Octant symmetry

In cis -fused decalones, two different conformations are possible, as in the steroid (f) and nonsteroid (g) conformers of Table 4. The octant rule predicts a strong negative Cotton effect for (f) and a strong positive Cotton effect for (g). Assuming chair conformations, nearly all of the ring atoms of (f) lie in octant symmetry planes (carbons 1, 2, 4, 7, 10 and the angular methyl) or have counterparts across the octant planes (carbons 3 and 5,... 

The unit cell of cubic diamond corresponds to a face-centered packing of carbon atoms. Aside from the four C atoms in the vertices and face centers, four more atoms are present in the centers of four of the eight octants of the unit cell. Since every octant is a cube having four of its eight vertices occupied by C atoms, an exact tetrahedral coordination results for the atom in the center of the octant. The same also applies to all other atoms — they are all symmetry-equivalent. In the center of every C-C bond there is an inversion center. As in alkanes the C-C bonds have a length of 154 pm and the bond angles are 109.47°. 

The shape of the third octant surface, XY, is of importance only in cases where perturbers lie in front of the carbonyl oxygen. The substituents lying in symmetry planes XZ and YZ may in fact give very small, but nonzero, contributions, as in chiral molecules the C = 0 symmetry-derived octant surfaces are distorted from planarity11. 

Cyclopentanones and hexahydrindanones are represented by octant projections of their twist form (C2 symmetry) (Figure 4) or an asymmetric envelope form. 

Although we considered only the octant with positive jc, y, and z, it follows from the form of (5.30) and (5.31) that they are the potentials in the neighboring octant with z negative. Moreover, the eightfold degeneracy of the ellipsoidal coordinates implies that the conditions (5.20) are satisfied. Thus, (5.30) and (5.31) give the potential at all points in space this fortunate result is a consequence of the fact that the particle has the same symmetry as the ellipsoidal coordinates. For particles with less symmetry we would have to attack the problem octant by octant. 

In magnetic resonance spectroscopy the octant problem does not arise, although ambiguity in assigning each symmetry-related tensor to the appropriate symmetry-related reaction site persists. 

FIGURE 5. Octant diagram for chiral olefins showing (a) intersecting symmetry planes xy, yz, xz, and (b) the corresponding front and rear octants viewed along the x-axix... 

Cristobalite can be represented by a face-centered cubic (ccp) arrangement of Si atoms in the Zn positions of the 3 2PT structure of ZnS with SiC>4 tetrahedra in four octants of the cube. This representation emphasizes the high symmetry of (3-cristobalite, the highest symmetry of the polymorphs of silica. 

To describe the contents of a unit cell, it is sufficient to specify the coordinates of only one atom in each equivalent set of atoms, since the other atomic positions in the set are readily deduced from space group symmetry. The collection of symmetry-independent atoms in the unit cell is called the asymmetric unit of the crystal structure. In the International Tables, a portion of the unit cell (and hence its contents) is designated as the asymmetric unit. For instance, in space group P2 /c, a quarter of the unit cell within the boundaries 0asymmetric unit. Note that the asymmetric unit may be chosen in different ways in practice, it is preferable to choose independent atoms that are connected to form a complete molecule or a molecular fragment. It is also advisable, whenever possible, to take atoms whose fractional coordinates are positive and lie within or close to the octant 0 < x < 1/2,0 < y < 1/2, and 0 < z < 1 /2. Note also that if a molecule constitutes the asymmetric unit, its component atoms may be related by non-crystallographic symmetry. In other words, the symmetry of the site at which the molecule is located may be a subgroup of the idealized molecular point group. 

Figure 7. (Left) Classical octant rule diagram (ref 1) for the ketone carbonyl n - x transition. Local symmetry-derived, orthogonal octant planes XZ and YZ divide all space into quadrants, and a non-symmetry-derived third nodal surface (A) is approximated by an orthogonal plane bisecting the C=0 bond. "Front" octants are those nearer an observer along the +Z axis, while "back" octants lie towards -Z. (Middle) Octant contribution signs that perturbers make in back and front octants. (Right) Revised octant rule [6] with octant planes XZ and YZ unchanged and the third nodal surface defined theoretically as a concave surface (B). [Reprinted with permission from ref. 7. Copyright ° 1986 American Chemical Society.]...

On the lowest level, using symmetry and nodal properties of the MOs of H2C=0 as a representative of the inherently symmetric carbonyl chromophore, an octant rule can be derived for the influence of a (static) perturber of the (n, n ) excitation The geometrical symmetry (C2v) of the H2C=0 unit with the two symmetry planes, yz and xz, leads to a quadrant rule. If the orbitals involved in the (n, n ) excitation of ketones are described as " = 2p and n = N 2p — 2pJ) (N being a normalization constant), the nodal plane (xy) of the virtual orbital generates the octant diagram for the contributions of substituents of chirally perturbed compounds (Figure 10). 

If viewed from the oxygen, most optically active carbonyl compounds have their substituents only in the rear octants. The appearance of the plane that separates the rear octants from the front octants is not determined by the symmetry of the isolated chromophore. Calculations have shown that it has approximately the shape depicted in Figure 3.4b. Many examples have verified the validity of the octant rule, but there are also cases where it is not applicable, at least not in its original, simple form. This is true for ketones with a cyclopropane ring in the a, jS-position and for fluorosubstituted ketones, for which the experimentally observed sign can be reproduced only if the perturbation due to the fluorine atom is assumed to be smaller than that due to the hydrogen atom. More recent detailed calculations solved some of these problems. (Cf. Charney, 1979.)... 

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