Tetrahedron symmetry

Stowasser B, Budt K-H, Jian-Qi L, Peyman A, Ruppert D. New hybrid transition state analog inhibitors of HIV protease with peripheric C2-symmetry. Tetrahedron Lett. 1992 33 6625-6628. 

The synthesis of [M4L6] 3 -metallacryptates (see Sect. 2.3) was realized by the combination of bis-bidentate ligands of two-fold symmetry (tetrahedron edges) and octahedral metal centers with three-fold symmetry (tetrahedron vertices) lib, 2b, 3, 141. An alternative approach to the formation of symmetric tetrahedral clusters has been described in which the octahedral metal centers (three-fold axis) occupy the vertices of the tetrahedron and trigonally symmetric tris-bidentate ligands (three-fold axis) occupy the faces of the equilateral triangles 1161. 

The symmetry of the structure we are looking for is imposed on the field 0(r) by building up the field inside a unit cubic cell of a smaller polyhedron, replicating it by reflections, translations, and rotations. Such a procedure not only guarantees that the field has the required symmetry but also enables substantial reduction of independent variables 0/ the function F (f)ij k )- For example, structures having the symmetry of the simple cubic phase are built of quadrirectangular tetrahedron replicated by reflection. The faces of the tetrahedron lie in the planes of mirror symmetry. The volume of the tetrahedron is 1 /48 of the unit cell volume. 

Figure 13.17 Molecular structure of some sulfides of arsenic, stressing the relationship to the AS4 tetrahedron (point group symmetry in parentheses).

When a tetrahedron is reflected or rotated, a structure indistinguishable from the original is obtained. Chemists call the tetrahedron a symmetric structure. Escher, in his painting Symmetry No. 20, shows a translational repeating symmetry. He flips or moves his objects, creating a pattern. 

Perhaps only slightly less common than octahedral symmetry is tetrahedral symmetry. We now examine the d orbital splitting in this environment. The story is much the same as above, except that it is now convenient to place the four point charges of the tetrahedron as shown in Fig. 3-6. Here ligands are put at alternate... 

Consider now spin-allowed transitions. The parity and angular momentum selection rules forbid pure d d transitions. Once again the rule is absolute. It is our description of the wavefunctions that is at fault. Suppose we enquire about a d-d transition in a tetrahedral complex. It might be supposed that the parity rule is inoperative here, since the tetrahedron has no centre of inversion to which the d orbitals and the light operator can be symmetry classified. But, this is not at all true for two reasons, one being empirical (which is more of an observation than a reason) and one theoretical. The empirical reason is that if the parity rule were irrelevant, the intensities of d-d bands in tetrahedral molecules could be fully allowed and as strong as those we observe in dyes, for example. In fact, the d-d bands in tetrahedral species are perhaps two or three orders of magnitude weaker than many fully allowed transitions. 

Fig. 17. Tetragonal symmetry projective view (above) and cut A-B (below) of a [LS],-layer formed by a sequence of [L,S] tetrahedrons (ideally PbFCl-t3q)e distorted CeSI-, FeOCl-, and NdSBr-types). (Redrawn from C. Dagron and F. Thevet, Ann. Ckim. 6, 67 (1971), Fig. 6, p. 77.)...

Suitably Substituted Adamantanes. Adamantanes bearing four different substituents at the bridgehead positions are chiral and optically active, and 14, for example, has been resolved. This type of molecule is a kind of expanded tetrahedron and has the same symmetry properties as any other tetrahedron. 

Preference for the chair transition state is a consequence of orbital-symmetry relationships Hoffmann, R. Woodward, R.B. J. Am. Chem. Soc., 1965, 87, 4389 Fukui, K. Fujimoto, H. Tetrahedron Lett, 1966, 251. 

The ligands of a tetrahedral complex occupy the comers of a tetrahedron rather than the comers of a square. The symmetry relationships between the d orbitals and these ligands are not easy to visualize, but the splitting pattern of the d orbitals can be determined using geometry. The result is the opposite of the pattern found in octahedral... 

Caesium chloride is not body-centered cubic, but cubic primitive. A structure is body centered only if for every atom in the position x, y, z there is another symmetry-equivalent atom in the position x+ j,y+ j,z+ j in the unit cell. The atoms therefore must be of the same kind. It is unfortunate to call a cluster with an interstitial atom a centered cluster because this causes a confusion of the well-defined term centered with a rather blurred term. Do not say, the 04 tetrahedron of the sulfate ion is centered by the sulfur atom. 

You say that your nonlinear molecule has the high symmetiy of a regular polyhedron, such as a tetrahedron, cube, octahedron, dodecahedron, icosahedron,... sphere. If it is a sphere, it is monatomic. On the other hand, if it is not monatomic, it has the symmetry of one of the Platonic solids (see the introduction to Chapter 8). 

Ermer, O. Tetrahedron 31, 1849 (1975). Added in proof Stanton, R.E., Mclver, J. W., J. Amer. Chem. Soc. 97, 3632 (1975), have recently applied symmetry considerations to the characterisation of transition states in the course of quantum chemical calculations (see also refs. 83 and 103). 

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