Subject: Summary: Chemical Softness by DFT

From: "Jerry C.C. Chan" <[email-masked]>
Date: Thu, 15 May 1997 09:09:16 +0200 (MES)

Dear CCLers,

	Thanks are due to all who response to my questions concerning
the chemical hardness.  I am happy that many experienced CCLers did share
with me their valuable expertise.

Sincere thanks,
Jerry

> 	Last week I posted a question concerning the determination of
> chemical hardness (or softness) of the metal centre of a TM complexes. 
> Maybe the way I asked the question was not specific enough, I just
> received two mails asking for the summary.  So I try to write down my
> thinking explicitly and hope that someone in the net, who could not
> tolerate the naiveness or even mistakes of my post, would kindly comment
> on it.  I will summarize. 
>  
> I want to determine the chemical hardness of the metal centre of a charged
> TM complexes.  The method that comes to my mind is to do a HF calculation 
> and get the MO spectrum.  Based on the Koopmans' theorem the hardness of 
> the complexes would be (E_LUMO - E_HOMO)/2.
> 
> Even my thinking is correct, HF calculation seems to be unreliable for TM 
> complexes and correlated methods are usually very expensive.  Therefore 
> last time I enquired the possibility of using DFT methods as an 
> alternative although Koopman's theorem does not hold for KS orbitals.
> 
> By the way, I have heard that Fukui function is somehow related to the
> chemical hardness but I don't have the recipe.  

++++++++++++++++++++++++++++++++
Stefan Fau <[email-masked]>
Pascal HEBANT <[email-masked]>
Xavier Assfeld <[email-masked]>
Hogue, Pat (AZ76) <[email-masked]>

R.G. Parr and W. Yang, "Density-Functional Theory of Atoms and
Molecules" (Oxford University Press, 1989, New York).

Ian Fleming, "Frontier Orbitals and Organic Chemical Reactions"

"Hardness and softness in Density Functional Theory in Chemical Hardness",  
K.D. Sen Ed., Coll. Structure and Bonding 80, (Springer-Verlag, Berlin,
1993).

R.G. Pearson:				J.Am. Chem. Soc. 85, 3533 ('63)
N.K. Ray, L. Samuels, R.G. Parr: 	J.Chem. Phys. 70, 3680 ('79)
M.S. Gopinathan, M.A. Whitehead:   	Isr. J. Chem. 19, 209 ('80)
J.P. Perdew, R.G. Parr, M. Levy, J.L. Balduz: 
                                     	Phys. Rev. Lett. 49, 1691 ('82)
R.G. Parr, R.G. Pearson:           	J. Am. Chem. Soc 105, 7512 ('83)
R.F. Nalewajski:                   	J. Am. Chem. Soc. 106, 944 ('84)
R.G. Parr, W. Yang:                	J. Am. Chem. Soc. 106, 4049 ('84)
W. Yang, W.J. Mortier:             	J. Am. Chem. Soc. 108, 5708 ('86)
R.G. Pearson:                      	J. Chem. Ed. 64, 561 ('87)
A.R. Orsky, M.A. Whitehead:        	Can. J. Chem. 65, 1970 ('87)
M. Berkowitz, R.G. Parr:           	J. Chem. Phys. 88, 2554 ('88)
R.G. Parr, P.K. Chattaraj:         	J. Am. Chem. Soc. 113, 1854 ('91)
P.K. Chattaraj, H. Lee, R.G. Parr: 	J. Am. Chem. Soc. 113, 1855 ('91)
D. Datta, S.N. Singh:              	JCS Dalton Trans. 1541, ('91)
J. Cioslowski, S.T. Mixon:         	J. Am. Chem. Soc. 115, 1084 ('93)
J. Cioslowski & B.B. Stefanov: 		J. Chem. Phys. 99, 5151 ('93)
R.G. Pearson:				Acc. Chem. Res. 26, 250 ('93)
P.K. Chattaraj, P.v.R. Schleyer:   	J. Am. Chem. Soc. 116, 1067 ('94)
F. Mendez, J.L. Gasquez:           	J. Am. Chem. Soc. 116, 9298 ('94)
W. Kohn, A.D. Becke, R.G. Parr: 	J. Phys. Chem. 100, 12974 ('96)

