Understanding Angular Overlap Method in Advanced Inorganic Chemistry

 
A
d
v
a
n
c
e
d
 
I
n
o
r
g
a
n
i
c
 
C
h
e
m
i
s
t
r
y
By 
Dr. 
MAHMOUD NAJIM
 
A
n
g
u
l
a
r
 
O
v
e
r
l
a
p
 
M
e
t
h
o
d
 
Molecular orbital diagram (Fig. 1)does 
not 
provide 
a 
complete understanding
of the theory of 
complexes 
or 
a 
method 
to 
quantify 
the energies 
involved 
and 
one
way 
to 
approach 
this 
problem is 
AOM 
which 
can 
illustrate 
a 
qualitative  
discussion
of the 
physical 
rationale 
for 
the
 
theory.
 
Fig. 
2 
showed that 
the 
interaction 
of 
two 
atomic 
orbitals 
having 
symmetry
appropriate 
for positive overlap 
to give 
a 
bonding 
and 
antibonding 
orbital 
with
energy 
determined 
by the 
overlap 
and the energies of the original
 
orbitals.
The degree of the 
overlap 
(S
ML
) 
will 
depend 
upon 
the angle 
between 
the
orbitals 
( 
Fig. 
3) 
which 
can be used 
as a 
basis of 
AOM to 
treat 
the 
coordination
compounds 
by 
MOT 
assumed 
to 
the 
following
 
equation:
S
ML 
= 
S
σ 
cosѲ 
……………………. 
1
The energy of 
interaction 
Ε 
is 
taken 
as:
Ε = 
βS
2 
………………………… 
..
 
2
ML
Were 
β 
is 
constant inversely 
proportional 
to 
the 
deference in 
energy 
between
the original 
orbitals. 
Because 
it is 
constant 
for given metal ion 
and set of 
ligands,  
so
only 
the angular 
dependence 
of the 
overlap 
integral 
need 
to 
be 
considered
 
.
 
For 
the 
overlap 
of 
d
z
2 
orbital 
with 
a 
ligand 
on the 
z 
axis(
Ѳ 
=0 
) 
and 
according
to 
the 
functions for 
d 
orbitals given in 
Table 
we may 
rewrite 
the 
above 
equation  
as
follows:
2
Ε = β 
S
σ 
f
(
Ѳ
, 
Φ 
) 
…………………
 
3
By 
using of the 
last equation 
and the 
Table 
considering 
the 
two
 
eg
orbitals 
of 
an 
octahedral 
set 
we 
can 
evaluate
 
S
ML
S
ML 
= ½ .4 
√3/2 
S
σ 
……………………..
 
4
 
Substitution of eq. 
4 on 
eq. 
2 
we 
can
 
have
 
 
Ε  =
 
3βS
 
2
 
………………………….
 
5
σ
 
The 
same 
result 
should 
be 
determined for 
the second 
eg orbital Fig. 
4
showed 
a comparison 
of 
the 
results 
from 
the 
AOM 
and 
CFT , these
 
methods
defined
 
that
10Dq 
= 
 
S
σ
2
 
The 
source 
of the 
splitting 
energy of the 
t
2
g 
and 
eg(eg*) 
for 
AOM 
and CFT
was referred 
to 
different 
causes. 
For 
CFT 
is 
the 
barycentre 
of the 
unpaired 
d
orbitals while 
it 
is 
the 
barycentre for 
the 
unpaired 
d 
orbitals 
and the 
ligand
 
σ
 
donor orbitals for AOM, 
so 
it becomes 
12βS 
2 
of MOSE 
immediately 
after
 
four
 
σ
 
σ
ligand electrons occupying 
the 
eg 
level 
for 
the 
metal 
( 3βS 
2 
for 
each
 
e)
 
without regard 
to 
d
 
electrons.
Changing 
from 
d
1
- 
d
3 
cause no 
change in 
the MOSE because 
t
2
g 
level
strictly non bonding 
with σ-only 
system, 
although high spin 
d
4 
will 
be less
stabilized 
(12- 3)βS 
2 
similar 
to 
weak field 
case
 
