add summary paragraph and random correction in Sec V
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@ -250,6 +250,14 @@ within this context.
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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 WFT, DFT, and QMC in order to create a new hybrid method with more attractive features and higher accuracy.
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In particular, we show here that one can combine CIPSI and KS-DFT via the range separation (RS) of the interelectronic Coulomb operator \cite{Sav-INC-96a,Toulouse_2004} to obtain accurate FN-DMC energies with compact multi-determinant trial wave functions.
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The present manuscript is organized as follows.
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In Sec.~\ref{sec:rsdft-cipsi}, we provide theoretical details about the CIPSI algorithm (Sec.~\ref{sec:CIPSI}) and range-separated DFT (Sec.~\ref{sec:rsdft}).
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Computational details are reported in Sec.~\ref{sec:comp-details}.
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In Sec.~\ref{sec:mu-dmc}, we discuss the influence of the range-separation parameter on the fixed-node error as well as the link between RS-DFT and Jastrow factors.
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Section \ref{sec:atomization} examines the performance of the present scheme for the atomization energies of the Gaussian-1 set of molecules.
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Finally, we draw our conclusion in Sec.~\ref{sec:conclusion}.
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Unless otherwise stated, atomic units are used.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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@ -259,6 +267,7 @@ Unless otherwise stated, atomic units are used.
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%====================
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\subsection{The CIPSI algorithm}
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\label{sec:CIPSI}
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%====================
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Beyond the single-determinant representation, the best
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multi-determinant wave function one can wish for --- in a given basis set --- is the FCI wave function.
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@ -538,7 +547,7 @@ The take-home message of this first numerical study is that RS-DFT trial wave fu
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This is a key result of the present study.
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%======================================================
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\subsection{Link between RS-DFT and Jastrow factors }
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\subsection{Link between RS-DFT and Jastrow factor}
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\label{sec:rsdft-j}
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%======================================================
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The data presented in Sec.~\ref{sec:fndmc_mu} evidence that, in a finite basis, RS-DFT can provide
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@ -645,7 +654,7 @@ This is yet another key result of the present study.
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%%% TABLE II %%%
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\begin{table}
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\caption{\ce{H2O}, double-zeta basis set. Integrated on-top pair density $\expval{ P }$
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\caption{\ce{H2O}, double-$\zeta$ basis set. Integrated on-top pair density $\expval{ P }$
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for $\Psi^J$ and $\Psi^\mu$ with different values of $\mu$.
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\titou{Please remove table and merge data in Fig. 4.}}
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\label{tab:table_on_top}
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@ -760,16 +769,16 @@ As a conclusion of the first part of this study, we can highlight the following
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Atomization energies are challenging for post-HF methods
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because their calculation requires a perfect balance in the
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description of atoms and molecules. Basis sets used in molecular
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calculations are atom-centered, so they are always better adapted to
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description of atoms and molecules. The mainstream one-electron basis sets employed in molecular
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calculations are atom-centered, so they are, by construction, better adapted to
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atoms than molecules and atomization energies usually tend to be
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underestimated by variational methods.
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In the context of FN-DMC calculations, the nodal surface is imposed by
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the trial wavefunction which is expanded on an atom-centered basis
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set, so we expect the fixed-node error to be also tightly related to
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the trial wavefunction which is expanded in an atom-centered basis
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set, so we expect the fixed-node error to be also intimately related to
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the basis set incompleteness error.
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Increasing the size of the basis set improves the description of
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the density and of electron correlation, but also reduces the
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the density and of the electron correlation, but also reduces the
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imbalance in the quality of the description of the atoms and the
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molecule, leading to more accurate atomization energies.
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@ -781,9 +790,9 @@ An extremely important feature required to get accurate
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atomization energies is size-consistency (or strict separability),
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since the numbers of correlated electron pairs in the isolated atoms
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are different from those of the molecules.
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The energy computed within density functional theory is size-consistent, and
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as it is a mean-field method the convergence to the complete basis set
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(CBS) limit is relatively fast. Hence, DFT methods are very well adapted to
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The energy computed within DFT is size-consistent, and
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as it is a mean-field method the convergence to the CBS limit
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is relatively fast. Hence, DFT methods are very well adapted to
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the calculation of atomization energies, especially with small basis
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sets. But going to the CBS limit will converge to biased atomization
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energies because of the use of approximate density functionals.
