199 lines
13 KiB
TeX
199 lines
13 KiB
TeX
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\section{Explicitly-correlated FCI methods}
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One of the most fundamental problem of conventional electronic structure methods is their slow energy convergence with respect to the size of the one-electron basis set.
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This problem was already spotted thirty years ago by Kutzelnigg \cite{Kutzelnigg85} who stated that
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\begin{quote}
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\textit{``traditional CI is not really bad, it only has difficulties to represent the wave function at those regions of configuration space where one interelectronic distance $r_{ij}$ approaches zero.''}
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\end{quote}
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To address this problem he proposed to introduce explicitly the correlation between electrons via the introduction of the interelectronic distance $\ree$ as a basis function \cite{Kutzelnigg91, Termath91, Klopper91a, Klopper91b, Noga94}.
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As mentioned in the previous chapter, this yields a prominent improvement of the energy convergence from $O(L^{-3})$ to $O(L^{-7})$ (where $L$ is the maximum angular momentum of the one-electron basis).
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This idea was later generalised to more accurate correlation factors $f_{12} \equiv f(\ree)$ \cite{Persson96, Persson97, May04, Tenno04b, Tew05, May05}.
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The resulting F12 methods achieve chemical accuracy for small organic molecules with relatively small Gaussian basis sets \cite{Tenno12a, Tenno12b, Hattig12, Kong12}.
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For example, as illustrated by Tew and coworkers, one can obtain, at the CCSD(T) level, quintuple-zeta quality correlation energies with a triple-zeta basis \cite{Tew07b}.
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Here, following Kutzelnigg's idea, we propose to introduce the explicit correlation between electrons within the CI method via a dressing of the CI matrix \cite{Huron73, Evangelisti83}.
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This method, involving effective Hamiltonian theory, has been shown to be successful in other scenarios \cite{Heully92}.
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Compared to other explicitly-correlated methods, this dressing strategy has the advantage of introducing the explicit correlation at a low computational cost.
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The present explicitly-correlated dressed CI method is completely general and can be applied to any type of truncated, full, or even selected CI methods \cite{Giner13, Scemama13a, Scemama13b, Scemama14, Giner15, Caffarel16}.
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However, for the sake of generality, we will discuss here the dressing of the FCI matrix.
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%----------------------------------------------------------------
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\subsection{Ansatz}
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%----------------------------------------------------------------
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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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\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 \QOp 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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\QOp = \IdOp - \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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\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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\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 $\ree$, 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}\,\ree + \order{\ree^2}.
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\end{equation}
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%----------------------------------------------------------------
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\subsection{Dressing}
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%----------------------------------------------------------------
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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 $\HOp \ket{\Psi} = E \ket{\Psi}$, 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}{\HOp}{F} - E] + \sum_{J \ne I} c_J H_{IJ} = 0,
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\end{equation}
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where $H_{IJ} = \mel{I}{\HOp}{J}$.
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Hence, we obtain the desired energy by diagonalising 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}{\HOp}{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}{\HOp}{F} = \sum_J t_J \qty[ \mel{I}{\HOp 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 $\mel{I}{\HOp}{F} = 0$, which demonstrates that our dressed CI method becomes exact in the limit of a complete one-electron basis.
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At this stage, two key comments are in order.
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First, as one may have realised, the coefficients $t_I$ are unknown.
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However, they can be set to ensure the correct electron-electron cusp conditions \cite{Tenno04a}.
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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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\end{equation}
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which can be easily solved using standard linear algebra packages.
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Second, because Eq.~\eqref{eq:DrH} depends on the CI coefficient $c_I$, one must iterate the diagonalisation process self-consistently until convergence of the desired eigenvalues of the dressed Hamiltonian $\oH$.
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At each iteration, we solve Eq.~\eqref{eq:tI} to obtain the coefficients $t_I$ and dress the Hamiltonian [see Eq.~\eqref{eq:DrH}].
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In practice, we initially start with a CI vector obtained by the diagonalisation of the undressed Hamiltonian, and convergence is usually reached within few cycles.
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This iteration process can be also embedded in the Davidson diagonalisation process, which is also an iterative process.
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For pathological cases, a DIIS-like procedure may be employed \cite{Pulay80, Pulay82}.
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%%% FIG 1 %%%
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\begin{figure}
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\centering
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\includegraphics[width=0.6\linewidth]{../Chapter5/fig/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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\subsection{Matrix elements}
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%----------------------------------------------------------------
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Compared to a conventional CI calculation, new matrix elements are required.
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The simplest of them $f_{IJ}$ --- required in Eqs.~\eqref{eq:IHF} and \eqref{eq:tI} --- can be easily computed by applying Slater-Condon rules \cite{Szabo}.
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They involve two-electron integrals over the geminal factor $f_{12}$.
