Linear Complexes

2009 ◽  
pp. 159-194
Author(s):  
Helmut Pottmann ◽  
Johannes Wallner
Keyword(s):  
Linear Complexes

scholarly journals Influence of Copper(I) Halides on the Reactivity of Aliphatic Carbodiimides

2020 ◽  
Vol 3 (1) ◽  
pp. 20
Author(s):  
Valentina Ferraro ◽  
Marco Bortoluzzi
Keyword(s):  
Density Functional Theory ◽  
Electronic Structure ◽  
General Formula ◽  
Electron Density ◽  
Density Functional ◽  
Functional Theory ◽  
Fukui Functions ◽  
Bond Lengths ◽  
Linear Complexes ◽  
Nitrogen Bond

The influence of copper(I) halides CuX (X = Cl, Br, I) on the electronic structure of N,N′-diisopropylcarbodiimide (DICDI) and N,N′-dicyclohexylcarbodiimide (DCC) was investigated by means of computational DFT (density functional theory) methods. The coordination of the considered carbodiimides occurs by one of the nitrogen atoms, with the formation of linear complexes having a general formula of [CuX(carbodiimide)]. Besides varying the carbon–nitrogen bond lengths, the thermodynamically favourable interaction with Cu(I) reduces the electron density on the carbodiimides and alters the energies of the (NCN)-centred, unoccupied orbitals. A small dependence of these effects on the choice of the halide was observable. The computed Fukui functions suggested negligible interaction of Cu(I) with incoming nucleophiles, and the reactivity of carbodiimides was altered by coordination mainly because of the increased electrophilicity of the {NCN} fragments.

scholarly journals XVII. On a new geometry of space

1865 ◽  
Vol 155 ◽  
pp. 725-791 ◽  
Keyword(s):  
Analytical Geometry ◽  
Infinite Space ◽  
Linear Complexes

I. On Linear Complexes of Right Lines . 1. Infinite space may be considered either as consisting of points or transversed by planes. The points, in the first conception, are determined by their coordinates, by x, y, z for instance, taken in the ordinary signification; the planes, in the second conception, are determined in an analogous way by their coordinates, introduced by myself into analytical geometry, by t, u, v for instance. The equation tx + uy + vz + 1 = 0 represents, in regarding x, y, z as variable and t, u, v as constant, a plane by means of its points. The three constants t, u, v are the coordinates of this plane. The same equation, in regarding t, u, v as variable, x, y, z as constant, represents a point by means of planes passing through it. The three constants are the coordinates of the point.

Geometrical Properties of Three Symmetric Matrices

1935 ◽  
Vol 31 (2) ◽  
pp. 174-182 ◽  
Author(s):  
H. W. Turnbull
Keyword(s):  
Explicit Form ◽  
Three Dimensions ◽  
Symmetric Matrices ◽  
Geometrical Properties ◽  
Cubic Surface ◽  
Quadric Surfaces ◽  
Quartic Curve ◽  
Polar Property ◽  
Linear Complexes ◽  
Image Position

In the early editions of the Geometry of Three Dimensions Salmon had stated that the equations of any three quadric surfaces could be simultaneously reduced to the sums of five squares. Such a reduction is not possible in general, but can be performed if and only if a certain combinant Λ, of the net of quadrics, vanishes. Algebraically the theory of such a net of quadrics is equivalent, as Hesse(2) showed, to that of a plane quartic curve: and the condition for the equation a quartic to be expressible to the sum of five fourth powers is equivalent to the condition Λ = 0(1). While Clebsch(3) was the first to establish this condition, Lüroth(4) gave it more explicit form by studying the quartic curvewhich satisfies the condition. Frahm(5) seems to have been the first to prove the impossibility of the above reduction of three general quadric surfaces, by remarking that the plane quartic curve obtained in Hesse's way from the locus of the vertices of cones of the net of quadrics would be a Lüroth quartic. Frahm further remarked that the three quadrics, so conditioned, could be regarded as the polar quadrics belonging to a cubic surface in ∞2 ways; but that for three general quadrics no such cubic surface exists. An explicit algebraical account of these properties was given by E. Toeplitz(6), who incidentally noticed that certain linear complexes associated with three general quadrics became special linear complexes when Λ = 0. This polar property of three quadrics in [3] was generalized to n dimensions by Anderson (7).

Ab initio research on third-order nonlinear optical properties of linear complexes [M(I)(PH3)2]+(M=Cu, Ag, Au)

2003 ◽  
Vol 137 (1-3) ◽  
pp. 1523-1524 ◽  
Author(s):  
Y.Q. Qiu ◽  
Z.M. Su ◽  
L.K. Yan ◽  
Y. Liao ◽  
M. Zhang ◽  
...  
Keyword(s):  
Optical Properties ◽  
Ab Initio ◽  
Nonlinear Optical ◽  
Nonlinear Optical Properties ◽  
Third Order ◽  
Linear Complexes
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