Molecular Symmetry and Group Theory

Vocabulary flashcards covering the fundamental symmetry operations, types of mirror planes, chirality rules, and the mathematical principles of group theory.

Symmetry Operation

An action that leaves a molecule in a configuration indistinguishable from its original orientation, performed on symmetry elements such as points, lines, or planes.

Identity ($E$)

A "do nothing" operation present in every object, necessary for mathematical group consistency; mathematically represented as $(x, y, z) \rightarrow (x, y, z)$.

Proper Rotation ($C_n$)

A rotation by $360^{\circ}/n$ around an axis where the z-axis remains unchanged in the rotation matrix.

Principal Axis

The axis of rotation with the highest value of $n$, which is assigned to the z-axis.

Reflection ($\sigma$)

An operation through a mirror plane categorized by its relation to the principal axis; for $\sigma_h$, the effect is $(x, y, z) \rightarrow (x, y, -z)$.

Inversion ($i$)

An operation where every point is projected through a center point to an equal distance on the opposite side; mathematically $(x, y, z) \rightarrow (-x, -y, -z)$.

Improper Rotation ($S_n$)

A composite operation consisting of a proper rotation ($C_n$) followed by a reflection ($\sigma_h$) perpendicular to that axis.

Vertical Mirror Plane ($\sigma_v$)

A mirror plane that contains the principal axis.

Horizontal Mirror Plane ($\sigma_h$)

A mirror plane that is perpendicular to the principal axis.

Dihedral Mirror Plane ($\sigma_d$)

A vertical plane that bisects the angle between two $C_2$ axes.

Successive Improper Rotation Operations

For an $S_n$ axis, if $n$ is even, the axis generates $n$ operations; if $n$ is odd, the axis generates $2n$ operations.

$S_1$ and $S_2$ Identities

Special symmetry identities where $S_1 = \sigma$ and $S_2 = i$.

Achiral

A molecule that possesses any improper rotation axis ($S_n$), including mirror planes ($\sigma$) and inversion centers ($i$).

Mathematical Group Criteria

A set of symmetry operations must satisfy four criteria: 1. Identity ($E$ is present), 2. Closure ($PQ$ is in the set), 3. Associativity ($P(QR) = (PQ)R$), and 4. Inverses.

Abelian Group

A group where operations commute ($PQ = QP$); an example is $H_2O$ ($C_{2v}$).

Non-Abelian Group

A group where the order of operations matters ($PQ \neq QP$); an example is $NH_3$ ($C_{3v}$).