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
- This page explores the role of group theory in chemistry to analyze molecular properties through symmetry operations, simplifying calculations for molecular orbitals, vibrations, and rotations. It employs the analogy of a Rubik's cube to illustrate symmetry.
- 12.2: Symmetry Elements and Operations Define the Point Groups
- This page discusses symmetry operations and elements in 3D space, including identity, rotation, reflection, inversion, and improper rotation, which help characterize molecular symmetry. It explains how to define molecular coordinate systems and classify molecules into point groups using notation systems like Schönflies and Hermann-Mauguin. The text relates symmetry to physical properties, such as chirality and dipole moments, showing how certain symmetries influence these characteristics.
- 12.3: Symmetry Operations Define Groups
- This page defines a mathematical group in relation to symmetry operations in chemistry, outlining criteria such as identity, closure, associativity, and reciprocality. It contrasts non-Abelian groups (non-commuting elements) with Abelian groups (commuting elements) and highlights the representation of symmetry operations by matrices for easier calculations.
- 12.4: Symmetry Operations as Matrices
- This page discusses how transformation matrices represent symmetry operations within groups, focusing on the \(C_{2v}\) point group and its key operations like identity, reflection, and n-fold rotation. It emphasizes that combining two elements must result in another group element, satisfying properties such as associativity and the existence of inverses.
- 12.5: Molecules can be Represented by Reducible Representations
- This page explores matrix representation for defining molecular properties, focusing on the symmetry of valence \(s\) orbitals in ammonia (\(\sf NH_3\)), classified under the \(C_{3v}\) point group. It details the selection of a suitable basis set and analyzes symmetry operations through derived matrices, revealing a reducible representation. It also suggests that alternative basis sets can be utilized to investigate other properties, including p orbital representations and molecular motions.
- 12.6: Character Tables Summarize the Properties of a Point Group
- This page discusses matrix representations of point groups and their role in molecular symmetry, including similarity transforms and the formation of character tables. It highlights the invariance of characters and includes the character table for the \(C_{3v}\) group.
- 12.7: Characters of Irreducible Representations
- This page discusses irreducible representations in group theory, focusing on their connection to molecular symmetry. It classifies various irreducible representations (A, B, E, T, etc.) by their symmetry traits and the transformations under symmetry operations. The complexities of irreducible representations in infinite groups are addressed, along with methods for their representation.
- 12.8: Using Symmetry to Solve Secular Determinants
- This page explores the conditions for zero integrals using group theory to assess symmetry, establishing that integrals remain invariant under symmetry operations. It examines secular equations related to molecular orbital energies and coefficients. Additionally, the text details the construction of molecular orbitals from atomic orbitals, emphasizing the use of symmetry-adapted linear combinations (SALCs) for bonding and antibonding orbitals, specifically in the context of \(NH_3\).
- 12.10: Molecular Motions of a Molecule can be Represented by a Reducible Reperesentation
- This page explains how to determine the normal vibrational modes of polyatomic molecules, specifically using group theory with \(H_2O\) as an example. It covers the \(3N\) Cartesian basis for molecular motions, focusing on the \(9\) elements of \(H_2O\).
- 12.11: Reducible Representations are Comprised of Irreducible Representations
- This page discusses the categorization of ammonia's motion using its \(C_{3v}\) reducible representation. By applying character tables, the irreducible representations are determined as \(3 A_1\), \(1 A_2\), and \(4 E\). It identifies translational modes as \(A_1 + E\) and rotational modes as \(A_2 + E\). The normal vibrational modes, after subtracting translational and rotational representations, are found to be \(2 A_1 + 2 E\).
- 12.12: Normal Modes of Vibrations Describe how Molecules Vibrate
- This page explains normal modes of vibration in molecules, highlighting diatomic and polyatomic structures. Diatomic molecules possess one vibrational mode, whereas polyatomic molecules display complex vibrations—stretching, bending, torsion—depending on their degrees of freedom. Non-linear molecules have \(3N - 6\) modes, while linear ones have \(3N - 5\).
- 12.13: Symmetry of Vibrations Describes their Spectroscopic Behavior
- This page summarizes vibrational transitions in molecules during infrared light absorption or emission, detailing the potential energy surface and the role of transition moment integrals and symmetry rules. It highlights infrared spectroscopy for detecting vibrational changes via dipole moments and introduces Raman scattering as a method for exploring hidden vibrational modes.
- 12.14: Symmetry Adapted Linear Combinations are the Sum over all Basis functions
- This page discusses the construction and application of Symmetry Adapted Linear Combinations (SALCs) for analyzing molecular vibrations and symmetries, particularly for water (H2O) in the C2V point group. It outlines two methods for constructing SALCs, emphasizes the role of group theory, and details the use of SALCs in spectroscopy and molecular geometry comparisons.
- 12.15: Molecular Orbitals can be Constructed on the Basis of Symmetry
- This page discusses molecular orbitals' formation in diatomic and polyatomic molecules through linear combinations of atomic orbitals (LCAO) and symmetry principles. It covers matrix representation construction for symmetry operations and provides specific steps for deriving symmetry-adapted linear combinations (SALCs), including those for water and ammonia.
- 12.E: Group Theory - The Exploitation of Symmetry (Exercises)
- This page covers homework exercises from McQuarrie and Simon's "Physical Chemistry," focusing on symmetry and molecular orbital theory. It addresses wave function normalization, Huckel determinant calculations, and eigenvalues related to molecular symmetry in groups like \(C_{2v}\) and \(C_{3v}\). Character tables for symmetry point groups \(C_i\), \(T_d\), and \(O_h\) are discussed, alongside the derivation of symmetry orbitals and irreducible representations.
- 12.T: Character Tables
- This page discusses classification groups in symmetry, including Nonaxial Groups (e.g., \(C_1\), \(C_s\), \(C_i\)) and Cyclic \(C_n\) Groups (e.g., \(C_2\) to \(C_7\)). It outlines molecular point groups \(C_{2h}\), \(C_{3h}\), \(D_{nh}\), and antiprismatic \(D_{nd}\) groups, detailing symmetry operations, character tables, and irreducible representations (e.g., \(A_g\), \(B_g\)).
- 12.T: Correlation Tables
- This page outlines molecular symmetry groups and their irreducible representations, covering point groups like \(C_{2v}\) to \(D_{\infty h}\). It details each group's symmetry operations and characterizes transformations, labeling irreducible representations such as \(A_1\), \(B_1\), and \(E\). This classification is crucial for comprehending the transformation of molecular states and their implications for molecular structures and behavior in chemistry.