Data table I
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This is BibTeX, Version 0.99d (TeX Live 2016/Debian)
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The top-level auxiliary file: CI-F12.aux
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The style file: aipnum4-1.bst
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@ -49,9 +49,9 @@
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% addresses
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\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}
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\begin{document}
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\begin{document}
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\title{Dressing the configuration interaction matrix with explicit correlation}
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\title{Dressing the configuration interaction matrix with explicit correlation}
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\author{Anthony Scemama}
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\affiliation{\LCPQ}
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@ -98,36 +98,36 @@ Atomic units are used throughout.
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Inspired by a number of previous research \cite{Shiozaki11}, our electronic wave function ansatz $\ket{\Psi} = \kD + \kF$ is simply written as the sum of a ``conventional'' part
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\begin{equation}
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\label{eq:D}
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\kD = \sum_{I} c_I \kI
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\kD = \sum_{I} c_I \kI
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\end{equation}
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composed by a linear combination of determinants $\kI$ of coefficients $c_I$ and an ``explicitly-correlated'' part
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\begin{equation}
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\label{eq:WF-F12-CIPSI}
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\kF = \sum_{I} t_I \hQ f \kI
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\kF = \sum_{I} t_I \hQ f \kI
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\end{equation}
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with coefficients $t_I$.
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The projector
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\begin{equation}
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\hQ = \hI - \sum_{I} \dyad{I}
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\hQ = \hI - \sum_{I} \dyad{I}
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\end{equation}
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ensures the orthogonality between $\kD$ and $\kF$, and
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\begin{equation}
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\label{eq:Ja}
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f = \sum_{i < j} \gamma_{ij} f_{ij}
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f = \sum_{i < j} \gamma_{ij} f_{ij}
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\end{equation}
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is a correlation factor, and
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\begin{equation}
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\gamma_{ij} =
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\begin{cases}
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1/2, & \text{for opposite-spin electrons},
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\\
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1/4, & \text{for same-spin electrons}.
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\end{cases}
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\gamma_{ij} =
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\begin{cases}
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1/2, & \text{for opposite-spin electrons},
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\\
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1/4, & \text{for same-spin electrons}.
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\end{cases}
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\end{equation}
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As first shown by Kato \cite{Kato51, Kato57} (and further elaborated by various authors \cite{Pack66, Morgan93}), for small $r_{12}$, the two-electron correlation factor $f_{12}$ in Eq.~\eqref{eq:Ja} must behave as
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\begin{equation}
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f_{12} = \gamma_{12}\,r_{12} + \order{r_{12}^2}.
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f_{12} = \gamma_{12}\,r_{12} + \order{r_{12}^2}.
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\end{equation}
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%----------------------------------------------------------------
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@ -136,23 +136,23 @@ As first shown by Kato \cite{Kato51, Kato57} (and further elaborated by various
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Our primary goal is to introduce the explicit correlation between electrons at low computational cost.
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Therefore, assuming that $\hH \ket{\Psi} = E \Psi$, one can write, by projection over $\bra{I}$,
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\begin{equation}
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c_I \qty[ H_{II} + c_I^{-1} \mel*{I}{\hH}{F} - E] + \sum_{J \ne I} c_J H_{IJ} = 0.
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c_I \qty[ H_{II} + c_I^{-1} \mel*{I}{\hH}{F} - E] + \sum_{J \ne I} c_J H_{IJ} = 0.
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\end{equation}
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where $H_{IJ} = \mel{I}{\hH}{J}$.
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Hence, we obtain the desired energy by diagonalizing the dressed Hamiltonian:
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\begin{equation}
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\label{eq:DrH}
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\oH_{IJ} =
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\begin{cases}
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H_{II} + c_I^{-1}\mel*{I}{\hH}{F}, & \text{if $I = J$},
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\\
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H_{IJ}, & \text{otherwise},
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\end{cases}
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\oH_{IJ} =
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\begin{cases}
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H_{II} + c_I^{-1}\mel*{I}{\hH}{F}, & \text{if $I = J$},
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\\
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H_{IJ}, & \text{otherwise},
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\end{cases}
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\end{equation}
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with
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\begin{equation}
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\label{eq:IHF}
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\mel{I}{\hH}{F} = \sum_J t_J \qty[ \mel{I}{\hH f}{J} - \sum_{K} H_{IK} f_{KJ} ],
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\mel{I}{\hH}{F} = \sum_J t_J \qty[ \mel{I}{\hH f}{J} - \sum_{K} H_{IK} f_{KJ} ],
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\end{equation}
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and $f_{IJ} = \mel{I}{f}{J}$.
