Mathematics of chemistry with focus on particular symmetries

Question

A student of mine came with a somewhat unusual request:

My question about group theory and if you have a book you could recommend for some foundation: specifically chemical applications of group theory with emphases on 3-center bonding, symmetry-based selection rules for cyclization, and 3D lattices and their symmetries.

This is not a math major, but, the student is rather inquisitive as is evident from their desire to study underlying mathematics to better understand chemistry.

I've studied from group theory in physics books (Wu Ki Tung, Howard Georgi, Greiner) and I'd like to read Cornwell's trifecta sometime. But, the specific interest of mine falls more in the realm of quarks and less in the realm of atomic or molecular physics aka Chemistry. Perhaps someone here as a good recommendation ? Probably a book which assumes little more than linear algebra is all that would be appropriate given this student only has had the calculus sequence.

Answer 1

I would like to recommend

Chemical Applications of Group Theory

by F. Albert Cotton.

Description

Retains the easy-to-read format and informal flavor of the previous editions, and includes new material on the symmetric properties of extended arrays (crystals), projection operators, LCAO molecular orbitals, and electron counting rules. Also contains many new exercises and illustrations.

Table of Contents

PRINCIPLES.

Definitions and Theorems of Group Theory.

Molecular Symmetry and the Symmetry Groups.

Representations of Groups.

Group Theory and Quantum Mechanics.

Symmetry-Adapted Linear Combinations.

APPLICATIONS.

Molecular Orbital Theory and Its Applications in Organic Chemistry.

Molecular Orbital Theory for Inorganic and Organometallic Compounds.

Ligand Field Theory.

Molecular Vibrations.

Crystallographic Symmetry.

Answer 2

I am not exactly sure that the book I am going to recommend covers all the topics you mention. However, 1) I am extremely fond of this book and 2) it is claimed to be written (the first part) for quantum chemists:

The first part was originally written for quantum chemists. It describes the correspondence, due to Frobenius, between linear representations and charac­ ters. This is a fundamental result, of constant use in mathematics as well as in quantum chemistry or physics. I have tried to give proofs as elementary as possible, using only the definition of a group and the rudiments of linear algebra. The examples (Chapter 5) have been chosen from those useful to chemists. Blockquote

Which is more important, the book is written by a mathematician and is very very much different from the usual exposition of the group theory in organic chemistry textbooks.

So, here is my suggestion:

Linear Representations of Finite Groups by Jean-Pierre Serre.

Answer 3

If the student is a chemist for sure he had already heard about Cotton´s book. So, he would be asking to a mathematician and would get as response what every chemist would tell him. If the student really wants to know the foundation he should head to mathematics works progressively. Cotton's book is more about chemistry than about group theory. For more completeness I will recommend another book that is fun to read, albeit unfortunately less known among chemists. My supervisor once told me that he still believes that the best book to understand symmetry in chemistry is Symmetry by McWeeny. It covers both molecules and solids. It is not as applied as Cotton but not as abstract as Serre. The research area of their authors is reflected in their books: Cotton was an inorganic chemist, McWeeny is a mathematical physicist, and Serre a pure mathematician. Additionally, it has a very low price.