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.

This part is concerned with variational theory prior to modern quantum mechanics. The exception, saved for Chapter 10, is electromagnetic theory as formulated by Maxwell, which was relativistic before Einstein, and remains as fundamental as it was a century ago, the first example of a Lorentz and gauge covariant field theory. Chapter 1 is a brief survey of the history of variational principles, from Greek philosophers and a religious faith in God as the perfect engineer to a set of mathematical principles that could solve practical problems of optimization and rationalize the laws of dynamics. Chapter 2 traces these ideas in classical mechanics, while Chapter 3 discusses selected topics in applied mathematics concerned with optimization and stationary principles.

In the quantum theory of interacting electrons, a physically correct theory of time dependence should in principle be formulated as a relativistic quantum field theory. The physical model is that of electrons, each characterized by a probability distribution over space-time events xi, ti, that interact indirectly through the quantized electromagnetic field. This theory is simplified for particular applications by neglecting true radiative effects of quantum electrodynamics, and by passing to the limit of large c, the velocity of light in vacuo.

For direct N-electron variational methods, the computational effort increases so rapidly with increasing N that alternative simplified methods must be used for calculations of the electronic structure of large molecules and solids. Especially for calculations of the electronic energy levels of solids (energy-band structure), the methodology of choice is that of independent-electron models, usually in the framework of density functional theory [189, 321, 90]. When restricted to local potentials, as in the local-density approximation (LDA), this is a valid variational theory for any N-electron system. It can readily be applied to heavy atoms by relativistic or semirelativistic modification of the kinetic energy operator in the orbital Kohn–Sham equations [229, 384].

This part is concerned with variational principles underlying field theories. Chapter 10 develops the nonquantized theory of interacting relativistic fields, emphasizing Lorentz and gauge invariant Lagrangian formalism. The theory of a classical nonabelian gauge field is carried to the point of proving gauge invariance and of deriving the local conservation law for field energy and momentum densities.

In 1926, Schrödinger [365] recognized that the variational theory of elliptical differential equations with fixed boundary conditions could produce a discrete eigenvalue spectrum in agreement with the energy levels of Bohr's model of the hydrogen atom. This conceptually startling amalgam of classical ideas of particle and field turned out to be correct. Within a few years, the new wave mechanics almost completely replaced the ad hoc quantization of classical mechanics that characterized the “old” quantum theory initiated by Bohr. Although the matrix mechanics of Heisenberg was soon shown to be logically equivalent, the variational wave theory became the standard basis of methodology in the physics of electrons.

The nonrelativistic Schrödinger theory is readily extended to systems of N interacting electrons. The variational theory of finite N-electron systems (atoms and molecules) is presented here. In this context, several important theorems that follow from the variational formalism are also derived.

Before undertaking the major subject of variational principles in quantum mechanics, the present chapter is intended as a brief introduction to the extension of variational theory to linear dynamical systems and to classical optimization methods. References given above and in the Bibliography will be of interest to the reader who wishes to pursue this subject in fields outside the context of contemporary theoretical physics and chemistry. The specialized subject of optimization of molecular geometries in theoretical chemistry is treated here in some detail.

Linear systems

Any multicomponent system whose dynamical behavior is governed by coupled linear equations can be modelled by an effective Lagrangian, quadratic in the system variables. Hamilton's variational principle is postulated to determine the time behavior of the system. A dynamical model of some system of interest is valid if it satisfies the same system of coupled equations.