Conductivity of a Conductor

Let’s consider the electrical conductivity of a material that is considered to be a conductor.
                             
Consider a single Na atom
The electron configruation of Na is 1s22s22p63s1.

                           The electron configuration of an isolated Na atom. 2 electrons in the 1s energy lever, 2 in the 2s, 6 in the 2p and 1 in the 3s.
                               
Now consider four Na atoms
Consider four Na atoms bonded together in a metallic bond to make a molecule:          
                        4 Na atoms next to each other each with the electron configuration showing the inner electrons as part of the individual atoms and the 3s electrons as part of the whole molecule.
                           
The 3s electrons from each Na atom are considered “delocalized”, i.e. part of the whole solid, and not associated with any one individual atom.  
The inner electrons (the electrons in the 1s, 2s and 2p orbitals) are essentially unchanged when part of the solid.

Question: 
If the 3s electrons are considered part of the whole solid, is the Pauli Exclusion Principle violated? (Since the 3s energy level can hold only 2 electrons.) 
Answer: 
No!  The 3s level will split into four distinct and slightly different energy levels to avoid violating the Pauli Exclusion Principle.
         
                                                      
Now consider lots of Na atoms
Consider millions of Na atoms bonded together in a solid:
                                                 
                Lots of Na atoms showing the inner electrons in their orbitals, unchanged from the isolated Na atom, and the electrons in the outer 3s orbitals splitting into distinct but very close energy levels and so that all together they make up a pseudocontinuous energy band
Again, the inner core electrons are not directly involved in the bonding and their energy diagrams remain essentially unchanged.
                         
Again the 3s energy levels of all of the atoms will split into distinct energy levels to avoid violating the Pauli Exclusion Principle. There will be as many distinct energy levels as there are Na atoms. These levels will be very close together and the difference between the highest and the lowest energy is small. Hence, we have a pseudocontinuous energy band (half-filled in this case). It is simply a narrow range of energy levels with millions (as many as there are Na atoms in the solid) of discrete energy levels in it.
 
Since this energy band is only half filled the electrons in it have a high mobility, since it doesn't take much for an electron to move from one discrete energy level to another. (Remember the levels are very close together.) SInce this band is produced from valence electrons, it is termed the valance band. 
             
Overall Conclusion on the Conductivity of Conductors:
Metals are good electrical conductors because their valence band is only partially filled and the electrons there are charge carriers. The detailed picture of the energy levels will look different for different conductors than the case shown here, but the result is the same: A partially filled valance band where the electrons are charge carriers and have high mobility.

                    Valance band of a conductor with the bottom half shaded to indicate that it is only half (or partially) filled.



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