📚 Molecular Orbitals | 分子轨道
Molecular orbital (MO) theory provides a powerful model for understanding covalent bonding, electronic structure, and magnetic properties of molecules. Unlike simpler models, it describes electrons as delocalised over the entire molecule, occupying molecular orbitals formed by the linear combination of atomic orbitals. This approach explains phenomena that valence bond theory cannot, such as the paramagnetism of oxygen and the relative energies of bonding in diatomic molecules. In IB Chemistry HL, MO theory is central to understanding bonding beyond Lewis structures.
分子轨道 (MO) 理论提供了一个强有力的模型，用于理解共价键、电子结构和分子的磁性。与较简单的模型不同，它将电子描述为离域在整个分子之上，占据由原子轨道线性组合形成的分子轨道。这一方法解释了价键理论无法解释的现象，例如氧气的顺磁性以及双原子分子中成键能量的相对差异。在 IB 化学 HL 中，分子轨道理论是理解超越路易斯结构的成键的核心。
1. Fundamentals of Molecular Orbital Theory | 分子轨道理论基础
Molecular orbital theory views a molecule as a unified entity with electrons moving under the influence of all nuclei. Atomic orbitals from the constituent atoms combine to form a new set of molecular orbitals, each with a specific energy and shape. The total number of molecular orbitals formed equals the number of atomic orbitals combined. Electrons fill these molecular orbitals according to the Aufbau principle, Hund’s rule, and the Pauli exclusion principle, just as they do in atoms.
分子轨道理论将分子视为统一的实体，电子在所有原子核的影响下运动。来自组成原子的原子轨道组合形成一组新的分子轨道，每个轨道具有特定的能量和形状。形成的分子轨道总数等于组合的原子轨道数量。电子按照构造原理、洪特规则和泡利不相容原理填充这些分子轨道，就像在原子中一样。
2. Linear Combination of Atomic Orbitals (LCAO) | 原子轨道线性组合 (LCAO)
Molecular orbitals are constructed by the linear combination of atomic orbitals (LCAO). When two atomic orbitals overlap, they can combine constructively (in phase) or destructively (out of phase). Constructive overlap produces a bonding molecular orbital with increased electron density between the nuclei, leading to a lower energy state. Destructive overlap generates an antibonding molecular orbital, which has a nodal plane between the nuclei and higher energy, destabilising the molecule if occupied.
分子轨道由原子轨道线性组合 (LCAO) 构成。当两个原子轨道重叠时，它们可以同相（相位相同）或异相（相位相反）组合。同相重叠产生成键分子轨道，原子核间电子密度增加，能量较低。异相重叠生成反键分子轨道，原子核间存在节面，能量较高，若被占据则使分子不稳定。
3. Bonding and Antibonding Orbitals | 成键与反键轨道
In a bonding orbital, the increased electron density between the positively charged nuclei provides an attractive force that binds the atoms. These orbitals are designated with a symbol such as σ (sigma) or π (pi) without an asterisk. Antibonding orbitals are marked with an asterisk (*) and contain a node that reduces electron density in the internuclear region. When electrons occupy antibonding orbitals, they counteract the stabilising effect of bonding electrons, weakening or preventing the bond.
在成键轨道中，带正电的原子核之间电子密度增加，提供的吸引力将原子结合在一起。这些轨道用 σ 或 π 符号标记，不带星号。反键轨道则标以星号 (*)，含有一个节面，降低了核间区域的电子密度。当电子占据反键轨道时，会抵消成键电子的稳定作用，削弱或阻止键的形成。
4. Sigma (σ) and Pi (π) Molecular Orbitals | σ 和 π 分子轨道
Sigma molecular orbitals are formed by head-on overlap of atomic orbitals along the internuclear axis. They are cylindrically symmetrical around this axis, with no nodal plane containing the bond axis. A σ orbital can be formed from s-s overlap, s-p end-on overlap, or p-p end-on overlap. Pi molecular orbitals result from the side-by-side overlap of parallel p orbitals. A π orbital possesses a nodal plane that contains the internuclear axis, and electron density lies above and below this plane.
