Molecular Symmetry and Group Theory Vocabulary

Vocabulary practice cards covering fundamental concepts of symmetry elements, symmetry operations, mathematical group theory rules, and representation theory.

Positive Rotation Direction

In symmetry operations, rotations considered positive are performed in the counter-clockwise direction.

Principal Axis

The highest-order (highest fold) rotation axis present in a molecule.

Subsidiary Axis of Symmetry

A lower-fold or lower-order rotation axis in a molecule relative to the principal axis.

Improper Axis of Symmetry ($S_n$)

A symmetry element comprising a proper rotation operation around an axis followed by reflection through a plane perpendicular to that axis.

Center of Symmetry ($i$)

A point within a molecule located at its center of mass through which every atom can be inverted to an equivalent position.

Identity Operation ($E$)

A symmetry operation that results in no change to the molecule; it is characteristic of every molecule and included for mathematical completeness.

Mathematical Group

A collection or set of elements (such as numbers, matrices, vectors, or symmetry operations) that obeys four postulates: closure, existence of identity, associativity, and existence of reciprocals.

Abelian Group

A mathematical group in which all elements commute with each other ($A \times B = B \times A$).

Non-Abelian Group

A mathematical group in which elements do not all commute with each other ($A \times B \neq B \times A$).

Generator of Symmetry Operations

The minimum number of symmetry operations combined with each other to yield the complete set of products in a point group.

Non-Axial Point Group

A category of point group in which no axis of symmetry is present (e.g., $C_1$, $C_s$, $C_i$).

Axial Point Group

A category of point group that contains a basic rotational or improper rotational generator along with additional symmetry generators.

Order of Group ($h$)

The total number of symmetry operations belonging to a point group.

Vertical Plane of Symmetry ($\text{σ}_v$)

A plane of symmetry containing the principal axis of symmetry.

Horizontal Plane of Symmetry ($\text{σ}_h$)

A plane of symmetry perpendicular to the principal axis of symmetry.

Dihedral Plane of Symmetry ($\text{σ}_d$)

A plane of symmetry containing the principal axis of symmetry and bisecting the angle between two $C_2$ axes.

Representation of Group

A set of matrices corresponding to single operations in a group that combine among themselves in a manner parallel to how the group elements combine.

Reducible Representation

A representation of higher dimension that can be reduced to representations of lower dimension.

Irreducible Representation

A representation of lower dimension that cannot be further reduced.

First Postulate of GOT

The Great Orthogonality Theorem postulate stating that the number of irreducible representations in a group is always equal to the number of classes.

Second Postulate of GOT

The Great Orthogonality Theorem postulate stating that the sum of the squares of the dimensions of all irreducible representations in a group equals the order of the group ($h$).

Third Postulate of GOT

The Great Orthogonality Theorem postulate stating that the sum of the squares of the characters of an irreducible representation equals the order of the group ($h$).

Fourth Postulate of GOT

The Great Orthogonality Theorem postulate stating that the characters of any two distinct irreducible representations in the same group are orthogonal to each other.