Symmetry adapted linear combinations SALCs

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. 

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

Various methods (described in Chapter 4) can be used to determine the symmetry of atomic orbitals in the point group of a molecule, i. e., to determine the irreducible representation of the molecular point group to which the atomic orbitals belong. There are two possibilities depending on the position of the atoms in the molecule. For a central atom (like O in H20 or N in NH3), the coordinate system can always be chosen in such a way that the central atom lies at the intersection of all symmetry elements of the group. Consequently, each atomic orbital of this central atom will transform as one or another irreducible representation of the symmetry group. These atomic orbitals will have the same symmetry properties as those basis functions in the third and fourth areas of the character table which are indicated in their subscripts. For all other atoms, so-called group orbitals or symmetry-adapted linear combinations (SALCs) must be formed from like orbitals. Several examples below will illustrate how this is done. 

Reduce each representation from Step 3 to its irreducible representations. This is equivalent to finding the symmetry of the group orbitals or the symmetry-adapted linear combinations (SALCs) of the orbitals. The group orbitals are then the combinations of atomic orbitals that match the symmetry of the irreducible representations. 

Strictly speaking the densities of the electronic cloud on the sites of the atomic nuclei, the so-called on-site density, need not be different for different components of a degenerate wave function. A simple counter-example is a orbital level in a cubic cluster. Let a,) denote a a-type atomic orbital on a given site i. The symmetry adapted linear combinations (SALC s) of these basis orbitals are given by (see Fig. 1) ... 

Figure 1 has been constructed from Fig. 8.13 of [13], as modified by Fig. 2.8 of [14] to illustrate how this approach is applied to Ti02. The 3d, 4s, and 4p atomic states of Ti, and the 2s and the 2p atomic states of O, are used to construct a set of symmetry adapted linear combinations (SALC s) of atomic orbitals that are consistent with the Oh symmetry group of a regular, or undistorted octahedron as in an ideal cubic rutile structure. It is important to understand that other atomic states, e.g., the Hf 5f state symmetries can be introduced as well, and that the introduction of... 

These defect state electronic structures will be compared with experimental data in Sect. 4, and use the symmetry adapted linear combinations, SALC s, description of [13] as for assigning spectroscopic defect state signatures to equivalent molecular orbital states. 

Nanocrystalline TiOi has been used as prototypical elemental TM oxide with a distorted rutile phase. Most important are the spectra in Fig. 7 (XAS), Fig. 8 (VUV SE), and Fig. 10 (SXPS), and the comparisons of 3d-state splittings in Figs. 9(a) and (b) These assignments for Ti 3d, 4s and 4p atomic features in these specta have been compared with the Symmetry Adapted Linear Combinations (SALC s) atomic states approach of FA. Cotton in [13], and the modification of this to include covalency effects, in. [14 and 17]. 

Recall from Section 2.11(a) Polyatomic molecular orbitals that molecular orbitals are formed as linear combinations of atomic orbitals of the same symmetry and similar in energy. We find first a linear combination of atomic orbitals on peripheral atoms (in this case H) and the coefficient c, and then combine the combinations of appropriate symmetry with atomic orbitals on the central atom. The linear combinations of atomic orbitals of peripheral atoms (also called symmetry-adapted linear combinations—SALC) are given after Table RS5.1 in the Resource Section 5. Thus, we have to find linear combinations in Djh point group that belong to symmetry classes Al , Al", and E. We also have to keep in mind that H atom has only one s orbital. 

By far, the theoretical approaches that experimental inorganic chemists are most familiar with and in fact use to solve questions quickly and qualitatively are the simple Hiickel method and Hoffmann s extended Hiickel theory. These approaches are used in concert with the application of symmetry principles in the building of symmetry adapted linear combinations (SALCs) or group orbitals. The ab initio and other SCF procedures outlined above produce MOs that are treated by group theory as well, but that type of rigor is not usually necessary to achieve good qualitative pictures of the character and relative orderings of the molecular orbitals. 

To find symmetry-adapted linear combinations (SALCs), follow the procedure described in Section 12.S(c). Refer to the table above that displays the transformations of the original basis orbitals. To find SALCs of a given symmetry species, take a column of the table, multiply each entry by the character of the species irreproducible representation, sum the terms in the column, and divide by the order of the group. For example, the characters of species A are 1, 1. 1.1, so the columns to be summed are identical to the columns in the table above. Each column sums to zero, so we conclude that there are no SALCs of Ai symmetry. (No surprise here the orbitals span only At and B].) An At SALC is obtained by multiplying the characters 1. 1, — 1. — 1 by the first column ... 

Use the different sets of AOs on the ligands as basis sets to generate reducible representations known as symmetry-adapted linear combinations (SALCs). Any basis function that is unchanged by a given symmetry operation will contribute I to the character, any basis function that transforms into the opposite of itself will contribute — I, and any basis function that is transformed into a basis function on a different li d will be an off-diagonal element and contribute 0 to the character. 

A molecular orbital diagram for a transition metal complex can be generated from the orbitals of the metal and the symmetry-adapted linear combinations (SALCs) of the orbitals of the ligands. The SALCs are typically illustrated on one side of the diagram, the orbitals of the metal on the other side, and the molecular orbitals that result from combining the... 

The electronic states must transform as one of the irreducible representations r of the molecular point group, and so linear combinations of the AOs in Fig. 7.1 must be found which transform as these representations. Such symmetry-adapted linear combinations (SALCs) may be obtained using the projection operator technique [3]. Application of the projection operator... 

Figure ,2 Symmetry-adapted linear combinations (SALCs) and molecular orbitals (MOs) generated from the Is AOs in a bent ABg molecule.

Assigning a particular molecular orbital to its symmetry species is like assigning a vibration to its symmetry species (Section 8.5) we need to consider the effects of the various symmetry operations on the sign and orientation of the wavefimction. One problem is that we do not necessarily know what a particular wavefunction looks like, and it is here that the symmetry adapted linear combination (SALC) approach to the construction of molecular orbitals is very useful. We begin by classifying the valence-shell atomic orbitals (AOs) of the constituent atoms, in the symmetry of the molecule, as the set of molecular orbitals (MOs) will have the same distribution of symmetry species as the set of contributing atomic orbitals. Then we can attempt to construct molecular orbitals, bearing in mind that an orbital of any particular symmetry species arises from combinations only of atomic orbitals of that same symmetry species. 

SYMMETRY ADAPTED LINEAR COMBINATIONS (SALC) WAVE EUNCTIONS... 

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