter, there is not a single U value

that describes suciently well all the properties of ceria, especially the U values that

are required for dierent structures (e.g. dierent surfaces, oxygen vacancies, and

atom/molecule adsorptions). The reason for this problem, regardless of the specic

method of Hubbard U employment, stems from the inadequate treatment of the in-

teractions of overlap electrons of Ce-4f and O-2p, in other words related with the

hopping interaction. As the focus of this thesis, we attempt to correct this controver-

sial Hubbard U dependence of the properties of ceria and related structures by means

of applying dierent U values to distinct Ce sites instead of scanning U values applied

indierently to all the Ce sites. We shall refer to this method as local Hubbard U in

the remaining of the text. Bulk, surface and Ce sites in the neighbor of the adsorbate

species are considered as distinct and corresponding U values applied are labelled as

Ub, Us, and Un, respectively (e.g. Fig. 3.4). Local Hubbard U dependence of the (111)

31

and (110) surface energies, CO and Au adsorption to various sites at these surfaces,

and the charge state of Au atoms at these adsorption sites are investigated. Change

of Hubbard U specically on certain Ce sites results in an energy change that is not

completely relevant, since the eects of the hopping potential, and change in charge

distribution that are the subject of the local Hubbard U investigation. Therefore, a

correction to the energies has to be made. For the calculation of surface energies,

Hubbard energy dierences between relaxed structures of Us 6= Ub and Us = Ub

∆EHub = EHub(Us 6= Ub)− EHub(Us = Ub) (3.2)

is used to account for this correction. In this equation EHub(Us 6= Ub) is the Hub-

bard energy of the system optimized with Us value dierent from the Ub value, and

EHub(Us = Ub) is the Hubbard energy of the system optimized with Us value being

equal to the Ub value. With this equation, the Hubbard energy gain upon change of

Hubbard potential U is found and subtracted from the total energy to nd the sur-

face energies that are corrected mostly by energy change through change in charge

distribution:

E′ = E −∆EHub (3.3)

Therefore, the equation for the surface energy calculation reads

E′surf = (Ef −∆EHub − Eb)/A− (Ei − Eb)/(2A) (3.4)

where E′surf is the corrected surface energy, Ef is the energy of the optimized structure,

Eb is the energy of the bulk where number of atoms is equal to the number of atoms in

the surface structure, Ei is the energy of the surface structure before optimization, and

A is the surface area. Correction of the energies by using Hubbard energy dierence

can be misleading. Change in Hubbard U results in charge redistribution and there is a

total energy change related with this including Hubbard energy change. Let us change

the Hubbard U value of a certain Ce site at the surface. There will be Hubbard energy

change and total energy change even without an adsorbate nearby. Therefore, energies

32

can be corrected with the total energy change of the system upon Hubbard U change

without the adsorbate in the system. It is expected that adsorption energies found

by local Hubbard U will be between the energies calculated with the Hubbard energy

correction and the total energy correction. Equations for the total energy correction

are as follows:

E′′ = E −∆ETot (3.5)

∆ETot = [E(Ub, Us, Un)− E(Ub, Un = Us)]Ad/Surface|opt\Ad (3.6)

where ∆ETot is the correction to the energy by using total energy dierence, the

label Ad/Surface|opt\Ad denotes that atom or molecule adsorbed surface structure

is optimized then the adsorbed atom or molecule is subtracted from the structure,

E(Ub, Us, Un) and E(Ub, Un = Us) are the total energies of the mentioned system with

the corresponding Hubbard U values. ∆ETot is a self-consistent eld calculation, in

other words there is no ionic relaxation.

Figure 3.4: Au adsorbed at O-top site on (111) surface of ceria.

