A systematic study on the structures and electronic properties of copper clusters has been performed using the density functional theory. In the calculation, there are many isomers near the ground state for small copp...A systematic study on the structures and electronic properties of copper clusters has been performed using the density functional theory. In the calculation, there are many isomers near the ground state for small copper clusters. Our results show that the three-dimensional isomers of copper clusters start from Cu7 cluster and then show a tendency to form more compact structures. The results of the formation energy and the second derivative of binding energy with duster size show that besides N = 8, N =11 is also a magic number. Furthermore, it is the first time to find that the ground state of 11-atom clusters is a biplanar structure as same as the 13-atom cluster. The clear odd-even alternation as cluster size for the formation energy indicates the stability of electronic close shell existed in the range studied.展开更多
This paper discusses "geometric property(T)". This is a property of metric spaces introduced in earlier works of the authors for its applications to K-theory. Geometric property(T) is a strong form of "...This paper discusses "geometric property(T)". This is a property of metric spaces introduced in earlier works of the authors for its applications to K-theory. Geometric property(T) is a strong form of "expansion property", in particular, for a sequence(Xn)of bounded degree finite graphs, it is strictly stronger than(Xn) being an expander in the sense that the Cheeger constants h(Xn) are bounded below.In this paper, the authors show that geometric property(T) is a coarse invariant,i.e., it depends only on the large-scale geometry of a metric space X. The authors also discuss how geometric property(T) interacts with amenability, property(T) for groups,and coarse geometric notions of a-T-menability. In particular, it is shown that property(T) for a residually finite group is characterised by geometric property(T) for its finite quotients.展开更多
文摘A systematic study on the structures and electronic properties of copper clusters has been performed using the density functional theory. In the calculation, there are many isomers near the ground state for small copper clusters. Our results show that the three-dimensional isomers of copper clusters start from Cu7 cluster and then show a tendency to form more compact structures. The results of the formation energy and the second derivative of binding energy with duster size show that besides N = 8, N =11 is also a magic number. Furthermore, it is the first time to find that the ground state of 11-atom clusters is a biplanar structure as same as the 13-atom cluster. The clear odd-even alternation as cluster size for the formation energy indicates the stability of electronic close shell existed in the range studied.
基金supported by the U.S.National Science Foundation(Nos.DMS1229939,DMS1342083,DMS1362772)
文摘This paper discusses "geometric property(T)". This is a property of metric spaces introduced in earlier works of the authors for its applications to K-theory. Geometric property(T) is a strong form of "expansion property", in particular, for a sequence(Xn)of bounded degree finite graphs, it is strictly stronger than(Xn) being an expander in the sense that the Cheeger constants h(Xn) are bounded below.In this paper, the authors show that geometric property(T) is a coarse invariant,i.e., it depends only on the large-scale geometry of a metric space X. The authors also discuss how geometric property(T) interacts with amenability, property(T) for groups,and coarse geometric notions of a-T-menability. In particular, it is shown that property(T) for a residually finite group is characterised by geometric property(T) for its finite quotients.
