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].

In this appendix we list some of the commonly needed physical constants in spectroscopy. We also calculate the emitted field intensity directly from the third-order response functions for standard experimental conditions and molecular properties. The purpose of this calculation is not to obtain a highly accurate result, but to serve as a (somewhat interesting) exercise of units. In the following, we use units of meters-kilograms-seconds (SI units) unless otherwise noted.

One of the strengths of 2D IR spectroscopy is the ability to quantitatively link experimental results to computer simulations, be it molecular dynamics simulations, quantum chemistry calculations, or ideally a combination of both on the level of mixed quantum mechanics/molecular mechanics (QM/MM) calculations. In the present chapter, we outline how such simulations are performed and present some examples with computer code that can be reproduced on a personal computer. We also describe more sophisticated models that have been developed. The motivation of the chapter is not to get the most accurate agreement with experiment, but to outline the essential concepts with working examples.

In this chapter we use the molecular dynamics simulation package Gromacs 3.3 [183] (which can be downloaded for free from http://www.gromacs. org), the quantum chemistry program Gaussian09 for electronic structure calculations [58], and simple Mathematica or C codes (together with Numerical Recipes routines [152]). All the relevant computer programs in this chapter can be downloaded from the book webpage (http://www.2d-ir-spectroscopy.com), so the reader has operational programs to start with which can then be modified at will. For each of the Mathematica programs, Matlab versions are available on the book webpage as well.

2D lineshapes: Spectral diffusion of water

Perhaps the most accurate quantities that can currently be modeled are 2D IR lineshapes.

The concepts outlined in the previous chapters lay a foundation from which new pulse sequences can be designed. We have covered two types of 2D IR spectra that can be generated with third-order pulse sequences, the so-called rephasing and non-rephasing pulse sequences. We start this chapter by presenting the third type of third-order 2D IR pulse sequence, which we term the two-quantum (2Q) pulse sequence, for reasons that will become apparent. We then improve upon this two-quantum pulse sequence by adding two more laser pulses to generate a fifth-order two-quantum coherence, which also enables 3D IR experiments. Purely absorptive 3D IR experiments are described as are transient 2D IR spectroscopies which are also fifth-order nonlinear experiments. Currently, 3D IR pulse sequences are largely unexplored, and the ones that have been implemented have only been applied to a few molecules. This chapter is written to illustrate some of the basic concepts to serve as a platform for more sophisticated experiments in the future.

Two-quantum pulse sequence

Third-order 2D IR spectra are generated from pulse sequences that interact three times with the sample. The electric field for one of the pulse interactions must be the complex conjugate of the other two, which gives rise to the third basic types of third-order 2D IR spectra: E*1E2E3, E1E*2E3, E1E2E.