含义Meaning
哈密顿量可写成可解部分加小扰动 \(\hat H = \hat H^{(0)} + \hat H'\) 时，按扰动级数逐阶修正能量和波函数。二阶修正对基态总是负的。
When \(\hat H = \hat H^{(0)} + \hat H'\) with a small perturbation, energies and wavefunctions are corrected order by order; the second-order correction to the ground state is always negative.
符号说明Symbols
| 符号Symbol | 含义 Meaning | 单位 Unit |
|---|---|---|
| \(\hat H'\) | 微扰哈密顿量perturbation | J |
| \(|n\rangle, E_n^{(0)}\) | 未微扰本征态与本征值unperturbed eigenstates and energies |
推导Derivation
引入参数 λ：\(\hat H = \hat H^{(0)} + \lambda\hat H'\)，把 \(E_n\) 与 \(|\psi_n\rangle\) 展开为 λ 的幂级数。
Introduce λ and expand \(E_n\), \(|\psi_n\rangle\) in powers of λ.
比较 λ¹ 项并左乘 \(\langle n|\)：\(E^{(1)} = \langle n|\hat H'|n\rangle\)。
First order: \(E^{(1)} = \langle n|\hat H'|n\rangle\).
一阶波函数 \(|n^{(1)}\rangle = \sum_{k\ne n}\frac{\langle k|\hat H'|n\rangle}{E_n^{(0)} - E_k^{(0)}}|k\rangle\)；代入 λ² 项得二阶能量。
The first-order wavefunction feeds the second-order energy.
应用：斯塔克效应、极化率、MP2 电子相关能（以 HF 为零级）。
Applications: Stark effect, polarisability, MP2 correlation (HF as zeroth order).
适用条件与局限Assumptions & limitations
要求扰动矩阵元远小于能级间隔；能级简并时须用简并微扰理论（先对角化简并子空间）。
Requires matrix elements ≪ level spacings; degenerate levels need degenerate perturbation theory.
级数不一定收敛（MPn 在某些体系发散）。
The series need not converge (MPn can diverge).