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Molecular-Orbital Theory
Introduction
Orbitals in molecules are not necessarily localized on atoms or between atoms as
suggested in the valence bond theory.
Molecular orbitals can also be formed the LCAO where more than two atomic orbitals
are used. (Technically, we can use all of the atomic orbitals in the LCAO.)
Linear combinations of orbitals result in the interference of waves. Both constructive and
destructive interference may result.
Within a diatomic molecule, N atomic orbitals on atom 1 and N atomic orbitals on atom 2
result in 2N molecular orbitals.
**In a practical sense, only those atomic orbitals that have similar energies and the
appropriate symmetry can combine to form molecular orbitals.**
These molecular orbitals are classified as bonding orbitals, antibonding orbitals or
nonbonding orbitals.
Types of Molecular Orbitals
Bonding Orbitals
For the linear combination of only two atomic orbitals, the bonding orbital is constructed
from adding the wavefunctions together.
MO  AO  A   AO  B
Thus, the probability density of the bonding orbital can be written as
 MO   AO  A    AO  B  2 AO  A   AO  B 
2
2
2
 A 2  B2  2AB
The 2AB term is the constructive interference of the overlap of the atomic orbitals.
- The interference results in an increase of electron density between the nuclei.
- Thus, the nuclei have greater attraction to each other, via their mutual attraction to
the increased electron density.
1s
1s
2px
2px
1s
2p
2
Aside: S   AB d is called overlap integral.
Antibonding Orbitals
For the linear combination of only two atomic orbitals, the antibonding orbital is
constructed from subtracting the wavefunctions together.
MO  AO  A   AO  B
Thus, the probability density of the bonding orbital can be written as
 MO   AO  A    AO  B   2 AO  A   AO  B 
2
2
2
 A 2  B2  2AB
The -2AB term is the destructive interference of the overlap of the atomic orbitals.
- The interference results in a decrease of electron density between the nuclei and
increase of electron density away from the nuclei.
- Thus, the nuclei are drawn away from each other.
1s
2px
1s
1s
2px
2p
The energy increase of the antibonding orbital is slightly higher than the energy decrease
1
E 
of the bonding orbital.
1 S
Example: Let S 
1
4
E 
1
1
1 4

 
1 S 1 1 5 5
4 4
E 
1
1
1 4

 
1 S 1 1 3 3
4 4
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Energy Level Diagrams of Diatomic Molecular Orbitals
Homonuclear Diatomic Molecules
Hydrogen
Bond order = ½ (# of bonding e- – # of antibonding e-) = number of bonds
The bond order for the hydrogen molecule is one.
Helium
The bond order for the helium molecule is zero, i.e., He2 does not exist.
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Lithium
Dilithium exists! (Though not as crystal!)
The bond order for the lithium molecule is one, i.e., 2 bonds – 1 antibond = 1 bond.
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Beryllium
Diberyllium does not exist.
The bond order for the beryllium molecule is zero,
- i.e., 2 bonds – 2 antibonds = 0 bond.
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Molecular Orbital Diagrams for Second Row Diatomic Molecules
B2 through N2
Notes:
For B through N, 2s and 2p are close enough in energy for hybridization to change
the order of the molecular orbital energies.
B2 bond is  bond?!? Also, B2 is paramagnetic.
- Molecular electronic structure also follows Hund’s rules.
C2 bonding is 2  bonds. C2 is diamagnetic.
Note carbide ion, C22- has triple bond.
N2 has triple bond. N2 is diamagnetic.
Q: Which is more stable, N2+ or N2-?
A: Antibond stronger than bond; therefore N2+ is predicted to be more stable.
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O2 through Ne2
Notes:
O2 bond is 1  and 1  bond. Also, O2 is paramagnetic.
F2 bonding is  bond. F2 is diamagnetic.
Bond order of Ne2 is zero. Ne atoms are not bound.
- Dimer does exist as van der Waals state.
- Ne2+ can exist.
Q: According to MO theory, Ne22+ should be stable ion. Why haven’t we heard of it
until now?
A: I don’t know.
Hypothesis: Neon is even more electronegative than fluorine.
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Heteronuclear Diatomic Molecules
Energy of atomic orbitals is affected by effective nuclear charge.
More electronegative atom will draw electron density to itself in a bonding orbital.
Consider carbon monoxide as an example.
C
O
C



C
O

O
C
O

Note: Most electropositive atom has most anti-bonding density.
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Term Symbols of Molecular States
Molecular Angular Momentum of Diatomic Molecules
Orbital Angular Momentum
The electron in a non-sigma bond experiences a cylindrically symmetric potential energy
in contrast to an electron in an atomic non-s orbital that has a spherically symmetric
potential energy.
The electron in a non-sigma bond has an angular momentum, L. However, because the
electric field from the nuclei and the other electrons is not spherically symmetric, the
angular is not well characterized.
However, for an electron in a non-sigma bond, the projection of the angular momentum
upon the internuclear axis, ML, is well characterized.
- To distinguish from the atomic ML, the molecular ML is given a new symbol, .
Thus, the electronic state of a diatomic molecule is partially characterized by .

