Eigenvalue Tables

Precisely the same method as for D4 can be used to construct generalized eigenvalue tables for all, except the cubic point groups. The generalized commutation relation, Eq. (7), holds for any n... 

The inversion operation commutes with all rotations and reflections and so simply leads to an extra eigenvalue. The operator 6h also commutes with both Cn and Q and so again leads to an extra eigenvalue. For all, except the cubic point groups, the construction of the eigenvalue table is then straightforward. Results for all the point groups are listed in Appendix A, in a way similar to Table 2 for D4. 

In this way all phasefactors are fixed and we obtain the eigenvalue table for Td, Table 4. 

The eigenvalue tables for the remaining pointgroups, Th, O and Oh are constructed in completely analogy with the table for Td and are also given in Appendix A. 

For the construction of the eigenvalue tables it is convenient to use the complex eigenvalues we have employed. Sometimes it is, however, practical to have eigenvalue tables with real elements only. These can be obtained as follows. Let us with 1) and... 

Table 5. Eigenvalue table for D4, half-integer spin...

We then get the eigenvalue table for T, Table 6. The last two pairs of the table transform into each other under the time-reversal operation which is why they are collected under the common representation G3/2. 

Table 7. Eigenvalue table for Td, halfinteger spin Representation X(Q) X( () 7(C3) 7(o)...

Although we have mainly talked about states it is clear that operators may just as well be classified by their behaviour under the various symmetry operations. Using the eigenvalue tables it is then straight forward to derive quantitative relations between matrix... 

The same results may also be formulated in terms of coupling coefficients. From the eigenvalue tables we can immediately see how a product of two states transforms under the various symmetry operations. Let us consider the product of two E states in the point group D3. We have the four product states EiEi), jExE-j), lE Ej) and lE-jE-i). 

Table 3.3 Scrambled LDM for acetic acid and resulting eigenvalues Table 3(a)...

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