clean diagrams

Manuscript/CI-F12.tex

@@ -109,6 +109,7 @@ Atomic units are used throughout.
 
 %----------------------------------------------------------------
 %\section{Ans{\"a}tz} : Ca prend un trema au pluriel seulement : Ans{\"a}tze
+% Ouais Toto! Style je parle allemand quoi! "J'ai ri a ta blague moi!!"
 \section{Ansatz}
 %----------------------------------------------------------------
 Inspired by a number of previous research (see Ref.~\onlinecite{Shiozaki11} and references therein), our electronic wave function \emph{ans{\"a}tz} 
@@ -215,15 +216,6 @@ Assuming, witout loss of generality that $\cD{0}$ is the largest coefficient $\c
         \end{cases}
 \end{equation}
 
-%%% FIG 1 %%%
-\begin{figure}
-    \includegraphics[width=\linewidth]{fig1}
-    \caption{
-    \label{fig:CBS}
-    Schematic representation of the various orbital spaces and their notation.
-    The arrows represent the three types of excited determinants contributing to the dressing term: the pure doubles $\ket*{_{ij}^{\alpha \beta}}$ (green), the mixed doubles $\ket*{_{ij}^{a \beta}}$ (magenta) and the pure singles $\ket*{_{i}^{\alpha}}$ (orange).}
-\end{figure}
-%%%    %%%
 
 %----------------------------------------------------------------
 \section{Matrix elements}
@@ -261,76 +253,6 @@ Substituting \eqref{eq:RI} into the first term of the right-hand side of Eq.~\eq
 where $\mD$ is the set of ``conventional'' determinants obtained by excitations from the occupied space $\qty{i}$ to the virtual one $\qty{a}$, and $\mC = \mA \setminus \mD$.
 Because $f$ is a two-electron operator, the way to compute efficiently Eq.~\eqref{eq:IHF-RI} is actually very similar to what is done within second-order multireference perturbation theory. \cite{Garniron17b}
 
