Irreducible representations labelling

The rotation-vibration-electronic energy levels of the PH3 molecule (neglecting nuclear spin) can be labelled with the irreducible representation labels of the group The character table of this group is given in table Al.4.10. 

The birth of crystal field theory is due to Bethe who in 1929 showed that open-shell energy levels in a crystalline environment could be associated with the irreducible representation labels of the site point group. Little experimental work was done at that time because... 

From Table C.6 we see that what we call irreducible representations are simply the distinct rhythms of pluses and minuses, which after squaring, give the fully symmetric behaviour. All the electronic states of pyrazine and its diprotonated derivative can be described by the irreducible representation labels Ag, Bi, Big, Bsg, Au, B u, Biut 3u-... 

Wavefunctions can be labeled with appropriate irreducible representation labels. Therefore, the wavefunction of the bonding molecular orbital for H2 would be labeled Ajg (that is, Another phrase that means irreducible representations is symmetry species. We say that the symmetry species of this wavefunction is A g. 

For a hydrogen atom whose symmetry is described by the Rh(3) point group, the Is orbital has the irreducible representation label The 2s orbital also has the irreducible representation label Df The electromagnetic (EM) radiation operator that causes an electron to go from the Is state to any other state has the irreducible representation Show that the integral... 

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. 

Therefore, the four sp hybrid orbitals can bond with any other molecular orbital or set of molecular orbitals that have either A or T2 irreducible representation labels in Tj symmetry. 

From the representation of the normal modes of a symmetric linear molecule shown in Figure 14.30, draw the changes in the vectors upon operation of each symmetry element and assign irreducible representation labels to the normal modes of CO2. You will have to use the character table in Appendix 3. 

The degeneracy of a vibration is related to its character of the identity element of its irreducible representation label. Doubly degenerate vibrations always have an irreducible representation label having Xe 2. Triply degenerate vibrations always have an irreducible representation label having Xe 3. There are no higher degeneracies for vibrations. 

Determine their irreducible representation labels. Which (if either) of these vibrational modes is expected to be IR-active for symmetry reasons, and why ... 

For polyatomic molecules, the point group has enough symmetry elements (or rather, classes, and so therefore irreducible representations) that the following statement is usually applicable The ground electronic state and the allowed excited states are usually of different irreducible representation labels. 

The original paper by R.S. Mulliken including his assignment of irreducible representation labels ... 

In the previous section we deduced irreducible representation labels for two of the vibrational modes of HjO by inspecting the molecular geometry and thinking about the possible movements of the atoms. This is easy enough for simple molecules but the representation for a particular atomic motion may not be so straightforward for more complex cases. Also, if there are many vibrational modes, it is unlikely that we would find them all by inspection. What is required is a general method to find the representations for all possible molecular vibrations. The approach we will use is first to identify all the irreducible representations that are present for a given basis and then interpret each of them in terms of combinations of the basis functions. The rest of this chapter is dedicated to the first part of this process and the second part is the subject of Chapter 6. 

In the following examples, the point group of a variety of complexes is used to derive the symmetry labels for the atomic orbitals (AOs) of the central atom. The central atom orbitals are at the intersection of all the symmetry elements in the point groups considered and so are never moved through space by an operation. However, they may be reorientated, and so we will work out the characters for each AO set (p, d) and then apply the reduction formula to find the appropriate irreducible representation labels. These results will be used in Chapter 7 when assembling MOs for some of the complexes, and there we will use the fact that the standard character tables have the p and d functions written in the rightmost columns. For the central atom, this means that we can simply read off the symmetry label from the table. 

You can check your assignments of irreducible representation labels to these diagrams with the right-hand side of the MO diagram shown in Figure 7.36. 

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