Group theory elementary applications

That chemistry and physics are brought together by mathematics is the raison d etre" of tbe present volume. The first three chapters are essentially a review of elementary calculus. After that there are three chapters devoted to differential equations and vector analysis. The remainder of die book is at a somewhat higher level. It is a presentation of group theory and some applications, approximation methods in quantum chemistry, integral transforms and numerical methods. 

Flurry, R. L. Jr., Symmetry Groups, Prentice-Hall, Englewood Cliffs, New Jersey, 1980. An excellent introduction to chemical applications of group theory. Our text assumes famil-arity with only the most elementary group theoretical ideas and notations. 

The study of molecular vibrations will be introduced by a consideration of the elementary dynamical principles applying to the treatment of small vibrations. In order that attention may be focused on the dynamical principles rather than on the technique of their application, this chapter vill employ only relativelj familiar and straightforward mathematical methods, and the illustrations will be simple. This will serve adequately as an introduction to the applications of quantum mechanics and group theory to the problem of molecular vibrations. Since, how-ever, these straightforward methods become cumbersome and impractical, even for simple molecules, equivalent but more powerful techniques u.sing matrix and vector notations will be discussed in Chap. 4. 

A brief introduction to group theory was presented, along with two elementary applications Determining whether a molecule could be polar and whether it could be optically active. 

In the next section, the principal ingredients involved in the ELF will be explained, and their relation with chemical concepts will be clarified. Then, a brief comparison of the ELF with other theoretical related tools, like the atoms in molecules model of Bader, will be done. Next, some elementary concepts from the mathematical theory of topological analysis will be in a rather crude way presented. After that, some applications, extensions and results will be discussed, focusing in particular on applications developed at our group. 

In the first, geometrical part, the crystal system, the crystal clasSf the translation group and the space group of the crystal under investigation are determined by systematic application of crystal structure theory in conjunction with tables. In addition the volume of the elementary cell, i.e. the smallest unit from which the whole crystal can be built up merely by parallel displacement, is established by calculation of its axes and angles. 

The static theory of atomic forces, which has been considered almost exclusively up to now with the methods of quantum mechanics, needs the addition of a dynamics of a chemical reaction. The collision methods, used by several groups, do not appear to be a useful method for the treatment of chemical processes (Section 1). In what follows, a dynamical theory for the simplest cases of bimolecular reactions is developed (Sections 2 and 3), in which the problem of chemistry is expressed immediately, and clearly discussed with the help of elementary examples (Section 4). Sections 5 and 6 contain a perturbation approach, whic converges for small and large collision velocities and allows for a relatively simple sqiproximation method for the reaction velocity of non-adiabatic processes. On the other hand the theory contains a quantitative description of the connection to the adiabatic reaction process in the limit of low velocities or separated characteristics of the potential. In this way it yields a conditional justification for the application of potential theoretical representations to chemical processes and at the same time a fixation of the limits of such idealisations. 

When asked in the early 1930s by Max von Laue what group theoretical result derived so far was the most important, Wigner replied the explanation of the Laporte rule (the concept of parity) and the quantum theory of vector addition (angular momentum). Partly in recognition of the power of these new theoretical tools, Wigner would receive the 1963 Physics Noble prize for his contributions to the theory of the atomic nucleus and the elementary particles, particularly through the discovery and application of fundamental symmetry principles. ... 

---