Subject: meaning of eigenvalues in dft

From: Uli Salzner <[email-masked]>
Date: Wed, 27 Mar 1996 10:19:19 -0330 (NST)

recently I posted a question concerning the meaning of eigenvalues in 
density functional theory. I received many very helpful references. 
Thanks to everybody who took the time to answer. Here is the original 
question and a summary of the answers:

Callaway and March (Sol. State Phys. 38, 135 (1984)) discuss the meaning 
or better "non-meaning" of eigenvalues in density functional methods. It 
seems that the band gap of a sufficiently large system can be calculated 
by taking the energy difference between the valence and the conduction 
band but that the difference between HOMO and LUMO in molecules can not 
be used in connection with Koopman's theorem. I am wondering whether this 
is the end of the story or whether it is possible for practical purposes 
to use HOMO and LUMO levels. Would that be a crude approximation or total 
garbage? 

I am especially interested in density functional methods as implemnented in 
GAUSSIAN 94, which lists the HOMO and LUMO levels in DFT just as for HF. It 
is quite tempting to use them and as far as I tried the results seem to be 
reasonable. What I would like to do is to compare solid state calculations 
(with Cerius2) and calculations for oligomers of increasing size and try to 
extrapolate towards the solid. I am only interested in systems at 0 
temperature where the occupation of the one-electron functions is either 
0 or 1.

I would be very thankful for any comments or related references. A 
summary of the answers will be posted.

*****************************************************************************

Dear Uli Salzner,

a good functional would give perfect ionization potential (IP). See Baerends et
al. I.J.Q.C. 52 (1994) 711 and Krieger et al. DFT ed. E.K.U. Gross and R.M.
Dreizler (1995) p 191, Plenum Press N.Y.

The bad IP shows the deficiencies of the funtionals. I've read a paper in which
the authors shifted simply the eigenvalues (e.g. subtracted 3 eV from the upper
eigenvalues), and got good agreement with the experiment (UPS). (?!)

I think some progress is expected in this field in the near future (may be on
the ICTCP II).

-- 

 Gabor I. Csonka	   e-mail: [email-masked]
 Lab.Chimie Theorique      Universite Henri Poincare
 B.P. 239                  54506 Vandoeuvre-les-Nancy FRANCE
 tel: +33-83.91.25.29      fax: +33-83.91.25.30

******************************************************************************

 i think that the work of Levy, and others 
actually shows that for the EXACT density,
and exchange-correlation functional the KS
orbital-energies will be -IP..... but of course any
approximation to these may not have any meaning.

Dr. Noj Malcolm
[email-masked]

******************************************************************************

> Callaway and March (Sol. State Phys. 38, 135 (1984)) discuss the meaning 
> or better "non-meaning" of eigenvalues in density functional methods. It 
> seems that the band gap of a sufficiently large system can be calculated 
> by taking the energy difference between the valence and the conduction 
> band but that the difference between HOMO and LUMO in molecules can not 
> be used in connection with Koopman's theorem. I am wondering whether this 
> is the end of the story or whether it is possible for practical purposes 
> to use HOMO and LUMO levels. Would that be a crude approximation or total 
> garbage? 

The question of how to compute excitation energies with DFT is not solved,
but useful discussions can be found, by people like Kohn, Theophilou (difficult
to read), Gross and Nagy. The idea of the Slater transition state is important
in this discussion.

A recent paper dealing with the subject is

Mel Levy, Excitation energies from density-functional orbital energies,
Physical Review A, Vol. 52, No. 6 (December 1995), pp. R4313-R4315.

The classical paper is

E. K. U. Gross, L. N. Oliveira and W. Kohn, Phys. Rev. A 37, 2805 (1988),

and two other papers in the same journal (following this one). Also have
a look at

Agnes Nagy, Exact ensemble exchange potentials for multiplets,
International Journal of Quantum Chemistry, Suppl. 29 (1995), pp. 297-301.

