Introduction

Multichannel quantum defect theory uses scattering methods to provide a uniform treatment of spectroscopic and fragmentation phenomena. It rests on the idea that the exchange and correlation interactions between an outer Rydberg electron and the positive ion core act over a relatively short range, so that the detached electron moves in a purely Coulomb field at larger distances. One therefore thinks, even in the bound state context, of the scattering effect of the non-Coulomb core on the Coulomb wavefunctions. Put in explicit terms this means that the outer parts of the Rydberg orbitals are solutions of the Coulomb equation, the phases of which are determined by matching to the inner wavefunction at the core boundary. There can also be more complicated situations, in which the non-Coulomb interactions lead to energy transfer from the core, which ‘auto-ionizes’ the detached electron from a bound to a continuum state. The general solutions are normalized to allow a uniform description at energies above and below the ionization limit. Further ramifications, which are deferred to a later chapter, allow the inclusion of simultaneous ionization and dissociation. Reviews that emphasize molecular aspects of the theory are given by Greene and Jungen [1] and Ross [2]. There is also a collection of seminal papers, edited by Jungen [3].

This exposition starts with a description of the properties of Coulomb wavefunctions at arbitrary energies, using definitions that provide a uniform description of both bound and continuum states.

Introduction

The properties of molecular Rydberg states are most commonly observed experimentally by photo-excitation and photo-ionization, and it is impossible to ignore the explosion of interest in multiphoton phenomena over the past twenty years. It is, however, beyond the scope of this book to attempt anything like a comprehensive treatment. Attention is therefore restricted to the weak field theory, in which light acts as a perturbation. Readers are referred to Lambropoulos and Smith for a fuller discussion [1]. Explicit results are restricted to one and two photons, leaving the reader to consult the literature for extension to n photons. This chapter concerns excitation between discrete bound states, either by single-photon absorption or n + 1 resonant multiphoton ionization (REMPI) [2, 3, 4].

The first of the following sections outlines the perturbation theory of one- and two-photon absorption, as initiated by Göppert-Mayer, and the extension to three-photon processes is indicated [5]. Aspects of the theory, such as the point group symmetries of the resulting dipole (n = 1), polarizability (n = 2) and hyperpolarizability (n = 3) operators are readily deduced in a Cartesian formulation [6]. However, the relevant angular momentum manipulations including the selection rules for the various n-photon linear and circular polarization possibilities are often most easily performed in a complex spherical tensor representation, which is outlined in Section 6.3 [6, 7]. There are also advantages, for resonant processes, in employing an alternative density matrix, which focuses on spatial characteristics of the excited angular momentum, rather than the overall excitation probability.

The quantum defect and the frame transformation approximation are the two most important components of the MQDT machinery. This chapter starts by examining the validity of the latter approximation. To put the classical argument in Chapter 1 into a quantum mechanical perspective, Section 4.1 demonstrates the insensitivity of the radial wavefunction accompanying energy changes of the order of typical vibrational and rotational energy intervals. Readers who expect to apply the transformation at the core boundary may be surprised to find that it remains valid over often quite a wide range of radial separations, Δr, which varies inversely with the magnitude of the rotational or vibrational energy transfer involved.

A typical transformation element takes the form of the projection, 〈i|α〉 of an uncoupled state |i〉 onto a coupled Born–Oppenheimer state |α〉, the form of which varies according to the nature of the relevant motion. For example, in the rotational case |α〉 = |∧〉 is a specified body-fixed angular momentum projection, while |i〉 = |N+〉 is the positive ion angular momentum after the Rydberg electron has been uncoupled from the molecular frame. Section 4.2 restricts attention to the simplest angular momentum coupling case, with applications chosen to illustrate the quantum defect description of topics in the spectroscopic literature, such as ∧-doubling and ℓ-uncoupling [1, 2, 3]. The angular momentum manipulations required to handle more complicated coupling cases are treated in Appendix C, which also includes an account of the relevant parity and symmetry considerations.

