refine 7-9

jeanwsr · Aug 10, 2020 · af57436 · af57436
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diff --git a/chap7/chap7.pdf b/chap7/chap7.pdf
diff --git a/chap7/chap7.tex b/chap7/chap7.tex
@@ -500,6 +500,22 @@ \subsection{Application of the Formalism to \ce{H_2} and \ce{HeH^+}}
 \varepsilon_{22}^\pm &= \varepsilon_1 \pm \sqrt{(\varepsilon_1 - \varepsilon_2)^2 + K_{12}^2} \notag\\
 &= \varepsilon_2 - \qty[(\varepsilon_2 - \varepsilon_1) \mp \sqrt{(\varepsilon_2 - \varepsilon_1)^2 + K_{12}^2}]
 \end{align}
+\subex{d.}
+\begin{align}
+\varepsilon_{22}^+ &= \varepsilon_2 - \qty[(\varepsilon_2 - \varepsilon_1) - \sqrt{(\varepsilon_2 - \varepsilon_1)^2 + K_{12}^2}] \notag\\
+&\approx \varepsilon_2 - \qty[\Delta - \sqrt{\Delta^2 + K_{12}^2}] \notag\\
+&= \varepsilon_2 - {^N}E_{\corr} \notag\\
+&= {^{N+1}}\mathscr{E}_0 - {^N}\mathscr{E}_0
+\end{align}
+\begin{align}
+\varepsilon_{22}^- &= \varepsilon_2 - \qty[(\varepsilon_2 - \varepsilon_1) + \sqrt{(\varepsilon_2 - \varepsilon_1)^2 + K_{12}^2}] \notag\\
+&\approx \varepsilon_2 + \qty[-(\varepsilon_2 - \varepsilon_1) -\Delta + \Delta - \sqrt{\Delta^2 + K_{12}^2}] \notag\\
+&= \varepsilon_2 - (\varepsilon_2 - \varepsilon_1) - \Delta - {^N}E_{\corr} \notag\\
+&= \varepsilon_2 - (\varepsilon_2 - \varepsilon_1) - \qty(\varepsilon_2 - \varepsilon_1 + \dfrac{1}{2}(J_{11} + J_{22}) - 2J_{12} + K_{12}) - {^N}E_{\corr} \notag\\
+&= 2\varepsilon_1 - \varepsilon_2 - \qty(\dfrac{1}{2}(J_{11} + J_{22}) - 2J_{12} + K_{12}) - {^N}E_{\corr} \notag\\
+&\approx 2\varepsilon_1 - \varepsilon_2 - J_{11} + 2J_{12} - K_{12}) - {^N}E_{\corr} \notag\\
+&= {^N}\mathscr{E}_0 - {^{N-1}}\mathscr{E}_1
+\end{align}