1

Introduction

This first chapter provides a short reader's guide, which may help to make the material presented in this book more easily accessible. Moreover, it serves to highlight the philosophy and intention of this book.

1.1 Philosophy of this Book

Relativistic quantum chemistry is the relativistic formulation of quantum mechanics applied to many-electron systems, i.e., to atoms, molecules and solids. It combines the principles of Einstein's theory of special relativity, which have to be obeyed by any fundamental physical theory, with the basic rules of quantum mechanics. By construction, it represents the most fundamental theory of all molecular sciences, which describes matter by the action, interaction, and motion of the elementary particles of the theory. In this sense it is important for physicists, chemists, material scientists, and biologists with a molecular view of the world. It is important to note that the energy range relevant to the molecular sciences allows us to operate with a reduced and idealized set of "elementary" particles. "Elementary" to chemistry are atomic nuclei and electrons. In most cases, neither the structure of the nuclei nor the explicit description of photons is required for the theory of molecular processes. Of course, this elementary level is not always the most appropriate one if it comes to the investigation of very large nanometer-sized molecular systems. Nevertheless it has two very convenient features:

For all issues relevant to the chemistry and physics of atoms, molecules, clusters, and solids only electromagnetic and - to a negligible extent - weak interactions, which are responsible for the radioactive ß-decay and the non-conservation of parity, contribute. The internal structure of hadrons, i.e., protons and neutrons built up by quarks governed by strong interactions and also gravitational forces, do not play any role and are therefore not covered by this presentation.

A main purpose of this book is to provide a structured and self-contained presentation of relativistic quantum chemistry as a semi-classical theory. We deem this necessary as there hardly exists any such contiguous and detailed presentation. The main reason appears to be the fact that quantum electrodynamics was developed in the 1930s and 1940s shortly after the advent of the new quantum theory. It was already clear in those days that quantum electrodynamics represents the fundamental theory of light and electrons, and hence the semi-classical theory was almost instantaneously abandoned. However, this most elegant and sophisticated theory of quantum electrodynamics is too abstract to be grasped immediately. The connection to classical physics, which has always been a guiding principle in quantum theory and even for the development of quantum electrodynamics, is seldom made. Instead, especially modern accounts start directly with the field-quantized formulation although the semi-classical theory is sufficient for chemistry as countless numerical studies in quantum chemistry demonstrated. This issue had already been noted by Dirac in his famous lectures on quantum theory, but he nevertheless changed the presentation of the electromagnetic quantum field theory in one of the later editions and gave up the semi-classical theory. However, because of the paramount importance of the semi-classical theory to chemistry, we derive this theory from the very basis, i.e., from classical electrodynamics. Of course, the transition to quantum electrodynamics is also made in this book, but it plays a minor role. This transition can be much better understood once the problems with the classical and the semi-classical theory have been worked out. Only then can one fully appreciate the emergence of concepts such as retarded electromagnetic interactions, magnetic spin-spin coupling or orbit-other-orbit interaction.

Because of the limited space available and the vast number of relativistic studies on atoms and molecules, we chose to accompany the derivation of the theory with a multitude of references to the original research literature in order to provide the reader with a topical overview of results for electronic systems.

Relativity adds a new dimension to quantum chemistry, which is the choice of the Hamiltonian operator. While the Hamiltonian of a molecule is exactly known in nonrelativistic quantum mechanics (if one focuses on the dominating electrostatic monopole interactions to be considered as being transmitted instantaneously), this is no longer the case for the relativistic formulation. Numerical results obtained by many researchers over the past decades have shown how Hamiltonians which capture most of the (numerical) effect of relativity on physical observables can be derived. Relativistic quantum chemistry therefore comes in various flavors, which are more or less well rooted in fundamental physical theory and whose relation to one another will be described in detail in this book. The new dimension of relativistic Hamiltonians makes the presentation of the relativistic many-electron theory very complicated, and the degree of complexity is far greater than for nonrelativistic quantum chemistry. However, the relativistic theory provides the consistent approach toward the description of nature: molecular structures containing heavy atoms can only be treated correctly within a relativistic framework. Prominent examples known to everyone are the color of gold and the liquid state of mercury at room temperature. Moreover, it must be understood that relativistic quantum chemistry provides universal theoretical means that are applicable to any element from the periodic table or to any molecule - not only to heavy-element compounds.

It is the nature of the subject that makes its presentation rather formal and requires some basic, mainly conceptual knowledge in mathematics and physics. However, only standard mathematical techniques (such as differential and integral calculus, matrix algebra) are required. More advanced subjects such as complex analysis and tensor calculus are occasionally also used. Furthermore, also basic knowledge of classical Newtonian mechanics and electrodynamics will be helpful to more quickly understand the concise but short review of these matters in the second chapter of this book.

Many (pseudo-)relativistic quantum chemical approaches provide methods which can be implemented in computer programs in a very efficient manner - an aspect which may be called a boundary condition of computational chemistry imposed on theoretical chemistry. Most of these approaches demand only as much computing time as their nonrelativistic analogs (or are more expensive by a constant but small factor). The quantum chemistry community has developed a certain working knowledge of the reliability of these relativistic methods, but their relation to one another sometimes remains unclear. In the light of the importance of relativistic methods, we therefore derive all methods from first principles and highlight their development to sophisticated computational tools in chemistry. In doing so, we shall understand which aspects of the rigorous theory of relativistic quantum mechanics survive in relativistic quantum chemistry; we learn and have to accept that relativistic effects in extended molecules are about the art of efficiently correcting numerical results such as energies for many-electron systems in an essentially nonrelativistic framework.

1.2 Short Reader's Guide

A book on a theoretical topic, which is based on fundamental physical theory but extends to the realm of experimental chemistry - such as the present one does - always faces difficulties, as it might be too formal for chemists while it could be too technical and specialized for non-experts and even physicists. Our main goal is the presentation of an almost complete derivation of the relativistic theory for many-electron systems as the fundamental theory of the molecular sciences. For this purpose, we have tried to introduce all essential concepts and ideas and derive all basic equations explicitly. As a consequence, parts of the book - such as the solution of the Dirac hydrogen atom - seem to be lengthy. However, many derivations cannot be easily found in such detail elsewhere in the literature (if at all). Also, the number of misconceptions can be rather large regarding issues of relativistic quantum chemistry, which is the reason why we try to provide a self-contained presentation of the theory. The resulting equations for many-particle systems cannot be solved analytically, and the derivation of working equations is strongly driven by the need to be able to solve them on modern computer hardware. Although it is not possible to delve deeply into how this is achieved in every...