含义Meaning
把多电子问题化为每个电子在其余电子平均场中运动的单电子方程，用自洽场（SCF）迭代求解。罗特汉（1951）将轨道按基函数展开，变为矩阵本征值问题，是现代量子化学程序的核心。
Turns the many-electron problem into one-electron equations in the mean field of the others, solved self-consistently (SCF). Roothaan (1951) expanded orbitals in a basis, giving a matrix eigenproblem at the heart of quantum-chemistry codes.
符号说明Symbols
| 符号Symbol | 含义 Meaning | 单位 Unit |
|---|---|---|
| \(\hat f\) | 福克算符Fock operator | J |
| \(\hat h\) | 单电子算符（动能 + 核吸引）one-electron operator | J |
| \(\hat J, \hat K\) | 库仑与交换算符Coulomb and exchange operators | J |
| \(\mathbf C\) | MO 系数矩阵MO coefficient matrix | 1 |
| \(\mathbf S\) | 基函数重叠矩阵basis-overlap matrix | 1 |
| \(\varepsilon_i\) | 轨道能orbital energies | J |
推导Derivation
取单个斯莱特行列式为试探函数，能量 \(E = \sum_i h_{ii} + \tfrac12\sum_{ij}(J_{ij} - K_{ij})\)。
Use one Slater determinant; its energy involves h, J and K integrals.
在轨道正交约束下（拉格朗日乘子）对轨道变分，得哈特里–福克方程。
Vary the orbitals subject to orthonormality (Lagrange multipliers) to obtain the HF equations.
福克算符依赖于占据轨道本身，所以要迭代：猜轨道 → 构造 F → 解 → 更新，直到自洽。
F depends on the orbitals, so iterate until self-consistent.
基组展开 \(\chi_i = \sum_\mu C_{\mu i}\phi_\mu\) 得罗特汉方程 FC = SCε。
Basis expansion gives FC = SCε.
库普曼斯定理：\(-\varepsilon_\text{HOMO}\) 近似等于第一电离能。
Koopmans: \(-\varepsilon_\text{HOMO}\) approximates the first ionisation energy.
适用条件与局限Assumptions & limitations
缺失电子相关能（通常占总能量约 1%，但对键能、反应能至关重要）；HF 极限能量高于真实值。
Misses correlation energy (~1 % of the total but crucial for bond and reaction energies).
结果受基组大小限制；后 HF 方法（MP2、CCSD(T)）或密度泛函理论用于提高精度。
Basis-set limited; post-HF (MP2, CCSD(T)) or DFT improve accuracy.