Subject: Summary: DFT and Koopmans' theorem

From: Kevin Gross <[email-masked]>
Date: Fri, 16 Feb 2001 11:39:31 -0500

Thank you for all the insightful responses regarding DFT and Koopmans'
theorem.

My question was simple:
>Does Koopmans' theorem still hold for MO energies obtained from DFT
>calculations? I'm curious because I've found that HOMO & LUMO energies
>obtained from HF calculations are much better regression descriptors
>than are the energies taken from identical B3LYP calcs. Any comments or
>references regarding this topic will be summarized.

To summarize most of the responses, Koopmans' theorem does not hold for
DFT.  Of course, DFT is not an MO method, so it doesn't make sense to
even talk about HOMOs or LUMOs (nonetheless, to avoid HOKSO/LUSKO
terminology, it is understood that in DFT, a HOMO energy refers to the
eigenvalue of the highest occupied Kohn-Sham orbital).  Apparently,
though, some interpretation of the Kohn-Sham orbital energies is
possible.  Being that B3LYP is a hybrid of HF and DFT, it seems as
though Koopmans' theorem does not apply to this specific case.

The individual responses are below.

Thanks,

Kevin Gross
Wright State University
Dayton, OH 45435

-----------------------------------------------------------------------
kevin - perhaps you should read "what do the kohn-sham orbitals and
eigenvalues mean?" in JACS, 1999, _121_, 3414-3420 by stowasser and
hoffman.

regards,

steve t
-----------------------------------------------------------------------
Kevin -

Koopmans' Theorem generally performs poorly when combined with DFT in my
experience.  The reason HF tends to work well is because HF ignores
correlation effects and Koopmans' Theorem ignores electronic relaxation
of the newly created cation.  We get 'lucky' in that the two tend to
balance out.  With DFT, the correlation is not neglected, so there is
nothing to balance out the poor approximation that Koopmans' Theorem
makes.

Dave

--
Dr. David J. Giesen
Eastman Kodak Company                           [email-masked]
2/83/RL MC 02216                                (ph) 1-716-58(8-0480)
Rochester, NY 14650                             (fax)1-716-588-1839
-----------------------------------------------------------------------
No, they don't. The fact that the B3LYP energies are not good
descriptors in a regression analysis is interesting, and I would offer
one word of caution before you consider this a firm conclusion:

Don't simply regress on 'HOMO energies'. Instead, regress on 'MO
energies' where you choose MO's of similar shape (this cautionary
statement applies to both HF and DFT energies).

DFT frequently orders MOs differently than HF, and you might find that
HOMO shapes are not correlated with each other (in which case one does
not expect a systematic variation in HOMO energy).

-Alan

====

Alan Shusterman
Department of Chemistry
3203 SE Woodstock Blvd
Reed College
Portland, OR 97202
503-771-1112, ext. 7699
-----------------------------------------------------------------------
    First B3LYP is not truly a KS-DFT method. It mixes KS-DFT and HF;
because of that I think that neither Janak theorem nor Koopmans' theorem
apply in B3LYP, but I'm not sure, I haven't checked.

   The question of Koopmans' theorem and KS-DFT is often asked, I recall
it's been discussed many times on CCL so it may be in the archives.

   In my opinion, Koopmans' theorem is not so important anyway:  it
relates a theoretical construct (HF orbital energies) to another
theoretical construct (the energy difference between the HF ground state
and a hypothetical state in which HF orbitals do not relax relative to
GS).  There IS something similar to a Koopmans' theorem in DFT, it's
Janak's theorem.  It states that:

(dE/dn_i) = e_i          (e_i = i'th orbital energy)

[Janak, Phys. Rev. B 18 (1978) 7165]

the derivative of the total energy w.r.t. to occupation number of
orbital "i" is exactly the orbital-energy (Kohn-Sham eigenvalue) "i".
This is true not only for the (unknown) exact exchange-correlation
potential but for all commonly used approximate XC potentials.  From
this formula one can in principle get ionization potentials by
integrating e_i over dn_i, between 1 and 0 (between 0 and 1 for an
electron affinity).  An approximation to this integral is to take the
value of the integrand at the midpoint,   e_i(n_i=1/2), and multiply by
the range of integration which is just (+/-)one (add/remove one
electron), which gives Slater's transition state approximation for IPs:

IP(Slater's TS) = - e_i(n_i=1/2)

[ Slater, Adv. Quantum Chem. 6 (1972) 1 ;
  Slater, "The SCF field for Molecules and Solids: Quantum Theory
   of Molecules and Solids", vol. 4, New York (McGraw Hill, 1974) ]

   Williams et al. refined Slater's formula, they got a better
approximation to the integral  for n_i from 1 to 0, see

[ Williams et al.. J. Chem. Phys. 63 (1975) 628 ]

    In a sense Janak's theorem is better than Koopmans' theorem because
it relates a theoretical construct to a real thing (if we integrate e_i
for n_i from 1 to 0, or from 0 to 1). IF ONE HAD THE EXACT
EXCHANGE-CORRELATION POTENTIAL, one would get the EXACT lowest
ionization potential by integrating Janak's formula; Slater's formula is
an approximation to that.