-The set $\mC$ is defined by two simple rules.
-First, because $f$ is a two-electron operator [and thanks to the matrix element $f_{AJ}$ in \eqref{eq:IHF-RI}], we know that the sum over $A$ is restricted to the singly- or doubly-excited determinants with respect to the determinant $\kJ$.
-Second, to ensure that $H_{IA} \neq 0$, $A$ must be connected to $\kI$, i.e.~differs from $\kI$ by no more than two spin orbitals.
-Three types of determinants have these two properties (see Fig.~\ref{fig:CBS}).: 
-i) the pure doubles $\hT_{ij}^{\alpha \beta}\ket*{I}$,
-ii) the mixed doubles $\hT_{ij}^{\alpha b}\ket*{I}$, and
-iii) the pure singles $\hT_{i}^{\alpha}\ket*{I}$.
-
-\alert{
-The matrix element between two determinants differing by a double excitation $\hT_{ij}^{kl}$ is given by
-\begin{equation}
-  \mel{I}{\hH f}{J} = \{ ij || kl \} - \sum_m \{ ijm || mkl \} \Delta_{mI} \Delta_{mJ}
-\end{equation}
-where 
-\begin{equation}
-\Delta_{mI} = \mel{I}{a_m^\dagger a_m}{I},
-\end{equation}
-\begin{equation}
-  \{ ijm || mkl \}  = \sum_{\alpha} \langle i j || \alpha m \rangle [ \alpha m || k l ]
-                      + \langle i j || m \alpha \rangle [ m \alpha || k l ],
-\end{equation}
-\begin{equation}
-  \{ ij || kl \}  = \sum_{\alpha \beta} \langle i j || \alpha \beta \rangle [ \alpha \beta || k l ] + \sum_m \{ ijm || mkl \} 
-\end{equation}
-The matrix element between two determinants differing by a single excitation $\hT_{i}^{k}$ is given by
-\begin{equation}
-  \mel{I}{\hH f}{J} = \sum_j \Delta_{jI} \Delta_{jJ} \qty( \{ ij || kj \}  - \sum_m \{ ijm || mkj \} \Delta_{mI} \Delta_{mJ} )
-\end{equation}
-and the diagonal terms are
-\begin{equation}
-  \mel{I}{\hH f}{I} = \sum_{ij} \Delta_{iI} \Delta_{jI} \qty( \{ ij || ij \}  - \sum_m \{ ijm || mij \} \Delta_{mI} )
-\end{equation}
-}
-
-
-Although $\mel{0}{\hH}{_{i}^{a}} = 0$, note that the Brillouin theorem does not hold in the CABS, i.e.~$\mel{0}{\hH}{_{i}^{\alpha}} \neq 0$.
-Here, we will eschew the generalized Brillouin condition (GBC) which set these to zero. \cite{Kutzelnigg91}
- 
-%\begin{gather}
-%    \mel*{0}{\hH}{_i^\alpha} = \mel{i}{h}{\alpha} + \sum_{j} \mel{ij}{}{\alpha j} 
-%    \\
-%    \mel*{0}{\hH}{_{ij}^{\alpha\beta}} = \mel{ij}{}{\alpha \beta} 
-%    \\
-%    \mel*{0}{\hH}{_{ij}^{a\beta}} = \mel{ij}{}{a \beta} 
-%\end{gather}
-%
-%\begin{gather}
-%    \mel*{_k^c}{\hH}{_i^\alpha} = \mel{c}{h}{\alpha} + \sum_{j} \mel{cj}{}{\alpha j} 
-%    \\
-%    \mel*{_k^c}{\hH}{_{ij}^{\alpha\beta}} = 0
-%    \\
-%    \mel*{_k^c}{\hH}{_{ij}^{a\beta}} = 
-%\end{gather}
-%
-%\begin{gather}
-%    \mel*{_{kl}^{cd}}{\hH}{_i^\alpha} = 
-%    \\
-%    \mel*{_{kl}^{cd}}{\hH}{_{ij}^{\alpha\beta}} = 
-%    \\
-%    \mel*{_{kl}^{cd}}{\hH}{_{ij}^{a\beta}} = 
-%\end{gather}
-%
-%\begin{gather}
-%    \mel*{_{klm}^{cde}}{\hH}{_i^\alpha} = 
-%    \\
-%    \mel*{_{klm}^{cde}}{\hH}{_{ij}^{\alpha\beta}} = 
-%    \\
-%    \mel*{_{klm}^{cde}}{\hH}{_{ij}^{a\beta}} = 
-%\end{gather}
-
 %----------------------------------------------------------------
 \section{Computational details}
 %----------------------------------------------------------------
@@ -485,6 +407,126 @@ This work was performed using HPC resources from GENCI-TGCC (Grant No.~2018-A004
 \end{acknowledgments}
 %----------------------------------------------------------------
 
