Chemical applications of topology and group theory

The 60 even permutations of the ligands in the five-coordinate complexes, ML ₅, form the alternating group A ₅, which is isomorphic with the icosahedral pure rotation group I. Using this idea, it is shown how a regular icosahedron can be used as a topological representation for isomerizations of the five-coordinate complexes, ML ₅, involving only even permutations if the five ligands L correspond either to the five nested octahedra with vertices located at the midpoints of the 30 edges of the icosahedron or to the five regular tetrahedra with vertices located at the midpoints of the 20 faces of the icosahedron. However, the 120 total permutations of the ligands in five-coordinate complexes ML ₅ cannot be analogously represented by operations in the full icosahedral point group I _h, since I _his the direct product I×C₂ whereas the symmetric group S ₅ is only the semi-direct product A ₅S₂. In connection with previously used topological representations on isomerism in five-coordinate complexes, it is noted that the automorphism groups of the Petersen graph and the Desargues-Levi graph are isomorphic to the symmetric group S ₅ and to the direct product S ₅×S ₂, respectively. Applications to various fields of chemistry are briefly outlined.     

Multiple Bonds between Transition Metals and “Bare” Main Group Elements: Links between Inorganic Solid State Chemistry and Organometallic Chemistry

The last two decades have seen a dramatic development in the study of metal-metal multiple bonds, particular successes being recorded in the field of organometallic chemistry. Syntheses designed to produce novel transition metal complexes with single, double, triple and quadruple metal-metal bonds occupy a most important place in such research, as also do reactivity studies. A striving to establish general principles has provided much of the motivation for such work, but one less obvious goal—the commercial application of the catalytic properties of metal-metal multiple bonding systems, in the medium and long term—should not be overlooked. All aspects of the investigations of metal-metal multiple bonds also apply to a particular class of compound that has, however, enjoyed little lime-light and thus deserves the present review: complexes with multiple bonds between transition metals and substituent-free (“bare”) main group elements. Although based mostly on accidental discoveries, the few noteworthy examples are now beginning to unfold general concepts of synthesis that are capable of being extended and thus are deserving of exploitation in preparative chemistry. The availability of further structural patterns exhibiting multiple bonds between transition metals and ligand-free main group elements might enable preparative organometallic chemistry to expand in a completely new direction (for instance by the stabilizing or activation of small molecules at the metal complex). This essay discusses the chemistry of complexes of bare carbon, nitrogen, and oxygen ligands (carbido-, nitrido-, and oxo-complexes) and their relationships to higher homologues from both a synthetic and a structural point of view.     

Möbius function and characteristic monomials for combinatorial enumeration

After the definitions of amplified representations and number-theoretical vectors, the markaracter table of a cyclic subgroup is converted into the corresponding Q-conjugacy character table. The conversion is shown to necessitate an interconversion matrix that contains M?bius functions as elements. Since the interconversion matrix gives characteristic monomials for cyclic groups, all the powers appearing in each of the characteristic monomials are shown to be integers. Characteristic monomials for finite groups are then built up by starting from those of cyclic groups. This procedure clarifies the fact that all the powers appearing in each characteristic monomial for finite groups are integers. The relationship between characteristic monomial tables and unit-subduced-cycle-index tables is discussed with respect to their application to isomer enumeration. Received: 15 July 1998 / Accepted: 16 December 1998 / Published online: 16 March 1999     

基团的电子效应与单取代苯对位^1H,^13C的化学位移

基团的电子效应与单取代苯对位１Ｈ、１３Ｃ的化学位移＊韩长日冯娇杨钟照平（海南师范学院化学系海口５７１１５８）关键词１Ｈ的化学位移１３Ｃ的化学位移基团电负性共轭效应引言在１Ｈ、１３Ｃ核磁共振谱中，化学位移值的大小主要取决于屏蔽作用的大小，而屏蔽作用的大...     

Synthese und Kristallstruktur von Y(HSO4)3-I und Y(HSO4)3 · H2O

Syntheses and Crystal Structures of Y(HSO₄)₃-I and Y(HSO₄)₃ · H₂O Lath shaped crystals of Y(HSO₄)-I are obtained by treatment of Y₂O₃ with conc. sulfuric acid at 200 °C. Y(HSO₄)₃-I crystallizes orthorhombic (Pbca, Z = 8, a = 1201.5(1), b = 953.76(8), c = 1650.4(1) pm, R_all = 0.0388). In the crystal structure Y³⁺ is coordinated by eight monodentate HSO₄^– groups. Colorless, plate like single crystals of Y(HSO₄)₃ · H₂O grew from a solution of Y₂O₃ in 85% sulfuric acid upon cooling. In the crystal structure of the triclinic compound (P1, Z = 2, a = 679.8(1), b = 802.8(2), c = 965.9(2) pm, α = 79.99(2)°, β = 77.32(2)°, γ = 77.50(2)°, R_all = 0.0264) Y³⁺ is surrounded by seven HSO₄^– groups and one molecule of water.     

