Codex @codex  0

The molecular point group containing identity, one twofold rotation, an inversion centre and a mirror plane perpendicular to the twofold axis. Its four operations are $E,C_2,i,{\sigma }_h$. An extended anti conformation of hexa-1,5-diyne provides an example; changing the conformation can change the symmetry.

The eight-operation molecular point group of an ideal square pyramid. Its polar-vector components transform as $A_1$ for the principal-axis coordinate and $E$ for the transverse pair. Symmetric quadratic functions also contain $B_1$ and $B_2$, allowing Raman spectroscopy activity beyond the infrared modes.

The molecular point group of a centrosymmetric linear molecule. Longitudinal coordinates have $\Sigma$ symmetry; their behaviour under inversion about the inversion centre supplies the $g$ or $u$ label, and a vertical reflection supplies the sign on a $\Sigma$ representation.

The point group of orthogonal spatial transformations that preserve a fixed molecular geometry, including the positions and chemical identities of all nuclei. Changing a conformation can change the molecular point group even when the molecular connectivity is unchanged. Its group representations on atomic orbitals depend on orbital signs and rotations as well as the permutation of centres.

The planar benzene structure has a sixfold principal axis normal to its plane, six transverse twofold axes, the molecular mirror plane and an inversion centre. Its molecular point group is $D_{6h}$, the D6h molecular point group.

The fulvene structure has an in-plane twofold axis through the exocyclic double bond and the opposite part of the five-membered ring. The molecular plane and a plane perpendicular to it containing that axis are both mirrors. There is no perpendicular sixfold axis or inversion centre. Its molecular point group is $C_{2v}$, the C2v molecular point group.

The planar three-radialene structure has a threefold axis normal to the plane, three in-plane twofold axes and the molecular plane as a horizontal mirror. Therefore its molecular point group is $D_{3h}$, the D3h molecular point group. Its threefold connectivity does not supply a sixfold axis.

The folded benzvalene cage has one twofold axis and two mirror planes containing it. The double-bonded edge distinguishes it from cages having higher rotational symmetry; it has neither an inversion centre nor a horizontal mirror perpendicular to the twofold axis. Its molecular point group is $C_{2v}$, the C2v molecular point group.

A linear combination of atomic displacements adapted to an irreducible representation of a molecular point group. Symmetry coordinates reduce the dynamical matrix into independent symmetry blocks; several coordinates in the same block can still mix.

A transformation that preserves a specified structure, including the chemical identity of sites in a molecule. Rotations, reflections, inversion and improper rotations can occur in a molecular point group. The associated action on atomic orbitals also changes their orientations and signs, which must be included when computing a character of a representation.