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%% LyX 0.10 (C) 1995 1996 by Matthias Ettrich and the LyX Team

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\begin{document}

For the TIPT of a 2 electrion atom, \( H_{0}=\frac{p_{1}^{2}}{2m}+\frac{p_{2}^{2}}{2m}-\frac{Ze^{2}}{r_{1}}-\frac{Ze^{2}}{r_{2}} \), \( H_{1}=\frac{e^{2}}{|r_{1}^{i}-r_{2}^{i}|} \). In general \( \frac{H_{1}}{H_{0}}\sim \frac{1}{Z} \). Oth order
solution is \( \psi (r_{1}^{i},r_{2}^{j})=\psi (r_{1}^{i})\psi (r_{2}^{i}) \) where \( \psi (r_{1}^{i})=R_{nl}(r)Y_{lm}(\widehat{r}) \) and \( E_{n}=\frac{-Z^{2}}{n^{2}}R_{y} \) where \( R_{y}=\frac{me^{4}}{2\hbar ^{2}}=\frac{1}{2}\alpha ^{2}m_{e}c^{2}=13.6eV \). For 2 electrons, \( E_{n}=-Z^{2}R_{y}(\frac{1}{n_{1}^{2}}+\frac{1}{n_{2}^{2}}) \). Lowest state
has degeneracy 0. For first excited state, degeneracy is 16.

\( \triangle 0^{(1)}=<\Psi _{0}|H_{1}|\Psi _{0}>=\int d^{3}r_{1}d^{3}r_{2}\frac{\rho (r_{1})\rho _{2}(r)}{|r^{i}_{1}-r_{2}^{i}|} \) for \( \rho (r^{i})=e\frac{e^{-2r/a}}{\pi a^{3}} \) for \( a=\frac{a_{0}}{Z} \). 

\begin{itemize}

\item Multipole expansion of Coulomb interaction: \( \frac{1}{|r_{1}^{i}-r_{2}^{i}|}=\sum _{l=0}^{\infty }\sum _{m=-l}^{m=l}\frac{4\pi }{2l+1}Y^{*}_{lm}(\widehat{r_{1}})Y_{lm}(\widehat{r_{2}})\frac{r_{min}^{l}}{r_{max}^{l+1}} \) where \( r_{min}=min(r_{1}+r_{2}) \) so \( \triangle 0^{(1)}=e^{2}\sum _{l,m}\frac{4\pi }{2l+1}[\int _{0}^{\infty }r_{1}^{2}dr_{1}\rho (r_{1})\int _{0}^{\infty }r_{2}^{2}dr_{2}\rho (r_{2})\frac{r_{min}^{l}}{r_{max}^{l+1}}][sphericalpart] \)

\end{itemize}

\( =e^{2}16\pi ^{2}\int _{0}^{\infty }r_{1}^{2}dr_{1}\rho (r_{1})\int _{0}^{\infty }r_{2}^{2}dr_{2}\rho (r_{2})\frac{r_{min}^{l}}{r_{max}^{l+1}} \)

\end{document}
