saving work in spin part
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@ -761,180 +761,7 @@ Increasing the size of the basis set improves the description of
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the density and of the electron correlation, but also reduces the
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imbalance in the description of atoms and
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molecule, leading to more accurate atomization energies.
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%============================
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\subsection{Size consistency}
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%============================
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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 molecule and its isolated atoms
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are different.
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The energies computed within DFT are size-consistent, and
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\titou{because it is a mean-field method the convergence to the CBS limit
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is relatively fast}. \cite{FraMusLupTou-JCP-15}
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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. \cite{Giner_2018,Loos_2019d,Giner_2020}
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\titou{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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Likewise, FCI is also size-consistent, but the convergence of
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the FCI energies towards 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. \cite{Kutzelnigg_1985,Kutzelnigg_1991} But ultimately the exact energy will be reached.
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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-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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QMC energies can be made size-consistent by extrapolating the
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FN-DMC energy to estimate the energy obtained with the FCI as a trial
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wave function.\cite{Scemama_2018,Scemama_2018b} Alternatively, the
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size-consistency error can be reduced by choosing the number of
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selected determinants such that the sum of the PT2 corrections on the
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fragments is equal to the PT2 correction of the molecule, enforcing that
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the variational potential energy surface (PES) is
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parallel to the perturbatively corrected PES, which is a relatively
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accurate estimate of the FCI PES.\cite{Giner_2015}
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Another source of size-consistency error in QMC calculations originates
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from the Jastrow factor. Usually, the Jastrow factor contains
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one-electron, two-electron and one-nucleus-two-electron terms.
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The problematic part is the two-electron term, whose simplest form can
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be expressed as in Eq.~\eqref{eq:jast-ee}.
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The parameter
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$a$ is determined by the electron-electron cusp condition, \cite{Kato_1957,Pack_1966} 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 \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 $b_{\ce{A}}$ 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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$b_A = b_B = b_{\ce{AB}}$.
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When pseudopotentials are used in a QMC calculation, it is of common
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practice to localize the non-local part of the pseudopotential on the
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complete trial wave function $\Phi$.
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If the wave function is not size-consistent,
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so will be the locality approximation. Within the DLA,\cite{Zen_2019} the Jastrow factor is
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removed from the wave function on which the pseudopotential is localized.
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The great advantage of this approximation is that the FN-DMC energy
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only depends on the parameters of the determinantal component. Using a
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non-size-consistent Jastrow factor, or a non-optimal Jastrow factor will
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not introduce an additional error in FN-DMC calculations, although it
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will reduce the statistical errors by reducing the variance of the
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local energy. Moreover, the integrals involved in the pseudopotential
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are computed analytically and the computational cost of the
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pseudopotential is dramatically reduced (for more detail, see
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Ref.~\onlinecite{Scemama_2015}).
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%%% TABLE III %%%
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\begin{table}
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\caption{FN-DMC energy (in hartree) using the VDZ-BFD basis set and the srPBE functional
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of the fluorine atom and the dissociated \ce{F2} molecule for various $\mu$ values.
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The size-consistency error is also reported.}
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\label{tab:size-cons}
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\begin{ruledtabular}
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\begin{tabular}{cccc}
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$\mu$ & \ce{F} & Dissociated \ce{F2} & Size-consistency error \\
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\hline
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0.00 & $-24.188\,7(3)$ & $-48.377\,7(3)$ & $-0.000\,3(4)$ \\
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0.25 & $-24.188\,7(3)$ & $-48.377\,2(4)$ & $+0.000\,2(5)$ \\
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0.50 & $-24.188\,8(1)$ & $-48.376\,9(4)$ & $+0.000\,7(4)$ \\
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1.00 & $-24.189\,7(1)$ & $-48.380\,2(4)$ & $-0.000\,8(4)$ \\
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2.00 & $-24.194\,1(3)$ & $-48.388\,4(4)$ & $-0.000\,2(5)$ \\
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5.00 & $-24.194\,7(4)$ & $-48.388\,5(7)$ & $+0.000\,9(8)$ \\
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$\infty$ & $-24.193\,5(2)$ & $-48.386\,9(4)$ & $+0.000\,1(5)$ \\
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\end{tabular}
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\end{ruledtabular}
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\end{table}
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%%% %%% %%% %%%
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In this section, we make a numerical verification that the produced
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wave functions are size-consistent for a given range-separation
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parameter.
