Symmetry great orthogonality theorem

An important consequence of group theory, i.e., the so-called great orthogonality theorem, is the basis of the equation for calculating the numbers w, of the vibrations belonging to a symmetry species i ... 

In his book Molecular Symmetry and Group Theory, Robert Carter introduces a tabular method for applying the great orthogonality theorem, which is illustrated... 

Reduction of Tj j using the great orthogonality theorem yields two IRRs Sg and By convention, one-electron MO diagrams use lower case symbols to represent the symmetries of the MOs. In addition, the subscripts and superscripts are usually dropped and replaced with the subscript b for the bonding MOs and an asterisk superscript for the antibonding MOs. Application of the projection operator yields... 

Use the great orthogonality theorem to determine the symmetries of the SALCs. The projection operator method can be used to determine the mathematical forms of the SALCs and their corresponding shapes. 

Therefore, this is a sum of two Aj, one A2, and two E symmetry species. This is how the great orthogonality theorem is applied to reduce character sets into their unique set of irreducible representations. 

Although equation 13.8 is somewhat general, it does not cover all cases. For example, in symmetry species that have E or T labels, multiplication of the irreducible representations yields a reducible representation that must be reduced using the great orthogonality theorem. In such cases, the integral is identically zero unless the reducible representation can be broken down into irreducible representations, one of which must be A (or whatever the totally symmetric representation is). Such a reducible representation is said to contain A. Mathematically, this is written as... 

FIGURE 13.26 Operation of the symmetry classes of T on the sp orbitals. The a, b, c, and ti labels are used only to keep track of the individual hybrid orbitals. The nrnnber of hybrid orbitals that do not move when a symmetry operation occurs is listed in the final coliunn. This set of mrmbers is the reducible representation F of the sp orbitals. The great orthogonality theorem is used to reduce F into its irreducible representation labels. 

Without using the great orthogonality theorem, reduce the given irreducible representation in Cgy symmetry. Does your answer make sense ... 

Determine if the following integrals can be nonzero If the molecular or atomic system has the given local symmetry. Use the great orthogonality theorem if necessary. 

The irreducible representations are from different point groups, and have different symmetry classes and orders. Therefore, the great orthogonality theorem does not apply. 

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