Modern spectroscopic techniques such as laser-induced fluorescence, resonance-enhanced multiphoton ionization (REMPI), cavity ringdown, and ZEKE are important tools in the physical and chemical sciences. These, and other techniques in electronic and photoelectron spectroscopy, can provide extraordinarily detailed information on the properties of molecules in the gas phase and see widespread use in laboratories across the world. Applications extend beyond spectroscopy into important areas such as chemical dynamics, kinetics, and analysis of complicated chemical systems such as plasmas and the Earth's atmosphere. This book aims to provide the reader with a firm grounding in the basic principles and experimental techniques employed in modern electronic and photoelectron spectroscopy. It is aimed particularly at advanced undergraduate and graduate level students studying courses in spectroscopy. However, we hope it will also be more broadly useful for the many graduate students in physical chemistry, theoretical chemistry, and chemical physics who encounter electronic and/or photoelectron spectroscopy at some point during their research and who wish to find out more.

There are already many books available describing the principles, experimental techniques, and applications of spectroscopy. However, our aim has been to produce a book that tackles the subject in a rather different way from predecessors. Students at the advanced undergraduate and early graduate levels should be in a position to develop their knowledge and understanding of spectroscopy through contact with the research literature.

A severe restriction of conventional photoelectron spectroscopy is its low resolution. The main limitation is instrumental resolution, particularly that caused by the electron energy analyser, as was discussed in Chapter 12. Resolving rotational structure is not a realistic prospect for conventional photoelectron spectroscopy but even vibrational structure may be difficult to resolve. In addition to the instrumental resolution must be added other factors such as rotational and Doppler broadening which, if they could be dramatically reduced, might make a sufficient difference to improve many photoelectron spectra. A potential solution is to combine conventional photoelectron spectroscopy with supersonic molecular beams. Supersonic expansions can produce dramatic cooling of rotational degrees of freedom and, if part of the expansion is skimmed into a second vacuum chamber, can be converted to a beam with a very narrow range of velocities. This is precisely the approach adopted by Wang et al. [1], the molecular beam being crossed at right angles by HeI VUV radiation (58.4 nm) to produce a near Doppler-free photoelectron spectrum. The resolution achieved is in the region of 12 meV (100 cm-1).

The ultraviolet photoelectron spectra of CO2, OCS, and CS2 in molecular beams are discussed here. These illustrate some of the important concepts involved in the interpretation of the photoelectron spectra of polyatomic molecules.

Molecular symmetry is of great importance in the discussion of spectroscopy. It helps to simplify the explanation of complex phenomena, such as molecular vibrations, and is an important aid in the derivation of electronic states and transition selection rules. It also simplifies the application of molecular orbital theory, which is often applied to assign or predict electronic spectra. In many cases, it provides strikingly simple answers to complicated questions.

In its original form, group theory is a rigorous mathematical subject. No attempt will be made here to be rigorous – the aim is simply to summarize the basics as they apply to symmetry, in light of which the spectroscopic applications of the theory can become clearer. Although the concepts introduced here might be valid for any object with symmetry elements, we will apply these only to molecules. This appendix is not intended to be a comprehensive account of point group symmetry and group theory. Instead the intention is to review some of the key principles required for applications in electronic spectroscopy. A newcomer to the subject of symmetry and group theory is first advised to consult an appropriate textbook on this topic, such as one of those listed in the Further Reading at the end of this appendix.

Symmetry elements and operations

We begin with two fundamental concepts, symmetry operations and symmetry elements. Symmetry operations are transformations that move the molecule such that it is indistinguishable from its initial position and orientation.

As described in Chapter 11, laser-induced fluorescence (LIF) spectroscopy is one of the simplest and yet most powerful tools for obtaining high resolution spectra. Its high sensitivity is particularly convenient for the investigation of extremely reactive molecules, such as free radicals and ions. In this Case Study we illustrate how LIF spectroscopy can be used to obtain important information on a small carbon cluster, the C3 molecule. The spectra presented were originally obtained by Rohlfing [1], who produced C3 by pulsed laser ablation of graphite. This is a violent method for vaporizing a solid and the plasma formed above the graphite surface will undoubtedly contain carbon atoms, clusters such as C2, C3, and various cations and anions. To reduce spectral congestion, the laser ablation source was combined with a supersonic nozzle to produce a cooled sample for spectroscopic probing.

The LIF spectrum was obtained by crossing the supersonic jet with a tunable pulsed laser beam and measuring the intensity of fluorescence as a function of laser wavelength. As discussed in Section 11.2, an LIF excitation spectrum is similar to an absorption spectrum but the signal intensity depends not only on the absorption probability, but also the fluorescence quantum yield of the upper state.

