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25
Manuscript/rsdft-cipsi-qmc.bib
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@ -692,3 +692,28 @@ note={Gaussian Inc. Wallingford CT}
publisher = {American Physical Society},
doi = {10.1103/PhysRevA.70.062505}
}
@misc{Scemama_2015,
author = {Scemama, Anthony and Giner, Emmanuel and
Applencourt, Thomas and Caffarel, Michel},
title = {{QMC using very large configuration interaction-type expansions}},
howpublished = {Pacifichem, Advances in Quantum Monte Carlo},
year = {2015},
month = {Dec},
doi = {10.13140/RG.2.1.3187.9766}
}
@article{Tenno_2004,
author = {Ten-no, Seiichiro},
title = {{Explicitly correlated second order perturbation theory: Introduction of a
rational generator and numerical quadratures}},
journal = {J. Chem. Phys.},
volume = {121},
number = {1},
pages = {117--129},
year = {2004},
month = {Jul},
issn = {0021-9606},
publisher = {American Institute of Physics},
doi = {10.1063/1.1757439}
}

117
Manuscript/rsdft-cipsi-qmc.tex
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@ -465,7 +465,7 @@ $\mu=\infty$.
\label{fig:overlap}
\end{figure}
This data confirms that RSDFT/CIPSI can give improved CI coefficients
This data confirms that RSDFT-CIPSI can give improved CI coefficients
with small basis sets, similarly to the common practice of
re-optimizing the wave function in the presence of the Jastrow
factor. To confirm that the introduction of RS-DFT has the same impact
@ -490,8 +490,7 @@ $1$. Nevertheless, a slight maximum is obtained for
$\mu=5$~bohr$^{-1}$.
\section{Atomization energy benchmarks}
\section{Atomization energies}
\label{sec:atomization}
Atomization energies are challenging for post-Hartree-Fock methods
@ -499,40 +498,40 @@ because their calculation requires a perfect balance in the
description of atoms and molecules. Basis sets used in molecular
calculations are atom-centered, so they are always better adapted to
atoms than molecules and atomization energies usually tend to be
underestimated.
underestimated with variational methods.
In the context of FN-DMC calculations, the nodal surface is imposed by
the trial wavefunction which is expanded on an atom-centered basis
set. So we expect the fixed-node error to be also related to the basis
set incompleteness error.
set. So we expect the fixed-node error to be also tightly related to
the basis set incompleteness error.
Increasing the size of the basis set improves the description of
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.
Another important feature required to get accurate atomization
energies is size-extensivity, since the numbers of correlated electrons
in the isolated atoms are different from the number of correlated
electrons in the molecule.
\subsection{Size-consistence}
An extremely important feature required to get accurate
atomization energies is size-extensivity, since the numbers of
correlated electron pairs in the isolated atoms are different from
those of the molecules.
In the context of selected CI calculations, when the variational energy is
extrapolated to the FCI energy\cite{Holmes_2017} there is obviously no
extrapolated to the FCI energy\cite{Holmes_2017} there is no
size-consistence error. But when the selected wave function is used
for as a reference for post-Hartree-Fock methods or QMC calculations,
there is a residual size-consistence error originating from the
truncation of the determinant space.
as a reference for post-Hartree-Fock methods (sCI+PT2 for instance)
or for QMC calculations, there is a residual size-consistence error
originating from the truncation of the determinant space.
% Invariance with m_s
QMC calculations can be made size-consistent by extrapolating the
QMC energies can be made size-consistent by extrapolating the
FN-DMC energy to estimate the energy obtained with the FCI as a trial
wave function.\cite{Scemama_2018,Scemama_2018b} Alternatively, the
size-consistence error can be reduced by choosing the number of
selected determinants such that the sum of the PT2 corrections on the
atoms is equal to the PT2 correction of the molecule, enforcing that
the variational dissociation potential energy surface (PES) is
the variational potential energy surface (PES) is
parallel to the perturbatively corrected PES, which is an accurate
estimate of the FCI PES.\cite{Giner_2015}
Another source of size-consistence error in QMC calculation may originate
Another source of size-consistence 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
@ -540,11 +539,12 @@ be expressed as
\begin{equation}
J_\text{ee} = \sum_i \sum_{j<i} \frac{a r_{ij}}{1 + b r_{ij}}.
\end{equation}
The parameter
$a$ is determined by cusp conditions, and $b$ is obtained by energy
or variance minimization.\cite{Coldwell_1977,Umrigar_2005}
One can easily see that this parameterization of the two-body
interation is not size-consistent. The dissociation of a
heteroatomic diatomic molecule $AB$ with a parameter $b_{AB}$
interation is not size-consistent: the dissociation of a
diatomic molecule $AB$ with a parameter $b_{AB}$
will lead to two different two-body Jastrow factors on each atom, each
with its own optimal value $b_A$ and $b_B$. To remove the
size-consistence error on a PES using this ansätz for $J_\text{ee}$,
@ -552,18 +552,21 @@ one needs to impose that the parameters of $J_\text{ee}$ are fixed:
$b_A = b_B = b_{AB}$.
