Symmetry-adapted linear combinations,

The preceding sections have shown how to use symmetry analysis on a basis of vectors to determine the irreducible representations for molecular vibrations. In building the reducible representation the basis vectors are treated as individual objects to which characters are assigned. This makes the total character for the basis easy to calculate by summing the results for each of the basis vectors in isolation. 

What is now required is a method which can be applied after the reduction process to give the collective motion of each irreducible representation in terms of the basis automatically. This is the job of the projection operator, which is explained in the next section. Here, the use of the basis to describe collective motions of the atoms of a molecule is discussed so that it is clear what the projection operator is aiming for. 

Collective functions formed from the basis which belong to a particular irreducible representation are referred to as symmetry adapted linear combinations (SALCs). In essence, each SALC can be thought of as a sum over all the basis functions  

To see how these SALCs conform to the symmetry irreducible representations, it is interesting to consider the affect of the C2V symmetry operations on the functions themselves. For example, after a C2 operation the two H atoms swap over and so do the basis vectors. However, the mode is a function of both vectors, and it is the symmetry of the overall function that determines the value of its character. Looking at the result for each function as a whole, there are still vectors at Hj and H2. For the symmetric mode, and ( 2 will have swapped but the function looks identical it is still bi + 2, so has a character of 1. In terms of the old basis, b has become 2 and vice versa. 

For the antisymmetric mode, and b2 would also be swapped by a C2 rotation, but the sign of 2 was opposite to b so the function changes sign, i.e. 

For linear combinations of wavefunctions, the symmetry species of the individual wavefunctions is very important. One ramification of symmetry considerations is that wavefunctions of different symmetry species do not combine to, say, make bonds. Because we are seeking linear combinations of atomic wavefunctions, this allows us to conclude that the only useful combinations for molecules will be of those atomic wavefunctions that belong to the same symmetry species of the molecule. The construction of symmetry-adapted linear combinations utilizes this simplifying idea. 

By keeping symmetry in mind, it is possible to construct appropriate combinations of atomic orbital wavefunctions to approximate molecular orbital wavefunctions that cover, or span, the entire molecule. The use of symmetry is the first real restriction we have placed on linear combinations, but it makes sense. After all, it serves 

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Determine which atomic orbitals will be used. 

Make a table that has each individual atomic orbital listed on one side (say, the left) and the symmetry operations listed on a perpendicular side. List the symmetry operations individually, not by class. There are h symmetry operations in the point group, where h is the order of the group. There should therefore be h entries for the symmetry operations. 

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FIGURE 21. Top Photoelectron spectrum of (Z,Z,Z)-cyclonona-l,4,7-triene 189. Bottom Diagram showing the homoconjugative a/n-interaction between the symmetry-adapted linear combinations of the 7T- and of the methylene cr-orbitals... 

The electronic wave functions are adequately described as antisymmetrized products of symmetry-adapted linear combinations of atomic orbitals. 

The determination of molecular orbitals in terms of symmetry-adapted linear combinations of atomic orbitals is analogous to the determination of normal vibrational modes by forming symmetry-adapted linear combinations of displacements. Both calculations are in reality the reduction of a representa-... 

In the derivation of molecular orbitals we started with individual orbitals and created symmetry-adapted linear combinations by solving a secular... 

IV. Projector Operators Symmetry Adapted Linear Combinations of Atomic Orbitals... 

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 ... 

The kind of functions we need may be called symmetry-adapted linear combinations (SALCs). It is the purpose of this chapter to explain and illustrate the methods for constructing them in a general way. The details of adaptation to particular classes of problems will then be easy to explain as the needs arise. 

Finally, new material and more rigorous methods have been introduced in several places. The major examples are (1) the explicit presentation of projection operators, and (2) an outline of the F and G matrix treatment of molecular vibrations. Although projection operators may seem a trifle forbidding at the outset, their potency and convenience and the nearly universal relevance of the symmetry-adapted linear combinations (SALC s) of basis functions which they generate justify the effort of learning about them. The student who does so frees himself forever from the tyranny and uncertainty of intuitive and seat-of-the-pants approaches. A new chapter which develops and illustrates projection operators has therefore been added, and many changes in the subsequent exposition have necessarily been made. 

Whatever method is used in practice to generate spin eigenfunctions, the construction of symmetry-adapted linear combinations, configuration state functions, or CSFs, is relatively straightforward. First, we note that all the methods we have considered involve 7V-particle functions that are products of one-particle functions, or, more strictly, linear combinations of such products. The application of a point-group operator G to such a product is... 

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