Definitions

All crystals and most molecules possess symmetry, which can be exploited to simplify the discussion of their physical properties. Changes from one configuration to an indistinguishable configuration are brought about by sets of symmetry operators, which form particular mathematical structures called groups. We thus commence our study of group theory with some definitions and properties of groups of abstract elements. All such definitions and properties then automatically apply to all sets that possess the properties of a group, including symmetry groups.

Binary composition in a set of abstract elements {gi}, whatever its nature, is always written as a multiplication and is usually referred to as “multiplication” whatever it actually may be. For example, if gi and gj are operators then the product gi gj means “carry out the operation implied by gj and then that implied by gi.” If gi and gj are both n-dimensional square matrices then gi gj is the matrix product of the two matrices gi and gj evaluated using the usual row × column law of matrix multiplication. (The properties of matrices that are made use of in this book are reviewed in Appendix A1.) Binary composition is unique but is not necessarily commutative: gi gj may or may not be equal to gj gi.

The photoionization spectrum of diphenylamine provides an unusual and interesting illustration of the Franck–Condon principle. Diphenylamine (DPA), illustrated in Figure 17.1, is a relatively large molecule to study by gas phase spectroscopy and it might be thought that the vibrational structure in its electronic spectra would be highly congested and difficult to interpret. After all, this is a molecule with 66 vibrational modes! However, it was shown in Section 7.2.3 that only totally symmetric modes generally need to be considered in interpreting electronic spectra. Also, there is the further simplification that not all of the totally symmetric modes need be Franck–Condon active, i.e. will give a significant progression. DPA is an excellent example of this, with the main structure arising from a single vibrational mode.

Before spectra are considered, the experimental procedure, carried out by Boogaarts and co-workers [1], will be outlined. Mass-analysed threshold ionization (MATI) spectroscopy was employed. This technique, which was briefly described in Section 12.6, is essentially the same as ZEKE spectroscopy but employs ion rather than electron detection. It has the advantage over ZEKE spectroscopy in that ions can be separated according to their mass, which in most cases enables the spectral carrier to be determined with confidence. By analogy with ZEKE spectroscopy, a cation ← neutral molecule electronic absorption spectrum is effectively obtained.

Depending on the resolution, a spectrum may consist of well-resolved discrete peaks, each of which is attributable to a single specific transition, or it may consist of broader bands that are actually composed of several unresolved transitions. In either case, the intensities will depend on a number of factors. The sensitivity of the spectrometer is crucial. So too is the concentration of the absorbing or emitting species. However, our interest in the remainder of this chapter is with the intrinsic transition probability, i.e. the part that is determined solely by the specific properties of the molecule. The key to understanding this is the concept of the transition moment.

Transition moments

Consider two pairs of energy levels, one pair in molecule A and one pair in a completely different molecule B. Assume for the sake of simplicity that the energy separation between the pair of levels is exactly (and fortuitously) the same for both molecules. Suppose that a sample of A is illuminated by a stream of monochromatic photons with the correct energy to excite A from its lower to its upper energy level. There will be a certain probability that a molecule is excited per unit time. Now suppose sample A is replaced with B, keeping the concentration and all other experimental conditions unchanged. In general the probability of photon absorption per unit time for B would be different from A, perhaps by a very large amount.

The study of molecular complexes in the gas phase provides important information on intermolecular forces and spectroscopy has played a vital role in this field. As an illustration, the complex formed between an aluminium atom and a water molecule is described here.

To obtain Al(H2O), it is necessary to bring together aluminium atoms and water molecules. Getting water into the gas phase is easy, but aluminium poses more of a problem since at ordinary temperatures the solid has a very low vapour pressure. An obvious solution is to heat the aluminium in an oven. However, the high temperature has a concomitant downside; if water is passed through (or near) the oven the high temperature will almost certainly prevent the formation of a weakly bound complex such as Al(H2O). Instead, the heat may allow the activation barriers to be exceeded for other reactions, leading to products such as the insertion species HAlOH.

