Irreducible representations, and

More generally, it is possible to combine sets of Cartesian displacement coordinates qk into so-called symmetry adapted coordinates Qrj, where the index F labels the irreducible representation and j labels the particular combination of that symmetry. These symmetry adapted coordinates can be formed by applying the point group projection operators to the individual Cartesian displacement coordinates. 

FIGURE 18. The symmetry-adapted, orthogonal linear combinations of the localized a-orbitals of norbornadiene 75 belonging to the irreducible representations / and 62 of the point group C2v- The A] and B2 combinations are the relay orbitals for through-bond interaction between jra and 7Tb which define, according to equation 56, the orbitals jt and jt... 

For practical applications these relationships lead to a number of important rules about irreducible representations and their characters ... 

A primitive idempotent is now defined as an idempotent belonging to a single irreducible representation, and with h — 1. A primitive idem-potent p, in the -adapted basis, has the form... 

Labeling these electronic energy levels in a proper way (irreducible representations) and determine their degeneracy. 

The representations that involve the lowest-dimension matrices are called irreducible representations and have a particular relevance in group theory. 

Here the R form the set of linear coordinate transformations that leave the nuclear framework invariant, yC are the characters associated with the fi ,-dimensional irreducible representation and g is the order of the point group, G. 

Methods for treating the factor group vibrations have been given by Davydov (25), as well as by Bhagavantam and Venkatarayudu (22). A simple analysis is possible through what is known as the correlation method (20, 26, 27) by which one is able to write the irreducible representations and thereby classify k=0 phonons directly and simply. The number of A = 0 phonons is 3 N, where N is taken to be the number of atoms in the entire unit cell. However, there are only 3 N-3 optically active phonons because the acoustic vibrations have approximately... 

Various schemes exist to try to reduce the number of CSFs in the expansion in a rational way. Symmetry can reduce the scope of the problem enormously. In die TMM problem, many of die CSFs having partially occupied orbitals correspond to an electronic state symmetiy other than that of the totally symmetric irreducible representation, and dius make no contribution to the closed-shell singlet wave function (if symmetry is not used before the fact, die calculation itself will determine the coefdcients of non-contributing CSFs to be zero, but no advantage in efdciency will have been gained). Since this application of group dieoiy involves no approximations, it is one of the best ways to speed up a CAS calculation. 

Now we prove the existence of the isotypic decomposition for finite-dimensional representations. Just as any natural number has a prime factorization, every finite-dimensional representation of a compact group has an isotypic decomposition. This decomposition tells us what irreducible representations appear as subrepresentations and what their multiplicities are. Note that Proposition 6.11 guarantees uniqueness as well, since the selection of irreducible representations and exponents are uniquely determined. 

In other words, the representations U of 5m(2) as differential operators on homogeneous polynomials in two variables are essentially the only finitedimensional irreducible representations, and they are classified by their dimensions. Unlike the Lie group 50(3), the Lie algebra sm(2) has infinitedimensional irreducible representations on complex scalar product spaces. See Exercise 8.10. 

Since we will continually be requiring the characters of the irreducible representations of the point groups, it is convenient to put them together in tables known as character tables- In the character table of a point group each row refers to a particular irreducible representation and, since the characters of operations of the same class are identical, only a single entry (C,) is made for all the operations of a given class. The columns are headed by a representative element from each class preceded by the number of elements or operations in that class gf. 

For example, the point group has three classes (and necessarily three irreducible representations) and its character table is shown in... 

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