SALCs combinations

The hydrogen group orbitals are referred to as symmetry adjusted linear combinations (SALC). Although their development will not be shown here, the molecular orbital diagrams for other tetrahedral molecules are similar. 

Back transformation into the original basis will generate the expected results for the le MOs (expected combinations of SALC-AOs). 

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. 

The most important and frequent use for projection operators is to determine the proper way to combine atomic wave functions on individual atoms in a molecule into MOs that correspond to the molecular symmetry. As pointed out in Chapter 5, it is essential that valid MOs form bases for irreducible representations of the molecular point group, we encounter the problem of writing SALCs when we deal with molecules having sets of symmetry-equiv-... 

The cyclopropenyl group, C3H3, is the simplest carbocycle with a delocalized n system and can serve as a prototype for this class of molecules. Let us see how the pn orbitals of the individual carbon atoms can be combined into MOs—or at least the immediate precursors of actual n MOs. Cyclic n systems will be discussed in general in Chapter 7, and this illustration is intended only to demonstrate the use of projection operators in making SALCs of AOs on different atoms. 

Suppose that, instead of writing the secular determinant from an n x n array of atomic orbitals, we use an n x n array of n orthonormal, linear combinations of the basis set orbitals. Suppose, furthermore—and this is the key—we require these linear combinations to be SALCs, that is, each one is required to be a function which forms a basis for an irreducible representation of the point group of the molecule. Then, as shown in Chapter 5, all integrals of the types... 

Combine the basis orbitals into linear combinations corresponding to each of the irreducible representations. These SALCs can always be constructed systematically by using the projection operator technique developed in Chapter 6. 

When the 10 pn orbitals are combined into SALCs, y l to y/,n, which are then listed in order of their symmetry types, namely,... 

The SALCs y/x through, 0 may then be combined into MOs. For the Au MOs, the simultaneous equations from which the secular equation originates are as follows ... 

Consider first the Ax SALC. It must have the same symmetry as the s orbital on atom A. This requires that it be everywhere positive and unchanged by all symmetry operations, and thus it must be gx + g2 + negative amplitude where it is negative. To match the p orbitals, on a 1 1 basis, the combinations clearly must be... 

Figure 8.3 By combining the central s orbital with the ligand orbital SALC to give either positive overlap or negative overlap, we get i//h and respectively. The expressions written for these are only meant to express this sign relationship the actual expressions for (/a and if/h contain different coefficients.

For the EK SALCs we require combinations that match the d and orbitals. The former has positive lobes along the z axis that have twice the amplitude of the toroidal negative region in the xy plane. To match this we need the following normalized SALC ... 

It should be noted that the displacement coordinates have been labeled and then combined into the -type symmetry coordinates in a particular way. The angles have been indexed so that 0, is opposite to rh This assures that 0, and the change therein, A0 are related to the molecular symmetry elements in the same way as are r, and Ar,. Then, when the SALCs are written, A0, and Ar, occupy corresponding positions in the expressions. Unless this is done the symmetry factorization will not work out. 

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. 

Symmetry adapted linear combinations (SALCs) of the determinants... 

S-S Sip Sop SALC SCF SO SOMO T tds tdt tfd TP UV xo 9m 9l 9 P Dithiolene or dithiolate chelate In-plane dithiolene S p orbital Out-of-plane dithiolene S p orbital Symmetry adapted linear combination Self-consistent held Sulfite oxidase Singly occupied molecular orbital Tesla (Trihuoromethyl)ethylenediselenato Toluene-2,3-dithiolate Bis(trihuoromethyl)-1,2-dithiete Trigonal prismatic Ultraviolet Xanthine oxidase Metal-based function Ligand-based function Symmetric in-plane dithiolene molecular orbital... 

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