+%----------------------------------------------------------------
+\appendix
+%----------------------------------------------------------------
+
+%----------------------------------------------------------------
+\section{Matrix elements}
+%----------------------------------------------------------------
+
+%The set $\mC$ is defined by two simple rules.
+%First, because $f$ is a two-electron operator [and thanks to the matrix element $f_{AJ}$ in \eqref{eq:IHF-RI}], we know that the sum over $A$ is restricted to the singly- or doubly-excited determinants with respect to the determinant $\kJ$.
+%Second, to ensure that $H_{IA} \neq 0$, $A$ must be connected to $\kI$, i.e.~differs from $\kI$ by no more than two spin orbitals.
+%Three types of determinants have these two properties (see Fig.~\ref{fig:CBS}).: 
+%i) the pure doubles $\hT_{ij}^{\alpha \beta}\ket*{I}$,
+%ii) the mixed doubles $\hT_{ij}^{\alpha b}\ket*{I}$, and
+%iii) the pure singles $\hT_{i}^{\alpha}\ket*{I}$.
+
+In this Appendix, we explain how to calculate the matrix elements 
+\begin{equation}
+	\Xi = \mel{I}{\hH f}{J} = \sum_{A} \mel{I}{\hH}{A} \mel{A}{f}{J}
+\end{equation} 
+within the RI approximation, and, in particular, the different cases that one can encounter.
+The determinants $\kI$ and $\kJ$ belong to the internal space, while $\kA$ belongs to the external space.
+Also, we recall that $\hH$ contains one- and two-electron terms, while $f$ only contains two-electro terms.
+Consequently, the external determinants correspond to single or double excitations only with respect to the internal determinants.
+The expression above can then be further decomposed as
+\begin{equation}
+	\Xi  
+	= \underbrace{\sum_{A_1} \mel{I}{\hH}{A_1} \mel{A_1}{f}{J}}_{\Xi_1}
+	+ \underbrace{\sum_{A_2} \mel{I}{\hH}{A_2} \mel{A_2}{f}{J}}_{\Xi_2}
+\end{equation} 
+where $\ket{A_1}$ and $\ket{A_2}$ are external determinants with one electron in the CABS (single excitation with respect to to internal determinant) and two electrons in the CABS (double excitation with respect to to internal determinant), respectively.
+We will look at the second term, $\Xi_2$, first.
+
+When there is two electrons in the CABS in $\kA$, this means that only double excitations can link $\kI$ and $\kA$ as well as $\kA$ and $\kJ$.
+Obviously, these two double excitations can be different.
+Therefore, we have 3 cases where $\kI$ and $\kJ$ differs by 0, 1 or 2 spinorbitals.
+
+\begin{equation}
+	 \Xi_2 =  \Xi_2^0 + \Xi_2^1 + \Xi_2^2
+\end{equation} 
+
+\begin{align}
+	 \Xi_2^2  & = \sum_{\alpha \beta} \langle i j || \alpha \beta \rangle [ \alpha \beta || k l ] = \{ ij || kl \}
+	 \\
+	 \Xi_2^1  & = \sum_m \{ im || ml \} 
+	 \\
+	 \Xi_2^0  & = \sum_{mn} \{ mn || mn \} 
+\end{align}
+
+The matrix element between two determinants differing by a double excitation $\hT_{ij}^{kl}$ is given by
+\begin{equation}
+  \mel{I}{\hH f}{J} = \{ ij || kl \} - \sum_m \{ ijm || mkl \} \Delta_{mI} \Delta_{mJ}
+\end{equation}
+where 
+\begin{equation}
+\Delta_{mI} = \mel{I}{a_m^\dagger a_m}{I},
+\end{equation}
+\begin{equation}
+  \{ ijm || mkl \}  = \sum_{\alpha} \langle i j || \alpha m \rangle [ \alpha m || k l ]
+                      + \langle i j || m \alpha \rangle [ m \alpha || k l ],
+\end{equation}
+\begin{equation}
+  \{ ij || kl \}  = \sum_{\alpha \beta} \langle i j || \alpha \beta \rangle [ \alpha \beta || k l ] + \sum_m \{ ijm || mkl \} 
+\end{equation}
+The matrix element between two determinants differing by a single excitation $\hT_{i}^{k}$ is given by
+\begin{equation}
+  \mel{I}{\hH f}{J} = \sum_j \Delta_{jI} \Delta_{jJ} \qty( \{ ij || kj \}  - \sum_m \{ ijm || mkj \} \Delta_{mI} \Delta_{mJ} )
+\end{equation}
+and the diagonal terms are
+\begin{equation}
+  \mel{I}{\hH f}{I} = \sum_{ij} \Delta_{iI} \Delta_{jI} \qty( \{ ij || ij \}  - \sum_m \{ ijm || mij \} \Delta_{mI} )
+\end{equation}
+
+%%% FIG 1 %%%
+\begin{figure}
+    \includegraphics[width=\linewidth]{fig1}
+    \caption{
+    \label{fig:CBS}
+    Schematic representation of the various orbital spaces and their notation.
+    The arrows represent the three types of excited determinants contributing to the dressing term: the pure doubles $\ket*{_{ij}^{\alpha \beta}}$ (green), the mixed doubles $\ket*{_{ij}^{a \beta}}$ (magenta) and the pure singles $\ket*{_{i}^{\alpha}}$ (orange).}
+\end{figure}
+%%%    %%%
+
+Although $\mel{0}{\hH}{_{i}^{a}} = 0$, note that the Brillouin theorem does not hold in the CABS, i.e.~$\mel{0}{\hH}{_{i}^{\alpha}} \neq 0$.
+Here, we will eschew the generalized Brillouin condition (GBC) which set these to zero. \cite{Kutzelnigg91}
+ 
+%\begin{gather}
+%    \mel*{0}{\hH}{_i^\alpha} = \mel{i}{h}{\alpha} + \sum_{j} \mel{ij}{}{\alpha j} 
+%    \\
+%    \mel*{0}{\hH}{_{ij}^{\alpha\beta}} = \mel{ij}{}{\alpha \beta} 
+%    \\
+%    \mel*{0}{\hH}{_{ij}^{a\beta}} = \mel{ij}{}{a \beta} 
+%\end{gather}
+%
+%\begin{gather}
+%    \mel*{_k^c}{\hH}{_i^\alpha} = \mel{c}{h}{\alpha} + \sum_{j} \mel{cj}{}{\alpha j} 
+%    \\
+%    \mel*{_k^c}{\hH}{_{ij}^{\alpha\beta}} = 0
+%    \\
+%    \mel*{_k^c}{\hH}{_{ij}^{a\beta}} = 
+%\end{gather}
+%
+%\begin{gather}
+%    \mel*{_{kl}^{cd}}{\hH}{_i^\alpha} = 
+%    \\
+%    \mel*{_{kl}^{cd}}{\hH}{_{ij}^{\alpha\beta}} = 
+%    \\
+%    \mel*{_{kl}^{cd}}{\hH}{_{ij}^{a\beta}} = 
+%\end{gather}
+%
+%\begin{gather}
+%    \mel*{_{klm}^{cde}}{\hH}{_i^\alpha} = 
+%    \\
+%    \mel*{_{klm}^{cde}}{\hH}{_{ij}^{\alpha\beta}} = 
+%    \\
+%    \mel*{_{klm}^{cde}}{\hH}{_{ij}^{a\beta}} = 
+%\end{gather}
+
+
+
 \bibliography{CI-F12,CI-F12-control}
 
