  We investigate the hyperfine energy levels and Zeeman splittings for
homonuclear alkali-metal dimers in low-lying rotational and vibrational states,
which are important for experiments designed to produce quantum gases of deeply
bound molecules. We carry out density-functional theory (DFT) calculations of
the nuclear hyperfine coupling constants. For nonrotating states, the
zero-field splittings are determined almost entirely by the scalar nuclear
spin-spin coupling constant. By contrast with the heteronuclear case, the total
nuclear spin remains a good quantum number in a magnetic field. We also
investigate levels with rotational quantum number N = 1, which have long-range
anisotropic quadrupole-quadrupole interactions and may be collisionally stable.
For these states the splitting is dominated by nuclear quadrupole coupling for
most of the alkali-metal dimers and the Zeeman splittings are considerably more
complicated.
