Chimica Inorganica 3
General Electronic Considerations of Metal-Ligand Complexes
Metal complexes are Lewis acid-base adducts formed between metal ions (the
acid) and ligands (the base).
Chimica Inorganica 3
The interaction of the frontier atomic (for single atom ligands) or molecular (for
many atom ligands) orbitals of the ligand and metal lead to bond formation,
Chimica Inorganica 3
More quantitatively, the interaction energy of stabilization and destabilization,
εσ and εσ*, respectively, is defined on the following energy level diagram,
Chimica Inorganica 3
Treating this problem within the LCAO framework comprising metal and
ligand orbitals yields,
! = c M" M + cL"L
and solving for the Hamiltonian,
Hˆ ! = E!
Hˆ - E ! = Hˆ - E c M! M + cL!L = 0
Chimica Inorganica 3
Left-multiplying by φM and φL yields the set of linear homogeneous equations,
cM ! M Hˆ ! E ! M + cL ! M Hˆ ! E ! L = 0
cM ! L Hˆ ! E ! M + cL ! L Hˆ ! E ! L = 0
which furnishes the secular determinant,
H MM ! E
H ML ! ESML
H ML ! ESML
H LL ! E
(
)
=
EM ! E
H ML ! ESML
H ML ! ESML
EL ! E
=0
2
2
E 2 1 ! SML
+ E ( 2H ML SML ! H MM ! H LL ) + H MM H LL ! H ML
=0
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Hypothesis n. 1 SML = 0
H ML = 0
E1 = H MM ; E2 = H LL
cM 1 = 1;cL1 = 0
cM 2 = 0;cL 2 = 1
Hypothesis n. 2 SML ! 0
i) H MM = H LL
ii) H MM ! H LL
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i) H MM = H LL = ! ; H ML = ! ;SML = S
(
)
E 2 1 ! S 2 + 2E ( ! S ! " ) + ! 2 ! " 2 = 0
E=
! (" S ! ! ) ±
! (! S ! " ) ±
( ! S ! " )2 ! (! 2 ! " 2 ) (1 ! S 2 )
(1 ! S )
2
=
( ! S )2 + ! 2 ! 2 " S! ! ! 2 + " 2 + (! S )2 ! ( ! S )2
(1 + S ) (1 ! S )
=
2
"
! ( ! S ! " ) + !2 ! S" + ! 2 + (" S )
! ( ! S ! " ) + (! S ! " ) ! (1 + S ) ! ! (1 + S ) ! ! "
$ E2 =
=
=
=
(1 + S ) (1 ! S )
(1 + S ) (1 ! S )
(1 + S ) (1 ! S )
(1 ! S )
$
#
2
$
! ( ! S ! " ) ! !2 ! S" + ! 2 + (" S )
! ( ! S ! " ) ! (! S ! " ) ! (1 ! S ) + ! (1 ! S ) ! + "
E
=
=
=
=
$ 1
(1 + S ) (1 ! S )
(1 + S ) (1 ! S )
(1 + S ) (1 ! S )
(1 + S )
%
Keep in mind that β < 0 and S > 0
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! !"
1
"
E
=
;
c
=
!c
=
A2
B2
$ 2 (1 ! S )
2 (1 ! S )
$
#
1
$ E1 = ! + " ; cA1 = cB1 =
$
(1 + S )
2 (1 + S )
%
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! !"
E2 =
1!S
(
)
ε
E1 =
! +"
1+S
(
ψ2
)
ϕ1
cM 2 = !cL 2 =
1
2 (1 ! S )
ϕ2
ψ1
cM 1 = cL1 =
1
2 (1 + S )
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ii) H MM ! H LL
ψ2
ϕM
H ML ! ESML
EL ! E
!
D
2
D
!E
2
!B
D
2
ϕL
ψ1
H ML ! ESML
H ML ! ESML = !B
!
D = EM ! EL
EM ! E
!B
D
!E
2
=0
" D
% "D
%
2
!
!
E
!
E
$#
'& $#
'& = B
2
2
D2
+ B 2 = E2
4
=0
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D2
+ B 2 = E2
4
H ML ! ESML = !B
ψ2
E-
B2
!
D
ϕM
E! = !B
!
D = EM ! EL
E+
D =0
E+ = B
if
D
2
if
D
2
B2
D
ψ1
ϕL
B << D
(
D2
D
B2
2
=±
1+ 4 2 !
