minor corrections here and there
This commit is contained in:
parent
392e231b4d
commit
6c5b90cf0e
@ -13,10 +13,19 @@
|
||||
]{hyperref}
|
||||
\urlstyle{same}
|
||||
|
||||
\newcommand{\ie}{\textit{i.e.}}
|
||||
\newcommand{\eg}{\textit{e.g.}}
|
||||
\newcommand{\alert}[1]{\textcolor{red}{#1}}
|
||||
\newcommand{\eg}[1]{\textcolor{blue}{#1}}
|
||||
\definecolor{darkgreen}{HTML}{009900}
|
||||
\usepackage[normalem]{ulem}
|
||||
\newcommand{\titou}[1]{\textcolor{red}{#1}}
|
||||
\newcommand{\trashPFL}[1]{\textcolor{red}{\sout{#1}}}
|
||||
\newcommand{\PFL}[1]{\titou{(\underline{\bf PFL}: #1)}}
|
||||
|
||||
\newcommand{\mc}{\multicolumn}
|
||||
\newcommand{\fnm}{\footnotemark}
|
||||
\newcommand{\fnt}{\footnotetext}
|
||||
\newcommand{\tabc}[1]{\multicolumn{1}{c}{#1}}
|
||||
|
||||
\newcommand{\EPT}{E_{\text{PT2}}}
|
||||
\newcommand{\EDMC}{E_{\text{FN-DMC}}}
|
||||
@ -93,20 +102,20 @@ sets because the constraints imposed by the FN approximation
|
||||
are less severe than the constraints imposed by the finite-basis
|
||||
approximation.
|
||||
|
||||
\alert{However, it is usually harder to control the FN error in DMC, and this might affect energy differences such as atomization energies.
|
||||
\titou{However, it is usually harder to control the FN error in DMC, and this might affect energy differences such as atomization energies.
|
||||
Moreover, improving systematically the nodal surface of the trial wave function can be a tricky job as there is no variational principle for the nodes.}
|
||||
|
||||
The qualitative picture of the electronic structure of weakly
|
||||
correlated systems, such as organic molecules near their equilibrium
|
||||
geometry, is usually well represented with a single Slater
|
||||
determinant. This feature is in part responsible for the success of
|
||||
density-functional theory (DFT) and coupled cluster.
|
||||
density-functional theory (DFT) and coupled cluster theory.
|
||||
DMC with a single-determinant trial wave function can be used as a
|
||||
single-reference post-Hatree-Fock method, with an accuracy comparable
|
||||
to coupled cluster.\cite{Dubecky_2014,Grossman_2002}
|
||||
The favorable scaling of QMC, its very low memory requirements and
|
||||
its adequacy with massively parallel architectures make it a
|
||||
serious alternative for high-accuracy simulations on large systems.
|
||||
serious alternative for high-accuracy simulations of large systems.
|
||||
|
||||
As it is not possible to minimize directly the FN-DMC energy with respect
|
||||
to the variational parameters of the trial wave function, the
|
||||
@ -115,7 +124,7 @@ finite-basis approximation.
|
||||
The conventional approach consists in multiplying the trial wave
|
||||
function by a positive function, the \emph{Jastrow factor}, taking
|
||||
account of the electron-electron cusp and the short-range correlation
|
||||
effects. The wave function is then re-optimized within Variational
|
||||
effects. The wave function is then re-optimized within variational
|
||||
Monte Carlo (VMC) in the presence of the Jastrow factor and the nodal
|
||||
surface is expected to be improved. Using this technique, it has been
|
||||
shown that the chemical accuracy could be reached within
|
||||
@ -125,13 +134,13 @@ 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,Caffarel_2016_2}
|
||||
When the basis set is increased, the trial wave function gets closer
|
||||
\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}
|
||||
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
|
||||
black-box way without needing any expertise in QMC.
|
||||
But this technique can't be applied to large systems because of the
|
||||
But 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
|
||||
@ -339,9 +348,9 @@ post-HF method of interest.
|
||||
|
||||
|
||||
All the calculations were made using BFD
|
||||
pseudopotentials\cite{Burkatzki_2008} with the associated double,
|
||||
triple and quadruple zeta basis sets (BFD-V$n$Z).
|
||||
CCSD(T) and DFT calculations were made with
|
||||
pseudopotentials\cite{Burkatzki_2008} with the associated double-,
|
||||
triple-, and quadruple-$\zeta$ basis sets (BFD-VXZ).