symbol
0

1

2

3

Each electron contributes to the total orbital angular momentum thus
  1  2 
Each  state is doubly degenerate except the  state that has one state.
In other words, E  E
A non-sigma bond electron has two directions to precess about the internuclear axis.
Spin Angular Momentum
The spin precesses about the internuclear axis since the magnetic moment of the electron
tries to align itself with the magnetic field created by the “nuclei orbiting about the
electron”.
- This phenomenon could be referred to as molecular spin-orbit coupling.
The projection of the spin, S, on the internuclear axis is MS.
- To distinguish from the atomic MS, the molecular MS is given a new symbol, .
- Note source of confusion:  can refer to a specific orbital angular
momentum state or it can refer to the general total
spin angular momentum.
Each electron contributes to the total spin angular momentum thus
  1  2 
Multiplicity = 2 + 1
The rules for multiplicity are the same as for atomic structure.
- filled orbital
multiplicity = 1 singlet
0
- half-filled orbital
multiplicity = 2 doublet
 1 2
- two half-filled orbitals   1
multiplicity = 3 triplet
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Introduction to Symmetry
Parity
g – gerade
symmetric for inversion through center
u – ungerade antisymmetric for inversion through center
Reflection
+ - reflection through plane containing internuclear axis is symmetric
– - reflection through plane containing internuclear axis is antisymmetric
Consider the symmetries of the sigma and pi bonding and antibonding orbitals.
Parity
Reflection
g
+
u
–
u
+
g
–
When two or more electrons are in a molecular state, the total symmetry of the state is
determined by multiplying the symmetries of the electrons together.
Multiplication rules for of parity and reflection symmetries
Parity
gg  g gu  ug  u u u  g
Reflection
                                   
Writing Term Symbols
Information available in a molecular term symbol
multiplicity
orbital angular momentum
2
 g
reflection symmetry
parity
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Writing an electron configuration as a term symbol.
Use B2 as an example.
The electron configuration for B2 can be written as (1s2)(1s*2)(2s2)(2s*2)(2p2)
As in the atomic case, closed subshells do not contribute to the total angular momentum.
Two cases for electrons in the 2p orbitals.
each electron is in different orbitals
  state
  1  2  1   1  0
electron spins can be parallel or antiparallel
  1  2  1 2  1 2  1 or   1  2  1 2 1 2  0
thus, possible terms are 3  g or 1 g
both electrons in the same orbital
  state
  1  2  1  1  2
electron spins can be antiparallel only (can’t violate Pauli exclusion principle)
  1  2  1 2 1 2  0
thus, possible terms are 1  g
Using Hund’s rules as with the atomic term symbols, the term with the highest
multiplicity is the lowest energy. Thus the ground state term of B2 is 3  g
Use C2 as another example.
The ground state electron configuration for C2 can be written as
(1s2)(1s*2)(2s2)(2s*2)(2p4). Thus the term for the ground state is 1 g .
However, consider excited state 3  u . What is the electronic configuration for lowest
possible excited state? How can we assign the molecular orbitals for the various
electrons?
The  state implies that  = 0. At this point, let us consider that a single electron has
been excited to a * orbital.
  0  1  2  0  1  1 2  1
Note that the above could be true if the electron is excited to a higher  orbital
as well.
multiplicity = 3 implies that   1  1  2  1 2
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The one electron remaining in the  orbital has an ungerade parity and negative
reflection symmetry.
Thus, the electron in the unknown orbital must be gerade parity and negative
reflection symmetry.
u ?  u  ?  g
     ?    

