diff --git a/HF/Loos-TCCM_HF.tex b/HF/Loos-TCCM_HF.tex
index 814d0b2..6d5e793 100644
--- a/HF/Loos-TCCM_HF.tex
+++ b/HF/Loos-TCCM_HF.tex
@@ -330,23 +330,23 @@
 
 \begin{frame}{Antisymmetry}
 	\begin{block}{Problem:}
-		\violet{\textit{``Show that, for a system of two fermions, the wave function vanishes when they are at the same point in space''}}
+		\violet{\textit{``Show that, for a system of two fermions, the wave function vanishes when they are at the same point in spin-space''}}
 	\end{block}
 	\pause
 	\begin{block}{Solution}
 		\blue{Indistinguishable} particles means 
 		\begin{equation}
-			\boxed{\abs{\Psi(x_1,x_2)}^2 = \abs{\Psi(x_2,x_1)}^2  \Rightarrow  \Psi(x_1,x_2) = \alert{\pm} \Psi(x_2,x_1)}
+			\boxed{\abs{\Psi(\bx_1,\bx_2)}^2 = \abs{\Psi(\bx_2,\bx_1)}^2  \Rightarrow  \Psi(\bx_1,\bx_2) = \alert{\pm} \Psi(\bx_2,\bx_1)}
 		\end{equation}
 		\\
 		\bigskip
 		\pause
-		\violet{Bosons} mean \violet{$\Psi(x_1,x_2) = \Psi(x_2,x_1)$} and \orange{Fermions} mean \orange{$\Psi(x_1,x_2) = -\Psi(x_2,x_1)$}
+		\violet{Bosons} mean \violet{$\Psi(\bx_1,\bx_2) = \Psi(\bx_2,\bx_1)$} and \orange{Fermions} mean \orange{$\Psi(\bx_1,\bx_2) = -\Psi(\bx_2,\bx_1)$}
 		\\
 		\bigskip
-		Let's put them at the same spot, i.e. \blue{$x = x_1 = x_2$}
+		Let's put them at the same spot, i.e. \blue{$\bx = \bx_1 = \bx_2$}
 		\begin{equation}
-			\text{\violet{For Fermions, }} \Psi(x,x) = - \Psi(x,x)	\quad 	\Rightarrow 	\quad 	\boxed{\Psi(x,x) = 0}
+			\text{\violet{For Fermions, }} \Psi(\bx,\bx) = - \Psi(\bx,\bx)	\qq{$\Rightarrow$} 	\boxed{\Psi(\bx,\bx) = 0}
 		\end{equation}
 		\alert{The wave function vanishes! $\quad 	\Rightarrow 	\quad$ This is called the Fermi hole!}
 	\end{block}
@@ -354,29 +354,29 @@
 
 \begin{frame}{Antisymmetry (Take 2)}
 	\begin{block}{Problem:}
-		\violet{\textit{``Given two one-electron functions $\chi_1(x)$ and $\chi_2(x)$, could you construct a two-electron (fermionic) wave function $\Psi(x_1,x_2)$?''}}
+		\violet{\textit{``Given two one-electron functions $\chi_1(\bx)$ and $\chi_2(\bx)$, could you construct a two-electron (fermionic) wave function $\Psi(\bx_1,\bx_2)$?''}}
 	\end{block}
 	\pause
 	\begin{block}{Solution}
 		A possible solution is
 		\begin{equation}
-			\Psi(x_1,x_2) = \chi_1(x_1) \chi_2(x_2) -  \chi_1(x_2) \chi_2(x_1)
+			\Psi(\bx_1,\bx_2) = \chi_1(\bx_1) \chi_2(\bx_2) -  \chi_1(\bx_2) \chi_2(\bx_1)
 		\end{equation}
 		This has been popularized by \blue{Slater}:
 		\begin{equation}
 			\boxed{
-			\Psi(x_1,x_2) = 
+			\Psi(\bx_1,\bx_2) = 
 			\begin{vmatrix}
-				\chi_1(x_1)	&	\chi_2(x_1)		\\
-				\chi_1(x_2)	&	\chi_2(x_2)		\\
+				\chi_1(\bx_1)	&	\chi_2(\bx_1)		\\
+				\chi_1(\bx_2)	&	\chi_2(\bx_2)		\\
 			\end{vmatrix}
-			= \chi_1(x_1) \chi_2(x_2) -  \chi_1(x_2) \chi_2(x_1)
+			= \chi_1(\bx_1) \chi_2(\bx_2) -  \chi_1(\bx_2) \chi_2(\bx_1)
 			}
 		\end{equation}
 		\alert{This is called a Slater determinant!}
 		\\
 		\bigskip
-		A wave function of the form \green{$\Psi(x_1,x_2) = \chi_1(x_1) \chi_2(x_2)$} is called a \green{Hartree product}
+		A wave function of the form \green{$\Psi(\bx_1,\bx_2) = \chi_1(\bx_1) \chi_2(\bx_2)$} is called a \green{Hartree product}
 	\end{block}
 \end{frame}
 
