Modifs toto
This commit is contained in:
parent
072585fb60
commit
5d2d010380
@ -1151,3 +1151,17 @@
|
||||
note = {}
|
||||
}
|
||||
|
||||
@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}
|
||||
}
|
||||
|
||||
@ -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}
|
||||
@ -156,13 +160,13 @@ 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}
|
||||
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
|
||||
\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.
|
||||
\manu{Nevertheless,} this technique cannot be applied to large systems because of the
|
||||
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
|
||||
@ -225,8 +229,8 @@ of the wave functions is required.
|
||||
\subsection{CIPSI}
|
||||
%====================
|
||||
Beyond the single-determinant representation, the best
|
||||
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
|
||||
multi-determinant wave function one can obtain in a given basis set is the FCI.
|
||||
FCI is 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
|
||||
@ -263,7 +267,7 @@ accuracy so all the CIPSI selections were made such that $\abs{\EPT} <
|
||||
\label{sec:rsdft}
|
||||
%=================================
|
||||
|
||||
\manu{The range-separated DFT (RS-DFT)} was introduced in the seminal work of Savin,\cite{Sav-INC-96a,Toulouse_2004}
|
||||
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}
|
||||
@ -279,7 +283,7 @@ 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 \manu{using a density functional}, and the long-range part within a WFT method like FCI in the present case.
|
||||
interaction 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 ($\mu = \infty$).
|
||||
@ -296,8 +300,8 @@ $\mathcal{F}^{\text{lr},\mu}$ 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}
|
||||
\manu{The exact ground state energy can be therefore obtained as a minimization
|
||||
over a multi-determinant wave function as follows}:
|
||||
The exact ground state energy can therefore be 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}
|
||||
@ -331,14 +335,12 @@ 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*}
|
||||
@ -422,10 +424,9 @@ 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}
|
||||
@ -434,8 +435,7 @@ 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}
|
||||
@ -476,10 +476,11 @@ 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 \manu{such as CI coefficients and molecular orbitals}.
|
||||
$\mu$, 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 $\Psi^\mu$}
|
||||
@ -501,19 +502,20 @@ 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$^{-1}$.
|
||||
\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 $\mu$.
|
||||
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
|
||||
RS-DFT energies obtained with these two functionals differ by several
|
||||
tens of m\hartree{}.
|
||||
|
||||
\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.}
|
||||
An other important aspect here regards 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 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 }
|
||||
@ -521,11 +523,12 @@ 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 $\Phi = e^J \Psi$,
|
||||
@ -533,18 +536,19 @@ 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 $\Psi^\mu$.
|
||||
@ -555,8 +559,27 @@ 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,
|
||||
@ -771,10 +794,7 @@ 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}
|
||||
|
||||
Loading…
Reference in New Issue
Block a user