  After the study of the three body molecular system H$_2^+$ ({\it J. Phys. B:
At. Mol. Opt. Phys.} {\bf 45} 065101), its isotopomer, the deuterium molecular
ion D$_2^+$ is studied. The three-body Schr\"odinger equation is solved using
the Lagrange-mesh method in perimetric coordinates. Energies and wave functions
for four vibrational states $v=0-3$ and bound or quasibound states for total
orbital momenta from 0 to 56 are calculated. The 1986 fundamental constant
$m_d=3670.483014\,m_e$ is used. The obtained energies have an accuracy from
about 13 digits for the lowest vibrational state to at least 9 digits for the
third vibrational excited state. Quadrupole transition probabilities per time
unit between those states over the whole rotational bands were calculated.
Extensive results are presented with six significant figures.
