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The DFT+U Method in the Framework of the Parallel Density Functional Code ParaGauss [Elektronische Ressource] / Raghunathan Ramakrishnan. Gutachter: Notker Rösch ; Manfred Kleber. Betreuer: Notker Rösch

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TECHNISCHE UNIVERSITÄT MÜNCHEN
Fachgebiet Theoretische Chemie
The DFT + U Method in the Framework of the Parallel Density
Functional Code ParaGauss
Raghunathan Ramakrishnan
Vollständiger Abdruck der von der Fakultät für Chemie der Technischen Universität München
zur Erlangung des akademischen Grades eines
Doktors der Naturwissenschaften (Dr. rer. nat.)
genehmigten Dissertation.
Vorsitzender: Univ.-Prof. Dr. K. Köhler
Prüfer der Dissertation:
1. Univ.-Prof. Dr. N. Rösch
2. Univ.-Prof. Dr. M. Kleber
Die Dissertation wurde am 22.02.2011 bei der Technischen Universität München eingereicht und
durch die Fakultät für Chemie am 15.03.2011 angenommen.Acknowledgments
The knowledge I gained during the course of my thesis work is very valuable to me and I am
very thankful to my supervisor, Prof. Dr. Notker Rösch, for giving me the opportunity to work
for my doctoral thesis in his research group and for his interest in this project. Prof. Rösch’s
constant advising in both scientific and non-scientific issues are very important to me and I think
that they came at the right time of my academic career. I wish to thank Prof. Dr. Manfred Kleber
for refereeing my fellowship reports and providing feedbacks.
It is a pleasure to thank Dr. Alexei Matveev for helping me improve my programming skills
and for various discussions. I am indebted to Dr. Sven Krüger for various discussions and helps
through out my project.
I would like to thank Dr. Sonjoy Majumder for his help during the initial phase of the project.
I am thankful to my office room mates, past and present, Dr. Sun Qiao, Ajanta Deka, Prof. B.
Dunlap, Dr. Ji Lai, Yin Wu, Thomas Martin Soini, Chun-Ran Chang, Dr. Kremleva Alena for
various helps and to give me enough space to make me feel at home.
I thank Frau. Ruth Mösch, Frau. Barbara Asam for various administrative helps. I thank
my colleagues Dr. Alena Ivanova, Dr. Alexander Genest, Dr. Amjad Mohammed Basha, Astrid
Nikodem, Dr. Benjami Martorell, David Tittle, Duygu Bas¸aran, Dr. Egor Vladimirov, George
Beridze, Gopal Dixit, Dr. Grigory Shamov, Dr. Grzegorz Jezierski, Dr. Hristiyan Aleksandrov,
Dr. Ilya Yudanov, Dr. Ion Chiorescu, Dr. Juan Santana, Dr. Lyudmila Moskaleva, Manuela
Metzner, Mayur Karwa, Miquel Huix i Rotllant, Dr. Miriam Häberle, Dr. Olga Zakharieva,
Ralph Koitz, Dr. Rupashree Shyama Ray, Dr. Sergey Bosko, Shane Parker, Siham Naima Derrar,
Dr. Virve Karttunen, Dr. Vladimir Nasluzov, Dr. Wilhelm Eger, and Zhijian Zhao for providing
a friendly working environment.
I thank the library, academic and non-academic staff members of Technische Universität
München for their excellent service. Thanks are also due to various land ladies and Hausmeis-
ter(in) for making life simple in Munich.
I am grateful to Prof. Krishnan Mangala Sunder for his interest in my academic progress and
for various discussions and helps for a very long time.
I thank my wife Shampa, my brother and my parents whose love, support and encouragement
enabled me to complete this work.
