Introduction

In the eigenvalue problem of a Hamiltonian in quantum mechanics, the eigenfunctions of the Hamiltonian are classified in terms of the unitary irreducible representations (unirreps) of the symmetry group G of the Hamiltonian. In constructing approximate eigenfunctions by LCAO-MOs of a molecule belonging to a certain symmetry group G, the corresponding problem is to find the irreducible basis sets of G constructed by the linear combinations of the atomic orbitals belonging to the equivalent atoms of the molecule. Such a set is called a set of symmetry-adapted linear combinations (SALC) of the equivalent basis functions or equivalent orbitals. A standard method for such a problem is the generating operator method introduced in Section 6.9: it generates the desired basis set from an appropriate basis function. This method is very general and powerful but it is often extremely laborious to use; Cotton (1990). It is so very formal that one has little feeling until one arrives at the final result, which often could simply be obtained by inspection.

For point groups and their extensions, there exists a simple direct method of constructing the SALC belonging to a unirrep of a symmetry group G. The method requires knowledge of the basis functions of a space vector r = (x, y, z) in three dimensions belonging to the unirrep. The basis sets are well known for all point groups; e.g., those for the point groups Tp and D3p are reproduced in Tables 7.1 and 7.2, respectively, from the Appendix.

This book is written for graduate students and professionals in physics, chemistry and in particular for those who are interested in crystal and magnetic crystal symmetries. It is mostly based on the papers written by the author over the last 20 years and the lectures given at Temple University. The aim of the book is to systematize the wealth of knowledge on point groups and their extensions which has accumulated over a century since Schönflies and Fedrov discovered the 230 space groups in 1895. Simple, unambiguous methods of construction for the relevant groups and their representations introduced in the book may overcome the abstract nature of the group theoretical methods applied to physical chemical problems.

For example, a unified approach to the point groups and the space groups is proposed. Firstly, a point group of finite order is defined by a set of the algebraic defining relations (or presentation) through the generators in Chapter 5. Then, by incorporating the translational degree of freedom into the presentations of the 32 crystallographic point groups, I have determined the 32 minimum general generator sets (MGGSs) which generate the 230 space groups in Chapter 13. Their representations follow from a set of five general expressions of the projective representations of the point groups given in Chapter 12. It is simply amazing to see that the simple algebraic defining relations of point groups are so very far-reaching.

Let G be a finite group and H be a subgroup of G. Then a representation of the group G automatically describes a representation of the subgroup H of G. Such a representation is called a representation of H subduced by a representation of G. Conversely, from a given representation of a subgroup H of G we can form a representation of the group G. Such a representation is called a representation of G induced by a representation of its subgroup H. The problem is to form the irreducible representations (irreps in short) of G from the irreps of its subgroup H. If the group G is finite and solvable (see Section 8.4.1), the problem of forming the irreps of G may be solved by a step-by-step procedure from the trivial irrep of the trivial identity subgroup. This method is possible, for example, for a crystallographic point group. An alternative approach is via the induced irreps of G from the so-called small representations of the little groups of the irreps of H. As a preparation, we shall discuss subduced representations first.

Subduced representations

Let G = {g} be a group and H = {h} be a subgroup of G. Let Γ(G) = {Γ(g); g ∈ G} be a representation of G, then it provides a representation of H by {Γ(h); h ∈ H}. This representation is called the subduced representation of Γ(G) onto H or the representation of H subduced by the representation Γ(G).

Introduction

Many-body theory has been very successful in the ab initio calculation of electron–atom scattering cross sections. Its application has made it possible to take into account the target atom polarization in the collision process without the introduction of a semi-empirical potential. The many-body approach and the diagrammatic technique associated with it have also permitted the illumination of hidden difficulties in the description of the electron-atom scattering process.

The first calculations of electron–hydrogen elastic scattering cross sections using the diagrammatic technique of many-body theory were performed by H. P. Kelly about thirty years ago. In that calculation, as well as in subsequent investigations involving helium, argon and xenon, the polarization of the target atom by slow and medium energy electrons was taken into account. The effects of the exchange between the projectile electrons and target electrons were also accounted for. Although, at first glance, the interaction between the incoming electrons and the atomic particles is of second order, the polarization of the target in these works included a number of higher order corrections. This inclusion has proven to be very important.

