From d6a67984b8a4149ce848526caf2211667f46fcdd Mon Sep 17 00:00:00 2001 From: Emmanuel Giner Date: Mon, 10 Aug 2020 15:42:24 +0200 Subject: [PATCH] minor modifs --- Manuscript/rsdft-cipsi-qmc.bib | 23 +++++++++++ Manuscript/rsdft-cipsi-qmc.tex | 71 +++++++++++++++++----------------- 2 files changed, 59 insertions(+), 35 deletions(-) diff --git a/Manuscript/rsdft-cipsi-qmc.bib b/Manuscript/rsdft-cipsi-qmc.bib index 2dc9707..21ddf73 100644 --- a/Manuscript/rsdft-cipsi-qmc.bib +++ b/Manuscript/rsdft-cipsi-qmc.bib @@ -1057,6 +1057,18 @@ Year = {1996}, Bdsk-Url-1 = {https://doi.org/10.1016/S1380-7323(96)80091-4}} +@incollection{Sav-INC-96a, + author = {A. Savin}, + title = {Beyond the Kohn-Sham Determinant}, + booktitle = {Recent Advances in Density Functional Theory}, + publisher = {World Scientific}, + address = {}, + editor = {D. P. Chong}, + pages = {129-148}, + year = {1996} + } + + @article{Toulouse_2004, Author = {Toulouse, Julien and Colonna, Fran{\c c}ois and Savin, Andreas}, Doi = {10.1103/PhysRevA.70.062505}, @@ -1128,3 +1140,14 @@ Version = {2.1.2}, Year = 2020, Bdsk-Url-1 = {https://doi.org/10.5281/zenodo.3677565}} + +@article{TouSavFla-IJQC-04, + author = {J. Toulouse and A. Savin and H.-J. Flad}, + title = {Short-range exchange-correlation energy of a uniform electron gas with modified electron-electron interaction}, + journal = {Int. J. Quantum Chem.}, + volume = {100}, + pages = {1047}, + year = {2004}, + note = {} +} + diff --git a/Manuscript/rsdft-cipsi-qmc.tex b/Manuscript/rsdft-cipsi-qmc.tex index 49d4703..7d13db3 100644 --- a/Manuscript/rsdft-cipsi-qmc.tex +++ b/Manuscript/rsdft-cipsi-qmc.tex @@ -19,6 +19,7 @@ \definecolor{darkgreen}{HTML}{009900} \usepackage[normalem]{ulem} \newcommand{\toto}[1]{\textcolor{blue}{#1}} +\newcommand{\manu}[1]{\textcolor{green}{#1}} \newcommand{\trashAS}[1]{\textcolor{blue}{\sout{#1}}} \newcommand{\titou}[1]{\textcolor{red}{#1}} \newcommand{\trashPFL}[1]{\textcolor{red}{\sout{#1}}} @@ -111,7 +112,7 @@ bound to the exact energy, and the latter is recovered only when the nodes of the trial wave function coincide with the nodes of the exact wave function. The polynomial scaling with the number of electrons and with the size -of the trial wave function makes the FN-DMC method particularly attractive. +of {\manu{in what sense is it polynomial?}the trial wave function makes the FN-DMC method particularly attractive. In addition, the total energies obtained are usually far below those obtained with the FCI method in computationally tractable basis sets because the constraints imposed by the FN approximation @@ -154,14 +155,14 @@ FN-DMC.\cite{Petruzielo_2012} Another approach consists in considering the FN-DMC method as a \emph{post-FCI method}. The trial wave function is obtained by approaching the FCI with a selected configuration interaction (SCI) -method such as CIPSI for instance.\cite{Giner_2013,Caffarel_2016_2} +method such as CIPSI for instance.\cite{Giner_2013,Giner_2015,Caffarel_2016_2} \titou{When the basis set is increased, the trial wave function gets closer to the exact wave function, so the nodal surface can be systematically improved.\cite{Caffarel_2016} WRONG} -This technique has the advantage that using FCI nodes in a given basis -set is well defined, so the calculations are reproducible in a +This technique has the advantage \manu{of using the} FCI nodes in a given basis +set \manu{, which is perfectly well defined and therefore makes the calculations} reproducible in a black-box way without needing any expertise in QMC. -But this technique cannot be applied to large systems because of the +\manu{Nevertheless,} this technique cannot be applied to large systems because of the exponential scaling of the size of the trial wave function. Extrapolation techniques have been used to estimate the FN-DMC energies obtained with FCI wave