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It is interesting to note that, in an infinite basis, we have $\mel{I}{\hH}{F} = 0$, which demonstrates that our dressed CI method becomes exact in the limit of a complete one-electron basis.
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@ -164,7 +164,7 @@ First, as one may have realized, the coefficients $t_I$ are unknown.
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\alert{This yields the following linear system of equations
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\begin{equation}
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\label{eq:tI}
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\sum_J (\delta_{IJ} + f_{IJ}) t_J = c_I,
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\sum_J (\delta_{IJ} + f_{IJ}) t_J = c_I,
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\end{equation}
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which can be easily solved using standard linear algebra packages.}
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@ -175,13 +175,13 @@ For pathological cases, a DIIS-like procedure may be employed \cite{Pulay82}.
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%%% FIG 1 %%%
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%\begin{figure}
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% \includegraphics[width=\linewidth]{fig1}
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% \caption{
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% \label{fig:CBS}
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% Schematic representation of the various orbital spaces and their notation.
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% The arrows represent the three types of excited determinants contributing to the dressing: the pure doubles $\ket*{_{ij}^{\alpha \beta}}$ (green), the mixed doubles $\ket*{_{ij}^{a \beta}}$ (magenta) and the pure singles $\ket*{_{i}^{\alpha}}$ (orange).}
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% \includegraphics[width=\linewidth]{fig1}
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% \caption{
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% \label{fig:CBS}
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% Schematic representation of the various orbital spaces and their notation.
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% The arrows represent the three types of excited determinants contributing to the dressing: the pure doubles $\ket*{_{ij}^{\alpha \beta}}$ (green), the mixed doubles $\ket*{_{ij}^{a \beta}}$ (magenta) and the pure singles $\ket*{_{i}^{\alpha}}$ (orange).}
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%\end{figure}
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%%% %%%
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%%% %%%
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%----------------------------------------------------------------
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\textit{Matrix elements.---}
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@ -203,17 +203,17 @@ This CBS is built as the union of the orbital basis set (OBS) $\qty{p}$ (divided
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In the CBS, one can write
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\begin{equation}
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\label{eq:RI}
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\hI = \sum_{A \in \mA} \dyad{A}{A}
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\hI = \sum_{A \in \mA} \dyad{A}{A}
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\end{equation}
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where $\mA$ is the set of all the determinants $\kA$ corresponding to electronic excitations from occupied orbitals $\qty{i}$ to the extended virtual orbital space $\qty{a} \cup \qty{\alpha}$.
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Substituting \eqref{eq:RI} into the first term of the right-hand side of Eq.~\eqref{eq:IHF}, one gets
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\begin{equation}
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\label{eq:IHF-RI}
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\begin{split}
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\mel{I}{\hH}{F}
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& = \sum_J t_J \qty[ \sum_{A \in \mA} H_{IA} f_{AJ} - \sum_{K \in \mD} H_{IK} f_{KJ} ]
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\\
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& = \sum_J t_J \sum_{A \in \mC} H_{IA} f_{AJ},
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\mel{I}{\hH}{F}
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& = \sum_J t_J \qty[ \sum_{A \in \mA} H_{IA} f_{AJ} - \sum_{K \in \mD} H_{IK} f_{KJ} ]
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\\
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& = \sum_J t_J \sum_{A \in \mC} H_{IA} f_{AJ},
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\end{split}
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\end{equation}
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where $\mD$ is the set of ``conventional'' determinants obtained by excitations from the occupied space $\qty{i}$ to the virtual one $\qty{a}$, and $\mC = \mA \setminus \mD$.
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@ -231,35 +231,35 @@ Although $\mel{0}{\hH}{_{i}^{a}} = 0$, note that the Brillouin theorem does not
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Here, we will eschew the generalized Brillouin condition (GBC) which set these to zero \cite{Kutzelnigg91}.