|
||||
CCSD(T) and KS-DFT calculations were made with
|
||||
\emph{Gaussian09},\cite{g16} using an unrestricted Hartree-Fock
|
||||
determinant as a reference for open-shell systems.
|
||||
|
||||
@ -372,36 +381,37 @@ fluctuations of the local energy.
|
||||
error}
|
||||
\label{sec:mu-dmc}
|
||||
\begin{table}
|
||||
\caption{Fixed-node energies and number of determinants in the water
|
||||
molecule and the fluorine dimer with different trial wave functions.}
|
||||
\caption{Fixed-node energies (in hartree) and number of determinants in \ce{H2O} and \ce{F2} with various trial wave functions.}
|
||||
\label{tab:h2o-dmc}
|
||||
\centering
|
||||
\begin{ruledtabular}
|
||||
\begin{tabular}{crlrl}
|
||||
& \multicolumn{2}{c}{BFD-VDZ} & \multicolumn{2}{c}{BFD-VTZ} \\
|
||||
$\mu$ & $\Ndet$ & $\EDMC$ & $\Ndet$ & $\EDMC$ \\
|
||||
\begin{tabular}{ccrlrl}
|
||||
& & \multicolumn{2}{c}{BFD-VDZ} & \multicolumn{2}{c}{BFD-VTZ} \\
|
||||
\cline{3-4} \cline{5-6}
|
||||
System & $\mu$ & $\Ndet$ & $\EDMC$ & $\Ndet$ & $\EDMC$ \\
|
||||
\hline
|
||||
& \multicolumn{4}{c}{H$_2$O} \\
|
||||
$0.00$ & $11$ & $-17.253\,59(6)$ & $23$ & $-17.256\,74(7)$ \\
|
||||
$0.20$ & $23$ & $-17.253\,73(7)$ & $23$ & $-17.256\,73(8)$ \\
|
||||
$0.30$ & $53$ & $-17.253\,4(2)$ & $219$ & $-17.253\,7(5)$ \\
|
||||
$0.50$ & $1\,442$ & $-17.253\,9(2)$ & $16\,99$ & $-17.257\,7(2)$ \\
|
||||
$0.75$ & $3\,213$ & $-17.255\,1(2)$ & $13\,362$ & $-17.258\,4(3)$ \\
|
||||
$1.00$ & $6\,743$ & $-17.256\,6(2)$ & $256\,73$ & $-17.261\,0(2)$ \\
|
||||
$1.75$ & $54\,540$ & $-17.259\,5(3)$ & $207\,475$ & $-17.263\,5(2)$ \\
|
||||
$2.50$ & $51\,691$ & $-17.259\,4(3)$ & $858\,123$ & $-17.264\,3(3)$ \\
|
||||
$3.80$ & $103\,059$ & $-17.258\,7(3)$ & $1\,621\,513$ & $-17.263\,7(3)$ \\
|
||||
$5.70$ & $102\,599$ & $-17.257\,7(3)$ & $1\,629\,655$ & $-17.263\,2(3)$ \\
|
||||
$8.50$ & $101\,803$ & $-17.257\,3(3)$ & $1\,643\,301$ & $-17.263\,3(4)$ \\
|
||||
$\infty$ & $200\,521$ & $-17.256\,8(6)$ & $1\,631\,982$ & $-17.263\,9(3)$ \\
|
||||
& \multicolumn{3}{c}{F$_2$} \\
|
||||
$0.00$ & $23$ & $-48.419\,5(4)$ \\
|
||||
$0.25$ & $8$ & $-48.421\,9(4)$ \\
|
||||
$0.50$ & $1743$ & $-48.424\,8(8)$ \\
|
||||
$1.00$ & $11952$ & $-48.432\,4(3)$ \\
|
||||
$2.00$ & $829438$ & $-48.441\,0(7)$ \\
|
||||
$5.00$ & $5326459$ & $-48.445(2)$ \\
|
||||
$\infty$ & $8302442$ & $-48.437(3)$ \\
|
||||
\ce{H2O}
|
||||
& $0.00$ & $11$ & $-17.253\,59(6)$ & $23$ & $-17.256\,74(7)$ \\
|
||||
& $0.20$ & $23$ & $-17.253\,73(7)$ & $23$ & $-17.256\,73(8)$ \\
|
||||
& $0.30$ & $53$ & $-17.253\,4(2)$ & $219$ & $-17.253\,7(5)$ \\
|
||||
& $0.50$ & $1\,442$ & $-17.253\,9(2)$ & $16\,99$ & $-17.257\,7(2)$ \\
|
||||
& $0.75$ & $3\,213$ & $-17.255\,1(2)$ & $13\,362$ & $-17.258\,4(3)$ \\
