Spin invariance
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@ 434,7 +434,7 @@ than the energies obtained with a single KohnSham determinant ($\mu=0$):


a gain of $3.2 \pm 0.6$~m\hartree{} at the doublezeta level and $7.2 \pm


0.3$~m\hartree{} at the triplezeta level are obtained for water, and


a gain of $18 \pm 3$~m\hartree{} for F$_2$. Interestingly, using the


RSDFTCIPSI trial wave function with a rangeseparation parameter


RSDFTCIPSI trial wave function with a rangeseparation parameter


$\mu=1.75$~bohr$^{1}$ with the doublezeta basis one can obtain for


water a FNDMC energy $2.6 \pm 0.7$~m\hartree{} lower than the energy


obtained with the FCI trial wave function. This can be explained by


@ 463,12 +463,12 @@ $\mu=\infty$.


\begin{figure}


\centering


\includegraphics[width=\columnwidth]{overlap.pdf}


\caption{Overlap of the RSDFT CI expansion with the


\caption{Overlap of the RSDFT CI expansion with the


CI expansion optimized in the presence of a Jastrow factor.}


\label{fig:overlap}


\end{figure}




This data confirms that RSDFTCIPSI can give improved CI coefficients


This data confirms that RSDFTCIPSI can give improved CI coefficients


with small basis sets, similarly to the common practice of


reoptimizing the trial wave function in the presence of the Jastrow


factor. To confirm that the introduction of srDFT has an impact on


@ 476,7 +476,7 @@ the CI coefficients similar to the Jastrow factor, we have made the


following numerical experiment. First, we extract the 200 determinants


with the largest weights in the FCI wave function out of a large CIPSI


calculation. Within this set of determinants, we diagonalize


selfconsistently the RSDFT Hamiltonian with different values of


selfconsistently the RSDFT Hamiltonian with different values of


$\mu$. This gives the CI expansions $\Psi^\mu$. Then, within the same


set of determinants we optimize the CI coefficients in the presence of


a simple one and twobody Jastrow factor. This gives the CI expansion


@ 504,7 +504,7 @@ atoms than molecules and atomization energies usually tend to be


underestimated with 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


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


@ 552,7 +552,7 @@ oneelectron, twoelectron and onenucleustwoelectron terms.


The problematic part is the twoelectron term, whose simplest form can


be expressed as


\begin{equation}


J_\text{ee} = \sum_i \sum_{j<i} \frac{a r_{ij}}{1 + b r_{ij}}.


J_\text{ee} = \sum_i \sum_{j<i} \frac{a\, r_{ij}}{1 + b\, r_{ij}}.


\end{equation}


The parameter


$a$ is determined by cusp conditions, and $b$ is obtained by energy


@ 606,12 +606,13 @@ Ref.~\onlinecite{Scemama_2015}).


In this section, we make a numerical verification that the produced


wave functions are sizeconsistent for a given rangeseparation


parameter.


We have computed the energy of the dissocited fluorine dimer, where


We have computed the 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:sizecons} shows that it is indeed the


case, so we can conclude that the proposed scheme provides


sizeconsistent FNDMC energies for all values of $\mu$.


sizeconsistent FNDMC energies for all values of $\mu$ (within


$2\times$ statistical error bars).






\subsection{Spininvariance}


@ 621,8 +622,8 @@ fragments. To get reliable atomization energies, it is important to


have a theory which is of comparable quality for openshell and


closedshell systems. A good test is to check that all the components


of a spin multiplet are degenerate.


FCI wave functions are invariant with respect to the spin quantum


number $m_s$, but the introduction of a


FCI wave functions have this property and give degenrate 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 samespin electron pairs are different from those


for the oppositespin pairs.\cite{Tenno_2004}


@ 632,9 +633,9 @@ is used.




Within DFT, the common density functionals make a difference for


samespin and oppositespin interactions. As DFT is a


singledeterminant theory, the functionals are designed to work in


with the highest value of $m_s$, and therefore different


values of $m_s$ lead to different energies.


singledeterminant 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 RSDFT, the determinantal expansions will be


impacted by this spurious effect, as opposed to FCI.




