How Chemists Use Group Theory Created by Anne K. Bentley, Lewis & Clark College and posted on VIPEr ( on March 26, Copyright Anne K. Bentley This work is licensed under the Creative Commons Attribution Non- commercial Share Alike License. To view a copy of this license visit

Why do chemists care about symmetry? It allows the prediction of chirality IR and Raman spectroscopy bonding

Which objects share the same symmetry as a water molecule?

How can we “quantify” symmetry?

Symmetry can be described by symmetry operations and elements. rotation, C n reflection, σ inversion, i improper rotation, S n identity, E

Objects that share the same set of symmetry elements belong to the same point group. = C 2v (E, C 2, two σ v )

The operations in a group follow the requirements of a mathematical group. Closure Identity Associativity Reciprocality if AB = C, then C is also in the group

Closure Identity Associativity Reciprocality AE = EA = A The operations in a group follow the requirements of a mathematical group.

Closure Identity Associativity Reciprocality (AB)C = A(BC) The operations in a group follow the requirements of a mathematical group. The C 2v point group is an Abelian group – ie, all operations commute (AB = BA). Most point groups are not Abelian.

Closure Identity Associativity Reciprocality AA -1 = E The operations in a group follow the requirements of a mathematical group. In the C 2v point group, each operation is its own inverse.

Closure Identity Associativity Reciprocality The operations in a group follow the requirements of a mathematical group.

Each operation can be represented by a transformation matrix. = transformation matrix original coordinates new coordinates – – Which operation is represented by this transformation matrix? –x –y z

The transformation matrices also follow the rules of a group. – – C2C – σ yz = – σ xz

Irreducible representations can be generated for x, y, and z C 2v E C 2 σ v(xz) σ v(yz) x y z 1–

A complete set of irreducible representations for a given group is called its character table. C 2v E C 2 σ v(xz) σ v(yz) x y z 1– ?

A complete set of irreducible representations for a given group is called its character table. C 2v E C 2 σ v(xz) σ v(yz) x y z 1– xy

More complicated molecules… ammonia, NH 3 C 3v methane, CH 4 T d

Applications of group theory IR spectroscopy Molecular orbital theory

Gases in Earth’s atmosphere nitrogen (N 2 )78% oxygen (O 2 ) 21% argon (Ar)0.93% carbon dioxide (CO 2 ) 400 ppm (0.04%)

carbon dioxide stretching modes not IR active IR active

Are the stretching modes of methane IR active? T d E 8C 3 3C 2 6S 4 6 σ d Γ Γ = A 1 + T 2

methane’s A 1 vibrational mode not IR active

methane T 2 stretching vibrations all at the same energy T 2 irreducible rep transforms as (x, y, z) together, they lead to one IR band

Molecular Orbital Theory How and why does something like this form?

Bonding Basics Atoms have electrons Electrons are found in orbitals, the shapes of which are determined by wavefunctions

Bonding Basics A bond forms between two atoms when their electron orbitals combine to form one mutual orbital. += + =

Bonding is Determined by Symmetry += += no bond forms bond forms + =

Use group theory to assign symmetries and predict bonding. SF 6 A 1g T 1u (and two more) T 2g EgEg and

Outer atoms are treated as a group. A 1g T 1u EgEg

Which types of bonds will form? central sulfur six fluorine A 1g A 1g T 1u T 1u T 2g E g E g

Concluding Thoughts

Recommended Resources Cotton, F. Albert. Chemical Applications of Group Theory, Wiley: New York, Carter, Robert L. Molecular Symmetry and Group Theory, Wiley: Harris, Daniel C. and Bertolucci, Michael D. Symmetry and Spectroscopy, Dover Publications: New York, Vincent, Alan, Molecular Symmetry and Group Theory, Wiley: New York,

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