The Importance Of Group Theory

Cataloging the symmetry of molecules is very useful.

Group Theory is a mathematical method by which aspects of a molecules symmetry can be determined.

The symmetry of a molecule reveals information about its properties (i.e., structure, spectra, polarity, chirality, etc…)

Clearly, the symmetry of the linear molecule A-B-A is different from A-A-B.

In A-B-A the A-B bonds are equivalent, but in A-A-B they are not.

However, important aspects of the symmetry of H2O and CF2Cl2 are the same. This is not obvious without Group Theory.

Symmetry Operations/Elements

A molecule or object is said to possess a particular operation if that operation when applied leaves the molecule unchanged.

Each operation is performed relative to a point, line, or

Identity

2. n-Fold Rotations

3. Reflection

4. Inversion

5. Improper n-Fold Rotation

1. Identity is indicated as E does nothing, has no effect all molecules/objects possess the identity operation, i.e., posses E.

E has the same importance as the number 1 does in multiplication (E is needed in order to define inverses).

2. n-Fold Rotations: Cn, where n is an integer rotation by 360°/n about a particular axis defined as the n-fold rotation axis.

C2 = 180° rotation, C3 = 120° rotation, C4 = 90° rotation, C5 = 72° rotation, C6 = 60° rotation, etc.

Rotation of H2O about the axis shown by 180° (C2) gives the same molecule back.

Therefore H2O possess the C2 symmetry element.

However, rotation by 90° about the same axis does not give back the identical molecule

Therefore H2O does NOT possess a C4 symmetry axis.

BF3 posses a C3 rotation axis of symmetry.

(Both directions of rotation must be considered)

This triangle does not posses a C3 rotation axis of

Reflection: s (the symmetry element is called a mirror plane or plane of symmetry)

If reflection about a mirror plane gives the same molecule/object back than there is a plane of symmetry (s).

If plane contains the principle rotation axis (i.e., parallel), it is a vertical plane (sv)

If plane is perpendicular to the principle rotation axis, it is a horizontal plane (sh)

If plane is parallel to the principle rotation axis, but bisects angle between 2 C2 axes, it is a diagonal plane (sd)

H2O posses 2 sv mirror planes of symmetry because they are both parallel to the principle rotation axis (C2)

XeF4 has two planes of symmetry parallel to the principle rotation axis: sv

XeF4 has two planes of symmetry parallel to the principle rotation axis and bisecting the angle between 2 C2 axes : sd

XeF4 has one plane of symmetry perpendicular to the principle rotation axis: sh

4. Inversion: i (the element that corresponds to this operation is a center of symmetry or inversion center)

The operation is to move every atom in the molecule in a straight line through the inversion center to the opposite side of the molecule.

Therefore XeF4 posses an inversion center at the Xe