33

34

CHAPTER 4

LOCAL HUBBARD U & CERIA

4.1 Literature Review of Ceria

Both LDA+U and GGA+U methods predict lattice constants of CeO2 and Ce2O3 with

Wannier-Boys projection through self-consistently determined U values (Sec. 2.9) in

good agreement with the experiment [8]. An underestimation of 0.03 Å in LDA+U and

overestimation of 0.07 Å in GGA+U are found in CeO2; in Ce2O3, the lattice constant

is underestimated by 0.05 Å in LDA+U and overestimated by the same amount in

GGA+U calculations. Loschen et al.[9] showed that the computed lattice parameter

and the eective U values are directly proportional for both LDA+U and GGA+U

methods when atomic projection is used. Since, GGA already overestimates the ex-

perimental lattice constant, application of U increases the discrepancy with the ex-

periment. Within the LDA+U method reports showed that an eective U value of

6.7 eV reproduces the experimental lattice constant of CeO2. Keating et al. reported

a value of 5.48 Å for the lattice constant with U=5 eV in the PBE(Perdew-Burke-

Ernzerhof)+U approximation [49]. Plata et al. reported for PW91(Perdew-Wang)+U

functional with eective U applied not only to Ce-4f but also to O-2p decreases lattice

constant of overestimated GGA value. In this work, U values of Uf=5 eV (applied to

Ce-4f orbitals) and Up=12 eV (applied to O-2p orbitals) result in a lattice parameter

of 5.45 Å , which is 5.48 Å at the same Uf value but setting Up=0 eV.

Fabris et al.[50] found that LDA predicts empty 4f gap states at 4 eV for defect-free

bulk CeO2 and a band gap of 5.6 eV with LDA and 5.7 eV with GGA. The experimental

value for the band gap is 6.0 eV [4]. Loschen et al.[9] reported that band gap is slightly

35

aected by the eective U value with atomic projection, but it is always smaller than

the measured value of 6 eV. For CeO2, as the eective U value increases, the band gap

decreases from 5.4 to 4.6 eV for both LDA+U and GGA+U functionals. At the same

time, the energy of the empty 4f states lying in the band gap shifts from 1.3 to 2.3 eV,

an improvement considering that the experimental value is 3 eV. Also a slight decrease

in 4f band width is found with the increase of eective U. For Ce2O3, an increase of

the eective U shifts the 4f gap states closer to the conduction band, and eventually

mixes them. Experimentally observed 4f gap states lies at 2.4 eV, which is calculated

as 1.7 eV (LDA+U) and 2.1 eV (GGA+U) with 5 eV of eective U. Work of Plata et

al.[51], which includes U correction also to O-2p states in PW91+U method, showed

that Up and O2p-Ce4f gap are directly proportional, and at 5 eV of U value for both

Ce4f and O2p resulting in O2p-Ce4f gap to be 2.3 eV underestimating experimental

range, 2.6-3.9 eV and 5.4 eV of O2p-Ce5d gap below the experimental range, 6-8 eV.

Since there is an overlap between Ce-4f and O-2p states, the choise of the method of

projection to the atomic states is crucial in application of Hubbard U method. The

calculated energy of reduction of CeO2 to Ce2O3 is not consistent with the experimental

value when atomic projection is applied with a self-consistent value of U. Wannier type

projection, however gives somewhat better agreement with experiment and also the

energy becomes independent of the applied U for Ceria and this means that there is no

hybridization between Ce-4f and other states (e.g. O-2p) [8]. Transformation energy

of CeO2 to Ce2O3 is found to be 2.02 eV with LDA and 0.82 eV with GGA, while the

experimental value is 1.97 eV. In this case LDA performs much better than GGA [50].

For the transformation energy of CeO2 to Ce2O3, a perfect agreement between theory

and experiment (3.94 eV) is obtained by using a value of 3 eV (GGA+U) and 6 eV

(LDA+U) of eective U with atomic projection [9]. Work of Plata et al.[51] employing

also U for O2p states in PW91+U found that formation energy of CeO2 and energy

of reduction to Ce2O3 are best described with Uf=5 eV and Up=5 eV U values, −9.04

eV (exp.: −10.44 eV) and 4.01 eV (exp.: 3.99 eV, PBE+U[52]: 2.03 eV), respectively.

