As we discussed in the last post , the free electron gas model is not infallible. Almost 30 years after the electron gas model was proposed, quantum chemistry changed the perspective of looking at bonding and thus a new approach to metallic bonding was stated.
Felix Bloch graduated in Zurich, Switzerland and later went to Werner Heisenberg for his doctorate. In 1928 , he started working on applying quantum mechanics to solids.Till then, the quantum theory was just applied to gases and so this was the first attempt to apply this theory to solids.He applied the Schrödinger’s equation to solids and he got a set of solutions.
The band theory was developed later by Walter Hietler and Fritz London , where they applied the quantum theory of solids(MOT ) to Avogadro number(6.023 ×1023) of atoms at a time -LCAO of large no of atomic orbitals(AOs) to give molecular orbitals(MOs).
We have already studied that two AOs form two MOs – one lower energy bonding molecular orbital(BMO) and one higher energy anti bonding molecular orbital(ABMO). When this same approach is applied to an ensemble of atomic orbitals, we get very closely spaced BMOs and ABMOs, which appear to be like a band from a distance. The energy gap between them is very less.The energy gap is so less that the levels appear to be in a continuum.
Half of this band comprises of BMOs → Bonding band / Valence bond and
half band is made of ABMOs→ Conduction band.
Thus, if N atomic orbitals combine to form N molecular orbitals, then N/2 will be bonding orbitals and N/2 will be anti bonding.
If there is only one electron in the valence shell of the atom, then the band will be half filled i.e all the electrons will pair up the valence band. The ABMOs/ conduction band will remain vacant .In this case, the energy gap between the highest occupied BMO and lowest unoccupied ABMO is so small , that with application of a little potential difference , electrons can be excited from the valence band to the vacant conduction band.
e.g. – Copper – [Ar] 3d10 4s1
The energy gap between the valence and conduction band is called as the band gap.
If there is are two electrons in the valence shell of the atom, then the band will be fully filled i.e all the electrons will pair up and fill all orbitals. The ABMOs will NOT remain vacant.
e.g. -Manganese [Ar] 3d5 4s2
In this case, however, there are vacant p-orbitals (4p in case of Mn) which form the p- band.The lower end of this vacant p-band lies a little lower in energy than the topmost end of the s- band(as shown in the figure below). Thus, electrons can be promoted to this p-band easily with application of little potential difference. This theory thus explains why metals are good conductors of electricity.
The band theory also gives a plausible explanation for the fact that metals have lustre , i.e they have a shiny appearance. When light is shone upon a metal , the energy of this light is sufficient to excite the electrons to the vacant band in the metal structure.These electrons cascade down to emit light, which appears as a shine on the metal.
We will study later, the applications of band theory in deciding whether a substance is insulator, conductor or a semiconductor. In the next post , however, we will continue with our current topic – Chemical bonding. After learning about primary bonds, now we will move on to study secondary bonding in species.
Be a perpetual student of life and keep learning…
Good day !
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References and Further Reading –