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small fix in 3.4 #46

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May 31, 2024
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small fix in 3.4 #46

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ce0ba06
small fix in Chap 3.4
maki49 May 31, 2024
c2479e8
fix spin explanation
maki49 May 31, 2024
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6 changes: 3 additions & 3 deletions Chaps/Chap3.tex
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Original file line number Diff line number Diff line change
Expand Up @@ -1161,7 +1161,7 @@ \subsection{闭壳层H-F：限制性自旋轨道}\label{sec3.4.1}
那么
\begin{alignat}{3}
f(\mathbf{r}_1)\psi_j(\mathbf{r}_1) &= h(\mathbf{r}_1)\psi_j(\mathbf{r}_1) &+ \sum_c\int\dd\omega_1\dd{x}_2\,\alpha^*(\omega_1)\chi_c^*(\mathbf{x}_1)\twoe\chi_c(\mathbf{x}_2)\alpha(\omega_1)\psi_j(\mathbf{r}_1)\notag\\
&&-\sum_c\int\dd\omega_1\dd{x}_2\,\alpha^*(\omega_1)\chi_c^*(\mathbf{x}_1)\twoe\chi_c(\mathbf{x}_1)\alpha(\omega_2)\psi_j(\mathbf{r}_2)\notag\\
&&-\sum_c\int\dd\omega_1\dd{x}_2\,\alpha^*(\omega_1)\chi_c^*(\mathbf{x}_2)\twoe\chi_c(\mathbf{x}_1)\alpha(\omega_2)\psi_j(\mathbf{r}_2)\notag\\
&=\epsilon_j\psi_j(\mathbf{r}_1)&
\end{alignat}
式中含$h(\mathbf{r}_1)$的那项中$\dd\omega_1$的积分已经预先积完,
Expand All @@ -1178,13 +1178,13 @@ \subsection{闭壳层H-F：限制性自旋轨道}\label{sec3.4.1}
& {}\quad + \sum_c^{N/2}\int\dd\omega_1\dd\omega_2\dd{r}_2\,\alpha^*(\omega_1)\psi_c^*(\mathbf{r}_2)\alpha^*(\omega_2)\twoe\psi_c(\mathbf{r}_2)\alpha(\omega_2)\alpha(\omega_1)\psi_j(\mathbf{r}_1)\notag\\
& {}\quad + \sum_c^{N/2}\int\dd\omega_1\dd\omega_2\dd{r}_2\,\,\alpha^*(\omega_1)\psi_c^*(\mathbf{r}_2)\beta^*(\omega_2)\twoe\psi_c(\mathbf{r}_2)\beta(\omega_2)\alpha(\omega_1)\psi_j(\mathbf{r}_1)\notag\\
& {}\quad - \sum_c^{N/2}\int\dd\omega_1\dd\omega_2\dd{r}_2\,\,\alpha^*(\omega_1)\psi_c^*(\mathbf{r}_2)\alpha^*(\omega_2)\twoe\psi_c(\mathbf{r}_1)\alpha(\omega_1)\alpha(\omega_2)\psi_j(\mathbf{r}_2)\notag\\
& {}\quad - \sum_c^{N/2}\int\dd\omega_1\dd\omega_2\dd{r}_2\,\alpha^*(\omega_1)\psi_c^*(\mathbf{r}_2)\beta^*(\omega_2)\twoe\psi_c(\mathbf{r}_2)\beta(\omega_1)\alpha(\omega_2)\psi_j(\mathbf{r}_1)\notag\\
& {}\quad - \sum_c^{N/2}\int\dd\omega_1\dd\omega_2\dd{r}_2\,\alpha^*(\omega_1)\psi_c^*(\mathbf{r}_2)\beta^*(\omega_2)\twoe\psi_c(\mathbf{r}_1)\beta(\omega_1)\alpha(\omega_2)\psi_j(\mathbf{r}_1)\notag\\
& = \epsilon_j\psi_j(\mathbf{r}_1)
\label{3.120}
\end{align}
现在可以进行对$\dd\omega_1$和$\dd\omega_2$的积分.
\autoref{3.120}中最后一项根据自旋正交性为零.
这反映自旋平行电子之间没有交换作用的事实.
这反映了交换作用只存在于自旋平行电子之间的事实.
两个库伦项是相等的,
那么可得
\begin{alignat}{3}
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