Oh, boy, oh, boy! Real scientific controversy!
I would like to take the opportunity to bore you yet again with
one of my sermons. This diatribe is ostensibly in response to
a request for comparing AM1 and PM3 charges. I'll come back to
that via a roundabout route. It is quite long and involved, so
want to save it and read it later.
Let me once again precede this message with the information that I
spent time with Michael Dewar at the U. of Texas as a post-doc.,
so my opinions are "tarnished" by that experience.
As a person whose major research emphasis is development of semi-
empirical parameters, I have had occasion to directly examine
the MNDO, AM1, and PM3 parameters in close detail. I am also
familiar with many of the principal people involved in the der-
ivation of these methods. The basic contention, that I will
hopefully support to some extent below, is that PM3 has left
chemistry behind in a quest for improved results. At first,
this statement may seem a bit strange, but stay with me.
1. The HF quantum mechanical method is based on a series of rel-
atively crude approximations that fortuitously give results
that are not too far off from experiment. If we operate on
the assumption that the model has chemical validity (and we
must to proceed further), it must be explainable in terms of
chemical trends and phenomena. In the present case, this
means that the parameter values themselves must be reasonable
in the context of other elements: the parameters must exhibit
PERIODICITY. It should be possible to derive a set of trial
parameters given an understanding of the implementation of
the scheme within the NDDO approximation and the values of
nearby elements. This has indeed formed the basis of the
Dewar approach to parameterization. For the great majority
of cases, the parameters from Dewar and coworkers follow
more or less periodic trends. The values are not exactly
periodic, but vary due to the crudity of the model and the
variety of chemistry described. (This variance is easy to
rationalize if one realizes that, for example, the AM1 sulfur
parameters must accomodate two valence states without the help
of d orbitals for handling the hypervalent case.)
2. The quantum mechanical model for MNDO and AM1 (and PM3) is
identical. The only differences in the methods are, that due
to limtations in computer time, the lighter elements in MNDO
had active two assumption:
A. The Slater orbital exponent for s and p orbitals were
close enough to be set equal (zetas and zetap).
B. The beta values for s and p orbitals of the lighter
elements are also set equal. These are used in a
function that results in the resonance integral, and is
hence responsible for bonding.
(For a detailed discussion of the MNDO/PM3/AM1 parameterization
model see: Dewar, M. J. S.; Thiel, W. J. Am. Chem. Soc. 1977,
99, 4907.) Both of these have been allowed to attain different
values in AM1 and PM3. Additionally, AM1 and PM3 added gaussian
functions to directly correct the core-core repulsion function.
It must be emphaiszed that gaussians are a purely empirical
correction, and are in programmer's terms, a PATCH. They hold
no chemical significance in and of themselves. The Dewar group
has traditionally used gaussians to correct for particular
types of molecules or effects. These include particular
bond energies (Al-Cl), hypervalency (P, S), minimal basis set,
and/or the lack of d orbitals. Gaussians must be added very
carefully, as they effect the energy in a direct manner
and will "drag" the other parameters during
procedure is best, followed by careful optimization. A stepwise
examination of the results after each parameterization run.
To summarize: Gaussians should not be part of the CHEMISTRY of
the system, but should act to correct the parameters for the
specific deficiencies mentioned above.
3. PM3 was parameterized using a vast amount of experimental data,
and many elements were parameterized simulateneously. For
the most part, AM1 and MNDO parameterization proceeded one
element at a time using only a SUBSET of the BEST data for a
parameterization basis. By carefully selecting the mole-
cules used in the basis set, it is possible to reproduce the
important and significant chemistry of an element. By using
ALL experimental data of any quality, the parameterization
becomes a prisoner to more easily obtained experimental data.
This is most evident in the plethora of data available for
halogenated molecules and the relative paucity of results for
organometallic compounds. Chemical judgement must be used
to balance the molecular basis against this type in influence.
When I joined the Dewar group in 1987, I moved into an office
with a sign that said "Human Factors" on the door. It took me
some time to realize that this was not a computer nerd
joke, but a philosophical statement.
The above points are both philosophical and practical. Given the
manner in which PM3 was parameterized and the results of the par-
ameterization, I am doubtful of its utility as a quantum chemical
model for general application. As an example of what can happen,
The parameters for aluminum are compared across the three methods.
I am using aluminum, because I developed the AM1 parameters for this
element and I am most familiar with it. Similar trends can be
found in many of the other PM3 elements.
