Geometrical equivalence

The geometrical equivalent of the deflation operation is a projection of a pattern of points into a subspace which is orthogonal to v,. If all elements of the residual... 

One learns from these molecular complexes that equivalent synthons can lead to virtually identical crystal structures. Synthons in, V and VI are chemically and geometrically equivalent though they originate from different molecules, a nitrile, an N-oxide and a nitro compound. These three synthons are used in crystal design in almost the same way. So, different molecules may yield similar crystal structures if they are capable of forming equivalent synthons. This is a powerful concept because it establishes a many-to-one correspondence between molecular and crystal structures. 

Gv( f) covering symmetry67. For orientations of B0 in the mirror plane S, the symmetry group of the spin Hamiltonian is < 9f = C2h(e2f). The direct product base of the nuclear spin functions of two geometrically equivalent nuclei reduces to two classes, containing six A-type and three B-type functions, respectively. Second order perturbation theory applied to H = UtHU, where U symmetrizes the base functions of the Hamil-... 

It should be noted that for geometrically equivalent nuclei a complicated ENDOR spectrum may be observed for arbitrary orientations of B0, if the hfs tensors are nearly... 

EI-EPR has also been applied in powder samples, where single crystal-like EPR spectra can be obtained for B0 observer fields which correspond to the orientations of principal values of ligand hfs tensors37. This is illustrated in Fig. 15 for a powder sample of dibenzene vanadium diluted into polycrystalline ferrocene. The hfs tensors of the twelve geometrically equivalent benzene protons are not coaxial with the g and the... 

In the corresponding copper complex all eight protons are geometrically equivalent within error limits. The metal-proton distances in Ag(TPP) calculated by the classical... 

Six proton tensors have been analyzed independently. The direction cosines indicate that the protons are geometrically equivalent in pairs, as expected from the crystal struc-... 

Working hypothesis that the chain monomeric units are geometrically equivalent [6]. 

The process of reflection by the real lattice cannot be visualized in terms of the reciprocal lattice but the condition for reflection by the real lattice (the Bragg equation) naturally has its jjrecise geometrical equivalent in terms of the reciprocal lattice. This is illustrated in Fig. 81, in which X Y represents the orientation of a set of crystal planes which we will suppose is in a reflecting position. Along the normal to this... 

The inequality of the apical and equatorial P-Cl distances in phosphorus pentachloride is not cast in doubt by this argument, since these directions are not geometrically equivalent even for equality of the dis-ances. 

Consider the planar methyl radical, H3C, which has three geometrically equivalent protons. Their spins combine to produce four distinct values for the local field ... 

The acetylacetone carbanion undergoes addition to the imine carbon atom of complex (92) but subsequent cyclization and deacylation processes occur (Scheme 42).212 The apical ammonia group is the most acidic and consequently is favoured to cyclize on to the carbonyl group. In the bis-(1,2-diaminoethane) complex related to the imine chelate (92), the two apical nitrogen atoms are no longer geometrically equivalent. Thus two products are formed when hydrogen cyanide is added to the complex and subsequent cyclization takes place (Scheme 43).213 However, the cyclization reaction is stereoselective. 

The axis of this macromolecule (i.e., the helix) is parallel to a crystallographic axis and all the monomer units occupy geometrically equivalent positions in relation to this axis. 

The existence of only one kind of CH2CI2 molecule means that all four positions surrounding the carbon atom are geometrically equivalent, which requires a tetrahedral coordination geometry. Unfortunately, this fact can only really be made convincing by inspecting a three-dimensional mechanical model of the molecule. 

There are well known examples of 6-coordinate central atoms with 1,2, and 4 lone pairs. Since the corners of an octahedron are geometrically equivalent, the distinction between axial and equatorial atoms that we had to consider in the 5-coordinate case no longer exists thus the horizontal rectangles you see in the sketches below have no special significance and could just as well be placed in either of the two possible vertical planes. 

We assume here that since two geometrically equivalent bonds are formed by each carbon, this atom must be sp-hybridized in acetylene. On each carbon, one sp hybrid bonds to a hydrogen and the other bonds to the other carbon atom, forming the c bond skeleton of the molecule. 

Using Equation 1.12, or its geometrical equivalent, calculate the coefficients for the frontier orbitals of the heptatrienyl cation. 

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