modifications in first part of results
This commit is contained in:
parent
c043fc80d3
commit
bce0c03c68
@ -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
|
||||
calculations to the FCI energy. In this work, we target the chemical
|
||||
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.
|
||||
The parameter $\mu$ controls the range of the separation, and allows
|
||||
to go continuously from the KS Hamiltonian ($\mu=0$) to
|
||||
the FCI Hamiltonian ($\mu \to \infty$).
|
||||
the FCI Hamiltonian ($\mu = \infty$).
|
||||
|
||||
To rigorously connect WFT and DFT, the universal
|
||||
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,
|
||||
$w_{\text{ee}}^{\text{lr},\mu=0}(r) = 0$, and thus
|
||||
RS-DFT reduces to standard KS-DFT and $\Psi^\mu$
|
||||
is the KS determinant. For $\mu\to\infty$, the long-range
|
||||
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
|
||||
@ -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 $\hat{H}^{\mu\,(k)}$
|
||||
parameterized using the current one-electron density $n^{(k)}$.
|
||||
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
|
||||
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
|
||||
$\hat{H}^{\mu\,(k,l)}$ 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
|
||||
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}
|
||||
@ -407,7 +407,7 @@ and correlation functionals defined in
|
||||
Ref.~\onlinecite{GolWerStoLeiGorSav-CP-06} (see also
|
||||
Refs.~\onlinecite{TouColSav-JCP-05,GolWerSto-PCCP-05}).
|
||||
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
|
||||
described in Ref.~\onlinecite{Applencourt_2018}.
|
||||
|
||||
@ -431,7 +431,8 @@ with a time step of $5 \times 10^{-4}$ a.u.
|
||||
|
||||
%%% TABLE I %%%
|
||||
\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}
|
||||
\centering
|
||||
\begin{ruledtabular}
|
||||
@ -461,9 +462,9 @@ with a time step of $5 \times 10^{-4}$ a.u.
|
||||
\begin{figure}
|
||||
\centering
|
||||
\includegraphics[width=\columnwidth]{h2o-dmc.pdf}
|
||||
\caption{Fixed-node energies of \ce{H2O} for different
|
||||
values of $\mu$, using the srLDA or srPBE density
|
||||
functionals to build the trial wave function.}
|
||||
\caption{Fixed-node energy of \ce{H2O} as a function
|
||||
of $\mu$ for various levels of theory to generate
|
||||
the trial wave function.}
|
||||
\label{fig:h2o-dmc}
|
||||
\end{figure}
|
||||
%%% %%% %%% %%%
|
||||
@ -481,31 +482,34 @@ parameters having an impact on the nodal surface fixed (\titou{such as ??}).
|
||||
\subsection{Fixed-node energy of $\Psi^\mu$}
|
||||
\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
|
||||
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 ($\mu=0$):
|
||||
a gain 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.
|
||||
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.
|
||||
Coming now to the nodes of the trial wave functions $\Psi^{\mu}$ with
|
||||
intermediate values of $\mu$, Fig.~\ref{fig:h2o-dmc} shows that
|
||||
a smooth behaviour is obtained:
|
||||
starting from $\mu=0$ (\textit{i.e.} the KS determinant),
|
||||
the FN-DMC error is reduced continuously until it reaches a minimum for an optimal value of $\mu$,
|
||||
and then the FN-DMC error raises until it reaches the $\mu=\infty$ limit (\textit{i.e.} the FCI wave function).
|
||||
For instance, with respect to the FN-DMC energy of the FCI trial wave function in the double zeta basis set,
|
||||
with the optimal value of $\mu$, one can obtain a lowering of the
|
||||
FN-DMC energy of $2.6 \pm 0.7$~m\hartree{}, with an optimal value at
|
||||
$\mu=1.75$~bohr$^{-1}$.
|
||||
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 $\mu$ (which is obviously basis set and functional dependent),
|
||||
and then the FN-DMC error raises until it reaches the $\mu=\infty$ limit (\ie, the FCI wave function).
|
||||
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$^{-1}$.
|
||||
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 $\mu$. Eventually, the nodes
|
||||
of the wave functions $\Psi^\mu$ obtained with the short-range
|
||||
LDA exchange-correlation functional give very similar FN-DMC energies with respect
|
||||
to those obtained with the short-range PBE functional, even if the
|
||||
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{}.
|
||||
|
||||
\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 }
|
||||
\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 $\Psi^J$.
|
||||
Then, we can easily compare $\Psi^\mu$ and $\Psi^J$ as they are developed
|
||||
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,
|
||||
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 $\Psi^J$.
|
||||
|
||||
%%% FIG 3 %%%
|
||||
@ -663,7 +667,7 @@ $\mu=0.5$ and $\mu=1$~bohr$^{-1}$, constently with the FN-DMC energies
|
||||
and the overlap curve.
|
||||
|
||||
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}),
|
||||
one can notice that the differences with respect to the usual Hamiltonian come
|
||||
from the non-divergent two-body interaction $\hat{W}_{\text{ee}}^{\text{lr},\mu}$
|
||||
@ -694,7 +698,7 @@ As a conclusion of the first part of this study, we can notice that:
|
||||
\item numerical experiments (overlap $\braket{\Psi^\mu}{\Psi^J}$,
|
||||
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,
|
||||
\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
|
||||
and transcorrelated approaches deal with an effective non-divergent
|
||||
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
|
||||
be expressed as
|
||||
\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}
|
||||
The parameter
|
||||
$a$ 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.
|
||||
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
|
||||
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
|
||||
$\EPT<1$~m\hartree{}, and the number of determinants exceeded
|
||||
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
|
||||
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
|
||||
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
|
||||
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
|
||||
|
Loading…
Reference in New Issue
Block a user