Die Deprotonierung silylierter Amido-Komplexe von Seltenerdelementen

Deprotonation Reactions of Silylated Amido Complexes of Rare Earth Elements The deprotonation of the rare earth element-tris(bistrimethylsilyl)amides Ln[N(SiMe₃)₂]₃ of scandium, ytterbium, and lutetium with sodium-bis(trimethylsilyl)amide in THF leads to the complexes [Na(THF)₃LnCH₂SiMe₂NSiMe₃{N(SiMe₃)₂}₂] [Ln = Sc ( 1 ), Yb ( 2 ), and Lu ( 3 )]. According to crystal structure analyses of 1 and 2 the metal atoms Sc and Yb are constituents of planar LnCSiN four-membered rings. At the same time, the C atom of the CH₂ group is coordinated with the sodium ion in a linear axis Ln–C–Na; the sodium ion obtains a distorted tetrahedral arrangement by three THF molecules. The equatorial positions of the methylene-C atom, which is coordinated in a trigonal bipyramidal fashion, are occupied by the two H atoms and the Si atom of the four-membered ring. 2.6-dimethylbenzoisonitrile can be inserted into the Yb–CH₂ bond of 2 and the new five-membered heterocylce YbNCSiN originates, the exocyclic CH₂ group of which enters into a C–C coupling with the centrosymmetric dimer 4 while the ytterbium undergoes reduction. At the same time, sodium-7-methyl indolate is formed, which together with [NaN(SiMe₃)₂(THF)₂] forms the centrosymmetric dimeric molecular aggregate [NaN(SiMe₃)₂(THF)₂Na(C₉H₁₆N)]₂ ( 5 ). 1 : Space group P2₁/n, Z = 8, lattice dimensions at –80 °C: a = 2941.4(2), b = 1205.5(1), c = 2952.4(3) pm; β = 113.455(8)°; R₁ = 0.0625. 2 : Space group P2₁/n, Z = 8, lattice dimensions at –80 °C: a = 2943.9(1), b = 1219.5(1), c = 2944.3(1) pm; β = 113.372(4)°; R₁ = 0.0361. 4 : Space group P 1, Z = 4, lattice dimensions at –80 °C: a = 1117.0(1), b = 1207.5(1), c = 1614.3(2) pm; α = 73.634(10)°, β = 82.091(10)°, γ = 74.391(10)°; R₁ = 0.0525. 5 : Space group P2₁/n, Z = 2, lattice dimensions at –80 °C: a = 1126.7(1), b = 1459.3(1), c = 1741.1(1) pm; β = 96.461(8)°; R₁ = 0.0458. Quantum chemical DFT calculations of the scandium model compound [Na(Me₂O)₃ScCH₂SiMe₂NSiH₃{N(SiH₃)₂}₂] ( 1 M ) give a very large negative charge at the pentacoordinated carbon atom of the four-membered ring that is concentrated in a lone-pair orbital which has mainly p character. The carbon atom interacts with the positively charged scandium atom mainly by Coulombic interactions.     

New Possible Applications of Heavy Main-Group Elements in Organic Synthesis

Aside from elements of the 2nd row, and one element of the 3rd row of the periodic system—Si, P, S, and Se, respectively, whose organoelement groups such as Me₃Si and Ph₃P^⊕ have proven useful in numerous organic syntheses—other elements of the 3rd as well as 4th and 5th row (Ge, As, Sn, Sb, Te, Pb, Bi) can also be used as components of synthetically useful organoelement groups, the elements As, Sn, and Pb, in particular, offering certain advntages over the others. Some of these organoelement groups are suitable equivalents for Li- or halogen-substituents attached to carbon; they stabilize carbanionic centers (minimum of this effect at the 3rd-row elements), and owing to their suitability as leaving groups in β-eliminations, also open up interesting synthetic possibilities. The thermally unduced syn- and silica-gel induced anti-elimination of Ph₃Sn, Ph₂Sb, Ph₃Pb, together with β-OH, are novel. With the newly synthesized compounds Ph_nEl—Ch₂—Li (El = Sn, Pb, As, Sb, Bi) and other α- and β-lithiated R_nEl- and Ph₂As(O)-reagents such organoelement groups can be introduced into organic compounds and exploited in organic and organoelement synthesis.     

Tame characters and ramification of finite flat group schemes

In this paper, for a complete discrete valuation field K of mixed characteristic (0,p) and a finite flat group scheme G of p-power order over OK, we determine the tame characters appearing in the Galois representation in terms of the ramification theory of Abbes and Saito, without any restriction on the absolute ramification index of K or the embedding dimension of G.     

Large orbit sizes in finite group actions

In this paper, we study the relations of the sizes of various sections of finite linear groups and the largest orbit size of the linear group actions. We also study various applications of those orbit theorems.