+++++++++++++++++++++++++++++++++++++++++
From: Liang Lou <[email-masked]>

There is an equivalence in DFT calculations which is called Slater's
transition state method. In this method, the EA and IP are approximated by
the values of an halfly occupied orbital. For example, for IP, you put
+0.5e on the molecule and run a full SCF. Then you check the eigenvalue
for the one-particle state with an 0.5 occupation number. This gives a
good estimate of the IP. For EA, charge the molecule with -0.5e. The
formal explanation of this "transition state" method was given by Janet
(coauthor of the book "calculation of electronic structure in metals"). It
is roughly as follows. The total energy of an N-electron system in LDA can
be written as E=sum[n_k * epsilon_k] + ..., where n_k is the occupation of
the eigenstate k and epsilon_k is the corresponding eigenvalue.  The
eigenvalue of the one-particle state k is simply epsilon_k=dE/dn_k. This
is an equivalence of the koopmans' theorem in LDA. The Slater's transition
state is a "finite-difference" approximation to the infinitesimal d(n_k).
The error of this method, from my experience, is in the range between
0.1-0.3eV, approximately the same as from a small-delta SCF calculation
(e.g., dE(EA) = E(N) - E(N+1)). 

+++++++++++++++++++++++
From: Rene Fournier <[email-masked]>

> although Koopman's theorem does not hold for KS orbitals.
   I would not worry about that.  First, Koopman's theorem
is not so great anyway.  It equates two theoretical constructs:
the energy difference between the GS and a HYPOTHETICAL excited
state with orbitals identical to those of the GS on one hand,
and the difference between the HOMO and LUMO HF orbital energies
on the other.

   Actually, I would say that Kohn-Sham DFT is THE IDEAL theory
for your problem.  It has two useful theorems:

  (a) the negative of the KS HOMO energy is equal to the TRUE
ionization energy of the system (relates a theoretical construct
to an observable).   CAVEAT: this theorem holds only in the limit
of an "exact" XC potential.  [ But how could one expect exact
calculation of observables in any approximate theory anyway?
At least, in KS-DFT, the framework for exact calculations of
this kind is there. ]

  (b) the derivative of the KS-DFT energy w.r.t. number of
electrons, whether the XC is exact or approximate, is precisely
equal to the energy of the highest partly occupied orbital
(Janak's theorem)

   The hardness is most conveniently defined as being half the
second derivative of the energy w.r.t. number of electrons;
the finite difference approximation to that is (I-A)/2; and
that in turn can be approximated as (E_LUMO - E_HOMO)/2 .
Thanks to Janak's theorem, you can get some derivatives (dE/dN)
simply by looking up orbital energies.

   There is an excellent discussion of hardness and Fukui function in
Parr and Yang's book "Density-Functional Theory of Atoms and Molecules"
(Oxford University Press, 1989, New York). 

   BTW, the Fukui function is the derivative of the electron
density at point r w.r.t. total number of electrons, N.  If the
sum of nuclei charges is M, then:

f(r) is roughly the density associated with the LUMO for N = M+delta
f(r) is roughly the density associated with the HOMO for N = M-delta
f(r) is roughly the average of the above two for N=M

++++++++++++++++++++++++++++++++++++
From: "N. Sukumar" <[email-masked]>

Hardness in DFT is given by the partial second derivative of the energy
functional with respect to the electron density, at constant external
potential, ie. (d^{2}E/d\{rho}^2)_v
See Parr & Yang's book and the references therein for details, including
relations to the Fukui functional (which is the mixed partial derivative
of the energy functional with respect to density annd external potential) :
Robert G. Parr & Weitao Yang, "Density Functional Theory of Atoms and
Molecules" (Oxford University Press, New York, 1989)
Jerzy Cioslowski in Florida has done some ab initio (Hartree-Fock level)
calculations (on small molecules) using the DFT definition of hardness,
without relying on the Koopman's theorem. Some references are :
J. Cioslowski & S. T. Mixon, J. Amer. Chem. Soc. 115, 1084 (1993)
J. Cioslowski & B. B. Stefanov, J. Chem. Phys. 99, 5151 (1993)
see also references therein (since the above two papers are primarily
concerned with BOND hardness).

