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59
Manuscript/rsdft-cipsi-qmc.bib
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@ -626,4 +626,63 @@ note={Gaussian Inc. Wallingford CT}
issn = {0021-9606},
publisher = {American Institute of Physics},
doi = {10.1063/1.1487829}
}
@article{Ludovicy_2019,
author = {Ludovicy, Jil and Mood, Kaveh Haghighi and
Lüchow, Arne},
title = {{Full Wave Function Optimization with Quantum Monte Carlo{---}A
Study of the Dissociation Energies of ZnO, FeO, FeH, and CrS}},
journal = {J. Chem. Theory Comput.},
volume = {15},
number = {10},
pages = {5221--5229},
year = {2019},
month = {Oct},
issn = {1549-9618},
publisher = {American Chemical Society},
doi = {10.1021/acs.jctc.9b00241}
}
@article{HaghighiMood_2017,
author = {Haghighi Mood, Kaveh and Lüchow, Arne},
title = {{Full Wave Function Optimization with Quantum Monte Carlo and Its Effect on
the Dissociation Energy of FeS}},
journal = {J. Phys. Chem. A},
volume = {121},
number = {32},
pages = {6165--6171},
year = {2017},
month = {Aug},
issn = {1089-5639},
publisher = {American Chemical Society},
doi = {10.1021/acs.jpca.7b05798}
}
@article{Scemama_2006,
author = {Scemama, Anthony and Filippi, Claudia},
title = {{Simple and efficient approach to the optimization of correlated wave functions}},
journal = {Phys. Rev. B},
volume = {73},
number = {24},
pages = {241101},
year = {2006},
month = {Jun},
issn = {2469-9969},
publisher = {American Physical Society},
doi = {10.1103/PhysRevB.73.241101}
}
@article{Filippi_2000,
author = {Filippi, Claudia and Fahy, Stephen},
title = {{Optimal orbitals from energy fluctuations in correlated wave functions}},
journal = {J. Chem. Phys.},
volume = {112},
number = {8},
pages = {3523--3531},
year = {2000},
month = {Feb},
issn = {0021-9606},
publisher = {American Institute of Physics},
doi = {10.1063/1.480507}
}

114
Manuscript/rsdft-cipsi-qmc.tex
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@ -103,14 +103,6 @@ The favorable scaling of QMC, its very low memory requirements and
its adequation with massively parallel architectures make it a
serious alternative for high-accuracy simulations on large systems.
Different molecular orbitals can be chosen to build the single Slater
determinant. Three main options are commonly chosen: Hartree-Fock (HF),
Kohn-Sham (KS) or natural (NO) orbitals of a correlated wave function.
The nodal surfaces obtained with the KS determinant are in general
better than those obtained with the HF determinant,\cite{Per_2012} and
of comparable quality to those obtained with a Slater determinant
built with NOs.\cite{Wang_2019}
As it is not possible to minimize directly the DMC energy with respect
to the variational parameters of the trial wave function, the
fixed-node approximation is much more difficult to control than the
@ -118,10 +110,11 @@ 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 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 DMC.\cite{Petruzielo_2012}
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
DMC.\cite{Petruzielo_2012}
Another approach consists in considering the DMC method as a
\emph{post-FCI method}. The trial wave function is obtained by
@ -138,8 +131,8 @@ exponential scaling of the size of the trial wave function.
Extrapolation techniques have been used to estimate the DMC energies
obtained with FCI wave functions,\cite{Scemama_2018} and other authors
have used a combination of the two approaches where highly truncated
CIPSI trial wave functions are re-optimized under the presence of a
Jastrow factor to keep the number of determinants
CIPSI trial wave functions are re-optimized in VMC under the presence
of a Jastrow factor to keep the number of determinants
small,\cite{Giner_2016} and where the consistency between the
different wave functions is kept by imposing a constant energy
difference between the estimated FCI energy and the variational energy
@ -189,41 +182,76 @@ within this context.
\section{Combining range-separated DFT with CIPSI}
\label{sec:rsdft-cipsi}
Starting from a Hartree-Fock determinant in a small basis set,
we have seen that we can systematically improve the trial wave
function in two directions. The first one is by increasing the
size of the atomic basis set, and the second one is by
increasing the determinant expansion towards the FCI limit.
