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Manuscript/rsdft-cipsi-qmc.tex
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@ -69,7 +69,7 @@
\begin{abstract} \begin{abstract}
By combining density-functional theory (DFT) and wave function theory (WFT) via the range separation (RS) of the interelectronic Coulomb operator, we obtain accurate fixed-node diffusion Monte Carlo (FN-DMC) energies with compact multideterminant trial wave functions. By combining density-functional theory (DFT) and wave function theory (WFT) via the range separation (RS) of the interelectronic Coulomb operator, we obtain accurate fixed-node diffusion Monte Carlo (FN-DMC) energies with compact multi-determinant trial wave functions.
These compact trial wave functions are generated via the diagonalization of the RS-DFT Hamiltonian. These compact trial wave functions are generated via the diagonalization of the RS-DFT Hamiltonian.
In particular, we combine here short-range correlation functionals with selected configuration interaction (SCI). In particular, we combine here short-range correlation functionals with selected configuration interaction (SCI).
As the WFT method is relieved from describing the short-range part of the correlation hole around the electron-electron coalescence points, the number of determinants in the trial wave function required to reach a given accuracy is significantly reduced as compared to a conventional SCI calculation. As the WFT method is relieved from describing the short-range part of the correlation hole around the electron-electron coalescence points, the number of determinants in the trial wave function required to reach a given accuracy is significantly reduced as compared to a conventional SCI calculation.
@ -124,7 +124,8 @@ the FN-DMC energy associated with a given trial wave function is an upper
bound to the exact energy, and the latter is recovered only when the bound to the exact energy, and the latter is recovered only when the
nodes of the trial wave function coincide with the nodes of the exact nodes of the trial wave function coincide with the nodes of the exact
wave function. wave function.
The trial wave function in FN-DMC is then a key ingredient which dictates via the quality of its nodal surface the accuracy of the resulting energy and properties. The trial wave function, which can be single- or multi-determinantal in nature depending on the type of correlation at play, is then a key ingredient dictating via the quality of its nodal surface the accuracy of the resulting energy and properties.
The polynomial scaling of its computational cost with respect to the number of electrons and with the size The polynomial scaling of its computational cost with respect to the number of electrons and with the size
of the trial wave function makes the FN-DMC method particularly attractive. of the trial wave function makes the FN-DMC method particularly attractive.
This favorable scaling, its very low memory requirements and This favorable scaling, its very low memory requirements and
@ -135,6 +136,19 @@ those obtained with the FCI method in computationally tractable basis
sets because the constraints imposed by the fixed-node approximation sets because the constraints imposed by the fixed-node approximation
are less severe than the constraints imposed by the finite-basis are less severe than the constraints imposed by the finite-basis
approximation. approximation.
However, because it is not possible to minimize directly the FN-DMC energy with respect
to the linear and non-linear parameters of the trial wave function, the
fixed-node approximation is much more difficult to control than the
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 bulk of the dynamical correlation.
%electron-electron cusp and the short-range correlation effects.
The trial 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. \cite{Umrigar_2005,Scemama_2006c,Umrigar_2007,Toulouse_2007,Toulouse_2008}
Using this technique, it has been shown that the chemical accuracy could be reached within
FN-DMC.\cite{Petruzielo_2012}
%However, it is usually harder to control the FN error in DMC, and this %However, it is usually harder to control the FN error in DMC, and this
@ -150,26 +164,14 @@ correlated systems, such as organic molecules near their equilibrium
geometry, is usually well represented with a single Slater geometry, is usually well represented with a single Slater
determinant. This feature is in part responsible for the success of determinant. This feature is in part responsible for the success of
DFT and coupled cluster theory. DFT and coupled cluster theory.
DMC with a single-determinant trial wave function can be used as a Likewise, DMC with a single-determinant trial wave function can be used as a
single-reference post-Hartree-Fock method, with an accuracy comparable single-reference post-Hartree-Fock method for weakly correlated systems, with an accuracy comparable
to coupled cluster.\cite{Dubecky_2014,Grossman_2002} to coupled cluster.\cite{Dubecky_2014,Grossman_2002}
This approach obviously fails in the presence of strong correlation, like in
As it is not possible to minimize directly the FN-DMC energy with respect transition metal complexes, low-spin open-shell systems, and covalent bond breaking situations which cannot be even qualitatively described by a single electronic configuration.
