G.E. Kimball & E.M. Loebl, J.Chem.Educ. 36(1959)233 discuss (on page 235) the molecular orbital treatment of Li_{2} in the form of an LCAOMO approximation. Summary: The linear combination of 1s, 2s (and 2p?) AO's undergoes drastic changes by antisymmetrization, leading to exchange and Coulomb interaction terms. The energy partition between these two categories is arbitrary and leaves the total energy invariant. By proper choice of components of the wavefunction one can minimize the exchange and maximize the Coulomb contributions. What then survives of the originally used linear combination of orbitals after the minimization of total energy resembles the electronic density distribution in Fig. 1 on p. 235, reproduced here. It shows  without scale given  the contours of three nonoverlapping spheres in a plane containing the Li nuclei, and is a template for a Kimball model calculation (outlined but not realized in the paper). 
I do not question the authors' arguments, but compute Li_{2} as Kimball model with variations. AFAIK no Kimball model computation of Li_{2} has been published. But LiH was treated by L.M.Kleiss(1952, loc.cit. p. 1012,37). The above figure seems to be a natural continuation, removing two H atoms from two LiH and combining the two half occupied Li(2s) clouds. 

The external Li2 after KimballLoebl (above 4) and the internal (above 5) are superimposed on the Li2 total density contour plot in the famous 'Pictorial Studies of Molecules' by Arnold C. Wahl, ANL7076, 1965. Scale: The innermost contour corresponds to 1 electron per bohr^3. The next 14 contours are always 1/2 of the density of the contour above: While the KimballLoebl model, left, needs more parametrization to resemble the experimental molecule, the internal model, right, requires only small adjustments of the original Kimball parameters. But, it is necessary to use the bonding cloud with a 2s kinetic parameter (H.R. Westerman, 1952, loc.cit. main page, has first treated free electron clouds with higher quantum numbers, ns and nporbitals, n <= 2). Which one shows the more natural representation of the LiLi bond ? It depends on what you try to understand. The right model is a pleasing development of what is known about the H_{2} molecule, with very little exchange correction, because overlap of the valence electrons with the 1s^{2}cores is so small. But, Li_{2} is far from similar to H_{2}! It is a strong tripole (a linear quadrupole), cannot be isolated but condenses to a metallic extended structure with brokenup electron pair if brought in contact with others. In the metal there are 8 or more clouds around each Li core producing a terrible conundrum of Pauli exclusion holes and huge exchange corrections with the model on the right. If this gets disentangled (by antisymmetrization) you end up much nearer to the left model, a Li^{+} in a sea of half filled external clouds connecting the whole structure and giving it metallic conductivity. Electrons can jump around the halfempty "cloud (anionsub)lattice", almost freely, a "Lithiumelectride salt". Solid state physicists talk about a half filled valence band, looking at the energylevels, but mean the same thing.
