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.

The mean field theories of phase transitions considered in the previous chapter explicitly or implicitly neglect fluctuations, i.e. local deviations of the order parameter from its most probable, macroscopic value. This neglect is epitomized in the basic assumption underpinning Landau's theory, i.e. that the order parameter is uniformly replaced by the value which minimizes the phenomenological Landau free energy; fluctuations around this most probable value do not contribute to the partition function (4.36). The fact that local deviations of the order parameter from its average are not built into mean field considerations means, a fortiori, that correlations between fluctuations occurring in different subsystems are neglected. Correlations on the molecular scale, as quantified by the density correlation functions or static structure factors introduced in section 3.4, dominate first-order phase transitions like freezing, and can be accounted for quantitatively within standard structural theories of dense fluids, like the mean spherical approximation (MSA) and Percus–Yevick (PY) approximation mentioned in chapter 3. The range of these spatial correlations is typically a few molecular diameters. However, upon approaching the critical point of a second-order phase transition, e.g. the liquid–vapour critical point, density fluctuations become correlated over mesoscopic scales, while the macrosopic response of the system (e.g. the compressibility) diverges. A quantitative description of large-scale fluctuations is a highly challenging problem which has only been understood since the 1970s in terms of the idea of scale invariance, one of the key concepts of this chapter, which may also be applied to other, apparently unrelated, ‘critical’ objects like polymers.

One of the most remarkable observations in physical sciences or, for that matter, of everyday life, is that most substances, with a well defined chemical composition, can exist in one of several states, exhibiting very different physical properties on the macroscopic scale; moreover one can transform the substance from one state (or phase) to another, simply by varying thermodynamic conditions, like temperature or pressure. In other words, a collection of N molecules, where N is typically of the order of Avogadro's number NA, will spontaneously assemble into macroscopic states of different symmetry and physical behaviour, depending on a limited number of thermodynamic parameters. The most common states are either solid or fluid in character, and are characterized by qualitatively different responses to an applied stress. At ambient temperatures, the solid states of matter are generally associated with the mineral world, while ‘soft’ matter, and in particular the liquid state, are more intimately related to life sciences. In fact it is generally accepted that life took its origin in the primordial oceans, thus underlining the importance of a full quantitative understanding of liquids. However, even for the simplest substances, there are at least two different fluid states, namely a low density ‘volatile’ gas (or vapour) phase, which condenses into a liquid phase of much higher density upon compression or cooling. For more complex substances, generally made up of highly anisotropic molecules or of flexible macromolecules, the liquid state itself exhibits a rich variety of structures and phases, often referred to as ‘complex fluids’.