12: Group Theory - The Exploitation of Symmetry
- Page ID
- 11807
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Symmetry can help resolve many chemistry problems and usually the first step is to determine the symmetry. If we know how to determine the symmetry of small molecules, we can determine symmetry of other targets which we are interested in. Usually, it is not only the symmetry of molecule but also the symmetries of some local atoms, molecular orbitals, rotations and vibrations of bonds, etc. that are important. For example, if the symmetries of molecular orbital wave functions are known, we can find out information about the binding. Also, by the selection rules that are associated with symmetries, we can explain whether the transition is forbidden or not and also we can predict and interpret the bands we can observe in Infrared or Raman spectrum. The qualitative properties of molecular orbitals can be obtained using symmetry from group theory (whereas their precise energetics and ordering have to be determined by a quantum chemical method). Group Theory is a branch of the mathematical field of algebra. In quantum chemistry, group theory can applied to ab initio or semi-empirical calculations to significantly reduce the computational cost. Symmetry operations and symmetry elements are two basic and important concepts in group theory. When we perform an operation to a molecule, if we cannot tell any difference before and after we do the operation, we call this operation a symmetry operation. This means that the molecule seems unchanged before and after a symmetry operation. As Cotton defines it in his book, when we do a symmetry operation to a molecule, every points of the molecule will be in an equivalent position.
- 12.1: The Exploitation of Symmetry Can Help Simplify Numerical Calculations
- Molecular symmetry can be used to simplify numerical calculations and provide qualitative information about vibrational modes, electronic structure, and more. The process of obtaining this information is based on group theory.
- 12.2: Symmetry Elements and Operations Define the Point Groups
- A symmetry operation is an action that leaves an object looking the same after it has been carried out. Each symmetry operation has a corresponding symmetry element, which is the axis, plane, line or point with respect to which the symmetry operation is carried out. The symmetry element consists of all the points that stay in the same place when the symmetry operation is performed. Point groups are defined by the total the symmetry operations that belong to that group.
- 12.3: Symmetry Operations Define Groups
- A mathematical group is defined as a set of elements (\(g_1\), \(g_2\), \(g_3\)...) together with a rule for forming combinations \(g_j\). The number of elements \(h\) is called the order of the group. For our purposes, the elements are the symmetry operations of a molecule and the rule for combining them is the sequential application of symmetry operations investigated in the previous section.
- 12.4: Symmetry Operations as Matrices
- We can carry out the various symmetry operations in group theory using transformation matrices. Transformation matrices are matrices that map one set of coordinates or functions onto another set.
- 12.6: Character Tables Summarize the Properties of a Point Group
- Now that we have learned how to create a matrix representation of a point group within a given basis, we will move on to look at some of the properties that make these representations so powerful in the treatment of molecular symmetry.
- 12.7: Characters of Irreducible Representations
- the character of a group representation is a function on the group that associates to each group element the trace of the corresponding matrix. The character carries the essential information about the representation in a more condensed form.
- 12.8: Using Symmetry to Solve Secular Determinants
- As we continue with this course, we will discover that there are many times when we would like to know whether a particular integral is necessarily zero, or whether there is a chance that it may be non-zero. We can often use group theory to differentiate these two cases. You will have already used symmetry properties of functions to determine whether or not a one-dimensional integral is zero. For example, cos(x) is an ‘even’ function (symmetric with respect to reflection through the origin),
- 12.10: Molecular Motions of a Molecule can be Represented by a Reducible Reperesentation
- Choosing a 3-D axis centered at each atom for our basis set, we can make a reducible representation of a molecule that represents the entire motion (rotational, vibrational, and translational) motion of a molecule. By choosing a specific basis set, we can represent nearly any symmetry of molecules.
- 12.11: Reducible Representations are Comprised of Irreducible Representations
- Reducible representations can be broken down into their irreducible representations using the tabular method. These represent the most fundamental symmetry properties of molecules. For molecular motion, these will represent how the molecule rotates, vibrates, and translates in space.
- 12.12: Normal Modes of Vibrations Describe how Molecules Vibrate
- Changing the bond length during a vibration in a polyatomic will affect the length of nearby bonds, so we cannot consider the vibrational motion of each bond in isolation. Instead, normal modes of vibration involve the concerted motion of groups of bonds. Group theory may be used to identify the symmetries of the vibrational modes of a molecule.
- 12.14: Symmetry Adapted Linear Combinations are the Sum over all Basis functions
- The construction of linear combinations of the basis of atomic movements allows the vibrations belonging to irreducible representations to be investigated. The wavefunction of these symmetry equivalent orbitals is referred to as Symmetry Adapted Linear Combinations, or SALCs.
- 12.E: Group Theory - The Exploitation of Symmetry (Exercises)
- These are homework exercises to accompany Chapter 12 of McQuarrie and Simon's "Physical Chemistry: A Molecular Approach" Textmap.
- 12.T: Character Tables
- Character tables represent all of the symmetry that a molecule has for a given point group. Character tables for the most common point groups are given.
- 12.T: Correlation Tables
- Correlation tables allow you to ascend or descend in symmetry between point groups. Commonly used correlation tables are given.