|
||||
& $1.00$ & $6\,743$ & $-17.256\,6(2)$ & $256\,73$ & $-17.261\,0(2)$ \\
|
||||
& $1.75$ & $54\,540$ & $-17.259\,5(3)$ & $207\,475$ & $-17.263\,5(2)$ \\
|
||||
& $2.50$ & $51\,691$ & $-17.259\,4(3)$ & $858\,123$ & $-17.264\,3(3)$ \\
|
||||
& $3.80$ & $103\,059$ & $-17.258\,7(3)$ & $1\,621\,513$ & $-17.263\,7(3)$ \\
|
||||
& $5.70$ & $102\,599$ & $-17.257\,7(3)$ & $1\,629\,655$ & $-17.263\,2(3)$ \\
|
||||
& $8.50$ & $101\,803$ & $-17.257\,3(3)$ & $1\,643\,301$ & $-17.263\,3(4)$ \\
|
||||
& $\infty$ & $200\,521$ & $-17.256\,8(6)$ & $1\,631\,982$ & $-17.263\,9(3)$ \\
|
||||
\\
|
||||
\ce{F2}
|
||||
& $0.00$ & $23$ & $-48.419\,5(4)$ \\
|
||||
& $0.25$ & $8$ & $-48.421\,9(4)$ \\
|
||||
& $0.50$ & $1743$ & $-48.424\,8(8)$ \\
|
||||
& $1.00$ & $11952$ & $-48.432\,4(3)$ \\
|
||||
& $2.00$ & $829438$ & $-48.441\,0(7)$ \\
|
||||
& $5.00$ & $5326459$ & $-48.445(2)$ \\
|
||||
& $\infty$ & $8302442$ & $-48.437(3)$ \\
|
||||
\end{tabular}
|
||||
\end{ruledtabular}
|
||||
\end{table}
|
||||
@ -515,7 +525,7 @@ the density and of electron correlation, but also reduces the
|
||||
imbalance in the quality of the description of the atoms and the
|
||||
molecule, leading to more accurate atomization energies.
|
||||
|
||||
\subsection{Size-consistency}
|
||||
\subsection{Size consistency}
|
||||
|
||||
An extremely important feature required to get accurate
|
||||
atomization energies is size-consistency (or strict separability),
|
||||
@ -589,12 +599,12 @@ Ref.~\onlinecite{Scemama_2015}).
|
||||
|
||||
%\begin{squeezetable}
|
||||
\begin{table}
|
||||
\caption{FN-DMC Energies of the fluorine atom and the dissociated fluorine
|
||||
dimer, and size-consistency error.}
|
||||
\caption{FN-DMC energies (in hartree) of the fluorine atom and the dissociated fluorine
|
||||
dimer, and size-consistency error. \titou{BASIS?}}
|
||||
\label{tab:size-cons}
|
||||
\begin{ruledtabular}
|
||||
\begin{tabular}{cccc}
|
||||
$\mu$ & F & Dissociated F$_2$ & Size-consistency error \\
|
||||
$\mu$ & \ce{F} & Dissociated \ce{F2} & Size-consistency error \\
|
||||
\hline
|
||||
0.00 & $-24.188\,7(3)$ & $-48.377\,7(3)$ & $-0.000\,3(4)$ \\
|
||||
0.25 & $-24.188\,7(3)$ & $-48.377\,2(4)$ & $+0.000\,2(5)$ \\
|
||||
@ -619,7 +629,7 @@ size-consistent FN-DMC energies for all values of $\mu$ (within
|
||||
$2\times$ statistical error bars).
|
||||
|
||||
|
||||
\subsection{Spin-invariance}
|
||||
\subsection{Spin invariance}
|
||||
|
||||
Closed-shell molecules often dissociate into open-shell
|
||||
fragments. To get reliable atomization energies, it is important to
|
||||
@ -644,7 +654,7 @@ So in the context of RS-DFT, the determinantal expansions will be
|
||||
impacted by this spurious effect, as opposed to FCI.
