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Pierre-Francois Loos 2020-08-08 22:21:36 +02:00
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Manuscript/rsdft-cipsi-qmc.tex
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@ -251,7 +251,7 @@ estimate of the FCI energy, using a fixed value of the PT2 correction
as a stopping criterion enforces a constant distance of all the as a stopping criterion enforces a constant distance of all the
calculations to the FCI energy. In this work, we target the chemical calculations to the FCI energy. In this work, we target the chemical
accuracy so all the CIPSI selections were made such that $\abs{\EPT} < accuracy so all the CIPSI selections were made such that $\abs{\EPT} <
1$ millihartree. 1$ m\hartree{}.
@ -279,7 +279,7 @@ The main idea behind RS-DFT is to treat the short-range part of the
interaction within KS-DFT, and the long-range part within a WFT method like FCI in the present case. interaction within KS-DFT, 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 The parameter $\mu$ controls the range of the separation, and allows
to go continuously from the KS Hamiltonian ($\mu=0$) to to go continuously from the KS Hamiltonian ($\mu=0$) to
the FCI Hamiltonian ($\mu \to \infty$). the FCI Hamiltonian ($\mu = \infty$).
To rigorously connect WFT and DFT, the universal To rigorously connect WFT and DFT, the universal
Levy-Lieb density functional \cite{Lev-PNAS-79,Lie-IJQC-83} is Levy-Lieb density functional \cite{Lev-PNAS-79,Lie-IJQC-83} is
@ -331,7 +331,7 @@ energy is obtained as
Note that, for $\mu=0$, \titou{the long-range interaction vanishes}, \ie, Note that, for $\mu=0$, \titou{the long-range interaction vanishes}, \ie,
$w_{\text{ee}}^{\text{lr},\mu=0}(r) = 0$, and thus $w_{\text{ee}}^{\text{lr},\mu=0}(r) = 0$, and thus
RS-DFT reduces to standard KS-DFT and $\Psi^\mu$ RS-DFT reduces to standard KS-DFT and $\Psi^\mu$
is the KS determinant. For $\mu\to\infty$, the long-range is the KS determinant. For $\mu = \infty$, the long-range
interaction becomes the standard Coulomb interaction, \ie, interaction becomes the standard Coulomb interaction, \ie,
$w_{\text{ee}}^{\text{lr},\mu\to\infty}(r) = r^{-1}$, and $w_{\text{ee}}^{\text{lr},\mu\to\infty}(r) = r^{-1}$, and
thus RS-DFT reduces to standard WFT and $\Psi^\mu$ is thus RS-DFT reduces to standard WFT and $\Psi^\mu$ is
@ -369,12 +369,12 @@ In the outer (macro-iteration) loop (red), at the $k$th iteration, a CIPSI selec
to obtain $\Psi^{\mu\,(k)}$ with the RS-DFT Hamiltonian $\hat{H}^{\mu\,(k)}$ to obtain $\Psi^{\mu\,(k)}$ with the RS-DFT Hamiltonian $\hat{H}^{\mu\,(k)}$
parameterized using the current one-electron density $n^{(k)}$. parameterized using the current one-electron density $n^{(k)}$.
At each iteration, the number of determinants in $\Psi^{\mu\,(k)}$ increases. At each iteration, the number of determinants in $\Psi^{\mu\,(k)}$ increases.
One exits the outer loop when the absolute energy difference between two successive macro-iterations $\Delta E^{(k)}$ is below a threshold $\tau_1$ that has been set to $10^{-3}$ hartree in the present study which is consistent with the CIPSI threshold (see Sec.~\ref{sec:comp-details}). One exits the outer loop when the absolute energy difference between two successive macro-iterations $\Delta E^{(k)}$ is below a threshold $\tau_1$ that has been set to $10^{-3}$ \hartree{} in the present study which is consistent with the CIPSI threshold (see Sec.~\ref{sec:comp-details}).
An inner (micro-iteration) loop (blue) is introduced to accelerate the An inner (micro-iteration) loop (blue) is introduced to accelerate the
convergence of the self-consistent calculation, in which the set of convergence of the self-consistent calculation, in which the set of
determinants in $\Psi^{\mu\,(k,l)}$ is kept fixed, and only the diagonalization of determinants in $\Psi^{\mu\,(k,l)}$ is kept fixed, and only the diagonalization of
$\hat{H}^{\mu\,(k,l)}$ is performed iteratively with the updated density $n^{(k,l)}$. $\hat{H}^{\mu\,(k,l)}$ is performed iteratively with the updated density $n^{(k,l)}$.
