Mulliken Population Analysis

1966 Nobel Prize motivation: "for Mullikenfundamental work concerning chemical bonds and the electronic structure of molecules by

the molecular orbital method"

The basics of quantum chemistry

• The n-electronic wave function

• probability of simultaneously finding ē 1 with spin ms1 in the volume dx1dy1dz1 at (x1,y1,z1)ē 2 with spin ms2 in the volume dx2dy2dz2 at (x2,y2,z2)and so on.

( ) 21 1 1 1 1,..., , ,..., ....n s sn n n nx z m m dx dy dz dx dy dzψ

ψ

One-electronic density

• The probability density ρ of finding an electron (ANY!!!) in the neighborhood of point (x,y,z) is

• In most cases - knowing the ρ is knowing the system!

A 1.000.000$ question –How does ρ look like?

1

2

2 2

( , , ) ... ( , , , ,..., , ,..., ) ...n

s

n s s nall m

x y z n x y z x z m m dx dzρ ψ= ∑ ∫ ∫

ˆ ( , , ) ( , , )

( , , )sm

A A x y z x y z dxdydz

Z e x y z dxdydz en

ρ

ρ

=

= =

∫∫∫∑∫∫∫

The Hartree-Fock case• The n-electronic wave function ψ in the case of

Hartree-Fock (HF) approximation:

• Home work (3 points bonus! ). Prove:

• nj is the “occupation number” (nj = 0,1,2)

1 2

1 2

1 2

(1) (1)... (1)(1,2,... ) det (2) (2)... (2)

...... ...( )( ) ( )...

n

HF n

n

n

nn n

φ φ φψ φ φ φ

φφ φ

=

( ) 2, ,HF j j

jx y z nρ φ=∑

The energy functional = density functional (W. Kohn)

• Exact WF:• nj is the “generalized occupation number”

(nj ≅ 0 or 1); φj – natural orbitals j=1,…,∞ • Kohn - Sham : E=E[Ψ]=∫Ψ*ĤΨdV =E[ρ]=?• HF: EHF[ρ]=T[ρ]+Vne[ρ]+(Vc[ρ]+Vex[ρ])• DFT: E[ρ]=T[ρ]+Vne[ρ]+(Vc[ρ]+Vex[ρ]+Vcor[ρ])

single-electron theory including correlation!

( ) 2, , j j

jx y z nρ φ=∑

MO-LCAO approximation• In the formula:

ρ is found as the sum the probability-densityfunctions of all MOs φj

• The MOLCAO approximation:

Thus

where m is the number of MOs; • and b – is the number of AOs

( ) ( ) 2, , , ,HF j j

jx y z n x y zρ φ=∑

1 1 2 21

...b

j sj s j j bj bs

c c c cφ χ χ χ χ=

= = + + +∑* * * *

1 1 1 1 1 1

m b b m b b

j j j j rj sj r s rs r sj r s j r s

n n c c Dρ φ φ χ χ χ χ= = = = = =

= = =∑ ∑∑∑ ∑∑

Density Matrix

• Crj – the contribution of r-AO to j-MO• The probability density associated with

one electron in φj is |φj |2

Normalization condition:

where the S’s are overlap integrals:

•

*

1

m

rs j rj sjj

D n c c=

= ∑

2 2 2 21 2| | 1 ... 2j j j j bj rj sj rs

r sdV c c c c c Sφ

<

= = + + + +∑∫

rs r s r sS dv dvχ χ= ∫ ∫

Mulliken population analysis

• An electron in the MO φj contributes :♦ nrj=njcrj

2 to the net population in AO χr, ♦ nr-s,j =2njcrjcsjSrs to the overlap population of χr and χs.

• Mulliken proposed a method that apportionsthe electrons of an n-electron molecule into :1. Net populations nr in the AOs;2. Overlap populations nr-s for all pairs of AOs.

2 2 2 21 2| | 1 ... 2j j j j bj rj sj rs

r sdV c c c c c Sφ

<

= = + + + +∑∫

, , and r r j r s r s jj j

n n n n− −= =∑ ∑

Mulliken characteristics• The sum of all the net and overlap populations

equals the total number of electrons in the molecule:

• Gross atomic (A) population :• Mulliken charge of atom A : ZA=enA• Mulliken’s matrix (Mrr=nr and Mrs=nr-s) could be

divided according to atomic indexes A, B, …Then number of blocks in the A-B part of the

matrix M defines bond orderbetween atoms A and B

• NOTE: Mrs = Drs*Srs

2

r r s j jr r s s j

n n n dV n dVρ φ−>

+ = = =∑ ∑∑ ∑∫ ∫( )

12A r r s

r A r s A s An n n −

∈ > ∈ ∈

= +∑ ∑ ∑

Bonding Mulliken population analysis example : C2H2

• CΞC bonding. There are two π orbitals composed of two 2px and two 2py atomic orbitals of the two C atoms and a σ bond composed of the 1s, 2s and 2pz orbitals.

• The Gaussian output. Density matrix Drs (C1-C2 part) C2\C1 1S 2S 2PZ 2PX 2PY

1S 0.04570 -0.12510 0.14378 0.00000 0.00000 2S -0.12510 0.24810 -0.28900 0.00000 0.00000

2PZ -0.14378 0.28900 -0.30554 0.00000 0.00000 2PX 0.00000 0.00000 0.00000 0.75822 0.00000 2PY 0.00000 0.00000 0.00000 0.00000 0.75822