氧化鎳(NiO)與氧化鈷(CoO)是一強關聯的系統且為標準的Mott絕緣體。非共振非彈性X光散射實驗中發現氧化鎳與氧化鈷的d-d軌域之躍遷呈現很強之動量轉換相依(q-dependence)的特性，且此激子(exciton)的能量分布在準粒子能隙(quasiparticle band gap)之中。為了研究此局域化的躍遷行為，我們發展了局域化、有對稱性、能量可解析之Wannier函數。經由對Wannier電子電洞對的Fourier轉換之解析，我們了解此強動量轉換相依的起源且發現一新的選擇定則。進一步，我們提出一個稱為TDLDA+U的方法來解Bethe-Salpeter方程式用以描述電子電洞對間的吸引力來得出此能量分布在能隙中的激子。經由此方法，我們發現氧化鎳中的激子強烈混成在Mott能隙中。在Hartree-Fock近似下，一個有趣的動量轉換相依現象將被討論。另一方面，鈦酸鉛(PbTiO3)為一鐵電性材料，我們利用密度泛函微擾理論(DFPT)研究其晶體振動之特性。經由計算零溫下鈦酸鉛的聲子頻譜與彈性波速，我們確定了鈦酸鉛在低溫下的鐵電結構。更重要的，我們詳細分析鈦酸鉛在立方(cubic)與四方(tetragonal)晶體結構下的橫向光頻(TO)與縱向光頻(LO)振動模式間的偶合。不像立方結構中所發現的橫向光頻與縱向光頻間之巨大分裂，四方結構呈現正常的橫向與縱向振動模式之偶合關係。根據此發現，我們重新解釋了在鈦酸鉛摻雜鈣原子(Pb1-xCaxTiO3)下所發現的橫向與縱向之光頻分裂行為。最後我們將討論實驗上所觀測到鈦酸鉛摻雜鈣原子所發生之結構相變的成因，並預測低溫下新結構相變之發生。 We investigate the Frenkel excitons in NiO and CoO, the standard prototype of Mott insulator, within the diagrammatic picture of density-density response function. Recently immobile excitons showing up in the Mott gap from d-d excitations of NiO and CoO found by non-resonant inelastic X-ray scattering possess strong q-dependence and large binding energies. To study the local excitations, a new localized, symmetry-respecting, energy-resolved Wannier function is developed to perform the calculation. A novel TDLDA+U method is proposed to solve the relevant Bethe-Salpeter equation and the attraction of electron-hole pairs of corresponding real-space Wannier functions is well included. Strongly hybridized Frenkel excitons of NiO are found in the Mott gap via this approach and the local Hartree-Fock interaction reveals an interesting q-dependence in NiO and CoO due to the effect of quantum interference of excitons. Electronic structure calculations from first principles in density functional perturbation theory are made to elucidate the lattice dynamic properties of tetragonal PbTiO3. At low temperatures, the mechanical stability of ferroelectric PbTiO3 is identified by obtaining full calculated phonon dispersion curves and sound velocities. Most importantly, a normal coupling characteristic between transverse optical (TO) and longitudinal optical (LO) phonon modes are found in the stable tetragonal PbTiO3, unlike the giant LO-TO splitting in cubic perovskites, as determined by studying correlation matrices. Based on this information, the essentials of the analysis of LO-TO splitting are emphasized to examine the effect of doping in mixed Pb1-xCaxTiO3. Finally, we discuss the mechanism of structural phase transition observed by experiment in Pb1-xCaxTiO3 and predict a new phase transition (cubic-orthorhombic) for cubic Pb0.5Ca0.5TiO3 at low temperatures.
