suggest that a insulating magnetic state is obtained for Cr or Fe doping, while the states are metallic for Ti or V doping cases. To understand the results, we first point out that, for all the cases, the dopants are nearly in the 3+ valence state, and we always obtain the high-spin state due to the large Hund’s rule coupling of 3 d transition metal ions. Thus the charge and spin degrees of freedom are nearly frozen. This will directly lead to the insulating state of Fe-doped samples, because the Fe 3+ has five 3 d electrons, favoring the d 5 $^{↑}$d 0 ↓ configuration in a high-spin state, resulting in a gap between the majority and minority spins. We now consider the orbitals for other dopants. The local environment of dopants, which substitute the Bi sites, is a octahedral formed by six nearest neighboring Se 2 − ions. Such a local crystal field splits the d -shell into t$_{2}$$_{g}$ and e$_{g}$ manifolds. This splitting is large enough to stabilize the t 3 ↑ 2 $_{g}$e 0 ↑ g t 0 ↓ 2 $_{g}$e 0 ↓ g configuration of Cr 3+ ion, resulting in a gap between the t$_{2}$$_{g}$ and e$_{g}$ manifolds. For the case of Ti or V doping, even the t$_{2}$$_{g}$ manifold is partially occupied, leading to the metallic state. It is important to note that, although the local density approximation in the density functional theory may underestimate the electron correlation effects, the inclusion of electron-electron interaction U (such as in LDA+ U method) should further enhanced the gap. The energy gain due to the spin polarization is about 0.9eV/Fe, 1.5eV/Cr, 0.7eV/V and 0.02eV/Ti, respectively, which are very large numbers except for the case of Ti substitution. From the spin splitting of p orbitals of the band electrons, we can estimate the effective ex- change coupling J$_{eff}$ between the local moments and the p electrons $^{34}$. The estimated value is around 2.7eV for Cr and 2.8eV for Fe in Bi$_{2}$Se$_{3}$. This exchange coupling is comparable to that in GaMnAs $^{20,34}$. Once the ferromagnetically ordered insulating state is achieved with a reasonable T$_{c}$ , here in this section, we show that QAH effect can be realized for the 2D thin films. In the thick slab geometry, the spatially-separated two pairs of surface states are well defined for the top and bottom surfaces respectively. However, with the reduction of the film thickness, quantum tunneling between the top and bottom surfaces becomes more and more pronounced, giving rise to finite mass term in the effective 2D model $^{19}$. In the basis of | + ↑〉 , |− ↓〉 , | + ↓〉 , |− ↑〉 , with ± referring to the bonding and anti-bonding combinations 9 V. THE QUANTIZED ANOMALOUS HALL EFFECT