From 0235c2c8f7e04f97e219278c191fb7157deabfc8 Mon Sep 17 00:00:00 2001 From: Anthony Scemama Date: Tue, 22 Sep 2020 12:58:18 +0200 Subject: [PATCH] Revision --- Response_Letter/CNRS_logo.pdf | Bin 0 -> 180517 bytes Response_Letter/ResponseLetter.bib | 57 + Response_Letter/UPS_letterhead.sty | 70 + Response_Letter/UPS_logo.pdf | Bin 0 -> 235750 bytes Revision/.gitignore | 12 + Revision/Makefile | 17 + Revision/algorithm.pdf | Bin 0 -> 54680 bytes Revision/density-mu.pdf | Bin 0 -> 33141 bytes Revision/g2-dmc-dz.pdf | Bin 0 -> 47717 bytes Revision/g2-dmc.pdf | Bin 0 -> 47835 bytes Revision/g2-ndet.pdf | Bin 0 -> 40329 bytes Revision/h2o-200-dmc.pdf | Bin 0 -> 6906 bytes Revision/h2o-dmc.pdf | Bin 0 -> 10060 bytes Revision/on-top-mu.pdf | Bin 0 -> 44185 bytes Revision/overlap.pdf | Bin 0 -> 5755 bytes Revision/rsdft-cipsi-qmc.bib | 1947 ++++++++++++++++++++++++++++ Revision/rsdft-cipsi-qmc.tex | 1198 +++++++++++++++++ 17 files changed, 3301 insertions(+) create mode 100644 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Rev. B}, + volume = {61}, + number = {4}, + pages = {2599--2612}, + year = {2000}, + month = {Jan}, + issn = {2469-9969}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevB.61.2599} +} + +@article{Sorella_1998, + author = {Sorella, Sandro}, + title = {{Green Function Monte Carlo with Stochastic Reconfiguration}}, + journal = {Phys. Rev. Lett.}, + volume = {80}, + number = {20}, + pages = {4558--4561}, + year = {1998}, + month = {May}, + issn = {1079-7114}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevLett.80.4558} +} + +@article{Hetherington_1984, + author = {Hetherington, J. H.}, + title = {{Observations on the statistical iteration of matrices}}, + journal = {Phys. Rev. A}, + volume = {30}, + number = {5}, + pages = {2713--2719}, + year = {1984}, + month = {Nov}, + issn = {2469-9934}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevA.30.2713} +} + +@article{Assaraf_2000, + author = {Assaraf, Roland and Caffarel, Michel and + Khelif, Anatole}, + title = {{Diffusion Monte Carlo methods with a fixed number of walkers}}, + journal = {Phys. Rev. E}, + volume = {61}, + number = {4}, + pages = {4566--4575}, + year = {2000}, + month = {Apr}, + issn = {2470-0053}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevE.61.4566} +} \ No newline at end of file diff --git a/Response_Letter/UPS_letterhead.sty b/Response_Letter/UPS_letterhead.sty new file mode 100644 index 0000000..a16887e --- /dev/null +++ b/Response_Letter/UPS_letterhead.sty @@ -0,0 +1,70 @@ +%ANU etterhead Yves +%version 1.0 12/06/08 +%need to be improved + + +\RequirePackage{graphicx} + +%%%%%%%%%%%%%%%%%%%%% DEFINE USER-SPECIFIC MACROS BELOW %%%%%%%%%%%%%%%%%%%%% +\def\Who {Anthony Scemama} +\def\What {Dr} +\def\Where {Universit\'e Paul Sabatier} +\def\Address {Laboratoire de Chimie et Physique Quantiques} +\def\CityZip {Toulouse, France} +\def\Email {scemama@irsamc.ups-tlse.fr} +\def\TEL {+33 5 61 55 73 39} +\def\URL {} % NOTE: use $\sim$ for tilde + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% MARGINS %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\textwidth 6in +\textheight 9.25in +\oddsidemargin 0.25in +\evensidemargin 0.25in +\topmargin -1.50in +\longindentation 0.50\textwidth +\parindent 5ex + +%%%%%%%%%%%%%%%%%%%%%%%%%%% ADDRESS MACRO BELOW %%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\address{ + \includegraphics[height=0.7in]{CNRS_logo.pdf} \hspace*{\fill}\includegraphics[height=0.7in]{UPS_logo.pdf} + \\ + \hrulefill + \\ + {\small \What~\Who\hspace*{\fill} Telephone:\ \TEL + \\ + \Where\hspace*{\fill} Email:\ \Email + \\ + \Address\hspace*{\fill} + \\ + \CityZip\hspace*{\fill} \URL} + } + +%%%%%%%%%%%%%%%%%%%%%%%%%%%% OTHER MACROS BELOW %%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%\signature{\What~\Who} + +\def\opening#1{\ifx\@empty\fromaddress + \thispagestyle{firstpage} + \hspace*{\longindendation}\today\par + \else \thispagestyle{empty} + {\centering\fromaddress \vspace{5\parskip} \\ +\today\hspace*{\fill}\par} + \fi + \vspace{3\parskip} + {\raggedright \toname \\ \toaddress \par}\vspace{3\parskip} + \noindent #1\par\raggedright\parindent 5ex\par + } + +%I 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eigenfunctions of the $\Hat{S}^2$ spin operator, as +described in Ref.~\onlinecite{Applencourt_2018}. + +QMC calculations have been performed with \textit{QMC=Chem},\cite{Scemama_2013} +in the determinant localization approximation (DLA),\cite{Zen_2019} +where only the determinantal component of the trial wave +function is present in the expression of the wave function on which +the pseudopotential is localized. Hence, in the DLA, the fixed-node +energy is independent of the Jastrow factor, as in all-electron +calculations. Simple Jastrow factors were used to reduce the +fluctuations of the local energy (see Sec.~\ref{sec:rsdft-j} for their explicit expression). +The FN-DMC simulations are performed with all-electron moves using the +stochastic reconfiguration algorithm developed by Assaraf \textit{et al.} +\cite{Assaraf_2000} with a time step of $5 \times 10^{-4}$ a.u. and a +projecting time of $1$ a.u. \alert{With such parameters, both the +time-step error and the bias due to the finite projecting time are +smaller than the error bars.} + +All the data related to the present study (geometries, basis sets, total energies, \textit{etc}) can be found in the {\SI}. + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\section{Influence of the range-separation parameter on the fixed-node error} +\label{sec:mu-dmc} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +%%% TABLE I %%% +\begin{table} + \caption{FN-DMC energy $\EDMC$ (in \hartree{}) and number of determinants $\Ndet$ in \ce{H2O} for various trial wave functions $\Psi^{\mu}$ obtained with the srPBE density functional.