Rotation triangular molecules

For example, in a plane triangular molecule such as BF3, each of the twofold symmetry axes lying in the plane can be carried into coincidence with each of the others by rotations of 27r/3 or 2 x 2nl3, which are symmetry operations. Thus all three twofold axes are said to be equivalent to one another. In a square planar AB4 molecule, there are four twofold axes in the molecular plane. Two of them, C2 and C2, lie along BAB axes, and the other two, C and Ci, bisect BAB angles. Such a molecule also contains four symmetry planes, each of which is perpendicular to the molecular plane and intersects it along one of the twofold axes. Now it is easy to see that C2 may be carried into C2 and vice versa, and that C2 may be carried into C2 and vice versa, by rotations about the fourfold axis and by reflections in the symmetry planes mentioned, but there is no way to carry C2 or C into either CJ or Cn or vice versa. Thus C2 and C2 form one set of equivalent axes, and and C form another. Similarly, two of the symmetry planes are equivalent to each other, but not to either of the other two, which are, however, equivalent to each other. 

A triatomic triangular molecule of the type AB2 becomes indistinguishable twice during the rotation of 360°, hence the axis is named C2. It has three such axes passing through each of the atoms and coincides with the centre of the molecule. Such molecules have a C3 axis (principal axis)... 

As an example, infrared spectroscopy has shown that the lowest stable hydration state for a Li-hectorite has a structure in which the lithium cation is partially keyed into the ditrigonal hole of the hectorite and has 3 water molecules coordinating the exposed part of the cation in a triangular arrangement (17), as proposed in the model of Mamy (J2.) The water molecules exhibit two kinds of motion a slow rotation of the whole hydration sphere about an axis through the triangle of the water molecules, and a faster rotation of each water molecule about its own C axis ( l8). A similar structure for adsorbed water at low water contents has been observed for Cu-hectorite, Ca-bentonite, and Ca-vermiculite (17). 

Data was taken in the electron energy range of 10-200 eV, but little sensitivity to the organic adsorbate is found above 100 eV. The observed diffraction pattern arises from three equivalent 120° — rotated domains of (2 X 2) unit cells. The optimum agree "ent between calculated and experimental intensity data for the metastable acetylene structure is achieved for an atop site coordination. The molecule is located at a z-distance of 2.5 A from the underlying surface platinum atom. However, the best agreement is obtained if the molecule is moved toward a triangular site, where there is a platinum atom in the second layer, by 0.25 A, as shown in Fig. 7.2. 

Fig. 24. Herringbone (a) and pinwiiccl (b) orientational ordering of uniaxial diatomic molecules on a triangular lattice. The heavy bars represent planar rotators and the circles denote vacancies, p is the degeneracy of the ordering. From Mourilsen (1985).

We illustrate this fact for the ammonia molecule, NH3. In its equilibrium conformation, the molecule is a triangular pyramid. Figure 9.4a shows the nuclear framework as viewed from the first octant of the coordinate system, and Fig. 9.4b shows the framework as viewed from the positive end of the z axis. The molecule is placed in the coordinate system in the conventional way, with the center of mass at the origin and the rotation axis of highest order (largest value of n) along the z axis. 

The left side of Fig. 28 demonstrates that at z-values of -8.5 and -5.5 A an octahedral hydration shell complex exists, where one comer points to the surface (which can be deduced from the fact that one water molecule contributes to the coordination number at cos 9 = 1). At distances of —7.0 and —4.5 A from the surface the octahedron remains stable but is rotated in such a way that two triangular planes are now parallel the surface. At -6.5 and -3.5 A, the first hydration shell consists of only five water molecules in a roughly pyramidal coordination with the basis of the pyramid again parallel to the surface. In all cases, the stmcture is more pronounced when the ion is closer to the surface. 

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