A range of functional theories in electronic structure theory can be formulated within a common symplectic geometric framework. The key object is the moment map for a Hamiltonian action of a compact Lie group on projective Hilbert space. After briefly recalling this formalism, we show how reduced variables relevant for electronic structure theory arise from suitable symmetry groups. We then focus on one-particle reduced density matrix functional theory (RDMFT), which is conceptually well suited to describe the properties of strongly correlated many electron systems. For spin-adapted RDMFT of spin-1/2 fermions, the orbital 1RDM is identified with the moment map for the action of U(d) on the fixed-spin sector of the N-fermion Hilbert space. Its admissible spectra therefore form a moment polytope, and we present new results on the symmetry-adapted pure and ensemble one-body N-representability problem that characterizes this domain.
In the second part of the talk, we introduce an ensemble formalism to target low-lying excited states within RDMFT. We show that the partial trace image of the convex hull of a suitable unitary orbit yields a convex polytope characterizing the set of admissible one-particle reduced density matrices (1RDMs). For these restricted ensembles of fixed spectra, the resulting spectral constraints on the eigenvalues of the 1RDM constitute mixed-state generalizations of Pauli's exclusion principle. We then discuss the key structural properties of the facet-defining inequalities of the corresponding moment polytopes and highlight their implications for applications in quantum chemistry.
About the Speaker:
Julia Liebert is a theoretical physicist and chemist working on foundational aspects of functional theories and developing new quantum information theoretic methods for applications in physical chemistry. She joined the Chemistry Department in Princeton in 2025 after receiving her PhD in physics from the Ludwig Maximilian University of Munich earlier that year. Her doctoral research was funded by a fellowship from the International Max Planck Research School of Quantum Science and Technology.
