3: An Introduction to Group Theory
- Page ID
- 419498
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Many problems in chemistry can be simplified based on the symmetry of molecules and/or the symmetries of atomic and molecular orbitals. Since this course will deal mostly in the mathematical models used to describe molecular motions (rotations and vibration) and the orbitals needed to describe the electronic structure of atoms and molecules, some introduction to the mathematics of symmetry is useful. The concepts discussed in this chapter will be used through the text to demonstrate how symmetry can be used to simplify the descriptions of atomic and molecular behavior.
- 3.1: Overview
- Group Theory is the mathematical theory associated with the mathematical properties of groups. In chemistry, group theory is the mathematics of symmetry.
- 3.2: Group Theory in Chemistry
- In Chemistry, group theory is useful in understanding the ramifications of symmetry within chemical bonding, quantum mechanics and spectroscopy.
- 3.3: Determining the Point Group for a Molecule- the Schoenflies notation
- The first step in determining the point group for a molecule is to determine the structure of the molecule. Once this is done, identify all of the symmetry elements the molecular structure possesses.
- 3.4: Multiplication Operation for Symmetry Elements
- Multiplication is fairly simple when it comes to symmetry operations. One simply applies the operations from right to left. Going back to the tennis racket example, it is fairly simple to visualize each symmetry element.
- 3.5: More Definitions- Order and Class
- An important definition is the order of a group.
- 3.6: Representations
- A representation is any mathematical construct that will reproduce the group multiplication table. In general, there are an infinite number of representations possible for a given group, however, most of them will be related through simple relationships, and thus can be constructed from (or reduced to) other representations.
- 3.7: The "Great Orthogonality Theorem"
- One thing that is important about irreducible representations is that they are orthogonal. This is the property that makes group theory so very useful in chemistry, because orthogonality makes integrals zero. It’s always easier to do the integrals when orthogonality tells us the result will be zero before doing any complicated math!
- 3.8: Character and Character Tables
- Most summaries of group theory do not give the full matrix specifications for each irreducible representation in each important point group. Rather, a very useful quantity is defined, called the character.
- 3.9: Direct Products
- The intensity of a transition in the spectrum of a molecule is proportional to the magnitude squared of the transition moment matrix element.