Hope this helps... Best regards,

Alain

--
Alain Kessi ([email-masked])
at Paul Scherrer Institut, Zuerich, Switzerland

*****************************************************************************

   Kohn-Sham eigenvalues do have meaning, but not the same as in HF. 
Moreover the full meaning of KS eigenvalue is not yet clear, but new 
discoveries are made. In particular, there is numerical evidence 
that certain differences of eigenvalues (orbital energies) are related
to the average of true singlet/triplet excitation energies in atoms,
provided that the exact V_xc potential is used  (work of A. Savin and
co-workers presented at the 1996 APS meeting).
    Here are some references about the meaning(s) of Kohn-Sham
eigenvalues:

1. Density Functional Theory of Atoms and Molecules, by Parr and
   Yang, section 7-6 (Oxford Press, 1990) : an overview.

2. Slater, "The SCF field for Molecules and Solids: Quantum Theory
   of Molecules and Solids", vol. 4, New York (McGraw Hill, 1974);
   based on semi-empirical X-alpha and thus a little out of the
   spirit of "modern" DFT, but still an excellent reference, with
   lucid description of the physics.

3. Slater, Adv. Quantum Chem. 6 (1972) 1 : describes
   "Slater's transition state (STS) method", where eigenvalues out of
   SCF calculations for configurations with non-integer occupation are
   used to approximate total energy differences.  (might be a practical
   way of getting better estimates of true HOMO-LUMO excitation energies
   than straight orbital energy differences).

4. Williams et al.. J. Chem. Phys. 63 (1975) 628: a generalization
   and refinement of the STS method.

5. Janak, Phys. Rev. B 18 (1978) 7165: shows that eigenvalues
   eps_i are equal to partial derivatives of the total energy
   w.r.t. the occupation numbers for any XC functional (including
   of course the unknown exact XC):
     eps_i = del(E) / del(n_i)

6. Perdew and Zunger, Phys. Rev. B 23 (1981) 5048: discusses a serious
   shortcoming of the LDA and a remedy, the Self-Interaction Correction
   (SIC).  In a scheme with SIC, the KS eigenvalues obtained are typically
   much closer to ionization potentials and electron affinities than in
   a LDA scheme.  Following the PZ 1981 paper, there is a number of
   SIC papers showing this, search for authors Harrison and Whitehead.

7. Perdew and Levy, Phys. Rev. Lett. 51 (1983) 1884 : the title
   is "Physical content of the exact KS orbital energies: band
   gaps and derivative discontinuities."

8. Fritsche et al. has suggested a simple correction to orbital energy
   differences to get better estimates of excitation energies in
   solids but it is not clear if that "works" for small molecules.
   see L. Fritsche Physica B 172 (1991) 7 and possibly more recent
   papers by the same author.

9. In another context, Malkin et al. (J. Am. Chem. Soc.
   116 (1994) 5898) proposed an equally simple, but very different,
   correction to orbital energy differences.  Although they intended
   this correction to improve sum-over-states (SOS) expressions, it
   could be tried as improvement to the HOMO-LUMO energy difference.

   Related to this:

   The total energy can be written as a sum of eigenvalues (those
of occupied orbitals) plus "other terms".   These other terms have
some qualitative meaning, therefore the sum of eigenvalues also
has qualitative meaning, see
10. Gopinathan and Whitehead, Israel J. Chem. 19 (1980) 209
11. J. Harris, Int. J. Quantum Chem. 13 (1979) 189.

   There are more references, including some very recent work.
I guess there is a lot more meaning to the KS eigenvalues than was
believed some years ago and that there is still important things
to be discovered, both on formal and numerical grounds.

   For your particular problem, many empirical schemes could be
tried:  straight orbital energy difference, STS, maybe some kind of
empirical SIC (the correct SIC scheme is fairly complicated and
compute-intensive), Fritsche's or Malkin's corrections.

    Regards,
               Rene Fournier     [email-masked]

****************************************************************************

    Dear Dr. Salzner;

    Dr. Levy of Tulane Univ. just presented a talk at the APS meeting
on precisely the meaning of HOMO-LUMO Kohn-Sham orbital energy differences,
and its relationship to the true excitation energy under certain assumptions.
Unfortunately my scribbled notes are grossly incomplete, but here is a
reference that you may want to check to learn about that:

AUTHOR(s):       Levy, Mel 
TITLE(s):        Excitation energies from density-functional orbital
                   energies.                                                  

           In:   Physical review.  A,  Atomic, molecular, and opt 
                 DEC 01 1995 v 52 n 6 
         Page:   R4313 

   Regards,
               Rene Fournier     [email-masked]