From e9f2120a30d28e4ff7de440fbd52bd62aaa3c51e Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Tue, 18 Aug 2020 15:55:15 +0200 Subject: [PATCH] saving work in spin part --- Manuscript/rsdft-cipsi-qmc.tex | 353 +++++++++++++++++---------------- 1 file changed, 178 insertions(+), 175 deletions(-) diff --git a/Manuscript/rsdft-cipsi-qmc.tex b/Manuscript/rsdft-cipsi-qmc.tex index 80f5179..9434444 100644 --- a/Manuscript/rsdft-cipsi-qmc.tex +++ b/Manuscript/rsdft-cipsi-qmc.tex @@ -761,180 +761,7 @@ Increasing the size of the basis set improves the description of the density and of the electron correlation, but also reduces the imbalance in the description of atoms and molecule, leading to more accurate atomization energies. - -%============================ -\subsection{Size consistency} -%============================ - -An extremely important feature required to get accurate -atomization energies is size-consistency (or strict separability), -since the numbers of correlated electron pairs in the molecule and its isolated atoms -are different. -The energies computed within DFT are size-consistent, and -\titou{because it is a mean-field method the convergence to the CBS limit -is relatively fast}. \cite{FraMusLupTou-JCP-15} -Hence, DFT methods are very well adapted to -the calculation of atomization energies, especially with small basis -sets. \cite{Giner_2018,Loos_2019d,Giner_2020} -\titou{But going to the CBS limit will converge to biased atomization -energies because of the use of approximate density functionals.} - -Likewise, FCI is also size-consistent, but the convergence of -the FCI energies towards the CBS limit is much slower because of the -description of short-range electron correlation using atom-centered -functions. \cite{Kutzelnigg_1985,Kutzelnigg_1991} But ultimately the exact energy will be reached. - -In the context of SCI calculations, when the variational energy is -extrapolated to the FCI energy \cite{Holmes_2017} there is no -size-consistency error. But when the truncated SCI wave function is used -as a reference for post-HF methods such as SCI+PT2 -or for QMC calculations, there is a residual size-consistency error -originating from the truncation of the wave function. - -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-consistency error can be reduced by choosing the number of -selected determinants such that the sum of the PT2 corrections on the -fragments is equal to the PT2 correction of the molecule, enforcing that -the variational potential energy surface (PES) is -parallel to the perturbatively corrected PES, which is a relatively -accurate estimate of the FCI PES.\cite{Giner_2015} - -Another source of size-consistency 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 -be expressed as in Eq.~\eqref{eq:jast-ee}. -The parameter -$a$ is determined by the electron-electron cusp condition, \cite{Kato_1957,Pack_1966} 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 -interaction is not size-consistent: the dissociation of a -diatomic molecule \ce{AB} with a parameter $b_{\ce{AB}}$ -will lead to two different two-body Jastrow factors, each -with its own optimal value $b_{\ce{A}}$ and $b_{\ce{B}}$. To remove the -size-consistency 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: -$b_A = b_B = b_{\ce{AB}}$. - -When pseudopotentials are used in a QMC calculation, it is of common -practice to localize the non-local part of the pseudopotential on the -complete trial wave function $\Phi$. -If the wave function is not size-consistent, -so will be the locality approximation. Within the DLA,\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 -only depends on the parameters of the determinantal component. Using a -non-size-consistent Jastrow factor, or a non-optimal Jastrow factor will -not introduce an additional 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 pseudopotential -are computed analytically and the computational cost of the -pseudopotential is dramatically reduced (for more detail, see -Ref.~\onlinecite{Scemama_2015}). - -%%% TABLE III %%% -\begin{table} - \caption{FN-DMC energy (in hartree) using the VDZ-BFD basis set and the srPBE functional - of the fluorine atom and the dissociated \ce{F2} molecule for various $\mu$ values. - The size-consistency error is also reported.