双语化学辞典Chem Lexicon
本科Undergraduate 群论与分子对称性 Group theory & molecular symmetry

振动自由度与振动模式对称性Vibrational degrees of freedom and mode symmetries

\[N_\text{vib} = 3N - 6\ (\text{非线形}),\quad 3N - 5\ (\text{线形}),\qquad \Gamma_\text{vib} = \Gamma_{3N} - \Gamma_\text{trans} - \Gamma_\text{rot}\]\[N_\text{vib} = 3N - 6\ (\text{non-linear}),\quad 3N - 5\ (\text{linear}),\qquad \Gamma_\text{vib} = \Gamma_{3N} - \Gamma_\text{trans} - \Gamma_\text{rot}\]
笛卡尔位移表示的特征标（+ 用于 Cₙ，− 用于 Sₙ、σ、i）character of the 3N Cartesian representation (+ for proper, − for improper operations)
\[\chi_{3N}(R) = N_\text{不动}(R)\times\left(\pm1 + 2\cos\theta_R\right)\]\[\chi_{3N}(R) = N_\text{unmoved}(R)\times\left(\pm1 + 2\cos\theta_R\right)\]

含义Meaning

N 原子分子有 3N 个运动自由度，扣除 3 个平动和 3 个（线形 2 个）转动即为振动模式数。用群论可求出每个模式的对称性，并判断红外（按 x, y, z 变换）和拉曼（按二次函数变换）活性。

A molecule of N atoms has 3N degrees of freedom; subtracting 3 translations and 3 (linear: 2) rotations gives the number of vibrations. Group theory gives each mode’s symmetry and its IR (x, y, z) and Raman (quadratic) activity.

符号说明Symbols

符号Symbol含义 Meaning单位 Unit
\(N\)原子数number of atoms1
\(N_\text{不动}(R)\)操作 R 下位置不变的原子数atoms left in place by R1
\(\theta_R\)该操作的转角（σ 对应 0°，i 对应 180°）rotation angle of R (σ: 0°, i: 180°)°

推导Derivation

  1. 每个原子三个笛卡尔位移，共 3N 个基函数构成可约表示 Γ₃N。

    3N Cartesian displacements form the representation Γ₃N.

  2. 只有不动原子贡献特征标；每个不动原子贡献 3×3 转动（反转）矩阵的迹 \(\pm1 + 2\cos\theta\)。

    Only unmoved atoms contribute, each with trace ±1 + 2cos θ.

  3. 约化 Γ₃N，再减去按 (x, y, z) 和 (Rx, Ry, Rz) 变换的不可约表示，得 Γ_vib。

    Reduce Γ₃N and subtract the irreps of (x,y,z) and (Rx,Ry,Rz).

  4. 有对称中心时，IR 活性（u）与拉曼活性（g）模式互斥。

    With an inversion centre IR (u) and Raman (g) activity are mutually exclusive.

适用条件与局限Assumptions & limitations

  • 给出模式数目和对称性，不给出频率；简并模式只算一个谱带。实际强度还取决于偶极矩/极化率导数大小。

    Gives number and symmetry, not frequencies; degenerate modes give one band; intensities depend on derivative magnitudes.

例题Worked example

H₂O（C₂ᵥ）：Γ_vib = 2A₁ + B₁，3 个模式都 IR 和拉曼活性。CO₂（D∞h）：Σg⁺（拉曼）+ Σu⁺ + Πu（IR），共 4 个。CH₄（Td）：A₁ + E + 2T₂，只有 T₂ 为 IR 活性（3019、1306 cm⁻¹）。

H₂O: 2A₁ + B₁, all IR and Raman active. CO₂: Σg⁺ (Raman) + Σu⁺ + Πu (IR). CH₄: A₁ + E + 2T₂; only T₂ is IR-active.