Symmetry Point Groups

Every object/molecule has a complete set of symmetry operations which together describe a molecules overall symmetry. This set of symmetry operations are refered to as a Point Group.

Schönflies notation is used to designate a molecules point group. (Crystallographers use a different notation called Hermann-Mauguin notation.)

Moleculer point groups fall into one of four general classes.

Non-rotational Groups (those with almost no symmetry)

Single Rotational Axis Groups (those with no C2 axes^ to the principle axis)

Dihedral Groups (those with C2 axes^ to the principle axis)

Cubic Groups (those with multiple high order axes, includes Octahedron, Tetrahedron and Icosohedron)

The symbols for these point groups are listed below with their associated operations
 
 
Category 
Symbol 
Symmetry Operations
Non-rotational
C
E (asymetric)
Cs 
E, sh
Ci 
E, i
 
Single-Axis 
Cn 
E, Cn, ....Cnn-1
(n = 2, 3, ......., ¥
Cnv 
E, Cn, ....Cnn-1, nsv (n/2 sv and n/2 sd if n even)
Cnh 
E, Cn, ....Cnn-1, sh
S2n 
E, S2n, ....S2n2n-1
C¥v 
E, C¥, ¥sv (non-centrosymetric linear)
Dihedral 
Dn 
E, Cn, ....Cnn-1, n ^ C2 
(n = 2, 3, ......., ¥
Dnd 
E, Cn, ....Cnn-1, S2n, ....S2n2n-1, n ^ C2, nsd 
Dnh 
E, Cn, ....Cnn-1, n ^ C2sh, nsv
D¥d 
E, C¥, S¥, ¥ ^ C2, ¥sv, sh, i (centrosymetric linear)
Cubic
T
E, 4C3, 4C32, 3C2, 3S4, 3S43, 6sd
O
E, 4C3, 4C32, 6C2, 3C4, 3C43, 3C2 (= C42), i, 3S4, 3S43, 4S6, 4S65, 3sh, 6sd
Ih 
E, 6C5, 6C52, 6C53, 6C54, 10C3, 10C32, 15C2, i, 6S10, 6S103, 6S107, 6S109, 10S6, 15s
 

 

 

4 Step Procedure for Determining Point Group Symmetry

Clearly from the long list of operations that may be present for each point group, the process of assigning groups by identifying each operation would be a time consuming task.

Fortunately, the process of determining a point group can be stream lined into a set of 4 fundemental questions conerning the presence or absence of key symmetry elements. These are as follows:

A flow chart can be used for systematically determining the symmetry of an object/molecule.

 

 

I. Special Groups

For these "cubic" groups there are lower symmetry subgroups (e.g. pure rotational ) of each type. These species are rarely encountered in a chemical context but are sometime useful in simplifying applications of group theory. Back

 

II. Principle Rotation Axis

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III. Perpendicular C2 axes

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IV. Horizontal and Vertical Mirror Planes

 

 

 

 

 

 

 

 

 

Schönflies Notation

Typical Schönflies point group symbols: C2h, C3v, D3h, D4d, Cs, Td

 

 

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Flow Chart for Systematic Determination of Point Group Symmetry


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