Merge branch 'master' of https://git.irsamc.ups-tlse.fr/scemama/RSDFT-CIPSI-QMC

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}
+}

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<j} \frac{a\, r_{ij}}{1 + b\, r_{ij}}.
-\end{equation}
+be expressed as in Eq.\eqref{eq:jast-ee}.
 The parameter
 $a$ is determined by cusp conditions, and $b$ is obtained by energy
 or variance minimization.\cite{Coldwell_1977,Umrigar_2005}