A solution to this apparent quandary is to make Al(H2O) by the laser ablation–supersonic jet method, which was mentioned briefly in Chapter 8 (see Section 8.2.3). Any involatile solid, including metals, can be vaporized by focussing a high intensity pulsed laser beam onto the surface of the solid.

Laser-induced fluorescence, resonance-enhanced multiphoton ionization, and cavity ringdown spectroscopic techniques offer ways of detecting electronic transitions without directly measuring light absorption. An alternative approach is possible if the excitation process leads to fragmentation of the original molecule. By monitoring one of the photofragments as a function of laser wavelength, a spectrum can be recorded. This is the basic idea behind photodissociation spectroscopy.

There are limitations to this approach. If photodissociation is slow, then the absorbed energy may be dissipated by other mechanisms, making photodissociation spectroscopy ineffective. It is also possible that some rovibrational energy levels in the excited electronic state will lead to fast photofragmentation whereas others will not. In this case there will be missing or very weak lines in the spectrum which, in a conventional absorption spectrum, may have been strong. Fast photofragmentation is clearly desirable on the one hand, but it can also be a severe disadvantage if it is too fast, since it may lead to serious lifetime broadening in the spectrum (see Section 9.1).

Despite the above limitations, photodissociation spectroscopy can provide important information. This is particularly true for relatively weakly bound molecules and complexes, since these have a greater propensity for dissociating. In this and the subsequent example the capabilities of photodissociation spectroscopy are illustrated by considering weakly bound complexes formed between a metal cation, Mg+, and rare (noble) gas (group 18) atoms.

This Case Study demonstrates some of the subtle arguments that can be employed in assigning vibrational features in electronic spectra. It also provides an illustration of how important structural information on a fairly complex molecule can be extracted. The original work was carried out by Gordon and Hollas using both direct absorption spectroscopy of 1,4-benzodioxan vapour and laser-induced fluorescence (LIF) spectroscopy in a supersonic jet [1]. The direct absorption spectra were of a room temperature sample and were therefore more congested than the jet-cooled LIF spectra. Nevertheless, the direct absorption data provided important information, as will be seen shortly. For the LIF experiments, both excitation and dispersed fluorescence methods were employed (see Section 11.2 for experimental details). Only a few selected aspects of the work by Gordon and Hollas are discussed here; the interested reader should consult the original papers for a more comprehensive account [1, 2].

Possible structures of 1,4-benzodioxan are shown in Figure 18.1. Assuming planarity of the benzene ring, there are three feasible structures that differ in the conformation of the dioxan ring. One possibility is that both C O bonds are displaced above (or equivalently below) the plane of the benzene ring yielding a folded structure with only a plane of symmetry (Cs point group symmetry).

Carbon monoxide was one of the first molecules studied by ultraviolet photoelectron spectroscopy [1]. A typical HeI spectrum is shown in Figure 13.1. The spectrum appears to be clustered into three band systems. The starting point for interpreting this spectrum is to consider the molecular orbitals of CO and the possible electronic states of the cation formed when an electron is removed.

Electronic structures of CO and CO+

Any student familiar with chemical bonding will almost certainly be able to construct a qualitative molecular orbital diagram for a diatomic molecule composed of first row atoms. Such a diagram is shown for CO in Figure 13.2. The orbital occupancy corresponds to the ground electronic configuration 1σ22σ23σ24σ21π45σ2. The σ MOs actually have σ+ symmetry but it is not uncommon to see the superscript omitted. Since all occupied orbitals are fully occupied, the ground state is therefore a 1Σ+ state and, since it is the lowest electronic state of CO, it is given the prefix X, i.e. X1Σ+, to distinguish it from higher energy 1Σ+ states of CO.

Consider the electronic states of the cation formed by removing an electron. If the electron is removed from the highest occupied molecular orbital (HOMO), the 5 orbital, then the cation will be in a 2Σ+ state.