Point group in the context of Orthogonal group

In geometry, a point reflection (also called a point inversion or central inversion) is a geometric transformation of affine space in which every point is reflected across a designated inversion center, which remains fixed. In Euclidean or pseudo-Euclidean spaces, a point reflection is an isometry (preserves distance). In the Euclidean plane, a point reflection is the same as a half-turn rotation (180° or π radians), while in three-dimensional Euclidean space a point reflection is an improper rotation which preserves distances but reverses orientation. A point reflection is an involution: applying it twice is the identity transformation.

An object that is invariant under a point reflection is said to possess point symmetry (also called inversion symmetry or central symmetry). A point group including a point reflection among its symmetries is called centrosymmetric. Inversion symmetry is found in many crystal structures and molecules, and has a major effect upon their physical properties.

In a tetrahedral molecular geometry, a central atom is located at the center with four substituents that are located at the corners of a tetrahedron. The bond angles are arccos(−⁠1/3⁠) = 109.4712206...° ≈ 109.5° when all four substituents are the same, as in methane (CH₄) as well as its heavier analogues. Methane and other perfectly symmetrical tetrahedral molecules belong to point group T_d, but most tetrahedral molecules have lower symmetry. Tetrahedral molecules can be chiral.

Pyrazine is a heterocyclic aromatic organic compound with the chemical formula C₄H₄N₂. It is a symmetrical molecule with point group D_2h. Pyrazine is less basic than pyridine, pyridazine and pyrimidine. It is a "deliquescent crystal or wax-like solid with a pungent, sweet, corn-like, nutty odour".

Pyrazine and a variety of alkylpyrazines are flavor and aroma compounds found in baked and roasted goods. Tetramethylpyrazine (also known as ligustrazine) is reported to scavenge superoxide anions and decrease nitric oxide production in human granulocytes.

In structural biology, a protomer is the structural unit of an oligomeric protein. It is the smallest unit composed of at least one protein chain. The protomers associate to form a larger oligomer of two or more copies of this unit. Protomers usually arrange in cyclic symmetry to form closed point group symmetries.

The term was introduced by Chetverin to make nomenclature in the Na/K-ATPase enzyme unambiguous. This enzyme is composed of two subunits: a large, catalytic α subunit, and a smaller glycoprotein β subunit (plus a proteolipid, called γ-subunit). At the time it was unclear how many of each work together. In addition, when people spoke of a dimer, it was unclear whether they were referring to αβ or to (αβ)₂. Chetverin suggested to call αβ a protomer and (αβ)₂ a diprotomer. Thus, in the work by Chetverin the term protomer was only applied to a hetero-oligomer and subsequently used mainly in the context of hetero-oligomers. Following this usage, a protomer consists of a least two different proteins chains. In current literature of structural biology, the term is commonly also applied to the smallest unit of homo-oligomers, avoiding the term "monomer".

In chemistry, octahedral molecular geometry, also called square bipyramidal, describes the shape of compounds with six atoms or groups of atoms or ligands symmetrically arranged around a central atom, defining the vertices of an octahedron. The octahedron has eight faces, hence the prefix octa. The octahedron is one of the Platonic solids, although octahedral molecules typically have an atom in their centre and no bonds between the ligand atoms. A perfect octahedron belongs to the point group O_h. Examples of octahedral compounds are sulfur hexafluoride SF₆ and molybdenum hexacarbonyl Mo(CO)₆. The term "octahedral" is used somewhat loosely by chemists, focusing on the geometry of the bonds to the central atom and not considering differences among the ligands themselves. For example, [Co(NH₃)₆], which is not octahedral in the mathematical sense due to the orientation of the N−H bonds, is referred to as octahedral.

The concept of octahedral coordination geometry was developed by Alfred Werner to explain the stoichiometries and isomerism in coordination compounds. His insight allowed chemists to rationalize the number of isomers of coordination compounds. Octahedral transition-metal complexes containing amines and simple anions are often referred to as Werner-type complexes.

In crystallography, a centrosymmetric point group contains an inversion center as one of its symmetry elements. In such a point group, for every point (x, y, z) in the unit cell there is an indistinguishable point (-x, -y, -z). Such point groups are also said to have inversion symmetry. Point reflection is a similar term used in geometry.Crystals with an inversion center cannot display certain properties, such as the piezoelectric effect and the frequency doubling effect (second-harmonic generation). In addition, in such crystals, one-photon absorption (OPA) and two-photon absorption (TPA) processes are mutually exclusive, i.e., they do not occur simultaneously, and provide complementary information.

The following space groups have inversion symmetry: the triclinic space group 2, the monoclinic 10-15, the orthorhombic 47-74, the tetragonal 83-88 and 123-142, the trigonal 147, 148 and 162-167, the hexagonal 175, 176 and 191-194, the cubic 200-206 and 221-230.