} - \label{tab:size-cons} - \begin{ruledtabular} - \begin{tabular}{cccc} - $\mu$ & \ce{F} & Dissociated \ce{F2} & Size-consistency error \\ - \hline - 0.00 & $-24.188\,7(3)$ & $-48.377\,7(3)$ & $-0.000\,3(4)$ \\ - 0.25 & $-24.188\,7(3)$ & $-48.377\,2(4)$ & $+0.000\,2(5)$ \\ - 0.50 & $-24.188\,8(1)$ & $-48.376\,9(4)$ & $+0.000\,7(4)$ \\ - 1.00 & $-24.189\,7(1)$ & $-48.380\,2(4)$ & $-0.000\,8(4)$ \\ - 2.00 & $-24.194\,1(3)$ & $-48.388\,4(4)$ & $-0.000\,2(5)$ \\ - 5.00 & $-24.194\,7(4)$ & $-48.388\,5(7)$ & $+0.000\,9(8)$ \\ - $\infty$ & $-24.193\,5(2)$ & $-48.386\,9(4)$ & $+0.000\,1(5)$ \\ - \end{tabular} - \end{ruledtabular} -\end{table} -%%% %%% %%% %%% - -In this section, we make a numerical verification that the produced -wave functions are size-consistent for a given range-separation -parameter. -We have computed the FN-DMC energy of the dissociated fluorine dimer, where -the two atoms are at a distance of 50~\AA. We expect that the energy -of this system is equal to twice the energy of the fluorine atom. -The data in Table~\ref{tab:size-cons} shows that it is indeed the -case, so we can conclude that the proposed scheme provides -size-consistent FN-DMC energies for all values of $\mu$ (within -twice the statistical error bars). - - -%============================ -\subsection{Spin invariance} -%============================ - -Closed-shell molecules often 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. A good test is to check that all the components -of a spin multiplet are degenerate, as expected from exact solutions. -FCI wave functions have this property and give degenerate energies with -respect to the spin quantum number $m_s$, but the multiplication by a -Jastrow factor introduces spin contamination if the parameters -for the same-spin electron pairs are different from those -for the opposite-spin pairs.\cite{Tenno_2004} -Again, when pseudopotentials are used this tiny error is transferred -in the FN-DMC energy unless the DLA is used. - -Within DFT, the common density functionals make a difference for -same-spin and opposite-spin interactions. As DFT is a -single-determinant theory, the density functionals are designed to be -used with the highest value of $m_s$, and therefore different values -of $m_s$ lead to different energies. -So in the context of RS-DFT, the determinantal expansions will be -impacted by this spurious effect, as opposed to FCI. - -%%% TABLE IV %%% -\begin{table} - \caption{FN-DMC energy (in hartree) for various $\mu$ values of the triplet carbon atom with - different values of $m_s$ computed with the VDZ-BFD basis set and the srPBE functional.} - \label{tab:spin} - \begin{ruledtabular} - \begin{tabular}{cccc} - $\mu$ & $m_s=1$ & $m_s=0$ & Spin-invariance error \\ - \hline - 0.00 & $-5.416\,8(1)$ & $-5.414\,9(1)$ & $+0.001\,9(2)$ \\ - 0.25 & $-5.417\,2(1)$ & $-5.416\,5(1)$ & $+0.000\,7(1)$ \\ - 0.50 & $-5.422\,3(1)$ & $-5.421\,4(1)$ & $+0.000\,9(2)$ \\ - 1.00 & $-5.429\,7(1)$ & $-5.429\,2(1)$ & $+0.000\,5(2)$ \\ - 2.00 & $-5.432\,1(1)$ & $-5.431\,4(1)$ & $+0.000\,7(2)$ \\ - 5.00 & $-5.431\,7(1)$ & $-5.431\,4(1)$ & $+0.000\,3(2)$ \\ - $\infty$ & $-5.431\,6(1)$ & $-5.431\,3(1)$ & $+0.000\,3(2)$ \\ - \end{tabular} - \end{ruledtabular} -\end{table} -%%% %%% %%% %%% - -In this section, we investigate the impact of the spin contamination -due to the short-range density functional on the FN-DMC energy. We have -computed the energies of the carbon atom in its triplet state -with BFD pseudopotentials and the corresponding double-$\zeta$ basis -set. The calculation was done with $m_s=1$ (3 spin-up electrons -and 1 spin-down electrons) and with $m_s=0$ (2 spin-up and 2 -spin-down electrons). - -The results are presented in Table~\ref{tab:spin}. -Although using $m_s=0$ the energy is higher than with $m_s=1$, the -bias is relatively small, more than one order of magnitude smaller -than the energy gained by reducing the fixed-node error going from the single -determinant to the FCI trial wave function. The highest bias, close to -2~m\hartree, is obtained for $\mu=0$, but the bias decreases rapidly -below 1~m\hartree{} when $\mu$ increases. As expected, with $\mu=\infty$ -there is no bias (within the error bars), and the bias is not -noticeable with $\mu=5$~bohr$^{-1}$. - -%============================ -\subsection{Benchmark} -%============================ +The size-consistency and the spin-invariance of the present scheme are discussed in Appendices \ref{app:size} and \ref{app:spin}. %%% FIG 