群论Group theory

群是满足封闭性、结合律、存在单位元和逆元的元素集合。分子的对称操作（恒等 E、旋转 Cₙ、反映 σ、反演 i、旋转反映 Sₙ）构成点群。可约表示可以分解为不可约表示，后者用马利肯符号标记（A、B 为一维，E 为二维，T 为三维；下标 g/u 表示对反演对称或反对称）。约化公式 nᵢ = (1/h)Σχ(R)χᵢ(R) 用于确定各不可约表示出现的次数。

A group is a set obeying closure and associativity with an identity and inverses. A molecule's symmetry operations, identity E, rotations Cₙ, reflections σ, inversion i and improper rotations Sₙ, form its point group. Reducible representations decompose into irreducible ones, labelled by Mulliken symbols (A and B one-dimensional, E two-dimensional, T three-dimensional; subscripts g/u for symmetric or antisymmetric under inversion). The reduction formula nᵢ = (1/h)Σχ(R)χᵢ(R) gives how many times each irreducible representation occurs.

群的概念由伽罗瓦在 1830 年代研究方程可解性时提出。1920–30 年代维格纳、贝特和外尔把群论引入量子力学，维格纳因对称性原理的贡献获 1963 年诺贝尔物理学奖。1929 年贝特用群论分析晶体场中 d 轨道的分裂。1963 年科顿出版《群论的化学应用》，使群论成为化学专业的标准课程。

Galois introduced the concept of a group in the 1830s while studying the solvability of equations. In the 1920s and 1930s Wigner, Bethe and Weyl brought group theory into quantum mechanics, and Wigner received the 1963 Nobel Prize in Physics for his work on symmetry principles. Bethe used group theory in 1929 to analyse the splitting of d orbitals in crystal fields, and Cotton's Chemical Applications of Group Theory (1963) made it a standard part of chemistry education.

群论用于：确定分子振动模式的数目和对称性，判断哪些模式红外活性（属于 x、y、z 的不可约表示）或拉曼活性（属于二次函数的不可约表示）；构造对称性匹配的分子轨道（SALC）；分析晶体场和配体场中 d 轨道的分裂；推导光谱选择定则；伍德沃德-霍夫曼轨道对称守恒规则；晶体学中的 230 种空间群。

Group theory is used to find the number and symmetry of vibrational modes and which are infrared-active (transforming as x, y or z) or Raman-active (transforming as quadratic functions); to build symmetry-adapted linear combinations of orbitals (SALCs); to analyse d-orbital splitting in crystal and ligand fields; to derive spectroscopic selection rules; in the Woodward–Hoffmann rules of orbital-symmetry conservation; and for the 230 space groups of crystallography.