  The structural, electronic and lattice dielectric properties of multiferroic
TbMn$_2$O$_5$ are investigated using density functional theory within the
generalized gradient approximation (GGA). We use collinear spin approximations
and ignore the spin-orbit coupling. The calculated structural parameters are in
excellent agreement with the experiments. We confirm that the ground state
structure of TbMn$_2$O$_5$ is of space group $Pb2_1m$, allowing polarizations
along the b-axis. The spontaneous electric polarization is calculated to be
1187 $nC\cdot$cm$^{-2}$. The calculated zone-center optical phonons frequencies
and the oscillator strengths of IR phonons agree very well with the
experimental values. We then derive an effective Hamiltonian to explain the
magnetically-induced ferroelectricity in this compound. Our results strongly
suggest that the ferroelectricity in TbMn$_2$O$_5$ is driven by the magnetic
ordering that breaks the the inversion symmetry, without invoking the
spin-orbit coupling.
