Point group symmetry defined

For all but the few smallest clusters, the number of possible structures is virtually unlimited. In order to be able to treat the larger systems, quite restrictive assumptions about their geometry has to be made. For those clusters where well-defined equilibrium structures do exist, these are likely to possess a non-trivial point-group symmetry (in many cases the highest possible symmetry). It therefore seemed justified to focus the study on high-symmetric systems. Symmetry can also be used to simplify the calculation of electronic structure, and reduces the number of geometrical degrees of freedom to be optimized. 

Point groups were defined as consisting of all the elements of symmetry possessed by a molecule and which intersect at a point. Such elements represent a group according to the rules by which they interact. 

The symmetry of a fabric or texture is defined statistically by Sander and expressed in terms of point group symmetry because translational symmetry is absent in petrology. 

A point group is defined as a set of symmetry operations that leave unmoved at least one point within the object to which these symmetry operations are being applied. No translation operations (that is, simple movements along a straight line) can be included in this description ... 

The hulk of a nanomaterial will refer to that portion that is not clearly defined as part of the surface, i.e., not within several atomic layers of the surface. This implies that 1-nm nanomaterials consist only of surface. Surface volume will refer to the volume of a nanocrystal or nanoparticle that is assumed to be part of the surface. The habit of a crystal is the particular external form that is presented within the options allowed by the point group symmetry. Common descriptive terms are tabular (tablet-like), cubic, acicular, and so forth. Occasionally, habits inconsistent with point group symmetry can occur from kinetic phenomena. 

The orientation of each symmetry element with respect to the three major crystallographic axes is defined by its position in the sequence that forms the symbol of the point group symmetry. The complete list of all 32 point groups is found in Table 1.8. 

In general, a point group symmetry operation is defined as a rotation or reflection of a macroscopic object such that, after the operation has been carried out, the object looks the same as it did originally. The macroscopic objects we consider here are models of molecules in their equilibrium configuration we could also consider idealized objects such as cubes, pyramids, spheres, cones, tetrahedra etc. in order to define the various possible point groups. 

For example, for a cube P-C + F = 8-12 + 6 = 2, as required. Similarly for an octahedron P-C + F = 6-12 + 8=2. The octahedron, moreover, is a geometrical dual of the cube, because the role of P and F are interchanged in the two structures. Thus the 8 ppints of the cube correspond to the 8 faces of the octahedron, and the 6 faces of the cube correspond to the 6 points of the octahedron. The duals share exactly the same value C (=12), and exactly, the same point group symmetry (Oh). For any case where the Descartes-Euler formula applies, duals are defined by the interchange of P and F, C held constant. Thus, Figure 1 indicates the dodecahedron and icosahedron are duals, and the tetrahedron is self-dual. 

Section 6.1 defines the point group symmetry operators and Section 6.2 gives additional detail about the point groups themselves. 

The ideal crystal is a rigid, three-dimensional array of molecules extending infinitely in all directions. This is the model used to evaluate the symmetry of a group of real atoms. The infinite extent of this array allows us to add new symmetry operations to our list of point group symmetry elements (Section 6.1). Previously, we counted only operations that leave the center of mass unchanged. However, the center of mass is not defined for an infinite number of atoms, so we can ignore that constraint now by adding translational symmetry elements to the list. 

Determine the point groups that define the symmetry of the following compounds, whose structures are illustrated in Figure 13.15. 

In Example 13.3e, we are assuming that resonance structures of NO3 are "averaging" out the symmetry to an overall D3h point group. If resonance weren t assumed, what point group would define the structure of NO3 ... 

The symmetry elements of a point group are defined with respect to a global axis system and so do not move under any of the operations of the group. 

Each tensor property has an intrinsic symmetry, which relates to the interchangeability of suffixes. However, the number of independent tensor components for a property also depends on the symmetry of the system it is describing. Thus the properties of an isotropic liquid, which has full rotational symmetry, can be defined in terms of a single independent coefficient. The number of independent components for a particular tensor property depends on the point group symmetry of the phase to which it refers. This is expressed by Neumann s principle which states that the symmetry elements of any physical property of a crystal must include the symmetry elements of the point group of the crystal. 

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