(Table).
σ
For 
various geometry 
it 
is 
easy 
to 
determine 
S
ML
 
according 
to 
the
 
Table
and by 
repeating 
the 
previous calculation 
we 
can
 
obtain:
 
2
E
Z
2  
= 
(
1/4
+ ¼+ 
¼
 
+
1/4
 
)βS
σ
 
……………
 
6
 
2
 
2 
=( 
¾+ 
3/4 
+
3/4
+
3/4
) 
βS
 
2
E
x 
 
-
 
y
 
σ
 
…………..
 
7
 
σ
 
For 
square planar 
the 
destabilizations 
of βS 
2 
for occupation 
of the
 
σ*Z
2
 
antibonding 
orbital 
and of 
3βS 
2 
for 
the 
occupation 
of 
σ*
 
2
 
σ
 
x
 
 
y
 
2
 
orbital
 
For 
the 
occupation 
of 
σ*
x
2 
– y
2 
orbital 
t
2 
and e 
levels in 
tetrahedral
 
symmetry
 
σ
 
E
xz 
= E
xy 
= 
E
yz 
= 4/3 βS
 
2
 
………….
 
8
 
E
x
2 
– y
2 
=
 
E
z
2
 
= 0 
……………..
 
9
 
This last 
equation 
is 
of 
considerable 
interest 
which 
can show 
that
Dq(tet)=4/9Dq(oct) 
as
 
below
2
 
2
4/3 
βS
σ 
( 
tetr.) 
/ 
3βS
σ 
( 
oct.) 
= 4/9 
………….
 
10
As 
a comparison between 
VBT, 
CFT 
and 
MOT 
we 
can refer 
to 
the
 
following:
1
Both 
CFT 
and 
MOT 
described coordination 
complexes 
by the 
existence 
of 
two 
sets
 
of
2
orbitals 
separated 
by 
an energy 
gaps, 
10Dq
 
=
 
3βS
σ
 
,
if 
the 
energy 
to 
pair 
the 
electrons
 
is
greater 
than 
this 
, high spin 
complexes 
will 
form 
otherwise 
low 
spin 
will
 
formed.
2
Both CFT 
and 
MOT 
showed that the 
visible 
spectra
 
of
 
the
 
complexes 
are 
attributed
to 
electronic 
transitions 
such as 
t
2
g 
→ eg* , 
but the 
fundamental 
assumptions 
were
different.
3
Both CFT 
and 
MOT 
describe 
complexes 
in terms 
of 
interactions 
metal 
orbitals 
and the
ligands 
and 
with 
the 
greater 
the 
interactions 
the 
greater 
of 
the amount 
of 
10Dq, 
and
then the 
greater 
metal- 
ligand 
bond not 
from 
the 
amounts 
of the 
electrostatic
 
effects.
4
VBT 
concentrate 
only on 
the 
formation 
of 
d
2
sp
3 
hybridization which 
agree 
with 
the
MOT 
description ( 
a
1
g, t
1
u 
and 
eg molecular
 
orbitals).
 
π- 
Bonding 
in metal
 
complexes
 
π 
- 
Bonding is very useful 
in 
explaining 
the 
stability 
of 
many complexes 
,
 
in
addition 
to 
σ-bonding 
formed 
between
 
M-L
Some ligands 
can formed 
π 
- 
Bonds 
when 
it consists 
appropriate 
orbitals
 
for
π 
- 
Bonding 
with 
a 
metal 
d
-orbitals
 
(
t
2
g
)
Three 
types of 
ligands 
can do
 
that:
1
Ligands 
with 
p
-orbitals perpindicular 
to 
the
 
σ-bond
2
Ligands 
with 
a 
d-orbital 
lying in 
a plane that 
include 
the 
metal
 
atom.
3
Ligands 
with 
a 
π* 
orbital 
lying 
in 
a plane that 
include 
the 
metal 
atom 
( 
Fig.
 
