From def20873190341a950bba683eca08d00f40b1755 Mon Sep 17 00:00:00 2001
From: Anthony Scemama <scemama@irsamc.ups-tlse.fr>
Date: Sun, 2 Aug 2020 16:25:57 +0200
Subject: [PATCH] Spin invariance

---
 Manuscript/rsdft-cipsi-qmc.tex | 37 +++++++++++++++++-----------------
 1 file changed, 19 insertions(+), 18 deletions(-)

diff --git a/Manuscript/rsdft-cipsi-qmc.tex b/Manuscript/rsdft-cipsi-qmc.tex
index fb25e9b..3439047 100644
--- a/Manuscript/rsdft-cipsi-qmc.tex
+++ b/Manuscript/rsdft-cipsi-qmc.tex
@@ -434,7 +434,7 @@ than the energies obtained with a single Kohn-Sham determinant ($\mu=0$):
 a gain of $3.2 \pm 0.6$~m\hartree{} at the double-zeta level and $7.2 \pm
 0.3$~m\hartree{} at the triple-zeta level are obtained for water, and
 a gain of $18 \pm 3$~m\hartree{} for F$_2$. Interestingly, using the
-RSDFT-CIPSI trial wave function with a range-separation parameter
+RS-DFT-CIPSI trial wave function with a range-separation parameter
 $\mu=1.75$~bohr$^{-1}$ with the double-zeta basis one can obtain for
 water a FN-DMC 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 RS-DFT CI expansion with the
     CI expansion optimized in the presence of a Jastrow factor.}
   \label{fig:overlap}
 \end{figure}
 
-This data confirms that RSDFT-CIPSI can give improved CI coefficients
+This data confirms that RS-DFT-CIPSI can give improved CI coefficients
 with small basis sets, similarly to the common practice of
 re-optimizing the trial wave function in the presence of the Jastrow
 factor. To confirm that the introduction of sr-DFT 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
-self-consistently the RSDFT Hamiltonian with different values of
+self-consistently the RS-DFT 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 two-body 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 FN-DMC calculations, the nodal surface is imposed by
 the trial wavefunction which is expanded on an atom-centered basis
-set. So we expect the fixed-node error to be also tightly related to
+set, so we expect the fixed-node 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 @@ 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
 \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 size-consistent for a given range-separation
 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: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 $\mu$.
+size-consistent FN-DMC energies for all values of $\mu$ (within
+$2\times$ statistical error bars).
 
 
 \subsection{Spin-invariance}
@@ -621,8 +622,8 @@ 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.
-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 same-spin electron pairs are different from those
 for the opposite-spin pairs.\cite{Tenno_2004}
@@ -632,9 +633,9 @@ 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 functionals are designed to work in
-with the highest value of $m_s$, and therefore different
-values of $m_s$ lead to different energies.
+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.
 
@@ -671,7 +672,7 @@ 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 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 Gaussian-1
 theory\cite{Pople_1989,Curtiss_1990} were chosen to test the quality
-of the RSDFT-CIPSI trial wave functions for energy differences.
+of the RS-DFT-CIPSI 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
-RSDFT-CIPSI      &  0           &  4.53        &  -1.66                  &  5.91        &  6.31        &  0.91       &  7.93        &  6.35        &  3.88       &  7.20        \\
+RS-DFT-CIPSI      &  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 @@ RSDFT-CIPSI      &  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@RSDFT-CIPSI  &  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@RS-DFT-CIPSI  &  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 RS-DFT: 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 RS-DFT 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