    I append an answer given by Kiet Nguyen to a related
question below, it contains useful references.

Cheers,
         Rene.
 --------------------------------------------------------------------
 | Rene Fournier                  | Bureau/Office  458  CCB         |
 | Chemistry,  York University    |      (416) 736 2100 Ext. 30687  |
 | 4700 Keele Street, North York  | FAX: (416) 736-5936             |
 | Ontario, CANADA   M3J 1P3      | e-mail: [email-masked]          |
 --------------------------------------------------------------------

#10  Kiet Nguyen,  [email-masked]

Although there is considerable controversy in the literature concerning
the meaning of Kohn-Sham orbitals,  I think they are far from
"meaningless".  This topic has been carefully analyzed and
reviewed.[1-5]   Parr and Yang stated that "... one should expect no
simple physical meaning for the Kohn-Sham orbital energies."[3]
However, using the Janak theorem which "provides a meaning for the
eigenvalues of the Kohn-Sham equation", they have connected the HOMO and
LUMO energies to the ionization potential and electron affinity,
respectively.[3]

[1]   E. J. Baerends and O. V. Gritsenko, J. Phys. Chem. A 101 (1997)
5383.
[2]   R. G. Parr and W. Yang, Density Functional Theory of Atoms and
Molecules (University Press,  Oxford, 1989).
[3]   J. P. Perdew, in: Density Functional Methods in Physics, edited by
R. M. Dreizler and J. da Providencia New York, 1985), p. 265
[4]   E. J. Baerends and O. V. Gritsenko, J. Phys. Chem. A 101 (1997)
5383.
[5] R. Stowasser and R.  Hoffman, J. Am. Chem. Soc. 121 (1999),  3414.

Kiet A. Nguyen
AFRL/MLPJ
Laser Hardened Materials Branch
Wright-Patterson AFB, OH  45433
Phone (937) 255-6671, Ext 3178
FAX   (937) 255-1128

==========
-----------------------------------------------------------------------
We touch this question briefly and refer to the relevant literature in
GI Csonka, BG Johnson Theor Chem Acc (1998) 99: 158-165 paper:

"Perdew, Parr et al. [1982 Phys. Rev. Lett.] showed that the exact KS
potential and its E(HOMO) change discontinuously as the electron number
N passes through an integer M. E(HOMO) is -I for N just below
M, -(I+A)/2 for N=M, and -A for N just
above M, where -A is the negative of the electron affinity." and "LSDA
and GGA, which miss the derivative discontinuity, effectively average
over it, providing an estimate of -(I+A)/2 for the E(HOMO)."

So Koopmans theorem does not hold for B3LYP.

For further details see the book of RG Parr and W Yang [1989] and
Perdew's work

I hope this helps,
with best regards

--
Gabor I. Csonka              Budapest University of Technology
FAX: (361) 463.36.42         Inorganic Chemistry Dept. Ch. Bldg
[email-masked]        H-1111, Bp. Szent Gellert ter 4
http://web.inc.bme.hu/~csonka/csg.html
http://www.ch.bme.hu/inc/csg.html
-----------------------------------------------------------------------
Dear Kevin:
I guess that with the true functional and within the Kohn-Sham scheme
the energy of the HOMO corresponds to the first ionization potential. I
don’t remember where is published the proff, If I find it i will send
you the reference.
Armando Navarro
Facultade de quimica
Departamento de quimica organica
Santiago de Compostela
Spain
-----------------------------------------------------------------------
No, Koopmans' theorem is derived with explicit reference to HF theory.
Regarding Kohn-Sham orbitals computed with modern DFT, the following
references should be of interst to you:

P. Politzer, F. Abu-Awwad: "A comparative analysis of Hartree-Fock and
Kohn.Sham orbital energies", Theor. Chem. Acc. 99, 83-87 (1998)

R. Stowasser, R. Hoffmann: "What Do the Kohn-Sham Orbitals and
Eigenvalues Mean?", J. Am. Chem. Soc. 121, 3414-3420 (1999)

Briefly: It is found that Kohn-Sham MO energies differ significantly
> from HF MO energies, but they tend to be linearly related.