My initial aim was to introduce the powerful but relatively under-used techniques of multichannel quantum defect theory (MQDT) to graduate students in atomic and molecular physics. The methods are particularly attractive in two ways. They provide an elegant, computationally tractable approach to the treatment of molecular Rydberg states, which invalidate the normal molecular assumption that the electronic motion is overwhelmingly rapid compared with other degrees of freedom. In addition the theory offers a unified description of the discrete molecular states below an ionization limit and those above in the ionization continuum. At the same time the novelty of the MQDT method makes it essential to point to the links with the familiar techniques of ‘normal’ molecular physics.

While writing, I realized that workers in two other fields would benefit from a more general treatment of molecular Rydberg states. In the first place there is a huge literature on electronic structure theory or ‘quantum chemistry’, which can, however, handle only the very lowest Rydberg states, owing to the very long range of the excited orbitals. A chapter has been written to show how the familiar quantum chemical techniques can be adapted to handle arbitrary members of the infinite Rydberg series. Secondly, to meet the demands of modern experiments, the chapters involving interaction with radiation take account of developments in the theoretical description of coherent multiphoton excitation and resonant multiphoton ionization.

The theory of photo-ionization owes much to the treatment of photo-excitation in the previous chapter, but there are significant differences. Most importantly the species is excited to a final state in which the electron is detached from the positive ion. The necessary boundary conditions resemble those for a scattering event, except that the partial waves are combined to produce outgoing plane wave motion in a particular target channel, instead of an incoming plane wave in the incident scattering channel. Confusingly the former are referred to as ‘incoming’ and the latter as ‘outgoing’ boundary conditions, because the amplitudes and phases are adjusted to ensure only incoming spherical waves in photo-fragmentation and outgoing spherical waves in scattering. Details are given in this chapter for the simple case of a single open ionization channel, leaving the multichannel boundary conditions to be treated in Appendix D.2. It is also shown in Section 7.1 how the spherical tensor machinery in Chapter 6 can be adapted to handle multiphoton ionization.

The theory is presented for a bulk sample with a random distribution of magnetic sub-levels, but the averaging over fragment sub-levels is more awkward than for a final bound state. Further complications come from possible changes in the angular momentum coupling regime between the parent neutral molecule and the resulting positive ion, details of which are covered in Appendix D.2. The following presentation is intended to combine the early results of Buckingham et al. with the formal ‘angular momentum transfer’ theory of Fano and Dill [1, 2, 3].

The previous chapter laid out the principles of multichannel quantum defect theory, showing in particular how knowledge of the quantum defects or scattering K-matrices are built into a unified theory of Rydberg spectroscopy and ionization dynamics. This chapter deals with the ab-initio determination of these quantum defects. We know from the discussion in Chapter 1 that they arise from interactions between the positive ion core and the Rydberg electron, which were seen to occur on a timescale far shorter than that of the molecular vibrations and rotations. It is therefore natural to compute them within the fixed nucleus Born–Oppenheimer approximation. Useful information on the lowest members of a given series may be obtained by traditional Hartree–Fock and configuration interaction techniques [1]. Carefully designed diffuse Rydberg orbitals are, however, required [2]. The resulting information is normally limited to the potential energy surfaces for principal quantum numbers n ≤ 4, from which it may be difficult to extract the desired forms of the quantum defects, as functions of the nuclear coordinates, particularly for polyatomic molecules. An alternative is to recognize that the distant outer parts of the Rydberg wavefunction may be expressed as Coulomb functions. Thus the ab-initio effort may be restricted to a finite volume, chosen to be large enough to allow a proper treatment of all Rydberg–core interactions [3, 4, 5, 6]. The inner and outer wavefunctions are then joined at the core boundary by a so-called R-matrix, from which the scattering K-matrix may be obtained directly, without reference to information on any potential energy surfaces.

