| Ab initio electronic structure methods provide a valuable tool for the study of the structure and properties of molecular systems. Due to the demanding computational costs of these methods, however, they can at present only be routinely applied to molecular systems containing up to hundreds of atoms. This thesis primarily addresses the issue of how to reduce the computational cost of the Kohn-Sham density functional theory (KS-DFT), one of the most popular ab initio methods, without comprising its accuracy.; The initial chapter introduces KS-DFT with an emphasis on its computational cost in an actual implementation. Four time-consuming pieces are identified, known as the Coulomb (J) problem, the Hartree-Fock exchange (K) problem, the exchange-correlation numerical integration problem and the diagonalization problem, respectively.; Due to the slow decaying of the Coulomb interaction, the Coulomb problem once dominated a KS-DFT calculation. This problem is addressed in Chapters 2, 3, and 4. A J matrix engine method is introduced in Chapter 2 which reduces the prefactor of the linear scaling for the formation of the Coulomb matrix in a ground state energy evaluation. Its extension to the Coulomb energy gradient evaluation, called the J force engine method, is presented in Chapter 3. Some preliminary discussions are presented in Chapter 4 concerning how to reduce the cost of the Coulomb part of the energy second derivatives evaluation.; The exchange problem can be solved with effective integral screening techniques, leading to LinK and related methods. In Chapter 5, we discuss the extension of the LinK method to reduce the cost of the exact (Hartree-Fock) exchange part of the energy second derivatives evaluation.; The diagonalization of the Fock matrix, which scales cubically with the system size, can potentially become the time-dominant step with the advent of efficient linear-scaling algorithms for all other components of a KS-DFT calculation. We propose a new indirect diagonalization method, which is called a curvy-step approach. The underlying sparse linear algebra tools are introduced in Chapter 6, the basic ideas of the curvy-step approach are presented in Chapter 7, and its application to KS-DFT calculations are reported in Chapter 8.; Very closely related to our work on the Coulomb and Hartree-Fock exchange parts of the energy second derivatives evaluation is our implementation of a spin-flip density functional theory (SF-DFT) described in Chapter 9. SF-DFT can be applied to treat the ground state and low-lying excited states, and encouraging results are obtained for diradicals and bond-breaking, which are hard problems due to the existence of strong non-dynamical correlation effects. |
