中文摘要：

计算钛空位（V_（Ti））和氧空位（V_O）共存情况下未掺杂金红石TiO_2的铁磁性.发现V_O可以产生局域磁矩,由它引起的自旋极化比V_（Ti）引起的更加局域,导致V_O之间的铁磁耦合作用弱于V_（Ti）之间的铁磁耦合作用.V_（Ti）之间的铁磁耦合在引入V_O之后进一步加强.V_O引入的电子调制两个分离的V_（Ti）之间的长程铁磁耦合.加入V_（Ti）之后,两个V_O的磁矩猝灭,当V_O的数量多于V_（Ti）的数量二倍时,V_O会对磁矩有贡献.结果与实验发现的V_O可以提高铁磁有序,并且总的磁矩会随着V_O数量的增多而增加的结果符合很好.

英文摘要：

Coexistence of Ti vacancies and O vacancies is calculated to study ferromagnetic ordering in undoped rutile TiO2 . It shows that O vacancy （ VO ） favors magnetic state in undoped TiO2 . Spin polarization induced by VO is more localized than that induced by VTi , which is the reason that FM coupling of two VO is weaker than that of two VTi . FM coupling of two VTi is strengthened as VO is introduced. Long-range FM coupling between two separate VTi is mediated by electrons introduced by VO . Magnetic moments induced by two VO quenched as VTi is introduced. VO contributes magnetic moment only if the number of VO is more than twice that of VTi . It is in quantitative agreement with experimental results that VO can promote ferromagnetic ordering, and magnetic moment increase with increase of the number of VO .