Introduction to Molecular Orbital Theory | 分子轨道理论入门

📚 Introduction to Molecular Orbital Theory | 分子轨道理论入门

Molecular Orbital (MO) Theory is one of the most powerful models in chemistry for understanding how atoms combine to form molecules. Unlike simpler models that treat electrons as localized between two atoms, MO theory describes electrons as spread across the entire molecule in delocalized orbitals.

分子轨道（MO）理论是化学中理解原子如何结合形成分子的最有力模型之一。与将电子视为定域在两个原子之间的简单模型不同，MO 理论将电子描述为在整个分子中离域分布的轨道中运动。

This theory explains phenomena that valence bond theory cannot, such as the paramagnetism of oxygen and the existence of molecular ions like He₂⁺. For IB Chemistry students, mastering MO theory is essential for understanding bonding, bond order, and magnetic properties.

该理论解释了价键理论无法解释的现象，例如氧气的顺磁性以及 He₂⁺ 等分子离子的存在。对于 IB 化学学生而言，掌握分子轨道理论是理解成键、键级和磁性质的关键。

1. Atomic Orbitals and Their Combination | 原子轨道及其组合

When two atomic orbitals overlap, they combine to form molecular orbitals. The number of molecular orbitals formed always equals the number of atomic orbitals that combine. This is known as the conservation of orbitals.

当两个原子轨道重叠时，它们组合形成分子轨道。形成的分子轨道数目始终等于组合的原子轨道数目，这称为轨道守恒原理。

‘Combination in phase’ produces a bonding molecular orbital (lower energy).

‘同相组合’产生成键分子轨道（能量较低）。
‘Combination out of phase’ produces an antibonding molecular orbital (higher energy).

‘异相组合’产生反键分子轨道（能量较高）。

This process can be represented mathematically as a linear combination of atomic orbitals (LCAO). The bonding orbital is stabilized by the constructive interference of wave functions, while the antibonding orbital is destabilized by destructive interference.

该过程可以用原子轨道的线性组合（LCAO）来表示。成键轨道通过波函数的相长干涉而稳定，反键轨道则因相消干涉而不稳定。

2. Bonding and Antibonding Orbitals | 成键与反键轨道

A bonding molecular orbital (σ or π) has a lower energy than the original atomic orbitals. Electrons placed here stabilize the molecule. The electron density is concentrated between the two nuclei, shielding the positive charges from each other.

成键分子轨道（σ 或 π）的能量低于原始原子轨道，电子填入此处可稳定分子。电子密度集中在两个原子核之间，屏蔽了彼此之间的正电荷排斥。

An antibonding molecular orbital (σ* or π*) has a higher energy. Electrons placed here destabilize the molecule. The electron density is localized behind each nucleus, and the nodal plane between the nuclei eliminates the shielding effect.

反键分子轨道（σ* 或 π*）具有较高的能量，电子填入此处会使分子不稳定。电子密度定域在核后方，核之间的节面消除了屏蔽效应。

Bonding: MO(σ) = ψₐ + ψᵦ | Antibonding: MO(σ*) = ψₐ − ψᵦ

3. Sigma (σ) and Pi (π) Orbitals | σ 与 π 轨道

Head-on (end-to-end) overlap of atomic orbitals produces sigma (σ) molecular orbitals. Sigma orbitals are cylindrical symmetric around the internuclear axis. They can form from s–s, s–p, or p–p head-on overlaps.

原子轨道的头对头（端对端）重叠产生 sigma（σ）分子轨道。σ 轨道绕核间轴呈圆柱对称，可由 s–s、s–p 或 p–p 的头对头重叠形成。

Sideways (lateral) overlap of p orbitals produces pi (π) molecular orbitals. π orbitals have a nodal plane containing the internuclear axis. Pi bonds are generally weaker than sigma bonds because the orbital overlap is less efficient.

p 轨道的侧向（横向）重叠产生 pi（π）分子轨道。π 轨道具有包含核间轴的节面。由于轨道重叠效率较低，π 键通常弱于 σ 键。

σ bond: s–s, s–p, p–p (axial) | π bond: p–p (lateral)

For p orbitals, we obtain one σ (from pz) and two π orbitals (from px and py) along each axis. The corresponding antibonding orbitals are labeled σ*, π*.

