The semi-empirical INDO method was used to study the electronic structures and the spectra of all of the 34 possible isomers of C78O based on C78 with group C2v. This calculation can simulate positions of an additiona...The semi-empirical INDO method was used to study the electronic structures and the spectra of all of the 34 possible isomers of C78O based on C78 with group C2v. This calculation can simulate positions of an additional oxygen atom in C78 and predict the spectroscopic characteristics of the isomers. The most stable geometry of C78O is the 73,78-C78O molecule with an epoxide structure. The added 73,78-bond is located between two hexagons （6-6） and is intersected by the shortest C2 axis in C78 with group C2v. Atomic orbitals of the oxygen atom play an important role in lowering HOMO energy of 73,78-C78O. Compared with C78 with group C2v, the blue-shift in the electronic absorption spectrum for 73,78-C78O was observed. The reason of the blue-shift effect was discussed, and the electronic transitions were assigned based on the theoretical calculations.展开更多
Eighteen possible isomers of C78（CH2）2 weTe investigated by the INDO method. It was indicated that the most stable isomer was 42,43,62,63-C78（CH2）2, where the -CH2 groups were added to the 6/6 bonds located at the...Eighteen possible isomers of C78（CH2）2 weTe investigated by the INDO method. It was indicated that the most stable isomer was 42,43,62,63-C78（CH2）2, where the -CH2 groups were added to the 6/6 bonds located at the same hexagon passed by the longest axis of C78 （C2v）, to form cyclopropane structures. Based on the most stable four geometries of C78（CH2）2 optimized at B3LYP/3-21G level, the first absorptions in the electronic spectra calculated with the INDO/CIS method and the IR frequencies of the C-C bonds on the carbon cage computed using the AM1 method were blue-shifted compared with those of C78 （C2v） because of the bigger LUMO-HOMO energy gap and the less conjugated carbon cage after the addition. The chemical shifts of ^13C NMR for the carbon atoms on the added bonds calculated at B3LYP/3-21G level were moved upfield thanks to the conversion from sp^2-C to sp^3-C.展开更多
Density functional theory （DFT） was applied at the B3LYP/6-31G^＊ level to investigate the relative stability of the five fullerene isomers as well as the anions of C78. Full geometry optimization was carried out an...Density functional theory （DFT） was applied at the B3LYP/6-31G^＊ level to investigate the relative stability of the five fullerene isomers as well as the anions of C78. Full geometry optimization was carried out and distributions of electrostatic potential were calculated. The results showed that the C78（D3h′） cage was the most stable for C78^q-（ q = 2, 4, 6）, and the potential minimum Vmin（r） inside the sphere of C78（Dzh′） was the biggest one among the five isomers of C78. So C78（Dzh′） were electrons from the scandium atoms that were easier to accept than the other four isomers. On the other hand, MEP maps inside the sphere of C78（Dzh′） had three minima near the three vertexes. Each vertex was formed by the intersection of the three hexagons. Our results allowed a possible explanation for the bonding between the scandium atoms and the fullerene cage.展开更多
Structures and stabilities of 22 possible isomers for C78O2 based on C78（C2v） were studied by INDO series of methods. The most stable geometry is that two oxygen atoms are added to 1,2-bond crossed by the shortest z...Structures and stabilities of 22 possible isomers for C78O2 based on C78（C2v） were studied by INDO series of methods. The most stable geometry is that two oxygen atoms are added to 1,2-bond crossed by the shortest z axis and 32,35-bond near the longest x axis in C78（C2v）,in agreement with the experiment of adducts for C78（C2v）. 1,2,32,35-C78O2 is more stable than 1,2,19,22-C78O2 by 0.12 kcal/mol and epoxide structures are formed in each isomer. Electronic spectra for C78O2 were investigated with INDO/CIS method. The reason of the blue-shift for UV absorptions of 1,2,32,35-C78O2 compared with that of C78（C2v） was discussed and electronic transition was theoretically assigned.展开更多
The structures and spectra of 20 possible isomers of C78（CH2）3 have been studied by using AMI, INDO/CIS and DFT methods. The results show that the most stable isomer is 1,2,3,4,5,6-C78（CH2）3 （A） with annulene st...The structures and spectra of 20 possible isomers of C78（CH2）3 have been studied by using AMI, INDO/CIS and DFT methods. The results show that the most stable isomer is 1,2,3,4,5,6-C78（CH2）3 （A） with annulene structures, where three -CH2 groups are added to the 6/6 bonds located at the same hexagon passed by the shortest axis of C78 （C2v）. Compared with that of C78 （C2v）, the first absorption in the electronic spectrum of C78（CH2）3 （A） iS blue-shifted because of its wider LUMO-HOMO energy gap. While the IR frequencies of the C-C bonds on the carbon cage are red-shifted owing to the formation of annulene structures and the extension of the conjugated system. The chemical shifts of the carbon atoms in 13C NMR spectra are moved upfield upon the addition.展开更多
For any vertex u ? V(G), let T N (u) = {u} ∪ {uυ|uυ ? E(G), υ ? υ(G)} ∪ {υ ? υ(G)|uυ ? E(G) and let f be a total k-coloring of G. The total-color neighbor of a vertex u of G is the color set C f(u) = {f(x) | ...For any vertex u ? V(G), let T N (u) = {u} ∪ {uυ|uυ ? E(G), υ ? υ(G)} ∪ {υ ? υ(G)|uυ ? E(G) and let f be a total k-coloring of G. The total-color neighbor of a vertex u of G is the color set C f(u) = {f(x) | x ? T N (u)}. For any two adjacent vertices x and y of V(G) such that C f(x) ≠ C f(y), we refer to f as a k-avsdt-coloring of G (“avsdt” is the abbreviation of “ adjacent-vertex-strongly-distinguishing total”). The avsdt-coloring number of G, denoted by χast(G), is the minimal number of colors required for a avsdt-coloring of G. In this paper, the avsdt-coloring numbers on some familiar graphs are studied, such as paths, cycles, complete graphs, complete bipartite graphs and so on. We prove Δ(G) + 1 ? χast(G) ? Δ(G) + 2 for any tree or unique cycle graph G.展开更多
文摘The semi-empirical INDO method was used to study the electronic structures and the spectra of all of the 34 possible isomers of C78O based on C78 with group C2v. This calculation can simulate positions of an additional oxygen atom in C78 and predict the spectroscopic characteristics of the isomers. The most stable geometry of C78O is the 73,78-C78O molecule with an epoxide structure. The added 73,78-bond is located between two hexagons （6-6） and is intersected by the shortest C2 axis in C78 with group C2v. Atomic orbitals of the oxygen atom play an important role in lowering HOMO energy of 73,78-C78O. Compared with C78 with group C2v, the blue-shift in the electronic absorption spectrum for 73,78-C78O was observed. The reason of the blue-shift effect was discussed, and the electronic transitions were assigned based on the theoretical calculations.