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@ -793,10 +802,10 @@ the FCI energies to the CBS limit is much slower because of the
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description of short-range electron correlation using atom-centered
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functions. But ultimately the exact energy will be reached.
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In the context of selected CI calculations, when the variational energy is
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extrapolated to the FCI energy\cite{Holmes_2017} there is no
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In the context of SCI calculations, when the variational energy is
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extrapolated to the FCI energy \cite{Holmes_2017} there is no
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size-consistency error. But when the truncated SCI wave function is used
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as a reference for post-Hartree-Fock methods such as SCI+PT2
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as a reference for post-HF methods such as SCI+PT2
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or for QMC calculations, there is a residual size-consistency error
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originating from the truncation of the wave function.
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@ -820,12 +829,12 @@ $a$ is determined by cusp conditions, and $b$ is obtained by energy
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or variance minimization.\cite{Coldwell_1977,Umrigar_2005}
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One can easily see that this parameterization of the two-body
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interaction is not size-consistent: the dissociation of a
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diatomic molecule $AB$ with a parameter $b_{AB}$
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diatomic molecule \ce{AB} with a parameter $b_{\ce{AB}}$
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will lead to two different two-body Jastrow factors, each
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with its own optimal value $b_A$ and $b_B$. To remove the
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size-consistency error on a PES using this ansätz for $J_\text{ee}$,
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with its own optimal value $b_{\ce{A}}$ and $b_{\ce{B}}$. To remove the
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size-consistency error on a PES using this ans\"atz for $J_\text{ee}$,
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one needs to impose that the parameters of $J_\text{ee}$ are fixed:
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$b_A = b_B = b_{AB}$.
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$b_A = b_B = b_{\ce{AB}}$.
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When pseudopotentials are used in a QMC calculation, it is common
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practice to localize the non-local part of the pseudopotential on the
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@ -926,7 +935,7 @@ impacted by this spurious effect, as opposed to FCI.
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In this section, we investigate the impact of the spin contamination
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due to the short-range density functional on the FN-DMC energy. We have
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computed the energies of the carbon atom in its triplet state
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with BFD pseudopotentials and the corresponding double-zeta basis
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with BFD pseudopotentials and the corresponding double-$\zeta$ basis
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set. The calculation was done with $m_s=1$ (3 spin-up electrons
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and 1 spin-down electrons) and with $m_s=0$ (2 spin-up and 2
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spin-down electrons).
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@ -991,18 +1000,18 @@ The 55 molecules of the benchmark for the Gaussian-1
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theory\cite{Pople_1989,Curtiss_1990} were chosen to test the
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performance of the RS-DFT-CIPSI trial wave functions in the context of
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energy differences. Calculations were made in the double-, triple-
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and quadruple-zeta basis sets with different values of $\mu$, and using
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natural orbitals of a preliminary CIPSI calculation.
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and quadruple-$\zeta$ basis sets with different values of $\mu$, and using
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NOs from a preliminary CIPSI calculation \titou{as a starting point}.
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For comparison, we have computed the energies of all the atoms and
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molecules at the DFT level with different density functionals, and at
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molecules at the KS-DFT level with various semi-local and hybrid density functionals [PBE, BLYP, PBE0, and B3LYP], and at
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the CCSD(T) level. Table~\ref{tab:mad} gives the corresponding mean
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absolute errors (MAE), mean signed errors (MSE) and standard
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deviations (RMSD). For FCI (RS-DFT-CIPSI, $\mu=\infty$) we have
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given extrapolated values at $\EPT\rightarrow 0$, and the error bars
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correspond to the difference between the energies computed with a
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two-point and with a three-point linear extrapolation.
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provided the extrapolated values at $\EPT \to 0$, and the error bars
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correspond to the difference between the energies \titou{computed with a
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two-point and with a three-point linear extrapolation}. \cite{Loos_2018a,Loos_2019,Loos_2020b,Loos_2020c}
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In this benchmark, the great majority of the systems are well
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In this benchmark, the great majority of the systems are weakly correlated and are then well
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described by a single determinant. Therefore, the atomization energies
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calculated at the DFT level are relatively accurate, even when
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the basis set is small. The introduction of exact exchange (B3LYP and
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@ -1014,10 +1023,10 @@ and FCI energies.