对于 p 轨道，沿每个轴我们获得一个 σ 轨道（由 pz 产生）和两个 π 轨道（由 px 和 py 产生）。相应的反键轨道标记为 σ*、π*。

4. Energy Level Diagrams | 能级图

The relative energies of molecular orbitals determine how electrons fill them. For homonuclear diatomic molecules of the second period, there are two possible ordering schemes depending on the energy gap between the 2s and 2p orbitals.

分子轨道的相对能量决定了电子的填充顺序。对于第二周期同核双原子分子，根据 2s 和 2p 轨道间的能隙大小，存在两种可能的排序方案。

Case 1 (O₂, F₂, Ne₂): The 2s–2p gap is large; the σ₂ₚ orbital lies above the π₂ₚ orbitals.

情况 1（O₂、F₂、Ne₂）： 2s–2p 能隙较大，σ₂ₚ 轨道的能量高于 π₂ₚ 轨道。

Case 2 (Li₂, Be₂, B₂, C₂, N₂): The 2s–2p gap is small; s–p mixing lowers the σ₂ₚ orbital below the π₂ₚ orbitals.

情况 2（Li₂、Be₂、B₂、C₂、N₂）： 2s–2p 能隙较小，s–p 混合使 σ₂ₚ 轨道的能量降低至 π₂ₚ 轨道之下。

O₂, F₂: σ₂ₛ < σ₂ₛ* < σ₂ₚ < π₂ₚ = π₂ₚ < π₂ₚ* = π₂ₚ* < σ₂ₚ*

B₂–N₂: σ₂ₛ < σ₂ₛ* < π₂ₚ = π₂ₚ < σ₂ₚ < π₂ₚ* = π₂ₚ* < σ₂ₚ*

5. Electron Filling Rules | 电子填充规则

Electrons fill molecular orbitals following the same three principles used for atomic orbitals: Aufbau principle (fill lowest energy first), Pauli exclusion principle (maximum two electrons per orbital with opposite spins), and Hund’s rule (degenerate orbitals are filled singly before pairing).

电子填充分子轨道遵循与原子轨道相同的三个原则：构造原理（先填充最低能量轨道）、泡利不相容原理（每个轨道最多容纳两个自旋相反的电子）以及洪特规则（简并轨道先单占后成对）。

These rules allow us to construct the electronic configuration of any diatomic molecule or ion. The configuration determines the bond order and magnetic behavior.

这些规则使我们能够构建任何双原子分子或离子的电子构型，构型决定键级和磁性行为。

6. Bond Order Calculation | 键级计算

Bond order (BO) is defined as half the difference between the number of bonding electrons and the number of antibonding electrons:

键级（BO）定义为成键电子数与反键电子数之差的一半：

Bond Order = (N_bonding − N_antibonding) / 2

For H₂, both electrons occupy the σ₁ₛ bonding orbital: BO = (2 − 0) / 2 = 1, indicating a single bond. For He₂, we would have BO = (2 − 2) / 2 = 0, predicting that He₂ does not exist.

对于 H₂，两个电子都占据 σ₁ₛ 成键轨道：BO = (2 − 0) / 2 = 1，表明为单键。对于 He₂，BO = (2 − 2) / 2 = 0，预测 He₂ 不存在。

A bond order of 1, 2, or 3 corresponds to single, double, and triple bonds respectively. Fractional bond orders (e.g., 1.5 in NO₃⁻) indicate resonance stabilization.

键级 1、2、3 分别对应单键、双键和三键。分数键级（如 NO₃⁻ 中的 1.5）表明存在共振稳定化。

7. Homonuclear Diatomic Molecules | 同核双原子分子

Let us examine a few key examples. N₂ has 10 valence electrons. The configuration is σ₂ₛ² σ₂ₛ*² π₂ₚ² π₂ₚ² σ₂ₚ², giving BO = (8 − 2) / 2 = 3. This explains nitrogen’s extremely strong triple bond and its chemical inertness.