functions,\cite{Scemama_2018} and other authors @@ -217,15 +218,15 @@ CBS limit, a fixed-node error necessarily remains because the single-determinant ansätz does not have enough flexibility to describe the nodal surface of the exact correlated wave function of a generic $N$-electron system. -If one wants to have to exact CBS limit, a multi-determinant parameterization +If one wants to recover the exact CBS limit, a multi-determinant parameterization of the wave functions is required. %==================== \subsection{CIPSI} %==================== Beyond the single-determinant representation, the best -multi-determinant wave function one can obtain is the FCI. FCI is -a \emph{post-Hartree-Fock} method, and there exists several systematic +multi-determinant wave function one can obtain \manu{in a given basis set} is the FCI. +FCI is \manu{the ultimate goal of} \emph{post-Hartree-Fock} methods, and there exists several systematic improvements between the Hartree-Fock and FCI wave functions: increasing the maximum degree of excitation of CI methods (CISD, CISDT, CISDTQ, \emph{etc}), or increasing the complete active space @@ -262,8 +263,8 @@ accuracy so all the CIPSI selections were made such that $\abs{\EPT} < \label{sec:rsdft} %================================= -Following the seminal work of Savin,\cite{Savin_1996,Toulouse_2004} -the Coulomb operator entering the interelectronic repulsion is split into two pieces: +\manu{The range-separated DFT (RS-DFT)} was introduced in the seminal work of Savin,\cite{Sav-INC-96a,Toulouse_2004} +where the Coulomb operator entering the electron-electron repulsion is split into two pieces: \begin{equation} \frac{1}{r} = w_{\text{ee}}^{\text{sr}, \mu}(r) @@ -278,7 +279,7 @@ where are the singular short-range (sr) part and the non-singular long-range (lr) part, respectively, $\mu$ is the range-separation parameter which controls how rapidly the short-range part decays, $\erf(x)$ is the error function, and $\erfc(x) = 1 - \erf(x)$ is its complementary version. The main idea behind RS-DFT is to treat the short-range part of the -interaction within KS-DFT, and the long-range part within a WFT method like FCI in the present case. +interaction \manu{using a density functional}, and the long-range part within a WFT method like FCI in the present case. The parameter $\mu$ controls the range of the separation, and allows to go continuously from the KS Hamiltonian ($\mu=0$) to the FCI Hamiltonian ($\mu = \infty$). @@ -295,8 +296,8 @@ $\mathcal{F}^{\text{lr},\mu}$ is a long-range universal density functional and $\bar{E}_{\text{Hxc}}^{\text{sr,}\mu}$ is the complementary short-range Hartree-exchange-correlation (Hxc) density functional. \cite{Savin_1996,Toulouse_2004} -One obtains the following expression for the ground-state -electronic energy +\manu{The exact ground state energy can be therefore obtained as a minimization +over a multi-determinant wave function as follows}: \begin{equation} \label{min_rsdft} E_0= \min_{\Psi} \qty{ \mel{\Psi}{\hat{T}+\hat{W}_\text{{ee}}^{\text{lr},\mu}+\hat{V}_{\text{ne}}}{\Psi} @@ -404,7 +405,7 @@ CCSD(T) and KS-DFT energies have been computed with All the CIPSI calculations have been performed with \emph{Quantum Package}.\cite{Garniron_2019,qp2_2020} We used the short-range version -of \titou{the local-density approximation (LDA)} and Perdew-Burke-Ernzerhof (PBE) \cite{PerBurErn-PRL-96} exchange +of the local-density approximation (LDA)\cite{Sav-INC-96a,TouSavFla-IJQC-04} and Perdew-Burke-Ernzerhof (PBE) \cite{PerBurErn-PRL-96} exchange and correlation functionals defined in Ref.~\onlinecite{GolWerStoLeiGorSav-CP-06} (see also Refs.~\onlinecite{TouColSav-JCP-05,GolWerSto-PCCP-05}). @@ -478,13 +479,13 @@ For this purpose, we consider a weakly correlated molecular system, namely the w molecule \titou{near its equilibrium geometry.