} + \label{tab:h2o-dmc} + \begin{ruledtabular} + \begin{tabular}{crlrl} + & \multicolumn{2}{c}{VDZ-BFD} & \multicolumn{2}{c}{VTZ-BFD} \\ + \cline{2-3} \cline{4-5} + $\mu$ & $\Ndet$ & $\EDMC$ & $\Ndet$ & $\EDMC$ \\ +\hline + $0.00$ & $11$ & $-17.253\,59(6)$ & $23$ & $-17.256\,74(7)$ \\ + $0.20$ & $23$ & $-17.253\,73(7)$ & $23$ & $-17.256\,73(8)$ \\ + $0.30$ & $53$ & $-17.253\,4(2)$ & $219$ & $-17.253\,7(5)$ \\ + $0.50$ & $1\,442$ & $-17.253\,9(2)$ & $16\,99$ & $-17.257\,7(2)$ \\ + $0.75$ & $3\,213$ & $-17.255\,1(2)$ & $13\,362$ & $-17.258\,4(3)$ \\ + $1.00$ & $6\,743$ & $-17.256\,6(2)$ & $256\,73$ & $-17.261\,0(2)$ \\ + $1.75$ & $54\,540$ & $-17.259\,5(3)$ & $207\,475$ & $-17.263\,5(2)$ \\ + $2.50$ & $51\,691$ & $-17.259\,4(3)$ & $858\,123$ & $-17.264\,3(3)$ \\ + $3.80$ & $103\,059$ & $-17.258\,7(3)$ & $1\,621\,513$ & $-17.263\,7(3)$ \\ + $5.70$ & $102\,599$ & $-17.257\,7(3)$ & $1\,629\,655$ & $-17.263\,2(3)$ \\ + $8.50$ & $101\,803$ & $-17.257\,3(3)$ & $1\,643\,301$ & $-17.263\,3(4)$ \\ + $\infty$ & $200\,521$ & $-17.256\,8(6)$ & $1\,631\,982$ & $-17.263\,9(3)$ \\ + \end{tabular} + \end{ruledtabular} +\end{table} +%%% %%% %%% %%% + +%%% FIG 2 %%% +\begin{figure} + \includegraphics[width=\columnwidth]{h2o-dmc.pdf} + \caption{FN-DMC energy of \ce{H2O} as a function + of $\mu$ for various trial wave functions $\Psi^{\mu}$ generated at different levels of theory. + The raw data can be found in the {\SI}.} + \label{fig:h2o-dmc} +\end{figure} +%%% %%% %%% %%% + +The first question we would like to address is the quality of the +nodes of the wave function $\Psi^{\mu}$ obtained for intermediate values of the +range separation parameter (\ie, $0 < \mu < +\infty$). +For this purpose, we consider a weakly correlated molecular system, namely the water +molecule at its experimental geometry. \cite{Caffarel_2016} +We then generate trial wave functions $\Psi^\mu$ for multiple values of +$\mu$, and compute the associated FN-DMC energy keeping fixed all the +parameters impacting the nodal surface, such as the CI coefficients and the molecular orbitals. + +%====================================================== +\subsection{Fixed-node energy of RS-DFT-CIPSI trial wave functions} +\label{sec:fndmc_mu} +%====================================================== +From Table~\ref{tab:h2o-dmc} and Fig.~\ref{fig:h2o-dmc}, where we report the fixed-node energy of \ce{H2O} as a function of $\mu$ for various short-range density functionals and basis sets, +one can clearly observe that relying on FCI trial +wave functions ($\mu = \infty$) give FN-DMC energies lower +than the energies obtained with a single KS determinant ($\mu=0$): +a lowering of $3.2 \pm 0.6$~m\hartree{} at the double-$\zeta$ level and $7.2 \pm +0.3$~m\hartree{} at the triple-$\zeta$ level are obtained with the srPBE functional. +Coming now to the nodes of the trial wave function $\Psi^{\mu}$ with +intermediate values of $\mu$, Fig.~\ref{fig:h2o-dmc} shows that +a smooth behavior is obtained: +starting from $\mu=0$ (\ie, the KS determinant), +the FN-DMC error is reduced continuously until it reaches a minimum +for an optimal value of $\mu$ (which is obviously basis set and functional dependent), +and then the FN-DMC error raises until it reaches the $\mu=\infty$ limit (\ie, the FCI wave function). +For instance, with respect to the fixed-node energy associated with the RS-DFT-CIPSI(srPBE/VDZ-BFD) trial wave function at $\mu=\infty$, +one can obtain a lowering of the FN-DMC energy of $2.6 \pm 0.7$~m\hartree{} +with an optimal value of $\mu=1.75$~bohr$^{-1}$. +This lowering in FN-DMC energy is to be compared with the $3.2 \pm +0.7$~m\hartree{} gain in FN-DMC energy between the KS wave function ($\mu=0$) +and the FCI wave function ($\mu=\infty$). When the basis set is improved, the +gain in FN-DMC energy with respect to the FCI trial wave function is reduced, +and the optimal value of $\mu$ is slightly shifted towards large $\mu$ as expected. +Last but not least, the nodes of the wave functions $\Psi^\mu$ obtained with the srLDA +functional give very similar FN-DMC energies with respect +to those obtained with srPBE, even if the +RS-DFT energies obtained with these two functionals differ by several +tens of m\hartree{}. +Accordingly, all the RS-DFT calculations are performed with the srPBE functional in the remaining of this paper. + +Another important aspect here is the compactness of the trial wave functions $\Psi^\mu$: +at $\mu=1.75$~bohr$^{-1}$, $\Psi^{\mu}$ has \textit{only} $54\,540$ determinants at the RS-DFT-CIPSI(srPBE/VDZ-BFD) level, while the FCI wave function contains $200\,521$ determinants (see Table \ref{tab:h2o-dmc}). Even at the RS-DFT-CIPSI(srPBE/VTZ-BFD) level, we observe a reduction by a factor two in the number of determinants between the optimal $\mu$ value and $\mu = \infty$. +The take-home message of this first numerical study is that RS-DFT-CIPSI trial wave functions can yield a lower fixed-node energy with more compact multi-determinant expansion as compared to FCI. +This is a key result of the present study. + +%====================================================== +\subsection{RS-DFT vs Jastrow factor} +\label{sec:rsdft-j} +%====================================================== +The data presented in Sec.~\ref{sec:fndmc_mu} evidence that, in a finite basis, RS-DFT can provide +trial wave functions with better nodes than FCI wave functions. +As mentioned in Sec.