这都是教科书上都有的内容，我只是整理整理，把一些约定也统一一下，方便以后写代码的时候参照。
1. 约定
1.1 约化矩阵约定
\[\langle J‘M‘ | s \sigma | J M \rangle = \langle J‘ || s || J \rangle (JM,s \sigma| J‘ M‘).
\]
1.2 时间反演算符约定
\[\tilde{b}_\beta = (-1)^{ b + \beta } b_{-\beta}.
\]
2. 单体算符的二次量子化
\[Q_{s\sigma} = \sum^N_{i=1} q(\vec{r}_i) = \sum_{\alpha, \beta} \langle \alpha | q | \beta \rangle
\alpha^\dagger \beta.
= \sum_{\alpha, \beta} \langle \alpha || q || \beta \rangle \frac{ [ j_\alpha ] }{ [t] } ( \alpha^\dagger \otimes \tilde{\beta} )_{s \sigma}.
\]
这个容易推导，我自己写了笔记。
3. E2 跃迁的约化矩阵元
根据 Lawson 的书 P435(电子版445页)，
\[\langle \alpha || Y_\lambda || \beta \rangle = (-1)^{l_\beta + l_\alpha + j_\beta - j_\alpha} \sqrt{ \frac{ (2\lambda+1)(2j_\beta+1) }{ 4\pi (2j_\alpha+1) } } ( j_\beta \frac{1}{2}; \lambda 0 | j_\alpha \frac{1}{2} ) \frac{ 1 + (-1)^{l_\alpha + l_\beta + \lambda} }{2} \int R_\alpha R_\beta r^2 dr.
\]
E2 跃迁的跃迁算符是
\[Q_{2\mu} = r^2 Y_{2\mu} = \sum_{\alpha \beta } q(\alpha \beta) ( \alpha^\dagger \otimes \tilde{b} )_{2 \mu}, \q(\alpha \beta) = \frac{[j_\alpha]}{\sqrt{5}} (-1)^{l_\beta + l_\alpha + j_\beta - j_\alpha} \sqrt{ \frac{ (2\lambda+1)(2j_\beta+1) }{ 4\pi (2j_\alpha+1) } } ( j_\beta \frac{1}{2}; \lambda 0 | j_\alpha \frac{1}{2} ) \frac{ 1 + (-1)^{l_\alpha + l_\beta + \lambda} }{2} \int R_\alpha R_\beta r^4 dr.
\]