} \cite{Caffarel_2016} We then generate trial wave functions $\Psi^\mu$ for multiple values of $\mu$, and compute the associated fixed-node energy keeping all the -parameters having an impact on the nodal surface fixed (\titou{such as ??}). +parameters having an impact on the nodal surface fixed \manu{such as CI coefficients and molecular orbitals}. %====================================================== \subsection{Fixed-node energy of $\Psi^\mu$} \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, +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 ($\mu=0$): @@ -499,18 +500,20 @@ for an optimal value of $\mu$ (which is obviously basis set and functional depen 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 FN-DMC/VDZ-BFD energy 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$^{-1}$. +with an optimal value of $\mu=1.75$~bohr$^{-1}$. +\manu{This lowering in FN-DMC energy is to be compared with the $3.2 \pm 0.7$~m\hartree{} of gain in FN-DMC energy between the KS wave function ($\mu=0$) and the FCI wave function ($\mu=\infty$)}. When the basis set is increased, 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 $\mu$. Eventually, the nodes -of the wave functions $\Psi^\mu$ obtained with the srLDA +value of $\mu$ is slightly shifted towards large $\mu$. +Last but not least, the nodes of the wave functions $\Psi^\mu$ obtained with the srLDA exchange-correlation functional give very similar FN-DMC energies with respect to those obtained with the srPBE functional, even if the RS-DFT energies obtained with these two functionals differ by several -tens of m\hartree{}. +tens of m\hartree{}. -\titou{The key fact here is that, at $\mu=1.75$~bohr$^{-1}$, $\Psi^{\mu}$ has \textit{only} $54\,540$ determinants at the srPBE/VDZ-BFD level, while the FCI wave function contains $200\,521$ determinants (see Table \ref{tab:h2o-dmc}). Even at the 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 here is that RS-DFT trial wave functions yield a lower fixed-node energy with more compact multideterminant expansion as compared to FCI.} +\manu{An other important aspect here regards the compactness of the trial wave functions $\Psi^\mu$:} +\titou{at $\mu=1.75$~bohr$^{-1}$, $\Psi^{\mu}$ has \textit{only} $54\,540$ determinants at the srPBE/VDZ-BFD level, while the FCI wave function contains $200\,521$ determinants (see Table \ref{tab:h2o-dmc}). Even at the 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 numerical study is that RS-DFT trial wave functions can yield a lower fixed-node energy with more compact multideterminant expansion as compared to FCI.} %====================================================== \subsection{Link between RS-DFT and Jastrow factors } @@ -519,7 +522,7 @@ The take-home message here is that RS-DFT trial wave functions yield a lower fix 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 function. Such behaviour can be directty compared to the common practice of -re-optimizing the multideterminant 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_2006c,Umrigar_2007,Toulouse_2007,Toulouse_2008} +re-optimizing the multideterminant 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_2006c,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. \titou{T2: maybe we should mention that we only reoptimize the CI coefficients as it is of common practice to re-optimize more than this.} @@ -644,7 +647,7 @@ report in Table~\ref{table_on_top} the integrated on-top pair density \begin{equation} \expval{ n_2(\br,\br) } = \int d\br \,\,n_2(\br,\br) \end{equation} -where $n_2(\br_1,\br_2$ is the two-body density [normalized to $N(N-1)$ where $N$ is the number of electrons] +where $n_2(\br_1,\br_2)$ is the two-body density [normalized to $N(N-1)$ where $N$ is the number of electrons] obtained for both $\Psi^\mu$ and $\Psi^J$. Then, in order to have a pictorial representation of both the on-top pair density and the density, we report in Fig.