Finding Reducible Representations

Using D2h symmetry and labeling the orbitals (fi-fi2) as shown above proceed by using the orbitals to define a reducible representation.which may be subsequently reduced to its irreducible components. Use projectors to find the SALC-AOs for these irreps. 

When we come to apply the results we have so far discovered to quantum mechanical situations, we will find that the application usually revolves around the reduction of some reducible representation for the point group concerned. We have already seen how to find out which irreducible representations appear in the reduction of a reducible representation, namely if we write... 

Hence we have a method of finding basis functions which belong to a given irreducible representation, if we are given some function space which produces a reducible representation. Notice that in addition to the 0M, the construction of P (eqn (7-6.6)) requires only the knowledge of the characters of the T representation. 

From Table 7-9,2 and using eqn (5-7,2) we can find the diagonal elements of the matrices which represent the 4h point group in the p-orbital basis and in the d-orbital basis. From these elements we get the characters of two reducible representations they are shown in Table 7-9,3, By applying eqn (7-4.2)... 

Given the characters x of a reducible representation T of the indicated point group 9 for the various classes of 9 in the order in which these classes appear in the character table, find the number of times each irreducible representation occurs in T. 

Consider the four functions of Problem 5.2 which form a basis for a reducible representation T of Using projection operators find the orthonormal basis functions which reduce T. Assume (/, / ) = 5y. 

The six 2p, a+omic orbitals fa,--- fa) form a basis for a reducible representation TAO of 9%, since by applying the usual techniques ( 5-7) we find that the transformation operators Om transform fa either into itself or the negative of itself or into one of the other five atomic orbitals or the negative of one of the others ... 

Except for this change, we find hyb(l ) in the same way as before. We note, however, that this time the direction of a vector may be reversed as the result of a symmetry operation and in such a case there will be a contribution of — 1 to the character of that operation. Furthermore, we immediately see that in carrying out the different symmetry operations, no vector perpendicular to the molecular plane is ever interchanged with one in the molecular plane and vice versa. This implies two things the representation rhyb is at once in a partially reduced form (the matrices are already in block form, each consisting of two blocks) and the vectors perpendicular to the molecular plane on their own form a basis for a reducible representation of (which we will call rby ) and the vectors in the molecular plane on their own also form a basis for a reducible representation of 9 (which we will call iy.T) necessarily... 

Since each MO belongs to some irreducible representation of the molecular point group, we must find linear combinations of the AOs that transform according to the irreducible representations group theory enables us to do this. A symmetry operation sends each nucleus either into itself or into an atom of the same type a symmetry operation will thus transform each AO into some linear combination of the AOs (9.63). Therefore (Section 9.6), the AOs form a basis for some representation TAO of the point group of the molecule. This representation (as any reducible representation) will be the direct sum of certain of the irreducible representations r r2,...,r (not necessarily all different) of the molecular point group ... 

To find the MOs for benzene, we choose a basis comprising a 2pz AO on each carbon atom and determine the characters of I, , the reducible representation generated by... 

It was quite easy to find the irreducible representation of Rz before, as the representation we worked out appeared to be an irreducible representation itself. In most cases, however, a reducible representation is found when the symmetry operations are applied to a certain basis. Now a simpler way will be shown (1) to describe the representation on a given basis without generating the matrices themselves and (2) to reduce them, if reducible, to irreducible representations. 

It may be asked, of course, whether this is the only way of decomposing the Ti representation. The answer is reassuring The decomposition of any reducible representation is unique. If we find a solution just by inspection of the character table, it will be the only one. Often this is the fastest and simplest way to decompose a reducible representation. 

The symmetry group of NH2D (ND2H) is the C2V group of tire permutations and permutation-inversions of the elements E, (12), E, and (12). By the same arguments as described above for NH3 we find that the symmetry coordinates for NH2D form the basis of a reducible representation ... 

To find the symmetry adapted combinations we first csonsider the application of the transformation operators Og to the 33 atomic orbitals. We see at once that for all symmetry operations Jt of the Pt, point group, the central atom M is left unchanged and consequently any will only transform metal orbitals into metal orbitals (or combinations of metal orbitals) and ligand orbitals into ligand orbitals (or combinations of ligand orbitals). Thus, we can immediately reduce (the reducible representation using aU 33 atomio orbitals) to the... 

For a trigonal planar molecule such as BF3, the orbitals likely to be involved in bonding are the 2s, and 2py orbitals. This can be confirmed by finding the reducible representation in the point group of vectors pointing at the three fluorines and reducing it to the irreducible representations. The procedure for doing this is outlined below. 

Find the reducible representation for the vectors, using the appropriate group and character table, and find the irreducible representations that combine to form the reducible representation. 

The atomic orbitals that fit the irreducible representations are those used in the hybrid orbitals. Using the symmetry operations of the D f, group, we find that the reducible representation... 

Find the reducible representation for all the a bonds, reduce it to its irreducible representations, and determine the sulfur orbitals used in bonding for SOCI2. 

For establishing how many irreducible representations of type i factor group are contained in the above described factor group reducible representation, we have to find the number... 

As we have already seen ( 6.2.5), the charaaer table gives us information on orbital symmetry properties. If the molecule contains a central atom, the symmetries of the orbitals of this atom are indicated in the last two columns of the table. However, the orbitals on non-central atoms, for example the Ish orbitals in H2O or NH3, are not individually bases for an irreducible representation (Tables 6.1 and 6.3). These AO form a basis for a reducible representation that can be decomposed into a sum of irreducible representations of the point group. Although the character table does not give the result immediately, it does enable us to find it by using the reduction formula. 

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