3 Molecular Orbital (MO) Theory
In MO theory electrons are not regarded as belonging to particular
bonds, but should be treated as spreading throughout the entire
molecule.
MO theory is more fully developed than VB theory and is widely
used in modern discussions of bonding.
3.1 The hydrogen molecule ion H2+
rA1, rB1 are the distances of
the electron from the two
nuclei and R is the distance
between the two nuclei.
SE:
2 2

1  V
2 me
V 
e2
1
1 1
( 
 )
4 0 rA1 rB1 R
Solving the SE: H=E
results in one-electron wavefunctions,
called molecular orbitals (MO).
The SE can be solved for H2+ but the wavefunctions are very
complicated functions. The solution cannot be extended to polyatomic
systems.
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3.1 Linear combinations of atomic orbitals (LCAO)
If an electron can be found in an atomic orbital (AO) belonging
to atom A and also in an AO belonging to atom B, then the
overall wavefunction is a superposition of the atomic orbitals:
=N(AB)
where for H2+ A denotes H1sA, B denotes H1sB and N is a
normalization factor.
An approximate MO formed from a linear combination of AOs
is called an LCAO-MO.
An MO with cylindrical symmetry around the internuclear axis
is called a s orbital (because it has zero orbital angular
momentum around the internuclear axis).
3.2 Shape of s orbitals
A general indication of the
shape of the boundary
surface of a s orbital.
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3.3 Bonding orbitals
Probability density of the electron in H2+: +2=N2 (A2 + 2AB+B2)
A2: probability density if the electron were confined to the atomic
orbital A
B2: probability density if the electron were confined to the atomic
orbital B
2AB: an extra contribution to the density (‘overlap density’)
The electron density calculated
from the linear combination of
atomic orbitals. Note the
accumulation of electron
density in the internuclear
region.
3.3 Bonding orbitals
A representation of the
constructive interference
that occurs when two H1s
orbitals overlap and form a
bonding s orbital.
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3.4 Antibonding orbitals
-2=N2 (A2 - 2AB+B2)
- corresponds to a higher energy than +. It is also a s
orbital. This 2s orbital is an example of an antibonding
orbital. If occupied it contributes to a reduction in the
cohesion between the two atoms.
A representation of the
destructive interference that
occurs when two H1s orbitals
overlap and form an antibonding
s* orbital.
3.4 Antibonding orbitals
The electron density
calculated from the linear
combination of atomic
orbitals. Note the
elimination of electron
density from the
internuclear region.
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3.4.1 Molecular potential energy curves
The calculated and
experimental molecular
potential energy curves
for a hydrogen moleculeion.
3.5 Bonding and antibonding effects
A partial explanation of the
origin of bonding and
antibonding effects.
(a) In a bonding orbital, the
nuclei are attracted to the
accumulation of electron
density in the internuclear
region.
(b) In an antibonding orbital
the nuclei are attracted to an
accumulation of electron
density outside the
internuclear region
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3.6 The structures of diatomic molecules
3.6.1 Molecular orbital energy level diagram of H2
E
A molecular orbital energy level
diagram constructed from the
overlap of H1s orbitals; the
separation of the levels
corresponds to that found at
the equilibrium bond length.
The ground electronic
configuration of H2 is obtained
by accommodating the two
electrons in the lowest
available orbital (the bonding
orbital).
3.6.2 Molecular orbital energy level diagram of He2
E
The ground electronic
configuration of the
hypothetical four-electron
molecule He2 has two
bonding electrons and two
antibonding electrons. It
has a higher energy than
the separated atoms, and
so is unstable.
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3.6.3 Period 2 diatomic molecules
Elementary concepts only include the orbitals of the valence
shell to form molecular orbitals (such as VB theory).
A general principle of MO theory is that all orbitals of the
appropriate symmetry contribute to a molecular orbital.
To form s orbitals, for example, all orbitals with cylindrical
symmetry about the internuclear axis are taken into account.
From the 2s and 2pz
orbitals four molecular
orbitals can be built.
The general form of the s orbitals that may be formed is
  c A2 s A2 s  cB 2 s B 2 s  c A2 pz A2 p z  cB 2 p z B 2 p z
From the four atomic orbitals four molecular orbitals of s
symmetry can be formed by an appropriate choice of the
coefficients c.
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3.6.4 s orbitals built from the overlap of p orbitals
If 2s and 2pz orbitals have distinctively different energies they
may be treated separately. The four s orbitals then fall into two
sets.
A schematic
representation of the
composition of bonding
and antibonding s
orbitals built from the
overlap of p orbitals.
3.6.5 p orbitals
The 2px and 2py orbitals of each atom are perpendicular
to the internuclear axis giving rise to a bonding and an
antibonding p orbital.
A schematic
representation of the
structure of p bonding
and antibonding
molecular orbitals
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3.6.6 Molecular orbital energy level diagram for
homonuclear diatomic molecules
E
In some cases, p orbitals
are less strongly bonding
than s orbitals because
their maximum overlap
occurs off-axis, suggesting
that the molecular orbital
diagram should be as
shown on the left (O2 and
F2).
3.6.7 Orbital energies of period 2 diatomics
The order of energies of the s and p orbitals varies along Period 2.
The variation of
the orbital
energies of
Period 2
homonuclear
diatomics.
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3.6.8 Orbital energies of period 2 diatomics
The order shown is
appropriate as far as N2.
The relative order is
controlled by the separation
of the 2s and 2p orbitals in
the atoms, which increases
across the group. The
consequent switch in order
occurs at about N2.
3.6.9 Bond order
A measure of the net bonding in a diatomic molecule is its bond
order, b:
b
1
(n  n*)
2
Where n is the number of electrons in bonding orbitals and n* is the
number in antibonding orbitals.
The bond order is useful for discussing the characteristics of bonds.
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3.6.10 The structures of homonuclear diatomic
molecules
Ground state electron
configuration of N2 is
1s2 2s*2 1p4 3s2 and bond
order is ½ (8-2)=3.
O2: 1s2 2s*2 3s2 1p4 2p*2
bond order: ½ (8-4)=2.
F2: 1s2 2s*2 3s2 1p4 2p*4
bond order: ½ (8-6)=1.
O2: The 2p*2 electrons occupy different orbitals and have parallel
spins! The net spin angular momentum is S=1(triplet state) and
oxygen should be paramagnetic (not revealed by VB theory).
3.6.11 Bond order and bond length
Typical bond lengths in
diatomic and polyatomic
molecules. For bonds of a
given pair of elements:
The greater the bond order,
the shorter the bond.
Similarly, the greater the
bond order, the greater the
bond strength.
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3.7 Heteronuclear diatomic molecules
The electron distribution in
the covalent bond between
the atoms is not evenly
shared resulting in a polar
bond. In HF, for example,
the F atom has a net
negative charge and the H
atom a partial positive
charge.
3.7.1 Electronegativity
The charge distribution in bonds is commonly discussed in terms
of the eletronegativity, , of the elements involved.
It is a measure of the power of an atom to attract electrons to itself
when it is part of a compound.
The greater the difference in electronegativities, the greater the
polar character of the bond.
Examples: HF: 1.78; CH (commonly regarded as almost nonpolar): 0.51
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3.7.1 Electronegativity
The charge distribution in
bonds is commonly
discussed in terms of the
eletronegativity, , of the
elements involved.
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