Symmetry Properties of Orbitals

Let us assume that some reducible representation may be written as the sum of a set of irreducible representations as in equation 4.23. 

Using the shorthand notation introduced in equations 4.15 and 4.16, all such products will be zero except that for j = i. 

Using this result for the hydrogen Is orbitals of NH3 (4.3) leads to a reduction of the reducible representation (see equations 4.22 and 4.29 along with Table 4.4) as 

A similar process may be employed for the hydrogen Is orbitals of water (4.4). 

Another example is shown in 4.5. Suppose that instead of OH2, the molecule of 

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A depiction of how molecular orbitals are formed from atomic orbitals of separated atoms and/or how these molecular orbitals correlate with atomic orbitals of the atoms united as the nuclei come together to form a molecular species. Such diagrams are especially useful in rationalizing the symmetry properties of orbitals. 

The symmetry properties of orbitals, and the pattern of electron occupation of the orbitals, is important in concerted reactions, that is, those that transform substrates directly into products without going through intermediates. A large group of concerted reactions called pericydic reactions have been widely studied for alkenes and... 

The interaction of the Cr atom with the trigonal set of CO groups is governed by the symmetry properties of orbitals under C3 rotations. The set of three a donor orbitals on the carbon atoms will give SALCs of a and e symmetry. It must be noted that in C3 symmetry there is no ex versus e2 distinction both sets of d orbitals, (dsz, dyz) and dxy) belong to the E... 

The atomic orbitals suitable for combination into hybrid orbitals in a given molecule or ion will he those that meet certain symmetry criteria. The relevant symmetry properties of orbitals can be extracted from character tables by simple inspection. We have already pointed out (page 60) that the p, orbital transforms in a particular point group in the same manner as an x vector. In other words, a px orbital can serve as a basis function for any irreducible representation that has "x" listed among its basis functions in a character table. Likewise, the pr and p. orbitals transform as y and vectors. The d orbitals—d d dy, d >, t, and d ,—transform as the binary products xy, xz, yr, x2 — y2, and z2, respectively. Recall that degenerate groups of vectors, orbitals, etc, are denoted in character tables by inclusion within parentheses. 

The idea that symmetry properties of orbitals might be important in determining the course of certain reactions apparently originated with L. J. Oosterhoff, quoted by E. Havinga andj. L. M. A. Schlatmann, Tetrahedron, 16, 146 (1961). 

The following sections present an empirical approach to applying the selection rules. The chapter continues with a basic introduction to the analysis of symmetry properties of orbitals and the application of orbital correlation diagrams to the relatively simply cyclobutene-butadiene interconversion it concludes with some examples of the frontier orbital approach to pericyclic reactions. 

HMO (Hiickel molecular orbital theory) is the simplest quantitative molecular orbital theory. It was developed in the 1930s by Erich Hiickel to describe planar hydrocarbons with conjugated jt bonds [13]. HMO is based on the idea of o — jt separation, treating rr electrons only. HMO calculations are the only ones that are practical to do without the aid of a computer, giving rather poor energies and orbital functions but faithfully reproducing the symmetry properties of orbitals. 

Box 3.1 The Symmetry Properties of Orbital and Spin Wave Functions... 

The truth table or multiplication table for the symmetry properties of orbital and spin wave functions is shown in Table 3.1. 

The arguments used by Woodward and Hoffmann (1969) are based on the symmetry properties of orbitals. Initially they employed the frontier orbital approach, as outlined above, but subsequently developed Longuet-Higgins and Abrahamson s method (1965) which utilized a group theoretical analysis. It is not in our interests, and neither is it necessary, to apply Group Theory in its full rigour in particular we shall use the simplified system of nomenclature developed previously (see p, 14). 

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