含义Meaning
d 电子进入八面体场中分裂的 \(t_{2g}\)（降低 0.4Δo）和 \(e_g\)（升高 0.6Δo）轨道后，相对球形场的能量变化。P 为每对额外成对电子的成对能。
Energy change of d electrons in the split \(t_{2g}\) (−0.4Δo) and \(e_g\) (+0.6Δo) levels relative to a spherical field; P is the pairing energy per extra pair.
符号说明Symbols
| 符号Symbol | 含义 Meaning | 单位 Unit |
|---|---|---|
| \(\Delta_\mathrm{o}\) | 八面体分裂能octahedral splitting | cm⁻¹, kJ·mol⁻¹ |
| \(n_{t_{2g}}, n_{e_g}\) | 两组轨道上的电子数electrons in each set | 1 |
| \(P\) | 电子成对能pairing energy | kJ·mol⁻¹ |
| \(m\) | 比自由离子多出的电子对数extra pairs relative to the free ion | 1 |
推导Derivation
六个配体沿 ±x、±y、±z 接近，指向配体的 \(d_{z^2}\)、\(d_{x^2-y^2}\)（\(e_g\)）受排斥升高，夹在轴间的 \(d_{xy}, d_{xz}, d_{yz}\)（\(t_{2g}\)）降低。
Ligands on ±x, ±y, ±z repel \(d_{z^2}, d_{x^2-y^2}\) (\(e_g\)) more than \(d_{xy}, d_{xz}, d_{yz}\) (\(t_{2g}\)).
重心规则：分裂前后平均能量不变，\(3E(t_{2g}) + 2E(e_g) = 0\)，且 \(E(e_g) - E(t_{2g}) = \Delta_\mathrm{o}\)，解得 \(-0.4\Delta_\mathrm{o}\) 与 \(+0.6\Delta_\mathrm{o}\)。
Barycentre rule: \(3E(t_{2g}) + 2E(e_g) = 0\) with \(E(e_g) - E(t_{2g}) = \Delta_\mathrm{o}\) gives −0.4Δo and +0.6Δo.
d⁴–d⁷ 有高、低自旋两种排布：\(\Delta_\mathrm{o} > P\) 时电子成对进入 \(t_{2g}\)（低自旋，强场），\(\Delta_\mathrm{o} < P\) 时先占 \(e_g\)（高自旋，弱场）。
d⁴–d⁷ have high- and low-spin options: \(\Delta_\mathrm{o} > P\) favours low spin (strong field), \(\Delta_\mathrm{o} < P\) high spin.
CFSE 解释第一过渡系 M²⁺ 水合焓、晶格能的“双峰”曲线（d³、d⁸ 附近最负，d⁰、d⁵、d¹⁰ 为零）。
CFSE explains the double-humped hydration and lattice enthalpies of M²⁺ (most negative near d³ and d⁸, zero at d⁰, d⁵, d¹⁰).
适用条件与局限Assumptions & limitations
晶体场模型把配体视为点电荷，忽略共价性；配体场理论 / 分子轨道理论更完整（π 给体、π 受体效应）。
Treats ligands as point charges; ligand-field/MO theory adds covalency and π effects.
Jahn–Teller 畸变（d⁹、高自旋 d⁴、低自旋 d⁷）会进一步分裂能级。
Jahn–Teller distortion (d⁹, HS d⁴, LS d⁷) splits levels further.
例题Worked example
[Fe(CN)₆]⁴⁻（d⁶ 低自旋）：CFSE = −2.4Δo + 2P；[Fe(H₂O)₆]²⁺（d⁶ 高自旋）：−0.4Δo。
[Fe(CN)₆]⁴⁻ (LS d⁶): −2.4Δo + 2P; [Fe(H₂O)₆]²⁺ (HS d⁶): −0.4Δo.