3.2: Group Theory in Chemistry
- Page ID
- 420484
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)In Chemistry, group theory is useful in understanding the ramifications of symmetry within chemical bonding, quantum mechanics and spectroscopy. The group elements we are concerned with are symmetry operations.
| Symbol | Operation | Description | Element | Mathematical example |
|---|---|---|---|---|
| E | identity | This is the “don’t do anything to it” operation | E. | \(E (x,y,z) = (x,y,z)\) |
| \(C_{n}\) | Proper rotation | This is an operation in which the object is rotated about an axis by an angle of \(\frac{2\pi }{n}\) radians. The axis will be referred to as the “\(C_{n}\) axis”. | \(C_{n}\). The axis with the largest value of n is designated the “principle rotation axis” and the z-axis is always assigned as lying along the principle rotation axis. |
\(C_{4}(x,y,z) = (y,-x,z)\) \(C_{2}(x,y,z) = (-x,-y,z)\) Etc. |
| \(\sigma\) | Reflection plane | This operation involves reflection of the object through a mirror plane. | \(\sigma_{v}\), \(\sigma_{d}\) or \(\sigma_{h}\). \(\sigma_{v}\) and \(\sigma_{d}\) contain the principle rotation axis, whereas \(\sigma_{h}\) planes are perpendicular to the principle rotation axis. |
\(\sigma_{v}(x,y,z) = (-x,y,z)\) (for reflection through the \(yz\) plane)\(\sigma_{h}(x,y,z) = (x,y,-z)\) \(\sigma_{d}(x,y,z) = (y,x,z)\) |
| \(i\) | Inversion center | This operation involves reflection trough a point. | i. The inversion center (if it exists) will always be located at the center of mass of a molecule. | \(i(x,y,z) = (-x,-y,-z)\) |
| \(S_{n}\) | Improper rotation | This operation involves a rotation through a \(C_{n}\) axis followed by reflection by a \(\sigma_{h}\) plane. | \(S_{n}\). |
A symmetry operation is a geometrical manipulation that leaves an object in a geometry that is indistinguishable from that which it had before the manipulation. There are five important types of symmetry operations with which we are concerned. Each type of operation has an associated symmetry element. Using standardized notation, these operations and elements can be summarized as follows.
A given molecule may have several of the above symmetry elements. The particular combination will define a group, and that group can be given a named based on the type of symmetry elements it contains. Further, all of the convenient wavefunctions that describe the vibrations, rotations and molecular orbitals of the molecule will be eigenfunctions of the symmetry elements, forcing some very useful mathematical properties upon the wavefunctions.
A tennis racquet has all of the same symmetry elements as a water molecule or a formaldehyde molecule. Let’s identify these symmetry elements and write out a group multiplication table for the group to which that particular set belongs.
The most obvious symmetry element is always the identity element (E). Every object possesses this symmetry element. Some objects are so asymmetrical that this is the only symmetry element they possess. Certainly, a tennis racquet possesses the symmetry element E.
The next most useful element to examine is the reflection plane. An object may or may not possess this type of symmetry. A tennis racquet has two vertical (\(\sigma_{v}\)) reflection planes. One is in the plane of the strings and the other is perpendicular to the face of the racquet. This happens often that an object has more than one of a given type of symmetry element. For our purposes, we will designate the plane that is perpendicular to the face of the racquet as \(\sigma_{v}\) and the one that is parallel to the face of the racquet as \(\sigma_{v}’\).
A tennis racquet possesses neither an inversion center (\(i\)) nor an improper rotation axis (\(S_{n}\)).
The set of symmetry elements that the object does possess (\(E\), \(C_{2}\), \(\sigma_{v}\) and \(\sigma_{v}'\)) define a group that goes by the label \(C_{2v}\). Any object that has these and only these symmetry elements is said to have \(C_{2v}\) symmetry. It is easy to demonstrate that the set of symmetry elements that define \(C_{2v}\) define a group.