When pseudopotentials are used in a QMC calculation, it is common
practice to localize the pseudopotential on the complete wave
function. If the wave function is not size-consistent, so will be the
locality approximation. Recently, the determinant localization
approximation was introduced.\cite{Zen_2019} This approximation
consists in removing the Jastrow factor from the wave function on
which the pseudopotential is localized.
practice to localize the non-local part of the pseudopotential on the
complete wave function (determinantal component and Jastrow).
If the wave function is not size-consistent,
so will be the locality approximation. Within, the determinant
localization approximation,\cite{Zen_2019} the Jastrow factor is
removed from the wave function on which the pseudopotential is localized.
The great advantage of this approximation is that the FN-DMC energy
within this approximation only depends on the parameters of the
determinantal component. Using a size-inconsistent Jastrow factor, or
a non-optimal Jastrow factor will not introduce an additional
size-consistence error in FN-DMC calculations, although it will
reduce the statistical errors by reducing the variance of the local energy.
only depends on the parameters of the determinantal component. Using a
size-inconsistent Jastrow factor, or a non-optimal Jastrow factor will
not introduce an additional size-consistence error in FN-DMC
calculations, although it will reduce the statistical errors by
reducing the variance of the local energy. Moreover, the
integrals involved in the pseudo-potential are computed analytically
and the computational cost of the pseudo-potential is dramatically
reduced (for more detail, see Ref.~\onlinecite{Scemama_2015}).
The energy computed within density functional theory is extensive, and
@ -572,18 +575,40 @@ as it is a mean-field method the convergence to the complete basis set
the calculation of atomization energies, especially with small basis
sets, but going to the CBS limit will converge to biased atomization
energies because of the use of approximate density functionals.
On the other hand, the convergence of the FCI energies to the CBS
limit will be slower because of the description of short-range electron
correlation with atom-centered functions, but ultimately the exact
energy will be reached.
\subsection{Spin-invariance}
Closed-shell molecules usually dissociate into open-shell
fragments. To get reliable atomization energies, it is important to
have a theory which is of comparable quality for open-shell and
closed-shell systems.
FCI wave functions are invariant with respect to the spin quantum
number $m_s$, but the introduction of a
Jastrow factor breaks this spin-invariance if the parameters
for the same-spin electron pairs are different from those
for the opposite-spin pairs.\cite{Tenno_2004}
Again, using pseudo-potentials this error is transferred in the DMC
calculation unless the determinant localization approximation is used.
To check that the RSDFT-CIPSI are spin-invariant, we compute the
FN-DMC energies of the ?? dimer with different values of the spin
quantum number $m_s$.
\subsection{Benchmark}
The 55 molecules of the benchmark for the Gaussian-1
theory\cite{Pople_1989,Curtiss_1990} were chosen to test the quality
of the RSDFT-CIPSI trial wave functions for energy differences.
%\begin{squeezetable}
\begin{table*}
\caption{Mean absolute error (MAE), mean signed errors (MSE) and
@ -665,7 +690,28 @@ We could have obtained
single-determinant wave functions by using the natural orbitals of a first
\section{Conclusion}
We have seen that introducing short-range correation via
a range-separated Hamiltonian in a full CI expansion yields improved
nodes, especially with small basis sets. The effect is similar to the
effect of re-optimizing the CI coefficients in the presence of a
Jastrow factor, but without the burden of performing a stochastic
optimization.
The proposed procedure provides a method to optimize the
FN-DMC energy via a single parameter, namely the range-separation
parameter $\mu$. The size-consistence error is controlled, as well as the
invariance with respect to the spin projection $m_s$.
Finding the optimal value of $\mu$ gives the lowest FN-DMC energies
within basis set. However, if one wants to compute an energy
difference, one should not minimize the
FN-DMC energies of the reactants independently. It is preferable to
choose a value of $\mu$ for which the fixed-node errors are well
balanced, leading to a good cncellation of errors. We found that a
value of $\mu=0.5$~bohr${^-1}$ is the value where the errors are the
smallest. Moreover, such a small value of $\mu$ gives extermely
compact wave functions, making this recipe a good candidate for
accurate calcultions of large systems with a multi-reference character.
@ -712,9 +758,8 @@ Simulation of Functional Materials.
\item basis set dependent: $\mu \rightarrow \infty$ when $\mathcal{B}\rightarrow \text{CBS}$
\item large wave functions
\end{itemize}
\begin{itemize}
\item plot $N_{det}$ en fonction de $\mu$
\end{itemize}
\item Invariance with m_s
\end{itemize}
\end{enumerate}