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%\begin{gather}
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% \mel*{0}{\hH}{_i^\alpha} = \mel{i}{h}{\alpha} + \sum_{j} \mel{ij}{}{\alpha j}
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% \\
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% \mel*{0}{\hH}{_{ij}^{\alpha\beta}} = \mel{ij}{}{\alpha \beta}
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% \\
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% \mel*{0}{\hH}{_{ij}^{a\beta}} = \mel{ij}{}{a \beta}
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% \mel*{0}{\hH}{_i^\alpha} = \mel{i}{h}{\alpha} + \sum_{j} \mel{ij}{}{\alpha j}
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% \\
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% \mel*{0}{\hH}{_{ij}^{\alpha\beta}} = \mel{ij}{}{\alpha \beta}
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% \\
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% \mel*{0}{\hH}{_{ij}^{a\beta}} = \mel{ij}{}{a \beta}
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%\end{gather}
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%
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%\begin{gather}
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% \mel*{_k^c}{\hH}{_i^\alpha} = \mel{c}{h}{\alpha} + \sum_{j} \mel{cj}{}{\alpha j}
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% \\
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% \mel*{_k^c}{\hH}{_{ij}^{\alpha\beta}} = 0
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% \\
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% \mel*{_k^c}{\hH}{_{ij}^{a\beta}} =
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% \mel*{_k^c}{\hH}{_i^\alpha} = \mel{c}{h}{\alpha} + \sum_{j} \mel{cj}{}{\alpha j}
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% \\
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% \mel*{_k^c}{\hH}{_{ij}^{\alpha\beta}} = 0
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% \\
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% \mel*{_k^c}{\hH}{_{ij}^{a\beta}} =
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%\end{gather}
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%
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%\begin{gather}
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% \mel*{_{kl}^{cd}}{\hH}{_i^\alpha} =
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% \\
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% \mel*{_{kl}^{cd}}{\hH}{_{ij}^{\alpha\beta}} =
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% \\
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% \mel*{_{kl}^{cd}}{\hH}{_{ij}^{a\beta}} =
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% \mel*{_{kl}^{cd}}{\hH}{_i^\alpha} =
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% \\
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% \mel*{_{kl}^{cd}}{\hH}{_{ij}^{\alpha\beta}} =
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% \\
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% \mel*{_{kl}^{cd}}{\hH}{_{ij}^{a\beta}} =
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%\end{gather}
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%
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%\begin{gather}
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% \mel*{_{klm}^{cde}}{\hH}{_i^\alpha} =
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% \\
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% \mel*{_{klm}^{cde}}{\hH}{_{ij}^{\alpha\beta}} =
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% \\
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% \mel*{_{klm}^{cde}}{\hH}{_{ij}^{a\beta}} =
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% \mel*{_{klm}^{cde}}{\hH}{_i^\alpha} =
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% \\
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% \mel*{_{klm}^{cde}}{\hH}{_{ij}^{\alpha\beta}} =
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% \\
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% \mel*{_{klm}^{cde}}{\hH}{_{ij}^{a\beta}} =
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%\end{gather}
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%----------------------------------------------------------------
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@ -267,81 +267,86 @@ Here, we will eschew the generalized Brillouin condition (GBC) which set these t
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%----------------------------------------------------------------
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In all the calculations presented below, we consider the following Slater-type correlation factor \cite{Tenno04a}
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\begin{equation}
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f_{12} = \frac{1 - \exp( - \la r_{12} )}{\la},
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f_{12} = \frac{1 - \exp( - \la r_{12} )}{\la},
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\end{equation}
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which is fitted using $N_\text{GG}$ Gaussian geminals fo computational convenience \cite{Persson96, Persson97, May04, Tenno04b, Tew05, May05}, i.e.
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\begin{equation}
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\exp( - \la r_{12} ) \approx \sum_{\nu=1}^{\NGG} a_\nu \exp( - \la_\nu r_{12}^2 ).
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\exp( - \la r_{12} ) \approx \sum_{\nu=1}^{\NGG} a_\nu \exp( - \la_\nu r_{12}^2 ).
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\end{equation}
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The coefficients $a_\nu$ can be found in Ref.~\onlinecite{Tew05} for various $\NGG$, but we consider $\NGG = 6$ in this study.
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All the calculations have been performed with Quantum Package \cite{QP}.
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%%% TABLE 1 %%%
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%%% TABLE 1 %%%
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\begin{table}
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\caption{
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\label{tab:atoms}
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FCI-F12, CIPSI and FCI total ground-state energy of the neutral atoms for $Z = 2$ to $10$ calculated with Dunning's cc-pVXZ basis set.