6a,b
and c
 
respectively)
The 
effect 
of the 
π 
bonding 
on the 
value 
of ∆o depends on 
whether 
the 
π
ligand orbitals 
act 
as 
electron donors 
or 
acceptors There are two principles 
to 
be
described 
, 
the 
first 
when 
atomic 
orbitals overlap strongly 
, 
they 
mix 
strongly 
,
and the 
resulting bonding molecular orbitals are significantly lower in 
energy and
the 
antibonding molecular orbitals are significantly 
higher 
in 
energy than the
atomic 
orbitals. Second 
when the 
two 
atomic 
orbitals 
have 
similar 
energies 
they
interact 
strongly 
, 
but with 
large 
different 
energies 
they 
mix 
only slightly even 
if
there 
overlap is
 
large.
 
(Fig. 
6) 
π 
Bonding between metal 
d
-orbital 
&
 
Ligand
a) 
Lp
π 
Mdπ 
, b) L
d
π – 
Mdπ 
, c) 
Lπ* 
-
 
Mdπ
 
1
The 
ligand 
with 
filled orbitals 
has no 
low 
energy 
vacant
 
π
 
orbitals
 
( 
like 
Cl-, H
2
O),
and 
they lie lower in 
energy than the 
partially 
filled 
d 
orbitals 
of 
the 
metals. They
form 
a 
molecular 
orbitals, bonding lower 
than the 
ligand orbitals 
and 
antibonding
lie above 
the energy 
of 
the d 
orbitals 
of 
the 
free 
metal 
ion.(Fig. 
7). 
As 
a 
result 
a
strong 
π- 
donor 
ligand 
interaction 
decrease 
∆o
 
.
2
A 
ligands that 
can accept electrons 
into 
their 
π 
orbitals 
, 
we will 
see 
that 
such 
ligand
lie 
higher 
in spectrachemical 
series 
and 
give 
rise 
to 
a 
large ligand 
field 
splitting
parameter. 
These 
ligands 
has 
filled 
π 
orbitals 
at 
lower 
energies 
than 
metal 
t
2
g
orbitals 
and 
low 
energy 
empty 
π 
orbitals that 
are 
available 
for
 
occupation.
The 
π-
acceptor orbitals 
are 
vacant 
antibonding orbitals 
on 
the 
ligand, 
as 
in 
CO
and 
N
2, 
and these 
orbitals lie above 
the 
metal 
d 
orbital in 
energy. 
When 
these
vacant 
π* 
ligand orbitals 
are 
close in 
energy 
to 
the 
metal 
t
2 
g 
orbitals 
and
 
the
metal-ligand 
π 
overlap is 
strong, 
a 
little 
e 
density 
will 
be 
delocalized 
from 
the 
metal
to 
the 
ligand 
. As 
example 
the 
π* 
orbital 
of 
CO 
has 
its 
larger 
amplititude on 
the
 
C
atom, 
while 
the 
full 
bonding 
π 
orbitals 
of 
CO is low 
in energy and 
is largely localized
on 
the O 
atom( 
as O 
is 
more electronegative 
than 
C), 
and 
will 
make 
CO behave 
as a
net 
π
 
acceptor.
Most ligands 
with 
π 
acceptor orbitals 
are higher 
in 
energy than the 
metal 
d
orbitals, 
they form molecular 
orbitals 
with 
the 
metal in which 
d 
orbitals character
are 
lowered in 
energy, 
and 
the 
net 
result is 
that 
∆o 
is 
increased by the 
π-acceptor
interaction
 
(Fig. 
7) The effect of the π bonding on the 
ligand 
field
spliting 
 
parameter
 
a)
 