A third direction of improvement exists. It is the path which connects
the Hartree-Fock determinant to the Kohn-Sham determinant, which
usually has a better nodal surface. This third path is obtained by
the so-called \emph{range-separated} DFT framework which splits the
electron-electron interaction in a long-range part and a short-range
part. HELP MANU!
In single-determinant DMC calculations, the degrees of freedom used to
reduce the fixed-node error are the molecular orbitals on which the
Slater determinant is built.
Different molecular orbitals can be chosen:
Hartree-Fock (HF), Kohn-Sham (KS), natural (NO) orbitals of a
correlated wave function, or orbitals optimized under the
presence of a Jastrow factor.
The nodal surfaces obtained with the KS determinant are in general
better than those obtained with the HF determinant,\cite{Per_2012} and
of comparable quality to those obtained with a Slater determinant
built with NOs.\cite{Wang_2019} Orbitals obtained in the presence
of a Jastrow factor are generally superior to KS
orbitals.\cite{Filippi_2000,Scemama_2006,HaghighiMood_2017,Ludovicy_2019}
The description of electron correlation within DFT is very different
from correlated methods.
In DFT, one solves a mean field problem with a modified potential
incorporating the effects of electron correlation, whereas in
correlated methods the real Hamiltonian is used and the
electron-electron interactions are considered.
Nevertheless, as the orbitals are one-electron functions,
the procedure of orbital optimization in the presence of the
Jastrow factor can be interpreted as a self-consistent field procedure
with an effective Hamiltonian,\cite{Filippi_2000} similarly to DFT.
So DFT can be viewed as a very cheap way of introducing the effect of
correlation in the parameters determining the nodal surface. But in
the general case, even at the complete basis set limit a fixed-node
error will remain because the single-determinant ans\"atz does not
have enough freedom to describe the exact nodal surface.
If one wants to have to exact CBS limit, a multi-determinant
parameterization of the wave functions is required.
\subsection{CIPSI}
\emph{Configuration interaction using a perturbative selection made
iteratively} (CIPSI)\cite{Huron_1973} belongs to the class of selected
CI methods, whose goal is to build compressed configuration interaction
(CI) wave functions with a controlled accuracy, together with an estimate
of the associated eigenvalue obtained with perturbation theory.
Beyond the single-determinant representation, the optimal wave
function is obtained with FCI. FCI is often presented as a
\emph{post-Hartree-Fock} method, which implies that there is a
continuous connection between the Hartree-Fock and FCI wave functions.
Selected CI methods take a very short path between the Hartree-Fock
determinant and the FCI wave function by increasing iteratively the
number of determinants on which the wave function is expanded and
producing relatively compact wave functions.
Using a conventional iterative diagonalization method such as
Davidson's,\cite{Davidson_1975} the size of the CI space is
pre-defined, and at each iteration the length of the vectors is equal
to the size of the CI space. Within selected CI methods, at each
iteration the size of the determinant space is increased such that
one keeps only the most important determinants that will lead to an
accurate description of the state of interest. The lowest eigenpairs
are extracted from the CI matrix expressed in this determinant
subspace, and the CI energy can be estimated by computing a
second-order perturbative correction (PT2) to the variational energy
computed in the complete CI space. The magnitude of the PT2 correction
is a measure of the distance to the exact eigenvalue, and is an
adjustable parameter to control the quality of the wave function.
Davidson's,\cite{Davidson_1975} the dimension of the configuration
interaction (CI) space is pre-defined, so the dimensions of the CI
vectors grow rapidly towards the size of the CI space. Within selected
CI methods, at each iteration the size of the determinant space is
increased such that one keeps only the most important determinants
that will lead to an accurate description of the state of
interest. The lowest eigenpairs are extracted from the CI matrix
expressed in this determinant subspace, and the CI energy can be
estimated by computing a second-order perturbative correction (PT2) to
the variational energy computed in the complete CI space. The
magnitude of the PT2 correction is a measure of the distance to the
exact eigenvalue, and is an adjustable parameter to control the
quality of the wave function.
Within the \emph{Configuration interaction using a perturbative
selection made iteratively} (CIPSI)\cite{Huron_1973} method, the PT2
correction is computed along with the determinant selection. So the
magnitude of the PT2 correction can be made the only parameter of the
algorithm, and we choose this parameter as the convergence criterion.
Considering that the perturbatively corrected energy is a reliable
estimate of the FCI energy, using a fixed value of the PT2 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 $|\EPT| < 1$~mE$_h$.
\subsection{Range-separated DFT in a nutshell}