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small fix in chap 2, 3 #42

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Merged
jeanwsr merged 1 commit into from
Apr 15, 2024
Merged

small fix in chap 2, 3 #42

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10 changes: 5 additions & 5 deletions Chaps/Chap2.tex
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Original file line number Diff line number Diff line change
Expand Up @@ -1770,7 +1770,7 @@ \subsection{矩阵元的一般规则}
\end{subequations}
\autoref{2.107}可以写成：
\begin{align}
\braket{K|\mathscr{H}|K}=&\sum_m^N\braket{m|h|m}+\frac{1}{2}\sum_m^N\sum_{n>m}^N\braket{mn||mn}\notag\\
\braket{K|\mathscr{H}|K}=&\sum_m^N\braket{m|h|m}+\sum_m^N\sum_{n>m}^N\braket{mn||mn}\notag\\
=&\sum_m[m|h|m]+\sum_m^N\sum_{n>m}^N[mm|nn]-[mn|nm]
\end{align}
反对称双电子积分的求和遍及$\ket{K}$中所有不同的占据自旋轨道对儿$\chi_m,\chi_n$.
Expand Down Expand Up @@ -2261,7 +2261,7 @@ \subsection{将自旋轨道转换成空间轨道}
60))
\begin{align}
\chi_1(\mathbf{x})\equiv \psi_1(\mathbf{x}) = \psi_1(\mathbf{\mathbf{r}})\alpha(\omega)\\
\chi_2(\mathbf{x})\equiv \bar{\psi}_1(\mathbf{x}) = \psi_1(\mathbf{\mathbf{r}})\alpha(\omega)
\chi_2(\mathbf{x})\equiv \bar{\psi}_1(\mathbf{x}) = \psi_1(\mathbf{\mathbf{r}})\beta(\omega)
\end{align}
将上式定义带入(2.
154),
Expand Down Expand Up @@ -2310,7 +2310,7 @@ \subsection{将自旋轨道转换成空间轨道}
157)中的最后一个积分为
\begin{align}
[{\psi_1\bar{\psi}_1|\bar{\psi}_1\psi_1}] = & \int\dd{r_1}\dd\omega_1\dd{r_2}\dd\omega_2\, \psi_1^*(\mathbf{r_1})\alpha^*(\omega_1)\psi_(\mathbf{r_1})\beta(\omega_1)r_{12}^{-1}\notag\\
& \times \psi_1(\mathbf{r_2})\alpha^(\omega_2)\psi_(\mathbf{r_2})\beta(\omega_2) = 0
& \times \psi_1^*(\mathbf{r_2})\beta^*(\omega_2)\psi_(\mathbf{r_2})\alpha(\omega_2) = 0
\end{align}
最后一步是由于$\braket{\alpha|\beta}=\braket{\beta|\alpha} = 0$.
一般来说,
Expand Down Expand Up @@ -2404,7 +2404,7 @@ \subsection{将自旋轨道转换成空间轨道}
\begin{align}
\sum_{a}^{N}\sum_{b}^{N}\chi_a\chi_b & = \sum_{a}^{N}\chi_a\sum_{b}^{N}\chi_b\notag\\
& = \sum_{a}^{N/2}(\psi_a + \bar{\psi}_a)\sum_{b}^{N/2}(\psi_b + \bar{\psi}_b) \notag\\
& = \sum_{a}^{N/2}\sum_{b}^{N/2}\psi_a\psi_b + \psi_a\bar{\psi}_b + \bar{\psi}{\psi} + \bar{\psi}_a\bar{\psi}_b
& = \sum_{a}^{N/2}\sum_{b}^{N/2}\psi_a\psi_b + \psi_a\bar{\psi}_b + \bar{\psi}_a{\psi}_b + \bar{\psi}_a\bar{\psi}_b
\end{align}
或单纯写出符号
\begin{equation}
Expand Down Expand Up @@ -2536,7 +2536,7 @@ \subsection{库伦积分、交换积分}
但若这两个电子自旋相反,
即系统此时的波函数为$\ket{{\psi}_1\bar{\psi}_2}$,
则此概率不为零.
那么一个合理的推测就是电子间的库伦排斥会使态$\ket{\bar{\psi}_1\bar{\psi}_2}$对应的能量比$\ket{\psi}_1\bar{\psi}_2$的能量低.
那么一个合理的推测就是电子间的库伦排斥会使态$\ket{\bar{\psi}_1\bar{\psi}_2}$对应的能量比$\ket{\psi_1\bar{\psi}_2}$的能量低.
利用式(2.
110),
$\ket{\psi_1\bar{\psi}_2}$(记为$\ket{\uparrow\downarrow}$)的能量就是
Expand Down
2 changes: 1 addition & 1 deletion Chaps/Chap3.tex
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Expand Up @@ -856,7 +856,7 @@ \subsection{轨道能量与Koopmans定理}
\mathrm{IP} & = ^{N-1}E_c - ^NE_0\notag\\
& = -\braket{c|h|c} - \frac{1}{2}\sum_{a[b\equiv c]}\braket{ab||ab}- \frac{1}{2}\sum_{b[a\equiv c]}\braket{ab||ab}\notag\\
& = -\braket{c|h|c} - \frac{1}{2}\sum_{a}\braket{ac||ac}- \frac{1}{2}\sum_{b}\braket{cb||cb}\notag\\
& = -\braket{c|h|c} - \frac{1}{2}\sum_{b}\braket{cb||cb}
& = -\braket{c|h|c} - \sum_{b}\braket{cb||cb}
\end{align}
将此结果与\autoref{3.75}中占据自旋轨道能量的定义做比较可以发现,
从$\chi_c$上挪去一个电子的电离能就是轨道能量$\epsilon_c$的负值
Expand Down