  Properties of the $\omega$ structure are investigated for 27 transition
elements from the viewpoints of thermodynamical and mechanical stability based
on first-principles calculations. The thermodynamical stability of the $\omega$
structure is compared with those for the body-centered cubic (BCC),
face-centered cubic (FCC), and hexagonal close-packed (HCP) structures.
Similarly to the case of those popular crystal structures, the occupation
number for $d$ orbitals is found to roughly determine relative energy and
volume of the nonmagnetic (NM) $\omega$ structure. For the group 4 elements
(Ti, Zr, and Hf), the $\omega$ structure is almost the lowest in energy among
the investigated crystal structures and is also mechanically stable. The
$\omega$ structure of the group 7 elements (Mn, Tc, and Re) is also
mechanically stable. The $\omega$ Fe is found to exhibit a complicated magnetic
state that is different from the ferromagnetic (FM) and NM ones. This magnetic
state is the most favorable among the investigated magnetic states. The
$\omega$ Fe in this magnetic state is also mechanically stable. Energies of
binary alloys composed of the elements in the group 4 and those in the groups 5
and 6 are estimated by linear interpolation, and most of the alloys show
concentration ranges where the $\omega$ structure is the lowest in energy among
the investigated crystal structures.
