Solving one-body ensemble N-representability problems with spin

The Pauli exclusion principle is fundamental to understanding electronic quantum systems. It namely constrains the expected occupancies of orbitals according to . In this work, we first refine the underlying one-body -representability problem by taking into account simultaneously spin symmetries and a potential degree of mixedness of the -electron quantum state. We then derive a comprehensive solution to this problem by using basic tools from representation theory, convex analysis and discrete geometry. Specifically, we show 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 . These constraints are independent of and the number of orbitals, while their dependence on is linear, and we can thus calculate them for arbitrary system sizes and spin quantum numbers. Our results provide a crucial missing cornerstone for ensemble density (matrix) functional theory.