  Accurate first-principles calculations for the energies, charge
distributions, and spin symmetries of many-electron systems are essential to
understand and predict the electronic and structural properties of molecules
and materials. Kohn-Sham density functional theory (KS-DFT) stands out among
electronic-structure methods due to its balance of accuracy and computational
efficiency. It is now extensively used in fields ranging from materials
engineering to rational drug design. However, to achieve chemically accurate
energies, standard density functional approximations in KS-DFT often need to
break underlying symmetries, a long-standing "symmetry dilemma". By employing
fragment spin densities as the main variables in calculations (rather than
total molecular densities as in KS-DFT), we present an embedding framework in
which this symmetry dilemma is resolved for the case of stretched molecules.
The spatial overlap between fragment densities is used as the main ingredient
to construct a simple, physically-motivated approximation to a universal
functional of the fragment densities. This 'overlap approximation' is shown to
significantly improve semi-local KS-DFT binding energies of molecules without
artificial symmetry breaking.