σ 分子轨道由原子轨道沿核间轴“头碰头”重叠形成。它们围绕该轴呈圆柱对称，不含包含键轴的节面。σ 轨道可由 s-s 重叠、s-p 端向重叠或 p-p 端向重叠形成。π 分子轨道由平行的 p 轨道侧向重叠产生。π 轨道具有一个包含核间轴的节面，电子密度分布在该平面的上方和下方。
σ: cylindrical symmetry | π: nodal plane along the bond axis
5. Molecular Orbital Diagrams for Homonuclear Diatomics | 同核双原子分子的分子轨道能级图
For homonuclear diatomic molecules of the first and second periods, MO diagrams illustrate the filling order of molecular orbitals. The relative energies of σ and π orbitals can change depending on the degree of s–p mixing. For early elements (Li₂ through N₂), the π₂ₚ orbitals are lower in energy than the σ₂ₚ orbital due to mixing of the 2s and 2p atomic orbitals. For O₂, F₂, and Ne₂, the σ₂ₚ orbital lies below the π₂ₚ orbitals because of increased nuclear charge reducing s–p mixing.
对于第一和第二周期的同核双原子分子，MO 能级图展示了分子轨道的填充顺序。σ 和 π 轨道的相对能量可能因 s-p 混杂程度而改变。对于前几个元素（Li₂ 至 N₂），由于 2s 和 2p 原子轨道的混杂，π₂ₚ 轨道的能量低于 σ₂ₚ 轨道。而对于 O₂、F₂ 和 Ne₂，由于核电荷增大减弱了 s-p 混杂，σ₂ₚ 轨道位于 π₂ₚ 轨道之下。
| Li₂ – N₂ order | σ₁ₛ < σ*₁ₛ < σ₂ₛ < σ*₂ₛ < π₂ₚ < σ₂ₚ < π*₂ₚ < σ*₂ₚ |
| O₂ – Ne₂ order | σ₁ₛ < σ*₁ₛ < σ₂ₛ < σ*₂ₛ < σ₂ₚ < π₂ₚ < π*₂ₚ < σ*₂ₚ |
6. s–p Mixing and Orbital Energy Crossover | s-p 混杂与轨道能量交错
The phenomenon of s– rbital (along the bond axis). This mixing lowers the energy of the σ₂ₛ and σ*₂ₛ orbitals, while raising the energy of the σ₂ₚ orbital above the π₂ₚ level. In O₂ and F₂, the larger 2s– ing, so the σ₂ₚ orbital remains lower than π₂ₚ, following a more straightforward ordering.
当 2s 和 2p 原子轨道之间的能量间距足够小而发生相互作用时，就会产生 s-p 混杂现象。在 Li₂、Be₂、B₂、C₂ 和 N₂ 中，2s 轨道与 2p_z 原子轨道（沿键轴）发生混杂。这种混杂降低了 σ₂ₛ 和 σ*₂ₛ 轨道的能量，同时将 σ₂ₚ 轨道的能量提升至 π₂ₚ 能级之上。在 O₂ 和 F₂ 中，较大的 2s-2p 能隙抑制了混杂，因此 σ₂ₚ 轨道保持在 π₂ₚ 之下，遵循更简单的填充顺序。
7. Electronic Configurations and Bond Order | 电子排布与键级
opriate energy order. Bond order is calculated as half the difference between the number of electrons in bonding orbitals and antibonding orbitals. A bond order greater than zero indicates a stable molecule; the higher the bond order, the stronger and shorter the bond. For example, N₂ has configuration (σ₁ₛ)²(σ*₁ₛ)²(σ₂ₛ)²(σ*₂ₛ)²(π₂ₚ)⁴(σ₂ₚ)², yielding a bond order of (8 – 2)/2 = 3, consistent with a triple bond.
双原子分子的电子排布通过将总价电子数按适当的能量顺序填入分子轨道来书写。键级按成键轨道电子数与反键轨道电子数之差的一半计算。键级大于零表示分子稳定；键级越高，键能越大、键长越短。例如，N₂ 的排布为 (σ₁ₛ)²(σ*₁ₛ)²(σ₂ₛ)²(σ*₂ₛ)²(π₂ₚ)⁴(σ₂ₚ)²，键级为 (8 - 2)/2 = 3，与三键相符。
键级 = (成键电子数 - 反键电子数) ÷ 2
8. Magnetic Properties: Paramagnetism and Diamagnetism | 磁性：顺磁性与反磁性
two unpaired electrons in the degenerate π*₂ₚ antibonding orbitals, following Hund’s rule. Valence bond theory fails to predict this property, as the Lewis structure O=O shows all electrons paired. Similarly, B₂ is paramagnetic with two unpaired electrons in π₂ₚ orbitals.
