空间群不可约表示的计算方法

The Calculation Method of the Irreducible Representation of the Space Group

在固体物理中,采用周期性边界条件,则晶体的哈密顿对称群是空间群G,电子的本征态是布洛赫函数,这些函数组成构成空间群G的不可约表示的基(简记IR基)。由平移群的不可约基诱导波矢群G(kω)的一个gl维表示,通过构造G(kω)表象群G′k,利用本征函数法将gl维表示约化,得到G′k的IR基,利用G′k可将求波矢群G(kω)的IR问题转化为求表象群G′k的IR问题。在{βσ|v-ω(βσ)}G布里渊区内选择一个波矢k∨,求出波矢群G(kω),将空间群按G(kω)作左陪集分解G=Σσ(kω),可由波矢群不可约表示诱导出空间群不可约表示,与传统群论的计算方法相比,该方法理论简明,方法简便,易于程序化。

In solid state physics,the symmetry group of the Hamiltonian of crystal is space group G by taking periodic border conditions.The eigenstate of the electronic is Bloch function,these functions form the basis of the irreducible representation of the space group.The g_l dimensional representation of the wave vector group is induced by the basis of the irreducible representation of the translation group.The g_l dimensional representation is reduced by making the representation group G_k′ of G_k.Using the eigenfunction method,we have the irreducible representation basis of the G_k′.So the problem of IR of G_k is translated the problem of IR of G_k′.In brillion zone,the wave vector kω is selected and G_k is calculated.The G is formed by the left coset of G_k,then the IR of the space group is induced.This theory is simple and clear and easy to be programmed.