Point group establishing

Assigning the molecule to a point group establishes all the symmetry elements for that distribution of atomic nuclei. Carrying out any of those operations on the nuclei leaves their distribution effectively unchanged. The probability densities 11/21 for the electronic and vibrational wave-functions must also be symmetric under these operations, because identical regions of the molecule should yield identical contributions to our... 

The following simple rules can be established for point groups ... 

In Fig. 7-9.1 the symmetry elements for the B point group are shown (see also Fig. 3-6,1) as well as our choice of xt y and axes (this choice establishes the orientation of the p- and d-orbitals). In Table 7-9.1 the corresponding character table is given in full. 

Close study of the angles, indices, and axial ratios long since made it clear that every crystalline substance has a structure built upon a space -lattice" characteristic rtf the substance. It lias been established that this is due to the regular arrangement of the alums, molecules, or ions composing the substance. As shown by Table I. the lattice structures of crystals may be classified into 32 symmetry classes (point groups), which are further divided into seven systems. This topic also is discussed under Mineralogy. 

We have therefore established a diagnostic test (called the Frobenius-Schur test) which classifies the IRs V of a point group according to the three cases listed after eq. (12). (Note See the warning at the beginning of this chapter.)... 

The crystal field theory. The basics of the CFT were introduced in the classical work by Bethe [150] devoted to the description of splitting atomic terms in crystal environments of various symmetry. The splitting pattern itself is established by considering the reduction in the symmetry of atomic wave functions while the spatial symmetry of the system goes down from the spherical (in the case of a free atom) to that of a point group of the crystal environment. It is widely described in inorganic chemistry textbooks (seee.g. [152]). 

Point-group symmetries not listed in Table 3-1 may easily be assigned the appropriate Schoenflies notation by analogy. Thus, e.g., C5v, C5h, C7, Cg, etc. can be established. Such symmetries may well occur among real molecules. 

Figure 3-5 shows a possible scheme for establishing the molecular point group that has been widely used to reliably establish molecular symmetries [9],... 

A fundamental property of the wave function is that it can be used as basis for irreducible representations of the point group of a molecule [13], This property establishes the connection between the symmetry of a molecule and its wave function. The preceding statement follows from Wigner s theorem, which says that all eigenfunctions of a molecular system belong to one of the symmetry species of the group [14],... 

Compliance with the octet rule in diamond could be shown simply by using a valence bond approach in which each carbon atom is assumed sp hybridized. However, using the MO method will more clearly establish the connection with band theory. In solids, the extended electron wave functions analogous to MOs ate called COs. Crystal orbitals must belong to an irreducible representation, not of a point group, but of the space group reflecting the translational periodicity of the lattice. 

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