~\ref{sec:SD}, such behavior can be directly compared to the common practice of +re-optimizing the multi-determinant part of a trial wave function $\Psi$ (the so-called Slater part) in the presence of the exponentiated Jastrow factor $e^J$. \cite{Umrigar_2005,Scemama_2006,Umrigar_2007,Toulouse_2007,Toulouse_2008} +Hence, in the present paragraph, we would like to elaborate further on the link between RS-DFT +and wave function optimization in the presence of a Jastrow factor. +For the sake of simplicity, the molecular orbitals and the Jastrow +factor are kept fixed; only the CI coefficients are varied. + +Let us then assume a fixed Jastrow factor $J(\br_1, \ldots , \br_\Nelec)$ (where $\br_i$ is the position of the $i$th electron and $\Nelec$ the total number of electrons), +and a corresponding Slater-Jastrow wave function $\Phi = e^J \Psi$, +where +\begin{equation} +\label{eq:Slater} + \Psi = \sum_I c_I D_I +\end{equation} +is a general linear combination of (fixed) Slater determinants $D_I$. +The only variational parameters in $\Phi$ are therefore the coefficients $c_I$ belonging to the Slater part $\Psi$. +Let us define $\Psi^J$ as the linear combination of Slater determinants minimizing the variational energy associated with $\Phi$, \ie, +\begin{equation} + \Psi^J = \argmin_{\Psi}\frac{ \mel{ \Psi }{ e^{J} \hat{H} e^{J} }{ \Psi } }{\mel{ \Psi }{ e^{2J} }{ \Psi } }. +\end{equation} +Such a wave function satisfies the generalized Hermitian eigenvalue equation +\begin{equation} + e^{J} \hat{H} \qty( e^{J} \Psi^J ) = E \, e^{2J} \Psi^J, +\label{eq:ci-j} +\end{equation} +but also the non-Hermitian transcorrelated eigenvalue problem\cite{BoyHan-PRSLA-69,BoyHanLin-1-PRSLA-69,BoyHanLin-2-PRSLA-69,Tenno_2000,Luo-JCP-10,YanShi-JCP-12,CohLuoGutDowTewAla-JCP-19} +\begin{equation} + \label{eq:transcor} + e^{-J} \hat{H} \qty( e^{J} \Psi^J) = E \, \Psi^J, +\end{equation} +which is much easier to handle despite its non-Hermiticity. +Of course, the FN-DMC energy of $\Phi$ depends only on the nodes of $\Psi^J$ as the positivity of the Jastrow factor makes sure that it does not alter the nodal surface. +In a finite basis set and with an accurate Jastrow factor, it is known that the nodes +of $\Psi^J$ may be better than the nodes of the FCI wave function. +Hence, we would like to compare $\Psi^J$ and $\Psi^\mu$. + +To do so, we have made the following numerical experiment. +First, we extract the 200 determinants with the largest weights in the FCI wave +function out of a large CIPSI calculation obtained with the VDZ-BFD basis. Within this set of determinants, +we solve the self-consistent equations of RS-DFT [see Eq.~\eqref{rs-dft-eigen-equation}] +for different values of $\mu$ using the srPBE functional. This gives the CI expansions of $\Psi^\mu$. +Then, within the same set of determinants we optimize the CI coefficients in the presence of +a simple one- and two-body Jastrow factor $e^J$ with $J = J_\text{eN} + J_\text{ee}$ and +\begin{subequations} +\begin{gather} + J_\text{eN} = - \sum_{A=1}^{\Nat} \sum_{i=1}^{\Nelec} \qty( \frac{\alpha_A\, r_{iA}}{1 + \alpha_A\, r_{iA}} )^2, +\label{eq:jast-eN} \\ + J_\text{ee} = \sum_{i < j}^{\Nelec} \frac{a\, r_{ij}}{1 + b\, r_{ij}}. +\label{eq:jast-ee} +\end{gather} +\end{subequations} +The one-body Jastrow factor $J_\text{eN}$ contains the electron-nucleus terms (where $\Nat$ is the number of nuclei) with a single parameter +$\alpha_A$ per nucleus. +The two-body Jastrow factor $J_\text{ee}$ gathers the electron-electron terms +where the sum over $i < j$ loops over all unique electron pairs. +In Eqs.~\eqref{eq:jast-eN} and \eqref{eq:jast-ee}, $r_{iA}$ is the distance between the $i$th electron and the $A$th nucleus while $r_{ij}$ is the interlectronic distance between electrons $i$ and $j$. +The parameters $a=1/2$ +and $b=0.89$ were fixed, and the parameters $\gamma_{\text{O}}=1.15$ and $\gamma_{\text{H}}=0.35$ +were obtained by energy minimization of a single determinant. +The optimal CI expansion $\Psi^J$ is obtained by sampling the matrix elements +of the Hamiltonian ($\mathbf{H}$) and overlap ($\mathbf{S}$) matrices in the +basis of Jastrow-correlated determinants $e^J D_i$: +\begin{subequations} +\begin{gather} +H_{ij} = \expval{ \frac{e^J D_i}{\Psi^J}\, \frac{\hat{H}\, (e^J D_j)}{\Psi^J} }, +\\ +S_{ij} = \expval{ \frac{e^J D_i}{\Psi^J}\, \frac{e^J D_j}{\Psi^J} }, +\end{gather} +\end{subequations} +and solving Eq.~\eqref{eq:ci-j}.\cite{Nightingale_2001} + +We can easily compare $\Psi^\mu$ and $\Psi^J$ as they are developed +on the same set of Slater determinants. +In Fig.~\ref{fig:overlap}, we plot the overlap +$\braket*{\Psi^J}{\Psi^\mu}$ obtained for water as a function of $\mu$ (left graph) +as well as the FN-DMC energy of the wave function +$\Psi^\mu$ as a function of $\mu$ together with that of $\Psi^J$ (right graph). + +%%% FIG 3 %%% +\begin{figure*} + \includegraphics[width=\columnwidth]{overlap.pdf} + \includegraphics[width=\columnwidth]{h2o-200-dmc.pdf} + \caption{Left: Overlap between $\Psi^\mu$ and $\Psi^J$ as a function of $\mu$ for \ce{H2O}. + Right: FN-DMC energy of $\Psi^\mu$ (red curve) as a function of $\mu$, together with + the FN-DMC energy of $\Psi^J$ (blue line) for \ce{H2O}. + The width of the lines represent the statistical error bars. + For these two trial wave functions, the CI expansion consists of the 200 most important + determinants of the FCI expansion obtained with the VDZ-BFD basis (see Sec.~\ref{sec:rsdft-j} for more details). The raw data can be found in the {\SI}.} + \label{fig:overlap} +\end{figure*} +%%% %%% %%% %%% + +As evidenced by Fig.~\ref{fig:overlap}, there is a clear maximum overlap between the two trial wave functions at $\mu=1$~bohr$^{-1}$, which +coincides with the minimum of the FN-DMC energy of $\Psi^\mu$. +Also, it is interesting to notice that the FN-DMC energy of $\Psi^J$ is compatible +with that of $\Psi^\mu$ for $0.5 < \mu < 1$~bohr$^{-1}$, as shown by the overlap between the red and blue bands. +This confirms that introducing short-range correlation with DFT has +an impact on the CI coefficients similar to a Jastrow factor. +This is another key result of the present study. + +%%% FIG 4 %%% +\begin{figure*} + \includegraphics[width=\columnwidth]{density-mu.pdf} + \includegraphics[width=\columnwidth]{on-top-mu.pdf} + \caption{One-electron density $n(\br)$ (left) and on-top pair + density $n_2(\br,\br)$ (right) along the \ce{O-H} axis of \ce{H2O} + as a function of $\mu$ for $\Psi^\mu$, and $\Psi^J$ (dashed + curve). + The integrated on-top pair density $\expval{P}$ is + given in the legend. + For all trial wave functions, the CI expansion consists of the 200 most important + determinants of the FCI expansion obtained with the VDZ-BFD basis (see Sec.