The term "frustration" in the context of magnetism was originally used by P. W. Anderson and quickly adopted for application to the description of spin glasses and later to very special lattice types, such as the kagomé. The original use of the term was to describe systems with competing antiferromagnetic interactions and is important in current condensed matter physics in areas such as the description of emergent magnetic monopoles in spin ice. Within molecular magnetism, at least two very different definitions of frustration are used. Here we report the synthesis and characterization of unusual ninemetal rings, using magnetic measurements and inelastic neutron scattering, supported by density functional theory calculations. These compounds show different electronic/magnetic structures caused by frustration, and the findings lead us to propose a classification for frustration within molecular magnets that encompasses and clarifies all previous definitions. ,4,7-triazacyclononane). One area where molecular magnetism might contribute is in spin frustration (3), which is important in condensed matter physics in areas such as spin ice (4) and spin liquids (5). Potential model systems would be odd-numbered metal rings where the metal array forms equilateral triangles, or regular pentagons, heptagons, and larger 2D rings. Alternatively 3D solids made from regular odd-numbered rings should also be frustrated, e.g., regular tetrahedra and octahedra contain triangular faces and regular icosahedra contain pentagonal faces.Studies of spin frustration in molecular systems are largely restricted to that of triangular trimetallic species (6, 7). There are a few reports of five-metal rings but no detailed magnetic studies of such compounds (8), with the exception of a {Cu 5 } cage (9). The only significant examples of larger frustrated molecules are studies of the {M 30 } Keplerates (10) and a recent report of a heptametallic {VO} 7 ring (11), where the studies are limited to low temperature magnetization. There is also a report of an {Fe 9 } complex that can, very approximately, be described as a nine-metal ring (12).A further difficulty is that precisely what is meant by frustration varies in this area. The strictest definition used in molecular magnetism, from Kahn (13), states that frustration must result in a spin degenerate ground state, typically a degenerate pair of S = 1/2 states. Surprisingly, this definition excludes all possible models built from integer spins, as in such cases (for example a regular pentagon of s = 1 spins), the ground total spin state is unique and has S = 0. This definition thus seems prohibitively strict. At the other extreme, the term frustration has been used to describe cases where there are competing antiferromagnetic interactions, even where those interactions result in a unique spin ground state that can be described using a classical picture, with spins represented as arrows pointing alternately up and down. This definition seems, if anything, too permissive.We have reported studies of...

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The sequences of ground states in frustrated antiferromagnetic rings with odd number of local spins characterized by a single bond defect or by arbitrary uniform couplings to an additional spin located at the center are determined. The sequences provide firm constraints on the total ground-state quantum numbers, which are more stringent than those arising from the Lieb-Mattis theorem for bipartite quantum spin systems. Apart from their theoretical importance, they suggest the possibility of tailoring a given class of the molecular nanomagnets with desired ground-state properties by tuning the relevant couplings. In particular, they predict the spin S = 1/2 ground state for the centered rings composed of the half-integer spins with approximately uniform interactions. They confirm the applicability of the recent classification of spin frustration in both types of molecular nanomagnets. The classification is also discussed in the classical limit for the first class of the rings, providing a direct picture of frustration types. The Lieb-Mattis energy-level ordering and an analog of the Lande band, i.e., the energy spectra properties simplifying the characterization of the rings using the bulk magnetic or NMR measurements, are briefly discussed

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