含义Meaning
非谐性使振动能级间隔随 v 线性减小；把相邻间隔对 v 作图外推到零，曲线下面积给出解离能。也解释了泛频带的位置略低于基频的整数倍。
Anharmonicity makes the spacing shrink linearly with v; extrapolating the spacings to zero, the area under the plot gives the dissociation energy. It also explains why overtones fall slightly below integer multiples of the fundamental.
符号说明Symbols
| 符号Symbol | 含义 Meaning | 单位 Unit |
|---|---|---|
| \(\tilde\omega_e\) | 谐振波数harmonic wavenumber | cm⁻¹ |
| \(x_e\) | 非谐常数anharmonicity constant | 1 |
| \(\Delta G_{v+1/2}\) | 相邻能级间隔spacing between v and v + 1 | cm⁻¹ |
推导Derivation
莫尔斯势的精确能级恰为两项式，\(\tilde\omega_ex_e = \tilde\omega_e^2/(4D_e)\)。
Morse levels are exactly the two-term expression with \(\tilde\omega_ex_e = \tilde\omega_e^2/4D_e\).
相邻差 \(G(v+1) - G(v)\) 线性减小，在 \(v_\text{max}\) 处降为零（解离）。
Spacings fall linearly to zero at dissociation.
对所有间隔求和（三角形面积）得 \(D_0\)。
Summing them (triangle area) gives \(D_0\).
基频 \(\tilde\omega_e - 2\tilde\omega_ex_e\)，一级泛频 \(2\tilde\omega_e - 6\tilde\omega_ex_e\)。
Fundamental \(\tilde\omega_e - 2\tilde\omega_ex_e\), first overtone \(2\tilde\omega_e - 6\tilde\omega_ex_e\).
适用条件与局限Assumptions & limitations
线性外推常高估解离能（高 v 处实际曲线弯曲），对离子性分子误差可达 20%。
Linear extrapolation often overestimates D₀ (curvature at high v), by up to 20 % for ionic molecules.
例题Worked example
HCl：ω̃ₑ = 2990.9、ω̃ₑxₑ = 52.8 cm⁻¹ → 基频 2885、泛频 5668 cm⁻¹（实测 2886、5668）。
HCl: fundamental 2885, overtone 5668 cm⁻¹ (observed 2886, 5668).