to the linear and non-linear parameters of the trial wave function, the In such cases, a viable alternative is to consider the FN-DMC method as a
fixed-node approximation is much more difficult to control than the ``post-FCI'' method. The multi-determinant trial wave function is then produced by
finite-basis approximation. approaching FCI with a SCI method such as CIPSI for instance.\cite{Giner_2013,Giner_2015,Caffarel_2016_2}
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
Monte Carlo (VMC) in the presence of the Jastrow factor and the nodal
surface is expected to be improved. \cite{Umrigar_2005,Scemama_2006c,Umrigar_2007,Toulouse_2007,Toulouse_2008}
Using this technique, it has been shown that the chemical accuracy could be reached within
FN-DMC.\cite{Petruzielo_2012}
Another approach consists in considering the FN-DMC method as a
``post-FCI'' method. The trial wave function is obtained by
approaching the FCI with a SCI method such as CIPSI for instance.\cite{Giner_2013,Giner_2015,Caffarel_2016_2}
When the basis set is enlarged, the trial wave function gets closer to When the basis set is enlarged, the trial wave function gets closer to
the exact wave function, so we expect the nodal surface to be the exact wave function, so we expect the nodal surface to be
improved.\cite{Caffarel_2016} improved.\cite{Caffarel_2016}
@ -194,12 +196,12 @@ applicable for large systems with a multi-configurational character is
still an active field of research. The present paper falls still an active field of research. The present paper falls
within this context. within this context.
The central idea of the present work, and the launch-pad for the remainder of this study, is that one can combine the various strengths of these three philosophies in order to create a hybrid method with more attractive properties. The central idea of the present work, and the launch-pad for the remainder of this study, is that one can combine the various strengths of these three philosophies in order to create a hybrid method with more attractive properties.
In particular, we show here that one can combine FB-FCI and KS-DFT via the range separation (RS) of the interelectronic Coulomb operator to obtain accurate FN-DMC energies with compact multideterminant trial wave functions. In particular, we show here that one can combine FB-FCI and KS-DFT via the range separation (RS) of the interelectronic Coulomb operator to obtain accurate FN-DMC energies with compact multi-determinant trial wave functions.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Combining CIPSI with RS-DFT} \section{Combining WFT and DFT}
\label{sec:rsdft-cipsi} \label{sec:rsdft-cipsi}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -326,7 +328,7 @@ $\hat{W}_\text{ee}^{\text{lr},\mu}$ the long-range
electron-electron interaction, electron-electron interaction,
$n_\Psi$ the one-electron density associated with $\Psi$, $n_\Psi$ the one-electron density associated with $\Psi$,
and $\hat{V}_{\text{ne}}$ the electron-nucleus potential. and $\hat{V}_{\text{ne}}$ the electron-nucleus potential.
The minimizing multideterminant wave function $\Psi^\mu$ The minimizing multi-determinant wave function $\Psi^\mu$
can be determined by the self-consistent eigenvalue equation can be determined by the self-consistent eigenvalue equation
\begin{equation} \begin{equation}
\label{rs-dft-eigen-equation} \label{rs-dft-eigen-equation}
@ -528,7 +530,7 @@ tens of m\hartree{}.
An other important aspect here regards the compactness of the trial wave functions $\Psi^\mu$: An other important aspect here regards the compactness of the trial wave functions $\Psi^\mu$:
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 $\mu = \infty$. 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 of this numerical study is that RS-DFT trial wave functions can yield a lower fixed-node energy with more compact multideterminant expansion as compared to FCI. The take-home message of this numerical study is that RS-DFT trial wave functions can yield a lower fixed-node energy with more compact multi-determinant expansion as compared to FCI.
%====================================================== %======================================================
\subsection{Link between RS-DFT and Jastrow factors } \subsection{Link between RS-DFT and Jastrow factors }
@ -537,7 +539,7 @@ The take-home message of this numerical study is that RS-DFT trial wave function
The data presented in Sec.~\ref{sec:fndmc_mu} evidence that, in a finite basis, RS-DFT can provide The data presented in Sec.~\ref{sec:fndmc_mu} evidence that, in a finite basis, RS-DFT can provide
trial wave functions with better nodes than FCI wave function. trial wave functions with better nodes than FCI wave function.
Such behaviour can be directly compared to the common practice of Such behaviour can be directly compared to the common practice of
re-optimizing the multideterminant part of a trial wave function $\Psi$ (the so-called Slater part) in the presence of the exponentiated Jastrow factor $e^J$. \cite{Umrigar_2005,Scemama_2006c,Umrigar_2007,Toulouse_2007,Toulouse_2008} re-optimizing the multi-determinant part of a trial wave function $\Psi$ (the so-called Slater part) in the presence of the exponentiated Jastrow factor $e^J$. \cite{Umrigar_2005,Scemama_2006c,Umrigar_2007,Toulouse_2007,Toulouse_2008}
Hence, in the present paragraph, we would like to elaborate further on the link between RS-DFT Hence, in the present paragraph, we would like to elaborate further on the link between RS-DFT
and wave function optimization in the presence of a Jastrow factor. and wave function optimization in the presence of a Jastrow factor.
For simplicity in the comparison, the molecular orbitals and the Jastrow For simplicity in the comparison, the molecular orbitals and the Jastrow