~\ref{sec:rsdft-j} for more details). The raw data can be found in the {\SI}.} + \label{fig:densities} +\end{figure*} +%%% %%% %%% %%% + +In order to refine the comparison between $\Psi^\mu$ and $\Psi^J$, we +report several quantities related to the one- and two-body densities of +$\Psi^J$ and $\Psi^\mu$ with different values of $\mu$. First, we +report in the legend of the right panel of Fig~\ref{fig:densities} the integrated on-top pair density +\begin{equation} + \expval{ P } = \int d\br \,n_2(\br,\br), +\end{equation} +obtained for both $\Psi^\mu$ and $\Psi^J$, +where $n_2(\br_1,\br_2)$ is the two-body density [normalized to $\Nelec(\Nelec-1)$]. +Then, in order to have a pictorial representation of both the one-body density $n(\br)$ and the on-top +pair density $n_2(\br,\br)$, we report in Fig.~\ref{fig:densities} +the plots of $n(\br)$ and $n_2(\br,\br)$ along one of the \ce{O-H} axis of the water molecule. + +From these data, one can clearly notice several trends. +First, the integrated on-top pair density $\expval{ P }$ decreases when $\mu$ increases, +which is expected as the two-electron interaction increases in +$H^\mu[n]$. +Second, Fig.~\ref{fig:densities} shows that the relative variations of the on-top pair density with respect to $\mu$ +are much more important than that of the one-body density, the latter +being essentially unchanged between $\mu=0$ and $\mu=\infty$ while the +former can vary by about 10$\%$ in some regions. +%TODO TOTO +In the high-density region of the \ce{O-H} bond, the value of the on-top +pair density obtained from $\Psi^J$ is superimposed with +$\Psi^{\mu=0.5}$, and at a large distance the on-top pair density of $\Psi^J$ is +the closest to that of $\Psi^{\mu=\infty}$. The integrated on-top pair density +obtained with $\Psi^J$ is $\expval{P}=1.404$, which nestles between the values obtained at +$\mu=0.5$ and $\mu=1$~bohr$^{-1}$, consistently with the FN-DMC energies +and the overlap curve depicted in Fig.~\ref{fig:overlap}. + +These data suggest that the wave functions $\Psi^{0.5 \le \mu \le 1}$ and $\Psi^J$ are close, +and therefore that the operators that produced these wave functions (\ie, $H^\mu[n]$ and $e^{-J}He^J$) contain similar physics. +Considering the form of $\hat{H}^\mu[n]$ [see Eq.~\eqref{H_mu}], +one can notice that the differences with respect to the usual bare Hamiltonian come +from the non-divergent two-body interaction $\hat{W}_{\text{ee}}^{\text{lr},\mu}$ +and the effective one-body potential $\hat{\bar{V}}_{\text{Hxc}}^{\text{sr},\mu}[n]$ which is the functional derivative of the Hxc functional. +The roles of these two terms are therefore very different: with respect +to the exact ground-state wave function $\Psi$, the non-divergent two-body interaction +increases the probability of finding electrons at short distances in $\Psi^\mu$, +while the effective one-body potential $\hat{\bar{V}}_{\text{Hxc}}^{\text{sr},\mu}[n_{\Psi^{\mu}}]$, +providing that it is exact, maintains the exact one-body density. +This is clearly what has been observed in +Fig.~\ref{fig:densities}. +Regarding now the transcorrelated Hamiltonian $e^{-J}He^J$, as pointed out by Ten-no,\cite{Tenno_2000} +the effective two-body interaction induced by the presence of a Jastrow factor +can be non-divergent when a proper two-body Jastrow factor $J_\text{ee}$ is chosen, \ie, the Jastrow factor must fulfill the so-called electron-electron cusp conditions. \cite{Kato_1957,Pack_1966} +There is therefore a clear parallel between $\hat{W}_{\text{ee}}^{\text{lr},\mu}$ in RS-DFT and $J_\text{ee}$ in FN-DMC. +Moreover, the one-body Jastrow term $J_\text{eN}$ ensures that the one-body density remains unchanged when the CI coefficients are re-optmized in the presence of $J_\text{ee}$. +There is then a second clear parallel between $\hat{\bar{V}}_{\text{Hxc}}^{\text{sr},\mu}[n]$ in RS-DFT and $J_\text{eN}$ in FN-DMC. +Thus, one can understand the similarity between the eigenfunctions of $H^\mu$ and the optimization of the Slater-Jastrow wave function: +they both deal with an effective non-divergent interaction but still +produce a reasonable one-body density. + +%============================ +\subsection{Intermediate conclusions} +\label{sec:int_ccl} +%============================ + +As conclusions of the first part of this study, we can highlight the following observations: +\begin{itemize} +\item With respect to the nodes of a KS determinant or a FCI wave function, + one can obtain a multi-determinant trial wave function $\Psi^\mu$ with a smaller + fixed-node error by properly choosing an optimal value of $\mu$. +\item The optimal $\mu$ value is system- and basis-set-dependent, and it grows with basis set size. +\item Numerical experiments (overlap $\braket*{\Psi^\mu}{\Psi^J}$, + one-body density, on-top pair density, and FN-DMC energy) indicate + that the RS-DFT scheme essentially plays the role of a simple Jastrow factor + by mimicking short-range correlation effects. This latter + statement can be qualitatively understood by noticing that both RS-DFT + and the trans-correlated approach deal with an effective non-divergent + electron-electron interaction, while keeping the density constant. +\end{itemize} + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\section{Energy differences in FN-DMC: atomization energies} +\label{sec:atomization} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +Atomization energies are challenging for post-HF methods +because their calculation requires a subtle balance in the +description of atoms and molecules. The mainstream one-electron basis sets employed in molecular electronic structure +calculations are atom-centered, so they are, by construction, better adapted to +atoms than molecules. Thus, atomization energies usually tend to be +underestimated by variational methods. +In the context of FN-DMC calculations, the nodal surface is imposed by +the determinantal part of the trial wave function which is expanded in the very same