Point group

The point group of a Euclidean crystallographic group is the group of its orthogonal linear parts, equivalently its quotient by the translation subgroup. It preserves the translation Euclidean lattice. In an arbitrary dimension it is a finite orthogonal group; for a plane crystallographic group this lattice preservation restricts the allowed rotation orders.

A physical property tensor must be invariant under every symmetry operation of the material's point group. For a polar rank-$r$ tensor, $T_{i_1\dots i_r}=R_{i_1{j}_1}\cdots R_{i_r{j}_r}{T}_{j_1\dots j_r}$. Taking $R=-I$ proves that an odd-rank polar tensor vanishes in a centrosymmetric material. A susceptibility of order $n$ has rank $n+1$, so local bulk even-order electric-dipole response is forbidden there. The name refers to Franz Ernst Neumann.

The spatial symmetry of a fixed molecular geometry. Its point group acts on atomic positions, orbitals and vibrational coordinates; these actions need not have the same group representation.

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.

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 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.

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.

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 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 six-operation molecular point group with a threefold proper rotation axis and three vertical mirror planes. Its rotational symmetry number is three because mirror operations do not count as proper spatial rotations.

The 24-operation molecular point group of a regular planar hexagon with equivalent sites. Its irreducible representations have inversion labels $g,u$; polar-vector components transform as $A_{2u}$ for $z$ and $E_{1u}$ for $(x,y)$.

The 24-operation molecular point group of an ideal regular tetrahedron, including improper operations. Its irreducible representations are $A_1,A_2,E,T_1,T_2$, with dimensions $1,1,2,3,3$. The polar-vector representation is $T_2$, the axial-vector representation is $T_1$, and symmetric quadratic functions transform as $A_1+E+T_2$.

The 15 Cartesian displacements of tetrahedral methane have characters $(15,0,-1,-1,3)$ on classes $(E,8C_3,3C_2,6S_4,6{\sigma }_d)$. Removing translations $T_2$ and rotations $T_1$ gives $(9,0,1,-1,3)$ and the stated decomposition. There are nine vibrational coordinates in four frequency families.

In a centrosymmetric molecule, a fundamental vibration cannot be both electric-dipole infrared active and ordinary Raman spectroscopy active. Dipole components are odd under inversion; polarizability components are even. This selection rule concerns modes, not accidental numerical equality of unrelated frequencies.

For a finite molecular point group, the multiplicity of irrep $\lambda$ in a representation with character $\chi$ is $n_{\lambda }=|G{|}^{-1}{\sum }_C|C|\chi (C){\chi }_{\lambda }(C{)}^{\ast }$. The sum is over conjugacy classes.

An orbital set whose span is invariant under every molecular symmetry operation. Completeness here means closure of the chosen representation space; it does not mean that a finite orbital set spans the entire electronic Hilbert space.

The full icosahedral molecular point group, including inversion, with 120 operations. Its single-valued orbital irreducible representations have dimensions 1, 3, 3, 4 and 5 in each inversion parity.

The sixteen-element molecular point group of an ideal square with a horizontal mirror plane. Inversion distinguishes gerade and ungerade irreducible representations.

The twelve-element molecular point group of an ideal trigonal bipyramid. Reflection in its equatorial plane distinguishes primed and double-primed irreducible representations.

An eight-element molecular point group with classes $E,2S_4,C_2,2C_2^{\prime },2{\sigma }_d$. The vector representation decomposes as $B_2+E$.

The four-element molecular point group with one twofold axis and two mirror planes containing that axis. Water has this symmetry at its equilibrium bent geometry.

A rotation, reflection, inversion or improper rotation that carries a fixed molecular geometry into an indistinguishable configuration. Products of such operations must also be symmetries.

A molecular point group containing identity, three mutually perpendicular twofold rotations, inversion, and three mirror reflections. In the conventional axis labelling, $z\sim B_{1u}$, $y\sim B_{2u}$ and $x\sim B_{3u}$; $yz\sim B_{3g}$ and $xz\sim B_{2g}$. The symmetry of a perpendicular atomic $p$ orbital includes its sign reversal when an operation reverses its axis, in addition to any permutation of atomic centres.

A molecular point group with identity, one twofold axis, and two mirror planes containing that axis. With the twofold axis along $z$, its one-dimensional irreducible representations have Cartesian bases $z\sim A_1$, $x\sim B_1$, and $y\sim B_2$. For formaldehyde in the $xz$ plane, the vibrational representation is $3A_1+2B_1+B_2$.

A rotation preserving a rank-two lattice has order $1,2,3,4$ or $6$. In a lattice basis its matrix is integral, while its trace is $2\cos \theta$. The only integral values in $[-2,2]$ are $-2,-1,0,1,2$, which give precisely those rotation orders. Finiteness of a symmetry group alone would not impose this restriction.