  Thanks to an algebraic duality property of reduced states, the Schmidt best
approximation theorems have important corollaries in the rigorous theory of
two-electron moleculae. In turn, the "harmonium mode" or "Moshinsky atom"
constitutes a non-trivial laboratory bench for energy functionals proposed over
the years (1964--today), purporting to recover the full ground state of the
system from knowledge of the reduced 1-body matrix. That model is usually
regarded as solvable, but some important aspects of it, in particular the exact
energy and full state functionals ---unraveling the "phase dilemma" for the
system--- had not been calculated heretofore. The solution is made plain here
by working with Wigner quasiprobabilities on phase space. It allows in
principle for a thorough discussion of the (de)merits of several approximate
functionals popular in the theoretical chemical physics literature. We focus on
Gill's "Wigner intracule" method for the correlation energy.