atom-centered basis +set. Thus, we expect the fixed-node error to be also intimately connected to +the basis set incompleteness error. +Increasing the size of the basis set improves the description of +the density and of the electron correlation, but also reduces the +imbalance in the description of atoms and +molecules, leading to more accurate atomization energies. +The size-consistency and the spin-invariance of the present scheme, +two key properties to obtain accurate atomization energies, +are discussed in Appendices \ref{app:size} and \ref{app:spin}, respectively. + +%%% FIG 6 %%% +\begin{squeezetable} +\begin{table*} + \caption{Mean absolute errors (MAEs), mean signed errors (MSEs), and + root mean square errors (RMSEs) with respect to the NIST reference values obtained with various methods and + basis sets. + All quantities are given in kcal/mol. The raw data can be found in the {\SI}.} + \label{tab:mad} + \begin{ruledtabular} + \begin{tabular}{ll ddd ddd ddd} + & & \mc{3}{c}{VDZ-BFD} & \mc{3}{c}{VTZ-BFD} & \mc{3}{c}{VQZ-BFD} \\ + \cline{3-5} \cline{6-8} \cline{9-11} +Method & $\mu$ & \tabc{MAE} & \tabc{MSE} & \tabc{RMSE} & \tabc{MAE} & \tabc{MSE} & \tabc{RMSE} & \tabc{MAE} & \tabc{MSE} & \tabc{RMSE} \\ +\hline +PBE & 0 & 5.02 & -3.70 & 6.04 & 4.57 & 1.00 & 5.32 & 5.31 & 0.79 & 6.27 \\ +BLYP & 0 & 9.53 & -9.21 & 7.91 & 5.58 & -4.44 & 5.80 & 5.86 & -4.47 & 6.43 \\ +PBE0 & 0 & 11.20 & -10.98 & 8.68 & 6.40 & -5.78 & 5.49 & 6.28 & -5.65 & 5.08 \\ +B3LYP & 0 & 11.27 & -10.98 & 9.59 & 7.27 & -5.77 & 6.63 & 6.75 & -5.53 & 6.09 \\ +\\ +CCSD(T) & \(\infty\) & 24.10 & -23.96 & 13.03 & 9.11 & -9.10 & 5.55 & 4.52 & -4.38 & 3.60 \\ +\\ +RS-DFT-CIPSI & 0 & 4.53 & -1.66 & 5.91 & 6.31 & 0.91 & 7.93 & 6.35 & 3.88 & 7.20 \\ + & 1/4 & 5.55 & -4.66 & 5.52 & 4.58 & 1.06 & 5.72 & 5.48 & 1.52 & 6.93 \\ + & 1/2 & 13.42 & -13.27 & 7.36 & 6.77 & -6.71 & 4.56 & 6.35 & -5.89 & 5.18 \\ + & 1 & 17.07 & -16.92 & 9.83 & 9.06 & -9.06 & 5.88 & & & \\ + & 2 & 19.20 & -19.05 & 10.91 & & & & & & \\ + & 5 & 22.93 & -22.79 & 13.24 & & & & & & \\ + & \(\infty\) & 23.63(4) & -23.49(4) & 12.81(4) & 8.43(39) & -8.43(39) & 4.87(7) & 4.51(78) & -4.18(78) & 4.19(20) \\ +\\ +DMC@ & 0 & 4.61(34) & -3.62(34) & 5.30(09) & 3.52(19) & -1.03(19) & 4.39(04) & 3.16(26) & -0.12(26) & 4.12(03) \\ + RS-DFT-CIPSI & 1/4 & 4.04(37) & -3.13(37) & 4.88(10) & 3.39(77) & -0.59(77) & 4.44(34) & 2.90(25) & 0.25(25) & 3.745(5) \\ + & 1/2 & 3.74(35) & -3.53(35) & 4.03(23) & 2.46(18) & -1.72(18) & 3.02(06) & 2.06(35) & -0.44(35) & 2.74(13) \\ + & 1 & 5.42(29) & -5.14(29) & 4.55(03) & 4.38(94) & -4.24(94) & 5.11(31) & & & \\ + & 2 & 5.98(83) & -5.91(83) & 4.79(71) & & & & & & \\ + & 5 & 6.18(84) & -6.13(84) & 4.87(55) & & & & & & \\ + & \(\infty\) & 7.38(1.08) & -7.38(1.08) & 5.67(68) & & & & & & \\ + & Opt. & 5.85(1.75) & -5.63(1.75) & 4.79(1.11) & & & & & & \\ + \end{tabular} + \end{ruledtabular} +\end{table*} +\end{squeezetable} +%%% %%% %%% %%% + +The atomization energies of the 55 molecules of the Gaussian-1 +theory\cite{Pople_1989,Curtiss_1990} were chosen as a benchmark set to test the +performance of the RS-DFT-CIPSI trial wave functions in the context of +energy differences. +Calculations were made in the double-, triple- +and quadruple-$\zeta$ basis sets with different values of $\mu$, and using +NOs from a preliminary CIPSI calculation as a starting point (see Fig.~\ref{fig:algo}). +\footnote{At $\mu=0$, the number of determinants is not equal to one because +we have used the natural orbitals of a preliminary CIPSI calculation, and +not the srPBE orbitals. +So the Kohn-Sham determinant is expressed as a linear combination of +determinants built with NOs. It is possible to add +an extra step to the algorithm to compute the NOs from the +RS-DFT-CIPSI wave function, and re-do the RS-DFT-CIPSI calculation with +these orbitals to get an even more compact expansion. In that case, we would +have converged to the KS orbitals with $\mu=0$, and the +solution would have been the PBE single determinant.} +For comparison, we have computed the energies of all the atoms and +molecules at the KS-DFT level with various semi-local and hybrid density functionals [PBE, \cite{PerBurErn-PRL-96} BLYP, \cite{Becke_1988,Lee_1988} PBE0, \cite{Perdew_1996} and B3LYP \cite{Becke_1993}], and at +the CCSD(T) level. \cite{Cizek_1969,Purvis_1982,Scuseria_1988,Scuseria_1989} Table~\ref{tab:mad} gives the corresponding mean +absolute errors (MAEs), mean signed errors (MSEs), and root mean square errors (RMSEs) +with respect to the NIST reference values as explained in Sec.~\ref{sec:comp-details}. +For FCI (RS-DFT-CIPSI, $\mu=\infty$) we have +provided the extrapolated values (\ie, when $\EPT \to 0$), and, although one cannot provide theoretically sound error bars, they +correspond here to the difference between the extrapolated energies computed with a +two-point and a three-point linear extrapolation. \cite{Loos_2018a,Loos_2019,Loos_2020b,Loos_2020c} + +In this benchmark, the great majority of the systems are weakly correlated and are then well +described by a single determinant. Therefore, the atomization energies +calculated at the KS-DFT level are relatively accurate, even when +the basis set is small. The introduction of exact exchange (B3LYP and +PBE) makes the results more sensitive to the basis set, and reduce the +accuracy. Note that, due to the approximate nature of the xc functionals, +the statistical quantities associated with KS-DFT atomization energies do not converge towards zero and remain altered even in the CBS limit. +Thanks to the single-reference character of these systems, +the CCSD(T) energy is an excellent estimate of the FCI energy, as +shown by the very good agreement of the MAE, MSE and RMSE of CCSD(T) +and FCI energies for each basis set. +The imbalance in the description of molecules compared +to atoms is exhibited by a very negative value of the MSE for +CCSD(T)/VDZ-BFD ($-23.96$ kcal/mol) and FCI/VDZ-BFD ($-23.49\pm0.04$ kcal/mol), which is