Orbital properties symmetry

FO—VB Fragment orbital valence bond. A VB method that uses fragment orbitals with symmetry properties. The method is used for understanding the structures of transition states and molecules. 

The left hand side has been written to indicate the parentage of the function i.e. the fact that it has been derived from the ionised state r a. S2 A 2. The first two quantities on the right hand side are coupling coefficients — the linear coefficients required to generate a state with the correct spin and orbital properties (16,17). Although the function written in Eq. (8) has the same spin and symmetry as the ground state, it is clearly not antisymmetric in all N electrons, since the IVth electron, which is the one added, occupies uniquely the shell r. In order to form a properly antisymmetric state, we must ensure that the i 7th electron can occupy all the shells, and also every spin-orbital component of each shell. This... 

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. 

Closed orbits from symmetry arguments) Give a simple proof that orbits are closed for the simple harmonic oscillator x = v, v = -x, using only the symmetry properties of the vector field. (Hint Consider a trajectory that starts on the v-axis at (0, ), and suppose that the trajectory intersects the x-axis at (x,0). Then use symmetry arguments to find the subsequent intersections with the v-axis and x-axis.)... 

A key to understanding the mechanisms of the concerted pericyclic reactions was the recognition by Woodward and Hoffmann that the pathway of such reactions is determined by the symmetry properties of the orbitals that are directly involved. Specifically, they stated the requirement for conservation of orbital symmetry. The idea that the symmetry of each participating orbital must be conserved during the reaction process dramatically transformed the understanding of concerted pericyclic reactions and stimulated much experimental work to test and extend their theory. The Woodward and Hoffmann concept led to other related interpretations of orbital properties that are also successful in predicting and interpreting the course of concerted... 

A firm mechanistic understanding of concerted cycloadditions had to await the formulation of the reaction mechanism within the framework of molecular orbital theory. Consideration of the molecular orbitals of reactants and products revealed that in some cases a smooth transformation of orbitals of the reactant to those of the product is possible. In other cases, reactions that appear feasible, if no consideration is given to the symmetry and spatial orientation of the orbitals, are found to require high-energy transition states when the orbital properties are considered in detail. Those considerations have permitted description of many types of cycloadditions as allowed or forbidden. As has been discussed (Part A, Chapter 10), the application of orbital-symmetry relationships permits a conclusion as to whether a given concerted reaction is or is not energetically feasible. In this chapter, the synthetic application of these reactions will be emphasized. The same orbital-symmetry relationships that are informative as to the feasibility of a given reaction often are predictive of features of the stereochemistry of cycloaddition products. This predictable stereochemistry is an attractive feature for synthetic purposes. 

Fig. 6.24 A ligand n group orbital of symmetry. This orbital has the same symmetry properties as a rotation about the z axis. It is the fact that the ligand n orbitals have been chosen to be oriented with respect to the z axes (see caption to Fig. 6.22) that makes this orbital, mathematically and pictorially, particularly simple. Its two partners in Table 6.5 have the same symmetry properties as rotations about the x and y axes, respectively.

Because both electrons reside in the a bonding orbital, an electron configuration of can be used to describe the ground electronic state of H2. (Because H2 is a homonuclear diatomic molecule, we add a subscript g to the label——to indicate the orbital s symmetry property with respect to the center of the molecule. Electrons in the antibonding orbital are labeled a, the u also referring to the orbital s symmetry properties. Symmetry will be discussed in the next chapter.) To emphasize that the electrons in the a orbital derive from Is electrons from H atoms, the more detailed (cTgls) label can also be used. 

Pentadienyl System 2, 4-pentadienyl system possesses five molecular orbitals. Their symmetry properties and electron occupancies are given in Fig. 2.11. 

We have looked at the orbital properties of the main group AH5 molecules in Chapter 11. Two basic structures are known, the square pyramid (17.14) and the trigonal bipyramid (17.15). The ideal square pyramid has C4V symmetry. As a result. 

The EAN rule can also be explained in terms of molecular orbital theory. Housing of 18 electrons requires nine molecular orbitals, the HOMO of which should preferably be bonding in character. To produce these molecular orbitals, the metal combines all its valence orbitals with the ligand orbitals. The symmetry of the complex and the acceptor-donor properties of the ligands determine whether or not the HOMO is bonding, nonbonding, or antibonding in character. 

In this chapter the symmetry properties of atomie, hybrid, and moleeular orbitals are treated. It is important to keep in mind that both symmetry and eharaeteristies of orbital energetics and bonding "topology", as embodied in the orbital energies themselyes and the interaetions (i.e., hj yalues) among the orbitals, are inyolyed in determining the pattern of moleeular orbitals that arise in a partieular moleeule. 

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