add summary paragraph and random correction in Sec V
This commit is contained in:
parent
b0f9e85015
commit
206f10b3eb
@ 250,6 +250,14 @@ within this context.




The central idea of the present work, and the launchpad 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.


In particular, we show here that one can combine CIPSI and KSDFT via the range separation (RS) of the interelectronic Coulomb operator \cite{SavINC96a,Toulouse_2004} to obtain accurate FNDMC energies with compact multideterminant trial wave functions.






The present manuscript is organized as follows.


In Sec.~\ref{sec:rsdftcipsi}, we provide theoretical details about the CIPSI algorithm (Sec.~\ref{sec:CIPSI}) and rangeseparated DFT (Sec.~\ref{sec:rsdft}).


Computational details are reported in Sec.~\ref{sec:compdetails}.


In Sec.~\ref{sec:mudmc}, we discuss the influence of the rangeseparation parameter on the fixednode error as well as the link between RSDFT and Jastrow factors.


Section \ref{sec:atomization} examines the performance of the present scheme for the atomization energies of the Gaussian1 set of molecules.


Finally, we draw our conclusion in Sec.~\ref{sec:conclusion}.


Unless otherwise stated, atomic units are used.




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


@ 259,6 +267,7 @@ Unless otherwise stated, atomic units are used.




%====================


\subsection{The CIPSI algorithm}


\label{sec:CIPSI}


%====================


Beyond the singledeterminant representation, the best


multideterminant wave function one can wish for  in a given basis set  is the FCI wave function.


@ 538,7 +547,7 @@ The takehome message of this first numerical study is that RSDFT trial wave fu


This is a key result of the present study.




%======================================================


\subsection{Link between RSDFT and Jastrow factors }


\subsection{Link between RSDFT and Jastrow factor}


\label{sec:rsdftj}


%======================================================


The data presented in Sec.~\ref{sec:fndmc_mu} evidence that, in a finite basis, RSDFT can provide


@ 645,7 +654,7 @@ This is yet another key result of the present study.




%%% TABLE II %%%


\begin{table}


\caption{\ce{H2O}, doublezeta basis set. Integrated ontop pair density $\expval{ P }$


\caption{\ce{H2O}, double$\zeta$ basis set. Integrated ontop pair density $\expval{ P }$


for $\Psi^J$ and $\Psi^\mu$ with different values of $\mu$.


\titou{Please remove table and merge data in Fig. 4.}}


\label{tab:table_on_top}


@ 760,16 +769,16 @@ As a conclusion of the first part of this study, we can highlight the following




Atomization energies are challenging for postHF methods


because their calculation requires a perfect balance in the


description of atoms and molecules. Basis sets used in molecular


calculations are atomcentered, so they are always better adapted to


description of atoms and molecules. The mainstream oneelectron basis sets employed in molecular


calculations are atomcentered, so they are, by construction, better adapted to


atoms than molecules and atomization energies usually tend to be


underestimated by variational methods.


In the context of FNDMC calculations, the nodal surface is imposed by


the trial wavefunction which is expanded on an atomcentered basis


set, so we expect the fixednode error to be also tightly related to


the trial wavefunction which is expanded in an atomcentered basis


set, so we expect the fixednode error to be also intimately 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


the density and of the 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.




@ 781,9 +790,9 @@ An extremely important feature required to get accurate


atomization energies is sizeconsistency (or strict separability),


since the numbers of correlated electron pairs in the isolated atoms


are different from those of the molecules.


The energy computed within density functional theory is sizeconsistent, and


as it is a meanfield method the convergence to the complete basis set


(CBS) limit is relatively fast. Hence, DFT methods are very well adapted to


The energy computed within DFT is sizeconsistent, and


as it is a meanfield method the convergence to the CBS limit


is relatively fast. Hence, DFT methods are very well adapted to


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.


@ 793,10 +802,10 @@ the FCI energies to the CBS limit is much slower because of the


description of shortrange electron correlation using atomcentered


functions. But ultimately the exact energy will be reached.




In the context of selected CI calculations, when the variational energy is


extrapolated to the FCI energy\cite{Holmes_2017} there is no


In the context of SCI calculations, when the variational energy is


extrapolated to the FCI energy \cite{Holmes_2017} there is no


sizeconsistency error. But when the truncated SCI wave function is used


as a reference for postHartreeFock methods such as SCI+PT2


as a reference for postHF methods such as SCI+PT2


or for QMC calculations, there is a residual sizeconsistency error


originating from the truncation of the wave function.




@ 820,12 +829,12 @@ $a$ 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 twobody


interaction is not sizeconsistent: the dissociation of a


diatomic molecule $AB$ with a parameter $b_{AB}$


diatomic molecule \ce{AB} with a parameter $b_{\ce{AB}}$


will lead to two different twobody Jastrow factors, each


with its own optimal value $b_A$ and $b_B$. To remove the


sizeconsistency error on a PES using this ansätz for $J_\text{ee}$,


with its own optimal value $b_{\ce{A}}$ and $b_{\ce{B}}$. To remove the


sizeconsistency error on a PES using this ans\"atz for $J_\text{ee}$,


one needs to impose that the parameters of $J_\text{ee}$ are fixed:


$b_A = b_B = b_{AB}$.


$b_A = b_B = b_{\ce{AB}}$.




When pseudopotentials are used in a QMC calculation, it is common


practice to localize the nonlocal part of the pseudopotential on the


@ 926,7 +935,7 @@ impacted by this spurious effect, as opposed to FCI.


