diff --git a/Manuscript/rsdft-cipsi-qmc.bib b/Manuscript/rsdft-cipsi-qmc.bib index 387e6c8..e3766c4 100644 --- a/Manuscript/rsdft-cipsi-qmc.bib +++ b/Manuscript/rsdft-cipsi-qmc.bib @@ -1366,3 +1366,20 @@ Title = {Short-range exchange-correlation energy of a uniform electron gas with modified electron-electron interaction}, Volume = {100}, Year = {2004}} + + + +@article{Nightingale_2001, + author = {Nightingale, M. P. and Melik-Alaverdian, Vilen}, + title = {{Optimization of Ground- and Excited-State Wave Functions and van der Waals + Clusters}}, + journal = {Phys. Rev. Lett.}, + volume = {87}, + number = {4}, + pages = {043401}, + year = {2001}, + month = {Jul}, + issn = {1079-7114}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevLett.87.043401} +} diff --git a/Manuscript/rsdft-cipsi-qmc.tex b/Manuscript/rsdft-cipsi-qmc.tex index d05eeb0..3f2af71 100644 --- a/Manuscript/rsdft-cipsi-qmc.tex +++ b/Manuscript/rsdft-cipsi-qmc.tex @@ -13,6 +13,8 @@ ]{hyperref} \urlstyle{same} +\DeclareMathOperator*{\argmin}{arg\,min} + \newcommand{\ie}{\textit{i.e.}} \newcommand{\eg}{\textit{e.g.}} \newcommand{\alert}[1]{\textcolor{red}{#1}} @@ -36,6 +38,8 @@ \newcommand{\EPT}{E_{\text{PT2}}} \newcommand{\EDMC}{E_{\text{FN-DMC}}} \newcommand{\Ndet}{N_{\text{det}}} +\newcommand{\Nelec}{N_{\text{elec}}} +\newcommand{\Nat}{N_{\text{atoms}}} \newcommand{\hartree}{$E_h$} \newcommand{\LCT}{Laboratoire de Chimie Th\'eorique (UMR 7616), Sorbonne Universit\'e, CNRS, Paris, France} @@ -164,9 +168,9 @@ 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,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} +\toto{When the basis set is enlarged, the trial wave function gets closer to +the exact wave function, so we expect the nodal surface to be +improved.\cite{Caffarel_2016} } This technique has the advantage of using the FCI nodes in a given basis set, which is perfectly well defined and therefore makes the calculations reproducible in a black-box way without needing any expertise in QMC. @@ -339,14 +343,12 @@ energy is obtained as E_0= \mel{\Psi^{\mu}}{\hat{T}+\hat{W}_\text{{ee}}^{\text{lr},\mu}+\hat{V}_{\text{ne}}}{\Psi^{\mu}}+\bar{E}^{\text{sr},\mu}_{\text{Hxc}}[n_{\Psi^\mu}]. \end{equation} -Note that, for $\mu=0$, \titou{the long-range interaction vanishes}, \ie, -$w_{\text{ee}}^{\text{lr},\mu=0}(r) = 0$, and thus -RS-DFT reduces to standard KS-DFT and $\Psi^\mu$ -is the KS determinant. For $\mu = \infty$, the long-range +Note that for $\mu=0$ the long-range interaction vanishes, \ie, +$w_{\text{ee}}^{\text{lr},\mu=0}(r) = 0$, and thus RS-DFT reduces to standard +KS-DFT and $\Psi^\mu$ is the KS determinant. For $\mu = \infty$, the long-range interaction becomes the standard Coulomb interaction, \ie, -$w_{\text{ee}}^{\text{lr},\mu\to\infty}(r) = r^{-1}$, and -thus RS-DFT reduces to standard WFT and $\Psi^\mu$ is -the FCI wave function. +$w_{\text{ee}}^{\text{lr},\mu\to\infty}(r) = r^{-1}$, and thus RS-DFT reduces +to standard WFT and $\Psi^\mu$ is the FCI wave function. %%% FIG 1 %%% \begin{figure*} @@ -430,10 +432,9 @@ 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. -The FN-DMC simulations are performed with the stochastic reconfiguration -algorithm developed by Assaraf \textit{et al.}, \cite{Assaraf_2000} -with a time step of $5 \times 10^{-4}$ a.u. -\titou{All-electron move DMC?} +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. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Influence of the range-separation parameter on the fixed-node error} @@ -442,8 +443,7 @@ with a time step of $5 \times 10^{-4}$ a.u. %%% TABLE I %%% \begin{table} - \caption{Fixed-node energy $\EDMC$ (in \hartree{}) and number of determinants $\Ndet$ in \ce{H2O} for various trial wave functions $\Psi^{\mu}$. - \titou{srPBE?}.} + \caption{Fixed-node energy $\EDMC$ (in \hartree{}) and number of determinants $\Ndet$ in \ce{H2O} for various trial wave functions $\Psi^{\mu}$ obtained with the sr-PBE density functional.} \label{tab:h2o-dmc} \centering \begin{ruledtabular} @@ -484,10 +484,11 @@ The first question we would like to address is the quality of the nodes of the wave function $\Psi^{\mu}$ obtained with an intermediate range separation parameter (\ie, $0 < \mu < +\infty$). For this purpose, we consider a weakly correlated molecular system, namely the water -molecule \titou{near its equilibrium geometry.} \cite{Caffarel_2016} +molecule 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 such as CI coefficients and molecular orbitals. +\toto{$\mu$, and compute the associated fixed-node energy keeping fixed all the +parameters such as the CI coefficients and molecular orbitals impacting the +nodal surface.