Chris Rodgers, St John’s College
Michaelmas: Week 2: FAA Tutorial Work
Q4. Why is the colour due to d-d transitions in a complex having Oh symmetry expected to be much less intense than for a complex with Td symmetry?
Application of ligand field to both octahedral and tetrahedral complexes predicts a splitting of the d-orbital degeneracy[1]. We expect to get an energy-level diagram something akin to the following:
It seems reasonable, therefore, to expect that all transition metal complexes will demonstrate absorption lines corresponding to d-d transitions. However, in practice we find that this is not the case. Some complexes show strong absorptions, others only show very weak ones. For example, while Td complexes show intense d-d absorptions, Oh complexes do not. We can explain this in terms of selection rules.
In order to demonstrate the idea of a selection rule, let us consider a somewhat simpler system – the Hydrogen atom. The energy levels of a Hydrogen atom may be shown on a Grottrian diagram as follows:
We recall that the 1s2p transition is allowed, whereas the 1s2s transition is “symmetry forbidden”. The reason for this is that the Hydrogen atom possesses a centre of inversion. This means that when we set up the transition, the time-dependent quantum-mechanical dipole operator for the incident light is used to get the absorption coefficient. (This is because the light is a dipole and it interacts with a dipole in the atom or molecule.) As this is odd, then we expect that the parity of the initial <n| and final |m> states must differ. Otherwise, the integral is zero. In other words, <n|dipole operator|m> = 0 if n,m both even or n,m both odd but is non-zero when n is even, m is odd or n is odd, m is even. This gives rise to the Laporte selection rule. (l=1).[2]
Dipolar interaction operator: H(1)(t) = - . E(t)
where is the dipole moment operator, which for a single electron of charge –e is:
= -er
where r is the position vector relative to the nucleus. The intensity of a transition from some state |n1l1m1> to another state |n2l2m2> is proportional to the square of the transition matrix element: <n2l2m2|H(1)|n1l1m1>.
Hence, intensity of transition |<n2l2m2||n1l1m1>|2 (which is the square of the transition dipole moment).
By group theory we find that the transition dipole moment is zero unless it is totally symmetrical under the symmetry operations of the system. Hence, we can consider inversion as a useful first test of this symmetry. Inversion is represented by r-r. By the definition of odd-ness and even-ness we see that for odd , then -. This corresponds to odd values of l e.g. (p,f,…) but for even l e.g. (s,d,…) +. i.e. is even.
We know that an integral over all space of an ODD function cancels with itself to give zero, whereas an EVEN function does not and gives a non-zero result. Hence, in our integrand above, the parity is (-1)l1 . (-1) . (-1)l2. Hence we see that l= odd is a requirement. As |l| is not greater than 1 – we get the Lapore selection rule l=1.
This carries through into the SALC[3] combinations of d-orbitals in the Oh and Td symmetry complexes. The same rule applies in that if there is a centre of inversion, then we require the initial and final states to have different parity. This means that d-d transitions are symmetry forbidden – i.e. the transition dipole moment is zero. However, looking at the symmetry operations of the point group Td we see that it does not have a centre of inversion. This means that this rule no-longer applies and d-d transitions are allowed in Td complexes. In Oh complexes, there is a centre of inversion, so d-d transitions are symmetry forbidden. This explains why the observed intensity of Td complexes is much higher than in Oh complexes.[4]
Q5 What is meant by vibronic coupling in the context of electronic (UV-vis) absorption spectroscopy.
However, this does not explain why we still observe some absorption due to d-d transitions in Oh complexes. There must be some process which is relaxing this symmetry selection rule. The most obvious way to do this is to break the Oh symmetry. This can happen when an essentially Oh complex vibrates. Some of the modes of vibration will move the ligands in such a way that there is no longer a centre of inversion. Imagine the modes of vibration along the ligand-metal axis for each ligand. These will result in the ligand-metal distances changing - somewhat at random. This vibration is considerably slower than an electronic transition – so to all intents and purposes the molecule may be considered to be stationary during the transition. Such an uncoordinated motion will mean that in a large number of molecules there will be some with asymmetric differences in their ligand-metal distances. These differences will be spread in a distribution. When this asymmetry is present, the above algrebra does not apply – meaning that in the molecules with temporary asymmetric arrangements there is the possibility of absorption. This results in a weak absorption. It also means that the absorption line is broadened somewhat because there is a distribution of different environments and hence dipoles and potential fields in the different asymmetric molecules. This weak relaxation of the Laporte selection rule is known as vibronic coupling because it arises from the interaction of vibrational modes with the electronic transition modes[5].
Q6. What is meant by polarisation in the context of electronic (UV-vis) absorption spectroscopy?
If we consider the orientation of the angular momentum of the electron that is being excited during absorption, we can see that it is determined by the value of ml. When the light photon arrives, it will also have a certain, specific orientation of its electric dipole – its polarisation. A normal source of light will not be polarised and will, therefore, contain photons with a random polarisation. During the interaction, the relative orientation of the photon dipole to the atomic or molecular dipole will be critical in determining the extent of the transition.
Consider plane polarised radiation with the electric field pointing in the z-direction. The interaction hamiltonian has the form:
-z E(t) = e z E(t)=e r cos E(t)
The interaction dipole moment is an integral over the electron’s coordinates r, , . The integral has the form:
This is zero unless ml2=ml1. Therefore, for z-polarised radiation the selection rule is ml=0. Similarly for xy polarised light, the selection rule is ml=1. This means that we will get different multiplicity of allowed transitions depending on the polarisation of the incident light, which will alter the relative intensity of the peaks with different l value. (It will also lead to different hyperfine structure in accordance with the selection rules above.) There is a similar effect with the magnetic dipole interactions, although this is much smaller in magnitude. There is an even smaller effect when light interacts with an electric quadrupole – e.g. a d-orbital. These last two effects are too small to be observed normally during UV-vis spectroscopy.
[1] The derivation of these results is standard, if a little lengthy. An example of this derivation may be found in Inorganic Chemistry 3rd ed., James E. Huheey, pp412 onwards. Although there is insufficient space to go into detail here, the argument is essentially as follows: consider a transition metal ion, with its degenerate d-orbitals. Now surround the ion with a uniform spherical shell of –ve charge at the distance of the ligands. This will raise the energy of all the d-orbitals similarly because it is spherically symmetric and will introduce electron-electron repulsion in all the d-orbitals. Now, let this shell of charge be condensed into points at the positions of the ligands. There will now be different amounts of repulsion on the different orbitals. These may be characterised on a symmetry basis – showing that three d-orbitals will move one way in energy and the other two will shift the other way about a “centre of gravity” for the energy of the symmetric shell case. The directions of these shifts depend on the geometry of the ligands.
[2] Mathematical derivation taken from Molecular Quantum Mechanics, Atkins.
[3] Symmetry Adapted Linear Combinations. See Group Theory for Chemists, G. Davidson.
[4] Another explanation of this phenomenon may be found in d-Block Chemistry, Mark J. Winter (OCP), §7.2 & §7.3.
[5]Inorganic Chemistry 3rd ed., James E. Huheey, page 455 paragraph 1.