The photoelectron spectroscopy of anions is, in many respects, directly analogous to the photoelectron spectroscopy of neutral molecules. However, an important difference is that an electron in the valence shell of an anion is much more weakly bound than in a neutral molecule. In fact there are some molecules, such as N2, that are unable to bind an additional electron at all. The binding energy of an electron in an anion, which is known as the electron affinity (EA), is the energy difference between the neutral molecule and the anion. The electron affinity is defined as a positive quantity if the anion possesses a lower energy than the neutral molecule, i.e. the electron is bound to the molecule and energy must be added to remove it.

The photoelectron spectrum of an anion, also known as the photodetachment spectrum, can provide information on both the anion and the neutral molecule. A good example of this is the photoelectron spectrum of, which was first recorded by Ervin, Ho, and Lineberger [1].

The experiment

The most common method for generating anions in the gas phase is an electrical discharge. Ervin et al. produced by a microwave (ac) discharge through a helium/air mixture. A variety of neutral and charged species would be expected under such conditions, including several possible anions and cations.

The Born–Oppenheimer approximation is an essential element without which the very notion of a potential energy surface would not exist (1). It also provides an example of how different coordinates can often be treated independently as a first approximation. This approach has far-reaching consequences, since it greatly simplifies the construction of partition functions in statistical mechanics. The approximation involves neglect of terms that couple together the electronic and nuclear degrees of freedom. The nuclear motion is then governed entirely by a single PES for each electronic state because the Schrödinger equation can be separated into independent nuclear and electronic parts. The simplest approach to the nuclear dynamics then leads to the normal mode approximation via successive coordinate transformations. These developments are treated in some detail, partly because the results are used extensively in subsequent chapters, and partly because they highlight important general principles, which can easily be extended to other situations. The consequences of breakdown in the Born–Oppenheimer approximation, and treatments of dynamics beyond the normal mode approach, are discussed in Section 2.4 and Section 2.5, respectively.

Independent degrees of freedom

The Schrödinger equation that we normally wish to solve in order to identify wavefunctions and energy levels is a partial differential equation if more than one coordinate is involved. The most common method of solution for such equations involves separation of variables (2).

In this chapter we consider the calculation of thermodynamic and dynamic properties using stationary points sampled from the PES. In this approach attention is focused on local minima and transition states of the PES, defined as stationary points with zero and one negative Hessian eigenvalues, respectively (Section 4.1), and theories are required for the local densities of states and minimum-to-minimum rate constants, as outlined in Section 7.1.1 and Section 7.2.1. There can be several reasons to employ such techniques. In particular, it may be possible to calculate approximate thermodynamic and dynamic properties much faster than for conventional Monte Carlo or molecular dynamics simulations. For example, the equilibrium between competing structures separated by large potential energy barriers may be difficult to treat even with techniques such as parallel tempering. This situation arises for Lennard-Jones clusters with nonicosahedral global potential energy minima (Section 6.7.1, Section 8.3), where finite size analogues of a solid–solid phase transition can be identified (1–3). Such transitions probably represent the most favourable case for application of the superposition approximation discussed in Section 7.1, because only a few low-lying minima make significant contributions to the partition functions at the temperatures of interest. Some results for these transitions are illustrated in Section 7.1.1.

Dynamical properties have been calculated using databases of minima and transition states using a master equation approach (Section 7.2.2) for a number of different systems (4–28).

In Chapter 4 we discussed features of the potential energy surface such as minima, transition states, pathways and branch points. In this chapter some of the methods employed to locate and characterise these features will be outlined, while Chapter 7 will deal with techniques to calculate thermodynamic and dynamic properties that make explicit use of stationary point information. Other methods for sampling the PES and for calculating thermodynamic and dynamic quantities of interest will be summarised in the present chapter. Depending on the conditions of interest it may also be important to know what lies at the very bottom of the PES, for this is where the system will be found at low temperature if the dynamics permit equilibrium to be attained. Locating this global potential energy minimum is important in many different fields, and a diverse range of approaches have been suggested. Some of these are considered in Section 6.7, where we also discuss why certain potential energy surfaces make the global minimum relatively easy or difficult to locate.

Finding local minima

The field of geometry optimisation involves the location of stationary points on the PES, be they minima, transition states or higher index saddles with more than one negative Hessian eigenvalue (Section 4.1). Many algorithms have been suggested, even for the simplest problem of finding a local minimum, and there are even methods designed to locate branch points (Section 4.4) (1) and conical intersections (Section 2.4.2) (2–4).