 \end{document}

Manuscript/Tikz/0_1_11.tikz

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Manuscript/Tikz/0_1_22.tikz

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Manuscript/Tikz/0_2_22.tikz

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Manuscript/Tikz/1_1_11.tikz

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Manuscript/Tikz/1_1_12.tikz

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Manuscript/Tikz/1_1_21.tikz

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Manuscript/Tikz/1_1_22.tikz

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Manuscript/Tikz/1_2_22.tikz

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Manuscript/Tikz/2_1_12.tikz

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Manuscript/Tikz/2_1_21.tikz

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Manuscript/Tikz/2_1_22.tikz

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Manuscript/Tikz/2_2_22.tikz

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Manuscript/Tikz/3_1_22.tikz

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Manuscript/Tikz/Electrons.tikz

@@ -26,13 +26,16 @@
 % \updn{0.}{0.}
 %
 
-
-
+% Electron symbol
+\newcommand{\upel}{$\uparrow$}
+\newcommand{\dnel}{$\downarrow$}
 
 % Colors
 \newcommand{\cabs}[1]{ { \color{lightgray}{#1} } }
 \newcommand{\obs}[1]{  { \color{black}{#1} } }
-\newcommand{\exc}[1]{  { \color{red}{#1} } }
+\newcommand{\exca}[1]{  { \color{red}{#1} } }
+\newcommand{\excb}[1]{  { \color{Cyan}{#1} } }
+\newcommand{\excc}[1]{  { \color{Green}{#1} } }
 \newcommand{\noexc}[1]{{ \color{black}{#1} } }
 
 % Spacing between the 3 diagrams
@@ -40,52 +43,83 @@
 \newcommand{\XA}{0.}
 \newcommand{\XJ}{1.}
 
-
 % Up electron
 \newcommand{\up}[2]{
   \draw [-,thick] (-0.2+#1,#2) -- (0.2+#1,#2);
-  \node at (#1-0.1,#2) {\noexc{$\uparrow$}};
+  \node at (#1-0.1,#2) {\noexc{\upel}};
 }
 
-\newcommand{\upr}[2]{
+\newcommand{\upa}[2]{
   \draw [-,thick] (-0.2+#1,#2) -- (0.2+#1,#2);
-  \node at (#1-0.1,#2) {\exc{$\uparrow$}};
+  \node at (#1-0.1,#2) {\exca{\upel}};
+}
+
+\newcommand{\upb}[2]{
+  \draw [-,thick] (-0.2+#1,#2) -- (0.2+#1,#2);
+  \node at (#1-0.1,#2) {\excb{\upel}};
+}
+
+\newcommand{\upc}[2]{
+  \draw [-,thick] (-0.2+#1,#2) -- (0.2+#1,#2);
+  \node at (#1-0.1,#2) {\excc{\upel}};
 }
 
 % Down electron
 \newcommand{\dn}[2]{
   \draw [-,thick] (-0.2+#1,#2) -- (0.2+#1,#2);
-  \node at (#1+0.1,#2) {\noexc{$\downarrow$}};
+  \node at (#1+0.1,#2) {\noexc{\dnel}};
 }
 