~\ref{fig:n1} and Fig.~\ref{fig:n2} @@ -653,7 +656,7 @@ $n_2(\br,\br)$ along one \ce{O-H} axis of the water molecule. From these data, one can clearly notice several trends. First, from Table~\ref{table_on_top}, we can observe that the overall -on-top pair density decreases when $\mu$ increases. This is expected +on-top pair density decreases when $\mu$ increases, which is expected as the two-electron interaction increases in $H^\mu[n]$. Second, the relative variations of the on-top pair density with $\mu$ are much more important than that of the one-body density, the latter @@ -665,7 +668,7 @@ pair density obtained from $\Psi^J$ is superimposed with $\Psi^{\mu=0.5}$, and at a large distance the on-top pair density is the closest to $\mu=\infty$. The integrated on-top pair density obtained with $\Psi^J$ lies between the values obtained with -$\mu=0.5$ and $\mu=1$~bohr$^{-1}$, constently with the FN-DMC energies +$\mu=0.5$ and $\mu=1$~bohr$^{-1}$, consistently with the FN-DMC energies and the overlap curve. These data suggest that the wave functions $\Psi^{0.5 \le \mu \le 1}$ and $\Psi^J$ are close, @@ -788,8 +791,7 @@ When pseudopotentials are used in a QMC calculation, it is common practice to localize the non-local part of the pseudopotential on the complete wave function (determinantal component and Jastrow). If the wave function is not size-consistent, -so will be the locality approximation. Within, the determinant -localization approximation,\cite{Zen_2019} the Jastrow factor is +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 @@ -826,7 +828,7 @@ 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 energy of the dissociated fluorine dimer, where +We have computed the \manu{FN-DMC} energy of the dissociated fluorine dimer, where the two atoms are at a distance of 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 it is indeed the @@ -843,15 +845,14 @@ 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-shell and closed-shell systems. A good test is to check that all the components -of a spin multiplet are degenerate. -FCI wave functions have this property and give degenrate energies with +of a spin multiplet are degenerate\manu{, as expected from exact solutions}. +FCI wave functions have this property and give degenerate energies with respect to the spin quantum number $m_s$, but the multiplication by a Jastrow factor introduces spin contamination if the parameters for the same-spin electron pairs are different from those for the opposite-spin pairs.\cite{Tenno_2004} Again, when pseudopotentials are used this tiny error is transferred -in the FN-DMC energy unless the determinant localization approximation -is used. +in the FN-DMC energy unless the DLA is used. Within DFT, the common density functionals make a difference for same-spin and opposite-spin interactions. As DFT is a @@ -895,7 +896,7 @@ Although using $m_s=0$ the energy is higher than with $m_s=1$, the bias is relatively small, 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 highest bias, close to -2~m\hartree, is obtained for $\mu=0$, but the bias decreases quickly +2~m\hartree, is obtained for $\mu=0$, but the bias decreases rapidly below 1~m\hartree{} when $\mu$ increases. As expected, with $\mu=\infty$ there is no bias (within the error bars), and the bias is not noticeable with $\mu=5$~bohr$^{-1}$. @@ -1084,7 +1085,7 @@ solution would have been the PBE single determinant. \section{Conclusion} %%%%%%%%%%%%%%%%%%%% -We have seen that introducing short-range correation via +\manu{In the present work} we have shown that introducing short-range correation via a range-separated Hamiltonian in a full CI expansion yields improved nodal surfaces, especially with small basis sets. The effect of sr-DFT on the determinant expansion is similar to the effect of re-optimizing