休克尔分子轨道法Hückel method

σ-π 分离，只处理垂直于分子平面的 p 轨道；库仑积分都取为 α；相邻原子的共振积分取为 β（α、β 均为负值），不相邻的取 0；重叠积分 S = 0（i ≠ j）。由此得到久期行列式 |H − ES| = 0。例如乙烯 E = α ± β；1,3-丁二烯 E = α ± 1.618β 和 α ± 0.618β；苯 E = α ± 2β、α ± β（各二重简并），6 个 π 电子填满三个成键轨道，总能量比三个孤立双键低 2β，即离域能。

σ and π are separated and only p orbitals perpendicular to the plane are treated; all Coulomb integrals equal α; resonance integrals between neighbours equal β (both negative) and are zero otherwise; overlap S = 0 for i ≠ j. This gives the secular determinant |H − ES| = 0. Ethylene has E = α ± β; 1,3-butadiene α ± 1.618β and α ± 0.618β; benzene α ± 2β and α ± β (each doubly degenerate), and its six π electrons in the three bonding orbitals lie 2β lower than three isolated double bonds, the delocalisation energy.

1930–1931 年德国物理化学家埃里希·休克尔提出这一方法，用于解释苯的稳定性，并由此导出 4n+2 芳香性规则：单环共轭体系中，最低轨道之上的轨道成对简并，只有 4n+2 个 π 电子时才能形成闭壳层。弗罗斯特圆（1953 年）给出了一个快速画出单环体系 HMO 能级的几何方法。

The German physical chemist Erich Hückel introduced the method in 1930–31 to explain benzene's stability and derived from it the 4n+2 rule: in monocyclic conjugated systems the orbitals above the lowest come in degenerate pairs, so a closed shell forms only with 4n+2 π electrons. Frost's circle (1953) gives a quick geometric way to draw HMO levels for monocycles.

HMO 法用于计算 π 电子密度、键级和自由价，预测共轭分子的亲电/亲核反应位点和紫外吸收趋势；1965 年伍德沃德-霍夫曼规则和福井谦一的前线轨道理论都以休克尔轨道为基础分析周环反应的轨道对称性。扩展休克尔法（霍夫曼，1963）把方法推广到包含 σ 电子的所有价轨道。

HMO theory gives π-electron densities, bond orders and free valences, predicts electrophilic and nucleophilic sites and UV absorption trends of conjugated molecules, and underlies the 1965 Woodward–Hoffmann rules and Fukui's frontier orbital analysis of orbital symmetry in pericyclic reactions. The extended Hückel method (Hoffmann, 1963) generalises it to all valence orbitals, σ as well as π.