-\newcommand{\dnr}[2]{
+\newcommand{\dna}[2]{
   \draw [-,thick] (-0.2+#1,#2) -- (0.2+#1,#2);
-  \node at (#1+0.1,#2) {\exc{$\downarrow$}};
+  \node at (#1+0.1,#2) {\exca{\dnel}};
+}
+
+\newcommand{\dnb}[2]{
+  \draw [-,thick] (-0.2+#1,#2) -- (0.2+#1,#2);
+  \node at (#1+0.1,#2) {\excb{\dnel}};
+}
+
+\newcommand{\dnc}[2]{
+  \draw [-,thick] (-0.2+#1,#2) -- (0.2+#1,#2);
+  \node at (#1+0.1,#2) {\excc{\dnel}};
 }
 
 % Up and Down electrons
 \newcommand{\updn}[2]{
   \draw [-,thick] (-0.2+#1,#2) -- (0.2+#1,#2);
-  \node at (#1+0.1,#2) {\noexc{$\downarrow$}};
-  \node at (#1-0.1,#2) {\noexc{$\uparrow$}};
+  \node at (#1+0.1,#2) {\noexc{\dnel}};
+  \node at (#1-0.1,#2) {\noexc{\upel}};
 }
 
-\newcommand{\updnrr}[2]{
+\newcommand{\updnab}[2]{
   \draw [-,thick] (-0.2+#1,#2) -- (0.2+#1,#2);
-  \node at (#1+0.1,#2) {\exc{$\downarrow$}};
-  \node at (#1-0.1,#2) {\exc{$\uparrow$}};
+  \node at (#1+0.1,#2) {\excb{\dnel}};
+  \node at (#1-0.1,#2) {\exca{\upel}};
 }
 
-\newcommand{\updnrb}[2]{
+\newcommand{\updncb}[2]{
   \draw [-,thick] (-0.2+#1,#2) -- (0.2+#1,#2);
-  \node at (#1+0.1,#2) {\noexc{$\downarrow$}}; 
-  \node at (#1-0.1,#2) {\exc{$\uparrow$}};
+  \node at (#1+0.1,#2) {\excb{\dnel}};
+  \node at (#1-0.1,#2) {\excc{\upel}};
 }
 
-\newcommand{\updnbr}[2]{
+\newcommand{\updna}[2]{
   \draw [-,thick] (-0.2+#1,#2) -- (0.2+#1,#2);
-  \node at (#1+0.1,#2) {\exc{$\downarrow$}}; 
-  \node at (#1-0.1,#2) {\noexc{$\uparrow$}};
+  \node at (#1+0.1,#2) {\noexc{\dnel}}; 
+  \node at (#1-0.1,#2) {\exca{\upel}};
+}
+
+\newcommand{\updnb}[2]{
+  \draw [-,thick] (-0.2+#1,#2) -- (0.2+#1,#2);
+  \node at (#1-0.1,#2) {\noexc{\upel}}; 
+  \node at (#1+0.1,#2) {\excb{\dnel}};
+}
+
+\newcommand{\updnc}[2]{
+  \draw [-,thick] (-0.2+#1,#2) -- (0.2+#1,#2);
+  \node at (#1+0.1,#2) {\noexc{\dnel}}; 
+  \node at (#1-0.1,#2) {\excc{\upel}};
 }
 
 % Empty orbital
@@ -93,8 +127,9 @@
   \draw [-,thick] (-0.2+#1,#2) -- (0.2+#1,#2);
 }
 
-
-
-
+% Determinant label
+\node at (\XI,-0.7) {$\ket{I}$};
+\node at (\XA,-0.7) {$\ket{\alpha}$};
+\node at (\XJ,-0.7) {$\ket{J}$};
 
 \newcommand{\XX}{0.}

Manuscript/Tikz/test.tex

@@ -1,5 +1,5 @@
-\documentclass[]{article}
-\usepackage{tikz,xcolor}
+\documentclass[usenames,dvipsnames]{standalone}
+\usepackage{physics,tikz}
 \begin{document}
 \begin{tabular}{cccc}
 \input{0_1_11.tikz} & \input{0_1_22.tikz} & \input{0_2_22.tikz} & \input{1_1_11.tikz} \\
@@ -12,4 +12,3 @@
 3 1 22
 \end{tabular}
 \end{document}
-