reduced by a factor of two +when going to the triple-$\zeta$ basis, and again by a factor of two when +going to the quadruple-$\zeta$ basis. + +This significant imbalance at the VDZ-BFD level affects the nodal +surfaces, because although the FN-DMC energies obtained with near-FCI +trial wave functions are much lower than the FN-DMC +energies at $\mu = 0$, the MAE obtained with FCI ($7.38\pm1.08$ kcal/mol) is +larger than the MAE at $\mu = 0$ ($4.61\pm0.34$ kcal/mol). +Using the FCI trial wave function the MSE is equal to the +negative MAE which confirms that the atomization energies are systematically +underestimated. This corroborates that some of the basis set +incompleteness error is transferred in the fixed-node error. + +Within the double-$\zeta$ basis set, the calculations could be performed for the +whole range of values of $\mu$, and the optimal value of $\mu$ for the +trial wave function was estimated for each system by searching for the +minimum of the spline interpolation curve of the FN-DMC energy as a +function of $\mu$. +This corresponds to the line labelled as ``Opt.'' in Table~\ref{tab:mad}. +The optimal $\mu$ value for each system is reported in the \SI. +Using the optimal value of $\mu$ clearly improves the +MAEs, MSEs, and RMSEs as compared to the FCI wave function. This +result is in line with the common knowledge that re-optimizing +the determinantal component of the trial wave function in the presence +of electron correlation reduces the errors due to the basis set incompleteness. +These calculations were done only for the smallest basis set +because of the expensive computational cost of the QMC calculations +when the trial wave function contains more than a few million +determinants. \cite{Scemama_2016} +At the RS-DFT-CIPSI/VTZ-BFD level, one can see that +the MAEs are larger for $\mu=1$~bohr$^{-1}$ ($9.06$ kcal/mol) than for +FCI ($8.43\pm0.39$ kcal/mol). +The same comment applies to $\mu=0.5$~bohr$^{-1}$ with the quadruple-$\zeta$ basis. + +%%% FIG 5 %%% +\begin{figure*} + \centering + \includegraphics[width=\textwidth]{g2-dmc.pdf} + \caption{Errors in the FN-DMC atomization energies (in kcal/mol) for various + trial wave functions $\Psi^{\mu}$ and basis sets. Each dot corresponds to an atomization + energy. + The boxes contain the data between first and third quartiles, and + the line in the box represents the median. The outliers are shown + with a cross. The raw data can be found in the {\SI}.} + \label{fig:g2-dmc} +\end{figure*} +%%% %%% %%% %%% + +Searching for the optimal value of $\mu$ may be too costly and time consuming, so we have +computed the MAEs, MSEs and RMSEs for fixed values of $\mu$. +As illustrated in Fig.~\ref{fig:g2-dmc} and Table \ref{tab:mad}, +the best choice for a fixed value of $\mu$ is +$0.5$ bohr$^{-1}$ for all three basis sets. It is the value for which +the MAE [$3.74(35)$, $2.46(18)$, and $2.06(35)$ kcal/mol] and RMSE [$4.03(23)$, +$3.02(06)$, and $2.74(13)$ kcal/mol] are minimal. Note that these values +are even lower than those obtained with the optimal value of +$\mu$. Although the FN-DMC energies are higher, the numbers show that +they are more consistent from one system to another, giving improved +cancellations of errors. +This is yet another key result of the present study, and it can be explained by the lack of size-consistency when one considers different $\mu$ values for the molecule and the isolated atoms. +This observation was also mentioned in the context of optimally-tune range-separated hybrids. \cite{Stein_2009,Karolewski_2013,Kronik_2012} + +%%% FIG 6 %%% +\begin{figure*} + \centering + \includegraphics[width=\textwidth]{g2-ndet.pdf} + \caption{Number of determinants for various trial wave + functions $\Psi^{\mu}$ and basis sets. Each dot corresponds to an atomization energy. + The boxes contain the data between first and third quartiles, and + the line in the box represents the median. The outliers are shown + with a cross. The raw data can be found in the {\SI}.} + \label{fig:g2-ndet} +\end{figure*} +%%% %%% %%% %%% + +The number of determinants in the trial wave functions are shown in +Fig.~\ref{fig:g2-ndet}. As expected, the number of determinants +is smaller when $\mu$ is small and larger when $\mu$ is large. +It is important to note that the median of the number of +determinants when $\mu=0.5$~bohr$^{-1}$ is below $100\,000$ determinants +with the VQZ-BFD basis, making these calculations feasible +with such a large basis set. At the double-$\zeta$ level, compared to the +FCI trial wave functions, the median of the number of determinants is +reduced by more than two orders of magnitude. +Moreover, going to $\mu=0.25$~bohr$^{-1}$ gives a median close to 100 +determinants at the VDZ-BFD level, and close to $1\,000$ determinants +at the quadruple-$\zeta$ level for only a slight increase of the +MAE. Hence, RS-DFT-CIPSI trial wave functions with small values of +$\mu$ could be very useful for large systems to go beyond the +single-determinant approximation at a very low computational cost +while ensuring size-consistency. +For the largest systems, as shown in Fig.~\ref{fig:g2-ndet}, +there are many systems for which we could not reach the threshold +$\EPT<1$~m\hartree{} as the number of determinants exceeded +10~million before this threshold was reached. +For these cases, there is then a +small size-consistency error originating from the imbalanced +truncation of the wave functions, which is not present in the +extrapolated FCI energies (see Appendix \ref{app:size}). + +%%%%%%%%%%%%%%%%%%%% +\section{Conclusion} +\label{sec:conclusion} +%%%%%%%%%%%%%%%%%%%% + +In the present work, we have shown that introducing short-range correlation via +a range-separated Hamiltonian in a FCI expansion yields improved +nodal surfaces, especially with small basis sets. The effect of short-range DFT +on the determinant expansion is similar to the effect of re-optimizing +the CI coefficients in the presence of a Jastrow factor, but without +the burden of performing a stochastic optimization. + +In addition to the intermediate conclusions drawn in Sec.