We have seen that the energy levels of a diatomic molecule in a
state may be written as
What is the situation when the electrons have non-zero orbital angular
momentum ? In this case, if the orbital angular momentum of
the nuclei is now denoted by , the total angular momentum
is given by
Because L is non-zero, the electronic potential field in which
the nuclei move is no longer really axisymmetric, and N is no
longer a good quantum number. However, we may approximate the main
effect of non-zero L on the nuclear wave functions by replacing
the centrifugal term
in the nuclear
wave equation by one in the expectation value
We now consider the rotational spectrum that should be emitted by a diatomic molecule. Classically, we expect that radiation could be emitted as a result of the rotation of the molecule if the molecule has a net electric dipole moment, which will be the case for example in a molecule with ionic bonding between different nuclei (e.g. CN or OH). On the other hand, a homonuclear molecule, one with two identical nuclei (H, N, O, etc), has no overall electric dipole because the two ends are identical. Thus we expect that such molecules may be inhibited from emitting dipole radiation due to rotation.
Now look at the rotational emission problem quantum mechanically,
first for a homonuclear molecule. In the dipole approximation, we
expect that the transition amplitude between two rotational states
(having the same vibrational and electronic wave functions) will be
proportional to the matrix element of
To summarize the effects of exchange of the nuclei:
The same conclusion applies to radiation in which the vibrational state (but not the electronic state) changes, i.e. to purely vibrational transitions. Such radiation also requires the rotational state to change by , so again exchange causes the sign of the matrix element to change. We conclude that a homonuclear molecule in the ground electronic state does not emit purely rotational or vibrational spectra by dipole radiation. Such molecules can only emit dipole radiation if the electronic state changes; they can also emit quadrupole radiation or magnetic dipole radiation, but this is much weaker than the suppressed dipole radiation would be.
Heteronuclear molecules can emit a purely rotational, or a
vibrational-rotational spectrum. From the dependence of the energy of
the eigenstates on , as
, we immediately see that
that spectral lines due to rotational transitions satisfying will have frequencies (for
To have spectral lines arise through the change of vibrational levels,
the matrix element
Now for a given pair of vibration levels and , the
transitions fall into two groups, those with (the R
branch) and those with (the P branch). The frequencies
for the R branch are given approximately by
For a molecule with , transitions with (the Q branch) are also possible. If the two values are essentially equal, all the Q branch lines occur at the frequency , which instead of being absent is quite strong.
Another important effect involving vibrational and rotational levels is Raman scattering. In this effect, a photon is scattered by the molecule, effectively by an absorption immediately followed by an emission to a state near the original one, so that the frequency of the scattered photon is changed slightly. The absorption changes by one, and the re-emission does so as well. Thus in the end, the final emission satisfies the selection rule . It is found that the Raman effect does not require a permanent electric dipole moment; the moment induced by the radiation field itself is enough to make the process occur. This means that the Raman effect provides a means of probing the vibrational-rotational levels in molecules such as O and N which normally have no intrinsic vibrational-rotational spectrum.
Electronic spectra arise from transitions in which the electronic state of the molecule changes - these are the transitions most nearly analogous to atomic transitions, and typically involve photons in the visible and ultraviolet parts of the spectrum. At low resolution, electronic spectra seem to be made up of series of more or less evenly space bands; at higher resolution, each band is made of many individual spectral lines.
To understand these spectra, recall that we have found that the energy
of a single level of electronic state , vibrational state , and
rotational level may be written as a sum of these three energies,
, so that the frequency of a particular
transition will be given by
Selection rules control which electronic states can make strong transitions with each other; these are somewhat complicated and will not be treated here. There is no selection rule on the difference , since the two vibrational levels are not formed in the same electronic potential well, and values of up to 5 or 6 are not uncommon. Since the rotational energies involve the same angular functions (the 's) in both states, they continue to observe the selection rule between two states, or for states with .
For a given pair of electronic levels , , each of the bands
seen at low resolution corresponds to a particular value of . Writing the part of the energy difference due to the vibrational
Each series of bands for given values of and will have
a large number of lines (fine structure) because of the rich
structure of rotational levels possessed by each vibrational
level. Two or three series of rotational lines will be present for
each pair of vibrational levels, corresponding to (the
P branch), (the R branch), and perhaps
(the Q branch). For transitions between two levels, the
frequency series for the P and R branches are given by
In diatomic molecules we actually have four different kinds of angular momentum that combine in different ways. These are the orbital angular momentum of the electrons L, the spin angular momentum of the electrons S, the nuclear rotational angular momentum N, and the nuclear spin I, which can almost always be neglected except for its influence on symmetries in homonuclear molecules (see below). These may combine in a variety of fairly complicated ways.
Hund identified some of the most common ways in which the angular momentum combines, or couples. To appreciate this phenomenon, it is useful to have first studied many-electron atoms, which you have not yet done, so we will simply summarize a sample situation.
One possible situation is Hund's case (a), when L couples
strongly to R, so that is a good quantum number
(i.e. is well-defined for each quantum state), and also couples
strongly to S, so that the projection of S on R,
, is also a good quantum number. Then the sum of these
two components can take on the values
Other cases arise as other kinds of coupling dominate.
Nuclei have spin due to the intrinsic spins of protons and neutrons, which like electrons have spins of . These spins combine (couple) to form the total spin of the nucleus, which may be different for different excited states. The ground state of the nucleus always has a definite spin; for example, H has spin 1/2, while O has spin 0. The nuclear spin can couple with other angular momenta, but as mentioned above this coupling has no direct effect of importance on molecular spectra.
There are however important effects of nuclear spin in homonuclear molecules due to the operation of the Pauli exclusion principle. We know that the exclusion principle requires that the total wave function of a system be antisymmetric under exchange of identical fermions, or symmetric under exchange of bosons. Since electrons have spin 1/2 and do not combine into aggregates, the wave function must always be antisymmetric under exchange of electrons, but since nuclei can act as bosons (if they have an even number of nuclei) or fermions (with an odd number of nucleons), both symmetries are possible for nuclear exchange. Let's look at one example of how this restriction affects molecular spectra of a homonuclear molecule.
We have seen that the total wavefunction (without nuclear spin) of a
homonuclear molecule (we add in the electron spin function )
This kind of effect in homonuclear molecules is very helpful in determining the spin of the nuclei, in spite of the fact that these nuclear spins have almost no interaction with the rest of the molecule.
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