Introduction

A molecule is an assembly of positively charged nuclei and negatively charged electrons that forms a stable entity through the electrostatic forces which hold it all together. Since all the particles which make up the molecule are moving relative to each other, a full mechanical description of the molecule is very complicated, even when treated classically. Fortunately, the overall motion of the molecule can be broken down into various types of motion, namely, translational, rotational, vibrational and electronic. To a good approximation, each of these motions can be considered on its own. The basis of this classification was established in a ground-breaking paper written by Born and Oppenheimer in 1927, just one year after the introduction of wave mechanics. The main objective of their paper was the separation of electronic and nuclear motions in a molecule. The physical basis of this separation is quite simple. Both electrons and nuclei experience similar forces in a molecular system, since they arise from a mutual electrostatic interaction. However, the mass of the electron, m, is about four orders-of-magnitude smaller than the mass of the nucleus M. Consequently, the electrons are accelerated at a much greater rate and move much more quickly than the nuclei. On a very short time scale (less than 10−16 s), the electrons will move but the nuclei will barely do so; as a first approximation, the nuclei can be regarded as being fixed in space.

Introduction

Much of the beauty of high-resolution molecular spectroscopy arises from the patterns formed by the fine and hyperfine structure associated with a given transition. All of this structure involves angular momentum in some sense or other and its interpretation depends heavily on the proper description of such motion. Angular momentum theory is very powerful and general. It applies equally to rotations in spin or vibrational coordinate space as to rotations in ordinary three-dimensional space.

All the laws of physics are easier to accept (and even to understand) when the underlying symmetry of the problem is appreciated. For example, classical Euclidean space is isotropic and a physical system is invariant to any rotation in this space. By this we mean that all the measurable properties of the system are unaffected by the rotation. An investigation of the behaviour of a quantum state under such rotations allows the properties of the state to be defined. These properties are most succinctly expressed as quantum numbers. Although quantum numbers are frequently used to label the eigenstates or eigenvalues of a molecule, they really carry information about the symmetry properties of the associated eigenfunctions.

In this chapter we give only a brief description of angular momentum theory, sufficient to introduce the various techniques required for the description of molecular energy levels and the transitions between them.

Introduction and experimental methods

Simple absorption spectrograph

The first direct observations of pure rotational transitions induced by microwave or millimetre-wave radiation date from about 1945. Research during the Second World War, particularly on radiation sources in these wavelength regions, led to rapid developments in this branch of molecular spectroscopy. However, the observation of rotational structure in electronic and vibrational spectra, which had been routine for at least twenty years previously, meant that much was already understood about molecular rotations. Indeed most of the theory was developed in the years immediately following the birth of modern quantum mechanics. We have already described much of this theory in earlier chapters, and we will draw upon the results obtained in this chapter. We have also described molecular beam resonance and maser techniques for studying rotational spectra, but in this chapter we describe somewhat more conventional experiments in which the absorption of microwave radiation by molecules in the bulk gas phase is detected directly. As we will see, some ingenious techniques have been developed in more recent years to enable the study of short-lived transient species, neutral or charged free radicals. Molecular radio astronomy is also, in part, a branch of microwave absorption or emission spectroscopy, and we shall deal with that in subsection 10.1.6. We will also describe, in the next chapter, double resonance experiments in which microwave radiation is combined simultaneously with a second source of radiation of either similar or quite different wavelength.

Introduction

Microwave magnetic resonance, which has often been called ‘gas phase electron resonance’, and far-infrared laser magnetic resonance have been extremely important techniques in the study of free radicals. These techniques depend upon the presence of a large magnetic moment for the species under investigation, because both rely on the ability to tune the energy levels with a magnetic field into resonance with fixed frequency radiation. Although historically the first study of the rotational levels and their fine structure involved fairly conventional swept-frequency microwave spectroscopy, applied to the OH radical, subsequent development of the subject depended initially on the success of magnetic resonance methods. Pure microwave spectroscopy of gaseous free radicals has now become almost routine, and many examples will be described in chapter 10. We have, after some deliberation, decided to present the exciting subject of magnetic resonance with due regard to its historical development. Nevertheless, this and the two following chapters should be taken together for a balanced view. Laser magnetic resonance techniques are still widely used, but microwave magnetic resonance has now been largely superseded by swept-frequency methods, without the presence of an external magnetic field. Zeeman effects can, of course, still be investigated, but they are not an essential part of the detection method.

