Transcript Slide 1

```Lecture 15
Molecular Bonding Theories
1) Molecular Orbital Theory
•
Considers all electrons in the field of all atoms constituting a polyatomic
species, so that all molecular orbitals (MO’s) are multicenter.
•
The most common way to build MO’s, yMO,j, is by using a linear
combination of all available atomic orbitals yAO, i (LCAO):
y MO, j   cijy AO,i
i
•
Consider dihydrogen molecule ion H2+ as an example. We can form two
molecular orbitals from two hydrogen’s atomic orbitals yA and yB :
yA yB
•
yg = y A + y B
yu = y A - y B
yA yB
The probability to find an electron on the yg orbital is then
N 2 y g2 dv  N 2 ( y A2 dv  2 y Ay B dv  y B2 dv )  N 2 (1  2S  1)  2 N 2 (1  S )  1
where S is the overlap integral for atomic orbitals yA and yB and N is
normalizing constant.
S > 0 corresponds to a bonding (yg) and S < 0 (yu) – to antibonding
situations.
2) Molecular Orbital Theory. Dihydrogen molecule
•
In dihydrogen molecule we have 2 electrons, 1 and 2. Assuming that 1 and 2
are independent one from another (one-electron approximation), we can get:
yg = [y A(1) + y B(1)][y A(2) + y B(2)]
yg = [y A(1)y A(2) + y B(1)y B(2)] + [y A(1)y B(2) + y A(2)y B(1)]
yu = [y A(1)y A(2) + y B(1)y B(2)] - [y A(1)y B(2) + y A(2)y B(1)]
•
The first term corresponds to ionic contribution and the second one – to covalent
contribution to the bonding (compare with VB theory).
AMPLITUDE (A.U.)

yu
yA
u
g

yB
yg
g < u
INTERNUCLEAR DISTANCE, BOHR
3) Atomic orbital overlap and covalent bonding
•
•
•
Interaction between atomic orbitals leads to formation of bonds only if the orbitals:
– 1) are of the same molecular symmetry;
– 2) can overlap well (see explanation of the overlap integral below);
– 3) are of similar energy (less than 20 eV energy gap).
Any two orbitals yA and yB can be characterized by the overlap integral,
S  y Ay B dV
Depending on the symmetry and the distance between two orbitals, their overlap integral S
may be positive (bonding), negative (antibonding) or zero (non-bonding interaction).

S>0
S<0
s g)
 s u)

*
p u)
*p g)
H H
P
H
CC
P
M
P
C
S=0
*
p g)

*p u)
*
(no symmetry
match)
4) Simplified MO diagram for homonuclear diatomic molecules
(second period)
•
Combination of 1s, 2s, 2px, 2py and
2pz atomic orbitals of two atoms A
of second row elements leads to ten
molecular orbitals.
10)
9)
8)
7)
6)
5)
4)
3)
2)
1)
*2p = y2pA – y2pB
*2py = y2pyA – y2pyB
*2px = y2pxA – y2pxB
2py = y2pyA + y2pyB
2px = y2pxA + y2pxB
2p = y2pA + y2pB
*2s = y2sA – y2sB
2s = y2sA + y2sB
*1s = y1sA – y1sB
1s = y1sA + y1sB
(3u)
(g)
(g)
(u)
(u)
(3g)
(2u)
(2g)
(1u)
(1g)
3u
g
2p
2p
u
3g
2u
2s
2s
2g
1u
1s
1s
1g
x2
x1
A
A
y1
y2
z
5) Orbital mixing and level inversion in homonuclear diatomic
molecules
•
•
Orbitals belonging to the same atom mix if all of the following is true: 1) they are of
the same symmetry; 2) they are of similar energy (less than 20 eV difference).
Note that there is no -orbitals of the same symmetry in diatomic homonuclear
molecules (g and u only). So, they energy levels will remain unaffected by mixing.
Before s-p mixing
After s-p mixing
3u
3u
mix
3g
mix
3g
1u
less bonding,
lone pair like
1u
2u
2u
2g
more antibonding
2g
less antibonding,
lone pair like
more bonding
6) MO diagrams of homonuclear diatomic molecules
•
Filling the resulting MO’s of homonuclear
diatomic molecules with electrons leads to the
following results:
A
3u
B
g
AB
# of e’s
Bond
order
#
unpair.
e’s
Bond
energy,
eV
Li2
6
1
0
1.1
Be2
8
0
0
-
B2
10
1
2
3.0
C2
12
2
0
6.4
N2
14
3
0
9.9
O2
16
2
2
5.2
F2
18
1
0
1.4
Ne2
20
0
0
-
Bond order = ½ (#Bonding e’s - #Antibonding e’s)
2p
3g
2p
u
2s
2u
2s
2g
1u
1s
1g
1s
7) Molecular Orbital Theory. Energy levels in N2 molecule
•
Photoelectron spectroscopy of simple molecules is an invaluable source of the
information about their electronic structure.
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The He-I photoelectron spectrum of gaseous N2 below proves that there is the - level
inversion in this molecule. It also allows identify bonding (peaks with fine vibronic
structure) and non-bonding MO (simple peaks) in it.
N
6u
N
2g
15.6
2p
16.7
5g
2p
1u
18.8
4u
2s
2s
3g
2u
E, eV
1s
1g
1s
```