Re: [Phys-l] Band splitting in Carbon (diamond)...

[Stefan Jeglinski <jeglin@4pi.com>]

>  >> The valence band
> > >  is ... "derived" from the
> > >  HOMO, "derived" from the outer occupied shell of the atom in
> > >  question. The conduction band is the set of lowest unoccupied
> > >  energies, "derived" from the LUMO, "derived" from the outer
> > >  unoccupied shell of the atom in question.
>
>   That just can't be right.  An example suffices to
>   prove the point:  gray tin is a semiconductor, while
>   white tin is a metal.  Same atomic levels, wildly
>   different bands structure.

Which I would argue must be ultimately "derived" from a difference in 
the way the Sn atoms must be bonding in the 2 allotropes (whatever 
that is - I do not know the details). I debated even bringing up the 
HOMO-LUMO concept and using the word "derive." I've found it to be 
useful as an intermediate "step" in thinking about bringing atoms 
together eventually to form a crystal, but like the tight-binding or 
free electron approximations, MO theory has its limits.

I don't in fact know whether constructing MOs to try to describe 
bonding in Sn, for example, makes any sense (that is, whether it 
produces a result that has any basis in fact), but I'm betting 
(perhaps incorrectly) that trying to fit experimental data for the 2 
allotropes (or doing some first principle QM calculation) to a MO 
approach would suggest 2 possible HOMO-LUMO configurations, 
ultimately analogous to 2 different band structures, one metallic and 
semiconductor.

If I recall my graduate days correctly, a MO is nothing more than a 
new basis consisting of some combination of atomic orbitals. The new 
basis can be used to better describe bonding in a molecule. As such, 
new energy levels are deduced, and 2 of them have the descriptors 
HOMO and LUMO.

>   In any case the point is that you can
>   tune the LUMO-HUMO splitting to be anything you
>   want (over a wide range) without changing the
>   atomic splitting.

I would think this is because the non-negligible MO basis 
coefficients (matrix elements?) can be numerous and/or complex (if 
calculated from first principles). That is to say, sure, the MO 
consists of some combination of atomic orbitals, but any atomic 
splitting is exceedingly difficult to straightforwardly relate to any 
LUMO-HOMO splitting.

>  For a semiconductor the LUMO-HUMO splitting

You've used "LUMO-HUMO" twice. Is that a typo or are you in fact 
referring to something called the "HUMO"?

Besides that, I wonder if we are talking past one another? I see MO 
as a useful concept "along the way" but I would not have guessed it 
to be relevant compared to band structure calculations. Yet you 
recommend thinking of a crystal as a macromolecule with HOMO and 
LUMO. I'll have to think about that more.

Another comment I made was stuck somewhere in cyberspace and just got 
posted. I would appreciate a critical review of it as well, although 
keep in mind, I'm well aware that my discussion was quite 
introductory :-)


Stefan Jeglinski