Experimental methods

Microwave magnetic resonance

Electron spin resonance (e.s.r.) spectroscopy, applied to free radicals in condensed phases, is a long established technique with several commercially available spectrometers.

Nuclear spins and magnetic moments

In the course of developing a Hamiltonian for diatomic molecules, we have so far introduced and discussed two nuclear properties. We considered at length the nuclear kinetic energy in chapter 2, and in chapter 3 we took account of the nuclear charge in considering the potential energy arising from the electrostatic interaction between electrons and nuclei. With respect to the electrostatic interaction, however, we have implicitly treated the nucleus as an electric monopole, and this assumption is re-examined in section 4.4. First, however, we consider another important property of many nuclei, namely their spin and the important magnetic interactions within a molecule which arise from the property of nuclear spin. The possibility that a nucleus may have a spin and an associated magnetic moment was first postulated by Pauli, following the observation of unexpected structure in atomic spectra. The first quantitative theory of the interaction between a nuclear magnetic moment and the ‘outer’ electrons of an atom was provided by Fermi, Hargreaves, Breit and Doermann and Fermi and Segrè. In the case of diatomic molecules with closed shell electronic states, the magnetic interaction of the nuclear moment with the magnetic angular momentum vector, an I · J coupling, was treated by a number of authors. The interaction between the nuclear electric quadrupole moment and the electronic charges, an interaction which has nothing to do with nuclear spins or magnetic moments, was treated by Bardeen and Townes.

Introduction

The Born–Oppenheimer approximation is an important linch pin in the description of molecular energy levels. It reveals the difference between electronic and nuclear motions in a molecule, as a result of which we expect the separation between different electronic states to be much larger than that between vibrational levels within an electronic state. An extension of these ideas shows that the separation between vibrational levels is correspondingly larger than the separation between the rotational levels of a molecule. We thus have a hierarchy of energy levels which reveals itself in the electronic, vibrational and rotational structure of molecular spectra. This gradation in the magnitude of the different types of quanta also provides the inspiration for an energy operator known as the effective Hamiltonian.

In this chapter we introduce and derive the effective Hamiltonian for a diatomic molecule. The effective Hamiltonian operates only within the levels (rotational, spin and hyperfine) of a single vibrational level of the particular electronic state of interest. It is derived from the full Hamiltonian described in the previous chapters by absorbing the effects of off-diagonal matrix elements, which link the vibronic level of interest to other vibrational and electronic states, by a perturbation procedure. It has the same eigenvalues as the full Hamiltonian, at least to within some prescribed accuracy.

Introduction

We have seen that the fine and hyperfine structure of vibration–rotation levels arises almost entirely from interactions involving electron spin, nuclear spin, and the rotational motion of the nuclei, with or without the additional presence of applied magnetic or electric fields. In this chapter we concentrate on the study of direct transitions between rotational, fine or hyperfine energy levels. Such transitions occur mainly in those regions of the spectrum which extend from the radiofrequency, through the microwave and millimetre wave, to the far infrared region. They are therefore transitions that involve very low energy photons, with the absorption or emission of very small amounts of energy. Specialised techniques have been developed to carry out spectroscopic studies in this frequency range. In particular one often does not attempt to detect the low energy photons directly, but to make use of indirect detection methods which rely on the energy level population transfer resulting from spectroscopic transitions. We shall describe a number of the indirect methods which have been employed.