*± B +
4
2
D
*
E± = )
2
2
"
%
"
%
D
B
D
B
*±
1
+
2
=
±
+
$# 2 D '&
*+ 2 $#
D 2 '&
(
E! = !
H ML ! EM SML
D B
!
= EM !
2 D
"EML
E+ = +
H ML ! ELSML
D B
+
= EL +
2 D
"EML
2
2
(
)
)
2
2
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Maclaurin Expansion
xn ! d n f $
f x =( # n&
n=0 n! " dx % x=0
()
'
02/1698 – 14/06/1746
,
1
+
x
1
x
,
/
2
f x = 1 + x 2 ) 1 + * . 1 + x 2 2x 1 + .. 1 + x 2
1! 2 0 x=0 2!
.B2
2
x =4 2
D
x2
4B 2
B2
f x )1+
= 1+
= 1+ 2 2
2!
2D 2
D
()
(
)
()
2
D2
D
B2
2
4± B +
=±
1+ 4 2 )
4
2
4
D
E± = 3
2
2
!
$
!
$
D
B
D
B
4±
1
+
2
=
±
+
2&
#" 2 D &%
4 2 #"
D
%
5
2
(
)
+ 12
+
(
2x 1 + x
2
2
)
+ 23
/
2x 1
1 + ...
10 x=0
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E-
E! = EM
H
(
!
ML
"EML
H MM
E+ = EL +
(
H LL
E+
! EM SML
H ML ! ELSML
"EML
)
)
2
2
The Wolfsberg-Hemholz approximation provides a value for HML, defined as
H ML = ( EM + EL ) SML
!" *
H ML
2
2
EM SML + EL SML ! EM SML )
EL SML )
(
(
=
=
"EML
"EML
!"
E
(
= 1.75
M
+ EL
2
)S
ML
2
2
EM SML + EL SML ! EL SML )
EM SML )
(
(
=
=
"EML
"EML
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The derivation highlights the following general rules for the construction of MO
diagrams,
(1)  M—L atomic orbital mixing is proportional to the overlap of the metal and ligand
orbital, i.e., SML
corollary A: only orbitals of correct symmetry can mix and ∴ give a nonzero
interaction energy (i.e. SML ≠ 0)
corollary B: σ interactions typically give rise to larger interaction energies than those
resulting from π interactions and π interactions are greater than δ interactions owing
to more directional bonding along the series SML (σ) > SML (π) > SML (δ)
(2) M–L atomic orbital mixing is inversely proportional to energy difference of
mixing orbitals (i.e. ΔEML).
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Another issue of interest for the construction of MOs is,
(3) The order of the EL and EM energy levels almost always is:
σ(L) < π(L) < nd < (n+1)s < (n+1)p
 
π*L
depending on the nature of the ligand
 
This energy ordering comes directly from Valence Orbital Ionization Energies
(VOIE) of metal and main group atoms and PES spectra of molecular ligands.
VOIE’s of metal atoms
Atom: 3dn–14s → 3dn–24s 3dn–14s → 3dn–1 3dn–14p → 3dn–1
3d
4s
4p
Sc
4.7
5.7
3.2
Ti
5.6
6.1
3.3
V
6.3
6.3
3.5
Cr
7.2
6.6
3.5
Mn
7.9
6.8
3.6
Fe
8.7
7.1
3.7
Co
9.4
7.3
3.8
Ni
10.0
7.6
3.8
Cu
10.7
7.7
4.0
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VOIE’s of ligand atomic orbitals and PES spectra of selected ligands:
Atom
1s
H
C
N
O
F
Si
P
S
Br
13.6
2s
2p
19.4
25.6
32.3
40.2
10.6
13.2
15.8
18.6
3s
3p
14.9
18.8
20.7
7.7
10.1
11.6
4s
24.1
4p
12.5
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PES energies of ligands are in eVs (note: a VOIE
is simply the opposite of the ionization energy)
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General observations:
(1) The s orbitals are generally too low in energy to participate in
bonding (ΔEML(σ) is very large)
(2) Filled p orbitals are the frontier orbitals, and they have VOIEs that
place them below the metal orbitals
(3) For molecular ligands, since the frontier orbitals comprise s and p
orbitals, here too filled ligand orbitals have energies that are stabilized
relative to the metal orbitals