7.5: Point Groups

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Jul 26, 2020
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- 7.4: Successive Operations
- 7.6: Character Tables - An Introduction

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Point groups are used to describe molecular symmetries and are a condensed representation of the symmetry elements a molecule may posses. This includes both bond and orbital symmetry. Knowing molecular symmetry allows for a greater understanding of molecular structure and can help to predict many molecular properties.

Introduction

Point groups are a quick and easy way to gain knowledge of a molecule. They not only contain a molecule's symmetry elements, but also give rise to a character table, which is a complete set of irreducible representations for a point group. A molecule's point group can be determined by either elucidating each symmetry element contained in a molecule or by properly using the Schreiber chart (see below).

Point groups usually consist of (but are not limited to) the following elements:

E - The identity operator. This operation leaves a molecule completely unchanged and exists for mathematical purposes.
C_n - The C_n proper axis of rotation is a 360/n° rotation that when performed leaves a molecule the same. A proper rotation with the highest value of n is known as the major axis of rotation.
σ - The mirror plane. The mirror plane can be described as a plane which produces a reflection of part of a molecule that is unnoticeable and can be labeled as either σ_h , σ_v , σ_d.
i - The inversion center. A molecule has a center of inversion if, when inverted, the molecule is unchanged.

See the section on symmetry elements for a more thorough explanation of each.

Each point group is associated with a specific combination of symmetry elements

Each point group has it's own combination of symmetry elements. Listed below are some of the many point groups and their respective symmetry elements, according to category, followed by a representative example.

Non axial groups

C₁: E C₁: E, i

C_n groups

C₂: E, C₂ (notice the major axis of rotation is the point group) C₃: E, C₃, C₃²

H2O2

H₂O₂ C₂

D_n groups

D₂: E C₂(z), C₂(y), C₂(x) D₃: E, 2C₃, 3C₂

C_nv groups

C_2v: E, C₂, σ_v(xz), σ_v'(yz) C_3v: E, 2C₃, 3σ_v

H2O

H₂O C_2v

C_nh groups

C_2h: E, C₂, i, σ_h C_3h: E, C₃, C₃², σ_h, S₃, S₃³

B(CO)3

B(OH)₃

D_^nh groups

D_2h: E

C2H4

C₂H₄ D_2h

How to determine a molecules point group

A molecule's point group can be determined by calculating all the symmetry elements of a molecule and matching them to a respective point group. This process, however, is greatly simplified when the Schreiber chart is used:

References

Housecroft, Catherine E.; Sharpe, Alan G., Inorganic Chemistry, 3^rd Ed. Pearson Education Limited 2008, Chapter 4
Koster, George F.; Wheeler, Robert G.; Dimmock, John O., The Properties of Thirty-Two Point Groups, The MIT Press 1963
Bertolucci, Michael D.; Harris, Daniel C., Symmetry and Spectroscopy: An Introduction to Vibrational and Electronic Spectroscopy, Dover Publications 1989

Outside Links

Point groups in 3-D

Problems

Determine the point group of BH₃ by calculating all it's symmetry elements then use the chart and determine which method is faster.
Determine the point groups of BH₃ and NH₃. Why is there a difference?
What is the point group of PPh₃?
Determine the point groups of CO₂ and H₂O and then compare them.
Propose a molecule with no symmetry. What is it's point group?

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Introduction

Each point group is associated with a specific combination of symmetry elements

How to determine a molecules point group

References

Outside Links

Problems

Contributors and Attributions

Introduction

Each point group is associated with a specific combination of symmetry elements

How to determine a molecules point group

References

Outside Links

Problems

Contributors and Attributions

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