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# 化学代写|无机化学代考Inorganic Chemistry代写|CHEM-UA711 Bands formed from overlapping atomic orbitals

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## 化学代写|无机化学代考Inorganic Chemistry代写|Bands formed from overlapping atomic orbitals

The central idea underlying the description of the electronic structure of solids is that the valence electrons supplied by the atoms spread through the entire structure. This concept is expressed more formally by making a simple extension of MO theory in which the solid is treated like an indefinitely large molecule. In solid-state physics, this approach is called the tight-binding approximation. The description in terms of delocalized electrons can also be used to describe nonmetallic solids. We therefore begin by showing how metals are described in terms of molecular orbitals. Then we go on to show that the same principles can be applied, but with a different outcome, to ionic and molecular solids.

## 化学代写|无机化学代考Inorganic Chemistry代写|Band formation by orbital overlap

The overlap of a large number of atomic orbitals in a solid leads to a large number of molecular orbitals that are closely spaced in energy and so form an almost continuous band of energy levels (Fig. 3.61). Bands are separated by band gaps, which are values of the energy for which there is no molecular orbital.

The formation of bands can be understood by considering a line of atoms, and supposing that each atom has an s orbital that overlaps the s orbitals on its immediate neighbours (Fig. 3.62). When the line consists of only two atoms, there is a bonding and an antibonding molecular orbital. When a third atom joins them, there are three molecular orbitals. The central orbital of the set is nonbonding and the outer two are at low energy and high energy, respectively. As more atoms are added, each one contributes an atomic orbital, and hence one more molecular orbital is formed. When there are $N$ atoms in the line, there are $N$ molecular orbitals. The orbital of lowest energy has no nodes between neighbouring atoms and the orbital of highest energy has a node between every pair of neighbours. The remaining orbitals have successively $1,2, \ldots$ internuclear nodes and a corresponding range of energies between the two extremes.

The total width of the band, which remains finite even as $N$ approaches infinity (as shown in Fig. 3.63), depends on the strength of the interaction between neighbouring atoms. The greater the strength of interaction (in broad terms, the greater the degree of overlap between neighbours), the greater the energy separation of the non-node orbital and the all-node orbital. However, whatever the number of atomic orbitals used to form the molecular orbitals, there is only a finite spread of orbital energies (as depicted in Fig. 3.63). It follows that the separation in energy between neighbouring orbitals must approach zero as $N$ approaches infinity, otherwise the range of orbital energies could not be finite. That is, a band consists of a countable number but near-continuum of energy levels.

## 化学代写|无机化学代考Inorganic Chemistry代写|Band formation by orbital overlap

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