![]() Consequently, only the lower half of the band is filled. Just as with atomic orbitals or molecular orbitals, the electrons occupy the lowest energy levels available. Recall, however, that each of the metal atoms we started with contained only a single electron in each s orbital, so there are only n electrons to place in the band. Each of the original s orbitals could contain a maximum of two electrons, so the band can accommodate a total of 2 n electrons. Because the band contains as many energy levels as molecular orbitals, and the number of molecular orbitals is the same as the number of interacting atomic orbitals, the band in Figure 8.6.1 contains n energy levels corresponding to the combining of s orbitals from n metal atoms. and is proportional to the strength of the interaction between orbitals on adjacent atoms: the stronger the interaction, the larger the bandwidth. The difference in energy between the highest and lowest energy levels is the bandwidth The difference in energy between the highest and lowest energy levels in an energy band. The continuous set of allowed energy levels shown on the right in Figure 8.6.1 is called an energy band The continuous set of allowed energy levels generated in band theory when the valence orbitals of the atoms in a solid interact with one another, thus creating a set of molecular orbitals that extend throughout the solid. The levels that are lowest in energy correspond to mostly bonding combinations of atomic orbitals, those highest in energy correspond to mostly antibonding combinations, and those in the middle correspond to essentially nonbonding combinations. The result is essentially a continuum of energy levels, as shown on the right in Figure 8.6.1, each of which corresponds to a particular molecular orbital extending throughout the linear array of metal atoms. For n = 30, there are still discrete, well-resolved energy levels, but as n increases from 30 to a number close to Avogadro’s number, the spacing between adjacent energy levels becomes almost infinitely small. The energy separation between adjacent orbitals decreases as the number of interacting orbitals increases. Molecular orbitals of intermediate energy have fewer nodes than the totally antibonding molecular orbital. The lowest-energy molecular orbital corresponds to positive overlap between all the atomic orbitals to give a totally bonding combination, whereas the highest-energy molecular orbital contains a node between each pair of atoms and is thus totally antibonding.Īs we saw in Chapter 5, the lowest-energy orbital is the completely bonding molecular orbital, whereas the highest-energy orbital is the completely antibonding molecular orbital. As n becomes very large, the energy separation between adjacent levels becomes so small that a single continuous band of allowed energy levels results. \)įigure 8.6.1 The Molecular Orbital Energy-Level Diagram for a Linear Arrangement of n Atoms, Each of Which Contains a Singly Occupied s Orbital.
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