π
 
acceptor
 
b) 
π
 
donor
 
The important 
rule on 
π 
bonding is clear 
on the 
order 
of the 
ligands in
 
the
spectrochemical 
series 
which 
can be 
divided 
as
 
below:
π -donor < weak π -donor < no π effects< π
 
–acceptor
I- < Br- < 
Cl-  
< 
F- 
 
<
 
H
2
O
 
< 
NH
2 
< 
PR
3 
<
 
CO
Tetrahedral 
species can be 
divided roughly 
into 
two broad
 
clases:
1
Oxo 
species in which 
the 
metal 
ion is 
high 
in oxidation 
state 
( > 
or 
=6)and 
be 
very
extensive 
π
 
bonding(
MnO
4
-,CrO
4
-
)
2
Complexes 
with lower 
oxidation 
state 
( 
+2,+3) 
and the 
ligands are halide 
ions,
amine 
or
 
RO-
Slide Note
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Exploring the Angular Overlap Method (AOM) in advanced inorganic chemistry provides a qualitative discussion on the physical rationale behind the theory of complexes. By considering the interaction of atomic orbitals and the degree of overlap, AOM offers insights into energy quantification in coordination compounds. This method, alongside Molecular Orbital Theory (MOT), helps illustrate the angular dependence of overlap integral, contributing to a comprehensive understanding of complex formation. Comparing AOM with Crystal Field Theory (CFT) reveals distinct approaches to analyzing the energies involved in coordination complexes.


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  1. Advanced Advanced Inorganic Inorganic Chemistry By Dr. MAHMOUD NAJIM Chemistry Angular Angular Overlap Overlap Method Method

  2. Molecular orbital diagram (Fig. 1)does not provide a complete understanding of the theory of complexes or a method to quantify the energies involved and one way to approach this problem is AOM which can illustrate a qualitative discussion of the physical rationale for the theory. Fig. 2 showed that the interaction of two atomic orbitals having symmetry appropriate for positive overlap to give a bonding and antibonding orbital with energy determined by the overlap and the energies of the originalorbitals. The degree of the overlap (SML) will depend upon the angle between the orbitals ( Fig. 3) which can be used as a basis of AOM to treat the coordination compoundsby MOT assumed to the following equation: SML= S cos . 1 The energy of interaction is takenas: = S2 ..2 ML Were is constant inversely proportional to the deference in energy between the original orbitals. Because it is constant for given metal ion and set of ligands, so only the angular dependence of the overlap integral need to be considered .

  3. For the overlap of dz2 orbital with a ligand on the z axis( =0 ) and according to the functions for d orbitals given in Table we may rewrite the above equation as follows: 2 = S f( , ) 3 By using of the last equation and the Table considering the two eg orbitals of an octahedral set we can evaluate SML SML = .4 3/2 S ..4 Substitution of eq. 4 on eq. 2 we canhave = 3 S2 . 5 The same result should be determined for the second eg orbital Fig. 4 showed a comparison of the results from the AOM and CFT , these methods defined that 10Dq = 3 S 2

  4. The source of the splitting energy of the t2g and eg(eg*) for AOM and CFT was referred to different causes. For CFT is the barycentre of the unpaired d orbitals while it is the barycentre for the unpaired d orbitals and the ligand donor orbitals for AOM, so it becomes 12 S 2 of MOSE immediately afterfour ligand electrons occupying the eg level for the metal ( 3 S 2 for eache) without regard to d electrons. Changing from d1- d3cause no change in the MOSE because t2g level strictly non bonding with -only system, although high spin d4will be less stabilized(12- 3) S2similar to weak field case(Table). For various geometry it is easy to determine SMLaccording to the Table and by repeating the previouscalculation we can obtain: 2 EZ2 = (1/4+ + +1/4 ) S 6 2 2 =( + 3/4 +3/4+3/4) S2 ..7 Ex - y For square planar the destabilizations of S 2 for occupation of the *Z2 antibonding orbital and of 3 S 2 for the occupation of * 2 y2 orbital x

