Mathematical Physics in Theoretical Chemistry
Book Preface
Theoretical chemistry provides a systematic account of the laws governing chemical phenomena in matter. It applies physics and mathematics to describe the structure and interaction of atoms and molecules, the fundamental units of matter. Through the end of the 19th century, chemistry remained predominantly a descriptive and empirical science.1 True, there had been developed by then a consistent quantitative foundation based on the notions of atomic and molecular weights, combining proportions, thermodynamic quantities, and the fundamental ideas of molecular stereochemistry. Chemistry was certainly far more rational than its ancient roots in alchemy but was still largely a collection of empirical facts about the behavior of matter. Immanuel Kant, in his Critique of Pure Reason, claimed that “in any special doctrine of nature there can be only as much proper science as there is mathematics therein.”2 This can serve as our philosophical rationalization for emphasizing mathematical methods (speciﬁcally the ﬁeld designated mathematical physics) in theoretical chemistry.
The developments of physics in the 20th century made all of chemistry explicable, in principle, by quantum mechanics. As summarized by Dirac: “The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difﬁculty is only that the exact application of these laws leads to equations much too complicated to be soluble” [2]. By its very nature, quantum mechanics is mathematical physics and thereby we establish the connection which is the theme of this volume. However, the loophole noted by Dirac, the existence of chemical problems too mathematically complex to be solved exactly, justiﬁes the survival of parts of chemistry as an empirical science. In this category are semiempirical concepts of chemical bonding and reactivity. This has also led to computational models promoting rational drug design. These have also stimulated applications of other branches of mathematics, for example, information theory and graph theory applied to the deﬁnition of various chemical indices.
The primary objective of theoretical chemistry is to provide a coherent account for the structure and properties of atomic and molecular systems. Techniques adapted from mathematics and theoretical physics are applied in attempts to explain and correlate the structures and dynamics of chemical systems. In view of the immense complexity of chemical systems, theoretical chemistry, in contrast to theoretical physics, generally uses more approximate mathematical techniques, often supplemented by empirical or semiempirical methods.
This volume begins with an introduction to the quantum theory for atoms and small molecules, expanding upon the original applications of mathematical physics in chemistry. This ﬁeld is now largely subsumed within a subdiscipline known as computational chemistry. Chapter 1 begins with an introduction to the HartreeFock method, which is the conceptual foundation for computational chemistry. Chapter 2 discusses the basis functions employed in these computations, now largely dominated by Gaussian functions. Chapter 3 describes some postHartreeFock methods, which seek to attain “chemical accuracy” in atomic and molecular computations, in particular, conﬁguration interaction, manybody perturbation theory, and coupledcluster theory. Chapter 10 discusses diagrammatic techniques borrowed from theoretical physics, which can enhance the efﬁciency of computations. Chapter 7 is an account of the development of personal computers and their applications to computational chemistry.
For larger molecules and condensed matter, alternative approaches, including density functional theory (Chapter 4) and quantum MonteCarlo (Chapter 6), are becoming popular computational methods. Some additional topics covered in this volume are vibrational partition functions (Chapter 5), singularity analysis of perturbation theories (Chapter 9), and chemical applications of graph theory (Chapter 8).
Finally, Chapter 11 introduces the principles of the quantum computer, which has the speculative possibility of exponential enhancement of computational power for theoretical chemistry, as well as many other applications.
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