saving work in appendix B

Manuscript/rsdft-cipsi-qmc.tex

@@ -706,7 +706,7 @@ and therefore that the operators that produced these wave functions (\ie, $H^\mu
 Considering the form of $\hat{H}^\mu[n]$ [see Eq.~\eqref{H_mu}],
 one can notice that the differences with respect to the usual bare Hamiltonian come
 from the non-divergent two-body interaction $\hat{W}_{\text{ee}}^{\text{lr},\mu}$
-and the effective one-body potential  $\hat{\bar{V}}_{\text{Hxc}}^{\text{sr},\mu}[n]$ which is the functional derivative of the Hartree-exchange-correlation functional.
+and the effective one-body potential $\hat{\bar{V}}_{\text{Hxc}}^{\text{sr},\mu}[n]$ which is the functional derivative of the Hartree-exchange-correlation functional.
 The roles of these two terms are therefore very different: with respect
 to the exact ground-state wave function $\Psi$, the non-divergent two-body interaction
 increases the probability to find electrons at short distances in $\Psi^\mu$,
@@ -716,11 +716,11 @@ This is clearly what has been observed in
 Fig.~\ref{fig:densities}.
 Regarding now the transcorrelated Hamiltonian $e^{-J}He^J$, as pointed out by Ten-no,\cite{Tenno_2000}
 the effective two-body interaction induced by the presence of a Jastrow factor
-can be non-divergent when a proper Jastrow factor is chosen, \ie, the Jastrow factor must fulfill the so-called electron-electron cusp conditions. \cite{Kato_1957,Pack_1966}
-\titou{T2: I think we are missing the point here that the one-body Jastrow mimics the  effective one-body potential which makes the one-body density fixed.
-The two-body Jastrow makes the interaction non-divergent like the non-divergent two-body interaction in RS-DFT.
-Therefore, the one-body terms take care of the one-body properties and the two-body terms take care of the two-body properties. QED.}
-Therefore, one can understand the similarity between the eigenfunctions of $H^\mu$ and the Slater-Jastrow optimization:
+can be non-divergent when a proper two-body Jastrow factor $J_\text{ee}$ is chosen, \ie, the Jastrow factor must fulfill the so-called electron-electron cusp conditions. \cite{Kato_1957,Pack_1966}
+There is therefore a clear parallel between $\hat{W}_{\text{ee}}^{\text{lr},\mu}$ in RS-DFT and $J_\text{ee}$ in FN-DMC.
+Moreover, the one-body Jastrow term $J_\text{eN}$ ensures that the one-body density remain unchanged when the CI coefficients are re-optmized in the presence of $J_\text{ee}$.
+There is then a second clear parallel between $\hat{\bar{V}}_{\text{Hxc}}^{\text{sr},\mu}[n]$ in RS-DFT and $J_\text{eN}$ in FN-DMC.
+Thus, one can understand the similarity between the eigenfunctions of $H^\mu$ and the Slater-Jastrow optimization:
 they both deal with an effective non-divergent interaction but still
 produce a reasonable one-body density.
 
@@ -828,7 +828,9 @@ 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
 PBE0) make the results more sensitive to the basis set, and reduce the
-accuracy.  Thanks to the single-reference character of these systems,
+accuracy.  Note that, due to the approximate nature of the xc functionals, 
+the MAEs associated with KS-DFT atomization energies do not converge towards zero and remain altered even in the CBS limit.
+Thanks to the single-reference character of these systems,
 the CCSD(T) energy is an excellent estimate of the FCI energy, as
 shown by the very good agreement of the MAE, MSE and RMSE of CCSDT(T)
 and FCI energies.
@@ -842,7 +844,7 @@ This significant imbalance at the VDZ-BFD level affects the nodal
 surfaces, because although the FN-DMC energies obtained with near-FCI
 trial wave functions are much lower than the single-determinant FN-DMC
 energies, the MAE obtained with FCI ($7.38\pm1.08$ kcal/mol) is
-larger than the \titou{single-determinant} MAE ($4.61\pm 0.34$ kcal/mol).
+larger than the \titou{single-determinant} MAE ($4.61\pm0.34$ kcal/mol).
 Using the FCI trial wave function the MSE is equal to the
 negative MAE which confirms that the atomization energies are systematically
 underestimated. This confirms that some of the basis set