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We have computed the FN-DMC energy of the dissociated fluorine dimer, where
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the two atoms are at a distance of 50~\AA. We expect that the energy
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of this system is equal to twice the energy of the fluorine atom.
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The data in Table~\ref{tab:size-cons} shows that it is indeed the
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case, so we can conclude that the proposed scheme provides
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size-consistent FN-DMC energies for all values of $\mu$ (within
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twice the statistical error bars).
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%============================
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\subsection{Spin invariance}
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%============================
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Closed-shell molecules often dissociate into open-shell
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fragments. To get reliable atomization energies, it is important to
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have a theory which is of comparable quality for open-shell and
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closed-shell systems. A good test is to check that all the components
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of a spin multiplet are degenerate, as expected from exact solutions.
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FCI wave functions have this property and give degenerate energies with
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respect to the spin quantum number $m_s$, but the multiplication by a
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Jastrow factor introduces spin contamination if the parameters
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for the same-spin electron pairs are different from those
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for the opposite-spin pairs.\cite{Tenno_2004}
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Again, when pseudopotentials are used this tiny error is transferred
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in the FN-DMC energy unless the DLA is used.
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Within DFT, the common density functionals make a difference for
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same-spin and opposite-spin interactions. As DFT is a
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single-determinant theory, the density functionals are designed to be
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used with the highest value of $m_s$, and therefore different values
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of $m_s$ lead to different energies.
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So in the context of RS-DFT, the determinantal expansions will be
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impacted by this spurious effect, as opposed to FCI.
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%%% TABLE IV %%%
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\begin{table}
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\caption{FN-DMC energy (in hartree) for various $\mu$ values of the triplet carbon atom with
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different values of $m_s$ computed with the VDZ-BFD basis set and the srPBE functional.}
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\label{tab:spin}
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\begin{ruledtabular}
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\begin{tabular}{cccc}
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$\mu$ & $m_s=1$ & $m_s=0$ & Spin-invariance error \\
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\hline
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0.00 & $-5.416\,8(1)$ & $-5.414\,9(1)$ & $+0.001\,9(2)$ \\
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0.25 & $-5.417\,2(1)$ & $-5.416\,5(1)$ & $+0.000\,7(1)$ \\
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0.50 & $-5.422\,3(1)$ & $-5.421\,4(1)$ & $+0.000\,9(2)$ \\
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1.00 & $-5.429\,7(1)$ & $-5.429\,2(1)$ & $+0.000\,5(2)$ \\
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2.00 & $-5.432\,1(1)$ & $-5.431\,4(1)$ & $+0.000\,7(2)$ \\
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5.00 & $-5.431\,7(1)$ & $-5.431\,4(1)$ & $+0.000\,3(2)$ \\
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$\infty$ & $-5.431\,6(1)$ & $-5.431\,3(1)$ & $+0.000\,3(2)$ \\
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\end{tabular}
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\end{ruledtabular}
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\end{table}
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%%% %%% %%% %%%
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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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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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The results are presented in Table~\ref{tab:spin}.
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Although using $m_s=0$ the energy is higher than with $m_s=1$, the
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bias is relatively small, more than one order of magnitude smaller
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than the energy gained by reducing the fixed-node error going from the single
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determinant to the FCI trial wave function. The highest bias, close to
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2~m\hartree, is obtained for $\mu=0$, but the bias decreases rapidly
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below 1~m\hartree{} when $\mu$ increases. As expected, with $\mu=\infty$
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there is no bias (within the error bars), and the bias is not
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noticeable with $\mu=5$~bohr$^{-1}$.
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%============================
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\subsection{Benchmark}
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%============================
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The size-consistency and the spin-invariance of the present scheme are discussed in Appendices \ref{app:size} and \ref{app:spin}.
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%%% FIG 6 %%%
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\begin{squeezetable}
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@ -1161,10 +988,186 @@ Challenge 2019-gch0418) and from CALMIP (Toulouse) under allocation
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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].