@ 671,7 +672,7 @@ bias is relatively small, more than one order of magnitude smaller


than the energy gained by reducing the fixednode 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 quickly


below 1~m\hartree when $\mu$ increases. As expected, with $\mu=\infty$


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$^{1}$.




@ 682,7 +683,7 @@ noticeable with $\mu=5$~bohr$^{1}$.




The 55 molecules of the benchmark for the Gaussian1


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


of the RSDFTCIPSI trial wave functions for energy differences.


of the RSDFTCIPSI trial wave functions for energy differences.








@ 704,7 +705,7 @@ B3LYP & 0 & 11.27 & 10.98 & 9.59


\hline


CCSD(T) & \(\infty\) & 24.10 & 23.96 & 13.03 & 9.11 & 9.10 & 5.55 & 4.52 & 4.38 & 3.60 \\


\hline


RSDFTCIPSI & 0 & 4.53 & 1.66 & 5.91 & 6.31 & 0.91 & 7.93 & 6.35 & 3.88 & 7.20 \\


RSDFTCIPSI & 0 & 4.53 & 1.66 & 5.91 & 6.31 & 0.91 & 7.93 & 6.35 & 3.88 & 7.20 \\


\phantom{} & 1/4 & 5.55 & 4.66 & 5.52 & 4.58 & 1.06 & 5.72 & 5.48 & 1.52 & 6.93 \\


\phantom{} & 1/2 & 13.42 & 13.27 & 7.36 & 6.77 & 6.71 & 4.56 & 6.35 & 5.89 & 5.18 \\


\phantom{} & 1 & 17.07 & 16.92 & 9.83 & 9.06 & 9.06 & 5.88 &  &  &  \\


@ 712,7 +713,7 @@ RSDFTCIPSI & 0 & 4.53 & 1.66 & 5.91


\phantom{} & 5 & 22.93 & 22.79 & 13.24 &  &  &  &  &  &  \\


\phantom{} & \(\infty\) & 23.62 & 23.48 & 12.81 &  &  &  &  &  &  \\


\hline


DMC@RSDFTCIPSI & 0 & 4.61(34) & 3.62(\phantom{0.}34) & 5.30 & 3.52(19) & 1.03(19) & 4.39 & 3.16(26) & 0.12(26) & 4.12 \\


DMC@RSDFTCIPSI & 0 & 4.61(34) & 3.62(\phantom{0.}34) & 5.30 & 3.52(19) & 1.03(19) & 4.39 & 3.16(26) & 0.12(26) & 4.12 \\


\phantom{} & 1/4 & 4.04(37) & 3.13(\phantom{0.}37) & 4.88 & 3.39(77) & 0.59(77) & 4.44 & 2.90(25) & 0.25(25) & 3.74 \\


\phantom{} & 1/2 & 3.74(35) & 3.53(\phantom{0.}35) & 4.03 & 2.46(18) & 1.72(18) & 3.02 & 2.06(35) & 0.44(35) & 2.74 \\


\phantom{} & 1 & 5.42(29) & 5.14(\phantom{0.}29) & 4.55 & 4.38(94) & 4.24(94) & 5.11 &  &  &  \\


@ 820,7 +821,7 @@ Simulation of Functional Materials.


\item Exponential increase of number of Slater determinants


\item Cite papiers RSDFT: there exists an hybrid scheme combining fast convergence wr to basis set (non divergent basis set) and short expansion in SCI (cite papier Ferté)


\item Question: does such a scheme provide better nodal quality ?


\item In RSDFT we cannot optimize energy with $\mu$ , but in FNDMC


\item In RSDFT we cannot optimize energy with $\mu$ , but in FNDMC


\item Factual stuffs: with optimal $\mu$, lower FNDMC energy than HF/KS/FCI


\begin{itemize}


\item less determinants $\Rightarrow$ large systems



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