|
||||
|
||||
\begin{table}
|
||||
\caption{FN-DMC Energies of the triplet carbon atom (BFD-VDZ) with
|
||||
\caption{FN-DMC energies (in hartree) of the triplet carbon atom (BFD-VDZ) with
|
||||
different values of $m_s$.}
|
||||
\label{tab:spin}
|
||||
\begin{ruledtabular}
|
||||
@ -687,38 +697,39 @@ noticeable with $\mu=5$~bohr$^{-1}$.
|
||||
|
||||
\begin{squeezetable}
|
||||
\begin{table*}
|
||||
\caption{Mean absolute error (MAE), mean signed errors (MSE) and
|
||||
standard deviations (RMSD) obtained with the different methods and
|
||||
\caption{Mean absolute errors (MAE), mean signed errors (MSE) and
|
||||
standard deviations (RMSD) obtained with various methods and
|
||||
basis sets.}
|
||||
\label{tab:mad}
|
||||
\begin{ruledtabular}
|
||||
\begin{tabular}{ll rrr rrr rrr}
|
||||
Method & \(\mu\) & \phantom{} & VDZ-BFD & \phantom{} & \phantom{} & VTZ-BFD & \phantom{} & \phantom{} & VQZ-BFD & \phantom{} \\
|
||||
\phantom{} & \phantom{} & MAE & MSE & RMSD & MAE & MSE & RMSD & MAE & MSE & RMSD \\
|
||||
\begin{tabular}{ll ddd ddd ddd}
|
||||
& & \mc{3}{c}{VDZ-BFD} & \mc{3}{c}{VTZ-BFD} & \mc{3}{c}{VQZ-BFD} \\
|
||||
\cline{3-5} \cline{6-8} \cline{9-11}
|
||||
Method & $\mu$ & \tabc{MAE} & \tabc{MSE} & \tabc{RMSD} & \tabc{MAE} & \tabc{MSE} & \tabc{RMSD} & \tabc{MAE} & \tabc{MSE} & \tabc{RMSD} \\
|
||||
\hline
|
||||
PBE & 0 & 5.02 & -3.70 & 6.04 & 4.57 & 1.00 & 5.32 & 5.31 & 0.79 & 6.27 \\
|
||||
BLYP & 0 & 9.53 & -9.21 & 7.91 & 5.58 & -4.44 & 5.80 & 5.86 & -4.47 & 6.43 \\
|
||||
PBE0 & 0 & 11.20 & -10.98 & 8.68 & 6.40 & -5.78 & 5.49 & 6.28 & -5.65 & 5.08 \\
|
||||
B3LYP & 0 & 11.27 & -10.98 & 9.59 & 7.27 & -5.77 & 6.63 & 6.75 & -5.53 & 6.09 \\
|
||||
\hline
|
||||
\\
|
||||
CCSD(T) & \(\infty\) & 24.10 & -23.96 & 13.03 & 9.11 & -9.10 & 5.55 & 4.52 & -4.38 & 3.60 \\
|
||||
\hline
|
||||
\\
|
||||
RS-DFT-CIPSI & 0 & 4.53 & -1.66 & 5.91 & 6.31 & 0.91 & 7.93 & 6.35 & 3.88 & 7.20 \\
|
||||
\phantom{} & 1/4 & 5.55 & -4.66 & 5.52 & 4.58 & 1.06 & 5.72 & 5.48 & 1.52 & 6.93 \\
|
||||
\phantom{} & 1/2 & 13.42 & -13.27 & 7.36 & 6.77 & -6.71 & 4.56 & 6.35 & -5.89 & 5.18 \\
|
||||
\phantom{} & 1 & 17.07 & -16.92 & 9.83 & 9.06 & -9.06 & 5.88 & --- & --- & --- \\
|
||||
\phantom{} & 2 & 19.20 & -19.05 & 10.91 & --- & --- & --- & --- & --- & --- \\
|
||||
\phantom{} & 5 & 22.93 & -22.79 & 13.24 & --- & --- & --- & --- & --- & --- \\
|
||||
\phantom{} & \(\infty\) & 23.62 & -23.48 & 12.81 & --- & --- & --- & --- & --- & --- \\
|
||||
\hline
|
||||
DMC@RS-DFT-CIPSI & 0 & 4.61(\phantom{0.}34) & -3.62(\phantom{0.}34) & 5.30(\phantom{0.}09) & 3.52(19) & -1.03(19) & 4.39(04) & 3.16(26) & -0.12(26) & 4.12(03) \\