 ?    
Since the * orbital has gerade parity and negative reflection symmetry, it is possible that
the 3  u implies the electron configuration (1s2)(1s*2)(2s2)(2s*2)(2p3)(2p*1).
However, ambiguity exists in the term symbol.
The electron configuration, (1s2)(1s*2)(2s2)(2s*2)(2p3)(3p*1), would also result
in a term of 3  u . To unambiguously assign electronic structure, more information is
needed.
The term symbols of molecular states are very convenient when studying electronic
spectroscopy. The selection rules of electronic spectroscopy are dependent on the
symmetry changes of the states. Thus having the term symbols will allow us to quickly
identify between which states transitions can occur.
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Molecular Orbitals of Polyatomic Molecules
The molecular orbitals of polyatomic molecules can be constructed from adding and
subtracting appropriate combinations of atomic orbitals. The atomic orbitals must have a
proper spatial symmetry in order for a net bond overlap to occur.
A Little Trick – Symmetry Adapted LCAO
Consider the construction of molecular orbitals for a polyatomic molecule like linear H3.
It is more difficult to consider linear combinations of three orbitals rather than two
orbitals. Therefore, we will be clever and consider only combinations of two orbitals at a
time.
H3 – Linear Triatomic Hydrogen
For H3, we will create bonding and antibonding orbitals from the atomic orbitals of
hydrogens 1 and 3. Then we will combine the molecular orbitals of hydrogens 1 and 3
with the atomic orbital of hydrogen 2 to make a complete set of molecular orbitals.
First add and subtract the 1s orbitals of hydrogens 1 and 3.
+
1s
1s
1s
1s
1s
-
1s
Note that since hydrogens 1 and 3 are far apart, very little overlap occurs. An energy
level diagram demonstrates that the energy changes at this point are minimal.
Now let us take each molecular orbital above and add the 1s orbital of hydrogen 2 to
each.
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Orbital 1
Note that orbital 1 bonds all three hydrogen nuclei.
No Interaction
There is no interaction between the 1s* and 1s since the amount of positive overlap
equal the amount of negative overlap. Thus no new orbital is formed.
Now take the molecular orbitals from hydrogens 1 and 3 and subtract the 1s orbital of
hydrogen 2 from each.
Orbital 3
Note that orbital 3 is antibonding between all three hydrogen nuclei.
No Interaction
Again the interaction between the 1s* and 1s is zero.
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If we have formed two molecular orbitals from three atomic orbitals so far, where is the
third orbital? The third orbital has the shape of the 1s* orbital.
Orbital 2
We note that orbital 2 does not draw nuclei together as a bonding orbital would do nor
does it draw nuclei apart as an antibonding orbital would do. Thus, orbital 2 is named a
nonbonding orbital.
The energy level diagram for these molecular orbitals is
Thus linear H3 has a bonding orbital (orbital 1), a nonbonding orbital (orbital 2) and an
antibonding orbital (orbital 3).
H2O – Water
To find the molecular orbitals for water we will take a similar tack as we did with H3.
First we will construct , and * orbitals from the hydrogen 1s orbitals.
Now let us consider the interaction of these orbitals with the five orbitals of oxygen, 1s,
2s, 2px, 2py and 2pz.
First we can consider that the 1s orbital of oxygen is much lower in energy than hydrogen
1s orbitals. Therefore, as a first approximation, we will consider that there is no
interaction with the 1s orbital of oxygen. The 1s orbital of oxygen is a nonbonding
orbital or a perhaps better described as a nonbonding core orbital.
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Next, consider the interaction with the 2s orbital. The energy of the 2s oxygen orbital is
still below the energy of the 1s hydrogen orbital; but, it is close enough for an interaction
to occur. The shapes of the orbitals are similar to the H3 orbitals but the water orbitals
are bent.
No Interaction
The 1s* orbital may be a nonbonding orbital or it may interact with another orbital.
Now consider the interaction with 2p orbitals. First we need to define a coordinate
system.
Z
Y
H
H
O
X
Now consider the interaction between the hydrogen orbitals and the 2py orbital.
Looking down from z direction.
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Note that any constructive interference that occurs is balanced by the destructive
interference that occurs. Thus, the effective interaction is zero. The lack of interaction
occurs whether we consider the hydrogen orbitals in the 1s or the 1s* configuration.
Therefore, the 2py orbital is a nonbonding orbital.
Now consider the interaction with the 2pz orbital. Let us add the 2pz orbital to the
hydrogen molecular orbitals.
No Interaction
We still don’t know if the 1s* orbital will be a nonbonding orbital or whether it will
interact with another orbital.
Also, note that we could subtract the 2pz orbital from the 1s orbital to yield a strictly
antibonding orbital.
Finally, consider the interaction with the 2px orbital.
No Interaction
Note that we could subtract the 2px orbital to yield a strictly antibonding orbital.
Finally we answer the question about the 1s* orbital. It will interact with the 2px orbital
to form a bonding and antibonding orbital.
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Hückel Model
Previous discussion has been a simplified view of MO theory. Procedures exist to take
all of the atomic orbitals and mix them together to form molecular orbitals. Examining
the procedure is beyond what we would like to consider here. However, we get a taste of
the procedure by examining what is known as the Hückel model.
The Hückel model is applicable to the pi system of conjugated molecules.
1. Construct wavefunction from a sum of p orbitals perpendicular to the molecular
plane.
  c1p1  c2p2  c3p3 
2. Assume that all overlap integrals are zero.
S  pi p j  ij  0
3. Assume H11  H22  H33 

Hii  pi H pi  
4. Assume that H12  H23  H34 
  for adjacent p orbitals
Hij  pi H p j  
5. Assume that H13  H14  H24 
 0 for nonadjacent p orbitals
6. Then construct a secular determinant.
Consider cyclobutadiene as an example.

0
   c1 
  E

 
E

0   c2 
 
0 
 0

E
   c3 

  
0

  E 
 
 c4 
E

0


0

E

0
0

E

0

E
7. Make solution of secular determinant easier by performing the substitution,
E
x

x 1 0 1
1 x 1 0
0
0 1 x 1
1
0 1
x
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8. Find the solutions of the secular determinant.
x 4  4x 2  0  x  0,0, 2, 2  E  , ,   2,   2
9. Find the eigenvectors of the secular determinant to find the molecular orbitals
For E    2
v  1 2 1 2 1 2 1 2 
For E  
v 1
For E  
v 0 1
For E    2
v  1 2 1 2 1 2 1 2


2 0 1
2 0
2 0 1
2
1
1
1
1
p1  p2  p3  p4
2
2
2
2
1
1

p 
p
2 1
2 3
1
1

p2 
p
2
2 4
1
1
1
1
  p1  p2  p3  p4
2
2
2
2


