Molecular symmetry selection rules

Atoms have complete spherical synnnetry, and the angidar momentum states can be considered as different synnnetry classes of that spherical symmetry. The nuclear framework of a molecule has a much lower synnnetry. Synnnetry operations for the molecule are transfonnations such as rotations about an axis, reflection in a plane, or inversion tlnough a point at the centre of the molecule, which leave the molecule in an equivalent configuration. Every molecule has one such operation, the identity operation, which just leaves the molecule alone. Many molecules have one or more additional operations. The set of operations for a molecule fonn a mathematical group, and the methods of group theory provide a way to classify electronic and vibrational states according to whatever symmetry does exist. That classification leads to selection rules for transitions between those states. A complete discussion of the methods is beyond the scope of this chapter, but we will consider a few illustrative examples. Additional details will also be found in section A 1.4 on molecular symmetry. 

We now turn to electronic selection rules for syimnetrical nonlinear molecules. The procedure here is to examme the structure of a molecule to detennine what synnnetry operations exist which will leave the molecular framework in an equivalent configuration. Then one looks at the various possible point groups to see what group would consist of those particular operations. The character table for that group will then pennit one to classify electronic states by symmetry and to work out the selection rules. Character tables for all relevant groups can be found in many books on spectroscopy or group theory. Ftere we will only pick one very sunple point group called 2 and look at some simple examples to illustrate the method. 

The theory of molecular symmetry provides a satisfying and unifying thread which extends throughout spectroscopy and valence theory. Although it is possible to understand atoms and diatomic molecules without this theory, when it comes to understanding, say, spectroscopic selection rules in polyatomic molecules, molecular symmetry presents a small barrier which must be surmounted. However, for those not needing to progress so far this chapter may be bypassed without too much hindrance. 

These selection rules are affected by molecular vibrations, since vibrations distort the symmetry of a molecule in both electronic states. Therefore, an otherwise forbidden transition may be (weakly) allowed. An example is found in the lowest singlet-singlet absorption in benzene at 260 nm. Finally, the Franck-Condon principle restricts the nature of allowed transitions. A large number of calculated Franck-Condon factors are now available for diatomic molecules. 

It has long been realised that infrared (IR) spectroscopy would be an ideal tool if applied in situ since it can provide information on molecular composition and symmetry, bond lengths and force constants. In addition, it can be used to determine the orientation of adsorbed species by means of the surface selection rule described below. However, IR spectroscopy does not possess the spatial resolution of STM or STS, though it does supply the simplest means of obtaining the spatially averaged molecular information. 

The most difficult problem we face in deciding to use a basis of hybrids which reflects the molecular symmetry is how do we choose such a basis in view of the enormous numerical difficulties involved in optimising the non-linear parameters in molecular calculations The real question is are there any rules for this choice, can the optimisation be done (at least approximately) once and for all The chemical evidence is for us — it is the most basic concept of the theory of valence that particular electronic sub-structures tend to be largely environment-independent. How can we select our basis to reflect this chemical fact ... 

Having learnt about the concerted reactions, we can now undertake the theory of these reactions. The development of the theory of concerted reactions has been due chiefly to the work of R.B. Woodward and R. Hoffmann. They have taken the basic ideas of molecular orbital theory and used them, mainly in a qualitative way, to derive selection rules which predict the stereochemical course of various types of concerted reactions. These rules are best understood in terms of symmetries of interacting molecular orbitals. Here are will see some of the most important theoretical approaches and see how they are interrelated. 

Beyond such electronic symmetry analysis, it is also possible to derive vibrational and rotational selection rules for electronic transitions that are El allowed. As was done in the vibrational spectroscopy case, it is conventional to expand i j (R) in a power series about the equilibrium geometry of the initial electronic state (since this geometry is more characteristic of the molecular structure prior to photon absorption) ... 

If we start with states of tt-symmetry (dashed lines) we find three distinct peaks in the XES spectra reflecting the occupied states. The 1 a2u and lelgTr-like orbitals are essentially intact from the gas phase, while the third state, labeled e2u, is not seen for the free molecule. Based on symmetry-selection rules, it can be shown that this state is derived from the lowest unoccupied molecular orbital (LUMO) e2utt -orbital that becomes slightly occupied upon adsorption. We anticipate a similar bonding mechanism as discussed in the previous section for adsorbed ethylene with the exception of a weaker rehybridization due to the extra stability in the -system from the aromatic character. 

Energy state Electronic configuration Molecular state symmetry Selection rule for polarization of transition dipole... 

The two-photon absorption spectroscopy can overcome the symmetry barrier imposed by the selection rule for angular momenta in the one-photon process. Thus, the technique is able to identify and assign molecular and atomic states which are not accessable to one-photon spectroscopy. 

Tirana afld comparison with (9.189) shows that the group-theory selection rules for electronic transitions are the same as for vibrational transitions, except that we must consider the symmetry species of the electronic wave functions, rather than the vibrational wave functions. One complication is that the molecular geometry may change on electronic excitation in this case, we use the point group of lower symmetry to classify the wave functions and determine the selection rules. 

For a symmetric top, the selection rules are such that we can determine only B0 [see (5.85)]. Knowledge of Ib°, the moment of inertia about a principal axis perpendicular to the symmetry axis, is not sufficient to determine the molecular structure, except for a diatomic molecule. To get added information, the microwave spectra of isotopically substituted spe-... 

Just as with vibronically allowed transitions, in symmetry groups in which all Cartesian axes are not equivalent (noncubic groups), it is found that, in general, transitions will be allowed only for certain orientations of the electric vector of the incident light. One class of compounds in which this phenomenon has been studied both theoretically and experimentally consists of trischelate compounds such as tris(acetylacetonato)M(III) and tris(oxalato)M(III) complexes. In these complexes the six ligand atoms form an approximately octahedral array but the true molecular symmetry is only Dy Since there is no center of symmetry in these molecules, the pure electronic selection rules might be expected to be dominant. 

Both infrared (IR) and Raman spectroscopy have selection rules based on the symmetry of the molecule. Any molecular vibration that results in a change of dipole moment is infrared active. For a vibration to be Raman active, there must be a change of polarizability of the molecule as the transition occurs. It is thus possible to determine which modes will be IR active, Raman active, both, or neither from the symmetry of the molecule (see Chapter 3). In general, these two modes of spectroscopy are complementary specifically, if a molecule has a center of symmetry, no [R active vibration is also Raman active. 

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