**Mulliken population analysis** Mulliken population analysis (MPA) can be used to calculate Mulliken charges which can be seen as partial atomic charges.

Consider an UHF wave function ψ = ∑

_{a} ψ

_{a} where the sum is over all N

_{e} electrons, each electron occupying one molecular orbital ψ

_{a}.

N

_{e} = ∑

_{a} <ψ

_{a} | ψ

_{a}>

We can write the MO as a linear combination of atomic orbitals (LCAO): ψ

_{a} = ∑

_{i}c

_{ia}Ф

_{i} where the sum is over the basisset of N atomic orbitals Ф

_{i} and where c

_{ia} is the coefficient in which atomic orbital Ф

_{i} participates in MO ψ

_{a}.

Inserting the LCAO we get:

N

_{e} = ∑

_{a} <∑

_{i}c

_{ia}Ф

_{i} | ∑

_{j}c

_{ja}Ф

_{j}>

N

_{e} = ∑

_{a}∑

_{i}∑

_{j} <c

_{ia}Ф

_{i} | c

_{ja}Ф

_{j}>

N

_{e} = ∑

_{a}∑

_{i}∑

_{j} c

_{ia}c

_{ja} <Ф

_{i} | Ф

_{j}>

N

_{e} = ∑

_{i}∑

_{j}∑

_{a} c

_{ia}c

_{ja} <Ф

_{i} | Ф

_{j}>

Introducing the density matrix D with elements:D

_{ij} = ∑

_{a}c

_{ia}c

_{ja} gives:

N

_{e} = ∑

_{i}∑

_{j}D

_{ij} <Ф

_{i} | Ф

_{j}>

Introducing the overlap matrix S with elements: S

_{ij} = <Ф

_{i} | Ф

_{j}> gives:

N

_{e} = ∑

_{i}∑

_{j} D

_{ij} S

_{ij} 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: P

_{ij} = D

_{ij}S

_{ij} gives:

N

_{e} = ∑

_{i}∑

_{j} P

_{ij} Summing the elements of the population matrix gives us N

_{e} 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 P

_{ij} = P

_{ji} this arbitrary division simplifies to simply a sum over a row. This leads to the definition of the gross orbital population (GOP) with elements GOP

_{i} = ∑

_{j} P

_{ij} resulting in:

N

_{e} = ∑

_{i}GOP

_{i} Now, we have used atom centered basis functions, atomic orbitals. With N

_{e} electrons being divided over the atomic orbitals Ф

_{i} with gross orbital populations GOP

_{i} 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 GAP

_{A}.

GAP

_{A} = ∑

_{i} GOP

_{i} The charge Q

_{A} 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 Z

_{A}:

Q

_{A} = Z

_{A} - GAP

_{A} The charges Q

_{A} 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.