Jens >--<

=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
JENS SPANGET-LARSEN         Phone: +45 4674 2000 (RUC)
Department of Chemistry            +45 4674 2710 (direct)
Roskilde University (RUC)   Fax:   +45 4674 3011
P.O.Box 260                 E-Mail: [email-masked]
DK-4000 Roskilde, Denmark   http://virgil.ruc.dk/~jsl/
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
-----------------------------------------------------------------------
Hi,

this is an interesting problem .. and I've been able to find a very nice
article to describe just that (Koopman's for DFT in terms of occupied
and virtual orbitals in comparison with HF) ..

look at: R Stowasser, R Hoffmann, JACS, 1999, 121, 3414 ..

hope this helps !!

bye
serge
-----------------------------------------------------------------------
Dear kevin,

the eigenvalues connected to the Kohn-Sham (KS)  orbitals do not have a
strict physical meaning. In KS theory there is no equivalent of
Koopmans' theorem, which could relate orbital energies to ionization
energies.
However, as a direct consequence of the long-range behaviour of the
charge density, the eigenvalue of the HOMO of the KS orbitals equals the
negative of the exact ionization energy. This, however, is only true if
the essentially exact exchange-correlation potential is used  . See A.
Savin, C.J. Umriger, X. Gonze, Chem. Phys. Lett. 288 (1998) 391 for
details.

Best regards

Klaus
-----------------------------------------------------------------------

---

From: "S.I.Gorelsky" <[email-masked]>
Date: Fri, 16 Feb 2001 13:31:57 -0500 (EST)

> even talk about HOMOs or LUMOs
...

I disagree strongly with such a statement, and I think I am not
alone. It is still make sense to talk about HOMOs and LUMOs. All
depends how and where you can talk about them. Please check this Roald
Hoffmann's paper

     R Stowasser, R Hoffmann, JACS, 1999, 121, 3414

Based on my own experince, I should say that the "molecular orbitals"
obtained from DFT are quite similar to the molecular orbitals from
standard MO-LCAO methods, and one can extract a lot of useful information
about molecular systems from analysis of their MOs even if the density
functional methods are used.

Sergey Gorelsky

York University, CANADA

---

From: Kevin Gross <[email-masked]>
Date: Fri, 16 Feb 2001 13:42:32 -0500

Perhaps I was unclear.  I am talking about HOMO/LUMO as being
meaningless from a semantic point of view.  DFT uses Kohn-Sham orbitals
to arrive at the electron density and so the highest occupied orbital is
perhaps better termed HOKSO for Highest Occupied Kohn-Sham Orbital.  I
thought the parenthetical comment you left out may have cleared this up.
In context, I said:

>Of course, DFT is not an MO method, so it doesn't make sense to
>even talk about HOMOs or LUMOs (nonetheless, to avoid HOKSO/LUKSO
>eigenvalue of the highest occupied Kohn-Sham orbital).

Sorry for the confusion.

Kevin Gross

----- Original Message -----
From: "S.I.Gorelsky" <[email-masked]>
To: "Kevin Gross" <[email-masked]>
Cc: <[email-masked]>
Sent: Friday, February 16, 2001 1:31 PM
Subject: Re: CCL:Summary: DFT and Koopmans' theorem

:
: > To summarize most of the responses, Koopmans' theorem does not hold
for
: > DFT.  Of course, DFT is not an MO method, so it doesn't make sense
to
: > even talk about HOMOs or LUMOs
: ...
:
: I disagree strongly with such a statement, and I think I am not
: alone. It is still make sense to talk about HOMOs and LUMOs. All
: depends how and where you can talk about them. Please check this Roald
: Hoffmann's paper
:
:      R Stowasser, R Hoffmann, JACS, 1999, 121, 3414
:
: Based on my own experince, I should say that the "molecular orbitals"
: obtained from DFT are quite similar to the molecular orbitals from
: standard MO-LCAO methods, and one can extract a lot of useful
information
: about molecular systems from analysis of their MOs even if the density
: functional methods are used.
:
: Sergey Gorelsky
:
: York University, CANADA
:
:

---

From: "Tapas Kar" <[email-masked]>
Date: Fri, 16 Feb 2001 14:07:24 -0700

I agree with Sergey. We compared local quantities such as bond indices, atomic charges, valences etc obtained from HF
and KS.
Please see J. Phys. Chem A 200, 104, 9953-9963. No appreciable difference is noticed except slightly enhanced covalency
by KS orbitals.

------------------------------------------------
"We owe a lot to the Indians, who taught
 us how to count, without which no worthwhile
scientific discovery could have been made."
- Albert Einstein -
------------------------------------------------
Tapas Kar, Ph. D
Department of Chemistry & Biochemistry
Utah State University
Logan, UT 84322-0300

Tel: 435-797-7230
Fax: 435-797-3390
Email: [email-masked]
       [email-masked]
----- Original Message -----
From: "S.I.Gorelsky" <[email-masked]>
To: "Kevin Gross" <[email-masked]>
Cc: <[email-masked]>
Sent: Friday, February 16, 2001 11:31 AM
Subject: CCL:Summary: DFT and Koopmans' theorem

> -= This is automatically added to each message by mailing script =-
> [email-masked] -- To Everybody  | [email-masked] -- To Admins
> [email-masked] -- HELP CHEMISTRY or HELP SEARCH
> [email-masked] -- archive search    |    Gopher: gopher.ccl.net 70
> Ftp: ftp.ccl.net  |  WWW: http://www.ccl.net/chemistry/   | Jan: [email-masked]
>
>
>
>
>