36

4.2 Results & Discussion

4.2.1 Bulk Ceria

Inclusion of Hubbard U applied by using atomic projection worsens the band gap

but at the same time improves the energy of gap states and the lattice constant of

bulk ceria structure with respect to the experimental values. At 4 eV of U at which

the experimental lattice constant is obtained, the band gap decreases by 0.2 eV and

the energy of the gap states increases by 0.2 eV (Table. 4.1), in parallel with other

theoretical accounts [9]. Ceria nanosheets with (110) termination are experimantally

observed to have an indirect band gap where the dierence between direct and indirect

band gap is found to be 0.3 eV [53]. Here dierences between direct and indirect band

gaps of 0.65 and 0.42 eV are found with 0 eV and 4 eV of U, respectively (Fig. 4.1). In

case of Wannier projection, far better improvements are obtained over both the band

gap and 4f gap state energy. Here the maximally localized Wannier projection method

is used since ceria has a composite band which is a result of the presence of Ce-4f and

O-2p degenerate states in the valence band (Fig. 4.1-c). The eect of Hubbard U

on the lattice constant vanishes as may be expected since the Ce-4f states are made

orthogonal to neighboring states of O sites. But this dramatic dierence is not only

caused by the hopping term and the lattice constant, the wrong treatment of the

overlap electrons in eective Hubbard model is recovered with the Wannier projection.

Table4.1: Bulk ceria. Hubbard U dependence of lattice constant, a0; band gap, Eg;peak energy of localized gap states of Ce-4f character, E4f . Starred value (4*) indicatesthe use of Wannier projection. Distances are in Å and energies are in eV units.

U a0 Eg E4f

0 5.36 5.0 2.41 5.39 5.1 2.42 5.39 5.0 2.63 5.40 5.0 2.63.5 5.41 4.8 2.54 5.41 4.8 2.65 5.42 4.7 2.74* 5.36 5.7 3.1

exp. [4] 5.41 6.0 4.0

37

(a)-4

-2

0

2

4

6

Γ X W K Γ L U W L K

E (

eV

)

(b)-4

-2

0

2

4

6

Γ X W K Γ L U W L K

E (

eV

)

(c)

Figure 4.1: Band structure of bulk ceria for (a) U=0 eV, (b) U=4 eV employed withatomic projection. (c) Total and projected density of states of bulk ceria at U=0 eV.Fermi level energy is set to zero.

4.2.2 (111) & (110) Surfaces of Ceria

With a Hubbard U of 4 eV (Ub) from the bulk structure calculations, dierent Hubbard

U values are applied to surface Ce sites (Us) in surface structure calculations. There

are two competing interactions. The rst is between electrons of the surface atoms and

sub-surface atoms, and the second is between electrons of the surface atoms only. The

distance between the surface and sub-surface layers (d) is directly proportional to Us.

Because of these competing interactions, surface energies of (111) and (110) surfaces

are found to be minimum at 6 eV of Us, where surface energies are corrected (E′s) as

mentioned before in Sec. 3.3. At 6 eV of Us, surface energy of the (110) surface is

found to be 1.30 J/m2, a decrease of 0.10 J/m2 (7%) and surface energy of (111) is

found to be 0.77 J/m2, a decrease of 0.19 J/m2 (20%) with respect to Us = Ub = 4 eV.

Change in (111) surface energy is about twice the change in (110) surface energy, and

this is probably related to the larger overlap of Ce-4f states at the (111) surface with

O-2p states compared to the (110) surface because Ce atoms are coordinated to four O

38

Table4.2: Energies of (111) and (110) surfaces of ceria with dierent U values appliedto outermost Ce sites. U value of 4 eV is applied to the Ce sites residing in bulk. d isthe distance between the outermost Ce atomic layer and the previous one in Å unit.Surface energies are in J/m2.

(110) (111)Us Es E′s d Es E′s d

3 1.22 1.50 1.667 0.68 1.13 3.0963.5 1.31 1.45 1.669 0.82 1.04 3.0984 1.40 1.40 1.670 0.96 0.96 3.1005 1.57 1.34 1.672 1.23 0.85 3.1046 1.74 1.30 1.675 1.47 0.77 3.1087 1.89 1.30 1.677 1.71 0.78 3.1138 2.02 1.33 1.678 1.92 0.82 3.1169 2.15 1.39 1.677 2.11 0.91 3.12110 2.67 1.48 1.675 2.29 1.06 3.123