@@ -562,34 +562,34 @@
 	\begin{itemize}
 		\item \orange{Coulomb operator}
 			\begin{equation}
-				 \cJ_{\green{i}}(\blue{1}) \ket{ \chi_{\purple{j}}(\red{2}) } 
-				 = \mel{ \chi_{\green{i}}(\blue{1}) }{ r_{\blue{1}\red{2}}^{-1} }{ \chi_{\green{i}}(\blue{1})  }  \ket{ \chi_{\purple{j}}(\red{2}) } 
-				 = \qty[ \int d\blue{\bx_1} \chi_{\green{i}}^*(\blue{\bx_1}) r_{\blue{1}\red{2}}^{-1} \chi_{\green{i}}(\blue{\bx_1}) ]  \ket{ \chi_{\purple{j}}(\red{\bx_2}) } 
+				 \cJ_{\green{j}}(\blue{1}) \ket{ \chi_{\purple{i}}(\blue{1}) } 
+				 = \mel{ \chi_{\green{j}}(\red{2}) }{ r_{\blue{1}\red{2}}^{-1} }{ \chi_{\green{j}}(\red{2})  }  \ket{ \chi_{\purple{i}}(\blue{1}) } 
+				 = \qty[ \int d\red{\bx_2} \chi_{\green{j}}^*(\red{\bx_2}) r_{\blue{1}\red{2}}^{-1} \chi_{\green{j}}(\red{\bx_2}) ]  \ket{ \chi_{\purple{i}}(\blue{\bx_1}) } 
 			\end{equation}
 		\item \orange{Coulomb matrix elements}
 			\begin{equation}
 			\begin{split}
-				 \cJ_{\green{i}\purple{j}}
-				 & = \mel{\chi_{\purple{j}}(\red{2})}{\cJ_{\green{i}}(\blue{1})}{\chi_{\purple{j}}(\red{2})} 
-				 = \mel{ \chi_{\green{i}}(\blue{1}) \chi_{\purple{j}}(\red{2}) }{r_{\blue{1}\red{2}}^{-1}}{ \chi_{\green{i}}(\blue{1}) \chi_{\purple{j}}(\red{2})} 
+				 \cJ_{\purple{i}\green{j}}
+				 & = \mel{\chi_{\purple{i}}(\blue{1})}{\cJ_{\green{j}}(\blue{1})}{\chi_{\purple{i}}(\blue{1})} 
+				 = \mel{ \chi_{\purple{i}}(\blue{1}) \chi_{\green{j}}(\red{2}) }{r_{\blue{1}\red{2}}^{-1}}{ \chi_{\purple{i}}(\blue{1}) \chi_{\green{j}}(\red{2})} 
 				 \\
-				 & = \iint \chi_{\green{i}}^*(\blue{\bx_1}) \chi_{\purple{j}}^*(\red{\bx_2}) r_{\blue{1}\red{2}}^{-1} \chi_{\green{i}}(\blue{\bx_1}) \chi_{\green{i}}(\red{\bx_2}) d\blue{\bx_1} d\red{\bx_2}
+				 & = \iint \chi_{\purple{i}}^*(\blue{\bx_1}) \chi_{\green{j}}^*(\red{\bx_2}) r_{\blue{1}\red{2}}^{-1} \chi_{\purple{i}}(\blue{\bx_1}) \chi_{\green{j}}(\red{\bx_2}) d\blue{\bx_1} d\red{\bx_2}
 			\end{split}
 			\end{equation}
 		\item \violet{(non-local) Exchange operator}
 			\begin{equation}
-				\cK_{\green{i}}(\blue{1}) \ket{ \chi_{\purple{j}}(\red{2}) } 
-				= \mel{ \chi_{\green{i}}(\blue{1}) }{ r_{\blue{1}\red{2}}^{-1} }{ \chi_{\purple{j}}(\blue{1})  }  \ket{ \chi_{\green{i}}(\red{2}) } 
-				= \qty[ \int \chi_{\green{i}}^*(\blue{\bx_1}) r_{\blue{1}\red{2}}^{-1} \chi_{\purple{j}}(\blue{\bx_1}) d\blue{\bx_1} ]  \ket{ \chi_{\green{i}}(\red{\bx_2}) } 
+				\cK_{\green{j}}(\blue{1}) \ket{ \chi_{\purple{i}}(\blue{1}) } 
+				 = \mel{ \chi_{\green{j}}(\red{2}) }{ r_{\blue{1}\red{2}}^{-1} }{ \chi_{\purple{i}}(\red{2})  }  \ket{ \chi_{\green{j}}(\blue{1}) } 
+				 = \qty[ \int d\red{\bx_2} \chi_{\green{j}}^*(\red{\bx_2}) r_{\blue{1}\red{2}}^{-1} \chi_{\purple{i}}(\red{\bx_2}) ]  \ket{ \chi_{\green{j}}(\blue{\bx_2}) } 
 			\end{equation}
 		\item \violet{Exchange matrix elements}
 			\begin{equation}
 			\begin{split}
 				\cK_{\green{i}\purple{j}} 
-				& = \mel{\chi_{\purple{j}}(\red{2})}{\cK_{\green{i}}(\blue{1})}{\chi_{\purple{j}}(\red{2})} 
-				= \mel{ \chi_{\green{i}}(\blue{1}) \chi_{\purple{j}}(\red{2})}{ r_{\blue{1}\red{2}}^{-1} }{ \chi_{\purple{j}}(\blue{1}) \chi_{\green{i}}(\red{2}) } 
-				\\
-				 & = \iint \chi_{\green{i}}^*(\blue{\bx_1}) \chi_{\purple{j}}^*(\red{\bx_2}) r_{\blue{1}\red{2}}^{-1} \chi_{\purple{j}}(\blue{\bx_1}) \chi_{\green{i}}(\red{\bx_2}) d\blue{\bx_1} d\red{\bx_2} 
+				 & = \mel{\chi_{\purple{i}}(\blue{1})}{\cK_{\green{j}}(\blue{1})}{\chi_{\purple{i}}(\blue{1})} 
+				 = \mel{ \chi_{\purple{i}}(\blue{1}) \chi_{\green{j}}(\red{2}) }{r_{\blue{1}\red{2}}^{-1}}{ \chi_{\green{j}}(\blue{1}) \chi_{\purple{i}}(\red{2})} 
+				 \\
+				 & = \iint \chi_{\purple{i}}^*(\blue{\bx_1}) \chi_{\green{j}}^*(\red{\bx_2}) r_{\blue{1}\red{2}}^{-1} \chi_{\green{j}}(\blue{\bx_1}) \chi_{\purple{i}}(\red{\bx_2}) d\blue{\bx_1} d\red{\bx_2} 
 			\end{split}
 			\end{equation}
 	\end{itemize}
@@ -634,19 +634,27 @@
 \end{frame}
 