Parameter AM1 MNDO PM3 Units
-----------------------------------------------------
Uss -24.353585 -23.807097 -24.845404 eV
Upp -18.363645 -17.519878 -22.264159 eV
zetas 1.516593 1.702885 au
} 1.444161
zetap 1.306347 1.073269 au
betas -3.866822 -0.594301 eV
} -2.670284
betap -2.317146 -0.956550 eV
alpha 1.976586 1.868834 1.521073 1/A
Gaussians:
Intensity #1 0.090000 - -0.473090 eV
Width #1 12.392443 - 1.915825 A^2
Position #1 2.050394 - 1.451728 A
Intensity #2 - - -0.154051 eV
Width #2 - - 6.005086 A^2
Position - - 2.51997 A
The point on the potential surface located by PM3 is
significantly different than that located by AM1. This is
immediately apparent from the large discrepancy between the Upp
values. These are the important one-electron energy values and
they have strong influence on the parameter hypersurface. Also,
the difference between Uss and Upp for both MNDO and AM1 is about
6 eV (roughly the same value as for all elements on this period
according to AM1/MNDO). This has been reduced to 2.5 eV in PM3.
The realdifficulty, however, is in the beta values. These parameters
are the two-center/one-electron resonance terms and are responsible
for bonding interactions between atoms. The PM3 values are
almost zero, resulting in the conclusion that there is very
little bonding between atoms of aluminum! (Note that the
AM1 values for betas and betap spread out around the single MNDO
value for beta. This suggests that the MNDO values were reasonable
and AM1 simply adds greater flexibility.) PM3 regains the bonding
interactions lost due to the low beta values with two strongly
attractive Gaussians spanning the bonding region.
One result of such difficulties as discussed above is poor
quantum mechanical descriptions of molecules. A prime example is
formamide. The PM3 parameters were developed with an eye toward
reproducing -NO2 compounds, a somewhat severe test of such a min-
imal basis set approach. Focusing on these types of compounds has
caused the parameters to perform somewhat oddly.
Below is a table listing the charge on atoms in
formamide as predicted by PM3 and AM1. For purposes of
comparison, also listed are charges from a Mulliken analysis at
the HF/6-31G* level and NBO charges at the same level. I have
chosen to present this data because it such a finely tuned
barometer of chemical significance. The fact of the matter is
that there is no real way to experimentally determine charge
routinely, so this should conform to your chemical intuition more
than anything else.
Method C O N H H(N)
----------------------------------------------
AM1 0.26 -0.37 -0.45 0.12 0.22
PM3 0.21 -0.37 -0.03 0.08 0.05
Mullik. 0.67 -0.70 -0.93 0.12 0.41
NBO 0.51 -0.55 -0.88 0.14 0.39
Note that AM1 is in general qualitative agreement with the NBO
(probably the most reliable in this set). The PM3 charge on
nitrogen is simply beyond any chemical reason. The carbon
and oxygen parameters appear to offer a good model. Examination
of the PM3 nitrogen parameters indicates several conceptual problems,
such as p orbitals that are more contracted than s orbitals and very
strong gaussians spanning the bonding range. The overall heat
of formation results for N are better in PM3 than in AM1, but the
cost appears to be loss of an accurate chemical description of the
electronic structure.
Thus, the essence of the difference between the two philosophies
can be directly stated: the theoretical basis for the
method is either accepted or denied. Significant approximations
are made to gain the speed advantage that semiempirical methods
enjoy over their ab initio quantum mechanical brethren. But both
the ab initio and semiempirical models are finally, in the end
based on the Hartree-Fock set of ideas. These ideas
possess theoretical rigor as regards solution of the Schrdinger
Equation. If one simply views the semiempirical parameters as
adjustables within a curve-fit scheme rather than as components
of a theoretical model, little faith resides in the meaning of
their final values. Simply put, the method of parameterization
described above and used so successfully with AM1 and MNDO
expresses confidence in the theory. With a firmer footing in
chemical reality, AM1 parameters are more likely to yield useful
results for situations not specifically included in the parameter-
ization.
End of Sermon. One of my colleagues that read this called it
"preachy".
I hope that it didn't come over too much that way.
Dr. James J.P. Stewart of course has arguments to
counter these and I hope that many of you are aware of them. Every-
one has an opinion and this is mine. I hope that the above discourse
is taken in the spirit of scientific disagreement as it is intended.
Andy Holder
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DR. ANDREW HOLDER
Assistant Professor of Computational/Organic Chemistry
Department of Chemistry || BITNET Addr: AHOLDER ( ( at ) )
UMKCVAX1
University of Missouri - Kansas City || Internet Addr: aholder ( ( at ) )
vax1.umkc.edu
Spencer Chemistry, Room 502 || Phone Number: (816) 235-2293
Kansas City, Missouri 64110 || FAX Number: (816) 235-1717
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