Ligand-Orbital Representations

First we construct six linearly independent molecular orbitals from the six ligand a functions. The trace of the six a functions under the symmetry operations of is 

From the character table 8-1 we see that the six orbitals behave as the combination alg, tlu, and eg. 

This ligand combination is normalized, neglecting overlap between the six a orbitals. Thus, we have for the complete alg molecular wave function, 

we see that the p orbitals and the linear combinations of a orbitals follow each other under the various symmetry operations. With dx2 y2 and dz2 we construct the a orbitals as follows The 

Hence these two linear combinations are the a molecular orbitals transforming as eg. Note that we have now used up all the a valence orbitals e.g., let us add up all the parts  

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M). These ligand orbitals form a basis for the reducible representation rx of which can be broken down as ... 

The symmetry adapted linear combinations of the eight ligand orbitals which form a basis for these irreducible representations are found in the same way as in the octahedral MX case. They are ... 

Tm may also be derived by reducing the representation spanned by the ligand orbitals in a cubic MLg molecule or configuration, as occurs, for example, when an ion such as In+ is a substitutional impurity in CsCl. 

We first treat those complexes that have only a metal-ligand bonding. Examples of this type of complexes include [M(H20)6] + and [M(NH3)6] +, where M + is a first transition series metal ion and a typical n value is either 2 or 3. A coordinate system for such a complex is shown in Fig. 8.8.1. The representation generated by the six ligand orbitals is... 

To be consistent, we shall adopt the simple orbital representations used above to describe coordinate bonding interactions. Cyclobutadiene, being a bidentate ligand system, is described by four bonding molecular orbitals. They are similar in symmetry to the general bidentate system described earlier (cf. Fig. 5) and are represented in Fig. 7. 

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... 

The TT orbitals are more difficult to see, but if the y axis of the ligand orbitals is chosen along the bond axis and the x and z axes are arranged to allow the C2 operation to work properly, the results in Table 10-10 are obtained. The reducible representation includes the E, T], and T2 irreducible representations. The Tj has no matching metal atom... 

For both the complexes considered the metal f-orbitals lie, on a one-electron picture, below the ligand orbitals of e3(0) symmetry and above those belonging to the a(a), ex(rr), and e2(5) representations. Consequently, as indicated in Section 1, the systems may be pictured as accommodating the 20 electrons of the two cyclo-octatetraenyl dianion rings in the a2u, alg, elu, e2u, and e2g, mainly ligand levels, with the one or two f-electrons of Cem or UIV respectively located in the mainly metal f, e3u level. Moreover, the a2u, eiu, and e2u mainly metal f-levels will all be anti-bonding in character, whilst the e3u f-orbital will be bonding. 

In a second step, overlap between the metal and ligand orbitals is taken into consideration and the molecular orbitals are constructed, taking into account that only group-symmetrical orbitals belonging to the same symmetry representation may interact. Since the ligand t and t2u orbitals have no counterpart on the metal, these orbitals remain nonbonding. The same is (to a first approxima-... 

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