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@ -1057,6 +1057,18 @@
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Year = {1996},
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Bdsk-Url-1 = {https://doi.org/10.1016/S1380-7323(96)80091-4}}
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@incollection{Sav-INC-96a,
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author = {A. Savin},
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title = {Beyond the Kohn-Sham Determinant},
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booktitle = {Recent Advances in Density Functional Theory},
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publisher = {World Scientific},
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address = {},
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editor = {D. P. Chong},
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pages = {129-148},
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year = {1996}
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}
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@article{Toulouse_2004,
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Author = {Toulouse, Julien and Colonna, Fran{\c c}ois and Savin, Andreas},
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Doi = {10.1103/PhysRevA.70.062505},
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@ -1128,3 +1140,14 @@
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Version = {2.1.2},
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Year = 2020,
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Bdsk-Url-1 = {https://doi.org/10.5281/zenodo.3677565}}
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@article{TouSavFla-IJQC-04,
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author = {J. Toulouse and A. Savin and H.-J. Flad},
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title = {Short-range exchange-correlation energy of a uniform electron gas with modified electron-electron interaction},
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journal = {Int. J. Quantum Chem.},
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volume = {100},
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pages = {1047},
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year = {2004},
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note = {}
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}
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@ -19,6 +19,7 @@
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\definecolor{darkgreen}{HTML}{009900}
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\usepackage[normalem]{ulem}
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\newcommand{\toto}[1]{\textcolor{blue}{#1}}
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\newcommand{\manu}[1]{\textcolor{green}{#1}}
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\newcommand{\trashAS}[1]{\textcolor{blue}{\sout{#1}}}
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\newcommand{\titou}[1]{\textcolor{red}{#1}}
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\newcommand{\trashPFL}[1]{\textcolor{red}{\sout{#1}}}
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@ -111,7 +112,7 @@ bound to the exact energy, and the latter is recovered only when the
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nodes of the trial wave function coincide with the nodes of the exact
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wave function.
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The polynomial scaling with the number of electrons and with the size
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of the trial wave function makes the FN-DMC method particularly attractive.
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of {\manu{in what sense is it polynomial?}the trial wave function makes the FN-DMC method particularly attractive.
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In addition, the total energies obtained are usually far below
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those obtained with the FCI method in computationally tractable basis
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sets because the constraints imposed by the FN approximation
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@ -154,14 +155,14 @@ FN-DMC.\cite{Petruzielo_2012}
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Another approach consists in considering the FN-DMC method as a
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\emph{post-FCI method}. The trial wave function is obtained by
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approaching the FCI with a selected configuration interaction (SCI)
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method such as CIPSI for instance.\cite{Giner_2013,Caffarel_2016_2}
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method such as CIPSI for instance.\cite{Giner_2013,Giner_2015,Caffarel_2016_2}
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\titou{When the basis set is increased, the trial wave function gets closer
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to the exact wave function, so the nodal surface can be systematically
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improved.\cite{Caffarel_2016} WRONG}
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This technique has the advantage that using FCI nodes in a given basis
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set is well defined, so the calculations are reproducible in a
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This technique has the advantage \manu{of using the} FCI nodes in a given basis
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set \manu{, which is perfectly well defined and therefore makes the calculations} reproducible in a
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black-box way without needing any expertise in QMC.
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But this technique cannot be applied to large systems because of the
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\manu{Nevertheless,} this technique cannot be applied to large systems because of the
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exponential scaling of the size of the trial wave function.
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Extrapolation techniques have been used to estimate the FN-DMC energies
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obtained with FCI wave functions,\cite{Scemama_2018} and other authors
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@ -217,15 +218,15 @@ CBS limit, a fixed-node error necessarily remains because the
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single-determinant ansätz does not have enough flexibility to describe the
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nodal surface of the exact correlated wave function of a generic $N$-electron
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system.