The inner loop is exited when the absolute energy difference between two successive micro-iterations $\Delta E^{(k,l)}$ is below a threshold $\tau_2$ that has been here set to $10^{-2} \times \tau_1$ hartree. The inner loop is exited when the absolute energy difference between two successive micro-iterations $\Delta E^{(k,l)}$ is below a threshold $\tau_2$ that has been here set to $10^{-2} \times \tau_1$ \hartree{}.
The convergence of the algorithm was further improved The convergence of the algorithm was further improved
by introducing a direct inversion in the iterative subspace (DIIS) by introducing a direct inversion in the iterative subspace (DIIS)
step to extrapolate the density both in the outer and inner loops. \cite{Pulay_1980,Pulay_1982} step to extrapolate the density both in the outer and inner loops. \cite{Pulay_1980,Pulay_1982}
@ -407,7 +407,7 @@ and correlation functionals defined in
Ref.~\onlinecite{GolWerStoLeiGorSav-CP-06} (see also Ref.~\onlinecite{GolWerStoLeiGorSav-CP-06} (see also
Refs.~\onlinecite{TouColSav-JCP-05,GolWerSto-PCCP-05}). Refs.~\onlinecite{TouColSav-JCP-05,GolWerSto-PCCP-05}).
The convergence criterion for stopping the CIPSI calculations The convergence criterion for stopping the CIPSI calculations
has been set to $\EPT < 10^{-3}$ hartree or $ \Ndet > 10^7$. has been set to $\EPT < 10^{-3}$ \hartree{} or $ \Ndet > 10^7$.
All the wave functions are eigenfunctions of the $\Hat{S}^2$ spin operator, as All the wave functions are eigenfunctions of the $\Hat{S}^2$ spin operator, as
described in Ref.~\onlinecite{Applencourt_2018}. described in Ref.~\onlinecite{Applencourt_2018}.
@ -431,7 +431,8 @@ with a time step of $5 \times 10^{-4}$ a.u.
%%% TABLE I %%% %%% TABLE I %%%
\begin{table} \begin{table}
\caption{Fixed-node energies $\EDMC$ (in hartree) and number of determinants $\Ndet$ in \ce{H2O} with various trial wave functions $\Psi^{\mu}$.} \caption{Fixed-node energy $\EDMC$ (in \hartree{}) and number of determinants $\Ndet$ in \ce{H2O} for various trial wave functions $\Psi^{\mu}$.
\titou{srPBE?}.}
\label{tab:h2o-dmc} \label{tab:h2o-dmc}
\centering \centering
\begin{ruledtabular} \begin{ruledtabular}
@ -461,9 +462,9 @@ with a time step of $5 \times 10^{-4}$ a.u.
\begin{figure} \begin{figure}
\centering \centering
\includegraphics[width=\columnwidth]{h2o-dmc.pdf} \includegraphics[width=\columnwidth]{h2o-dmc.pdf}
\caption{Fixed-node energies of \ce{H2O} for different \caption{Fixed-node energy of \ce{H2O} as a function
values of $\mu$, using the srLDA or srPBE density of $\mu$ for various levels of theory to generate
functionals to build the trial wave function.} the trial wave function.}
\label{fig:h2o-dmc} \label{fig:h2o-dmc}
\end{figure} \end{figure}
%%% %%% %%% %%% %%% %%% %%% %%%
@ -481,31 +482,34 @@ parameters having an impact on the nodal surface fixed (\titou{such as ??}).
\subsection{Fixed-node energy of $\Psi^\mu$} \subsection{Fixed-node energy of $\Psi^\mu$}
\label{sec:fndmc_mu} \label{sec:fndmc_mu}
%====================================================== %======================================================
From Table~\ref{tab:h2o-dmc} and Fig.~\ref{fig:h2o-dmc}, 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,
one can clearly observe that relying on FCI trial one can clearly observe that relying on FCI trial
wave functions ($\mu \to \infty$) give FN-DMC energies lower wave functions ($\mu = \infty$) give FN-DMC energies lower
than the energies obtained with a single KS determinant ($\mu=0$): than the energies obtained with a single KS determinant ($\mu=0$):
a gain of $3.2 \pm 0.6$~m\hartree{} at the double-$\zeta$ level and $7.2 \pm a lowering of $3.2 \pm 0.6$~m\hartree{} at the double-$\zeta$ level and $7.2 \pm
0.3$~m\hartree{} at the triple-zeta level are obtained. 0.3$~m\hartree{} at the triple-$\zeta$ level are obtained.
