Symmetry-adapted linear

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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