how & why discrete energy levels of atoms changes into energy bands when they form molecule?

Question

If we are considering one atom,it has discrete energy levels.If we are considering a crystal or a molecule comprising of more than one atom ,it has energy bands.
How and for what purpose the trasfer from discrete energy level to energy bands occures?Why in molecule,it can't be like energy levels?
consider Si crystal..it is coposed of identical Si atoms..having identical energyl evels..but when they form the crystal,we are getting a range of energy ..not discrete..even though no two electron can occupy all the four quantumstates the same(pauli's exclusion principle)..,is it not possible to be like discrete energy levels..?

Answer 1

Electrons always have discrete energy levels, no matter whether they are in isolated atoms, in a multi-atom molecule, or in an extended lattice of atoms in a solid. The difference among these cases are the sizes of the energy differences separating the different energy levels available to the electrons in the system.

In any multi-atom system in which the atoms are bonded to one another each atom contributes at least one atomic orbital to the system of molecular orbitals in the system. For an N-atom system, there are at least N molecular orbitals (and N energy levels, some of which may be degenerate) produced by combinations of the atomic orbitals. There is a very large number of atoms present in a typical macroscopic sample of a solid. For example, for a 1 mg sample of Si, there are ~6*10^23 atoms/mol * 0.001 gm/23 gm/mol = 2.6*10^19 atoms. If each Si atom only contributes one atomic orbital to the system of molecular orbitals, there will be 2.6*10^19 available molecular orbitals, each with a characteristic energy. There are so many orbitals that the energy differences between them become very, very small, and the allowable energy states "blend together" and become almost indistingushable. These clusters of closely spaced energy levels are what form the band structure of solids.

See sources for pictures, which help visualize this concept.

Answer 2

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RE:
how & why discrete energy levels of atoms changes into energy bands when they form molecule?
If we are considering one atom,it has discrete energy levels.If we are considering a crystal or a molecule comprising of more than one atom ,it has energy bands.
How and for what purpose the trasfer from discrete energy level to energy bands occures?Why in molecule,it can't be like energy...

Answer 3

Dear Bipin,
The low lying electrons in an atom are not perturbed much by being in an agglomeration of atoms because their wavefunctions are not affected.
The valency electrons however lie in a potential which is composed of a linear combination of potentials from all the other atoms. This means that the energy levels can not be the same as in the isolated atom.
In most university physics courses you do classical Mechanics Course and in particular you do the solution of many equal masses attached to identical springs. This gives rise to solutions to coupled systems which shows what is happening at the quantum level of many bodied systems.
Unfortunately we can not solve these complex quantum equations and fall back on the analogy with the classical system in explaining energy bands and gaps.
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