+++++++++++++++++++++++++++++++
From: Oliver Warschkow  <[email-masked]>

You have asked about hardness/softness calculations. 
I dont have any experiences in such calculations myself,
but there is a recent review by Kohn, Becke and Parr
(J.Phys.Chem (1996), 100, 12974) where hardness,softness
and fukui-functions in the framework of DFT are discussed.  
The explicit expression given in there (eq.3.4) is

     hardness = (del mu / del N) at const. V

where mu is the chemical potential (or fermi leve Ef) 
of the system. So, it probably boils down to running a 
two DFT calculations on the system with slightly 
(that is fractionally) different number of electrons and 
to see how the HOMO orbital energy (or Ef if you use 
a Fermi-Dirac orbital occupation scheme) changes. Well, that 
would be my guess how to do it. You are right in
that you are probably better off in doing DFT instead of
HF on TM systems and according to the review, concepts
like hardness,softness etc are quite natural to DFT.

+++++++++++++++++++++++++++++
From: "Jack A. Smith" <[email-masked]>

  I think a good reference for you would be "Density Functional Theory of
Atoms and Molecules" by Parr and Yang (Oxford Press, 1989), particularly
chapters 4 & 5.  You'll see that the KS orbitals of DFT are actually more
directly related to the chemical potential, hardness, Fukui functions, etc.
than canonical HF orbitals [Grand Canonical HF, on the other hand, can be
viewed as a natural extension of HF to DFT which includes proper exchange
(see J. Linderberg, IJQC 12:supp 1, p267) but no extra correlation, and
whose orbitals allow similar interpretation as do KS orbitals].

-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
From: Liang Lou <[email-masked]>

> Thank you very much for your reply.  Since I always work on closed shell
> system, I would like to know if I understand your suggestion correctly: 
>
> If I have a +3 TM complexes, in G94 I can simply change the the charge
> from +3 to +3.5 and the multiplicity from 1 to 2 and then do a DFT
> calculation.  No special keywords are needed.  Repeat the calculation
> with the charge of +2.5.  I may have difficulties to get the SCF
> convergence for these +3.5 and +2.5 systems but using vshift should be
> able to solve it (though I don't have any experience on it). 

As far as I know, Gaussian is not particularly strong for metals,
especially in the sense of the efficiency of basis sets. Other DFT program
packages could do better (with either highly optimized Gaussian bases or
Slater type bases, or numerical bases).  I guess the level shifting would
result in artificial values for orbital energy near the separation of HOMO
and LUMO.  When calculating electron removal energies, IP and EA, the
system always changes spin multiplicity and there are always associated
uncertainties. Usually, the orbital energy values do not change in the
first few decimal places after he total energy has converged to under,
say, 10**(-4). Therefore, higher convergence of total energy will not
affect comparing with experiment.

---

From: Carolina Reyes <[email-masked]>
Date: Thu, 15 May 1997 13:12:52 -0700 (PDT)

PLEASE LET YOUR COLLEAGUE KNOW THAT IF S/HE IS GOING TO ACCESS EMAIL IN A 
LAB ENVIRONMENT, THEY NEED TO CLOSE THEIR MAIL SESSION AFTERWARD.  NO ONE 
IS IN TOO MUCH OF A RUSH TO PROTECT THEIR PRIVATE AND RESEARCH 
CORRESPONDENCE.  THANKS FOR YOUR ASSISTANCE IN POINTING THIS OUT TO YOUR 
FRIEND.

On Thu, 15 May 1997, Jerry C.C. Chan wrote:

> -------This is added Automatically by the Software--------
> -- Original Sender Envelope Address: [email-masked]
> -- Original Sender From: Address: [email-masked]
> [email-masked]: Everybody | [email-masked]: Coordinator
> [email-masked]: HELP CHEMISTRY or HELP SEARCH | Gopher: www.ccl.net 73
> Anon. ftp: www.ccl.net   | [email-masked] -- archive search
>              Web: http://www.ccl.net/chemistry.html 
> 
>