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

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:

I. Presence or Absence of Special Groups:

II. Presence or Absence of Proper Rotation Axes:

III. Presence or Absence of Perpendicular C₂ Axes:

IV. Presence or Absence of Vertical or Horizontal Mirror Planes:

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

Is the molecule linear? If yes, it belongs to either

C_¥v : no i, sh or ^C₂ (unsymmetrical)

D¥d: i, s_h and ^C₂ present (symmetrical)

Are there multiple high order axes present? If yes, there are three cases :

6 C₅ axes: I_h (only need to find 2)

6 C₄ axes: O_h (assumes no C₅ axes)

4 C₃ axes: T_d (assumes no C₄ or C₅ axes present)

Does the molecule have a rotation axis? If no, it belongs to one of the low-symmetry groups:

C_i: possesses only an inversion center

C_s: possesses only a mirror plane (s only)

C₁: has no symmetry

If a principle axis is present, the point group will designated either C or D according to what additional symmetry elements are present

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Is there a C₂ axis perpendicular to the principle axis?

Yes, then the molecule belongs to one of the dihedral "D" point groups
D_nh, D_nd, or D_n based on the presence of any additional mirror symmetry

No: then the molecule belongs to one of the single axis "C" point groups
C_nh, C_nv, or C_n based on the presence of any additional mirror symmetry

This is perhaps one of the most difficult questions, yet most important questions when assigning point group symmetry to a molecule/object. Note that the C₂ must pass through the center of the molecule

One method of identifying the possibility of having a ^C₂ axis, is to check to see if the "top" and "bottom" halves of the molecule (along the C_n axis) are identical. For a ^C₂ axis to be present, they must be the same.

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The final series of questions involves the presence and location of any mirror planes within the molecule

Does the molecule possess a horizontal mirror plane (s_h, ^ to C_n)?

If yes, then the point group is C_nh or D_nh

If no, then check for vertical mirrors. These planes will contain the principle axis. If the vertical planes are present the point group is C_nv or D_nd

If no mirror planes are present the point group is a pure rotational group and has the symbol C_n or D_n

Note that if a s_h is present, you don't need to check for vertical mirror planes beause the sh takes precedence.
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Typical Schönflies point group symbols: C_2h, C_3v, D_3h, D_4d, C_s, T_d

• The subscript number represents the order of the principle rotation axis
• The letter D or C indicates the presence (D_n) or absence (C_n) of a C₂ axis ^ to the major axis
• The subscript letters h, v, or d, correspond to the presence of a horizontal mirror plane (h), or vertical mirror planes (v for C and d for D). (h takes precedence over v or d)
• Groups with multiple high-order axes have special letter symbols corresponding to T for tetrahedral (T_d), O for octahedral (O_h), and I for Icosohedral (I_h)
• Groups without a principle rotation axis adopt the letter C followed by a subscript; s for mirror only (Cs), i inversion only (C_i), and 1 for no symmetry at all (C₁)

S2n is reserved for groups with only an Sn axis present.

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