Radiofrequency spectroscopy, in particular, is frequently combined with molecular beam techniques, or other methods using gas pressures which are low enough to remove the effects of molecular collisions. There are two main reasons for this. First, experiments which depend upon population transfer can only be successful if collisional relaxation or equilibration is absent.

Introduction

Double resonance spectroscopy involves the simultaneous use of two spectroscopic radiation sources, often of quite different wavelengths. Figure (a) illustrates the simplest example of many possible variations. High-frequency electronic excitation (f1) is combined with microwave or radiofrequency radiation (f2); the objective is usually to observe and measure the lower frequency spectrum by making use of the sensitivity advantages provided by the higher frequency radiation. Detection of the fluorescence intensity from the intermediate state E2 provides a monitor of the population of the state. The lower frequency transition f2 changes the population of E2, and hence changes the fluorescence intensity. Many of the experiments to be described in this chapter depend upon this simple scheme. Such experiments have been extremely valuable, particularly in the study of short-lived species such as neutral free radicals, molecular ions, or metastable excited electronic states. Their success usually depends on prior knowledge and study of the high-frequency spectrum, as we shall see. In other cases, however, the two radiation sources may be of similar wavelengths; microwave/microwave double resonance, for example, has proved to be a powerful method for confirming otherwise uncertain spectroscopic assignments.

As is often the case, the initial experiments were developed by atomic spectroscopists. Figure (b) illustrates an example from atomic physics, described by Brossel and Bitter. Mercury atoms are excited by a mercury lamp from the 1S ground state to the 3P1 excited state, in the presence of a small applied magnetic field.

Introduction

In chapter 3 we derived a Hamiltonian to describe the electronic motion in a diatomic molecule, starting from first principles. In our case, the first principles were the Dirac equation for a single particle, and the Breit equation for two interacting particles. We pointed out that even at this level our treatment was a compromise because it did not include quantum electrodynamics explicitly. Nevertheless we concluded the chapter with a rather complete and complicated Hamiltonian, and added yet more complications in chapter 4 with the inclusion of nuclear spin effects. In the next chapter, chapter 7, we will show how terms in the ‘true’ Hamiltonian may be reduced to ‘effective’ Hamiltonians designed to handle the particular cases which arise in spectroscopy. We will make extensive use of angular momentum theory, described in chapter 5, to describe the electronic and nuclear dynamics in diatomic molecules, and the interactions with applied magnetic and electric fields. The experimental study of these dynamical effects is dealt with at length in chapters 8 to 11. We will be classifying these studies according to molecular electronic states, and demonstrating how the high-resolution spectroscopic methods described probe the structural details of these electronic states. That, indeed, is one of the main purposes of spectroscopy.

Before we proceed to these details we must describe some aspects of the theory of the electronic and vibrational states of diatomic molecules.

The Dirac equation

The analysis of molecular spectra requires the choice of an effective Hamiltonian, an appropriate basis set, and calculation of the eigenvalues and eigenvectors. The effective Hamiltonian will contain molecular parameters whose values are to be determined from the spectral analysis. The theory underlying these parameters requires detailed consideration of the fundamental electronic Hamiltonian, and the effects of applied magnetic or electrostatic fields. The additional complications arising from the presence of nuclear spins are often extremely important in high-resolution spectra, and we shall describe the theory underlying nuclear spin hyperfine interactions in chapter 4. The construction of effective Hamiltonians will then be described in chapter 7.

In this section we outline the steps which lead to a wave equation for the electron satisfying the requirements of the special theory of relativity. This equation was first proposed by Dirac, and investigation of its eigenvalues and eigenfunctions, particularly in the presence of an electromagnetic field, leads naturally to the property of electron spin and its associated magnetic moment. Our procedure is to start from classical mechanics, and then to convert the equations to quantum mechanical form; we obtain a relativistically-correct second-order wave equation known as the Klein–Gordon equation. Dirac's wave equation is linear in the momentum operator and is so constructed that its eigenvalues and eigenfunctions are also solutions of the Klein–Gordon equation.