钇铁石榴石中的钇原子在被重金属原子镥、铋等取代后可获得显著提升的磁光特性，因此在高性能光学器件具有重要的应用价值。本文使用第一性原理计算研究了钇铁石榴石及其镥、铋取代产物的结构和电子结构、介电函数张量、光电导张量、磁光法拉第旋转以及磁光法拉第椭圆率随能量的变化，并结合晶体场理论对所得结果进行了系统分析。

通过分析钇铁石榴石的介电函数及光电导吸收峰，我们确定了不同峰的电子跃迁类型，从而确定磁光法拉第旋转及椭圆率上对应峰的性质。计算进一步揭示镥、铋取代钇原子对应的取代产物的磁光法拉第旋转及椭圆率相较钇铁石榴石有显著提升，且吸收峰的个数也有增加，与已有的实验结果高度吻合。结合吸收峰性质的分析发现镥、铋纯取代钇原子的体系中，取代原子半径越大，体系中的铁氧键长越长，铁氧之间杂化强度越小，八面体铁和四面体铁的晶体场劈裂能越小；在镥、铋混合取代钇原子的体系中，混合取代引起的晶格畸变体现在铁氧键长和键角的变化上，致使八面体铁和四面体铁的能带部分退简并，进一步降低晶体场劈裂能。结合Tanabe-Sugano能级图，计算结果证实镥、铋取代钇显著降低了其中铁的基态与激发态的能量差，揭示了光学跃迁增强的原因。

由于石榴石制备过程中不可避免的氧空位的存在，本文系统研究了不同浓度和位置的氧空位对镥、铋取代钇铁石榴石体系的磁光性质的影响。通过计算不同氧空位下的铁氧键长并根据价键模型判断氧空位导致的铁的化合价的变化：与氧空位直接相邻的铁从正三价降为正二价，不相邻的铁的价位无明显变化；计算不同氧空位下的八面体铁和四面体铁的分波态密度确定氧空位导致的晶体场劈裂能的变化：钇铁石榴石的八面体铁的晶体场劈裂能减小，四面体铁的晶体场劈裂能变化不明显，因此氧空位对钇铁石榴石光学跃迁有微小增强作用。

本文的研究对于进一步理解和分析钇铁石榴石及其镥、铋取代产物在磁光观测中的行为提供了可靠的理论参考。 

The magneto-optical properties of yttrium iron garnet (YIG) are significantly enhanced by substituting yttrium by heavy metallic atoms such as bismuth and lutetium. These properties attract much attention for the potential applications in high performance optical devices and the typical characteristics are the giant Faraday rotation angle after the substitution. In this thesis, first-principles calculations are carried out to study the electronic structure and magneto-optical parameters of YIG and the substituted garnets including the dielectric function tensor, optical conductivity, magneto-optical Faraday rotation angle, Faraday ellipticity, etc.

By varying the concentration of bismuth and lutetium substitution, we systematically analyze the influence on the crystal field splitting for both octahedral and tetrahedral iron, the optical transition and its corresponding magneto-optical properties. In addition, we also study the physical properties induced by the oxygen vacancies, which are inevitable in the materials. The Faraday rotation is remarkably enhanced by substituting bismuth, which has large spin-orbit coupling. The latter leads to strong double transition and charge transfer transition. A larger substitutional atom and/or longer iron-oxygen bond results in a smaller hybridization of iron and oxygen and a smaller energy of crystal field splitting. The lattice distortion due to the substitution results in lifting degeneracy of energy levels of iron, which determine the magnetic properties of the garnets. In the presence of oxygen vacancies, the valence of iron can be derived from the valence bond model while the extra electrons lead to a decrease in the crystal field splitting of the octahedrons in YIG. Our results provide theoretical references for understanding the experimental measurement of magneto-optical properties of YIG and its substitutions.