To explore stable polynuclear magnetic superhalogens, we perform an unbiased structure search for polynuclear iron-based systems based on pseudohalogen ligand CN using the CALYPSO method in conjunction with density functional theory. The superhalogen properties, magnetic properties, and thermodynamic stabilities of neutral and anionic Fe(CN) and Fe(CN) clusters are investigated. The results show that both of the clusters have superhalogen properties due to their electron affinities (EAs) and that vertical detachment energies (VDEs) are significantly larger than those of the chlorine element and their ligand CN. The distribution of the extra electron analysis indicates that the extra electron is aggregated mainly into pseudohalogen ligand CN units in Fe(CN) and Fe(CN) cluster. These features contribute significantly to their high EA and VDE. Besides superhalogen properties, these two anionic clusters carry a large magnetic moment just like the FeF cluster. Additionally, the thermodynamic stabilities are also discussed by calculating the energy required to fragment the cluster into various smaller stable clusters. It is found that Fe(CN) is the most favorable fragmentation product for anionic Fe(CN) and Fe(CN) clusters, and both of the anions are less stable against ejection of Fe atoms than Fe(CN).
A class of finite-dimensional Cartan-type Lie superalgebras H(n,m) over a field of prime characteristic is studied in this paper. We first determine the derivation superalgebra of H(n,m). Then we obtain that H(n,m) is restrictable and it is an extension of the Lie superalgebra [Formula: see text]. Finally, we prove that H(n,m) is isomorphic to a subalgebra of the restricted Hamiltonian Lie superalgebra [Formula: see text].
In this paper, we study Heisenberg vertex algebras over fields of prime characteristic. The new feature is that the Heisenberg vertex algebras are no longer simple unlike in the case of characteristic zero. We then study a family of simple quotient vertex algebras and we show that for each such simple quotient vertex algebra, irreducible modules are unique up to isomorphism and every module is completely reducible. To achieve our goal, we also establish a complete reducibility theorem for a certain category of modules over Heisenberg algebras.
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