Coming now to the nodes of the trial wave functions $\Psi^{\mu}$ with Coming now to the nodes of the trial wave functions $\Psi^{\mu}$ with
intermediate values of $\mu$, Fig.~\ref{fig:h2o-dmc} shows that intermediate values of $\mu$, Fig.~\ref{fig:h2o-dmc} shows that
a smooth behaviour is obtained: a smooth behaviour is obtained:
starting from $\mu=0$ (\textit{i.e.} the KS determinant), starting from $\mu=0$ (\ie, the KS determinant),
the FN-DMC error is reduced continuously until it reaches a minimum for an optimal value of $\mu$, the FN-DMC error is reduced continuously until it reaches a minimum
and then the FN-DMC error raises until it reaches the $\mu=\infty$ limit (\textit{i.e.} the FCI wave function). for an optimal value of $\mu$ (which is obviously basis set and functional dependent),
For instance, with respect to the FN-DMC energy of the FCI trial wave function in the double zeta basis set, and then the FN-DMC error raises until it reaches the $\mu=\infty$ limit (\ie, the FCI wave function).
with the optimal value of $\mu$, one can obtain a lowering of the For instance, with respect to the FN-DMC/VDZ-BFD energy at $\mu=\infty$,
FN-DMC energy of $2.6 \pm 0.7$~m\hartree{}, with an optimal value at one can obtain a lowering of the FN-DMC energy of $2.6 \pm 0.7$~m\hartree{}
$\mu=1.75$~bohr$^{-1}$. with an optimal value of $\mu=1.75$~bohr$^{-1}$.
When the basis set is increased, the gain in FN-DMC energy with 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 respect to the FCI trial wave function is reduced, and the optimal
value of $\mu$ is slightly shifted towards large $\mu$. Eventually, the nodes value of $\mu$ is slightly shifted towards large $\mu$. Eventually, the nodes
of the wave functions $\Psi^\mu$ obtained with the short-range of the wave functions $\Psi^\mu$ obtained with the srLDA
LDA exchange-correlation functional give very similar FN-DMC energies with respect exchange-correlation functional give very similar FN-DMC energies with respect
to those obtained with the short-range PBE functional, even if the to those obtained with the srPBE functional, even if the
RS-DFT energies obtained with these two functionals differ by several RS-DFT energies obtained with these two functionals differ by several
tens of m\hartree{}. tens of m\hartree{}.
\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 $\mu = \infty$.
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.}
%====================================================== %======================================================
\subsection{Link between RS-DFT and Jastrow factors } \subsection{Link between RS-DFT and Jastrow factors }
\label{sec:rsdft-j} \label{sec:rsdft-j}
@ -548,9 +552,9 @@ Then, within the same set of determinants we optimize the CI coefficients in the
a simple one- and two-body Jastrow factor. This gives the CI expansion $\Psi^J$. 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 Then, we can easily compare $\Psi^\mu$ and $\Psi^J$ as they are developed
on the same Slater determinant basis. on the same Slater determinant basis.
In figure~\ref{fig:overlap}, we plot the overlaps In Fig.~\ref{fig:overlap}, we plot the overlaps
$\braket{\Psi^J}{\Psi^\mu}$ obtained for water and the fluorine dimer, $\braket{\Psi^J}{\Psi^\mu}$ obtained for water and the fluorine dimer,
and in figure~\ref{dmc_small} the FN-DMC energy of the wave functions and in Fig.~\ref{dmc_small} the FN-DMC energy of the wave functions
$\Psi^\mu$ together with that of $\Psi^J$. $\Psi^\mu$ together with that of $\Psi^J$.
%%% FIG 3 %%% %%% FIG 3 %%%
@ -663,7 +667,7 @@ $\mu=0.5$ and $\mu=1$~bohr$^{-1}$, constently with the FN-DMC energies
and the overlap curve. and the overlap curve.
These data suggest that the wave functions $\Psi^{0.5 \le \mu \le 1}$ and $\Psi^J$ are close, These data suggest that the wave functions $\Psi^{0.5 \le \mu \le 1}$ and $\Psi^J$ are close,
and therefore that the operators that produced these wave functions (\textit{i.e.} $H^\mu[n]$ and $e^{-J}He^J$) contain similar physics. and therefore that the operators that produced these wave functions (\ie, $H^\mu[n]$ and $e^{-J}He^J$) contain similar physics.
