The full-parameter geometry optimization of cationic （ S ） - BINAP - Ru （ Ⅱ ） halide complex was performed by DFT method using B3LYP, PW91 and PBE potentials with several basis sets. PW91 with 3-21G / SDD basis s...The full-parameter geometry optimization of cationic （ S ） - BINAP - Ru （ Ⅱ ） halide complex was performed by DFT method using B3LYP, PW91 and PBE potentials with several basis sets. PW91 with 3-21G / SDD basis sets is found to be the most suitable method with consideration of both precision and efficiency. The dihedral angles （ θ ） of the binaphthyl or biphenyl with different phosphorus ligand - Ru （ Ⅱ ） halide complexes were found changing from 59.9 to 79.3 degree, while the natural bite angle （ βn ） of those complexes only changes from 87.4 to 90.3 degree. It is different from the common view of asymmetric organic chemists＇ that θ directly influences βn.展开更多
Necessary and sufficient conditions are studied that a bounded operator Tx =（x1^＊x, x2^＊x,…） on the space e∞, where xn^＊∈e∞^＊, is lower or upper semi-Fredholm; in particular, topological properties of the se...Necessary and sufficient conditions are studied that a bounded operator Tx =（x1^＊x, x2^＊x,…） on the space e∞, where xn^＊∈e∞^＊, is lower or upper semi-Fredholm; in particular, topological properties of the set {x1^＊, x2^＊ …} are investigated. Various estimates of the defect d（T） = codim R（T）, where R（T） is the range of T, are given. The case of xn^＊ = dnxtn^＊,where dn ∈ R and xtn^＊ 〉 0 are extreme points of the unit ball Be∞^＊, that is, tn ∈ βN, is considered. In terms of the sequence {tn}, the conditions of the closedness of the range R（T） are given and the value d（T） is calculated. For example, the condition {n ： 0 〈 ｜da｜ 〈 δ}= θ for some 5 is sufficient and if for large n points tn are isolated elements of the sequence {tn}, then it is also necessary for the closedness of R（T） （tn0 is isolated if there is a neighborhood U of tno satisfying tn ∈ U for all n ≠ n0）. If {n ： ｜dn｜ 〈 δ} = θ, then d（T） is equal to the defect δ{tn} of {tn}. It is shown that if d（T） = ∞ and R（T） is closed, then there exists a sequence {An} of pairwise disjoint subsets of N satisfying XAn ∈ R（T）.展开更多
基金the financial support by the National Natural Science Foundation of China(No.20373014).
文摘The full-parameter geometry optimization of cationic （ S ） - BINAP - Ru （ Ⅱ ） halide complex was performed by DFT method using B3LYP, PW91 and PBE potentials with several basis sets. PW91 with 3-21G / SDD basis sets is found to be the most suitable method with consideration of both precision and efficiency. The dihedral angles （ θ ） of the binaphthyl or biphenyl with different phosphorus ligand - Ru （ Ⅱ ） halide complexes were found changing from 59.9 to 79.3 degree, while the natural bite angle （ βn ） of those complexes only changes from 87.4 to 90.3 degree. It is different from the common view of asymmetric organic chemists＇ that θ directly influences βn.
文摘Necessary and sufficient conditions are studied that a bounded operator Tx =（x1^＊x, x2^＊x,…） on the space e∞, where xn^＊∈e∞^＊, is lower or upper semi-Fredholm; in particular, topological properties of the set {x1^＊, x2^＊ …} are investigated. Various estimates of the defect d（T） = codim R（T）, where R（T） is the range of T, are given. The case of xn^＊ = dnxtn^＊,where dn ∈ R and xtn^＊ 〉 0 are extreme points of the unit ball Be∞^＊, that is, tn ∈ βN, is considered. In terms of the sequence {tn}, the conditions of the closedness of the range R（T） are given and the value d（T） is calculated. For example, the condition {n ： 0 〈 ｜da｜ 〈 δ}= θ for some 5 is sufficient and if for large n points tn are isolated elements of the sequence {tn}, then it is also necessary for the closedness of R（T） （tn0 is isolated if there is a neighborhood U of tno satisfying tn ∈ U for all n ≠ n0）. If {n ： ｜dn｜ 〈 δ} = θ, then d（T） is equal to the defect δ{tn} of {tn}. It is shown that if d（T） = ∞ and R（T） is closed, then there exists a sequence {An} of pairwise disjoint subsets of N satisfying XAn ∈ R（T）.