} %====================================================== \subsection{Fixed-node energy of $\Psi^\mu$} @@ -509,10 +510,11 @@ and then the FN-DMC error raises until it reaches the $\mu=\infty$ limit (\ie, t 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$^{-1}$. -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 $\mu$. +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 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 $\mu$. 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 @@ -529,11 +531,12 @@ The take-home message of this numerical study is that RS-DFT trial wave function %====================================================== 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 +Such behaviour can be directly 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} 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.} +\toto{For simplicity in the comparison, the molecular orbitals and the Jastrow +factor are kept fixed: only the CI coefficients are modified.} Let us assume a fixed Jastrow factor $J(\br_1, \ldots , \br_N)$, and a corresponding Slater-Jastrow wave function $\Phi = e^J \Psi$, @@ -541,18 +544,19 @@ where $\Psi = \sum_I c_I D_I$ is a general linear combination of Slater determin The only remaining variational parameters in $\Phi$ are therefore the Slater part $\Psi$. Let us call $\Psi^J$ the linear combination of Slater determinant minimizing the variational energy \begin{equation} - \Psi^J = \text{argmin}_{\Psi}\frac{ \mel{ \Psi }{ e^{J} H e^{J} }{ \Psi } }{\mel{ \Psi }{ e^{2J} }{ \Psi } }. + \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 $\Psi^J$ satisfies the generalized hermitian eigenvalue equation +Such a wave function $\Psi^J$ satisfies the generalized Hermitian eigenvalue equation \begin{equation} - e^{J} H e^{J} \Psi^J = E e^{2J} \Psi^J, + 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{many_things} +but also the non-Hermitian \manu{transcorrelated eigenvalue problem \cite{many_things} MANU:CITATIONS} \begin{equation} \label{eq:transcor} - e^{-J} H e^{J} \Psi^J = E \Psi^J, + e^{-J} \hat{H} \qty( e^{J} \Psi^J) = E \Psi^J, \end{equation} -which is much easier to handle despite its non-hermicity. +which is much easier to handle despite its non-Hermiticity. Of course, the FN-DMC energy of $\Phi$ depends therefore only on the nodes of $\Psi^J$. In a finite basis set and with a quite accurate Jastrow factor, it is known that the nodes of $\Psi^J$ may be better than that of the FCI wave function, and therefore, we would like to compare $\Psi^J$ and $\Psi^\mu$. @@ -563,8 +567,27 @@ function out of a large CIPSI calculation. Within this set of determinants, we solve the self-consistent equations of RS-DFT [see Eq.~\eqref{rs-dft-eigen-equation}] with different values of $\mu$. This gives the CI expansions $\Psi^\mu$. Then, within the same set of determinants we optimize the CI coefficients in the presence of -a simple one- and two-body Jastrow factor. This gives the CI expansion $\Psi^J$. -Then, we can easily compare $\Psi^\mu$ and $\Psi^J$ as they are developed +a simple one- and two-body Jastrow factor \toto{$e^J$ of the form $\exp(J_{eN} + J_{ee})$ with +\begin{eqnarray} + J_\text{eN} & = & - \sum_{A=1}^{\Nat} \sum_{i=1}^{\Nelec} \left( \frac{\alpha_A\, r_{iA}}{1 + \alpha_A\, r_{iA}} \right)^2 +\label{eq:jast-eN} \\ + J_\text{ee} & = & \sum_{i=1}^{\Nelec} \sum_{j=1}^{i-1} \frac{a\, r_{ij}}{1 + b\, r_{ij}}. \label{eq:jast-ee} +\end{eqnarray} +$J_\text{eN}$ contains the electron-nucleus terms with a single parameter +$\alpha_A$ per atom, and $J_\text{ee}$ contains the electron-electron terms +where the indices $i$ and $j$ loop over all electrons. The parameters $a=1/2$ +and $b=0.89$ were fixed, and the parameters $\gamma_O=1.15$ and $\gamma_H=0.35$ +were obtained by energy minimization with a single determinant. +The optimal CI expansion $\Psi^J$ is obtained by sampling the matrix elements +of the Hamiltonian ($\mathbf{H}$) and the overlap ($\mathbf{S}$), in the +basis of Jastrow-correlated determinants $e^J D_i$: +\begin{eqnarray} +H_{ij} & = & \left \langle \frac{e^J D_i}{\Psi^J}\, \frac{\hat{H}\, (e^J D_j)}{\Psi^J} \right \rangle \\ +S_{ij} & = & \left \langle \frac{e^J D_i}{\Psi^J}\, \frac{e^J D_j}{\Psi^J} \right \rangle +\end{eqnarray} +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 Slater determinant basis. In Fig.~\ref{fig:overlap}, we plot the overlaps $\braket*{\Psi^J}{\Psi^\mu}$ obtained for water, @@ -779,10 +802,7 @@ 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 -\begin{equation} - J_\text{ee} = \sum_{i