文摘Eighteen possible isomers of C78（CH2）2 weTe investigated by the INDO method. It was indicated that the most stable isomer was 42,43,62,63-C78（CH2）2, where the -CH2 groups were added to the 6/6 bonds located at the same hexagon passed by the longest axis of C78 （C2v）, to form cyclopropane structures. Based on the most stable four geometries of C78（CH2）2 optimized at B3LYP/3-21G level, the first absorptions in the electronic spectra calculated with the INDO/CIS method and the IR frequencies of the C-C bonds on the carbon cage computed using the AM1 method were blue-shifted compared with those of C78 （C2v） because of the bigger LUMO-HOMO energy gap and the less conjugated carbon cage after the addition. The chemical shifts of ^13C NMR for the carbon atoms on the added bonds calculated at B3LYP/3-21G level were moved upfield thanks to the conversion from sp^2-C to sp^3-C.
基金Project supported by the Department of Education of Liaoning Province (2024201057)
文摘Density functional theory （DFT） was applied at the B3LYP/6-31G^＊ level to investigate the relative stability of the five fullerene isomers as well as the anions of C78. Full geometry optimization was carried out and distributions of electrostatic potential were calculated. The results showed that the C78（D3h′） cage was the most stable for C78^q-（ q = 2, 4, 6）, and the potential minimum Vmin（r） inside the sphere of C78（Dzh′） was the biggest one among the five isomers of C78. So C78（Dzh′） were electrons from the scandium atoms that were easier to accept than the other four isomers. On the other hand, MEP maps inside the sphere of C78（Dzh′） had three minima near the three vertexes. Each vertex was formed by the intersection of the three hexagons. Our results allowed a possible explanation for the bonding between the scandium atoms and the fullerene cage.
文摘Structures and stabilities of 22 possible isomers for C78O2 based on C78（C2v） were studied by INDO series of methods. The most stable geometry is that two oxygen atoms are added to 1,2-bond crossed by the shortest z axis and 32,35-bond near the longest x axis in C78（C2v）,in agreement with the experiment of adducts for C78（C2v）. 1,2,32,35-C78O2 is more stable than 1,2,19,22-C78O2 by 0.12 kcal/mol and epoxide structures are formed in each isomer. Electronic spectra for C78O2 were investigated with INDO/CIS method. The reason of the blue-shift for UV absorptions of 1,2,32,35-C78O2 compared with that of C78（C2v） was discussed and electronic transition was theoretically assigned.
文摘The structures and spectra of 20 possible isomers of C78（CH2）3 have been studied by using AMI, INDO/CIS and DFT methods. The results show that the most stable isomer is 1,2,3,4,5,6-C78（CH2）3 （A） with annulene structures, where three -CH2 groups are added to the 6/6 bonds located at the same hexagon passed by the shortest axis of C78 （C2v）. Compared with that of C78 （C2v）, the first absorption in the electronic spectrum of C78（CH2）3 （A） iS blue-shifted because of its wider LUMO-HOMO energy gap. While the IR frequencies of the C-C bonds on the carbon cage are red-shifted owing to the formation of annulene structures and the extension of the conjugated system. The chemical shifts of the carbon atoms in 13C NMR spectra are moved upfield upon the addition.
基金the National Natural Science Foundation of China (Grant Nos. 10771091, 10661007)
文摘For any vertex u ? V(G), let T N (u) = {u} ∪ {uυ|uυ ? E(G), υ ? υ(G)} ∪ {υ ? υ(G)|uυ ? E(G) and let f be a total k-coloring of G. The total-color neighbor of a vertex u of G is the color set C f(u) = {f(x) | x ? T N (u)}. For any two adjacent vertices x and y of V(G) such that C f(x) ≠ C f(y), we refer to f as a k-avsdt-coloring of G (“avsdt” is the abbreviation of “ adjacent-vertex-strongly-distinguishing total”). The avsdt-coloring number of G, denoted by χast(G), is the minimal number of colors required for a avsdt-coloring of G. In this paper, the avsdt-coloring numbers on some familiar graphs are studied, such as paths, cycles, complete graphs, complete bipartite graphs and so on. We prove Δ(G) + 1 ? χast(G) ? Δ(G) + 2 for any tree or unique cycle graph G.
