Subject: Koopmans' theorem, heat, and light

From: Christopher Cramer <[email-masked]>
Date: Wed, 21 Feb 2001 09:44:01 -0600 (CST)

The ongoing discussion has generated more of the second and less of the
third with respect to the first.

If I may:

Koopmans' "theorem" states that, if you take a closed-shell wave function
expressed as a single Slater determinant, and either remove an electron
> from the highest occupied orbital or add an electron to the lowest virtual
orbital, then the new singly occupied orbital is stable with respect to any
subsequent variation involving the unoccupied orbitals. This is a rigorous
mathematical proof, and as such qualifies as a theorem.

Note that, the theorem does NOT say that the remaining DOUBLY occupied
orbitals are stable with respect to variation. 

Nevertheless, Koopmans' "approximation" says, let's compare the energies of
the original closed-shell and newly formed open-shell determinants where
all doubly-occupied MOs are identical. The difference in the energies is
exactly the orbital energy of the singly-occupied orbital. Since that
orbital is either the HOMO or the LUMO, depending on whether we ionized or
attached, these energies are also called -IP and EA, respectively.

As has correctly been pointed out in an earlier post, Koopmans'
approximation works much better for IP because failure to account for
relaxation effects (which stabilize the radical cation) are offset by
failure to account for electron correlation (which preferentially 
stablizes the closed-shell system as it has more electrons) and finite
basis set limitations. EAs are usually very bad from Koopmans' approximation.

To be explicit, the relaxation effect is the energy gained from allowing
the doubly-occupied orbitals to be variationally optimized in the 
open-shell system (and, of course, the singly-occupied simultaneously).
Determining an IP or EA by this method is sometimes called a "delta-SCF"
approach. In the absence of including electron correlation effects and
saturating the basis set, this may or may not improve on the Koopmans'
estimate.

As for DFT, it has been shown, again with complete rigor, that the energy
of the highest Kohn-Sham orbital for a closed-shell system is PRECISELY 
the negative of the ionization potential WHEN THE EXACT FUNCTIONAL IS USED. 
As everyone knows, that exact functional remains somewhat elusive, so the 
actual quality of KS eigenvalues for estimating IPs is variable, depending 
on choice of approximate functional.

Thanks for participating in this pedagogical moment.

Chris

-- 

Below are our temporary addresses during our sabbatical
year in Barcelona (July 23, 2000 - August 1, 2001).

Chris at Univ. of Barcelona             Cramer Family in Barcelona
---------------------------             --------------------------
c/o Professor Javier Luque              Via Augusta 228 4/3
Fac. Farmacia, Dept. Fis. Quim.         08021 Barcelona
Universitat de Barcelona                SPAIN
Joan XXIII, s/n                         Phone:  (34) 93 209 4776
08028 Barcelona
SPAIN                 
Phone (my mobile): (34) 62 043 1176

Permanent data:

Christopher J. Cramer
University of Minnesota
Department of Chemistry
207 Pleasant St. SE
Minneapolis, MN 55455-0431
--------------------------
Phone:  (612) 624-0859 || FAX:  (612) 626-2006
[email-masked]
http://pollux.chem.umn.edu/~cramer