Multiplicity rotation axes

Many molecules and other objects have multiple rotation axes. For example, snowflakes (Figure 4.2) exhibit complex shapes that are nearly always hexagonal and approximately planar. The line through the center of the flake perpendicular to the plane of the flake contains a twofold (C2) axis, a threefold (C3) axis, and a sixfold (Cg) axis. Rotations by 240° (C ) and 300° (Q ) are also symmetry operations of the snowflake. 

A further possibility for multiple rotation axes is to have more than one candidate for the principal axis. For example, the highest order axis for ethene is a twofold axis, but there are three nonequivalent C2 axes, as shown in Figure 1.10. The choice of principal axis is now arbitrary, and it is usual to assign each axis a Cartesian label (X,T or Z) so that they can be referred to explicitly. 

Axial Groups with Multiple Rotation Axes D , D a and Z) h... 

Figure 4 The multiple rotation axes in the octahedron and cube...

The square matrix A x transforms the vector x into a vector y by the product y=Ax. Multiplication by the matrix A associates two vectors from the Euclidian space fR" and therefore corresponds to a geometric transformation in this space. A is a geometric operator. Non-square matrices would associate vectors from Euclidian spaces with different dimensions. The ordered combination of geometric transformations, such as multiple rotations and projections, can be carried out by multiplying in the right order the vector produced at each stage by the matrix associated with the next transformation. 

Torsional interactions. In addition to all the terms described above, we often introduce to the force field a torsional tetm Ax-y-z-w (l cos nm) for each torsional angle co showing how V changes when a rotation to about the chemical bond YZ, in the sequence X-Y-Z-W of chemieai bonds, takes place (n is the multiplicity of the energy barriers per single turn ). Some rotational barriers already result from the van der Waals interaction of the X and W atoms, but in practice, the barrier heights have to be corrected by the torsional potentials to reproduce experimental values. 

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