~\ref{sec:int_ccl}, +we have shown that varying the range-separation parameter $\mu$ and approaching +RS-DFT-FCI with CIPSI provides a way to adapt the number of +determinants in the trial wave function, leading to +size-consistent FN-DMC energies. +We propose two methods. The first one is for the computation of +accurate total energies by a one-parameter optimization of the FN-DMC +energy via the variation of the parameter $\mu$. +The second method is for the computation of energy differences, where +the target is not the lowest possible FN-DMC energies but the best +possible cancellation of errors. Using a fixed value of $\mu$ +increases the (size-)consistency of the trial wave functions, and we have found +that $\mu=0.5$~bohr$^{-1}$ is the value where the cancellation of +errors is the most effective. +Moreover, such a small value of $\mu$ gives extremely +compact wave functions, making this recipe a good candidate for +the accurate description of the whole potential energy surfaces of +large systems. If the number of determinants is still too large, the +value of $\mu$ can be further reduced to $0.25$~bohr$^{-1}$ to get +extremely compact wave functions at the price of less efficient +cancellations of errors. + +%%%%%%%%%%%%%%%%%%%%%%%% +\begin{acknowledgments} +A.B was supported by the U.S.~Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division, as part of the Computational Materials Sciences Program and Center for Predictive Simulation of Functional Materials. +This work was performed using HPC resources from GENCI-TGCC (Grand +Challenge 2019-gch0418) and from CALMIP (Toulouse) under allocation +2020-18005. +Funding from \textit{``Projet International de Coop\'eration Scientifique''} (PICS08310) and from the \textit{``Centre National de la Recherche Scientifique''} is acknowledged. +This study has been (partially) supported through the EUR grant NanoX No.~ANR-17-EURE-0009 in the framework of the \textit{``Programme des Investissements d'Avenir''}. +\end{acknowledgments} +%%%%%%%%%%%%%%%%%%%%%%%% + +%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\section*{Data availability} +%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +The data that support the findings of this study are openly available in Zenodo at \url{http://doi.org/10.5281/zenodo.3996568}. + +%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\appendix +%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +%============================ +\section{Size consistency} +\label{app:size} +%============================ + +An extremely important feature required to get accurate +atomization energies is size-consistency (or strict separability), +since the numbers of correlated electron pairs in the molecule and its isolated atoms +are different. + +KS-DFT energies are size-consistent, and because xc functionals are +directly constructed in complete basis, their convergence with respect +to the size of the basis set is relatively fast. \cite{FraMusLupTou-JCP-15,Giner_2018,Loos_2019d,Giner_2020} +Hence, DFT methods are very well adapted to +the calculation of atomization energies, especially with small basis +sets. \cite{Giner_2018,Loos_2019d,Giner_2020} +However, in the CBS, KS-DFT atomization energies do not match the exact values due to the approximate nature of the xc functionals. + +Likewise, FCI is also size-consistent, but the convergence of +the FCI energies towards the CBS limit is much slower because of the +description of short-range electron correlation using atom-centered +functions. \cite{Kutzelnigg_1985,Kutzelnigg_1991,Hattig_2012} +Eventually though, the exact atomization energies will be reached. + +In the context of SCI calculations, when the variational energy is +extrapolated to the FCI energy \cite{Holmes_2017} there is no +size-consistency error. But when the truncated SCI wave function is used +as a reference for post-HF methods such as SCI+PT2 +or for QMC calculations, there is a residual size-consistency error +originating from the truncation of the wave function. + +QMC energies can be made size-consistent by extrapolating the +FN-DMC energy to estimate the energy obtained with the FCI as a trial +wave function.\cite{Scemama_2018,Scemama_2018b} Alternatively, the +size-consistency error can be reduced by choosing the number of +selected determinants such that the sum of the PT2 corrections on the +fragments is equal to the PT2 correction of the molecule, enforcing that +the variational potential energy surface (PES) is +parallel to the perturbatively corrected PES, which is a relatively +accurate estimate of the FCI PES.\cite{Giner_2015} + +Another source of size-consistency error in QMC calculations originates +from the Jastrow factor. Usually, the Jastrow factor contains +one-electron, two-electron and one-nucleus-two-electron terms. +The problematic part is the two-electron term, whose simplest form can +be expressed as in Eq.~\eqref{eq:jast-ee}. +The parameter +$a$ is determined by the electron-electron cusp condition, \cite{Kato_1957,Pack_1966} and $b$ is obtained by energy +or variance minimization.\cite{Coldwell_1977,Umrigar_2005} +One can easily see that this parameterization of the two-body +interaction is not size-consistent: the dissociation of a +diatomic molecule \ce{AB} with a parameter $b_{\ce{AB}}$ +will lead to two different two-body Jastrow factors, each +with its own optimal value $b_{\ce{A}}$ and $b_{\ce{B}}$. To remove the +size-consistency error on a PES using this ans\"atz for $J_\text{ee}$, +one needs to impose that the parameters of $J_\text{ee}$ are fixed, \ie, +$b_{\ce{A}} = b_{\ce{B}} = b_{\ce{AB}}$. + +When pseudopotentials are used in a QMC calculation, it is of common +practice to localize the non-local part of the pseudopotential on the +complete trial wave function $\Phi$. +If the wave function is not size-consistent, +so will be the locality approximation. Within the DLA,\cite{Zen_2019} the Jastrow factor is +removed from the wave function on which the pseudopotential is localized. +The great advantage of this approximation is that the FN-DMC energy +only depends on the parameters of the determinantal component. Using a +non-size-consistent Jastrow factor, or a non-optimal Jastrow factor will +not introduce an additional error in FN-DMC calculations, although it +will reduce the statistical errors by reducing the variance of the +local energy. Moreover, the integrals involved in the pseudopotential +are computed analytically and the computational cost of the +pseudopotential is dramatically reduced (for more details, see +Ref.