Hello. In this video,
we're going to talk in more detail about atomic
molecular transitions. It turns out not all
radiative transitions between different quantum
states are allowed. This arises because of this
R squared term that we mentioned in the previous video, the transition probability. Both the value of
the dipole moment and the overlap between the wave functions
of the two states determine how strong
the transition will be. This leads to so-called
selection rules, rules about whether or not transitions are
going to occur, how strong they are going to be. So if the Einstein coefficients
are reasonably large, the transition is
said to be allowed. Otherwise, it's forbidden. Of course, in reality, there's a full range of
possibilities from very strong. The sodium D-lines are
a classic example. If you ever seen
yellowish streetlight, it's probably a
sodium streetlight. Sodium molecules are excited by an electric discharge
and then they spontaneously radiate around 580 nanometers in the yellow region of
the visible spectrum. So those are definitely
allowed transitions. But on the other hand,
for many molecules, there are transitions or there are pairs of quantum
states between which radiative transitions
are forbidden or at least would
be extremely weak. For atomic transitions, there's no restriction on the change in the principle quantum number. However, there are
restrictions on angular momentum
and total momentum. So Delta L can only be
zero or plus or minus one. Delta L of the valence electron has to be plus or minus one. Delta J has to be
either zero or plus or minus one with the exception that transitions
between two states with J equals zero are forbidden. The little transition
diagram on the right shows the sodium transitions, the so-called D-lines of sodium. Well, they're 5,890 and 5,896
angstroms respectively. They satisfy these
selection rules. So next, we discuss
molecular transitions. Because of Born-Oppenheimer,
we can write the wave function for a diatomic molecule as the product of the
electronic vibrational and rotational wave functions. So the transition
probability can be written as shown in
the equation below. The reason that the rotational wave
functions are factored out is they are not affected
by the dipole moment. So you have two integrals there. When electronic and vibrational
states don't change, then that expression
reduces to the first one in this slide where R_e is called the permanent
electric dipole moment. Only if it's non-zero will there be an allowed transition. The rotational
selection rules are determined by the rotational
overlap integral, therefore. That leads to the
rule Delta J equals plus or minus 1 for
the rigid rotator so that the lines only appear
at each Delta J equals 1. So you get a series of lines at increasing energies because you have a line that goes
from J equals 0-1, and that one that goes
from J equals 1-2, and so on and so forth. For more realistic molecules, you'll pick up a
more rich spectrum. So if we now add in
ro-vibrational transitions, then the transition probability becomes shown in
the equation given there where we have assumed that we can write the first integral as
shown there with R_e. We've assumed that
it's a little bit at least of a function of R, the distance between
the two atoms in the diatomic molecule. Because the wave
functions are orthogonal, the first integral will disappear unless R_e is a
function of R. So if we expanded R_e in a power series
like the equation below, then the first term will be a constant and then the second will be a derivative times
the distance or the difference between the
equilibrium R and the actual R. So you can write the transition probability
as shown in this slide. For a harmonic oscillator, the first term is zero except
Delta V plus or minus one. So the transitions are at an energy given by the
following two equations. The Sigma nought is
the energy difference between the two
vibrational levels. Then the second two terms
are the energies of the energy in the excited
vibrational state minus the energy in the
ground vibrational state. So this next slide shows a
cartoon that illustrates this. So the cartoon on the left shows an energy-level diagram
with two vibrational levels and then vertical arrows that illustrate where
absorption lines would be or absorptions. So for example, the
left series of lines, the first one starts
at J equals 0 and V equals 0 vibrational state, and goes to J equals 1 and the V equals 1
vibrational state. Then each line does
the similar thing but starting at the next
higher rotational state. Those form a series of lines in a spectra similar
to the spectra on the right called a branch and in this case,
the so-called R-branch. In other words, if you have J, it goes to J plus 1 in a
ro-vibrational spectrum. We'll see this is true in an
electronic spectra as well, that's called an R-branch. On the other side of
the left-hand diagram, we have vertical
lines that go from a value of J to a
value of J minus 1. The first one goes from J equals 1 to J equals 0 and so on. They form a band or
series of lines, and they're called the P-branch. So the P-branch is
for lines in which Delta J is minus one and that is represented by the right-hand side of the spectra shown as
the right cartoon. On this spectrum, there's
something called the Q-branch which, under
normal circumstances, will not appear
because that would represent a Delta J equals 0. Only a non-harmonic rigid rotator molecules
will that appear, but it does appear. So this again just illustrates
the branch concept, and there are branch in the P-branch for
ro-vibrational transition. Only Delta J plus or
minus 1 is allowed and the two branches are given here with a
little illustration. For a harmonic oscillator, only Delta V equals 1 is allowed. But for an anharmonic oscillator, the vibrational overlap integral allows Delta V equal
plus or minus 1, plus or minus 2, etc. So you pick up different bands and some of them are
shown on the right here. The left is just a diatomic
potential energy diagram showing the harmonic
versus the more realistic. But the diagram on the right shows several of the
additional branches. The fundamental is when
you're going a Delta V equal 1 but then you have what's called the first overtone, Delta V equals 2. Second overtone,
Delta V equals 3. If you go from an
excited state to another excited state
like V equals 1-2, that's called a hot band. So you can pick up a lot of structure when you have
