[Dangos]

I'm currently in AP chemistry learning about properties of solids and liquids. One of the concepts in the chapter is about conductivity. The textbook explains that metallic solids like iron are conductive because the difference in energy between the filled and unfilled molecular orbitals is extremely small, whereas network solids like diamond are not conductive because the difference in energy between the filled and unfilled molecular orbitals are large.
Although I understand that the difference in energy between filled and unfilled molecular orbitals helps determine the conductivity of a solid, I am unsure about what determines the difference in energy between the filled and unfilled molecular orbitals.

[AdiDex]

Basically Two Things -:

1. Overlap Efficiency (Technical term is Overlap Integral) between Atomic Orbitals ---> More will be the overlap , more will be the difference between resulting  Bonding and non-bonding orbitals .

2. Similarity in Energies of Starting Orbitals ---> If Two orbitals has similar energy , then the overlap will be better , i.e. The difference between resulting bonding and non-bonding  will be better .

[Burner]

I'm currently in AP chemistry learning about properties of solids and liquids. One of the concepts in the chapter is about conductivity. The textbook explains that metallic solids like iron are conductive because the difference in energy between the filled and unfilled molecular orbitals is extremely small, whereas network solids like diamond are not conductive because the difference in energy between the filled and unfilled molecular orbitals are large.
Although I understand that the difference in energy between filled and unfilled molecular orbitals helps determine the conductivity of a solid, I am unsure about what determines the difference in energy between the filled and unfilled molecular orbitals.

I don't think the terms used here are very accurate. 'Atomic orbitals' should be used instead of 'molecular orbitals'. (Of course, more precise term would be 'valence band' and 'conduction band'). Also, because of this, there shouldn't be much relationship between conductivity and bonding/antibonding orbitals.

[Corribus]

Trying to understand conductivity of solids using atomic orbitals is a little bit like trying to understand Shakespeare starting with the alphabet. Technically possible, but missing many in-between steps.

What you really need is band theory, which models how the energies of many nearby atomic orbitals combine to form many molecular orbitals with varying energy values.  If you understand how two atomic orbitals combine to form two molecular orbitals with higher and lower energy, you are well on your way. Now imagine an very large (infinite limit) number of atomic orbitals combining to form a very large number of molecular orbitals. Rather than talking about individual orbitals at this point, it is more practical to speak of bands that are characterized by many molecular orbitals that are almost the same all blending together. The large number of electrons are then filled into these bands. Depending on whether the bands are fully filled or not with electrons is what determines whether the material is an electrical insulator, conductor, or semiconductor. To some extent you can predict this for solids based on inspection of the atomic orbitals of individual atoms, but not always.

You may find some more about band theory here: http://hyperphysics.phy-astr.gsu.edu/hbase/Solids/band.html

You may also find this video helpful: https://www.youtube.com/watch?v=zdmEaXnB-5Q

EDIT: Lol, having actually watched the video, I see it leaps into a similar problem with referring to atomic orbitals as being synonymous with bands. This must be a common thing in high school chemistry courses. I don't really like it, because it's not correct, but I guess such inaccuracies are part and parcel for teaching concepts like this at the high school level.  No wonder high school students are confused. If I find a better video I'll post it here.

[Enthalpy]

[...] what determines the difference in energy between the filled and unfilled molecular orbitals.

I don't know of any simple rule, method nor heuristic that would predict from its composition whether a solid is a conductor, a semiconductor or an insulator.

It's not a mere consequence of the elements' properties, as the same element or compound can be a semiconductor or an insulator depending on the allotrope. Pressure matters too: for instance hydrogen becomes metallic.

To some extent, atoms with very different electronegativity tend to make an insulating ceramic rather than a metal, but there are many exceptions like SnO₂ or AlN which are used as semiconductors, or like α-Sn (grey tin) which isn't metallic.

As well, bigger atoms tend to make a smaller bandgap. Compare (diamond) C, Si, Ge. Or Ga_1-xAl_xAs, GaAs...
http://www.ioffe.ru/SVA/NSM/Semicond/

But materials often have several valleys in their conduction band, and the elements' identity and proportion act differently on these valleys, putting one or an other lower, or even changing the crystal's structure. So while one might have expected BN to have a wider gap than AlN, it's not the case.

Experiment is the real answer here too. Computer simulations have nice successes in predicting them by computing wave functions.

One should note that materials called insulators 30 years ago are now used as semiconductors: MgO, Si, AlN...

I have no worry to use "molecular orbitals" as a synonym for "bands". It's the same as saying "one molecule" for a piece of metal or semiconductor, which I feel correct since chemical bonds link the atoms together. Though, when studying such huge molecules, you shouldn't imagine that a bond is local to an atom pair, even in an insulator: this concept is wrong but fruitful in chemical reactions, it's wrong and misleading for semiconductors.