Molecular
Orbital Theory
Valence Bond Model
vs. Molecular Orbital Theory
Because arguments based on atomic orbitals focus on the
bonds formed between valence electrons on an atom, they are
often said to involve a valence-bond theory.
The valence-bond model can't adequately explain the fact
that some molecules contains two equivalent bonds with a bond
order between that of a single bond and a double bond. The
best it can do is suggest that these molecules are mixtures,
or hybrids, of the two Lewis structures that can be written
for these molecules.
This problem, and many others, can be overcome by using a
more sophisticated model of bonding based on molecular
orbitals. Molecular orbital theory is more powerful than
valence-bond theory because the orbitals reflect the geometry
of the molecule to which they are applied. But this power
carries a significant cost in terms of the ease with which
the model can be visualized.

Forming Molecular
Orbitals
Molecular orbitals are obtained by combining the atomic
orbitals on the atoms in the molecule. Consider the H2
molecule, for example. One of the molecular orbitals in this
molecule is constructed by adding the mathematical functions
for the two 1s atomic orbitals that come together to
form this molecule. Another orbital is formed by subtracting
one of these functions from the other, as shown in the figure
below.

One of these orbitals is called a bonding molecular
orbital because electrons in this orbital spend most of
their time in the region directly between the two nuclei. It
is called a sigma (
)
molecular orbital because it looks like an s orbital
when viewed along the H-H bond. Electrons placed in the other
orbital spend most of their time away from the region between
the two nuclei. This orbital is therefore an antibonding,
or sigma star (
*),
molecular orbital.

The
bonding molecular orbital concentrates electrons in the
region directly between the two nuclei. Placing an electron
in this orbital therefore stabilizes the H2
molecule. Since the
*
antibonding molecular orbital forces the electron to spend
most of its time away from the area between the nuclei,
placing an electron in this orbital makes the molecule less
stable.
Electrons are added to molecular orbitals, one at a time,
starting with the lowest energy molecular orbital. The two
electrons associated with a pair of hydrogen atoms are placed
in the lowest energy, or
bonding, molecular orbital, as shown in the figure below.
This diagram suggests that the energy of an H2
molecule is lower than that of a pair of isolated atoms. As a
result, the H2 molecule is more stable than a pair
of isolated atoms.


Using the
Molecular Orbital Model to Explain Why Some Molecules Do Not
Exist
This molecular orbital model can be used to explain why He2
molecules don't exist. Combining a pair of helium atoms with
1s2 electron configurations would produce a
molecule with a pair of electrons in both the
bonding and the
*
antibonding molecular orbitals. The total energy of an He2
molecule would be essentially the same as the energy of a
pair of isolated helium atoms, and there would be nothing to
hold the helium atoms together to form a molecule.
The fact that an He2 molecule is neither more
nor less stable than a pair of isolated helium atoms
illustrates an important principle: The core orbitals on an
atom make no contribution to the stability of the molecules
that contain this atom. The only orbitals that are important
in our discussion of molecular orbitals are those formed when
valence-shell orbitals are combined. The molecular orbital
diagram for an O2 molecule would therefore ignore
the 1s electrons on both oxygen atoms and concentrate
on the interactions between the 2s and 2p
valence orbitals.

Molecular
Orbitals of the Second Energy Level
The 2s orbitals on one atom combine with the 2s
orbitals on another to form a
2s
bonding and a
2s*
antibonding molecular orbital, just like the
1s
and
1s*
orbitals formed from the 1s atomic orbitals. If we
arbitrarily define the Z axis of the coordinate system
for the O2 molecule as the axis along which the
bond forms, the 2pz orbitals on the
adjacent atoms will meet head-on to form a
2p
bonding and a
2p*
antibonding molecular orbital, as shown in the figure below.
These are called sigma orbitals because they look like s
orbitals when viewed along the oxygen-oxygen bond.