an anharmonic oscillator. So that's ro-vibrational
transitions. For electronic transitions, the transition rules are relaxed and that leads to
additional possibilities, additional complexities,
if you will. Just a little cartoon here that shows two electronic states, and an expansion of the
vibrational and rotational level, and gives you some
numbers and idea of the order of magnitude of
things, electronics is big. That vibration was a
middle rotational, there's a little Delta energies. Then in the spectral regions, so electronic transitions
are typically visible to UV. Vibrational or
ro-vibrational transitions are in the infrared, and rotational vibrations
are in the microwave. So if we look at
electronic transitions and look at the electronic
transition probability, then we get what we
had shown before. Again, where R_e is the
electric transition moment. If R_e doesn't vary
very strongly with r, then we can pull it out and get the electronic
portion times a vibrational integral or overlap integral in a rotational
overlap integral. Those two overlap integrals are called the
Franck-Condon factor for both vibration and the line string
factor for rotation. If you have the wave functions clearly you can calculate them. So the electronics
selection rules depend on those integrals. So for electronic transitions, Delta Lambda has to be zero or plus or minus one and
Delta S has to be zero. Depending on spin-orbit coupling, other rules may apply. So here's a cartoon that shows the Franck-Condon
principle. The two electronic states, one above the other with their
potential energy curves. If they are shifted laterally in the
nuclear coordinate with respect to each other, then if I try to
undergo a transition from a lower electronic state to a higher electronic state, then those wave functions
may or may not overlap, and that's going to
have an impact on the allowed intensity
of the lines. The cartoon on the right shows what absorption
might be for those various vibrational
bands and what fluorescence might look like when you
excite from zero to six, zero to five, zero
to four and so on, we'll get into fluorescence
in a later video. So the Franck-Condon
factor will be a maximum where the maximums in the two wave functions
overlap each other in nuclear distance. So for electronic transitions, Delta J equals plus or
minus 1 holds normally. However, if the
angular momentum of either electronic
state is non-zero, then Delta J equals
zero is allowed. This allows for a Q branch
which I mentioned earlier. So this next slide shows what the R branch or Q
branch and the P branch are. The R branch is when
the excited state J is one integer larger
than the ground state. The Q branch is when J does
not change in the transition. The P branch is when the excited J is one less
than the ground state J, and it shows the value of the overlap integrals for those line strength factor for those are
proportionality of it. So an example of a diatomic
molecule that is received, take tens of study, including by me early
in my career, is OH. OH is an important radical in combustion chemistry and in
other applications as well. Chemical processing of
electronic components and so on. Lots of different
applications for OH that makes it interesting. On the left-hand side is an energy level
diagram for OH showing the X double Pi state in
the A doublet Sigma state. So these are two
electronic states. They are bound states. They have a well and the potential energy
distribution and they have different vibrational
and rotational levels. Then on the right is a spectrum over a relatively small portion of the absorptions spectrum. Spectrum is shown in
terms of transmittance, which is one minus absorption. So the absorption lines go down because if
there's no absorption, it would be 100
percent transmission, and it shows experimental
data and a fit of that data which is done
by regression techniques. Now, if you have non harmonic conditions and
other non-ideal behavior, then you pick up a bunch
of other branches. So it can be N, O, S, T additional branches
in addition to P, Q and R branches. These are branches that
allow different Delta J. So minus three, minus two, minus one, zero, one, two, three etc. Here's an example. This is OH, this shows
transitions involving P, R, O branches and Q branches. These contribute
to the OH spectrum that we saw a couple
of slides ago. I'm going to demonstrate
what one of these spectrum looks like a little
bit better using this computer program
called LIFbase. It only runs on Windows
and you can download it yourself if you have
Windows and run it. So this is a program that was invented at Stanford
Research Institute to simulate mainly
fluorescent spectrum but you can also simulate absorption and emission spectra. It's just for a few
molecules: OH, OD, NO, and CH. There's more here. There's a few more
diatomic molecules, but we'll look at OH here. You can set up the conditions. For example, we're going to do an absorption spectra here. We're going to start
at 3050 angstroms or 305 nanometers and go to 3150 angstroms or 315 nanometers. We would typically select
a particular temperature. In this case, we've selected 1500 degrees Kelvin
because where you're going to see OH is in flames and other high temperature
conditions. On the left, shows the spectrum
that you're going to get versus wavelength. On the right, it shows the rotational
population distribution for OH in the ground
state because, so this is going from the A
state to the B state of OH. I believe if I do the
right thing here, I can show the line assignments meaning where the band heads are and where the
particular lines are. I just clicked on Vibration. At the top of the graph, you can see zero, zero, there because we're basically
looking at the zeros, zero. Zero, zero vibrational
band of OH. This is electronic remember, so we're going from the ground vibrational state
in the X state to the ground vibrational
state in the B state. Now, if I select
Rotational and we get a very rich notation here. You see this is the P1, P2, Q1, Q2, and so on. It's a little rich. You can actually go in
and select only one. So I select Q1 for example. So this gives the lines that
are part of the Q1 spectrum. So this gives you an idea of how you might analyze spectra. There's many computer codes out there to help you do
this with basis, just one. This was designed for
a specific purpose. It's just cute for
illustrating these ideas. So that's it for this video. Thanks for listening
and have a great day.