Subject: CCL:summary: solid state/band structures

From: Keith Refson <[email-masked]>
Date: Wed, 01 Oct 1997 12:16:20 +0100 (BST)

Ulrike Salzner writes:
 
 > There seems to be a considerable gap between the physicists and chemists 
 > working in this field. 

That is certainly true!  

Can I add one comment which might avoid misunderstandings.  The term
"bandstructure calculation" would usually refer to a computation of
the eigenvalues (and wavefunctions), ie spectroscopy only.  The term
"total energy calculation" is often used to subsume all QM
calculations of the total energy, forces, optimization etc.  One
reason for making the distinction is that good bandstructure methods
(such as KKR, GW) can not compute total energies(*), and good total
energy methods (ie DFT) are poor for bandstructures!

 > Following the chemical literature little meaning 
 > is assigned to DFT orbital energies and, to my knowledge, there is no 
 > physical justification do interpret them as IPs and EAs.

Well, hmm.  It's not quite true that Kohn-sham eigenstates have no
physical significance.  They are "quasiparticle" states and their
eigenvalues are formally the derivatives of total energy with respect
to occupation.  However there is no formal way of constructing a
many-body wave-function from Kohn-Sham eigenstates so they are NOT
orbitals in the same sense as Hartree-Fock orbitals.  

*BUT* and surprisingly at first, because of the lack of formal
justification, the LDA Kohn-Sham eigenstate spectrum *sometimes* does
look *very* like the true (using advanced methods) excitation
spectrum, with the proviso that the band gap (in insulators) is
underestimated by 30-50%.  The proviso is that this applies to
weakly-correlated systems only.  In a strongly-correlated system (
narrow d-band transition metal compounds) such as NiO the LSDA/GGA band
structure is badly wrong.

 > 
 > However, there are all these DFT band structure codes which determine 
 > band gaps. 

Formally they must compute eigenvalues in order to evaluate the total
energy.  That doesn't mean you have to believe the eigenvalues of
excited states!

 > The band gap problem but not the fundamental question whether 
 > DFT eigenenergies should be used at all has been discussed. 

Oh yes it has!  At every electronic structure meeting I have ever been
to! 

 >  it appeared to me that DFT orbital energies have 
 > physical meening although the matter is by far not settled.

I think the question is *what* physical meaning and in which
circumstances.

There's a very nice paper by Uwe Schonberger, Phys Rev B 52 (1995)
8788-8793.  He does GW calculations on MgO and compares the band
structure with LSDA and with other methods.  What is *very*
interesting is that the LSDA band structure is quantitatively and
qualitatively correct apart from too narrow a band gap.  In other
words, if you simply shift the LSDA excited states upwards by 3 eV
they match the GW ones very well.   This  suggests that the
narrow gap is a fault of the LSDA rather than some (hypothetical?)
exact DFT.  I think it was Perdew who showed there must be a
discontinuity in the exchange-correlation potential at the fermi
energy in exact DFT which is not found in the LSDA (+GGA)
approximations.

One thing I have wondered but never investigated is whether the
bandgap error is systematic enough to make a correction for.  Does
anyone know whether this has been looked at seriously?

 > 
 > I am trying to obtain band gaps by extrapolating oligomer HOMO-LUMO gaps 
 > using DFT and run into considerable problems with reviewers (most likely 
 > chemists) who usually point out that DFT can not be used for this.

I would hope that physicists would give you a hard time for this too!

 > As far as I understand band structure calculations, the band gaps are  
 > analogous to HOMO-LUMO gaps in molecules. 

Almost exactly.  Except that you must specify (or assume) the
G-vectors at the band edges because of the dispersion.

 > Is there a fundamental difference between the 
 > DFT eigenenergies of molecules and of solids? Any comments would be 
 > greatly appreciated.

No.  Except for the existence of dispersive bands at all! Take a look
at Peter Bloechl's PAW calculation of a Ferrocene molecule or Graham
Acklands calculations of small liquid crystal molecules using
plane-waves and pseudopotentials!

 > 
 > Concerning the comment on ab initio codes in Georg Schreckenbach's summary: 
 > there is also a Hartree-Fock solid state program in Erlangen/Germany. 
 > This program seems to be able to do MP2 corrections. Moreover, W. 
 > Foerner (also formerly in Erlangen) et al. published a paper in J. Chem. 
 > Phys., 1997, 106, pp. 10249 on coulped cluster theory applied to polymers. 

I wasn't aware of these.  Are there any more details?

One new code I ought to add to the list is SIESTA, which is a periodic
boundary-conditions, DFT code with a local-orbital basis set.  It does
forces, optimizations MD and is apparently order N in scaling.  It is
being developed in Madrid and we are hoping to import it to the UK to
try out some large-scale calculations on minerals.  See
P. Ordejon et al Phys. Rev B51 (1995) 1456-1476 and Phys. Rev. B53
(1996) 10441-10444

(*) I had better put this in in case I am contradicted.  I do expect
this comment to be out of date soon.  There is a plethora of work on
postLDA and post-density-functional methods.  I recently saw a paper
claiming total energies from KKR methods and there is certainly work
done in Materials Science at Oxford using LDA+Hubbard U which seems to
give good bandstructures for NiO and UO2 surfaces!  It's an exciting
time in the field and it will be interesting to see what methods we
are all using in 5 years!

Keith Refson