Local spin of atoms in molecules: relation to electron localization and delocalization

Following the approach of Clark and Davidson, we partition the total squared spin of a molecule, S^2, into atomic, S₀^2, and interatomic, S₀ S₁, components within the framework of the Quantum Theory of Atoms in Molecules (QTAIM) and algebraically establish, and numerically analyze the relationship of these contributions to standard QTAIM measures of electron localization and delocalization for a set of molecules described at various levels of theory. Although this relationship is already known, it remains underutilized in QTAIM, warranting a dedicated examination in this work. Furthermore, we subdivide the atomic (or local) and bonding (or interatomic) spin terms into finer contributions, each associated with a specific electron population of an arbitrary atom or fragment of the molecule. This finer partition is achieved by using the theory of open quantum systems (OQSs). Our results reinforce our confidence in the proposed partitioning scheme for S^2, since it preserves the physical meaning of the components without resorting to mathematically equivalent but physically ambiguous transformations that have been frequently used in the literature. Several examples in simple molecular systems are also examined.