Solving one-body ensemble N-representability problems with spin

The Pauli exclusion principle is fundamental to understanding electronic quantum systems, imposing constraints on the expected occupancies n_iof orbitals φ_i, such that 0 ≤ n_i≤ 2. In this work, we refine the underlying onebody N -representability problem by incorporating spin symmetries and a potential degree of mixedness w of the N -electron quantum state. Employing basic tools from representation theory, convex analysis, and discrete geometry, we derive a comprehensive solution to this problem. Specifically, we demonstrate that the set of admissible orbital one-body reduced density matrices is fully characterized by linear spectral constraints on the natural orbital occupation numbers, defining a convex polytope Σ_N,S(w) ⊂ [0, 2]^d. These constraints are independent of the magnetization M and the number d of orbitals, while their dependence on N and the total spin S is linear, and we thus calculate them for arbitrary system sizes and spin quantum numbers. Our results provide a crucial missing cornerstone for ensemble density (matrix) functional theory.