Considering the form of $\hat{H}^\mu[n]$ (see Eq.~\eqref{H_mu}), Considering the form of $\hat{H}^\mu[n]$ (see Eq.~\eqref{H_mu}),
one can notice that the differences with respect to the usual Hamiltonian come one can notice that the differences with respect to the usual Hamiltonian come
from the non-divergent two-body interaction $\hat{W}_{\text{ee}}^{\text{lr},\mu}$ from the non-divergent two-body interaction $\hat{W}_{\text{ee}}^{\text{lr},\mu}$
@ -694,7 +698,7 @@ As a conclusion of the first part of this study, we can notice that:
\item numerical experiments (overlap $\braket{\Psi^\mu}{\Psi^J}$, \item numerical experiments (overlap $\braket{\Psi^\mu}{\Psi^J}$,
one-body density, on-top pair density, and FN-DMC energy) indicate one-body density, on-top pair density, and FN-DMC energy) indicate
that the RS-DFT scheme essentially plays the role of a simple Jastrow factor, that the RS-DFT scheme essentially plays the role of a simple Jastrow factor,
\textit{i.e.} mimicking short-range correlation effects. The latter \ie, mimicking short-range correlation effects. The latter
statement can be qualitatively understood by noticing that both RS-DFT statement can be qualitatively understood by noticing that both RS-DFT
and transcorrelated approaches deal with an effective non-divergent and transcorrelated approaches deal with an effective non-divergent
electron-electron interaction, while keeping the density constant. electron-electron interaction, while keeping the density constant.
@ -764,7 +768,7 @@ one-electron, two-electron and one-nucleus-two-electron terms.
The problematic part is the two-electron term, whose simplest form can The problematic part is the two-electron term, whose simplest form can
be expressed as be expressed as
\begin{equation} \begin{equation}
J_\text{ee} = \sum_i \sum_{j<i} \frac{a\, r_{ij}}{1 + b\, r_{ij}}. J_\text{ee} = \sum_{i<j} \frac{a\, r_{ij}}{1 + b\, r_{ij}}.
\end{equation} \end{equation}
The parameter The parameter
$a$ is determined by cusp conditions, and $b$ is obtained by energy $a$ is determined by cusp conditions, and $b$ is obtained by energy
@ -997,7 +1001,7 @@ when the trial wave function is expanded on more than a few million
determinants. determinants.
At the RS-DFT-CIPSI level, we can remark that with the triple-zeta At the RS-DFT-CIPSI level, we can remark that with the triple-zeta
basis set the MAE are larger for $\mu=1$~bohr$^{-1}$ than for the basis set the MAE are larger for $\mu=1$~bohr$^{-1}$ than for the
FCI. For the largest systems, as shown in figure~\ref{fig:g2-ndet} FCI. For the largest systems, as shown in Fig.~\ref{fig:g2-ndet}
there are many systems which did not reach the threshold there are many systems which did not reach the threshold
$\EPT<1$~m\hartree{}, and the number of determinants exceeded $\EPT<1$~m\hartree{}, and the number of determinants exceeded
10~million so the calculation stopped. In this regime, there is a 10~million so the calculation stopped. In this regime, there is a
@ -1024,7 +1028,7 @@ $\mu=0.5$~bohr$^{-1}$ with the quadruple-zeta basis set.
Searching for the optimal value of $\mu$ may be too costly, so we have Searching for the optimal value of $\mu$ may be too costly, so we have
computed the MAD, MSE and RMSD for fixed values of $\mu$. The results computed the MAD, MSE and RMSD for fixed values of $\mu$. The results
are illustrated in figure~\ref{fig:g2-dmc}. As seen on the figure and are illustrated in figure~\ref{fig:g2-dmc}. As seen on the figure and
in table~\ref{tab:mad}, the best choice for a fixed value of $\mu$ is in Table~\ref{tab:mad}, the best choice for a fixed value of $\mu$ is
0.5~bohr$^{-1}$ for all three basis sets. It is the value for which 0.5~bohr$^{-1}$ for all three basis sets. It is the value for which
the MAE (3.74(35), 2.46(18) and 2.06(35) kcal/mol) and RMSD (4.03(23), the MAE (3.74(35), 2.46(18) and 2.06(35) kcal/mol) and RMSD (4.03(23),
3.02(06) and 2.74(13)~kcal/mol) are minimal. Note that these values 3.02(06) and 2.74(13)~kcal/mol) are minimal. Note that these values