It’s Professor Dave, let’s revisit molecular orbital theory.
We’ve learned a bit about molecular orbital theory, and the ways in which it differs from
valence bond theory.
But let’s expand on this knowledge now, so that we can understand a phenomenon called
s-p mixing.
But before we get into that, let’s take some time to review the basics.
First, let’s compare valence bond theory and molecular orbital theory.
Valence bond theory considers covalent bonds as being localized between a pair of atoms,
resulting from the overlap of atomic orbitals or hybrid atomic orbitals.
This overlap results in sigma or pi bonds, and we can use this theory to predict molecular shape.
However there are simple molecules that do not conform to this theory, like the diatomic
oxygen molecule.
Given this Lewis structure, where each oxygen atom has a full octet, we would expect oxygen
to be diamagnetic, and weakly repel a magnetic field.
However it does in fact exhibit paramagnetism, as liquid oxygen is attracted to a magnetic field.
To explain this, we have to turn to molecular orbital theory, which differs from valence
bond theory in that electrons are considered delocalized throughout the molecule, and atomic
orbitals combine to form molecular orbitals, which are also delocalized throughout the molecule.
These can be sigma and pi bonding orbitals, or sigma star and pi star antibonding orbitals.
This theory ends up being better equipped to predict the arrangement of electrons in
a molecule in certain situations.
This is due to a process called the linear combination of atomic orbitals.
In the modern physics course, we talked a little bit about the wave function, and molecular
orbitals are combinations of atomic orbital wave functions.
Just the way the wave function describes the behavior of an electron within an atom, molecular
orbitals describe the behavior of an electron within a molecule, and just like atomic orbitals
they are filled when they contain two electrons of opposite spin.
This theory requires that we strongly consider the wave nature of the electron.
In the classical physics series, we talked a lot about waves and interference patterns,
so to get the full picture of wave behavior in order to better understand molecular orbitals,
check out this tutorial now.
Otherwise, we simply need to understand that when two atomic orbital wave functions are
combined, like these two s orbitals, there can be constructive interference, where the
peaks line up with the other peaks.
This causes the amplitude to increase, which results in a bonding molecular orbital, like
this sigma-s orbital.
Most of the electron density is between the nuclei, which strengthens the bond.
There can also be destructive interference, where the peaks line up with troughs.
This causes the amplitude to drop to zero, resulting in an antibonding orbital, like
this sigma-s star orbital.
Here there is a node between the nuclei, which weakens the bond.
The bonding orbital is of lower energy, and will thus be filled before the antibonding orbital.
For this to make more sense, we simply need to understand that these images of waves are
just two-dimensional analogies to help us visualize the type of overlap.
Electrons are three-dimensional objects, so the waves are in three dimensions, and constructive
interference thus results in regions of greater electron density, while destructive interference
results in region of no electron density, which are depicted by these bonding and antibonding
orbitals.
We can do the same thing with p orbitals, utilizing direct overlap, which will yield
sigma-p-x and sigma-p-x star molecular orbitals, or lateral overlap, yielding pi and pi star
molecular orbitals from the overlap of py and pz atomic orbitals.
Orbital number is always conserved, which is why the linear combination of two atomic
orbitals will always result in two molecular orbitals, one bonding and the other antibonding.
We can depict these linear combinations with molecular orbital diagrams, as we have seen.
The dashed lines show how the atomic orbitals on the sides combine to form the molecular
orbitals in the middle, such as we see here for hydrogen, with two 1s orbitals combining
to form one sigma-s orbital and one sigma-s star orbital.
If we go up to the electrons in the n = 2 shell, we can see the 2s orbitals combining
in precisely the same way, and then the 2p orbitals combining as follows, with the sigma
bonding orbital and then pi bonding orbitals, followed by the corresponding antibonding orbitals.
The good news is that these orbitals are still filled according to the Aufbau principle and
Hund’s rule, like we already know.
This means that we fill them from lower energy to higher, according to Aufbau, and we will
put one electron in each orbital for a set of degenerate orbitals before doubling them
up, according to Hund.
We’ve already learned in a previous tutorial how to fill in these MO diagrams for certain
diatomic molecules, like O2, and how to use them to calculate bond order.
Here’s that diagram one more time, along with the calculation for bond order, with
ten bonding electrons, including the sigma-1s bonding electrons, and six antibonding electrons,
including the sigma-1s star antibonding electrons.
Ten minus six is four, and four over two is two.
This explains the double bond in diatomic oxygen.
But we also get an explanation for the paramagnetism of the oxygen molecule.
We have two unpaired electrons in these antibonding orbitals, which does predict paramagnetism
for this molecule.
This is an example of the success of molecular orbital theory, as no other theory can account
for this behavior.
Now that we are up to speed, let’s go over one other thing.
The molecular orbitals that result from the combination of the 2p orbitals exhibit this
order in O2, but this will change for other homonuclear diatomic molecules.
Let’s look at this diagram, which depicts the relevant orbital energies for all the
potential homonuclear diatomic molecules in the second period, from lithium to neon.
As we go to the right in the period, effective nuclear charge increases, so the atomic radii
contract, and the orbitals get lower in energy.
On the left, the pi-2p bonding orbitals are lower in energy than the sigma-2p bonding
orbital, but on the right, the sigma-2p bonding orbital dips below, and thus will be filled first.
This discrepancy is due to a phenomenon called s-p mixing, which influences the energies
of these molecular orbitals.
This happens when the sigma-s and sigma-p wavefunctions combine in such a way that the
sigma s orbital becomes more stable and the sigma-p orbital becomes less stable.
As a result, we see the sigma-s and sigma-s star orbitals going lower in energy, and the
sigma-p and sigma-p star going up in energy, so much so that the sigma-p orbital now sits
above the pi-p orbitals.
Molecules like O2 and F2 have negligible s-p mixing, because the 2p orbitals in those atoms
have some paired electrons.
This raises the energy of the 2p orbitals, making them too dissimilar to the energy of
the 2s orbital to allow mixing to occur.
But going to the left on the period, where p orbitals have unpaired electrons or are
vacant, the 2p and 2s orbitals are of similar enough energies that s-p mixing will be prevalent,
and the order of the molecular orbitals will switch.
While this is an interesting phenomenon, if we are only using these diagrams to calculate
bond order, it won’t matter, because it is just the order of bonding orbitals that
is being rearranged, so it won’t affect a bond order calculation.
Nevertheless, we have hopefully reached a more sophisticated understanding of molecular orbitals.