我们来分析几个关键示例。N₂ 有 10 个价电子，构型为 σ₂ₛ² σ₂ₛ*² π₂ₚ² π₂ₚ² σ₂ₚ²，键级 BO = (8 − 2) / 2 = 3。这解释了氮气极强三键及其化学惰性。

O₂ has 12 valence electrons. The configuration is σ₂ₛ² σ₂ₛ*² σ₂ₚ² π₂ₚ² π₂ₚ² π₂ₚ*¹ π₂ₚ*¹, giving BO = (8 − 4) / 2 = 2. Importantly, the two π* electrons are unpaired (Hund’s rule), making O₂ paramagnetic.

O₂ 有 12 个价电子，构型为 σ₂ₛ² σ₂ₛ*² σ₂ₚ² π₂ₚ² π₂ₚ² π₂ₚ*¹ π₂ₚ*¹，键级 BO = (8 − 4) / 2 = 2。重要的是，两个 π* 电子未配对（洪特规则），使 O₂ 具有顺磁性。

Molecule	Valence e⁻	Configuration	BO	Magnetic Property
Li₂	2	σ₂ₛ²	1	Diamagnetic
B₂	6	σ₂ₛ² σ₂ₛ*² π₂ₚ¹ π₂ₚ¹	1	Paramagnetic
C₂	8	σ₂ₛ² σ₂ₛ*² π₂ₚ² π₂ₚ²	2	Diamagnetic
N₂	10	σ₂ₛ² σ₂ₛ*² π₂ₚ² π₂ₚ² σ₂ₚ²	3	Diamagnetic
O₂	12	σ₂ₛ² σ₂ₛ² σ₂ₚ² π₂ₚ² π₂ₚ² π₂ₚ¹ π₂ₚ*¹	2	Paramagnetic
F₂	14	σ₂ₛ² σ₂ₛ² σ₂ₚ² π₂ₚ² π₂ₚ² π₂ₚ² π₂ₚ*²	1	Diamagnetic

8. Heteronuclear Diatomic Molecules | 异核双原子分子

For heteronuclear molecules like CO or NO, the atomic orbitals of the two atoms have different energies. The more electronegative atom contributes lower-energy atomic orbitals, and its orbital character dominates the bonding MOs.

对于 CO 或 NO 等异核分子，两个原子的原子轨道能量不同。电负性较大的原子贡献能量较低的原子轨道，其轨道特征在成键 MO 中占主导。

CO is isoelectronic with N₂, having 10 valence electrons. The MO configuration is similar, and CO also has a bond order of 3. However, the orbitals are polarized: the bonding orbitals have more C character and the antibonding orbitals have more O character.

CO 与 N₂ 等电子，有 10 个价电子，MO 构型相似，CO 的键级也是 3。然而，轨道是极化的：成键轨道具有更多的 C 特征，反键轨道具有更多的 O 特征。

The concept of polarity in MO theory can be quantified by the mixing coefficient. A larger coefficient means greater contribution from that atomic orbital to the molecular orbital.

MO 理论中的极性概念可通过混合系数来量化，系数越大表示该原子轨道对分子轨道的贡献越大。

9. Predicting Stability and Existence | 预测稳定性和存在性

Molecules with a bond order greater than zero are predicted to be stable. A bond order of zero means the molecule cannot form. This explains why He₂, Be₂, and Ne₂ do not exist as stable species.

键级大于零的分子被预测为稳定。键级为零意味着分子无法形成。这解释了为什么 He₂、Be₂ 和 Ne₂ 不能以稳定物种存在。

Molecular ions can also be analyzed. For example, He₂⁺ has 3 electrons: σ₁ₛ² σ₁ₛ*¹, giving BO = (2 − 1) / 2 = 0.5. It is less stable than He₂⁺ with more bonding character but has been detected experimentally.

分子离子同样可以分析。例如，He₂⁺ 有 3 个电子：σ₁ₛ² σ₁ₛ*¹，键级 BO = (2 − 1) / 2 = 0.5。尽管弱于完整键，但实验上已检测到该物种。

Removing an electron from O₂ to form O₂⁺ increases the bond order from 2 to 2.5 and shortens the bond length. Likewise, adding an electron to form O₂⁻ decreases the bond order to 1.5 and lengthens the bond.

