Orbitals and States

An orbital is an energy level available to an electron in an atom or molecule. Since atoms and molecules usually contain many electrons, there are many orbitals, and they form the rungs of an energy ladder. Each rung can accept two electrons, but it can also take just one electron or remain altogether empty. 

Orbitals are however only mathematical concepts and the observable properties of atoms and molecules are their energy states. Such a state must be defined by the distribution —the wavefunction —of all the nuclei and all the electrons of the molecule. This would be a formidable problem and in practice the wavefunctions of the heavy nuclei are separated from those of the light electrons. 

This separation is known as the Born-Oppenheimer approximation. This assumes that the nuclei (being thousands of times heavier than electrons) remain motionless while electrons move around them. 

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Because of the pyramidal shape in these excited states the orbitals and states may be reclassified according to the Q point group (Table A. 1 in Appendix A). 

In a concerted reaction, orbital and state symmetry is conserved throughout the course of the reaction. Thus a symmetric orbital in butadiene must transform into a symmetric orbital in cyclobutene and an antisymmetric orbital must transform into an antisymmetric orbital. In drawing the correlation diagram, molecular orbitals of one symmetry on one side of the diagram are connected to orbitals of the same symmetry on the other side, while observing the noncrossing rule. 

Construct orbital and state diagrams for the following processes ... 

Fig. 5. Energies of jr-electron orbitals and states of ethylene as a function of twist angle... 

Fig. 6. Energies of orbitals and states of a single-bonded A2 molecule as a function of the bond length...

FIGURE3.24. Orbital and state correlations for the homolytic cleavage of radicals or ion radicals. Adapted from Figure 2 of reference 36, with permission from the American Chemical Society. 

We first examine the relationships between electron structure and the emission and absorption spectroscopy of metal complexes. Transition metal complexes are characterized by partially filled d orbitals. To a large measure the ordering and occupancy of these orbitals determines emissive properties. Figure 4.2 shows an orbital and state diagram for a representative octahedral MX6 d6 metal complex where M is the metal and X is a ligand that coordinates or binds at one site. The octahedral crystal field of the ligands splits the initially degenerate five atomic d-orbitals by an amount... 

Figure 4.2. Simplified schematic orbital and state diagrams for a rf metal in an octahedral environment showing d and n bonding and rr andbonding (n ) orbitals. A strong crystal field is assumed so that the t2 levels are filled. Ligand to metal charge transfer states are ignored.

Fig. Z The three types of radical electrocyclic reaction, as defined by Bischof [8] and Haselbach et al. [9] The notation A-C is taken from Ref [9]. Reactions of type C are essentially unknown, whereas tyi s A and B are both orbital- and state-symmetry forbidden...

Problem 8.28 (a) Apply the MO theory to the allyl system (cf. Problem 8.26). Indicate the relative energies of the molecular orbitals and state if they are bonding, nonbonding, or antibonding, (b) Insert the electrons for the carbocation C,H, the free radical C,H, and the carbanion CjH, and compare the relative energies of these three species. 

The MOs and electronic states of carbene have been discussed in Chapter 7. The orbital and state correlation diagrams for addition of CH2 to ethylene is shown in Figure 14.9. The Walsh bonding picture for the MOs of cyclopropane requires that the and a MOs of the ethylene also be included in the diagram. The a2 and least-motion pathway preserves a vertical plane of symmetry (as well as the other elements of the C2v point group), and the... 

Decide on the basis of orbital and state correlation diagrams whether or not the reaction is photolytically allowed. 

The photochemical dimerization of unsaturated hydrocarbons such as olefins and aromatics, cycloaddition reactions including the addition of 02 ( A ) to form endoperoxides and photochemical Diels-Alders reaction can be rationalized by the Woodward-Hoffman Rule. The rule is based on the principle that the symmetry of the reactants must be conserved in the products. From the analysis of the orbital and state symmetries of the initial and final state, a state correlation diagram can be set up which immediately helps to make predictions regarding the feasibility of the reaction. If a reaction is not allowed by the rule for the conservation of symmetry, it may not occur even if thermodynamically allowed. 

The numerical evaluation of the energies of orbitals and states is fundamentally a matter of making quantum mechanical computations. As indicated in Chapter 1, quantum mechanics per se is not the subject of this book, and indeed we have tried in general to avoid any detailed treatment of methods for solving the wave equation, emphasis being placed on the properties that the wave functions must have purely for reasons of symmetry and irrespective of their explicit analytical form. However, this discussion of the symmetry aspects of ligand field theory would be artificial and unsatisfying without some... 

Figure 7 Energy orderings of the orbitals and states of some [CrX(NH3)5]2+ complexes (X is Cl, Br, NCS) (from ref. 35)...

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