 \begin{frame}{Permutation symmetry}
-	\begin{block}{Permutation symmetry in physicist notation}
+	\begin{block}{Permutation symmetry in physicts' notations}
 		\begin{equation}
 			\braket{ij}{kl} 
 			= \braket{\chi_i \chi_j}{\chi_k \chi_l} 
 			= \iint \chi_i^*(\bx_1) \chi_j^*(\bx_2) \frac{1}{r_{12}} \chi_k(\bx_1) \chi_l(\bx_2) d\bx_1 d\bx_2
 		\end{equation}
-		\begin{align}
-			\qq*{\red{Complex-valued integrals:}} & 
-			\braket{ij}{kl} = \braket{ji}{lk} = \braket{kl}{ij}^*
-			\\
-			\qq*{\green{Real-valued integrals:}} & 
-			\braket{ij}{kl} = \braket{ji}{kl} = \braket{ij}{lk} = \braket{ji}{lk} = \braket{kl}{ij} = \braket{lk}{ij} = \braket{kl}{ji} = \braket{lk}{ji}
-		\end{align}
+		\begin{equation}
+			\qq*{\red{Complex-valued integrals:}} 
+			\braket{ij}{kl} = \braket{ji}{lk} = \braket{kl}{ij}^* = \braket{lk}{ji}^*
+		\end{equation}
+	\end{block}
+	\begin{block}{Permutation symmetry in chemists' notations}
+		\begin{equation}
+			[ij|kl] 
+			= [\chi_i \chi_j|\chi_k \chi_l]
+			= \iint \chi_i^*(\bx_1) \chi_j(\bx_1) \frac{1}{r_{12}} \chi_k^*(\bx_2) \chi_l(\bx_2) d\bx_1 d\bx_2
+		\end{equation}
+		\begin{equation}
+			\qq*{\green{Real-valued integrals:}} 
+			[ij|kl] = [ji|kl] = [ij|lk] = [ji|lk] = [kl|ij] = [lk|ij] = [kl|ji] = [lk|ji]
+		\end{equation}
 	\end{block}
 \end{frame}
 
@@ -664,7 +672,7 @@
 	\end{block}
 	\begin{block}{\orange{Case 2 = differ by one spinorbital}: $\ket{K} = \ket{\ldots m n \ldots}$ and $\ket{L} = \ket{\ldots p n \ldots}$}
 		\begin{equation}
-			\mel{K}{\cO_1}{L} = \sum_m \mel{m}{h}{p}
+			\mel{K}{\cO_1}{L} = \mel{m}{h}{p}
 		\end{equation}
 	\end{block}
 	\begin{block}{\red{Case 3 = differ by two spinorbitals}: $\ket{K} = \ket{\ldots m n \ldots}$ and $\ket{L} = \ket{\ldots p q \ldots}$}