LDA 1.35 0.94LDAa 1.35 1.00GGAa 1.01 0.60

LDA+Ub 1.41 1.02PBE+Ub 1.06 0.71PBE0[54] 0.81 0.72PW91+Uc 0.53 0.45HSE06c 0.91 0.81

a other lit.b U=5.3 eV and Wannier proj.[8]c U=5 eV [55]

atoms in the (111) surface and to two O atoms in the (110) surface that are out of the

atomic layer of surface Ce atoms (Fig. 1.2). The distance between the layers of surface

Ce atoms and sub-surface Ce atoms decreases by a small amount of 0.005 and 0.008

Å at Us = 6 eV with respect to Us = 4 eV for (110) and (111) surfaces, rescpectively.

An energy shift of about 0.5 eV in empty states is found upon increase of Us from 4 to

6 eV for the (111) surface structure. With the same increase of Us value only minor

changes are found for (110) surface structure (Fig. 4.2,4.3). Increasing Us produces

density of states that are closer to the result of the calculation with Us = Ub = 4 eV

employed with Wannier projection for the (111) surface structure (Fig. 4.4). However,

this result is not found for the (110) surface which indicates that increasing Us does

not mimic the use of Wannier projection. From these results, it can be infered that

eects of the local Hubbard U are related with the structural changes which are absent

in calculations of Hubbard U employed with Wannier projection.

39

(a) 0

5

10

15

20

25

-20 -15 -10 -5 0

DO

S

E (eV)

U=3U=4U=6

(b) 0

20

40

60

80

100

120

0 1 2 3 4 5 6

DO

S

E (eV)

U=3U=4U=6

Figure 4.2: Comparison of (a) valence and (b) conduction density of states for ceria(110) surface found with Us values of 3, 4, and 6 eV. Ub=4 eV.

(a) 0

5

10

15

20

25

30

35

-20 -15 -10 -5 0

DO

S

E (eV)

U=3U=4U=6

(b) 0

20

40

60

80

100

120

140

160

0 1 2 3 4 5 6

DO

S

E (eV)

U=3U=4U=6

Figure 4.3: Comparison of (a) valence and (b) conduction density of states for ceria(111) surface found with Us values of 3, 4, and 6 eV. Ub=4 eV.

0

50

100

150

200

250

300

350

-6 -4 -2 0 2 4 6 8 10

DO

S

E (eV)

bulk

(110) surface

(111) surface

Figure 4.4: DOS of ceria bulk, (111), and (110) surfaces calculated at U = 4 eVemployed with Wannier projection.

40

4.2.3 CO Adsorption on (111) & (110) Surfaces of Ceria

(a) (b)

Figure 4.5: Optimized structures of CO at (a) Ce-top site (labeled as Un) of ceria (111)surface, (b) Ce-top site of ceria (110) surface. Big grey and small red balls representCe and O atoms, respectively. Atoms reside at surface are darker in color.

Energies of CO adsorption at (111) and (110) surfaces are overestimated in LDA with

respect to GGA and hybrid functionals. When corrected with Hubbard energy change

(Eq. 3.2), a signicant decrease in these overestimated energies upon increasing U

values of neighboring Ce sites (Un) are found. CO adsorption energy to Ce-top site

at (111) surface (Fig. 4.5) is found to be 0.25 eV at Us=6 and Un=7 with a drop

of about 40% with respect to Us=4 and Un=4 (Table. 4.3). When the total energy

correction is included, the energy change becomes 0.01 eV indicating the contribution

of lattice relaxation. A similar energy change has been reported when Hubbard U is

applied with Wannier projection. The energy of adsorption is found to be 0.17 eV

whitin the GGA+U approximation [15]. The CO distance to the nearest neighbor Ce

site is increased by an amount of 0.016 Å.