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The corresponding cc-pVXZ\_OPTRI auxiliary basis is used as CABS.}
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\begin{ruledtabular}
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\begin{tabular}{lcccc}
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Atom & cc-pVXZ & FCI-F12 & CIPSI & FCI \\
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\begin{tabular}{lcccd}
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Atom & $N$ & FCI-F12 & CIPSI & \text{FCI} \\
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\hline
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\ce{He} & D & & & $-2.887\,595$\footnotemark[1] \\
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& T & & & $-2.900\,232$\footnotemark[1] \\
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& Q & & & $-2.902\,411$\footnotemark[1] \\
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& 5 & & & $-2.903\,152$\footnotemark[1] \\
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& 6 & & & $-2.903\,432$\footnotemark[1] \\
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& $\infty$ & & & $-2.903\,724$\footnotemark[2] \\
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\hline
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\ce{Li} & D & & & \\
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& T & & & \\
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& Q & & & \\
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& 5 & & & \\
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& $\infty$ & & & $-7.478\,060$\footnotemark[3] \\
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\hline
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\ce{Be} & D & & & \\
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& T & & & \\
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& Q & & & \\
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& 5 & & & \\
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& $\infty$ & & & $-14.667\,356$\footnotemark[4] \\
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\hline
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\ce{B} & D & & & \\
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& T & & & \\
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& Q & & & \\
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& 5 & & & \\
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& $\infty$ & & & $-24.653\,866$\footnotemark[5] \\
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\ce{He} & D & & & -2.887\,595 \footnotemark[1] \\
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(cc-pV$N$Z) & T & & & -2.900\,232 \footnotemark[1] \\
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& Q & & & -2.902\,411 \footnotemark[1] \\
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& 5 & & & -2.903\,152 \footnotemark[1] \\
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& 6 & & & -2.903\,432 \footnotemark[1] \\
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& $\infty$ & & & -2.903\,724 \footnotemark[2] \\
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\hline
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\ce{C} & D & & & \\
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& T & & & \\
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& Q & & & \\
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& 5 & & & \\
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& $\infty$ & & & $-37.840\,129$\footnotemark[6] \\
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\ce{Li} & D & & & -7.466\,025 (FCI) \\
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(cc-pCV$N$Z) & T & & & -7.474\,251 (FCI) \\
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& Q & & & -7.476\,373 (CIPSI) \\
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& $\infty$ & & & -7.478\,060 \footnotemark[3] \\
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\hline
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\ce{N} & D & & & \\
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& T & & & \\
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& Q & & & \\
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& 5 & & & \\
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& $\infty$ & & & $ 54.588917$\footnotemark[7] \\
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\hline
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\ce{O} & D & & & \\
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& T & & & \\
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& Q & & & \\
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& 5 & & & \\
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& $\infty$ & & & $-75.066892$\footnotemark[7] \\
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\ce{Be} & D & & & -14.651\,833 (FCI) \\
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(cc-pCV$N$Z) & T & & & -14.662\,369 (CIPSI) \\
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& Q & & & -14.665\,566 (CIPSI) \\
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& $\infty$ & & & -14.667\,356 \footnotemark[4] \\
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& $\infty$ & & & -14.667\,39 (TOTO) \\
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\hline
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\ce{F} & D & & & \\
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& T & & & \\
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& Q & & & \\
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& 5 & & & \\
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& $\infty$ & & & $-99.733\,424$\footnotemark[7] \\
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\ce{B} & D & & & -24.619\,099 (CIPSI) \\
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(cc-pwCV$N$Z) & T & & & -24.643\,222 (CIPSI) \\
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& Q & & & -24.650\,331 (CIPSI) \\
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& 5 & & & -24.652\,309 (CIPSI) \\
