Bonding and Antibonding(metals insulators)

[Bassalisk]

When we stack orbitals we get bands. Bands result in bonding and anti-bonding. Now for Na his s band is half full resulting in the top half to be conductive shell of anti-bonding orbitals.

Take Mg for instance. So his s band is completely full. Why does it conduct ? Because, as I learned, p band partially overlaps with s band thus enabling electrons to jump into conductive band.

Carbon band, after sp3 hybridization(I do not completely understand this principle) makes an energy gap. Why?

Why don't we have this energy gap in metals? Why doesn't this conducting band overlap in insulators too?

 

[cgk]

Carbon band, after sp3 hybridization(I do not completely understand this principle) makes an energy gap. Why?

For this question it is very hard to give an answer which is more concrete than "because the Schroedinger equation says so". Different arrangements of atoms lead to different realizable electronic structures of the total system, and in most cases the concrete outcome is hard to predict based on physical principles alone. (Although, if you know the answer, it is often possible to come up with reasons for the electronic arrangements in retrospect). It's a complex interplay of atomic properties, atom-atom interactions, and large-scale multibody effects.

Note that the actual question is "why is there a gap at ambient pressure and room temperature". For example, highly compressed hydrogen solids become metallic, and also for your diamond you may be able to significantly change its electronic properties by outside effects.

Why don't we have this energy gap in metals? Why doesn't this conducting band overlap in insulators too?

Materials are metallic *because* they don't have the gap, or they are insulating *because* they do have the gap. That's because the non-presence of the gap is required to allow the ellectronic mechanism behind metallic conductance to work.

 

[Bassalisk]

Basically it goes down to Schrödinger anyway... I will try to dig it out, thanks

 

[DrDu]

This you should have learned in your chemistry class, already.
While basically everything goes down to Schroedinger, that explains nothing.
The main difference between a metal like sodium and a non-metal like carbon of relevance here is their difference in electronegativity and the relatively low number of valence electrons per atom in the case of metals.
This means that a metal atom will allow for much more nearest neighbours than it has valence electrons. These electrons may roam quasi freely among the atoms. A little thought shows that an independent motion of the electrons will lead to strongly fluctuating charges on the atoms, i.e. the bonding has a considerable ionic character. This does not matter in the case of metals as their electronegativity ( the difference in energy of a cation and an anion) is low.
In the case of a non-metal the electronegativity is high which means that the bonds have to be strongly covalent. A description by MO or band theory itself is not so good to describe bonding in these substances, e.g. even for such an inocent molecule like F_2 no bond results from an MO treatment.
The strong repulsion of the filled atomic orbitals will lead to the formation of only just enough covalent bonds to pair up the valence electrons.
Note that there are two important factors: One is the difference in electronic structure, the other one is the resulting difference in crystal structure, namely some closest packed structure in the case of metals versus some molecular or complex lattice structure in the case of non-metals.

 

[asheg]

This you should have learned in your chemistry class, already.
While basically everything goes down to Schroedinger, that explains nothing.
The main difference between a metal like sodium and a non-metal like carbon of relevance here is their difference in electronegativity and the relatively low number of valence electrons per atom in the case of metals.
This means that a metal atom will allow for much more nearest neighbours than it has valence electrons. These electrons may roam quasi freely among the atoms. A little thought shows that an independent motion of the electrons will lead to strongly fluctuating charges on the atoms, i.e. the bonding has a considerable ionic character. This does not matter in the case of metals as their electronegativity ( the difference in energy of a cation and an anion) is low.
In the case of a non-metal the electronegativity is high which means that the bonds have to be strongly covalent. A description by MO or band theory itself is not so good to describe bonding in these substances, e.g. even for such an inocent molecule like F_2 no bond results from an MO treatment.
The strong repulsion of the filled atomic orbitals will lead to the formation of only just enough covalent bonds to pair up the valence electrons.
Note that there are two important factors: One is the difference in electronic structure, the other one is the resulting difference in crystal structure, namely some closest packed structure in the case of metals versus some molecular or complex lattice structure in the case of non-metals.

I'm not sure that one can find out crystal properties from electronegativity. For example, Diamond and Graphite made of carbon atoms, one is insulator and the other one is conductor. I think we should solve Schrödinger equation and fill valence electrons in the band and see if there will be a gap or not.

 

[DrDu]

I'm not sure that one can find out crystal properties from electronegativity. For example, Diamond and Graphite made of carbon atoms, one is insulator and the other one is conductor. I think we should solve Schrödinger equation and fill valence electrons in the band and see if there will be a gap or not.

Of course EN is not enough. But it does already a good job to separate metals from non-metals and to get a qualitative understanding of the difference.
It also is a good criterion to distinguish when an effective one-electron theory i.e. simple band picture is qualitatively correct and when electronic correlation effects are important.
Furthermore note that Graphite is a very bad conductor and I would certainly not classify it as a metal.
My main point is that band theory is very usefull to describe metals and maybe semiconductors, especially on a computational basis. It is also a starting point for more sophisticated calculations but it may not be the best model to describe qualitatively bonding, especially in insulators.

The other point I wanted to make, especially with respect to the original question, is that carbon on one hand side and metals like sodium or magnesium have very different crystal structures with few neighbours in the first case and many neighbours in the second, which is the main reason that the band structure has a gap in case of carbon (in diamond at least) and no gap in case of the metals.
A nice discussion of bonding in terms of solid state concepts can be found in
Walter A Harrison, Electronic structure and the properties of solids: The physics of the chemical bond (Dover, Paperback)

 

[Bassalisk]

Thanks will check that book out !