Enzymes are biological catalysts. Recall that a catalyst is a substance that accelerates a chemical reaction without itself being consumed by the reaction. Enzyme catalysis holds a special place in the history of chemistry. The law of mass action was enunciated in the nineteenth century in a thermodynamic context by Guldberg and Waage. By the early twentieth century, the law of mass action was supported by a growing mass of both thermodynamic and kinetic evidence. Nevertheless, some reactions had complex rate laws which were not obviously compatible with mass action. Enzyme catalysis posed such a difficulty. In fact, there was some debate as to whether enzyme molecules even had to come in contact with the reactant molecules to accelerate the reaction. In 1902, Victor Henri showed, with some help from Max Bodenstein, that enzyme kinetics is in fact compatible with the law of mass action. The complex rate law obeyed by enzyme-catalyzed reactions was simply a manifestation of a complex mechanism. Although debates with respect to the mode of action of enzymes persisted for some time, it was at least clear from this point on that enzyme catalysis was compatible with the then-emerging theory of chemical kinetics. This is the vein we will pick up in this chapter. Our study of enzyme kinetics will particularly emphasize the derivation of rate laws and the analysis of data from initial rate experiments.

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.