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\appendix
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%============================
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\section{Size consistency}
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\label{app:size}
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%============================
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%%% TABLE III %%%
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\begin{table}
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\caption{FN-DMC energy (in hartree) using the VDZ-BFD basis set and the srPBE functional
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of the fluorine atom and the dissociated \ce{F2} molecule for various $\mu$ values.
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The size-consistency error is also reported.}
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\label{tab:size-cons}
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\begin{ruledtabular}
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\begin{tabular}{cccc}
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$\mu$ & \ce{F} & Dissociated \ce{F2} & Size-consistency error \\
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\hline
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0.00 & $-24.188\,7(3)$ & $-48.377\,7(3)$ & $-0.000\,3(4)$ \\
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0.25 & $-24.188\,7(3)$ & $-48.377\,2(4)$ & $+0.000\,2(5)$ \\
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0.50 & $-24.188\,8(1)$ & $-48.376\,9(4)$ & $+0.000\,7(4)$ \\
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1.00 & $-24.189\,7(1)$ & $-48.380\,2(4)$ & $-0.000\,8(4)$ \\
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2.00 & $-24.194\,1(3)$ & $-48.388\,4(4)$ & $-0.000\,2(5)$ \\
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5.00 & $-24.194\,7(4)$ & $-48.388\,5(7)$ & $+0.000\,9(8)$ \\
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$\infty$ & $-24.193\,5(2)$ & $-48.386\,9(4)$ & $+0.000\,1(5)$ \\
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\end{tabular}
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\end{ruledtabular}
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\end{table}
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%%% %%% %%% %%%
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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 molecule and its isolated atoms
|
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are different.
|
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The energies computed within DFT are size-consistent, and
|
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\titou{because it is a mean-field method the convergence to the CBS limit
|
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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}
|
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\titou{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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Likewise, FCI is also size-consistent, but the convergence of
|
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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.
|
||||
|
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QMC energies can be made size-consistent by extrapolating the
|
||||
FN-DMC energy to estimate the energy obtained with the FCI as a trial
|
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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
|
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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}
|
||||
|
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Another source of size-consistency error in QMC calculations originates
|
||||
from the Jastrow factor. Usually, the Jastrow factor contains
|
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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).
|
||||
|
||||
%============================
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\section{Spin invariance}
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\label{app:spin}
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%============================
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%%% TABLE IV %%%
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\begin{table}
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\caption{FN-DMC energy (in hartree) for various $\mu$ values of the triplet carbon atom with
|
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different values of $m_s$ computed with the VDZ-BFD basis set and the srPBE functional.}
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\label{tab:spin}
|
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\begin{ruledtabular}
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\begin{tabular}{cccc}
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$\mu$ & $m_s=1$ & $m_s=0$ & Spin-invariance error \\
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\hline
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0.00 & $-5.416\,8(1)$ & $-5.414\,9(1)$ & $+0.001\,9(2)$ \\
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0.25 & $-5.417\,2(1)$ & $-5.416\,5(1)$ & $+0.000\,7(1)$ \\
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0.50 & $-5.422\,3(1)$ & $-5.421\,4(1)$ & $+0.000\,9(2)$ \\
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1.00 & $-5.429\,7(1)$ & $-5.429\,2(1)$ & $+0.000\,5(2)$ \\
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2.00 & $-5.432\,1(1)$ & $-5.431\,4(1)$ & $+0.000\,7(2)$ \\
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5.00 & $-5.431\,7(1)$ & $-5.431\,4(1)$ & $+0.000\,3(2)$ \\
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$\infty$ & $-5.431\,6(1)$ & $-5.431\,3(1)$ & $+0.000\,3(2)$ \\
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\end{tabular}
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\end{ruledtabular}
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\end{table}
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%%% %%% %%% %%%
|
||||
|
||||
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).
|
||||
|
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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$^{-1}$.
|
||||
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\titou{T2: what do you conclude from this section? What value of $m_s$ do you use to compute the atoms?}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\bibliography{rsdft-cipsi-qmc}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\end{document}
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Reference in New Issue