Finally, I gratefully acknowledge Deutscher Akademischer Austausch Dienst for awarding
me a fellowship to carry out my doctoral research in Germany.Dedicated to my teachersContents
1 Introduction 1
1.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Overview of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Kohn–Sham Density Functional Theory 9
2.1 The Kohn–Sham Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1.2 The Kohn–Sham Approach . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1.3 Approximate Exchange-Correlation Functionals . . . . . . . . . . . . . 14
2.2 Generalization and Interpretation of Kohn–Sham Theory . . . . . . . . . . . . . 16
2.2.1 Non-Integer Orbital Occupation Numbers . . . . . . . . . . . . . . . . . 16
2.2.2 Scaling Relations for Density Functionals . . . . . . . . . . . . . . . . . 17
2.2.3 Orbital Energies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3 Self-interaction Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3.1 Conditions for Exact Exchange-Correlation Functionals . . . . . . . . . 19
2.3.2 Self-Interaction Cancellation by Semi-Local Functionals . . . . . . . . . 21
2.3.3 Manifestations of the Self-Interaction Error . . . . . . . . . . . . . . . . 27
2.3.4 Self-Interaction Correction Schemes . . . . . . . . . . . . . . . . . . . . 30
3 The DFT + U Methodology 33
3.1 The DFT + U Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.1.1 The DFT + U Functional form . . . . . . . . . . . . . . . . . . . . . . . 35
3.1.2 Orbital Occupation Matrix . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.2 DFT + U Hamiltonian Matrix and Analytic Gradients . . . . . . . . . . . . . . . 43
3.2.1 Some Useful Expressions . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.2.2 DFT + U Hamiltonian Correction Matrix . . . . . . . . . . . . . . . . . 45
3.2.3 DFT + U Analytic Gradients . . . . . . . . . . . . . . . . . . . . . . . . 46
iii CONTENTS
3.3 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.3.1 Intermediate Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.3.2 Parallelization of Gradient Computation . . . . . . . . . . . . . . . . . . 48
3.4 FLL-DFT + U corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4 Computational Details 55
4.1 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.2 Effective Onsite-Coulomb Parameter . . . . . . . . . . . . . . . . . . . . . . . 56
5 DFT + U Application to Lanthanides 61
5.1 Lanthanide Trifluorides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.1.1 Role of 4 f Orbitals in the Bonding of LuF . . . . . . . . . . . . . . . . 633
5.1.2 Role of 5d Orbitals in the Bonding of LnF . . . . . . . . . . . . . . . . 713
5.2 Ceria Nanoparticles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.2.1 Model Nanoparticles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.2.2 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . 77
6 DFT + U Application to Actinides 83
6.1 Uranyl Dication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
6.1.1 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6.1.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
6.2 Penta Aqua Uranyl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
6.2.1 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . 95
6.2.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6.3 Uranyl Monohydroxide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6.3.1 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . 106
6.3.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
7 Summary and Outlook 123
A Basis sets 127
Bibliography 137List of abbreviations
B3LYP Becke 3-parameter Lee–Yang–Parr
BP Becke–Perdew
CCSD(T) Coupled-Cluster with Single and Double and Perturbative Triple excitations
DF Density Functional
DFT Density Functional Theory
DKH Douglas–Kroll–Hess
FF Fitting Function
FOJT First-Order Jahn-Teller
GGA Generalized Gradient Approximation
HF Hartree–Fock
HFS Hartree–Fock–Slater
HK Hohenberg–Kohn
HOMO Highest Occupied Molecular Orbital
IP Ionization Potential
KS Kohn–Sham
LCAO Linear Combination of Atomic Orbitals
LCGTO Linear Combination of Gaussian-Type Orbitals
LUMO Lowest Unoccupied Molecular Orbital
LDA Local Density Approximation
MO Molecular Orbital
MP2 Second-order Møller–Plesset perturbation theory
MGGA Meta Generalized Gradient Approximation
PBE Perdew–Burke–Ernzerhof
PBE0 Zero parameter hybrid functional based on PBE
PBEN Hammer, Hansen, and Nørskov revision of PBE
PW91 Perdew–Wang, year 1991
RECP Relativistic Effective-Core Potential
iiiiv
RPA Random Phase Approximation
SCF Self-Consistent Field
SD Slater–Dirac
SI Self-Interaction
SIC Self-Interaction Correction
SIE Self-Interaction Error
SOJT Second-Order Jahn-Teller
TPSS Tao–Perdew–Staroverov–Scuseria
UHF Unrestricted Hartree–Fock
VWN Vosko–Wilk–Nusair
XC Exchange-Correlation
ZORA Zero-Order Regular Approximation