含义Meaning
把可约表示 Γ 分解为不可约表示的直和 \(\Gamma = \bigoplus_i n_i\Gamma_i\)，给出每个不可约表示出现的次数。分子轨道对称性、振动模式分析都依赖它。点群页面的“约化”工具即按此计算。
Decomposes a reducible representation into irreducible ones, \(\Gamma = \bigoplus n_i\Gamma_i\), giving how often each occurs — the workhorse of MO symmetry and vibrational analysis. The reduce tool on each point-group page uses it.
符号说明Symbols
| 符号Symbol | 含义 Meaning | 单位 Unit |
|---|---|---|
| \(\chi_\Gamma(C)\) | 可约表示的特征标character of the reducible representation | 1 |
| \(n_i\) | 不可约表示 i 出现次数（非负整数）multiplicity of irrep i | 1 |
推导Derivation
可约表示的特征标是各不可约表示特征标之和：\(\chi_\Gamma(C) = \sum_j n_j\chi_j(C)\)。
Reducible characters are sums of irreducible ones.
两边乘 \(g_C\chi_i(C)^*\) 并对类求和，用正交关系 \(\sum_Cg_C\chi_j\chi_i^* = h\delta_{ij}\)。
Multiply by \(g_C\chi_i^*\), sum over classes and use orthogonality.
只剩 \(j = i\) 项：\(\sum_C g_C\chi_\Gamma\chi_i^* = hn_i\)。
Only j = i survives.
适用条件与局限Assumptions & limitations
结果必须是非负整数；出现分数说明 Γ 的特征标算错了。
Results must be non-negative integers; fractions mean an error in Γ.
例题Worked example
H₂O（C₂ᵥ）中两个 O–H 伸缩：Γ = (2, 0, 2, 0) → n(A₁) = (2+0+2+0)/4 = 1，n(B₁) = (2−0+2−0)/4 = 1，即 A₁ ⊕ B₁（取 xz 为分子平面）。
Water’s two O–H stretches: Γ = (2, 0, 2, 0) → A₁ ⊕ B₁ (molecule in the xz plane).