Energy of CO adsorption to the Ce-top site at the (110) surface (Fig. 4.5) is decreased

by an amount of 0.17 eV (47%) after Hubbard energy correction is applied at Us=6

and Un=7 eV with respect to Us=4 and Un=4 eV (Table. 4.4). At these U values,

the calculated adsorption energy of 0.19 eV constitutes a signicant improvement if

the value of 0.19 eV of PBE+U and the B3LYP value 0.10 eV are considered accurate

values. B3LYP is especially shown to produce the best values for CO adsorption while

PBE+U overestimates this value [15, 17, 18, 19]. The distance between CO and

41

Table4.3: LDA+U energy of CO adsorption on Ce-top site at (111) surface. Ub=4 eV.E′ and E′′ are the corrected energies with the Hubbard and total energy dierences,respectively. d is the distance of C to nearest neighbor Ce atom. Energies are in eV.

Us Un Eads E′′ads (E′ads) d

3.0 2.5 0.68 0.44 (0.47) 2.7153.0 3.0 0.44 0.44 (0.44) 2.7153.0 3.5 0.21 0.44 (0.39) 2.7163.5 3.0 0.67 0.44 (0.48) 2.7163.5 3.5 0.44 0.44 (0.44) 2.7163.5 4.0 0.21 0.44 (0.39) 2.7174.0 3.5 0.66 0.44 (0.49) 2.7194.0 4.0 0.44 0.44 (0.44) 2.7194.0 4.5 0.22 0.45 (0.38) 2.7206.0 4.0 1.26 0.44 (0.71) 2.7306.0 6.0 0.43 0.43 (0.43) 2.7346.0 6.5 0.24 0.43 (0.34) 2.7356.0 7.0 0.06 0.43 (0.25) 2.7356.0 8.0 -0.28 0.43 (0.04) 2.744

PW91[14] 0.17PBEa 0.18

PBE+Ua 0.17PW91+Ub 0.26

a U=4.5 eV.[15]b U=5 eV.[16]

Table4.4: LDA+U energy of CO adsorption on Ce-top side at (110) surface. Ub = 4eV. d is the distance of C to nearest neighbor Ce atom. Energies are in eV.

Us Un Eads E′′ads (E′ads) d

3.0 3.0 0.36 0.36 (0.36) 2.8343.0 3.5 0.13 0.36 (0.32) 2.8343.5 3.5 0.36 0.36 (0.36) 2.8343.5 4.2 0.05 0.37 (0.29) 2.8344.0 4.0 0.36 0.36 (0.36) 2.8364.0 4.5 0.14 0.36 (0.31) 2.8366.0 6.0 0.37 0.37 (0.37) 2.8346.0 6.5 0.18 0.37 (0.28) 2.8346.0 7.0 -0.004 0.32 (0.19) 2.8346.0 8.0 -0.35 0.37 (-0.02) 2.8327.0 7.0 0.37 0.37 (0.37) 2.832PBEa 0.16

PBE+Ua 0.19PW91[17, 14] 0.18PW91+Ub 0.21

B3LYP[17, 18] 0.10

a U=4.5 eV.[15]b U=5 eV.[16]

42

nearest neighbor Ce site decreases by 0.002 Å and the adsorption energy change is

positive, namely 0.01 eV at Us=6 and Un=7. These ndings are the opposite of what

is found for the (111) surface since surface Ce sites are located at the outermost atomic

layer in the (110) surface and therefore increasing U value at this site results in pulling

out of the layer.

4.2.4 Au Adsorption on (111) Surface of Ceria

(a) (b)

Figure 4.6: Optimized structures of Au at (a) O-top site of ceria (111) surface, (b)O-bridge site of ceria (110) surface.

Similar investigations are carried out for Au adsorption. In addition, Au charge state

is computed with Bader charge analysis [56]. Au adsorption energy on the O-top site

of the (111) surface (Fig. 4.6 a) is found to be 1.37 eV with LDA+U functional (Table.

4.5). Increasing the Us from 4 eV to 6 eV resulted in a decrease of 0.02 eV in the

adsorption energy. The charge state of Au is found to be −0.02 at both 4 and 6 eV

of Us values (Table. 4.5). Increasing Un value to 7 eV, keeping Us at 6 eV, decreases

the adsorption energy to 0.80 eV to which only Hubbard energy correction is included.

Again this energy drop can be considered as an improvement if the 0.6-1.0 eV range

of adsorption energies calculated with PBE and hybrid functionals reported so far are

considered to be the best approximations. The charge state of Au is found to be +0.004

at Un=7 eV while the corresponding value at Un=6 eV is −0.02. Eect of increasing

Un by 1 eV has little eect on the charge state in LDA, but this may not be the case

with PBE which is not investigated here.