如果分子含有未成对电子，它就具有顺磁性，会被磁场吸引。如果所有电子均已配对，则呈反磁性，会被磁场微弱排斥。分子轨道理论解释了为什么 O₂ 具有顺磁性：其电子排布 (σ₂ₚ)²(π₂ₚ)⁴(π*₂ₚ)² 在简并的 π*₂ₚ 反键轨道中有两个未成对电子，符合洪特规则。价键理论无法预测这一性质，因为路易斯结构 O=O 显示所有电子均已配对。类似地，B₂ 在 π₂ₚ 轨道中有两个未成对电子，也是顺磁性的。
9. Homonuclear Diatomic Molecules: Key Examples | 同核双原子分子关键实例
H₂: (σ₁ₛ)², bond order 1, diamagnetic. He₂: (σ₁ₛ)²(σ*₁ₛ)², bond order 0, does not exist. Li₂: core electrons plus (σ₂ₛ)², bond order 1, diamagnetic. Be₂ has bond order 0. B₂: (π₂ₚ)², bond order 1, paramagnetic. C₂: (π₂ₚ)⁴, bond order 2, diamagnetic. N₂: triple bond, diamagnetic. O₂: bond order 2, paramagnetic. F₂: bond order 1, diamagnetic. Ne₂: bond order 0. These trends demonstrate how filling antibonding orbitals progressively weakens bonding.
H₂：(σ₁ₛ)²，键级 1，反磁性。He₂：(σ₁ₛ)²(σ*₁ₛ)²，键级 0，不能存在。Li₂：内层电子加上 (σ₂ₛ)²，键级 1，反磁性。Be₂ 键级为 0。B₂：(π₂ₚ)²，键级 1，顺磁性。C₂：(π₂ₚ)⁴，键级 2，反磁性。N₂：三键，反磁性。O₂：键级 2，顺磁性。F₂：键级 1，反磁性。Ne₂：键级 0。这些趋势表明，填充反键轨道会逐步削弱成键。
10. Heteronuclear Diatomic Molecules | 异核双原子分子
In heteronuclear diatomic molecules like CO or NO, the atomic orbitals of the more electronegative atom lie lower in energy. The resulting molecular orbitals are not symmetrically distributed; bonding orbitals have greater contribution from the more electronegative atom, while antibonding orbitals are weighted toward the less electronegative element. CO is isoelectronic with N₂, having a (σ)²(σ*)²(π)⁴(σ)² configuration, bond order 3, and a small dipole moment due to the asymmetry. NO has a bond order of 2.5 and is paramagnetic, with one unpaired electron in a π* orbital.
在 CO 或 NO 等异核双原子分子中，电负性较高的原子的原子轨道能量较低。形成的分子轨道并非对称分布；成键轨道中电负性较高原子的贡献更大，而反键轨道则更偏向电负性较低的元素。CO 与 N₂ 是等电子体，电子排布为 (σ)²(σ*)²(π)⁴(σ)²，键级为 3，并因不对称而具有较小的偶极矩。NO 的键级为 2.5，具有顺磁性，π* 轨道中有一个未成对电子。
11. Comparison with Valence Bond Theory | 与价键理论的比较
Valence bond theory describes a chemical bond as the overlap of localised atomic orbitals, with resonance used to explain delocalisation. Molecular orbital theory inherently treats electrons as delocalised across the entire molecule. MO theory naturally explains the paramagnetism of O₂ and the fractional bond orders of species like He₂⁺. It also provides a more consistent treatment of excited states and spectroscopy. However, valence bond theory often gives a more intuitive picture of molecular geometry and hybridisation.
价键理论将化学键描述为定域原子轨道的重叠，并用共振解释离域现象。分子轨道理论则固有地将电子视为在整个分子中离域。MO 理论自然地解释了 O₂ 的顺磁性以及 He₂⁺ 等物种的非整数键级。它还对激发态和光谱提供了更一致的解释。然而，价键理论通常能更直观地展示分子几何构型和杂化。
12. Applications and Summary | 应用与总结
Molecular orbital theory is essential for predicting bond stability, bond length, magnetic behaviour, and electronic spectra. It gives insight into why certain molecules exist while others do not, and it underpins advanced topics in inorganic and physical chemistry. In the IB HL syllabus, mastering MO diagrams for first- and second-period diatomics, calculating bond orders, and linking electronic configuration to magnetic properties are key skills. By embracing the delocalised nature of electrons, MO theory offers a complete and predictive model for chemical bonding.
分子轨道理论对于预测键的稳定性、键长、磁行为和电子光谱至关重要。它能够揭示为什么某些分子可以存在而另一些不能，并为无机化学和物理化学中的进阶主题奠定基础。在 IB HL 教学大纲中，掌握第一和第二周期双原子分子的 MO 能级图、计算键级以及将电子排布与磁性联系起来是关键技能。通过接纳电子的离域特性，MO 理论为化学键提供了一个完整且具有预测性的模型。
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