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\begin{document}

Here's some atomic physics perturbation theory formulas:

\( <n_{1}l_{1}n_{2}l_{2}LM|H_{1}|n_{3}l_{3}n_{4}l_{4}LM> \)

\( =\sum _{m_{1},m_{2},m_{3},m_{4}}=<l_{1}l_{2}m_{1}m_{2}|LM><l_{3}l_{4}m_{3}m_{4}|LM>* \)

\( <n_{1}l_{1}m_{1}n_{2}l_{2}m_{2}|H_{1}|n_{3}l_{3}m_{3}n_{4}l_{4}m_{4}> \)

The last integral is: \( =\sum _{lm}\int r_{1}^{2}dr_{1}\int r_{2}^{2}dr_{2}R_{(nl)_{1}}(r_{1})R_{(nl)_{2}}(r_{2})\frac{r_{min}^{l}}{r^{l+1}_{max}}R_{(nl)_{3}}(r_{1})R_{(nl)_{4}}(r_{2}) \)

\( *\frac{e^{2}4\pi }{2l+1}(\sqrt{\frac{(2l_{1}+1)(2l+1)}{(2l_{3}+1)4\pi }}<l_{1}l00|l_{3}0><l_{1}lm_{1}m|l_{3}m_{3}>=\int Y^{*}_{(lm)_{1}}Y^{*}_{(lm)}Y^{*}_{(lm)_{3}})* \)something similar

\end{document}
