含义Meaning
电子跃迁远快于核运动，跃迁时核构型几乎不变（竖直跃迁）。各振动带的强度正比于上下态振动波函数重叠积分的平方（弗兰克–康登因子），决定电子光谱的振动精细结构。
Electronic transitions are much faster than nuclear motion, so they are vertical. Each vibronic band’s intensity is proportional to the squared overlap of vibrational wavefunctions (the Franck–Condon factor).
符号说明Symbols
| 符号Symbol | 含义 Meaning | 单位 Unit |
|---|---|---|
| \(\mu_{el}\) | 电子跃迁偶极矩electronic transition dipole | C·m |
| \(\chi_v\) | 振动波函数vibrational wavefunction | |
| \(\langle\chi_{v'}|\chi_{v''}\rangle\) | 振动重叠积分vibrational overlap integral | 1 |
推导Derivation
玻恩–奥本海默近似下 \(\Psi = \psi_{el}\chi_v\)，跃迁矩 \(\langle\Psi'|\hat\mu|\Psi''\rangle\)。
With \(\Psi = \psi_{el}\chi_v\), take the transition moment.
假设电子跃迁矩随核坐标变化很小（康登近似），可提到积分外：\(\mu_{el}\langle\chi_{v'}|\chi_{v''}\rangle\)。
Assuming \(\mu_{el}\) barely varies with geometry (Condon), it factors out.
若激发态键长与基态相同，强度集中在 0–0 带；键长变化越大，强度分布越宽，最强带偏向高 v′。
Equal geometries concentrate intensity in the 0–0 band; larger displacements spread it to higher v′.
适用条件与局限Assumptions & limitations
康登近似对弱的、对称禁阻跃迁失效（赫兹伯格–特勒振动耦合）。
Fails for weak, symmetry-forbidden transitions (Herzberg–Teller coupling).