Properties of Molecular Orbitals

Because both R and are in units of A, it is unnecessary to consider any unit conversions. Using the expressions above, 

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FIGURE 12.18 Radial plots of the molecular orbitals and their probabilities  

FIGURE 12.20 Aqualitative molecular orbital diagram for Hj is very similar to that of Hj, except for the presence of the second electron with a spin opposite that of the first. 

Bond order is qualitatively related to the strength and number (that is, single, double, triple) of bonds between atoms in a molecule. 

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It is to be noted that there is nothing fundamentally sacred in nodal properties of molecular orbitals. They only appear at the heart of the argument since it is couched in MO terms. Other equivalent approaches are possible, e.g., in VB terms 9>U>14) and in those other concepts will appear to be of primary importance. The choice of the MO theory is merely a matter of convenience, which is particularly pronounced in photochemical applications. 

The symmetry properties of molecular orbitals of cyclohexadiene are given in the following... 

SYMMETRY PROPERTIES OF MOLECULAR ORBITALS CORRELATION DIAGRAMS... 

The essence of the model is to build up any reaction profile through the qualitative mixing of appropriate electronic configurations. Thus, just as the properties of molecular orbitals (energy, geometry) may be readily appreciated by analysis of the atomic orbitals from which they are derived, the essence of a reaction profile may be more readily comprehended by understanding the building blocks from which it is constructed. 

Similarly, many people have used the concept of localization energy to account for the rates, and sites, of electrophilic substitution. The localization energy is a calculated value of the endothermicity in a reaction and is therefore part of an argument based on product development control. The plot of localization energy against rate constant is also a good straight line.60 This is no place to try to estimate the relative success of these two approaches they are obviously related in some deep-seated property of molecular orbitals.61... 

One important aspect of MO theory— the concept of symmetry-correct molecular orbitals— not only makes it possible to explain this experimental result but, in fact, requires it. A fimdamental property of molecular orbitals is that they have the full symmetry of the basis set of atomic orbitals used to generate the molecular orbitals. This means that the molecular orbitals must be either symmetric or antisymmetric with respect to the symmetry operations provided for by the symmetry group of the atomic orbitals.If we consider each C-H bond formed by overlap of an sp orbital on a carbon atom with a Is orbital on a hydrogen atom to be an MO, then clearly each of these MOs lacks the full symmetry of the basis set of s and p atomic orbitals. 

As a prelude to further analysis, it is useful to review one important property of molecular orbitals. As noted in Chapter 1, symmetry-correct molecular orbitals must be either symmetric or antisymmetric with respect to the full symmetry of the basis set of atomic orbitals that are used to construct the molecular orbitals. In the analysis of orbital symmetries, we will need to consider only the number of molecular S5nnmetry elements that are sufficient to distinguish between allowed and forbidden pathways. Also, it is not necessary to consider here the minor perturbation of molecular orbital symmetry that results from isotopic or alkyl substitution. In other words, to a first approximation the basis set orbitals of any conjugated diene are considered to be the same as those for 1,3-butadiene. Figtue 11.13 shows the... 

FIGURE 2.3 Symmetry properties of molecular orbitals of cyclobutene and butadiene. 

Other combinations of reactants such as two k electrons with two ti electrons or four K electrons with four k electrons do not react under thermal conditions, but readily react if they are irradiated with visible or ultraviolet light. These rules of reactivity in cycloadditions were discovered by Robert B. Woodward and Roald Hoffmann on the basis of quantum-mechanical analysis and the symmetry properties of molecular orbitals. These rules are known as the Woodward-Hoffmann rules. 

HaU, G. G., and J. Lennard-Jones. 1950. The molecular orbital theory of chemical valency. III. Properties of molecular orbitals. Proceedings of the Royal Society of London. Series A 202 155-165. 

From this point, we restrict our discussion to quantum mechanical calculations. Quantum mechanics gives us electronic structure, and electronic strucmre, in effect, gives us chemistry. This approach therefore allows us to follow chemical reaction profiles that involve bond-making/breaking processes, and to calculate thermodynamic properties, along with the properties of molecular orbitals, electron densities, and just about any type of spectroscopic property you can think of. It is no small wonder, then, that computational chemistry is now an essential technique to aid and guide the interpretation of experimental results. 

Each eigenvector is a collection of coefficients for the LCAO that form the corresponding MO. Using this information, a visualization of the shape and directional properties of molecular orbitals becomes possible. Reactivity is related to electronic redistribution and, in particular, to electron exchanges between HOMO and LUMO of reacting partners from the topology and symmetry properties of these frontier orbitals it is sometimes possible to obtain a qualitative prediction on the stereochemical course of chemical reactions [7]. 

Tabular representation of S3munetry properties of molecular orbitals of 1, 3, 5-hexatriene ... 

One can construct other indices for benzene character of benzenoid hydrocarbons, which we will designate as B in order to differentiate them from the index B. For example, one can consider the Kekule index, which assigns to individual Kekule valence structures an index derived from local properties of molecular orbitals, and take the average over all Kekule structures. Even though for many Kekule structures this index is bigger than the Kekule index of benzene, the average Kekule index appears smaller for polycyclic benzenoid hydrocarbons than it is for benzene. However, for an index to reflect benzene character, one expects certain trends among structurally related benzenoids to be satisfied, such as... 

The shapes of the orbitals are shown in Fig. 1.21, Note that the number of nodes increases as the energy increases, a general property of molecular orbitals. 

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