The adsorption energy of Au to O-bridge site at (111) surface (Fig. 4.6) is found to

be 1.51 eV using Us and Un values of 4 eV (Table. 4.6). At Us=6 eV, the adsorption

43

Table4.5: LDA+U energy of adsorption and charge state (qAu) of Au on O-top site on(111) surface of ceria. d is the distance of Au to nearest neighbor O atom. Ub=4 eV.Energies are in eV.

Us Un Eads E′′ads (E′ads) qAu d

3.0 3.0 1.38 1.38 (1.38) -0.01 2.0313.0 3.5 0.68 1.38 (1.25) -0.01 2.0303.0 6.0 -2.46 1.35 (0.24) -0.03 2.0224.0 4.0 1.37 1.37 (1.37) -0.02 2.0324.0 4.5 0.71 1.36 (1.24) -0.02 2.0326.0 3.0 5.21 1.37 (2.55) -0.01 2.0316.0 5.5 1.94 1.35 (1.61) -0.04 2.0296.0 6.0 1.35 1.35 (1.35) -0.02 2.0256.0 6.5 0.79 1.35 (1.09) -0.003 2.0236.0 7.0 0.24 1.35 (0.80) +0.004 2.0206.0 8.0 -0.78 1.34 (0.21) +0.008 2.018

LDA+Ua 1.27 -0.05PW91+U[26, 28] 0.96 +0.32

PW91+Ub 0.66 -0.04PW91+U[57] 0.88PBE[57] 0.79BLYP[57] 0.63

a U=5 eV.[26, 27]b geometry of LDA+U with U=5, GGA with U=3 eV.[27]c U=5 eV is used in all the other DFT+U studies

Table4.6: LDA+U energy of adsorption and charge state (qAu) of Au on O bridge siteat (111) surface of ceria. d is the distance of Au to nearest neighbor Ce atom. Ub=4eV. Energies are in eV.

Us Un Eads E′′ads (E′ads) qAu d

3.0 3.0 1.53 1.53 (1.53) -0.05 3.0053.0 3.5 1.30 1.53 (1.48) -0.05 2.9903.0 6.0 0.27 1.54 (1.18) -0.07 2.9784.0 4.0 1.51 1.51 (1.51) -0.06 3.0144.0 4.5 1.30 1.51 (1.47) -0.06 3.0246.0 3.0 2.78 1.50 (1.86) -0.07 2.9796.0 6.0 1.49 1.49 (1.49) -0.08 3.0026.0 6.5 1.30 1.49 (1.40) -0.08 3.0256.0 7.0 1.12 1.49 (1.31) -0.08 3.032

LDA+Ua 1.91LDA+U[27] 1.47PW91+U[58] 1.17

PW91+U[27, 28] 1.15 +0.34PW91+U[59] 1.87PW91+Ub 0.43 +0.33

a U=6 and free energies included.[59]b geometry of LDA+U with U=5, GGA with U=3 eV.[27]c U=5 eV is used in all the other DFT+U studies

44

energy drops by 0.02 eV which is the same amount found for the adsorption to O-top

site. The distance between Au and the nearest neighbor Ce atom decreased by 0.012

Å. At this Us=6 eV value, increasing Un from 6 eV to 7 eV results in a 0.18 eV drop

in adsorption energy which is corrected with Hubbard energy dierence. The distance

is increased by 0.018 Å with respect to the case where Un = Us = 4 eV. This nal

increase of the U value of neighboring Ce sites had no eect on the charge state of Au,

which is found to be −0.08, in contrast to the adsorption at O-top site. This apparent

insensitivity of the properties of Au adsorption to O-bridge site towards Hubbard U

and structure is also evident in other works. For example, the charge state of Au is

found same (+0.34 and +0.33) with PW91+U in both structures which are found by

GGA and LDA (Table. 4.6).