The most interesting points of a potential energy surface are usually the stationary (or critical) points where the gradient vanishes. The geometrical definitions of local minima and transition states on a potential energy surface are given in Section 4.1. Here we also explain how the definition of a transition state is related to alternative interpretations based on free energy or dynamical considerations. Symmetry properties of steepest-descent pathways are then examined in Section 4.2, and a classification scheme for rearrangement mechanisms is presented in Section 4.3. The symmetry restrictions imposed by these results upon possible reaction pathways are illustrated with a number of examples. Branch points and quantum tunnelling are considered in Section 4.4 and Section 4.5, with emphasis on the symmetry properties of the pathways involved. The invariance of the potential energy surface, stationary points and pathways to coordinate transformations is discussed in Section 4.6. Finally, in Section 4.7, we investigate the origin of zero Hessian eigenvalues, which reveals a fundamental difference between the translational and rotational degrees of freedom (1).

Classification of stationary points

A stationary point on a PES is a nuclear configuration where all the forces vanish, i.e. every component of the gradient vector is zero, ∂V(X)/∂Xα = 0 for 1 ≤ α ≤ 3N. Here, and subsequently, we will drop the ‘e’ subscript from V, which served to remind us in Chapter 2 that the potential energy surface describes the variation of the electronic energy with the nuclear coordinates within the Born–Oppenheimer approximation.

In this final chapter we consider applications of energy landscape theory to structural glasses and supercooled liquids. The ultimate objective of this approach is to understand and predict how the glass transition and associated phenomena depend upon details of the underlying potential energy surface. The large number of different models proposed to explain the glass transition must partly reflect different ways of expressing similar ideas, as well as the fundamental importance of the problem (1, 2). An overview of some of these theoretical methodologies is given in Section 10.1. Detailed comparisons between theory and experiment for properties such as dielectric loss (3, 4) or light-scattering spectra (5, 6) of a molecular glass former clearly present a significant challenge, and hence discrimination between different models is relatively hard.

The most popular systems for computer simulations of structural glass formers are described in Section 10.2. Surveys of local minima and transition states, including theoretical approaches based on the superposition framework, are treated in Section 10.3 and Section 10.4, and results for model potential energy surfaces are summarised in Section 10.5. The influence of the system size on the PES is analysed in Section 10.6, where properties of the configuration space are compared with the scaling laws expected for random networks.

This chapter discusses potential and free energy surfaces for molecules of biological interest, ranging from small peptides to proteins. Computer simulations and protein structure prediction are described in Section 9.1 and Section 9.2, respectively. Some theoretical aspects of protein folding are discussed in Section 9.3, and an introduction to the random energy model and the principle of minimal frustration is provided in Section 9.4. Two-dimensional free energy surfaces are considered in Section 9.5, with examples ranging from lattice and off-lattice bead representations to results obtained from biased sampling (Section 6.5.1) of all-atom models with explicit solvent.

Lattice models generally take a coarse-grained view of protein structure, as well as restricting the configuration space to a grid. The potential energy surface is also discretised: the catchment basins and transition states of a continuous PES are absent. These features are recovered in continuum bead models, where each amino acid is still represented by a single centre, but the configuration space is not restricted to a grid. One such model is discussed in detail in Section 9.6. Disconnectivity graphs for all-atom representations of two small molecules, IAN and NATMA, are analysed in Section 9.7 and Section 9.8, and both free energy and potential energy surfaces are considered for polyalanine peptides in Section 9.9.

While free energy surfaces have been calculated for all-atom protein representations, including explicit solvent, detailed analysis of potential energy surfaces has usually focused on smaller systems, particularly on the formation of elements of secondary structure.

For an N-atom system, including models of bulk material with N atoms in a periodically repeated supercell, the potential energy is a 3N-dimensional function. To refer to a potential energy hypersurface we must embed the function in a 3N + 1 dimensional space where the extra dimension corresponds to the ‘height’ of the surface.

There are two immediate problems with trying to use such a high-dimensional function in calculations. The first is that it is hard to visualise, and the second is that the number of interesting features, such as local minima, tends to grow exponentially with N. In this chapter we first consider how the number of stationary points grows with the size of the system (Section 5.1), and then discuss how the PES can be usefully represented in graphical terms. Simply plotting the energy as a function of one or two coordinates for a high-dimensional function is usually not very enlightening, and can be rather misleading. A very different approach to reducing the 3N + 1 dimensions down to just two uses the idea of monotonic sequences (Section 5.2), and was introduced by Berry and Kunz (1,2). Subsequently, the utility of disconnectivity graphs was recognised by Becker and Karplus (3), and a number of examples have been presented, as discussed in Section 5.3.