分子轨道
量子化学维基，人人都可编辑的量子化学百科全书。
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在 量子化学中, 分子中的电子的量子态, 例如分子中电子的哈密顿算符的本征态, 可以展开(见 组态相互作用 展开和 基矢量) 成为 电子 波函数的反对称乘积(Slater行列式) 的线性组合. 这些单电子函数的空间部分称为分子轨道 (MO). 若还考虑电子的自旋部分, 则称之为分子自旋轨道.
目前计算化学的许多方法都是由计算体系的分子轨道开始的。一个分子轨道描述一个电子在由体系的原子核以及其他电子平均分布所形成的电场中的运动行为。在两个电子占据同一个轨道的情况中，泡利原理要求它们自旋相反。
目录 |
示意性讨论
对分子结构的示意讨论(并不绝对精确，但是定性有用)，分子轨道可以由"原子轨道线性组合"方法获得拟设 (这里使用了杂化轨道的概念).
在这种方法中，分子轨道被表示成原子轨道的线性组合，就像每个原子有自己的轨道一样。
分子轨道的原子轨道线性组合的近似是由John Lennard-Jones在1929年提出的。 他那篇突破性的论文展示了如何从量子原理获得氟和氧分子的分子结构。这种分子轨道定性的方法可谓是量子化学的曙光。
分子轨道的一些性质:
- 分子轨道的数目等于用于线性展开的原子轨道数。
- 如果分子有某种对称性，那么分子轨道的简并性(也就是说具有相同的原子能量)在线性组合过程中也必须符合(称作 对称性匹配的自旋轨道 (SO))，它属于对称群的有限群表示。
- 属于某个群表示的分子轨道的数目等于属于这个群表示的自旋匹配的原子轨道数目
- 在特定的有限群表示中,原子能级越接近，对称性匹配的原子轨道杂化得越好。
例子
H2
这里氢气分子可以作为一个简单的例子， H2, 它有两个氢原子，分别称为H' and H"。它们原子轨道的基态称作 1s'和1s",由分子的对称性可以看到，它们不符合对称. 然而下面的对称性匹配原子轨道符合：
| 1s' - 1s" | 反对称组合: 映射操作之后符号相反，其他对称操作作用不变 |
|---|---|
| 1s' + 1s" | 对称组合: 任何对称操作符号都不变 |
The symmetric combination (called a bonding orbital) is lower in energy than the basis orbitals, and the antisymmetric combination (called an antibonding orbital) is higher. Because the H2 molecule has two electrons, they can both go in the bonding orbital, making the system lower in energy (and hence more stable) than two free hydrogen atoms. This is called a covalent bond.
H3
On the other hand, consider the molecule of H3 (belonging to the C2v group), with the atoms labelled H, H' (the atom along the axis of symmetry), H". Then we would expect three linear combinations:
- 1s - 1s' + 1s" Symmetric Anti Bonding (2 nodal surfaces perpendicular to the bonds)
- 1s - 1s" Antisymmetric Non bonding (1 nodal surface along the axis of symmetry)
- 1s + 1s' + 1s" Symmetric Bonding (0 nodal surface)
Two electrons occupy the symmetric bonding bonding orbital and the third one occupy the non bonding orbital.
Rare gases
Now let's move to larger atoms. Considering a hypothetical molecule of He2, since the basis set of atomic orbitals is the same as in the case of H2, we find that both the bonding and antibonding orbitals are filled, so there is no energy advantage to the pair. HeH would have a slight energy advantage, but not as much as H2 + 2 He, so the molecule exists only a short while. In general, we find that atoms such as He that have completely full energy shells rarely bond with other atoms. (In fact there is not a single stable molecule containing He, Ne or Ar except short-lived Van der Waals complexes.)
Inner shells
Inner shell orbitals should not be included in the LCAO expansion. Molecular structure relies on the outermost (valence) electrons of the atoms, which are usually of comparable energy.
Ionic bonds
- see main article ionic bond
When the energy difference between the atomic orbitals of two atoms is quite large, one atom's orbitals contribute almost entirely to the bonding orbitals, and the other's almost entirely to the antibonding orbitals. Thus, the situation is effectively that some electrons have been transferred from one atom to the other. This is called a (mostly) ionic bond.
More quantitative approach
To obtain quantitative values for the molecular energy levels, one needs to have molecular orbitals which are such that the configuration interaction (CI) expansion converges fast towards the full CI limit. The most common method to obtain such functions is the Hartree-Fock method which expresses the molecular orbitals as eigenfunctions of the Fock operator. One usually solves this problem by expanding the molecular orbitals as linear combinations of gaussian functions centered on the atomic nuclei (see linear combination of atomic orbitals and basis set (chemistry)). The equation for the coefficients of these linear combinations is a generalized eigenvalue equation known as the Roothaan equations which are in fact a particular representation of the Hartree-Fock equation.
See also
External links
- Visualizations of some atomic and molecular orbitals (Note: These visualisations run only on Apple Mac.)
- Java molecular orbital viewer shows orbitals of hydrogen molecular ion.wp:Molecular orbital
