Symmetry and 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 plane - called a symmetry element.

There are 5 kinds of operations

1. 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 symmetry.

XeF4 is square planar.

It has four DIFFERENT C2 axes

It also has a C4 axis coming out of the page called the principle axis because it has the largest n.

By convention, the principle axis is in the z-direction

 

3. 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 atom.

 

5. Improper Rotations: Sn

n-fold rotation followed by reflection through mirror plane perpendicular to rotation axis

Note: n is always 3 or larger because S1 = s and S2 = i.

These are different, therefore this molecule

does not posses a C3 symmetry axis.

This molecule posses the following symmetry elements: C3, 3 sd, i, 3 ^ C2, S6. There is no C3 or sh.

Eclipsed ethane posses the following symmetry elements: C3, 3 sv, 3 ^ C2, S3, sh. There is no S6 or i.

Compiling all the symmetry elements for staggered ethane yields a Symmetry Group called D3d.

Compiling all the symmetry elements for eclipsed ethane yields a Symmetry Group called D3h.

Symmetry group designations will be discussed in detail shortly

 

To be a group several conditions must be met:

1. Any result of two or more operations must produce the same result as application of one operation within the group.

i.e., the group multiplication table must be closed

Consider H2O which has E, C2 and 2 sv's.

i.e.,

of course

etc…

The group multiplication table obtained is therefore:

 

E

C2

sv

s'v

E

E

C2

sv

s'v

C2

C2

E

s'v

sv

sv

sv

s'v

E

C2

s'v

s'v

sv

C2

E

Note: the table is closed, i.e., the results of two operations is an operation in the group.

 

2. Must have an identity ()

 

3. All elements must have an inverse

i.e., for a given operation () there must exist an operation () such that

 

Classification of the Symmetry of Molecules

Certain symmetry operations can be present simultaneously, while others cannot.

There are certain combinations of symmetry operations which can occur together.

Symmetry Groups combine symmetry operations that can occur together.

Symmetry groups contain elements and there mathematical operations.

For example, one of the symmetry element of H2O is a C2-axis. The corresponding operation is rotation of the molecule by 180 about an axis.

 

Point Groups

Low Symmetry Groups

C1: only E

Cs: E and s only

Ci: E and i only

  • Cn, Cnv, Cnh Groups

    Cn: E and Cn only

    C2:

    C3:

     

    Cnv: E and Cn and n sv's

    C2v: E, C2, 2 sv H2O

    C3v: E, C3, 3 sv NH3

    C v: E, C , sv HF, HCN

    Cnh: E and Cn and sh (and others as well)

    C2h: E, C2, sh, i

    Dn, Dnv, Dnh Groups

    Dn: E, Cn, n C2 axes ^ to Cn

    D3: E, C3, 3 ^ C2

    [Co(en)3]3+

    Dnh: E, Cn, n C2 axes ^ to Cn, sh

    D3h: E, C3, 3 ^ C2, sh

  • D3h: E, C3, 3 ^ C2, sh

  • eclipsed ethane

    D6h: E, C6, 6 ^ C2, sh

  • D h: E, C , ^ C2, sh

  • H2

     

    Dnd: E, Cn, n C2 axes ^ to Cn,

  • D3d: E, C3, 3 ^ C2, 3 sd

  • staggered ethane

    Sn Group

    S2n: E, Cn, S2n (no mirror planes)

    S4, S6, S8, etc. (Note: never S3, S5, etc.)

    S4: E, C2, S4

    High Symmetry Cubic Groups, Td, Oh, Ih

    Td: E, 8 C3, 3 C2, 6 S4, 6 sd

    Tetrahedral structures

    No need to identify all the symmetry elements - simply recognize Td shape.

    methane, CH4

    Oh: E, 8 C3, 6 C2, 6 C4, i, 6 S4, 8 S6, 3 sh, 6 sd

    Octahedral structures

    No need to identify all the symmetry elements - simply recognize Oh shape.

    Ih: E, 12 C5, 20 C3, 15 C2, i, 12 S10, 20 S6, 15 s

    Icosahedron

    Other rare high symmetry groups are T, Th, O, and I