The formation and development of quantum theory in the first half of the 20^th century has led to a revolution in our understanding of pure and applied physics. Quantum theory has nowadays demonstrated a surprisingly accurate and predictive power in modern science and engineering. In this study, an important branch of quantum theory, density functional theory (DFT), is applied to studies of TiO₂ and doped TiO₂, which have shown considerable applications in industry.

The first chapter is an introduction to the theoretical background of DFT, in which a large quantity of efforts are focused on the analysis of exchange-correlation energy and how to approximate it by using local density approximation (LDA), generalized gradient approximation (GGA), and LDA+U, where the U is the Hubbard coefficient. This is followed in the second chapter by a discussion of practical implementations of the DFT-based calculations. We primarily introduce linearized augmented plane wave (LAPW) and augmented plane wave plus local orbital (APW+LO) methods, both of which are applied in our calculations. In chapter 3, we briefly introduce some fundamental properties of TiO₂ and its applications in industry. Chapters 4 through 8 are divided into two categories.

Chapters 4 through 6 are mainly concerned with insights into the mechanism of optical excitation in anatase TiO₂. Chapters 7 and 8 are concerned with TiO₂-based dilute magnetic semiconductors (DMS).

Chapter 4 presents detailed calculations on pristine TiO₂, including the structural optimization, density of states (DOS), band structure, and optical properties. Our calculations involve both bulk and slab TiO ₂, presenting reasonable results without considering inherent drawbacks of the calculation methods involved. Calculations on slab TiO₂ provide insight to account for the particular property of TiO₂ in nanoscale particles where a significant fraction of atoms are on the surface. In chapter 5, we investigate effects of the non-metal dopants such as N, C, and S on the electronic structure of TiO₂ host. They are all favorable dopants in making the original band gap of TiO₂ small enough for visible light excitation, although the mechanisms behind them are diverse. In Chapter 6, metal dopants in the TiO₂ host are extensively studied, including lanthanide elements such as Nd, Ce and Er, and transition metals such as Pt. Lanthanide 4f electrons and transition metal 5d electrons are intensively explored with LDA+U approach to account for their strong correlation. The findings are that the 4f or 5d states can function as donor or provide an intra-shell excitation, both of which are mechanisms for reducing the optical band gap.

In chapter 7, we calculate the electronic structure of an oxygen vacancy in anatase TiO₂. There are important findings. The states of a single oxygen vacancy could act as a shallow donor. However, the states of some vacancy pair structures are partially spin-polarized. In chapter 8, electronic structure of Co and V dopants are explored within LDA and LDA+U. It is found that they are all spin-polarized and the coupling between Co or V with oxygen vacancy are a favorable factor in inducing room temperature ferromagnetism in TiO₂ host. The origin of room temperature magnetism of TiO₂-based DMS is explored by taking Cr-doped TiO₂ as a special example. Cr 3d states are found to be situated within the band gap of the TiO₂ host and completely spin-polarized to generate its magnetism. The interaction of Cr ion without a mediating oxygen vacancy is too small to account for the occurrence of room temperature ferromagnetism. However, the ferromagnetic interactions in a structure with an oxygen vacancy residing between two Cr ions are strong enough to stablilize the ferromagnetic phase above room temperature.