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& $\infty$ & & & -24.653\,866 \footnotemark[5] \\
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& $\infty$ & & & -24.653\,90 (TOTO) \\
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\hline
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\ce{Ne} & D & & & \\
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& T & & & \\
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& Q & & & \\
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& 5 & & & \\
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& $\infty$ & & & $-128.937\,274$\footnotemark[7] \\
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\ce{C} & D & & & -37.792\,466 (CIPSI) \\
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(cc-pwCV$N$Z) & T & & & -37.829\,847 (CIPSI) \\
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& Q & & & -37.839\,816 (CIPSI) \\
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& 5 & & & -37.842\,731 (CIPSI) \\
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& $\infty$ & & & -37.840\,129 6 \\
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& $\infty$ & & & -37.845\,0 (TOTO) \\
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\hline
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\ce{N} & D & & & \\
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(cc-pwCV$N$Z) & T & & & \\
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& Q & & & \\
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& 5 & & & \\
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& $\infty$ & & & -54.588\,917 \footnotemark[7] \\
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& $\infty$ & & & -54.589\,3 (TOTO) \\
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\hline
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\ce{O} & D & & & \\
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(cc-pwCV$N$Z) & T & & & \\
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& Q & & & \\
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& 5 & & & \\
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& $\infty$ & & & -75.066\,892 \footnotemark[7] \\
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& $\infty$ & & & -75.067\,4 (TOTO) \\
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\hline
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\ce{F} & D & & & \\
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(cc-pwCV$N$Z) & T & & & \\
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& Q & & & \\
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& 5 & & & \\
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& $\infty$ & & & -99.733\,424 \footnotemark[7] \\
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& $\infty$ & & & -99.734\,1 (TOTO) \\
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\hline
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\ce{Ne} & D & & & \\
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(cc-pwCV$N$Z) & T & & & \\
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& Q & & & \\
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& 5 & & & \\
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& $\infty$ & & & -128.937\,274 \footnotemark[7] \\
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& $\infty$ & & & -128.938\,3 (TOTO) \\
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\end{tabular}
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\end{ruledtabular}
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\end{ruledtabular}
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\footnotetext[1]{Reference \onlinecite{Kong12}}
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\footnotetext[2]{Reference \onlinecite{Nakashima07}}
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\footnotetext[3]{Reference \onlinecite{Puchalski09}}
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@ -360,23 +365,23 @@ CIPSI, FCI-F12 i-FCIQMC and FCI total ground-state energy of the \ce{H2}, \ce{F2
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The corresponding cc-pVXZ\_OPTRI auxiliary basis is used as CABS.}
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\begin{ruledtabular}
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\begin{tabular}{lccccc}
|
||||
Molecule & cc-pVXZ & \mcc{CIPSI} & \mcc{FCI-F12} & \mcc{i-FCIQMC} & \mcc{FCI} \\
|
||||
Molecule & cc-pVXZ & \mcc{CIPSI} & \mcc{FCI-F12} & \mcc{i-FCIQMC} & \mcc{FCI} \\
|
||||
\hline
|
||||
\ce{H2} & D & & & \\
|
||||
& T & & & \\
|
||||
& Q & & & \\
|
||||
& 5 & & & \\
|
||||
& $\infty$ & & & & $-1.174\,476$\footnotemark[1] \\
|
||||
\ce{H2} & D & & & \\
|
||||
& T & & & \\
|
||||
& Q & & & \\
|
||||
& 5 & & & \\
|
||||
& $\infty$ & & & & $-1.174\,476$\footnotemark[1] \\
|
||||
\hline
|
||||
\ce{F2} & D & $-199.099\,28$\footnotemark[2] & & $-199.099\,41(9)$\footnotemark[3] \\
|
||||
& T & $-199.296\,5$\footnotemark[2] & & $-199.297\,7(1)$\footnotemark[3] \\
|
||||
\ce{F2} & D & $-199.099\,28$\footnotemark[2] & & $-199.099\,41(9)$\footnotemark[3] \\
|
||||
& T & $-199.296\,5$\footnotemark[2] & & $-199.297\,7(1)$\footnotemark[3] \\
|
||||
\hline
|
||||
\ce{H2O} & D & $-76.282\,136$\footnotemark[4] & & & $-76.282\,865$\footnotemark[5] \\
|
||||
& T & $-76.388\,287$\footnotemark[4] & & & $-76.390\,158$\footnotemark[5] \\
|
||||
& Q & $-76.419\,324$\footnotemark[4] & & & $-76.421\,148$\footnotemark[5] \\
|
||||
& 5 & $-76.428\,550$\footnotemark[4] & & & $-76.431\,105$\footnotemark[5] \\
|
||||
\ce{H2O} & D & $-76.282\,136$\footnotemark[4] & & & $-76.282\,865$\footnotemark[5] \\
|
||||
& T & $-76.388\,287$\footnotemark[4] & & & $-76.390\,158$\footnotemark[5] \\
|
||||
& Q & $-76.419\,324$\footnotemark[4] & & & $-76.421\,148$\footnotemark[5] \\
|
||||
& 5 & $-76.428\,550$\footnotemark[4] & & & $-76.431\,105$\footnotemark[5] \\
|
||||
\end{tabular}
|
||||
\end{ruledtabular}
|
||||
\end{ruledtabular}
|
||||
\footnotetext[1]{Reference \onlinecite{Pachucki10}}
|
||||
\footnotetext[2]{Reference \onlinecite{Giner15}}
|
||||
\footnotetext[3]{Reference \onlinecite{Cleland12}}
|
||||
|
||||
Loading…
Reference in New Issue
Block a user