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If one wants to have to exact CBS limit, a multi-determinant parameterization
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If one wants to recover the exact CBS limit, a multi-determinant parameterization
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of the wave functions is required.
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%====================
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\subsection{CIPSI}
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%====================
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Beyond the single-determinant representation, the best
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multi-determinant wave function one can obtain is the FCI. FCI is
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a \emph{post-Hartree-Fock} method, and there exists several systematic
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multi-determinant wave function one can obtain \manu{in a given basis set} is the FCI.
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FCI is \manu{the ultimate goal of} \emph{post-Hartree-Fock} methods, and there exists several systematic
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improvements between the Hartree-Fock and FCI wave functions:
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increasing the maximum degree of excitation of CI methods (CISD, CISDT,
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CISDTQ, \emph{etc}), or increasing the complete active space
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@ -262,8 +263,8 @@ accuracy so all the CIPSI selections were made such that $\abs{\EPT} <
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\label{sec:rsdft}
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%=================================
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Following the seminal work of Savin,\cite{Savin_1996,Toulouse_2004}
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the Coulomb operator entering the interelectronic repulsion is split into two pieces:
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\manu{The range-separated DFT (RS-DFT)} was introduced in the seminal work of Savin,\cite{Sav-INC-96a,Toulouse_2004}
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where the Coulomb operator entering the electron-electron repulsion is split into two pieces:
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\begin{equation}
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\frac{1}{r}
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= w_{\text{ee}}^{\text{sr}, \mu}(r)
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@ -278,7 +279,7 @@ where
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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.
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The main idea behind RS-DFT is to treat the short-range part of the
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interaction within KS-DFT, and the long-range part within a WFT method like FCI in the present case.
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interaction \manu{using a density functional}, and the long-range part within a WFT method like FCI in the present case.
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The parameter $\mu$ controls the range of the separation, and allows
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to go continuously from the KS Hamiltonian ($\mu=0$) to
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the FCI Hamiltonian ($\mu = \infty$).
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@ -295,8 +296,8 @@ $\mathcal{F}^{\text{lr},\mu}$ is a long-range universal density
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functional and $\bar{E}_{\text{Hxc}}^{\text{sr,}\mu}$ is the
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complementary short-range Hartree-exchange-correlation (Hxc) density
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functional. \cite{Savin_1996,Toulouse_2004}
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One obtains the following expression for the ground-state
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electronic energy
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\manu{The exact ground state energy can be therefore obtained as a minimization
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over a multi-determinant wave function as follows}:
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\begin{equation}
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\label{min_rsdft} E_0= \min_{\Psi} \qty{
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\mel{\Psi}{\hat{T}+\hat{W}_\text{{ee}}^{\text{lr},\mu}+\hat{V}_{\text{ne}}}{\Psi}
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@ -404,7 +405,7 @@ CCSD(T) and KS-DFT energies have been computed with
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All the CIPSI calculations have been performed with \emph{Quantum
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Package}.\cite{Garniron_2019,qp2_2020} We used the short-range version
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of \titou{the local-density approximation (LDA)} and Perdew-Burke-Ernzerhof (PBE) \cite{PerBurErn-PRL-96} exchange
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of the local-density approximation (LDA)\cite{Sav-INC-96a,TouSavFla-IJQC-04} and Perdew-Burke-Ernzerhof (PBE) \cite{PerBurErn-PRL-96} exchange
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and correlation functionals defined in
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Ref.~\onlinecite{GolWerStoLeiGorSav-CP-06} (see also
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Refs.~\onlinecite{TouColSav-JCP-05,GolWerSto-PCCP-05}).
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@ -478,13 +479,13 @@ For this purpose, we consider a weakly correlated molecular system, namely the w
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molecule \titou{near its equilibrium geometry.} \cite{Caffarel_2016}
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We then generate trial wave functions $\Psi^\mu$ for multiple values of
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$\mu$, and compute the associated fixed-node energy keeping all the
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parameters having an impact on the nodal surface fixed (\titou{such as ??}).