~\onlinecite{Scemama_2015}). + +In this section, we make a numerical verification that the produced +wave functions are size-consistent for a given range-separation +parameter. +We have computed the FN-DMC energy of the dissociated fluorine dimer, where +the two atoms are separated by 50~\AA. We expect that the energy +of this system is equal to twice the energy of the fluorine atom. +The data in Table~\ref{tab:size-cons} shows that this is indeed the +case, so we can conclude that the proposed scheme provides +size-consistent FN-DMC energies for all values of $\mu$ (within +twice the statistical error bars). + +%%% TABLE III %%% +\begin{table} + \caption{FN-DMC energy (in \hartree{}) using the VDZ-BFD basis set and the srPBE functional + of the fluorine atom and the dissociated \ce{F2} molecule for various $\mu$ values. + The size-consistency error is also reported.} + \label{tab:size-cons} + \begin{ruledtabular} + \begin{tabular}{cccc} + $\mu$ & \ce{F} & Dissociated \ce{F2} & Size-consistency error \\ + \hline + 0.00 & $-24.188\,7(3)$ & $-48.377\,7(3)$ & $-0.000\,3(4)$ \\ + 0.25 & $-24.188\,7(3)$ & $-48.377\,2(4)$ & $+0.000\,2(5)$ \\ + 0.50 & $-24.188\,8(1)$ & $-48.376\,9(4)$ & $+0.000\,7(4)$ \\ + 1.00 & $-24.189\,7(1)$ & $-48.380\,2(4)$ & $-0.000\,8(4)$ \\ + 2.00 & $-24.194\,1(3)$ & $-48.388\,4(4)$ & $-0.000\,2(5)$ \\ + 5.00 & $-24.194\,7(4)$ & $-48.388\,5(7)$ & $+0.000\,9(8)$ \\ + $\infty$ & $-24.193\,5(2)$ & $-48.386\,9(4)$ & $+0.000\,1(5)$ \\ + \end{tabular} + \end{ruledtabular} +\end{table} +%%% %%% %%% %%% + +%============================ +\section{Spin invariance} +\label{app:spin} +%============================ + +Closed-shell molecules often dissociate into open-shell +fragments. To get reliable atomization energies, it is important to +have a theory which is of comparable quality for open- and +closed-shell systems. A good check is to make sure that all the components +of a spin multiplet are degenerate, as expected from exact solutions. + +FCI wave functions have this property and yield degenerate energies with +respect to the spin quantum number $m_s$. +However, multiplying the determinantal part of the trial wave function by a +Jastrow factor introduces spin contamination if the Jastrow parameters +for the same-spin electron pairs are different from those +for the opposite-spin pairs.\cite{Tenno_2004} +Again, when pseudopotentials are employed, this tiny error is transferred +to the FN-DMC energy unless the DLA is enforced. + +The context is rather different within KS-DFT. +Indeed, mainstream density functionals have distinct functional forms to take +into account correlation effects of same-spin and opposite-spin electron pairs. +Therefore, KS determinants corresponding to different values of $m_s$ lead to different total energies. +Consequently, in the context of RS-DFT, the determinant expansion is impacted by this spurious effect, as opposed to FCI. + +In this Appendix, we investigate the impact of the spin contamination on the FN-DMC energy +originating from the short-range density functional. We have +computed the energies of the carbon atom in its triplet state +with the VDZ-BFD basis set and the srPBE functional. +The calculations are performed for $m_s=1$ (3 spin-up +and 1 spin-down electrons) and for $m_s=0$ (2 spin-up and 2 +spin-down electrons). + +The results are reported in Table~\ref{tab:spin}. +Although the energy obtained with $m_s=0$ is higher than the one obtained with $m_s=1$, the +bias is relatively small, \ie, more than one order of magnitude smaller +than the energy gained by reducing the fixed-node error going from the single +determinant to the FCI trial wave function. The largest spin-invariance error, close to +$2$ m\hartree{}, is obtained for $\mu=0$, but this bias decreases quickly +below $1$ m\hartree{} when $\mu$ increases. As expected, with $\mu=\infty$ +we observe a perfect spin-invariance of the energy (within the error bars), and the bias is not +noticeable for $\mu=5$~bohr$^{-1}$. + +Hence, at the FN-DMC level, the error due to +the spin invariance with RS-DFT-CIPSI trial wave functions is below the +chemical accuracy threshold, and is not expected to be problematic for the +comparison of atomization energies. + +%%% TABLE IV %%% +\begin{table} + \caption{FN-DMC energy (in \hartree{}) for various $\mu$ values of the triplet carbon atom with + different values of $m_s$ computed with the VDZ-BFD basis set and the srPBE functional. + The spin-invariance error is also reported.} + \label{tab:spin} + \begin{ruledtabular} + \begin{tabular}{cccc} + $\mu$ & $m_s=1$ & $m_s=0$ & Spin-invariance error \\ + \hline + 0.00 & $-5.416\,8(1)$ & $-5.414\,9(1)$ & $+0.001\,9(2)$ \\ + 0.25 & $-5.417\,2(1)$ & $-5.416\,5(1)$ & $+0.000\,7(1)$ \\ + 0.50 & $-5.422\,3(1)$ & $-5.421\,4(1)$ & $+0.000\,9(2)$ \\ + 1.00 & $-5.429\,7(1)$ & $-5.429\,2(1)$ & $+0.000\,5(2)$ \\ + 2.00 & $-5.432\,1(1)$ & $-5.431\,4(1)$ & $+0.000\,7(2)$ \\ + 5.00 & $-5.431\,7(1)$ & $-5.431\,4(1)$ & $+0.000\,3(2)$ \\ + $\infty$ & $-5.431\,6(1)$ & $-5.431\,3(1)$ & $+0.000\,3(2)$ \\ + \end{tabular} + \end{ruledtabular} +\end{table} +%%% %%% %%% %%% + +%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\bibliography{rsdft-cipsi-qmc} +%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\end{document} +