Download 5.04, Principles of Inorganic Chemistry II Lecture 16

5.04, Principles of Inorganic Chemistry II
MIT Department of Chemistry
Lecture 16: Angular Overlap Method (AOM) and M-L Diatomics
We will begin by “solving” the Hamiltonean by a simple method, afforded by the
Wolfsberg-Hemholtz approximation
from Wolfsberg-Hemholtz
approx
E+
E+ EM
EM
(H
+
ML
EMSML
2
)
EML
M
2
EML
EL
L
2
2
2
E S
= L ML = SML
EML
E
2
2
E S
x = L ML = SML
EML
b
E EL
(H
+
ML
EL SML
2
)
EML
E+ and E- are roots of secular determinant
describing the LCAO wave function,
= CMM + CLL
Thus by determining the overlap integral, SML, the relative energies of the metal
and ligand based orbitals may be obtained… it is this energy difference that is
measured by electronic spectroscopy.
5.04, Principles of Inorganic Chemistry II
Lecture 16
Page 1 of 3
Recall that the overall wave function is a product of radial and angular functions
n,,m = R n, (r) Y,m (, )
radial
angular
1
2
3
2z n 1 / 2 2+1
• L n+ () • R n, (r) = e
3
na
0 2n n ! (
)
[( ) ]
where =
2z
r
na0
Laguerre polynomial
1
d m
1 im
2 + 1 + m ! 1
2
Y,m (, ) = 1
• •
•
sin •
e
m
m
2
m ! 2 ! sin d cos 2 ,m ()
m ()
(
(
( )
)
)
(
)
In the AOM Method, the overlap integral is also factored into a radial and
angular product…
SML = S(r) F(,)
S(r)
F(,)
M
l
L
cos l
L
2
L
3cos -1
l
e-r
M
L
54.73˚
r
0
90
cos sin l
L
Under the condition of a fixed M-L distance, the overlap integral, SML, will
depend only on F(,) 0
5.04, Principles of Inorganic Chemistry II
45
45
0
90
Lecture 16
Page 2 of 3
90
To begin, let’s determine the energy of the d-orbitals for a M-L diatomic. There
are three types of interactions
z
z
z
y
y
x
x
y
2
x
y
x
2
y
y
x
2
( ) ( )
x
()
( ) (
2
)
()
E dz2 = SML = • F , = •1 = e E dyz = E dxz = SML = e E dxy = E dx2 y2 = SML = e
the metal orbital bears
the antibonding
interaction… hence dz2 is
the (M-L)* interaction
above is for L ligands…etc
metal orbitals are
destablized by e. For acceptors, the metal will be
stabilized by e.
SML() is small compared
to SML() or SML(). Thus
overlap integral and
associated energy is
ignored in most AOM
treatments.
( )
()
( )
destablized by e (the
corresponding L orbital is
stabilized by ’2•1 = e’)
The overall energy level diagrams for the M-L diatomic are:
donor
acceptor
-only
2
dz
e
dxz
2
dx -y
2
2
dz
L*
e
dyz
e
dyz
dxy
e
dxz,dyz
2
dz
e'
2
dz
dxz
2
dx -y , dxy
2
dxy
e
2 2
dx -y
2
2
dx -y , dxy
dxz,dyz
L (px, py)
e'
L (pz)
e'
L
e'
L
e'
Of course, more than 1 ligand in a typical coordination compound. The power of
AOM is that the e and e energies are additive. Thus MO energy levels of the
coordination compound are determined by simply summing e and e for each
M(d)-L interaction.
5.04, Principles of Inorganic Chemistry II
Lecture 16
Page 3 of 3