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Their computation has been thoroughly studied in the literature in the last thirty years \cite{Kutzelnigg91, Klopper92, Persson97, Klopper02, Manby03, Werner03, Klopper04, Tenno04a, Tenno04b, May05, Manby06, Tenno07, Komornicki11, Reine12, GG16}.
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These can be more or less expensive to compute depending on the choice of the correlation factor.
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As shown in Eq.~\eqref{eq:IHF}, the present explicitly-correlated CI method also requires matrix elements of the form $\mel{I}{\HOp f}{ J}$.
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These are more problematic, as they involve the computation of numerous three-electron integrals over the operator $\ree^{-1}f_{13}$, as well as new two-electron integrals \cite{Kutzelnigg91, Klopper92}.
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We have recently developed recurrence relations and efficient upper bounds in order to compute these types of integrals \cite{3ERI1, 3ERI2, 4eRR, IntF12}.
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However, we will also explore a different route here.
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We propose to compute them using the RI approximation \cite{Kutzelnigg91, Klopper02, Valeev04, Werner07, Hattig12}, which requires a complete basis set (CBS).
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This CBS is built as the union of the orbital basis set (OBS) $\qty{p}$ (divided as occupied $\qty{i}$ and virtual $\qty{a}$ subspaces) augmented by a complementary auxiliary basis set (CABS) $\qty{\alpha}$, such as $ \qty{p} \cap \qty{\alpha} = \varnothing$ and $\braket{p}{\alpha} = 0$ \cite{Klopper02, Valeev04} (see Fig.~\ref{fig:CBS}).
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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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\IdOp = \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}{\HOp}{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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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{PT2}.
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Although $\mel{0}{\HOp}{_{i}^{a}} = 0$, note that the Brillouin theorem does not hold in the CABS, i.e.~$\mel{0}{\HOp}{_{i}^{\alpha}} \neq 0$.
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Here, we will eschew the generalized Brillouin condition (GBC) which set these to zero \cite{Kutzelnigg91}.
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%----------------------------------------------------------------
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\subsection{An illustrative example}
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%----------------------------------------------------------------
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%%% FIG 1 %%%
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\begin{figure}
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\centering
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\includegraphics[width=0.45\linewidth]{../Chapter5/fig/glo1}
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\includegraphics[width=0.45\linewidth]{../Chapter5/fig/glo2}
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\caption{
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\label{fig:Glo}
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$\Ec$ for two electrons on a surface of a glome of unit radius as a function of the maximum angular momentum of the basis set $L$.
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For each calculation, the maximum angular momentum of the auxiliary basis is set to $L_\text{RI} = 3 L$.}
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\end{figure}
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%%% %%%
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To illustrate this method, we have computed the correlation energy of two electrons on a surface of a unit glome --- system we have presented earlier in the memoir --- as a function of the maximum angular momentum of the basis set $L$.
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For each calculation, the maximum angular momentum of the auxiliary (or RI) basis is set to $L_\text{RI} = 3 L$.
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Note that here, because of the form of the exact wave function presented earlier, we used a correlation factor $f_{12} = r_{12}$.
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Various methods have been considered:
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\begin{enumerate}
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\item the conventional FCI method which obviously corresponds to CISD here.
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\item FCI-PT2 in which we compute a second-order Epstein-Nesbet perturbative correction using the auxiliary basis.
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\item the FCI-F12 method where the explicitly-correlated basis function is treated variationally (i.e.~no dressing).
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\item the dressed FCI-F12 method presented here in which the energy is computed by projection and the dressing term is computed explicitly.
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\item the same dressed FCI-F12 method where the dressing term is computed with the help of the auxiliary basis.
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\end{enumerate}
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The results are depicted in Fig.~\ref{fig:Glo}.
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Few comments are in order:
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\begin{itemize}
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\item As expected, the convergence of the conventional FCI method is miserable.
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\item Treating the explicitly-correlated basis function variationally yields the fastest convergence but it requires the computation of expensive and numerous three- and four-electron integrals.
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\item the dressed FCI-F12 method significantly improves the convergence of the energy.
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With a relative small number of basis functions, one can reach sub-millihartree accuracy.
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However, the energy is not variational as it is calculated via projection.
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\item the PT2 correction allows to recover a significant fraction of the missing correlation energy.
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However, it does not produce a wave function one can use as a trial wave function for QMC calculations.
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\item the RI approximation induces a large error but still improve upon the conventional FCI method.
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Therefore, we believe that one should try to compute explicitly the three-electron integrals required in the dressed FCI-F12 method.
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\end{itemize}
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In regards to these results, we believe that the present dressed FCI-F12 method may be an interesting alternative for producing accurate and compact trial wave functions for DMC calculations.
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We hope to be able to consider more realistic systems in the near future as well as studying the nodal surfaces of these explicitly-correlated wave functions.
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