The structures of complexes [MⅡ2Cl4L2] and [MⅢ2Cl7L]- （M = Mo, Re; L = Ph2Ppy, （Ph2P）2py） were calculated by using density functional theory （DFT） PBE0 method. Based on the optimized geometries, the natural bo...The structures of complexes [MⅡ2Cl4L2] and [MⅢ2Cl7L]- （M = Mo, Re; L = Ph2Ppy, （Ph2P）2py） were calculated by using density functional theory （DFT） PBE0 method. Based on the optimized geometries, the natural bond orbital （NBO） analyses were carried out to study the nature of Re-Re and Mo-Mo bonds. The conclusions are as follows： the M-M distances in two-Ph2Ppy or （Ph2P）2py complexes [MⅡ2Cl4L2] are shorter than those in mono-Ph2Ppy or （Ph2P）2py complexes [MⅢ2Cl7L]- due to the double bridged N-C-P interactions. For singlet of all complexes, there is ReⅢ-ReⅢ or MoⅡ-MoⅡ quadruply bond in complex [Re2Cl7L]- or [Mo2Cl4L2], while only ReⅡ-ReⅡ or MoⅢ-MoⅢ triply bond in complex [Re2Cl4L2] or [Mo2Cl7L]-. The most stable spin state of 2 and 6, triplet, only contains triple ReⅢ-ReⅢ bond. Because the LPCl → BD＊Re-Re delocalizations weaken the Re-Re bond, the distance of ReⅢ-ReⅢ quadruple bonds in [Re2Cl7L]- is slightly longer than that of ReⅡ-ReⅡ triple bonds in [Re2Cl4L2]. Moreover, due to the delocalizations from the lone pair electrons of the remaining P’ atom to the M-M antibonding orbitals, the M-M distance in （Ph2P）2py complexes is slightly longer than that in Ph2Ppy complexes.展开更多
It is known that any m-Möbius transformation is an ordinary Möbius transformation in every one of its variables when the other variables do not take the values a and 1/a, where a is a parameter defin...It is known that any m-Möbius transformation is an ordinary Möbius transformation in every one of its variables when the other variables do not take the values a and 1/a, where a is a parameter defining the respective m-Möbius transformation. For ordinary Möbius transformations having distinct fixed points, the multiplier associated with one of these points completely characterizes the nature of that transformation, i.e. it tells us if it is elliptic, hyperbolic or loxodromic. The purpose of this paper is to show that fixed points exist also for m-Möbius transformations and multipliers associated with them can be computed as well. As in the classical case, the values of those multipliers describe completely the nature of the transformations. The method we used was that of a thorough study of the coefficients of the variables involved, with which occasion we discovered surprising symmetries. These were the results allowing us to prove the main theorem regarding the fixed points of a m-Möbius transformation, which is the key to further developments. Finally we were able to illustrate the geometric aspects of these transformations, making the whole theory as intuitive as possible. It was as opening a window into a space of several complex variables. This allows us to prove that if a bi-Möbius transformation is elliptic or hyperbolic in z<sub>1</sub> at a point z<sub>2</sub> it will remain the same on a circle or line passing through z<sub>2</sub>. This property remains true when we switch z<sub>1</sub> and z<sub>2</sub>. The main theorem, dealing with the fixed points of an arbitrary m-Möbius transformation made possible the extension of this result to these transformations.展开更多
基金the Natural Science Foundation of Guangdong Province (9151063101000037)Ministry of Education and Guangdong Province (2010B090400184)+2 种基金Program of Talent Introduction of Guangdong Province (C10133)Science and Technology Program of Guangzhou City (2011J4300063)Science and Technology Program of Guangdong Province (2010B060900007)
文摘The structures of complexes [MⅡ2Cl4L2] and [MⅢ2Cl7L]- （M = Mo, Re; L = Ph2Ppy, （Ph2P）2py） were calculated by using density functional theory （DFT） PBE0 method. Based on the optimized geometries, the natural bond orbital （NBO） analyses were carried out to study the nature of Re-Re and Mo-Mo bonds. The conclusions are as follows： the M-M distances in two-Ph2Ppy or （Ph2P）2py complexes [MⅡ2Cl4L2] are shorter than those in mono-Ph2Ppy or （Ph2P）2py complexes [MⅢ2Cl7L]- due to the double bridged N-C-P interactions. For singlet of all complexes, there is ReⅢ-ReⅢ or MoⅡ-MoⅡ quadruply bond in complex [Re2Cl7L]- or [Mo2Cl4L2], while only ReⅡ-ReⅡ or MoⅢ-MoⅢ triply bond in complex [Re2Cl4L2] or [Mo2Cl7L]-. The most stable spin state of 2 and 6, triplet, only contains triple ReⅢ-ReⅢ bond. Because the LPCl → BD＊Re-Re delocalizations weaken the Re-Re bond, the distance of ReⅢ-ReⅢ quadruple bonds in [Re2Cl7L]- is slightly longer than that of ReⅡ-ReⅡ triple bonds in [Re2Cl4L2]. Moreover, due to the delocalizations from the lone pair electrons of the remaining P’ atom to the M-M antibonding orbitals, the M-M distance in （Ph2P）2py complexes is slightly longer than that in Ph2Ppy complexes.
文摘It is known that any m-Möbius transformation is an ordinary Möbius transformation in every one of its variables when the other variables do not take the values a and 1/a, where a is a parameter defining the respective m-Möbius transformation. For ordinary Möbius transformations having distinct fixed points, the multiplier associated with one of these points completely characterizes the nature of that transformation, i.e. it tells us if it is elliptic, hyperbolic or loxodromic. The purpose of this paper is to show that fixed points exist also for m-Möbius transformations and multipliers associated with them can be computed as well. As in the classical case, the values of those multipliers describe completely the nature of the transformations. The method we used was that of a thorough study of the coefficients of the variables involved, with which occasion we discovered surprising symmetries. These were the results allowing us to prove the main theorem regarding the fixed points of a m-Möbius transformation, which is the key to further developments. Finally we were able to illustrate the geometric aspects of these transformations, making the whole theory as intuitive as possible. It was as opening a window into a space of several complex variables. This allows us to prove that if a bi-Möbius transformation is elliptic or hyperbolic in z<sub>1</sub> at a point z<sub>2</sub> it will remain the same on a circle or line passing through z<sub>2</sub>. This property remains true when we switch z<sub>1</sub> and z<sub>2</sub>. The main theorem, dealing with the fixed points of an arbitrary m-Möbius transformation made possible the extension of this result to these transformations.
