Subject: G94 LSDA confusion

From: [email-masked] (Ross Dickson)
Date: Tue, 5 Dec 95 11:50:35 EST

Ulf Ryde writes:
> 1. Gaussian does not give the same B3LYP energy as IBM:s Mulliken.
>    Who is right?

Ross Nobes replies:
>The answer is that both are right!  The Gaussian definition of B3LYP
>uses the VWN local correlation functional;  most other programs use
>what Gaussian terms VWN5.

I think this is worth amplifying.  Those of you who started using
density functional theory when Gaussian brought it out will want to 
pay attention.  Sorry if this takes a few moments:

The most common local spin-density approximation (LSDA) functional
used these days is that of Vosko, Wilk and Nusair (VWN).  Everybody
knows that.  Unfortunately, what some people don't realize is that
there's more than one "VWN functional" out there.  Look in your
Gaussian 94 (or G92/DFT) manual and you'll see them both.

In their famous paper [S. H. Vosko, L. Wilk, and M.  Nusair, Can. J.
Phys. 58(8), 1200 (1980)] VWN describe a number of different
functionals as they search for the "best fit" to the homogeneous
electron gas correlation energy -- i.e., the LSDA correlation
functional.  Along the way they include parameters for fitting their
Pade approximant expression to an RPA description of the electron gas,
a well-established-but-not-state-of-the-art approximation.  Then in a
table caption (!) they give the parameters which fit the same form to
Ceperley & Alder's quantum Monte Carlo calculations on the electron
gas.  The Ceperley & Alder data is considered the "last word" on
LSDA correlation by most experts in the field, so of these two sets
of parameters, the latter ones are preferred.

Unfortunately, the folks at Gaussian chose to implement the former as
the default LSDA correlation functional -- though the latter functional
*is* available under the keyword "VWN5".  I haven't put a lot of effort
into determining why they did things this way, but it may have
something to do with the fact that in Appendix E of Parr & Yang's
"Density Functional Theory of Atoms and Molecules", the RPA-fitting
parameters are given "as the most accurate available for the
uniform-gas correlation energy".

If you don't know whether to believe Robert Parr and the Gaussian
programmers or this post-doc from Canada on this point, you have a
couple of choices:  (1) You can read the VWN paper yourself -- and good
luck, it's not an easy read.  You'll understand why there's some
confusion.  (2) If you have access to G94, you can appeal to a third
party:  John Perdew, author of a couple of LSDA-fits of his own.
Perdew & Zunger's 1981 fit [Phys. Rev. B 23(10), 5048 (1981)] to the
same Ceperley & Alder quantum Monte Carlo data is included in Gaussian
under the keyword "PL".  If you run Gaussian jobs on, say, the hydrogen
atom, 6-31G*, you will find:

			E (hartrees)
		"VWN"	-.493937
		"VWN5"	-.476044
		"PL"	-.476230

This clearly indicates that "VWN5" and "PL" are fits to the same data,
while "VWN" is something else.  Unfortunately, what Gaussian has called
"VWN5" is what such people as John Perdew and Axel Becke call "LSDA",
while what Gaussian calls "LSDA" or "VWN" is something they don't use
anymore.

How much difference does all this make?  I don't know.  Not much, I
suspect.  Despite the startling difference in total energies above, the
dissociation energy of the hydrogen molecule differs by only 0.2
kcal/mol among the three LSDA methods.  But this inconsistency in
naming between Gaussian and every other density functional program in
the business is potentially quite troublesome.

Ross M. Dickson, Ph.D.                      [email-masked]
Dept. of Chemistry, Queen's University at Kingston, Ontario, Canada