含义Meaning
比抛物线更真实的双原子分子势能曲线：在平衡键长附近近似谐振，远处趋于解离能 \(D_e\)，可精确求解得到非谐振动能级。
A more realistic diatomic potential than a parabola: harmonic near \(r_e\), approaching the dissociation energy \(D_e\) at large r, and exactly solvable for anharmonic levels.
符号说明Symbols
| 符号Symbol | 含义 Meaning | 单位 Unit |
|---|---|---|
| \(D_e\) | 势阱深度（从极小值算的解离能）well depth | J, eV |
| \(D_0 = D_e - \tfrac12h\nu\) | 由零点能级算的解离能dissociation energy from the zero-point level | J |
| \(a\) | 势阱宽度参数range parameter | m⁻¹ |
| \(r_e\) | 平衡键长equilibrium bond length | m |
推导Derivation
r → ∞ 时 \(V \to D_e\)；r = r_e 时 V = 0 且 \(dV/dr = 0\)。
V → D_e at large r; V = 0 and dV/dr = 0 at r_e.
在 \(r_e\) 处展开：\(V \approx D_ea^2(r - r_e)^2\)，与 \(\tfrac12k_e(r-r_e)^2\) 比较得 \(a = \sqrt{k_e/2D_e}\)。
Expanding at r_e gives \(D_ea^2(r-r_e)^2\), so \(a = \sqrt{k_e/2D_e}\).
薛定谔方程解析解：\(E_v = h\nu(v + \tfrac12) - \frac{(h\nu)^2}{4D_e}(v + \tfrac12)^2\)，能级随 v 增大而变密。
Exact levels \(E_v = h\nu(v+\tfrac12) - \frac{(h\nu)^2}{4D_e}(v+\tfrac12)^2\) crowd together with v.
适用条件与局限Assumptions & limitations
短程排斥部分（r < r_e）不够陡；精确工作用 RKR 反演势或从头算势能曲线。
Too soft at short range; accurate work uses RKR or ab initio curves.
例题Worked example
H₂：D_e = 4.75 eV，ν̃ = 4401 cm⁻¹，零点能 0.27 eV，D₀ = 4.48 eV。
H₂: D_e = 4.75 eV, ν̃ = 4401 cm⁻¹, ZPE 0.27 eV, D₀ = 4.48 eV.