From:	Jan Reimers <wolfe!resrch!janr>
Date:	Tue, 28 Jan 97 14:43:01 PST
Subject:	Mulliken population analysis

Douglas A. Smith asked:
>The question:  Can people please explain the concept of having a balanced
>basis set and the dangers inherent in an unbalanced set?  In particular, why
>can I not simply use a large basis, including extra split valance functions,
>polarization and diffuse functions, only on the atoms I need them on, and a
>smaller, more compact basis set on the "unimportant" atoms to my chemical
>question?  This would certainly save time and resources during the calculation.

In response, Per-Ola Norrby wrote:

>        I'll give it a try, and hopefully I can do it in chemists language
>without loosing too much accuracy.
>
>        The basic problem is that all basis sets are incomplete.  There is
>no way you can use a finite number of basis functions to describe the
>electron density completely.  You can get fairly close, but at a high cost.
>Now, what happens to an atom with an incomplete basis?  It has some
>electron density that could be described better if it could use some
>additional basis functions.  Now, if there are unused basis functions on a
>neighboring atom, there is always SOME way that a linear combination of
>those can be used to stabilize the electron density on the original atom
>further.  Thus, the electron sharing between atoms is exaggerated and the
>bonds look stronger than they actually are.

This depends on how you determine the charge on each atom, and
on how you determine bond strength.  One must admit that adding more
basis functions anywhere (lets ignore numerical presicion problems and
near singular matricies for the moment) will give an improved approximation
to the true (or HF) charge density.  RFW Bader has shown us how to
extract atomic charges and bond strengths (or bond orders) from the
charge density.  Using Baders definitions, these quantites (and many others)
should improve as the approximation to the charge density improves.  So
one avoids the apparent paradox where augmenting the basis set decreases
the quality of the answers you get.

	It should be no surprise that this works because Baders definitions
of atomic charge and bond order are derived from the least action principle,
as applied to a real physical quantity; the charge density.  In fact Baders
definitions, should really be called THE definitions.

	Things like Mulliken population analysis are hopelessly tied to
the basis set, our approximation method for solving the SE.  These numbers
cannot possibly be considered "real physically observable quantities".
If you modify your basis set, you can expect to get "funny" numbers
from the population analysis.  This is just a demonstration that the
numbers are meaningless, nothing else!

>        Now, if all atoms have very few basis functions, there aren't too
>many unused functions that can be used by the neighbors, so the errors
>(basis set deficiency and superposition errors, BSDE and BSSE) partially
>cancel.  However, if one atom has a very small basis set and the neighbor
>many diffuse and polarization functions, you may get into a situation where
>a very substantial part of the electron density of the first atom is
>described by basis functions on the second.  If you try to do a Mulliken
>analysis on such a system, you get weird results.  The electron density in
>that region will also most probably be skewed, causing all kinds of
>distortions.

Yes exactly, but I interepret this as a definiciency in the polulation
analysis method, rather than as a basis set problem.

>        You CAN get away with things like this, if you are careful to do
>only comparisons between very similar systems, where the effect stays
>constant.  Naturally, you can get away with ANYTHING as long as you fulfill
>that requirement :-)

Or better yet, just ask physically meaningful questions of the wave function
and charge density.

>        Per-Ola Norrby

+--------------------------------------+-------------------------------------+
| Jan N. Reimers,  Research Scientist  | Sorry, Don't have time to write the |
| Moli Energy (1990) Ltd. B.C. Canada  | usual clever stuff in this spot.    |
| janr -x- at -x- molienergy.bc.ca                |
|
+--------------------------------------+-------------------------------------+



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