Regarding the charge state of Au atom adsorbed at various sites, there seems to be a

need for further explanation. There is a lattice constant dependence of the charge state

of Au at O-top site of the (111) surface that is revealed in the other works [26, 27],

and this dependence seems to be shifted to the Hubbard U, since relative position of

the O atom that resides between Au and nearest neighbor Ce atoms depends on the

Hubbard U at this site. This dependence of the charge state on Hubbard U is not

found for the O-bridge site, and also in the mentioned works, which showed the lattice

constant dependence, it is shown that this site is insensitive to the lattice constant. In

short, Hubbard U and lattice constant dependence of the charge state of Au together

indicates that Hubbard U dependence of the charge state of Au at these sites does not

directly stem from the interactions of the electrons that are shared between Ce and Au

if any, but from structural distortions that Hubbard U causes. Thus, this conclusion

sheds more light on the so far controversially claimed dependence of the charge state

of Au on the Hubbard U and it should be noted that the value of the Hubbard U can

not be optimized with respect to atom or molecule adsorption properties and that the

projection method is of crucial importance for the Hubbard U employment.

45

46

CHAPTER 5

CONCLUSION

The simplied form of the Hubbard model, in other words the eective Hubbard U

where the hopping potential is set to zero, in order to make the U value universal,

introduced diculty in the treatment of overlap electrons. So far, two methods, atomic

and Wannier projection, are used to calculate occupation of the Ce-4f orbitals, both

of which have drawbacks. Here, ceria is studied with DFT+U method where eect

of the Hubbard U, especially the eect of the hopping term on the lattice relaxation,

surface energies, electronic structure of the surface structures, CO and Au adsorption

energies and charge state of Au, are investigated indirectly using eective Hubbard

U. Eect of the projection that is used to calculate the Ce-4f orbital occupancies is

reported earlier and here also electronic structure of bulk ceria and surface energies

obtained with Wannier projection are reported. Lattice constant and (111) surface

energy becomes independent of Hubbard U upon using Wannier projection, but the

electronic structures are signicantly improved. However, with Wannier projection,

eect of the Hubbard U on lattice relaxation concomitantly eects on the energies and

electronic structures are missed. Since eective Hubbard U does not include hopping

term especially when employed with atomic projection, bulk, surface, and adsorbate

neighboring Ce sites are considered to be distinct for the Hubbard U application,

therefore dierent U values are applied to these sites.

Surface energies acquired minimum values, after the correction that is required to

reveal the eect of the hopping term J, at 6 eV of U applied to surface Ce sites

where 4 eV of U value is applied to bulk Ce sites. Drop in surface energies are about

7% and 20% for the (110) and (111) surfaces, therefore the dierence between these

47

surface energies increased. The eect on the electronic structures of these surfaces are

in parallel with surface energies, in other words the electronic structure of the (110)

surface is less aected than the (111) surface. For the (111) surface, energy of the 4f

gap states shifted towards conduction band by about 0.5 eV.

CO adsorption to (111) and (110) surfaces is found to be indirectly sensitive to U

applied to the nearest neighbor site. Because CO is not bonded strongly to Ce-top sites,

the eect of U is only through structural distortions over the surface upon adsorption.

Although the eect is indirect, the energy of adsorption is low therefore even small

distortions leads to energy changes depending on U, applied only to single site, that

can be 47% of the adsorption energy. This shows the signicance of Hubbard U even

for weak adsorptions or distortions. The eects on the atomic forces are low, about

0.02 eV.

Au adsorbed (111) surface structures exhibited similar tendencies as CO adsorbed

surfaces. The debated issue about the charge state of Au is investigated. Hubbard

U, concomitantly the lattice constant dependence of the charge state of Au had been

reported in the literature, and here site specic eective Hubbard U results also are in

parallel with those ndings. Charge state of Au adsorbed to O-top site is aected by

surface and neighbor U values, Us and Un. For this site, charge state is changed from

−0.02 to ∼+0.01, and this positive charge state is the rst that is reported in LDA

approximation.

In conclusion, eect of the hopping term and the interaction of overlap electrons are

important, even for weak atom or molecule adsorptions. Therefore, a better account

of the interaction of overlap electrons, in other words a better projection method is

needed for the calculation of occupancies used in Hubbard U.

48

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