The subsequent application of many-body theory to the scattering of electrons from highly polarizable atoms led to the development of methods which permitted the incorporation of the corresponding corrections non-perturbatively. A connection between elastic scattering and negative ion formation has also allowed one to use many-body techniques for the calculation of electron affinities.

Introduction

The process of single photoionization occurs when an atom or molecule absorbs a photon and ejects a single electron. Photoionization studies of multi-electronic systems can provide excellent portraits of the many-body effects that lie within both the initial target state and the final state consisting of the ion plus the photoelectron. Important examples of many-body effects include autoionizing resonances, giant shape resonances, relaxation, and polarization. A common element of all of these effects is their prominence near ionization thresholds. In this chapter, we will examine the many-body effects that are present in atomic single photoionization problems within the framework of many-body perturbation theory (MBPT).

We will refer to the corrections to a one-electron approximation (such as a Hartree-Fock approximation) as correlation effects. Although a one-electron approximation is capable of describing many of the gross properties of photoionization in atoms, a scheme for including correlation effects will be necessary in order to describe many of the processes that are mentioned above.

There has been considerable development in experimental techniques to study photoionization over the past few decades. Synchrotron radiation has been used to measure total photoabsorption cross sections over a wide range of energies for many atomic species. Additionally, photoelectron spectroscopy has been used to partition total cross sections into channel cross sections. Methods for the measurement of the angular distribution asymmetry parameter, β(ω), and spin-polarization parameters have also been developed. (See chapter seven by S. Manson for a description of the β asymmetry parameter.)

Introduction

In recent years, experimental studies on low and intermediate energy (e, 2e) processes have accumulated large amounts of triply differential cross section data. These (e, 2e) results, in which the energies and angles of both of the outgoing electrons produced in the electron-impact ionization process are specified, display strong electron-correlation effects. Owing to the difficulty involved in describing precisely various electron correlations, in particular, the Coulomb interaction between the two final-state continuum electrons, only approximate theoretical treatments have been carried out. At present, theoretical understanding of these data and the underlying effects are far from complete.

The near-threshold energy dependence of two electrons escaping from a positive ion has been studied theoretically by many authors using a number of methods. These studies cover the threshold behavior of the total and the differential cross sections for electron-impact ionization of atoms and ions.

In the early 1950's, Wannier applied to this problem the idea that the near-threshold energy-dependence of a reaction could be derived by investigating only the long-range interactions among its final products, without having a detailed knowledge about a small “reaction zone,” the size of which is of the order of magnitude of the Bohr radius. He revealed the importance of the configuration r1 = −r2 for the double escape of slow electrons from a positive ion by using methods of classical mechanics.

Hugh Padraic Kelly died on June 29, 1992 after a brave and lengthy struggle against cancer.

Hugh was a graduate of Harvard University, receiving an AB degree in 1953. He continued on to UCLA where he was awarded an M.Sc. degree in 1954. He served in the Marine Corps for three years before returning to graduate school at Berkeley. He worked there with Kenneth Watson, receiving his Ph.D. degree in 1963 and proceeding on to a postdoctoral fellowship with Keith Brueckner at the University of California, San Diego, where he began his seminal work on many-body theory. He was appointed to the faculty of the University of Virginia in 1965. He was a distinguished administrator, serving the University as Chairman of the Department of Physics, as Dean of the Faculty and as Provost.

Hugh was a very special person. He was from his first research paper the leader in the application of many-body perturbation theory in atomic and molecular physics using diagrammatic techniques. He was renowned internationally not only for his brilliant researches but also for his extraordinary personal qualities. He was modest, unassuming, always supportive of others. He had an abundance of creative ideas, which he freely shared. Hugh was the least competitive of people. He saw science as a joint enterprise in which he participated with his friends and students.