In this section, we investigate the impact of the spin contamination


due to the shortrange density functional on the FNDMC energy. We have


computed the energies of the carbon atom in its triplet state


with BFD pseudopotentials and the corresponding doublezeta basis


with BFD pseudopotentials and the corresponding double$\zeta$ basis


set. The calculation was done with $m_s=1$ (3 spinup electrons


and 1 spindown electrons) and with $m_s=0$ (2 spinup and 2


spindown electrons).


@ 991,18 +1000,18 @@ The 55 molecules of the benchmark for the Gaussian1


theory\cite{Pople_1989,Curtiss_1990} were chosen to test the


performance of the RSDFTCIPSI trial wave functions in the context of


energy differences. Calculations were made in the double, triple


and quadruplezeta basis sets with different values of $\mu$, and using


natural orbitals of a preliminary CIPSI calculation.


and quadruple$\zeta$ basis sets with different values of $\mu$, and using


NOs from a preliminary CIPSI calculation \titou{as a starting point}.


For comparison, we have computed the energies of all the atoms and


molecules at the DFT level with different density functionals, and at


molecules at the KSDFT level with various semilocal and hybrid density functionals [PBE, BLYP, PBE0, and B3LYP], and at


the CCSD(T) level. Table~\ref{tab:mad} gives the corresponding mean


absolute errors (MAE), mean signed errors (MSE) and standard


deviations (RMSD). For FCI (RSDFTCIPSI, $\mu=\infty$) we have


given extrapolated values at $\EPT\rightarrow 0$, and the error bars


correspond to the difference between the energies computed with a


twopoint and with a threepoint linear extrapolation.


provided the extrapolated values at $\EPT \to 0$, and the error bars


correspond to the difference between the energies \titou{computed with a


twopoint and with a threepoint linear extrapolation}. \cite{Loos_2018a,Loos_2019,Loos_2020b,Loos_2020c}




In this benchmark, the great majority of the systems are well


In this benchmark, the great majority of the systems are weakly correlated and are then well


described by a single determinant. Therefore, the atomization energies


calculated at the DFT level are relatively accurate, even when


the basis set is small. The introduction of exact exchange (B3LYP and


@ 1014,10 +1023,10 @@ and FCI energies.


The imbalance of the quality of description of molecules compared


to atoms is exhibited by a very negative value of the MSE for


CCSD(T) and FCI/VDZBFD, which is reduced by a factor of two


when going to the triplezeta basis, and again by a factor of two when


going to the quadruplezeta basis.


when going to the triple$\zeta$ basis, and again by a factor of two when


going to the quadruple$\zeta$ basis.




This large imbalance at the doublezeta level affects the nodal


This large imbalance at the VDZBFD level affects the nodal


surfaces, because although the FNDMC energies obtained with nearFCI


trial wave functions are much lower than the singledeterminant FNDMC


energies, the MAE obtained with FCI (7.38~$\pm$ 1.08~kcal/mol) is


@ 1027,7 +1036,7 @@ negative MAE which confirms that all the atomization energies are


underestimated. This confirms that some of the basisset


incompleteness error is transferred in the fixednode error.




Within the doublezeta basis set, the calculations could be done for the


Within the double$\zeta$ basis set, the calculations could be done for the


whole range of values of $\mu$, and the optimal value of $\mu$ for the


trial wave function was estimated for each system by searching for the


minimum of the spline interpolation curve of the FNDMC energy as a


@ 1042,7 +1051,7 @@ These calculations were done only for the smallest basis set


because of the expensive computational cost of the QMC calculations


when the trial wave function is expanded on more than a few million


determinants.


At the RSDFTCIPSI level, we can remark that with the triplezeta


At the RSDFTCIPSI 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 Fig.~\ref{fig:g2ndet}


there are many systems which did not reach the threshold


@ 1051,7 +1060,7 @@ $\EPT<1$~m\hartree{}, and the number of determinants exceeded


small sizeconsistency error originating from the imbalanced


truncation of the wave functions, which is not present in the


extrapolated FCI energies. The same comment applies to


$\mu=0.5$~bohr$^{1}$ with the quadruplezeta basis set.


$\mu=0.5$~bohr$^{1}$ with the quadruple$\zeta$ basis set.






%%% FIG 5 %%%


@ 1098,13 +1107,13 @@ Fig.~\ref{fig:g2ndet}. As expected, the number of determinants


is smaller when $\mu$ is small and larger when $\mu$ is large.


It is important to remark that the median of the number of


determinants when $\mu=0.5$~bohr$^{1}$ is below 100~000 determinants


with the quadruplezeta basis set, making these calculations feasilble


with such a large basis set. At the doublezeta level, compared to the


with the quadruple$\zeta$ basis set, making these calculations feasilble


with such a large basis set. At the double$\zeta$ level, compared to the


FCI trial wave functions the median of the number of determinants is


reduced by more than two orders of magnitude.


Moreover, going to $\mu=0.25$~bohr$^{1}$ gives a median close to 100


determinants at the doublezeta level, and close to 1~000 determinants


at the quadruplezeta level for only a slight increase of the


determinants at the double$\zeta$ level, and close to 1~000 determinants


at the quadruple$\zeta$ level for only a slight increase of the


MAE. Hence, RSDFTCIPSI trial wave functions with small values of


$\mu$ could be very useful for large systems to go beyond the


singledeterminant approximation at a very low computational cost


@ 1123,6 +1132,7 @@ solution would have been the PBE single determinant.




%%%%%%%%%%%%%%%%%%%%


\section{Conclusion}


\label{sec:conclusion}


%%%%%%%%%%%%%%%%%%%%




In the present work, we have shown that introducing shortrange correlation via



Loading…
Reference in New Issue
Block a user