  5. For the occupation of *x2 y2 orbital t2 and e levels in tetrahedral symmetry Exz = Exy = Eyz = 4/3 S2 .8 Ex2 y2 =Ez2= 0 ..9 This last equation is of considerable interest which can show that Dq(tet)=4/9Dq(oct) as below 2 2 4/3 S ( tetr.) / 3 S ( oct.) = 4/9 . 10 As a comparison between VBT, CFT and MOT we can refer to the following: 1 Both CFT and MOT described coordination complexes by the existence of two sets of orbitals separated by an energy gaps, 10Dq = 3 S ,if the energy to pair the electrons is greater than this , high spin complexes will form otherwise low spin will formed. 2Both CFT and MOT showed that the visible spectra of the to electronic transitions such as t2g eg* , but the fundamental assumptions were different. 3Both CFT and MOT describe complexes in terms of interactions metal orbitals and the ligands and with the greater the interactions the greater of the amount of 10Dq, and then the greater metal- ligand bond not from the amounts of the electrostaticeffects. 4VBT concentrate only on the formation of d2sp3 hybridization which agree with the MOT description ( a1g, t1u and eg molecular orbitals). 2 complexes are attributed

  6. - Bonding in metal complexes - Bonding is very useful in explaining the stability of many complexes , in addition to -bonding formed betweenM-L Some ligands can formed - Bonds when it consists appropriate orbitals for - Bonding with a metal d-orbitals (t2g) Three types of ligands can do that: 1 Ligands with p-orbitals perpindicular to the -bond 2 Ligands with a d-orbital lying in a plane that include the metal atom. 3 Ligands with a * orbital lying in a plane that include the metal atom ( Fig. 6a,b and c respectively) The effect of the bonding on the value of o depends on whether the ligand orbitals act as electron donors or acceptors There are two principles to be described , the first when atomic orbitals overlap strongly , they mix strongly , and the resulting bonding molecular orbitals are significantly lower in energy and the antibonding molecular orbitals are significantly higher in energy than the atomic orbitals. Second when the two atomic orbitals have similar energies they interact strongly , but with large different energies they mix only slightly even if there overlap islarge.

  7. (Fig. 6) Bonding between metal d-orbital & Ligand a) Lp Md , b) Ld Md , c) L * -Md

  8. 1 The ligand with filled orbitals has no low energy vacant orbitals and they lie lower in energy than the partially filled d orbitals of the metals. They form a molecular orbitals, bonding lower than the ligand orbitals and antibonding lie above the energy of the d orbitals of the free metal ion.(Fig. 7). As a result a strong - donor ligand interaction decrease o . 2 A ligands that can accept electrons into their orbitals , we will see that such ligand lie higher in spectrachemical series and give rise to a large ligand field splitting parameter. These ligands has filled orbitals at lower energies than metal t2g orbitals and low energy empty orbitals that are available for occupation. The -acceptor orbitals are vacant antibonding orbitals on the ligand, as in CO and N2, and these orbitals lie above the metal d orbital in energy. When these vacant * ligand orbitals are close in energy to the metal t2 g orbitals andthe metal-ligand overlap is strong, a little e density will be delocalized from the metal to the ligand . As example the * orbital of CO has its larger amplititude on the C atom, while the full bonding orbitals of CO is low in energy and is largely localized on the O atom( as O is more electronegative than C), and will make CO behave as a net acceptor. Most ligands with acceptor orbitals are higher in energy than the metal d orbitals, they form molecular orbitals with the metal in which d orbitals character are lowered in energy, and the net result is that o is increased by the -acceptor interaction ( like Cl-, H2O),

  9. (Fig. 7) The effect of the bonding on the ligand field spliting parameter a) acceptor b) donor

  10. The important rule on bonding is clear on the order of the ligands in the spectrochemical series which can be divided asbelow: -donor < weak -donor < no effects< acceptor I- < Br- < Cl- < F- < H2O < NH2 < PR3 < CO Tetrahedral species can be divided roughly into two broad clases: 1 Oxo species in which the metal ion is high in oxidation state ( > or =6)and be very extensive bonding(MnO4-,CrO4-) 2 Complexes with lower oxidation state ( +2,+3) and the ligands are halide ions, amine or RO-

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