The nature of Rydberg states

The nature of atomic Rydberg states is well described by Gallagher, though with less emphasis on theory [1]. Those of molecules are severely complicated by the additional nuclear degrees of freedom, in a way that gives them quite different properties from those treated in most spectroscopic texts [2, 3, 4, 5]. The essential difference is that established spectroscopic theory is rooted in the Born–Oppenheimer approximation, whereby the frequencies of the electronic motion are assumed to be so high compared with the vibrational and rotational ones that the nuclear motions may be treated as moving under an adiabatic electronic energy (or potential energy) surface. In addition the vibrational frequency usually far exceeds that of the rotations, which means that every vibrational state has an approximate rotational constant. Such considerations provide the basis for a highly successful systematic theory. Modern ab-initio methods allow the calculation of very reliable potential energy surfaces and there are a variety of efficient methods for diagonalizing the resulting Hamiltonian matrix within a functional or numerical basis. Electronically non-adiabatic interactions between a small number of electronic states can also be handled by this matrix diagonalization approach, even including fragmentation processes, if complex absorbing potentials are added to the molecular Hamiltonian.

The difficulty in applying such techniques to highly excited molecular electronic states is that the Rydberg spectrum of every molecule includes 100 electronic states with principal quantum number n = 10, separated from the n = 11 manifold by only 100 cm-1, which is small compared with most vibrational spacings and comparable to rotational spacings for small hydride species.

The huge spatial extension of atomic and molecular Rydberg states makes them amenable to manipulation in a variety of ways. One type of experiment involves the creation of a time-dependent wavepacket, which may be manipulated by subsequent light pulses to control the outcome of the fragmentation products [1]. Interesting intensity recurrences and revivals are also observed as leading and trailing elements of the wavepacket interfere with each other. The response to electric fields is also experimentally important in the field-ionization detection of highly excited species and in the technique of high-resolution pulsed-field zero-kinetic energy (ZEKE-PFI) spectroscopy [2, 3]. This chapter concentrates on these two topics, but the reader should be aware of the quasi-Landau response to magnetic fields, particularly at field strengths such that the Landau frequencies are comparable to those of hydrogenic orbits, because the Rydberg scaling properties make them ideal candidates for investigating ‘quantum chaos’ [2, 4].

Rydberg wavepackets

Despite the well-known equivalence between the time-dependent and time independent pictures for conservative systems (i.e. those with time-independent Hamiltonians), the ability to create and manipulate Rydberg wavepackets offers novel insights into the underlying dynamics. Here we concentrate on three aspects of the time-dependent theory. The first shows that the familiar level structure of the hydrogen atom leads to a surprisingly intricate pattern of recurrences and revivals arising from interference between different components of the spreading wavepacket. Revivals of a different type are seen to occur in molecules as a result of the stroboscopic beats between the frequencies of rotational and electronic motion that were described in Section 4.2.4.

The analytical forms for a variety of rotational frame transformations are given here. In view of the diversity of angular momentum coupling schemes, attention is first restricted to diatomic molecules, within the framework of Hund's coupling cases [5], which differ according to the relative importance of three factors – the electronic energy splitting between different ∧ components, the strength of spin–orbit coupling, and the rotational energy-level separations. The relative values of these three quantities allow six possibilities, each of which has a characteristic form for its parity-adapted wavefunction, although Hund himself only covered cases (a)–(d). This discussion is restricted to situations in which the Rydberg electron in a neutral molecule, which conforms to case (a), (b) or (c), is uncoupled from the molecular axis, to leave the positive ion in the same case as the parent molecule. Such excitations correspond to transformations of the type (a)→(e), (b)→(d) and (c) → (e′). The first of these has been most fully described by Jungen and Raseev [6]. The second is discussed in its simplest form in Chapter 4.2, along lines pioneered by Fano [7]. A fuller account, applicable to species with open shell cores, is given below. The final (c)→(e′) case, which has as yet found no application in the literature, is mentioned for completeness, but not treated in detail.

The final section includes results for the rotational frame transformation for asymmetric tops, in the absence of spin, which goes beyond earlier work [8], by employing permutation inversion symmetry [9].