6 %%% \begin{squeezetable} @@ -1161,10 +988,186 @@ Challenge 2019-gch0418) and from CALMIP (Toulouse) under allocation The data that support the findings of this study are available within the article and its {\SI}, and are openly available in [repository name] at \url{http://doi.org/[doi]}, reference number [reference number]. +%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\appendix +%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +%============================ +\section{Size consistency} +\label{app:size} +%============================ + +%%% TABLE III %%% +\begin{table} + \caption{FN-DMC energy (in hartree) using the VDZ-BFD basis set and the srPBE functional + of the fluorine atom and the dissociated \ce{F2} molecule for various $\mu$ values. + The size-consistency error is also reported.} + \label{tab:size-cons} + \begin{ruledtabular} + \begin{tabular}{cccc} + $\mu$ & \ce{F} & Dissociated \ce{F2} & Size-consistency error \\ + \hline + 0.00 & $-24.188\,7(3)$ & $-48.377\,7(3)$ & $-0.000\,3(4)$ \\ + 0.25 & $-24.188\,7(3)$ & $-48.377\,2(4)$ & $+0.000\,2(5)$ \\ + 0.50 & $-24.188\,8(1)$ & $-48.376\,9(4)$ & $+0.000\,7(4)$ \\ + 1.00 & $-24.189\,7(1)$ & $-48.380\,2(4)$ & $-0.000\,8(4)$ \\ + 2.00 & $-24.194\,1(3)$ & $-48.388\,4(4)$ & $-0.000\,2(5)$ \\ + 5.00 & $-24.194\,7(4)$ & $-48.388\,5(7)$ & $+0.000\,9(8)$ \\ + $\infty$ & $-24.193\,5(2)$ & $-48.386\,9(4)$ & $+0.000\,1(5)$ \\ + \end{tabular} + \end{ruledtabular} +\end{table} +%%% %%% %%% %%% + +An extremely important feature required to get accurate +atomization energies is size-consistency (or strict separability), +since the numbers of correlated electron pairs in the molecule and its isolated atoms +are different. +The energies computed within DFT are size-consistent, and +\titou{because it is a mean-field method the convergence to the CBS limit +is relatively fast}. \cite{FraMusLupTou-JCP-15} +Hence, DFT methods are very well adapted to +the calculation of atomization energies, especially with small basis +sets. \cite{Giner_2018,Loos_2019d,Giner_2020} +\titou{But going to the CBS limit will converge to biased atomization +energies because of the use of approximate density functionals.} + +Likewise, FCI is also size-consistent, but the convergence of +the FCI energies towards the CBS limit is much slower because of the +description of short-range electron correlation using atom-centered +functions. \cite{Kutzelnigg_1985,Kutzelnigg_1991} But ultimately the exact energy will be reached. + +In the context of SCI calculations, when the variational energy is +extrapolated to the FCI energy \cite{Holmes_2017} there is no +size-consistency error. But when the truncated SCI wave function is used +as a reference for post-HF methods such as SCI+PT2 +or for QMC calculations, there is a residual size-consistency error +originating from the truncation of the wave function. + +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-consistency error can be reduced by choosing the number of +selected determinants such that the sum of the PT2 corrections on the +fragments is equal to the PT2 correction of the molecule, enforcing that +the variational potential energy surface (PES) is +parallel to the perturbatively corrected PES, which is a relatively +accurate estimate of the FCI PES.\cite{Giner_2015} + +Another source of size-consistency 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 +be expressed as in Eq.~\eqref{eq:jast-ee}. +The parameter +$a$ is determined by the electron-electron cusp condition, \cite{Kato_1957,Pack_1966} 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 +interaction is not size-consistent: the dissociation of a +diatomic molecule \ce{AB} with a parameter $b_{\ce{AB}}$ +will lead to two different two-body Jastrow factors, each +with its own optimal value $b_{\ce{A}}$ and $b_{\ce{B}}$. To remove the +size-consistency 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: +$b_A = b_B = b_{\ce{AB}}$. + +When pseudopotentials are used in a QMC calculation, it is of common +practice to localize the non-local part of the pseudopotential on the +complete trial wave function $\Phi$. +If the wave function is not size-consistent, +so will be the locality approximation. Within the DLA,\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 +only depends on the parameters of the determinantal component. Using a +non-size-consistent Jastrow factor, or a non-optimal Jastrow factor will +not introduce an additional 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 pseudopotential +are computed analytically and the computational cost of the +pseudopotential is dramatically reduced (for more detail, see +Ref.