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parameters having an impact on the nodal surface fixed \manu{such as CI coefficients and molecular orbitals}.
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%======================================================
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\subsection{Fixed-node energy of $\Psi^\mu$}
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\label{sec:fndmc_mu}
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%======================================================
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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,
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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,
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one can clearly observe that relying on FCI trial
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wave functions ($\mu = \infty$) give FN-DMC energies lower
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than the energies obtained with a single KS determinant ($\mu=0$):
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@ -499,18 +500,20 @@ for an optimal value of $\mu$ (which is obviously basis set and functional depen
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and then the FN-DMC error raises until it reaches the $\mu=\infty$ limit (\ie, the FCI wave function).
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For instance, with respect to the FN-DMC/VDZ-BFD energy at $\mu=\infty$,
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one can obtain a lowering of the FN-DMC energy of $2.6 \pm 0.7$~m\hartree{}
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with an optimal value of $\mu=1.75$~bohr$^{-1}$.
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with an optimal value of $\mu=1.75$~bohr$^{-1}$.
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\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 ($\mu=\infty$)}.
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When the basis set is increased, the gain in FN-DMC energy with
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respect to the FCI trial wave function is reduced, and the optimal
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value of $\mu$ is slightly shifted towards large $\mu$. Eventually, the nodes
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of the wave functions $\Psi^\mu$ obtained with the srLDA
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value of $\mu$ is slightly shifted towards large $\mu$.
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Last but not least, the nodes of the wave functions $\Psi^\mu$ obtained with the srLDA
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exchange-correlation functional give very similar FN-DMC energies with respect
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to those obtained with the srPBE functional, even if the
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RS-DFT energies obtained with these two functionals differ by several
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tens of m\hartree{}.
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tens of m\hartree{}.
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\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 $\mu = \infty$.
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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.}
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\manu{An other important aspect here regards the compactness of the trial wave functions $\Psi^\mu$:}
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\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 $\mu = \infty$.
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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.}
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%======================================================
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\subsection{Link between RS-DFT and Jastrow factors }
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@ -519,7 +522,7 @@ The take-home message here is that RS-DFT trial wave functions yield a lower fix
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The data presented in Sec.~\ref{sec:fndmc_mu} evidence that, in a finite basis, RS-DFT can provide
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trial wave functions with better nodes than FCI wave function.
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Such behaviour can be directty compared to the common practice of
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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}
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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}
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Hence, in the present paragraph, we would like to elaborate further on the link between RS-DFT
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and wave function optimization in the presence of a Jastrow factor.
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\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.}
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@ -644,7 +647,7 @@ report in Table~\ref{table_on_top} the integrated on-top pair density
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\begin{equation}
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\expval{ n_2(\br,\br) } = \int d\br \,\,n_2(\br,\br)
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\end{equation}
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where $n_2(\br_1,\br_2$ is the two-body density [normalized to $N(N-1)$ where $N$ is the number of electrons]
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where $n_2(\br_1,\br_2)$ is the two-body density [normalized to $N(N-1)$ where $N$ is the number of electrons]
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obtained for both $\Psi^\mu$ and $\Psi^J$.
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Then, in order to have a pictorial representation of both the on-top
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pair density and the density, we report in Fig.~\ref{fig:n1} and Fig.~\ref{fig:n2}
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@ -653,7 +656,7 @@ $n_2(\br,\br)$ along one \ce{O-H} axis of the water molecule.
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From these data, one can clearly notice several trends.
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First, from Table~\ref{table_on_top}, we can observe that the overall
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on-top pair density decreases when $\mu$ increases. This is expected
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on-top pair density decreases when $\mu$ increases, which is expected
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as the two-electron interaction increases in $H^\mu[n]$.
