Symmetry Theory in Molecular Physics with Mathematica

This is a new kind of tutorial book. It is written using the logic engine Mathematica, which permits concrete exploration and development of every concept involved in Symmetry Theory. The book may be read in the hand or on a computer screen with Mathematica running behind it. It is intended for students of chemistry and molecular physics who need to know mathematical group theory and its applications, either for their own research or for understanding the language and concepts of their field. The book culminates with chapters that use permutaion groups to analyze flexible molecules, a topic which is on the frontier of current research and is not covered in any commonly adopted textbook. The author started working with Mathematica as soon as it appeared in 1988, bringing over ten years of near daily experience with Mathematica to this book.

Culminates with chapters that use permutation groups to analyze flexible molecules, a topic which is on the frontier of current research and is not covered in any commonly adopted textbook Makes use of modern methods of Mathematica to develop the subject of group theory as applied to molecular structure and to automate the complicated and tedious calculations involved Begins with careful definitions of symmetry and group, and then proceeds to an explicit proof that symmetry transforms always come in groups - the basic explanation of why group theory helps with the study of symmetry Includes supplementary material: sn.pub/extras

Autorentext
W.M. McClain started working with Mathematica as soon as it appeared in 1988, bringing over ten years of nearly daily experience with Mathematica to this book. He has written many research papers that use Mathematica and has also used group theory throughout his 20-year research career in nonlinear spectroscopy. He published the first group theoretic analysis of nonlinear tensors in vibronic sprectroscopy, regarded by many as a landmark paper.

Klappentext

Prof. McClain has indeed produced "a new kind of tutorial book." It is written using the logic engine Mathematica, which permits concrete exploration and development of every concept involved in Symmetry Theory. The book may be read in your hand, or on a computer screen with Mathematica running behind it. It is intended for students of chemistry and molecular physics who need to know mathematical group theory and its applications, either for their own research or for understanding the language and concepts of their field. The book has three major parts:

Part I begins with the most elementary symmetry concepts, showing how to express them in terms of matrices and permutations. These are then combined into mathematical groups. Many chemically important point groups are constructed and kept in a Mathematica package for easy reference. No other book gives such easy access to the groups themselves. The automated group construction machinery allows you to tabulate new groups that may be needed in research, such as permutation groups that describe flexible molecules.

In Part II, mathematical group theory is presented with motivating questions and experiments coming first, and theorems that answer those questions coming second. You learn to make representations of groups based on any set of symmetric objects, and then to make Mathematica operators that carry out rep construction as a single call. Automated construction of representations is offered by no other book. Part II follows a reconstructed trail of questions, clues and solid results that led Issai Schur to a complete proof of the Great Orthogonality.

In Part III, the projection operators that flow from the Great Orthogonality are automated and applied to chemical and spectroscopic problems, which are now seen to fall within a unified intellectual framework. The topics include chemical bonding in symmetric molecules, molecular vibrations and rigorous reasoning about quantum mechanical matrix elements. As a concrete example of the enormous power of the automated projectors, the tensor operators for two- and three- photon processes are projected under all tabulated groups. All the machinery presented is general, and will work with new groups that you may construct. Finally, there is machinery that accepts as input the multiplication table of any group and returns as output its character table. This will be of great use to spectroscopists who deal with flexible molecules belonging to permutation groups, which are too numerous even for a Mathematica package.

Inhalt
A tutorial on notebooks.- A basic tutorial.- The meaning of symmetry.- Axioms of group theory.- Several kinds of groups.- The fundamental theorem.- The multiplication table.- Molecules.- The point groups.- Euler rotation matrices.- Lie#x2019;s axis-angle rotations.- Recognizing matrices.- to the character table.- The operator MakeGroup.- Product groups.- Naming the point groups.- Tabulated representations of groups.- Visualizing groups.- Subgroups.- Lagrange#x2019;s Theorem.- Classes.- Symmetry and quantum mechanics.- Transformation of functions.- Matrix representations of groups.- Similar representations.- The MakeRep operators.- Reducible representations.- The MakeUnitary operator.- Schur#x2019;s reduction.- Schur#x2019;s First Lemma.- Schur#x2019;s Second Lemma.- The Great Orthogonality.- Character orthogonalities.- Reducible rep analysis.- The regular representation.- Projection operators.- Tabulated bases for representations.- Quantum matrix elements.- Constructing SALCs.- Hybridorbitals.- Vibration analysis.- Multiple symmetries.- One-photon selection rules.- Two-photon tensor projections.- Three-photon tensor projections.- Class sums and their products.- Make a character table.