含义Meaning
任何试探波函数算出的能量期望值都不低于真实基态能量。于是可以调节参数使能量最小，逼近真实解——几乎所有量子化学计算方法的基础。
The energy expectation value of any trial function is never below the true ground-state energy, so minimising it over parameters approaches the true solution — the basis of most quantum-chemical methods.
符号说明Symbols
| 符号Symbol | 含义 Meaning | 单位 Unit |
|---|---|---|
| \(\tilde\psi\) | 试探波函数trial wavefunction | |
| \(E_0\) | 真实基态能量exact ground-state energy | J |
| \(H_{ij}, S_{ij}\) | 哈密顿矩阵元与重叠积分Hamiltonian and overlap matrix elements | J, 1 |
推导Derivation
把 \(\tilde\psi\) 按真实本征函数展开：\(\tilde\psi = \sum c_n\psi_n\)。
Expand the trial function in exact eigenfunctions.
\(E[\tilde\psi] = \sum|c_n|^2E_n / \sum|c_n|^2\)，而 \(E_n \ge E_0\)。
Then \(E[\tilde\psi] = \sum|c_n|^2E_n/\sum|c_n|^2\) with \(E_n \ge E_0\).
所以 \(E[\tilde\psi] - E_0 = \sum|c_n|^2(E_n - E_0)/\sum|c_n|^2 \ge 0\)。
Hence the difference is non-negative.
线性变分：\(\tilde\psi = \sum c_j\phi_j\)，对 \(c_i\) 求极值得久期方程，解出的最低根是基态能量上限（LCAO-MO 的来源）。
Linear variation leads to the secular equation; its lowest root bounds the ground state — the origin of LCAO-MO.
适用条件与局限Assumptions & limitations
给出能量上限，但能量好不代表其他性质（偶极矩等）同样准确。
Gives an energy upper bound; other properties may be less accurate.
激发态需与更低态正交才有类似结论。
Excited states need orthogonality to lower states.
例题Worked example
氢原子用高斯试探函数 \(e^{-\alpha r^2}\)：最优 E = −0.424 Eₕ（真实 −0.5 Eₕ），高于真实值。
H atom with a Gaussian \(e^{-\alpha r^2}\): best E = −0.424 Eₕ, above the exact −0.5 Eₕ.