  To advance the foundation of one-particle reduced density matrix functional
theory (1RDMFT) we refine and relate some of its fundamental features and
underlying concepts. We define by concise means the scope of a 1RDMFT, identify
its possible natural variables and explain how symmetries could be exploited.
In particular, for systems with time-reversal symmetry, we explain why there
exist six equivalent universal functionals, prove concise relations among them
and conclude that the important notion of $v$-representability is relative to
the scope and choice of variable. All these fundamental concepts are then
comprehensively discussed and illustrated for the Hubbard dimer and its
generalization to arbitrary pair interactions $W$. For this, we derive by
analytical means the pure and ensemble functionals with respect to both the
real- and complex-valued Hilbert space. The comparison of various functionals
allows us to solve the underlying $v$-representability problems analytically
and the dependence of its solution on the pair interaction is demonstrated.
Intriguingly, the gradient of each universal functional is found to always
diverge repulsively on the boundary of the domain. In that sense, this key
finding emphasizes the universal character of the fermionic exchange force,
recently discovered and proven in the context of translationally-invariant
one-band lattice models.
