The Necessary and Sufficient Conditions of Separability for Multipartite Pure States

In this paper we present the necessary and sufficient conditions of separability for multipartite pure states. These conditions are very simple, and they don't require Schmidt decomposition or tracing out operations. We also give a necessary condition for a local unitary equivalence class for a bipartite system in terms of the determinant of the matrix of amplitudes and explore a variance as a measure of entanglement for multipartite pure states.Comment: Submitted to PRL in Sep. 2004, the paper No is LV9637. Submitted to SIAM on computing, in Jan., 2005, the paper No. is SICOMP 44687. Under reviewing no

The effects of halogen elements on the opening of an icosahedral B12 framework

The fully halogenated or hydrogenated B12X12 (X = H, F, Cl, Br and I) clusters are confirmed to be icosahedral. On the other hand, the bare B12 cluster is shown to have a planar structure. A previous study showed that a transformation from an icosahedron to a plane happens when 5 to 7 iodine atoms are substituted. Later, the transition was confirmed to be seven iodine substitutions based on an infrared spectroscopy study. In this study, we investigated the effects of different halogen atoms on the opening of the B12 icosahedral cage by means of density functional theory calculations. We found that the halogen elements do not have significant effects on the geometries of the clusters. The computed IR spectra show similar representative peaks for all halogen substituted clusters. Interestingly, we found a blue-shift in the IR spectra with the increase in the mass of the halogen atoms. Further, we compared the Gibbs free energies at different temperatures for different halogen atoms. The results show that the Gibbs free energy differences between open and close structures of B12X7 become larger when heavier halogen atoms are present. This interesting finding was subsequently investigated by energy decomposition analysis