~\onlinecite{Scemama_2015}). + +In this section, we make a numerical verification that the produced +wave functions are size-consistent for a given range-separation +parameter. +We have computed the FN-DMC energy of the dissociated fluorine dimer, where +the two atoms are at a distance of 50~\AA. We expect that the energy +of this system is equal to twice the energy of the fluorine atom. +The data in Table~\ref{tab:size-cons} shows that it is indeed the +case, so we can conclude that the proposed scheme provides +size-consistent FN-DMC energies for all values of $\mu$ (within +twice the statistical error bars). + +%============================ +\section{Spin invariance} +\label{app:spin} +%============================ + +%%% TABLE IV %%% +\begin{table} + \caption{FN-DMC energy (in hartree) for various $\mu$ values of the triplet carbon atom with + different values of $m_s$ computed with the VDZ-BFD basis set and the srPBE functional.} + \label{tab:spin} + \begin{ruledtabular} + \begin{tabular}{cccc} + $\mu$ & $m_s=1$ & $m_s=0$ & Spin-invariance error \\ + \hline + 0.00 & $-5.416\,8(1)$ & $-5.414\,9(1)$ & $+0.001\,9(2)$ \\ + 0.25 & $-5.417\,2(1)$ & $-5.416\,5(1)$ & $+0.000\,7(1)$ \\ + 0.50 & $-5.422\,3(1)$ & $-5.421\,4(1)$ & $+0.000\,9(2)$ \\ + 1.00 & $-5.429\,7(1)$ & $-5.429\,2(1)$ & $+0.000\,5(2)$ \\ + 2.00 & $-5.432\,1(1)$ & $-5.431\,4(1)$ & $+0.000\,7(2)$ \\ + 5.00 & $-5.431\,7(1)$ & $-5.431\,4(1)$ & $+0.000\,3(2)$ \\ + $\infty$ & $-5.431\,6(1)$ & $-5.431\,3(1)$ & $+0.000\,3(2)$ \\ + \end{tabular} + \end{ruledtabular} +\end{table} +%%% %%% %%% %%% + +Closed-shell molecules often dissociate into open-shell +fragments. To get reliable atomization energies, it is important to +have a theory which is of comparable quality for open- and +closed-shell systems. A good test is to check that all the components +of a spin multiplet are degenerate, as expected from exact solutions. + +FCI wave functions have this property and yield degenerate energies with +respect to the spin quantum number $m_s$. +However, multiplying the determinantal part of the trial wave function by a +Jastrow factor introduces spin contamination if the Jastrow parameters +for the same-spin electron pairs are different from those +for the opposite-spin pairs.\cite{Tenno_2004} +Again, when pseudopotentials are employed, this tiny error is transferred +to the FN-DMC energy unless the DLA is enforced. + +The context is rather different within DFT. +Indeed, mainstream density functionals have distinct functional forms to take +into account correlation effects of same-spin and opposite-spin electron pairs. +Therefore, KS determinants corresponding to different values of $m_s$ lead to different total energies. +Consequently, in the context of RS-DFT, the determinant expansions is impacted by this spurious effect, as opposed to FCI. + +In this Appendix, we investigate the impact of the spin contamination on the FN-DMC energy +originating from the short-range density functional. We have +computed the energies of the carbon atom in its triplet state +with the VDZ-BFD basis set and the srPBE functional. +The calculations are performed for $m_s=1$ (3 spin-up +and 1 spin-down electrons) and for $m_s=0$ (2 spin-up and 2 +spin-down electrons). + +The results are reported in Table~\ref{tab:spin}. +Although the energy obtained with $m_s=0$ is higher than the one obtained with $m_s=1$, the +bias is relatively small, \ie, more than one order of magnitude smaller +than the energy gained by reducing the fixed-node error going from the single +determinant to the FCI trial wave function. The largest bias, close to +$2$ m\hartree, is obtained for $\mu=0$, but this bias decreases quickly +below $1$ m\hartree{} when $\mu$ increases. As expected, with $\mu=\infty$ +we observe a perfect spin-invariance of the energy (within the error bars), and the bias is not +noticeable for $\mu=5$~bohr$^{-1}$. + +\titou{T2: what do you conclude from this section? What value of $m_s$ do you use to compute the atoms?} + %%%%%%%%%%%%%%%%%%%%%%%%%%%% \bibliography{rsdft-cipsi-qmc} %%%%%%%%%%%%%%%%%%%%%%%%%%%% \end{document} -