在此論文中，我們利用第一原理計算研究多鐵材料Cu3Mo2O9的磁性、電子態及多鐵性質。我們發現在此系統中，電子與電子間的庫倫排斥力必須被考慮，以致於導帶與價帶間能隙能夠被良好地描述。由於晶體結構所導致的幾何不穩定性，系統的磁結構尚未在實驗測量中被確定。在我們的理論計算當中得到的磁結構與Vilminot等研究人員根據實驗結果猜測出的非線性反鐵磁結構類似。交換作用與自旋軌道耦合間的爭競決定了電子自旋方向的傾斜。計算所得到的交換作用係數與實驗結果吻合良好。利用Berry’s phase計算，我們得到了系統自發電極化的理論值，其強度與實驗量測值在同一個數量級。然而，在我們計算中得到的電極化方向(平行於b軸)與實驗(平行於c軸)不符。此外，我們發現一磁結構之理論電極化方向與實驗相符，然而其磁結構之對稱性與實驗不符。目前，尚未有第一原理計算研究此氧化物，我們希望此論文能夠對同樣有興趣研究此材料的研究人員有所幫助。

In this thesis, we used the ab initio method to study a multiferroic oxide Cu3Mo2O9. The correlations of electrons must be considered in this system so that a reasonable energy gap can be obtained. Due to the geometric frustration of magnetic structure caused by crystal structure, the ground state spin configuration in this system still has not been determined experimentally. We found some spin configurations similar to the non-collinear anti-ferromagnetic spins configuration suggested by Vilminot et al.. Competition between exchange interactions and spin-orbit coupling effect determines the canting of spins on Cu atoms. The calculated exchange parameters agree with the experimental results well. By using Berry phase calculations, we obtained the theoretical value of spontaneous electric polarization. The strength of polarization in our results is in the same order of results of experiments. However, the direction of electric polarization we found (along b-axis) is different from the experimental measurements (along c-axis). We have found a spin configuration that the theoretical electric polarization of the state agrees with the experimental results. However, the symmetry of the spin configuration does not satisfy the conditions suggested by results of the neutron diffraction experiment. And, spins on neighboring Cu2 and Cu3 do not form a singlet dimer. Since there still is no ab initio calculation studying this oxide, we hope that our studies can help those who are also interested in this material.

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