  Using maximally localized Wannier functions obtained from DFT calculations,
we derive an effective Hubbard Hamiltonian for a bilayer of Sr$_3$Cr$_2$O$_7$,
the $n=2$ member of the Ruddlesden-Popper Sr$_{n+1}$Cr$_n$O$_{3n+1}$ system.
The model consists of effective $t_{2g}$ orbitals of Cr in two square lattices,
one above the other. The model is further reduced at low energies and two
electrons per site, to an effective Kugel-Khomskii Hamiltonian that describes
interacting spins 1 and pseudospins 1/2 at each site describing spin and
orbitals degrees of freedom respectively. We solve this Hamiltonian at zero
temperature using pseudospin bond operators and spin waves. Our results confirm
a previous experimental and theoretical study that proposes spin ordering
antiferromagnetic in the planes and ferromagnetic between planes, while
pseudospins form vertical singlets, although the interplane separation is
larger than the nearest-neighbor distance in the plane. We explain the physics
behind this rather unexpected behavior.
