Subject: Mulliken Populations Summary

From: [email-masked] (Magda Wajrak)
Date: Sat, 1 Feb 97 23:45:59 WST

Dear All,

Thank you so much for all who replied to my question regarding Mulliken
Populations. Overall the conclusion seems to be that Mulliken Populations
are very basis set depedent, thus what you put in is what you will get out.
They should only be used as a 'rough' guide, for more accurate results
different methods should be used, such as NBO and others.

Regarding my problem, I am not interested in the exact popultion, I was
only using it as a guide, but I was very surprised how Cadpac and G94
gave such different population analysis results even with the same basis
set.

Below are the responses. Thank you again.

**************************************************************************

Original Question:

Dear Computer Chemists,

I have a question regarding Mulliken Populations, I appologise if this is
an obvious question, but I haven't got much experience with mulliken populations,
so I am not sure what is going on.

How reliable are Mulliken Populations? The reason I ask is because, recently
I ran two exactly the same jobs one using G94 and another using Cadpac and when
I checked the charge distribution it was completely different.
The final geometry and energies were the same, so I am confused.

Also sometimes when I ran a job (water+metal) using G94 I get very strange
charge distribution and other times it is as expected.

Thank you for your time.

Magda Wajrak

***********************************************************************
Answer 1:

Dear Magda,

Mulliken Population analysis yield atomic charges that reflect mostly
properties of the basis sets used rather than the actual distribution
itself. Because of the MO description of Quantum Calculation, the
molecule's electron density is divided into net popoulations
and overlap populations.
According to Mulliken's gross populations in the individual AO, the overlap
population
is equally divided between two AOs. In fact, there is no chemical ground
for doing this.

There have been suggested other definitions of charges :
Bader charge : R.F.W. Bader, Atoms in molecules. A quantum theory
(Clarendon Press,
Oxford, 1990)
Cioslowski's charge : J. Cioslowski, J. Am. Chem. Soc. 1989, 111, 8333.

I hope it would help.

Thanks,

Cheol Ho Choi
PhD Student
Dept. of Chem.
Georgetown Univ.

Answer 2:

        It is known that the Mulliken population analysis is a breakdown
for basis sets having diffuse functions [J. Baker, Theor.Chim.Acta, 68,
221(1985)].

Masao Masamura
Preventive Dentistry
Okayama University Dental School
Shikata-cho, 2-5-1
Okayama 700
Japan
FAX: 81-86-225-3724 
e-mail: [email-masked] 

Answer 3:

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.    |
| [email-masked]                |                                     |
+--------------------------------------+-------------------------------------+

Answer 5:

Hi!

It would seem tha there is an argument for relating charge distribution in
a 'real' property.  In the case of macro-molecular systems this can be
done by aproximating point charges to reproduce the gradients or force
constants produced by an scf.

As the gradients and force constants can then be confirmed by comparason
to experiment - this would seem a logical approach.

Best wishes

Alex

 -------------------------------------------------------------------
|Alexander J Turner         [email-masked]                  |
|Post Graduate              |http://www.bath.ac.uk/~chpajt/home.html|
|School of Chemistry        |+144 1225 8262826 ext 5137             |
|University of Bath         |                                       |
|Bath, Avon, U.K.           |Field: QM/MM modeling                  |
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Answer 6:

Hi,

there is another method of determining atomic charges, the NBO analysis of 
Reed, Weinhold and coworkers. The results quite independent of the basis 
set. Furthermore, the NBO analysis gives bond orders and the best Lewis 
stucture for the molecule. The NLMO part gives information on 
(hyper)conjugation.

Review: Reed, Curtiss and Weinhold: JCP 1988, 88, p.899

Stefan
__________________________________________________________
Stefan Fau,               [email-masked]

FB Chemie der Philipps-Universitaet Marburg,
Hans-Meerwein-Str.
D-35032 Marburg

Answer 7:

Hi,
why does no one uses ( or is even aware of ) the population analysis
method of Davidson, Roby, and Ahlrichs ? In my opinion, it is the
cheapest one ( compared to Bader or NBO ) with the highest
interpretation potential ( no one should use population charges
to model the electrostatic potential of a molecule. These are quiet
different things !!! ). Here are the references :

	(1)	Ernest R. Davidson
		Electronic Population Analysis of Molecular
		Wavefunctions
		J. Chem. Phys. 46 (1967) 3320-3324

	(2)	Keith R. Roby
		Quantum theory of chemical valence concepts
		I. Definition of the charge on an atom in a
		molecule and of occupation numbers for electron
		density shared between atoms
		Mol. Phys. 27 (1974) 81-104

	(3)	Rolf Heinzmann and Reinhart Ahlrichs
		Population Analysis Based on Occupation Numbers
		of Modified Atomic Orbitals (MAOs)
		Theoret. Chim. Acta 42 (1976) 33-45

	(4)	D. W. J. Cruickshank, F.R.S.,
		and Elizabeth J. Avramides
		The Interpretation of Molecular Wave Functions:
		The Development and Application of Roby's Method
		for Electron Population Analysis
		Phil. Trans. R. Soc. Lond. A 304 (1982) 533-565

	(5)	Claus Ehrhardt and Reinhart Ahlrichs
		Population analysis based on occupation numbers
		II. Relationship between shared electron numbers
		and bond energies and characterization of
		hypervalent contributions
		Theor. Chim. Acta 68 (1985) 231-245

Ciao,
Heinz
---
Dr. Heinz Schiffer		Phone   ++49-69-305-2330
Hoechst CR&T			Fax     ++49-69-305-81162
Scientific Computing, G864	Email   [email-masked]
65926 Frankfurt am Main		        [email-masked]

Answer 8:

For a fairly systematic comparison of basis set and electron correlation
dependence of Mulliken, APT, NBO, and CHELP charges, see:

F. De Proft, J. M. L. Martin, and P. Geerlings, ``On the performance
of density functional methods for describing atomic populations, dipole
moments and infrared intensities'' Chemical Physics Letters 
250, 393--401 (1996). 

In a nutshell, Mulliken has the most severe basis set dependence but the
weakest correlation dependence, while the opposite is true for APT. 
Furthermore, B3LYP APT charges are in excellent agreement with QCISD
ones with the same basis set. In effect, B3LYP APT charges with a basis
set of at least split-valence plus polarization quality will be close
to converged.

Topological (Bader) charges are furthermore considered in 

P. Geerlings, F. De Proft, and J. M. L. Martin, ``Density-Functional
Theory Concepts and Techniques for Studying Molecular Charge Distributions
and Related Properties'', in Density-Functional methods in
chemistry (eds. J. Seminario and P. Politzer), Elsevier, 1996

Sincerely,
Jan M.L. Martin
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