mulliken population analysis - tau
TRANSCRIPT
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