12: Group Theory - The Exploitation of Symmetry
- Page ID
- 11807
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.
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(rating)- 12.4: Symmetry Operations as Matrices
- 12.2: Symmetry Elements and Operations Define the Point Groups
- 12.3: Symmetry Operations Define Groups
- 12.6: Character Tables Summarize the Properties of a Point Group
- 12.7: Characters of Irreducible Representations
- 12.9: Generating Operators
- 12.5: The \(C_{3v}\) Point Group
- 12.8: Using Symmetry to Solve Secular Determinants
- 12.E: Group Theory - The Exploitation of Symmetry (Exercises)
- 12.1: The Exploitation of Symmetry Can Help Simplify Numerical Calculations
- 12.12: Normal Modes of Vibrations Describe how Molecules Vibrate
- 12.T: Character Tables
- 12.11: Reducible Representations are Comprised of Irreducible Representations
- 12.T: Correlation Tables
- 12.10 Molecular Motions of a Molecule can be Represented by a Reducible Reperesentation
Recently updated
(date updated)- 12.T: Character Tables
- 12.10 Molecular Motions of a Molecule can be Represented by a Reducible Reperesentation
- 12.12: Normal Modes of Vibrations Describe how Molecules Vibrate
- 12.13: Symmetry of Vibrations Describes their Spectroscopic Behavior
- 12.6: Character Tables Summarize the Properties of a Point Group
- 12.1: The Exploitation of Symmetry Can Help Simplify Numerical Calculations
- 12.15: Molecular Orbitals can be Constructed on the Basis of Symmetry
- 12.14: Symmetry Adapted Linear Combinations are the Sum over all Basis functions
- 12.11: Reducible Representations are Comprised of Irreducible Representations
- 12.5: The \(C_{3v}\) Point Group
- 12.T: Correlation Tables
- 12.7: Characters of Irreducible Representations
- 12.4: Symmetry Operations as Matrices
- 12.2: Symmetry Elements and Operations Define the Point Groups
- 12.3: Symmetry Operations Define Groups
Recently added
(date created)- 12.13: Symmetry of Vibrations Describes their Spectroscopic Behavior
- 12.15: Molecular Orbitals can be Constructed on the Basis of Symmetry
- 12.14: Symmetry Adapted Linear Combinations are the Sum over all Basis functions
- 12.12: Normal Modes of Vibrations Describe how Molecules Vibrate
- 12.10 Molecular Motions of a Molecule can be Represented by a Reducible Reperesentation
- 12.T: Correlation Tables
- 12.T: Character Tables
- 12.11: Reducible Representations are Comprised of Irreducible Representations
- 12.9: Generating Operators
- 12.8: Using Symmetry to Solve Secular Determinants
- 12.7: Characters of Irreducible Representations
- 12.6: Character Tables Summarize the Properties of a Point Group
- 12.5: The \(C_{3v}\) Point Group
- 12.4: Symmetry Operations as Matrices
- 12.3: Symmetry Operations Define Groups