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fix: correct subscript in eq 3.52 matrix A and Fock operator in eq 3.72 #76

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jeanwsr merged 1 commit into from
May 24, 2026
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fix: correct subscript in eq 3.52 matrix A and Fock operator in eq 3.72 #76

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22 changes: 11 additions & 11 deletions Chaps/Chap3.tex
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Original file line number Diff line number Diff line change
Expand Up @@ -569,10 +569,10 @@ \subsection{正则H-F方程}
\begin{align}
\mathbf{A} =
\begin{pmatrix}
\chi_a(1) & \chi_2(1) & \cdots & \chi_a(1) & \cdots & \chi_N(1) \\
\chi_a(2) & \chi_2(2) & \cdots & \chi_a(2) & \cdots & \chi_N(2) \\
\chi_1(1) & \chi_2(1) & \cdots & \chi_a(1) & \cdots & \chi_N(1) \\
\chi_1(2) & \chi_2(2) & \cdots & \chi_a(2) & \cdots & \chi_N(2) \\
\vdots & \vdots & & \vdots & & \vdots \\
\chi_a(N) & \chi_2(N) & \cdots & \chi_a(N) & \cdots & \chi_N(N)
\chi_1(N) & \chi_2(N) & \cdots & \chi_a(N) & \cdots & \chi_N(N)
\end{pmatrix}
\end{align}
以使波函数$\ket{\Psi_0}$就是该矩阵的归一化行列式：
Expand All @@ -585,22 +585,22 @@ \subsection{正则H-F方程}
\begin{align}
\mathbf{A}' = \mathbf{AU} & =
\begin{pmatrix}
\chi_a(1) & \chi_2(1) & \cdots & \chi_N(1) \\
\chi_a(2) & \chi_2(2) & \cdots & \chi_N(2) \\
\chi_1(1) & \chi_2(1) & \cdots & \chi_N(1) \\
\chi_1(2) & \chi_2(2) & \cdots & \chi_N(2) \\
\vdots & \vdots & & \vdots \\
\chi_a(N) & \chi_2(N) & \cdots & \chi_N(N)
\chi_1(N) & \chi_2(N) & \cdots & \chi_N(N)
\end{pmatrix}
\begin{pmatrix}
U_{11} & U_{12} & \cdots & U_{1N} \\
U_{11} & U_{12} & \cdots & U_{1N} \\
U_{21} & U_{22} & \cdots & U_{2N} \\
\vdots & \vdots & & \vdots \\
U_{N1} & U_{N2} & \cdots & U_{NN}
\end{pmatrix} \notag\\
& = \begin{pmatrix}
\chi_a'(1) & \chi_2'(1) & \cdots & \chi_N'(1) \\
\chi_a'(2) & \chi_2'(2) & \cdots & \chi_N'(2) \\
\chi_1'(1) & \chi_2'(1) & \cdots & \chi_N'(1) \\
\chi_1'(2) & \chi_2'(2) & \cdots & \chi_N'(2) \\
\vdots & \vdots & & \vdots \\
\chi_a'(N) & \chi_2'(N) & \cdots & \chi_N'(N)
\chi_1'(N) & \chi_2'(N) & \cdots & \chi_N'(N)
\end{pmatrix}
\end{align}
而且由于
Expand Down Expand Up @@ -744,7 +744,7 @@ \subsection{轨道能量与Koopmans定理}
用\autoref{3.16}的Fock算符表达式可知,
轨道能量可以写作：
\begin{align}
\epsilon_i & = \braket{\chi_i|h|\chi_i} = \braket{\chi_i|h+\sum_b(\scr{J}_b-\scr{K}_b)|\chi_i}\notag\\
\epsilon_i & = \braket{\chi_i|f|\chi_i} = \braket{\chi_i|h+\sum_b(\scr{J}_b-\scr{K}_b)|\chi_i}\notag\\
& = \braket{\chi_i|h|\chi_i} + \sum_b\braket{\chi_i|\scr{J}_b|\chi_i} - \braket{\chi_i|\scr{K}_b|\chi_i}\notag\\
& = \braket{i|h|i} + \sum_b\braket{ib|ib} - \braket{ib|bi}\notag\\
& = \braket{i|h|i} + \sum_b\braket{ib||ib}
Expand Down
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