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Mulliken population analysis (Read 4425 times)
Gerrit-Jan Linker
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Mulliken population analysis
31.07.08 at 13:45:00
 
Mulliken population analysis
 
Mulliken population analysis (MPA) can be used to calculate Mulliken charges which can be seen as partial atomic charges.
 
Ne = ∑aa | ψa>
where the sum is over all Ne electrons, each electron occupying one molecular orbital ψa.
 
We can write the MO as a linear combination of atomic orbitals (LCAO): ψa = ∑iciaФi
where the sum is over the basisset of N atomic orbitals Фi and where cia is the coefficient in which atomic orbital Фi participates in MO ψa.
 
Inserting the LCAO we get:
 
Ne = ∑a <∑iciaФi | ∑jcjaФj>
Ne = ∑aij <ciaФi | cjaФj>
Ne = ∑aij ciacjai | Фj>
Ne = ∑ija ciacjai | Фj>
 
Introducing the density matrix D with elements:Dij = ∑aciacja gives:
 
Ne = ∑ijDiji | Фj>
 
Introducing the overlap matrix S with elements: Sij = <Фi | Фj> gives:
 
Ne = ∑ij Dij Sij
 
Note that the sums over i and j run over the N basis functions of the LCAO expansion.
 
Introducing the population matrix P with elements: Pij = DijSij gives:
 
Ne = ∑ij Pij
 
Summing the elements of the population matrix gives us Ne electrons.  
 
In the Mulliken population analysis (MPA) we first divide the electrons over the atomic basis functions. This is done by taking the diagonal elements of the population matrix P and then dividing the off-diagonal elements to the two basis functions involved. Since Pij = Pji this arbitrary division simplifies to simply a sum over a row. This leads to the definition of the gross orbital population (GOP) with elements GOPi = ∑j Pij resulting in:
 
Ne = ∑iGOPi
 
Now, we have used atom centered basis functions, atomic orbitals. With Ne electrons being divided over the atomic orbitals Фi with gross orbital populations GOPi it remains to sum the populations of all the atomic orbitals centered on a particular atom A to get the gross atom population on that atom GAPA.
 
GAPA = ∑i GOPi
 
The charge QA is defined as the difference of this gross atom population, GAP, and the number of electrons of this atom in free space, which is the atomic number ZA:
 
QA = ZA - GAPA
 
The  charges QA are called Mulliken charges.  
 
The problem with this approach is the arbitrary division of the off diagonal elements of the population matrix. This leads to charge separations that are often exaggerated especially when large basissets are used.
 
There are other techniques to compute atomic charges. A new method is the LoProp analysis by Gagliardi, Lindh ad Karlstrom.
 
MPA Reference:
R.S. Mulliken, J Chem Phys, 23, 1833 (1955)
http://apps.isiknowledge.com/full_record.do?product=UA&search_mode=GeneralSe arch&qid=7&SID=S2ha2MiN9NIkIKFmgCg&page=1&doc=4&colname=WOS&cacheurlFromRightClick=no
 
References:
Mulliken population analysis
http://en.wikipedia.org/wiki/Mulliken_population_analysis
LoProp method
http://www.oraxcel.com/cgi-bin/yabb2/YaBB.pl?num=1217510095/0#0
Orbital interaction Theory of Organis Chemistry, 2nd edition, Arvi Rauk.
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« Last Edit: 31.10.16 at 10:34:53 by Gerrit-Jan Linker »  

Gerrit-Jan Linker
Linker IT Software
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Gerrit-Jan Linker
YaBB Administrator
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Posts: 75
Re: Mulliken population analysis
Reply #1 - 07.08.08 at 09:49:53
 
MPA results and basisset dependency
 
I calculated MPA charges on an S atom of EDOTTF for different basissets using a HF calculation. Results
 
BasissetCharge
STO-3G0.22
3-21G0.67
4-31G0.45
6-31G0.59
6-31G*0.33
6-31G*0.33

 
The value calculated at minimal basisset STO-3G is reasonable and is only reproduced going beyond 6-31G. The charge is high compared to the charge calculated with the LoProp method of -0.07 with a 6-31G** basisset.
 
See also:
LoProp method  
http://www.oraxcel.com/cgi-bin/yabb2/YaBB.pl?num=1217510095/0#0  
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« Last Edit: 07.08.08 at 10:01:20 by Gerrit-Jan Linker »  

Gerrit-Jan Linker
Linker IT Software
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