Subject: The Reality Behind Semiempirical Parameters

From: [email-masked] (Charles Martin)
Date: Fri, 29 Apr 1994 14:46:20 -0600 (CDT)

More on Semi-empirical methods...

I would like to expand on a point made by E. G. Zoebisch
in her detailed description of the development of AM1.
Specifically, she claims that it is not possible for
semiempirical parameters to be ``wrong.''  Lets consider
why this may or may not be true, in the context of
effective Hamiltonian theory.  I hope this sparks some
discussion about how to improve the current semiempirical
models.

First, most semiempirical methods are constructed
using the Hartree Fock (HF) Hamiltonian as a base, or, at
best, using the valence shell subblock of the
full CI Hamiltonians.  These valence shell equations,
however, do not reproduce experiment to chemical accuracy,
and thus semiempirical methods employ empirical data
to compensate for the lack of accuracy of the valence shell
CI equations.  (Additionally, semiempirical methods include
data on heats of formation, a property which can not be
described by the full CI Hamiltonian.  This poses additional
theoretical difficulties I will not discuss here)
If, however, one does not employ the HF model as the base
for semiempirical methods, but rather employs an exact,
valence only theory, then it would be possible to make
some factual statements about the semiempirical parameters.

It is, in fact, possible to derive an exact effective valence shell
Hamiltonian (Hv) which (as shown by Freed):

(1) can be derived from the full CI equations and, in principle,
reproduces the full CI state energies exactly for
the valence-like states of a molecule.

(2) produces, in practice (via ab initio perturbation theory),
very accurate energies for all valence-like states

(3) contains parameters which are ab initio analogs of
the standard semi-empirical model effective valence-shell Hamiltonians.

(4) We may write Hv as a sum of many-electron operators:

Hv = Ec + U(1) + V(1,2) + W(1,2,3) + X(1,2,3,4) + ...

Where U(1) is an effective 1 electron operator,
V(1,2) is an effective 2 electron operator, etc.

The additional 3, 4, etc.  electron operators take into
account correlation effects (orbital polarization, core-valence
correlations, etc.) which are neglected in the HF model,
but which are in the full CI equations.

If we consider Hv as the proper (or at least a better) starting
place for understanding semiempirical methods, then we can
make some factual statements about semiempirical parameters:

(1) For small systems, such as atoms and diatomics, it is possible
to evaluate the Hv parameters  from experimental data.  In other
words, some Hv parameters are observables.  For example, it is
possible to obtain the most of the semiempirical pi-electron parameters
directly from experimental data.  For larger molecules,
linear combinations of Hv parameters are observables.

Note that the above arguments do not hold for the Hartree Fock(HF)
model.  The HF model is not equivalent to the full Schr\"odinger
equation--the Hv equations are!

(2) Generally for a given molecule, there are more Hv parameters than
experimental data points, and thus the Hv equations lead to an
overdetermined set of equations.

(3) Most semiempirical models are cast in terms of the
ZDO approximation, the NDO, etc.  In fact, the true
Hv contains additional 3- and 4-electron parameters
which are either (a) neglected or (b) averaged into
the semiempirical 1- and 2-electron terms

(4) For pi-electron systems, the 3- and 4-electron parameters
are only a few tenths of an eV, whereas for all-valence-electron
methods, these non-classical parameters are several eV in magnitude.

(5) The HF matrix elements are, of course, transferable between
molecules.   The true 1- and 2-electron Hv parameters display
remarkable transferability for pi-electron systems as well
(to within a few tenths of an eV or less)

Thus, for pi-electron systems, it is meaningful to ask what
are the ab initio values of the pi-electron (or PPP) parameters,
if one keeps in mind that
(a) you are neglecting the 3- and 4-electron terms (which
is not bad for large pi-electron systems), as well as
the other
(b) you understand the size of the parameters you are neglecting
and how they will affect the properties you are attempting to
describe.  For example, we have only examined spectra so far,
which is what PPP methods are generally used for.
(c) you consider the whole valence shell CI, and not just
single excitations.  By reducing the size of the valence shell
CI, the true parameters might have to be ``renormalized'' for that
subspace as well. A variety of other workers have considered
this problem.
(d) it is noted that the standard PPP values are in fact
somewhat different from the true values (beta is about 0.5 to 1 eV
off, gamma is also, etc.)  We are currently investigating this.

For all-valence-electron models, however, the situation is
far more complex than the PPP model might lead on to expect!
The standard semiempirical methods (MNDO, INDO, etc.)
use parameters which appear to contain the true 3- and 4-electron
parameters in an average sense. Since this averaging is
somewhat arbitrary, this can help explain why different
parameterization are needed for different properties.
The theory of all-valence-electron methods is still a wide
open and very complex problem.

Chuck  Martin