含义Meaning
只处理共轭体系 π 电子的最简 LCAO 方法（1931）：每个 p 轨道库仑积分 α，相邻原子共振积分 β，忽略重叠。能写出能级、轨道系数、键级和电荷。
The simplest LCAO treatment of π electrons (1931): Coulomb integral α per p orbital, resonance integral β between neighbours, overlap neglected. Gives energies, coefficients, bond orders and charges.
符号说明Symbols
| 符号Symbol | 含义 Meaning | 单位 Unit |
|---|---|---|
| \(\alpha\) | 库仑积分（≈ 碳 2p 电离能的负值）Coulomb integral | J |
| \(\beta\) | 共振积分（< 0，约 −75 kJ/mol）resonance integral (< 0, ≈ −75 kJ/mol) | J |
| \(p_{rs} = \sum_i n_ic_{ir}c_{is}\) | π 键级π bond order | 1 |
推导Derivation
π MO 写作 \(\psi = \sum c_r p_r\)，代入线性变分得久期方程。
Write \(\psi = \sum c_r p_r\) and apply linear variation.
休克尔近似把矩阵变成拓扑（邻接）矩阵，令 \(x = (\alpha - E)/\beta\)。
The Hückel approximations turn H into the adjacency matrix; set \(x = (\alpha - E)/\beta\).
乙烯：\(\begin{vmatrix}x & 1\\ 1 & x\end{vmatrix} = 0\) ⇒ \(x = \pm1\)，\(E = \alpha \pm \beta\)。
Ethene: \(x = \pm1\), \(E = \alpha \pm \beta\).
丁二烯：\(x^4 - 3x^2 + 1 = 0\) ⇒ \(E = \alpha \pm 1.618\beta, \alpha \pm 0.618\beta\)，\(E_\pi = 4\alpha + 4.472\beta\)，离域能 \(0.472\beta\)。
Butadiene: \(E = \alpha \pm 1.618\beta, \alpha \pm 0.618\beta\); delocalisation energy \(0.472\beta\).
苯：\(E = \alpha \pm 2\beta, \alpha \pm \beta\ (\times 2)\)，离域能 \(2\beta\)。
Benzene: delocalisation energy \(2\beta\).
适用条件与局限Assumptions & limitations
忽略重叠、电子间排斥和 σ 体系；β 的数值随性质（热化学、光谱）不同而不同。
Neglects overlap, electron repulsion and σ electrons; β depends on the property fitted.
杂原子需用 \(\alpha + h\beta\)、\(k\beta\) 参数；更好的方法有 PPP、扩展休克尔、从头算。
Heteroatoms need h, k parameters; better: PPP, extended Hückel, ab initio.