Recently, thickness-dependent band structure of molecular beam epitaxy grown ultrathin films Bi$_{2}$Se$_{3}$ was investigated by in-situ angle-resolved photoemission spectroscopy $^{28}$. An energy gap was first observed exper- imentally in the surface states of Bi$_{2}$Se$_{3}$ below the thick- ness of six quintuple layers, which confirms theoretical prediction as a finite size effect $^{30–33}$. Table II shows the fitting parameters to the ARPES data of Bi$_{2}$Se$_{3}$ thin films 28 using the energy spectra for- mula [Eq. (56)]. For the films with thickness ranging from 2 QL to 5 QL, all of them satisfy sgn(Δ B ) > 0 and ˜ V < ℏ v$_{F}$ √ | Δ / 2 B | , hence the films are possibly in the QSH regime. We identify that only 2 QL, 3 QL, and 4 QL belong to the nontrivial case for potential QSH effect. 5 QL is an exceptional case that the fitted parameters B and D do not satisfy the existence condition of an edge states solution $^{33}$. The condition of B $^{2}$< D2 will lead to the band gap closing for a large k . However, it is under- stood that the model is only valid near the Γ point, and the fitting parameters are limited to the case of small k . And the band gap was measured clearly for the film of 5QL. It was previously predicted, using the parameters from the first-principles calculation $^{19}$, that the gap Δ should oscillate as a function of the film thickness $^{30–32}$. However, this oscillation is not reflected in the measured results. In the quantum Hall effect the Chern number of the bulk states has an explicit correspondence to the num- ber of edge states in an open boundary condition $^{39}$. In topological insulator or QSH system, the Z$_{2}$ topological invariant has also a relation to the number of helical edge states $^{40}$. As a supplementary support to the above con- clusion, we demonstrated the presence of edge states in a periodic boundary condition along the x-direction and an open boundary condition (say along y-direction) im- posed in a geometry of strip of the thin film by means of numerical calculation. Using the parameters in Table I, we have concluded that a stripe of 2 - 4 QL will ex- hibit helical edge states. More specifically, we present the energy dispersion for 4QL in Fig. 6. There is a doubly- degenerate Dirac point inside the gap of the 2D surface states for 4 QL in consistence with the results obtained in the above sections. We derived two-dimensional effective continuous mod- els for the surface states and thin films of three- dimensional topological insulators (3DTI). A gapless Dirac cone was confirmed for the surface states of a 3DTI. For a thin film, the coupling between opposite topologi- cal surface states in space opens an energy gap, and the Dirac cone evolves into a gapped Dirac hyperbola. The thin film may break the top-bottom symmetry. For ex- ample, the thin film grows on a substrate, and possesses the structural inversion asymmetry (SIA). This SIA leads to a Rashba-like coupling and energy splitting in the mo- mentum space. It also leads to asymmetric distributions of states along film growth direction. The ARPES measurements on Bi$_{2}$Se$_{3}$ films have demonstrated that the surface spectra opens a visible en- ergy gap when the thickness is below 6QLs. 28 The energy gap was observed to be a function of the thickness of thin film, and in a good consistence with theoretical prediction as a finite size effect of the thickness of thin film. The Rashba-like splitting was measured clearly in the thin film of 2 to 6 QLs. This can be explained very well from the inclusion of the SIA. Since the thin film was grown on a SiC substrate and the other surface is exposed to the vacuum this fact results in the SIA in the thin film. Another direct evidence to support the SIA is the sig- nal intensity pattern of the energy spectra of ARPES. Layers E 0 D Δ B v F | ˜ V | (QL) (eV) (eV ˚ A $^{2}$) (eV) (eV ˚ A $^{2}$) (10 $^{5}$m/s) (eV) 2 -0.470 -14.4 0.252 21.8 4.47 0 3 -0.407 -9.7 0.138 18.0 4.58 0.038 4 -0.363 -8.0 0.070 10.0 4.25 0.053 5 -0.345 -15.3 0.041 5.0 4.30 0.057 6 -0.324 -13.0 0 0 4.28 0.068 −0.1 −0.05  0  0.05  0.1 −0.05  0  0.05  0.1 E (eV) k$_{x}$(Å −1 ) 4QL FIG. 6: (Color online) The energy spectra by the tight bind- ing calculation for 4 QL of Bi 2 Se 3 thin films with the width along y -direction 200 nm. The parameters are given in table II. TABLE II: Fitting parameters to the Bi 2 Se 3 thin films, using the energy spectra Eq. (56) from our effective model.[adopted from Ref. $^{28}$] C. Thin film Bi 2 Se 3 and QSH effect D. QSH effect of SIA and the edge states VI. CONCLUSIONS 10