Symmetry, Group Theory and Applications

Vocabulary practice flashcards covering symmetry operations, elements, group theory definitions, character tables, and spectroscopic selection rules.

Symmetry Operation

An operation which, when applied to an object, leaves it indistinguishable from the original object.

Symmetry Element

The geometrical entity—such as a point, line/axis, or plane—with respect to which a symmetry operation is carried out.

Identity ($E$)

The symmetry operation consisting of doing nothing ($P(x, y, z) \rightarrow P'(x, y, z)$), which exists for every object and is required to complete mathematical groups.

Proper Rotation ($C_n$)

A symmetry operation consisting of a rotation through an angle of $\frac{360^\circ}{n}$ about an axis of symmetry, where $n$ is the order of the axis.

Principal Axis

The rotation axis with the highest order (largest value of $n$) in a molecule, conventionally aligned along the $z$-axis.

Vertical Mirror Plane ($\sigma_v$)

A mirror plane that contains the principal rotation axis of the molecule.

Horizontal Mirror Plane ($\sigma_h$)

A mirror plane that is perpendicular to the principal rotation axis (or defines the plane of the molecule if no principal axis exists).

Dihedral Mirror Plane ($\sigma_d$)

A special case of a vertical mirror plane that bisects the angle between two $C_2$ axes that lie perpendicular to the principal axis (or bisects bond angles).

Inversion Centre ($i$)

A centre of symmetry through which each point of an object is projected through the centre to an equal distance on the opposite side ($P(x, y, z) \rightarrow P'(-x, -y, -z)$).

Improper Rotation ($S_n$)

A composite symmetry operation consisting of a proper rotation ($C_n$) through $\frac{360^\circ}{n}$ followed by a reflection in a plane perpendicular to the rotation axis ($\sigma_h$).

Chiral Molecule (Symmetry Definition)

A molecule that cannot be superimposed on its mirror image, which is only possible if it lacks all improper rotation operations ($S_n$), including $i$ ($S_2$) and $\sigma$ ($S_1$).

Abelian Group

A group in which all symmetry operations commute with one another ($PQ = QP$), resulting in a group multiplication table that is symmetric across the main diagonal.

Non-Abelian Group

A group in which at least some symmetry operations do not commute ($PQ \neq QP$), leading to an asymmetric multiplication table across the main diagonal.

Point Group

The complete set of symmetry operations exhibited by an isolated molecule, all of which leave at least one common point in the molecule unchanged.

Group Order ($h$)

The total number of symmetry operations contained within a mathematical or molecular point group.

Character ($\chi$)

The trace (sum of the diagonal elements) of the matrix representing a given symmetry operation within a specific representation.

Mulliken Symbols

Labels for symmetry species: $A$ and $B$ denote 1D representations, $E$ denotes 2D (doubly degenerate) representations, and $T$ (or $F$) denotes 3D (triply degenerate) representations.

Reducible Representation

A representation whose matrices can be decomposed or reduced into a direct sum of simpler, lower-dimensional representations of the group.

Irreducible Representation (Irrep)

A representation whose dimension cannot be reduced any further; all one-dimensional representations are irreducible by definition.

Symmetry Adapted Linear Combinations (SALCs)

Linear combinations of basis functions (such as atomic orbitals or nuclear displacements) constructed to transform as a specific irreducible representation of the molecular point group.

Reduction Formula

The formula $n_\Gamma = \frac{1}{h}\sum_R \chi_\Gamma(R)\chi(R)$ used to determine the integer number of times an irreducible representation $\Gamma$ appears in a reducible representation.

Projection Formula

The mathematical formula $\sum_R \chi_\Gamma(R)(R\phi_i)$ applied to a starting basis function $\phi_i$ to generate an unnormalized SALC of symmetry $\Gamma$.

Class

A set of symmetry operations related to each other by similarity transformation (conjugation), all having identical characters in any given representation.

Integral Vanishing Rule

The theorem stating that an integral over a function is zero unless the integrand forms a basis for or contains the totally symmetric irreducible representation of the point group.

Walsh Diagram

A correlation diagram tracking how molecular orbital energies change as the geometry of a molecule undergoes a systematic distortion, such as linear to bent.

Term Symbol

A notation of the form $^{2S+1}\Gamma$ describing both the spin multiplicity ($2S+1$) and overall spatial irreducible representation ($\Gamma$) of an electronic state.

Electric Dipole Moment Selection Rule

A permanent molecular dipole moment component $\mu_\alpha$ is non-zero only if the Cartesian coordinate $\alpha$ ($x, y,$ or $z$) transforms as the totally symmetric representation of the point group.

Electronic Transition Allowed Condition

A transition between electronic states with symmetries $\Gamma_i$ and $\Gamma_f$ is dipole-allowed if at least one Cartesian coordinate ($x, y,$ or $z$) transforms as $\Gamma_i \otimes \Gamma_f$.

IR Active Vibration Criterion

A fundamental vibrational mode can absorb infrared radiation if and only if its vibrational coordinate transforms under the same irreducible representation as $x, y,$ or $z$.

Raman Active Vibration Criterion

A fundamental vibrational mode is Raman active if and only if its vibrational coordinate transforms under the same irreducible representation as quadratic Cartesian terms ($x^2, y^2, z^2, xy, xz,$ or $yz$).

Mutual Exclusion Principle

The rule establishing that for any molecule with a centre of inversion ($i$), no vibrational mode can be both infrared (IR) active and Raman active.