Inorganic Chemistry

1. Point Group
- 1.1. Schönflies Notation
  - 1.1.1. Symmetry Operations
  - 1.1.2. Point Groups
- 1.2. Hermann-Mauguin Notation
  - 1.2.1. Symmetry Operations
  - 1.2.2. Point Groups
2. Solid State Chemistry
3. See Also
4. References

1. Point Group

1.1. Schönflies Notation

1.1.1. Symmetry Operations

Cₙ Rotation: n-fold rotation
σ Reflection
- σₕ Horizontal Reflection: reflection with respect to the plane perpendicular to the principal axis.
- σᵥ Vertical Reflection: reflection with respect to the plane parallel to the principal axis.
i Inversion
S₂ₙ Improper Rotation: Rotation about an axis and reflection along the axis.
- From German Spiegel 'mirror'

1.1.2. Point Groups

\( C_1 \) Only identity
\( C_i \) Identity and inversion
\( C_s \) Identity and improper rotation
\( C_{\infty v} \) Linear along an axis and vertical reflections
\( D_{\infty h} \) Linear along an axis and horizontal reflections
\( C_{n} \) (Cyclic) Only rotations along a single axis are possible
\( C_{nv} \) (Pyramidal) Rotations and vertical reflections
\( C_{nh} \) (Reflection) Rotations and horizontal reflection
\( D_n \) (Dihedral) Rotations and 2n C₂ rotations perpendicular to principal axis.
\( D_{nh} \) (Prismatic) Dihedral and horizontal reflection
\( D_{nd} \) (Antiprismatic) Dihedral and dihedral (vertical) reflections
\( S_n \) Dihedral and improper rotations
\( T, T_h, T_d \) (Tetrahedral)
\( O, O_h \) (Octahedral)
\( I, I_h \) (Icosahedral)

1.2. Hermann-Mauguin Notation

International Notation
- It is the standard in International Tables For Crystallography.

1.2.1. Symmetry Operations

n Rotation: n-fold rotation
m Reflection
- n/m the mirror plane is perpendicular to the n-fold rotation axis,
n̄ Rotoinversion: rotation by 2π/n and inversion
- 1̄ is the pure inversion

1.2.2. Point Groups

The symmetries in primary, secondary, tertiary directions are listed in order. The secondary directions are the all equivalent directions perpendicular to the primary direction, and tertiary directions are all equivalent directions between secondary directions. Note that the tertiary directions only occur when the rotational symmetry around the primary direction is even.

When the rotational symmetry is more than 10-fold, paranthesis is used to make it clear that it is not rotational symmetries along two directions.

Translation from Schönflies notation

\( n \): \( C_n \)
\( nm, nmm \): \( C_{nv} \). one \( m \) if \( n \) is odd, two \( m \) if \( n \) is even.
\( \tfrac{n}{m} \): \( C_{nh} \)
\( n2, n22 \): \( D_n \)
\( \bar{n}\tfrac{2}{m} \): \( D_{nd} \)
\( \tfrac{n}{m}\tfrac{2}{m}\tfrac{2}{m} \): \( D_{nh} \)