含义Meaning
两函数乘积的表示的特征标是各自特征标之积。积分只有在被积函数的直积包含全对称表示时才可能非零——这是所有选择定则和轨道是否“允许”相互作用的判据。
The character of a product representation is the product of characters. An integral can be non-zero only if the direct product of the integrand contains the totally symmetric representation — the criterion behind every selection rule and orbital-mixing rule.
符号说明Symbols
| 符号Symbol | 含义 Meaning | 单位 Unit |
|---|---|---|
| \(\Gamma_O\) | 算符（如 x, y, z 或 xy）的表示representation of the operator | |
| \(A_1\) | 全对称表示totally symmetric representation |
推导Derivation
对称操作不改变积分值（只是换坐标）：\(\int f\,d\tau = \int \hat Rf\,d\tau\)。
A symmetry operation leaves the integral unchanged.
对所有 R 平均：\(\int f = \frac1h\sum_R\int\hat Rf = \int\hat P^{A_1}f\)，只有 f 的全对称分量有贡献。
Averaging over R projects out the totally symmetric part.
非简并不可约表示的直积仍是不可约表示，\(\Gamma\otimes\Gamma = A_1\)；简并的需再用约化公式分解。
For non-degenerate irreps Γ⊗Γ = A₁; degenerate products need reduction.
适用条件与局限Assumptions & limitations
“对称允许”只说明积分可能非零，不保证其大小；严格为零只在对称性要求下才成立。
Symmetry-allowed means possibly non-zero, not large.
例题Worked example
Td 中 A₁ → T₂ 的电偶极跃迁：A₁ ⊗ T₂(x,y,z) ⊗ T₂ 含 A₁，允许；d–d 跃迁在 Oh 中 g ⊗ u ⊗ g = u，禁阻（拉波特规则）。
In Td, A₁ → T₂ is dipole-allowed; in Oh, g → g transitions are Laporte-forbidden.