The 2px orbitals on one atom interact
with the 2px orbitals on the other to form
molecular orbitals that have a different shape, as shown in
the figure below. These molecular orbitals are called pi
(
)
orbitals because they look like p orbitals when viewed
along the bond. Whereas
and
*
orbitals concentrate the electrons along the axis on which
the nuclei of the atoms lie,
and
*
orbitals concentrate the electrons either above or below this
axis.

The 2px atomic orbitals combine to
form a
x
bonding molecular orbital and a
x*
antibonding molecular orbital. The same thing happens when
the 2py orbitals interact, only in this
case we get a
y
and a
y*
antibonding molecular orbital. Because there is no difference
between the energies of the 2px and 2py
atomic orbitals, there is no difference between the energies
of the
x
and
y
or the
x*
and
y*
molecular orbitals.
The interaction of four valence atomic orbitals on one
atom (2s, 2px, 2py
and 2pz) with a set of four atomic orbitals
on another atom leads to the formation of a total of eight
molecular orbitals:
2s,
2s*,
2p,
2p*,
x,
y,
x*,
and
y*.
There is a significant difference between the energies of
the 2s and 2p orbitals on an atom. As a result,
the
2s
and
*2s
orbitals both lie at lower energies than the
2p,
2p*,
x,
y,
x*,
and
y*
orbitals. To sort out the relative energies of the six
molecular orbitals formed when the 2p atomic orbitals
on a pair of atoms are combined, we need to understand the
relationship between the strength of the interaction between
a pair of orbitals and the relative energies of the molecular
orbitals they form.
Because they meet head-on, the interaction between the 2pz
orbitals is stronger than the interaction between the 2px
or 2py orbitals, which meet edge-on. As a
result, the
2p
orbital lies at a lower energy than the
x
and
y
orbitals, and the
2p*
orbital lies at higher energy than the
x*
and
y*
orbitals, as shown in the figure below.

Unfortunately an interaction is missing from this model.
It is possible for the 2s orbital on one atom to
interact with the 2pz orbital on the other.
This interaction introduces an element of s-p mixing,
or hybridization, into the molecular orbital theory. The
result is a slight change in the relative energies of the
molecular orbitals, to give the diagram shown in the figure
below. Experiments have shown that O2 and F2
are best described by the model in the figure above, but B2,
C2, and N2 are best described by a
model that includes hybridization, as shown in the figure
below.


Bond Order
The number of bonds between a pair of atoms is called the bond
order. Bond orders can be calculated from Lewis
structures, which are the heart of the valence-bond model.
Oxygen, for example, has a bond order of two.

When there is more than one Lewis structure for a
molecule, the bond order is an average of these structures.
The bond order in sulfur dioxide, for example, is 1.5
the average of an S-O single bond in one Lewis structure and
an S=O double bond in the other.

In molecular orbital theory, we calculate bond orders by
assuming that two electrons in a bonding molecular orbital
contribute one net bond and that two electrons in an
antibonding molecular orbital cancel the effect of one bond.
We can calculate the bond order in the O2 molecule
by noting that there are eight valence electrons in bonding
molecular orbitals and four valence electrons in antibonding
molecular orbitals in the electron configuration of this
molecule. Thus, the bond order is two.

Although the Lewis structure and molecular orbital models
of oxygen yield the same bond order, there is an important
difference between these models. The electrons in the Lewis
structure are all paired, but there are two unpaired
electrons in the molecular orbital description of the
molecule. As a result, we can test the predictions of these
theories by studying the effect of a magnetic field on
oxygen.
Atoms or molecules in which the electrons are paired are diamagnetic
repelled by both poles of a magnetic. Those that have one or
more unpaired electrons are paramagnetic
attracted to a magnetic field. Liquid oxygen is attracted to
a magnetic field and can actually bridge the gap between the
poles of a horseshoe magnet. The molecular orbital model of O2
is therefore superior to the valence-bond model, which cannot
explain this property of oxygen.