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The imbalance of the quality of description of molecules compared
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to atoms is exhibited by a very negative value of the MSE for
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CCSD(T) and FCI/VDZ-BFD, which is reduced by a factor of two
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when going to the triple-zeta basis, and again by a factor of two when
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going to the quadruple-zeta basis.
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when going to the triple-$\zeta$ basis, and again by a factor of two when
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going to the quadruple-$\zeta$ basis.
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This large imbalance at the double-zeta level affects the nodal
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This large imbalance at the VDZ-BFD level affects the nodal
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surfaces, because although the FN-DMC energies obtained with near-FCI
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trial wave functions are much lower than the single-determinant FN-DMC
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energies, the MAE obtained with FCI (7.38~$\pm$ 1.08~kcal/mol) is
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@ -1027,7 +1036,7 @@ negative MAE which confirms that all the atomization energies are
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underestimated. This confirms that some of the basis-set
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incompleteness error is transferred in the fixed-node error.
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Within the double-zeta basis set, the calculations could be done for the
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Within the double-$\zeta$ basis set, the calculations could be done for the
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whole range of values of $\mu$, and the optimal value of $\mu$ for the
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trial wave function was estimated for each system by searching for the
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minimum of the spline interpolation curve of the FN-DMC energy as a
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@ -1042,7 +1051,7 @@ These calculations were done only for the smallest basis set
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because of the expensive computational cost of the QMC calculations
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when the trial wave function is expanded on more than a few million
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determinants.
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At the RS-DFT-CIPSI level, we can remark that with the triple-zeta
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At the RS-DFT-CIPSI level, we can remark that with the triple-$\zeta$
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basis set the MAE are larger for $\mu=1$~bohr$^{-1}$ than for the
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FCI. For the largest systems, as shown in Fig.~\ref{fig:g2-ndet}
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there are many systems which did not reach the threshold
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@ -1051,7 +1060,7 @@ $\EPT<1$~m\hartree{}, and the number of determinants exceeded
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small size-consistency error originating from the imbalanced
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truncation of the wave functions, which is not present in the
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extrapolated FCI energies. The same comment applies to
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$\mu=0.5$~bohr$^{-1}$ with the quadruple-zeta basis set.
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$\mu=0.5$~bohr$^{-1}$ with the quadruple-$\zeta$ basis set.
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%%% FIG 5 %%%
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@ -1098,13 +1107,13 @@ Fig.~\ref{fig:g2-ndet}. As expected, the number of determinants
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is smaller when $\mu$ is small and larger when $\mu$ is large.
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It is important to remark that the median of the number of
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determinants when $\mu=0.5$~bohr$^{-1}$ is below 100~000 determinants
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with the quadruple-zeta basis set, making these calculations feasilble
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with such a large basis set. At the double-zeta level, compared to the
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with the quadruple-$\zeta$ basis set, making these calculations feasilble
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with such a large basis set. At the double-$\zeta$ level, compared to the
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FCI trial wave functions the median of the number of determinants is
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reduced by more than two orders of magnitude.
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Moreover, going to $\mu=0.25$~bohr$^{-1}$ gives a median close to 100
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determinants at the double-zeta level, and close to 1~000 determinants
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at the quadruple-zeta level for only a slight increase of the
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determinants at the double-$\zeta$ level, and close to 1~000 determinants
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at the quadruple-$\zeta$ level for only a slight increase of the
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MAE. Hence, RS-DFT-CIPSI trial wave functions with small values of
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$\mu$ could be very useful for large systems to go beyond the
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single-determinant approximation at a very low computational cost
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@ -1123,6 +1132,7 @@ solution would have been the PBE single determinant.
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%%%%%%%%%%%%%%%%%%%%
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\section{Conclusion}
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\label{sec:conclusion}
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%%%%%%%%%%%%%%%%%%%%
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In the present work, we have shown that introducing short-range correlation via
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