In Phys. Rev. Lett. 2021, 127, 023001 a reduced density matrix functional theory (RDMFT) was proposed for calculating energies of selected eigenstates of interacting many-Fermion systems.Here, we develop a solid foundation for this so-called w-RDMFT and present the details of various derivations. First, we explain how a generalization of the Ritz variational principle to ensemble states with fixed weights w in combination with the constrained search would lead to a universal functional of the one-particle reduced density matrix. To turn this into a viable functional theory, however, we also need to implement an exact convex relaxation. This general procedure includes Valone's pioneering work on ground state RDMFT as the special case w = (1,0, •••). Then, we work out in a comprehensive manner a methodology for deriving a compact description of the functional's domain. This leads to a hierarchy of generalized exclusion principle constraints which we illustrate in great detail. By anticipating their future pivotal role in functional theories and to keep our work self-contained, several required concepts from convex analysis are introduced and discussed.

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From a geometric point of view, Pauli's exclusion principle defines a hypersimplex. This convex polytope describes the compatibility of 1-fermion and N -fermion density matrices, therefore it coincides with the convex hull of the pure N -representable 1-fermion density matrices. Consequently, the description of ground state physics through 1-fermion density matrices may not necessitate the intricate pure state generalized Pauli constraints. In this article, we study the generalization of the 1-body N -representability problem to ensemble states with fixed spectrum w, in order to describe finite-temperature states and distinctive mixtures of excited states. By employing ideas from convex analysis and combinatorics, we present a comprehensive solution to the corresponding convex relaxation, thus circumventing the complexity of generalized Pauli constraints. In particular, we adapt and further develop tools such as symmetric polytopes, sweep polytopes, and Gale order. For both fermions and bosons, generalized exclusion principles are discovered, which we determine for any number of particles and dimension of the 1-particle Hilbert space. These exclusion principles are expressed as linear inequalities satisfying hierarchies determined by the non-zero entries of w. The two families of polytopes resulting from these inequalities are part of the new class of so-called lineup polytopes.