  We perform a systematic investigation of the electronic properties of the
$^2\Sigma^+$ ground state of Li-alkaline-earth dimers. These molecules are
proposed as possible candidates for quantum simulation of lattice-spin models.
We apply powerful quantum chemistry coupled-cluster methods with large basis
sets to calculate potential energies, permanent dipole moment, and electron
density distributions for LiBe, LiMg, LiCa, LiSr, and LiYb. Our results reveal
a surprising irregularity in the dissociation energy and bond length with an
increase in the reduced mass of the molecule. At the same time the permanent
dipole moment at the equilibrium separation has the smallest value between 0.01
a.u. and 0.1 a.u. for the heaviest (LiSr and LiYb) molecules and increases to
1.4 a.u. for the lightest (LiBe), where 1 a.u. is one atomic unit of dipole
moment. We consider our study of the $^2\Sigma^+$ molecules a first step
towards a comprehensive analysis of their interactions in an optical trap.
