Covering group

Table 2.1.3 Multiple reaction monitoring of amino acids for their tandem mass spectrometry quantitation. In daily practise not all mentioned amino acids are measured in one run, but a set of ten dedicated evaluation programs has been developed, covering groups of amino acids associated with groups of disorders. Amino acids presented in italics indicate stable-isotope-labeled internal standards ... 

I have left group theory as the last chapter. If it is desired to cover group theory earlier in the course, Sections 9.1-9.8 of the group-theory chapter can be studied prior to Chapter 6 on molecular vibrations, and Sections 9.9-9.12 read in conjunction with Chapter 6. 

Table 16.8. Multiplication table for the covering group G of G = C2V showing the isomorphism of G with D4.

Table 16.9. Character table of the abstract group Gg, o/ t/ze covering group Cf and its isomorph D4. showing the corresponding classes ...

Since most of the nonrigid molecules treated so far may be described by a SRM whose covering group is the improper group C[, the internal isometric group is treated first. 

Typical examples for SRMs with proper covering group are listed in Table 3. 

The semidirect product of the two groups (2.90) and (2.92) may be considered as the analogue of the permutation-inversion groups1 for molecules with non-trivial covering groups ( )... 

Analogously for SRMs with proper covering group g) and primitive period transformations, the full isometric group Wig) =Jrg) ( ) is homomorphic to the permutation-inversion group... 

To complete the discussion on the symmetry of QRMs and NRMs it is interesting to point out the analogy between the covering group < e of the restructure of a QRM and the isometric group St %) of the SRM associated with a NRM... 

Determination of r NCf (F) for SRMs with Proper Covering Group ( )... 

Rotations in a vector space of three orbitals are described by the group SO(3) of orthogonal 3x3 matrices with determinant +1. To embed the octahedral rotation group in this covering group one needs a matrix representation of O which also consists of orthogonal and unimodular 3x3 matrices. Such a matrix representation is sometimes called the fundamental vector representation of the point group. In the case of O the fundamental vector representation is Ti and not T2. Indeed the 7] matrices are unimodular, i.e. have determinant +1, while the determinants of the T2 matrices are equal to the characters of the one-dimensional representation A2. 

A2. F. A. Cotton and G. Wilkinson, Advanced Inorganic Chemistry, 3rd ed. Wiley (Interscience), New York, 1972. The main group elements are covered group by group and there are concise sections on organometallic compounds in most chapters. 

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