|
||||
\phantom{} & 1/4 & 4.04(\phantom{0.}37) & -3.13(\phantom{0.}37) & 4.88(\phantom{0.}10) & 3.39(77) & -0.59(77) & 4.44(34) & 2.90(25) & 0.25(25) & 3.745(5) \\
|
||||
\phantom{} & 1/2 & 3.74(\phantom{0.}35) & -3.53(\phantom{0.}35) & 4.03(\phantom{0.}23) & 2.46(18) & -1.72(18) & 3.02(06) & 2.06(35) & -0.44(35) & 2.74(13) \\
|
||||
\phantom{} & 1 & 5.42(\phantom{0.}29) & -5.14(\phantom{0.}29) & 4.55(\phantom{0.}03) & 4.38(94) & -4.24(94) & 5.11(31) & --- & --- & --- \\
|
||||
\phantom{} & 2 & 5.98(\phantom{0.}83) & -5.91(\phantom{0.}83) & 4.79(\phantom{0.}71) & --- & --- & --- & --- & --- & --- \\
|
||||
\phantom{} & 5 & 6.18(\phantom{0.}84) & -6.13(\phantom{0.}84) & 4.87(\phantom{0.}55) & --- & --- & --- & --- & --- & --- \\
|
||||
\phantom{} & \(\infty\) & 7.38(1.08) & -7.38(1.08) & 5.67(\phantom{0.}68) & --- & --- & --- & --- & --- & --- \\
|
||||
\phantom{} & Opt. & 5.85(1.75) & -5.63(1.75) & 4.79(1.11) & --- & --- & --- & --- & --- & --- \\
|
||||
& 1/4 & 5.55 & -4.66 & 5.52 & 4.58 & 1.06 & 5.72 & 5.48 & 1.52 & 6.93 \\
|
||||
& 1/2 & 13.42 & -13.27 & 7.36 & 6.77 & -6.71 & 4.56 & 6.35 & -5.89 & 5.18 \\
|
||||
& 1 & 17.07 & -16.92 & 9.83 & 9.06 & -9.06 & 5.88 & & & \\
|
||||
& 2 & 19.20 & -19.05 & 10.91 & & & & & & \\
|
||||
& 5 & 22.93 & -22.79 & 13.24 & & & & & & \\
|
||||
& \(\infty\) & 23.62 & -23.48 & 12.81 & & & & & & \\
|
||||
\\
|
||||
DMC@ & 0 & 4.61(34) & -3.62(34) & 5.30(09) & 3.52(19) & -1.03(19) & 4.39(04) & 3.16(26) & -0.12(26) & 4.12(03) \\
|
||||
RS-DFT-CIPSI & 1/4 & 4.04(37) & -3.13(37) & 4.88(10) & 3.39(77) & -0.59(77) & 4.44(34) & 2.90(25) & 0.25(25) & 3.745(5) \\
|
||||
& 1/2 & 3.74(35) & -3.53(35) & 4.03(23) & 2.46(18) & -1.72(18) & 3.02(06) & 2.06(35) & -0.44(35) & 2.74(13) \\
|
||||
& 1 & 5.42(29) & -5.14(29) & 4.55(03) & 4.38(94) & -4.24(94) & 5.11(31) & & & \\
|
||||
& 2 & 5.98(83) & -5.91(83) & 4.79(71) & & & & & & \\
|
||||
& 5 & 6.18(84) & -6.13(84) & 4.87(55) & & & & & & \\
|
||||
& \(\infty\) & 7.38(1.08) & -7.38(1.08) & 5.67(68) & & & & & & \\
|
||||
& Opt. & 5.85(1.75) & -5.63(1.75) & 4.79(1.11) & & & & & & \\
|
||||
\end{tabular}
|
||||
\end{ruledtabular}
|
||||
\end{table*}
|
||||
@ -860,7 +871,7 @@ large systems.
|
||||
|
||||
|
||||
|
||||
%%---------------------------------------
|
||||
%%
|
||||
\begin{acknowledgments}
|
||||
This work was performed using HPC resources from GENCI-TGCC (Grand
|
||||
Challenge 2019-gch0418) and from CALMIP (Toulouse) under allocation
|
||||
|
Loading…
Reference in New Issue
Block a user