从 O₂ 移除一个电子形成 O₂⁺ 会使键级从 2 增加到 2.5，键长缩短。同样，增加一个电子形成 O₂⁻ 会使键级降至 1.5，键长增大。

10. Magnetic Properties | 磁性特征

MO theory uniquely explains magnetic behavior. A molecule with all electrons paired is diamagnetic (repelled by a magnetic field). A molecule with unpaired electrons is paramagnetic (attracted by a magnetic field).

MO 理论独特地解释了磁性行为。所有电子均配对的分子是抗磁性的（被磁场排斥）。具有未配对电子的分子是顺磁性的（被磁场吸引）。

Oxygen is paramagnetic, which is experimentally verifiable: liquid oxygen is attracted to a strong magnet. Valence bond theory with double bonds (all electrons paired) cannot explain this observation, while MO theory successfully predicts it.

氧气是顺磁性的，这可以通过实验验证：液氧会被强磁铁吸引。价键理论认为氧气是双键（所有电子配对）无法解释这一观察结果，而 MO 理论成功预测了该现象。

B₂ is also paramagnetic due to two unpaired electrons in degenerate π₂ₚ orbitals. Molecules like N₂, with all electrons paired, are diamagnetic.

B₂ 也因为简并 π₂ₚ 轨道中有两个未配对电子而呈顺磁性。像 N₂ 这样所有电子都配对的分子是抗磁性的。

11. MO Theory vs. Valence Bond Theory | MO 理论与价键理论的比较

Valence bond (VB) theory emphasizes localized bonds formed by overlapping atomic orbitals with paired electrons. It is intuitive for predicting molecular geometry and hybridization. However, it fails to explain O₂ paramagnetism and involves resonance for molecules like benzene.

价键（VB）理论强调由原子轨道重叠和电子配对形成的定域键，在预测分子几何构型和杂化方面直观有效。然而，它无法解释 O₂ 的顺磁性，并且对于苯等分子需要借助共振概念。

MO theory treats electrons as delocalized over the entire molecule. It naturally explains magnetic properties, bond order, and spectroscopic transitions. However, it is less intuitive for geometry prediction and harder to visualize.

MO 理论将电子视为在整个分子中离域，自然而然地解释了磁性、键级和光谱跃迁。但在几何预测上不够直观，且难以形象化。

In practice, chemists use both models. VB theory is preferred for explaining molecular shape and hybridization in organic chemistry; MO theory is essential for spectroscopy, magnetism, and reactions involving frontier orbitals.

实践中，化学家两者并用。VB 理论适合解释有机化学中的分子形状和杂化；MO 理论在光谱学、磁性以及涉及前线轨道的反应中不可或缺。

12. Applications and Exam Tips | 应用与考试要点

In IB Chemistry, MO theory appears in the Structure and Bonding topic. You should be able to draw energy level diagrams for second-period homonuclear diatomics, calculate bond order, and predict magnetic properties.

在 IB 化学中，MO 理论出现在结构与成键主题中。你应该能够绘制第二周期同核双原子分子的能级图、计算键级并预测磁性。

‘Always count electrons carefully before filling the MO diagram.’

‘填充 MO 图之前务必仔细数清电子数目。’
‘Remember the s–p mixing order for B₂, C₂, and N₂.’

‘记住 B₂、C₂ 和 N₂ 的 s–p 混合排序。’
‘Use Hund’s rule for degenerate π orbitals.’

‘对简并 π 轨道要运用洪特规则。’
‘Link bond order to bond length, bond energy, and stability.’

‘将键级与键长、键能和稳定性关联起来。’

Common exam questions ask you to explain why O₂ is paramagnetic or why He₂ does not exist. Practice by writing the full MO configuration for each species and calculating the bond order explicitly.

常见考题要求你解释为何 O₂ 为顺磁性或为何 He₂ 不存在。建议练习写出每个物种的完整 MO 构型并明确计算键级。

Mastering these concepts will not only earn you marks but also deepen your understanding of chemical bonding at a fundamental level.

掌握这些概念不仅能帮你得分，更能加深你对化学键合的本质理解。

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Introduction to Molecular Orbital Theory | 分子轨道理论入门