The magnetic and ferroelectric (FE) properties of TbMnO3 were investigated on the basis of relativistic density functional theory (DFT) calculations. We show that, due to spin-orbit coupling, the spin-spiral plane of TbMnO3 can be either the bc-or ab-plane, but not the ac-plane. As for the mechanism of FE polarization, our work reveals that the "pure electronic" model by Katsura, Nagaosa and Balatsky is inadequate in predicting the absolute direction of FE polarization. Our work indicates that to determine the magnitude and the absolute direction of FE polarization in spin-spiral states, it is crucial to consider the displacements of the ions from their centrosymmetric positions.PACS numbers: 75.80.+q,75.30.Gw, Recent studies on magnetic ferroelectric (FE) materials have shown that electric polarization can be significantly modified by the application of a magnetic field [1,2,3,4,5,6,7]. Perovskite TbMnO 3 with a spinspiral magnetic order is a prototypical multiferroic compound with a gigantic magnetoelectric effect [2]. Currently, there are two important issues concerning the FE polarization of TbMnO 3 . One concerns the origin of FE polarization. Model Hamiltonians studies of spin-spiral multiferroic compounds have provided two different pictures. In the Katsura-Nagaosa-Balatsky (KNB) model [5], the hybridization of electronic states induced by spinorbit coupling (SOC) leads to a FE polarization of the charge density distribution even if the ions are not displaced from their centrosymmetric positions. In contrast, the model study by Sergienko and Dagotto [7] concluded that oxygen ion displacements from their centrosymmetric positions are essential for the FE polarization in multiferroic compounds [8]. When carried out with the ions kept at their centrosymmetric positions, density functional theory (DFT) calculations [9] for the spin-spiral states of LiCuVO 4 predict FE polarizations that agree reasonably well in magnitude with experiment [10], which is in apparent support of the KNB model [5]. It is, therefore, important to check which model, the KNB or the "ion displacement" model, is relevant for the FE polarization in TbMnO 3 . The other issue concerns the spin-spiral plane of TbMnO 3 . Under a magnetic field, the spin-spiral plane of TbMnO 3 can be either the bc-plane or the abplane, but not the ac-plane. To explain this observation, it is necessary to probe the magnetic anisotropy of the Mn 3+ ion. The magnetic anisotropy of the Tb 3+ ion might be also relevant for the magnetoelectric effect, as suggested by Prokhnenko et al. [11].In this Letter, we investigated these issues on the basis of DFT calculations and found that the consideration of the ion displacements is essential for the FE polarizations in the spin-spiral state of TbMnO 3 , and the KNB model can be erroneous even for predicting the absolute direction of FE polarization. The absence of the ac-plane spin-spiral in TbMnO 3 is explained by the magnetic anisotropy of the Mn 3+ ion.Our calculations were based on DFT plus the on-site repulsion U m...