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Second, the relative variations of the on-top pair density with $\mu$
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are much more important than that of the one-body density, the latter
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@ -665,7 +668,7 @@ pair density obtained from $\Psi^J$ is superimposed with
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$\Psi^{\mu=0.5}$, and at a large distance the on-top pair density is
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the closest to $\mu=\infty$. The integrated on-top pair density
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obtained with $\Psi^J$ lies between the values obtained with
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$\mu=0.5$ and $\mu=1$~bohr$^{-1}$, constently with the FN-DMC energies
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$\mu=0.5$ and $\mu=1$~bohr$^{-1}$, consistently with the FN-DMC energies
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and the overlap curve.
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These data suggest that the wave functions $\Psi^{0.5 \le \mu \le 1}$ and $\Psi^J$ are close,
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@ -788,8 +791,7 @@ When pseudopotentials are used in a QMC calculation, it is common
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practice to localize the non-local part of the pseudopotential on the
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complete wave function (determinantal component and Jastrow).
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If the wave function is not size-consistent,
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so will be the locality approximation. Within, the determinant
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localization approximation,\cite{Zen_2019} the Jastrow factor is
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so will be the locality approximation. Within, the DLA,\cite{Zen_2019} the Jastrow factor is
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removed from the wave function on which the pseudopotential is localized.
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The great advantage of this approximation is that the FN-DMC energy
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only depends on the parameters of the determinantal component. Using a
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@ -826,7 +828,7 @@ Ref.~\onlinecite{Scemama_2015}).
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In this section, we make a numerical verification that the produced
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wave functions are size-consistent for a given range-separation
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parameter.
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We have computed the energy of the dissociated fluorine dimer, where
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We have computed the \manu{FN-DMC} energy of the dissociated fluorine dimer, where
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the two atoms are at a distance of 50~\AA. We expect that the energy
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of this system is equal to twice the energy of the fluorine atom.
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The data in table~\ref{tab:size-cons} shows that it is indeed the
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@ -843,15 +845,14 @@ Closed-shell molecules often dissociate into open-shell
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fragments. To get reliable atomization energies, it is important to
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have a theory which is of comparable quality for open-shell and
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closed-shell systems. A good test is to check that all the components
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of a spin multiplet are degenerate.
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FCI wave functions have this property and give degenrate energies with
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of a spin multiplet are degenerate\manu{, as expected from exact solutions}.
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FCI wave functions have this property and give degenerate energies with
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respect to the spin quantum number $m_s$, but the multiplication by a
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Jastrow factor introduces spin contamination if the parameters
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for the same-spin electron pairs are different from those
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for the opposite-spin pairs.\cite{Tenno_2004}
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Again, when pseudopotentials are used this tiny error is transferred
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in the FN-DMC energy unless the determinant localization approximation
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is used.
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in the FN-DMC energy unless the DLA is used.
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Within DFT, the common density functionals make a difference for
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same-spin and opposite-spin interactions. As DFT is a
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@ -895,7 +896,7 @@ Although using $m_s=0$ the energy is higher than with $m_s=1$, the
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bias is relatively small, more than one order of magnitude smaller
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than the energy gained by reducing the fixed-node error going from the single
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determinant to the FCI trial wave function. The highest bias, close to
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2~m\hartree, is obtained for $\mu=0$, but the bias decreases quickly
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2~m\hartree, is obtained for $\mu=0$, but the bias decreases rapidly
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below 1~m\hartree{} when $\mu$ increases. As expected, with $\mu=\infty$
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there is no bias (within the error bars), and the bias is not
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noticeable with $\mu=5$~bohr$^{-1}$.
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@ -1084,7 +1085,7 @@ solution would have been the PBE single determinant.
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\section{Conclusion}
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%%%%%%%%%%%%%%%%%%%%
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We have seen that introducing short-range correation via
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\manu{In the present work} we have shown that introducing short-range correation via
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a range-separated Hamiltonian in a full CI expansion yields improved
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nodal surfaces, especially with small basis sets. The effect of sr-DFT
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on the determinant expansion is similar to the effect of re-optimizing
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