diff --git a/Manuscript/rsdft-cipsi-qmc.bib b/Manuscript/rsdft-cipsi-qmc.bib index de2df48..30e318e 100644 --- a/Manuscript/rsdft-cipsi-qmc.bib +++ b/Manuscript/rsdft-cipsi-qmc.bib @@ -1,13 +1,35 @@ %% This BibTeX bibliography file was created using BibDesk. %% http://bibdesk.sourceforge.net/ -%% Created for Pierre-Francois Loos at 2020-08-17 12:55:12 +0200 +%% Created for Pierre-Francois Loos at 2020-08-18 10:07:35 +0200 %% Saved with string encoding Unicode (UTF-8) +@article{Pack_1966, + Author = {{R. T. Pack and W. Byers-Brown}}, + Date-Added = {2020-08-18 09:51:18 +0200}, + Date-Modified = {2020-08-18 09:51:34 +0200}, + Journal = {J. Chem. Phys.}, + Pages = {556}, + Title = {{Cusp conditions for molecular wavefunctions}}, + Volume = {45}, + Year = {1966}} + +@article{Kato_1957, + Author = {T. Kato}, + Date-Added = {2020-08-18 09:50:39 +0200}, + Date-Modified = {2020-08-18 09:50:46 +0200}, + Doi = {10.1002/cpa.3160100201}, + Journal = {Comm. Pure Appl. Math.}, + Pages = {151}, + Title = {{On the eigenfunctions of many-particle systems in quantum mechanics}}, + Volume = {10}, + Year = {1957}, + Bdsk-Url-1 = {https://doi.org/10.1002/cpa.3160100201}} + @article{Loos_2015b, Author = {Loos, Pierre-Fran{\c c}ois and Bressanini, Dario}, Date-Added = {2020-08-17 12:54:30 +0200}, @@ -1487,8 +1509,9 @@ Year = {2004}, Bdsk-Url-1 = {https://doi.org/10.1063/1.1757439}} -@article{Ten-no2000Nov, +@article{Tenno_2000, Author = {Ten-no, Seiichiro}, + Date-Modified = {2020-08-18 09:50:06 +0200}, Doi = {10.1016/S0009-2614(00)01066-6}, Issn = {0009-2614}, Journal = {Chem. Phys. Lett.}, diff --git a/Manuscript/rsdft-cipsi-qmc.tex b/Manuscript/rsdft-cipsi-qmc.tex index 8deb310..d0e425d 100644 --- a/Manuscript/rsdft-cipsi-qmc.tex +++ b/Manuscript/rsdft-cipsi-qmc.tex @@ -568,7 +568,7 @@ Such a wave function satisfies the generalized Hermitian eigenvalue equation e^{J} \hat{H} \qty( e^{J} \Psi^J ) = E \, e^{2J} \Psi^J, \label{eq:ci-j} \end{equation} -but also the non-Hermitian transcorrelated eigenvalue problem\cite{BoyHan-PRSLA-69,BoyHanLin-1-PRSLA-69,BoyHanLin-2-PRSLA-69,Ten-no2000Nov,Luo-JCP-10,YanShi-JCP-12,CohLuoGutDowTewAla-JCP-19} +but also the non-Hermitian transcorrelated eigenvalue problem\cite{BoyHan-PRSLA-69,BoyHanLin-1-PRSLA-69,BoyHanLin-2-PRSLA-69,Tenno_2000,Luo-JCP-10,YanShi-JCP-12,CohLuoGutDowTewAla-JCP-19} \begin{equation} \label{eq:transcor} e^{-J} \hat{H} \qty( e^{J} \Psi^J) = E \, \Psi^J, @@ -647,7 +647,7 @@ This is yet another key result of the present study. \begin{table} \caption{\ce{H2O}, double-zeta basis set. Integrated on-top pair density $\expval{ P }$ for $\Psi^J$ and $\Psi^\mu$ with different values of $\mu$. - \titou{Please remove table and merge data in Fig. 5.}} + \titou{Please remove table and merge data in Fig. 4.}} \label{tab:table_on_top} \begin{ruledtabular} \begin{tabular}{cc} @@ -697,52 +697,56 @@ From these data, one can clearly notice several trends. First, from Table~\ref{tab:table_on_top}, we can observe that the overall on-top pair density decreases when $\mu$ increases, which is expected as the two-electron interaction increases in $H^\mu[n]$. -Second, the relative variations of the on-top pair density with $\mu$ +Second, Fig.~\ref{fig:densities} shows that the relative variations of the on-top pair density with respect to $\mu$ are much more important than that of the one-body density, the latter being essentially unchanged between $\mu=0$ and $\mu=\infty$ while the former can vary by about 10$\%$ in some regions. %TODO TOTO In the high-density region of the \ce{O-H} bond, the value of the on-top pair density obtained from $\Psi^J$ is superimposed with -$\Psi^{\mu=0.5}$, and at a large distance the on-top pair density is +$\Psi^{\mu=0.5}$, and at a large distance the on-top pair density of $\Psi^J$ is the closest to $\mu=\infty$. The integrated on-top pair density obtained with $\Psi^J$ lies between the values obtained with -$\mu=0.5$ and $\mu=1$~bohr$^{-1}$, consistently with the FN-DMC energies +$\mu=0.5$ and $\mu=1$~bohr$^{-1}$ (see Table~\ref{tab:table_on_top}), consistently with the FN-DMC energies and the overlap curve depicted in Fig.~\ref{fig:overlap}. These data suggest that the wave functions $\Psi^{0.5 \le \mu \le 1}$ and $\Psi^J$ are close, and therefore that the operators that produced these wave functions (\ie, $H^\mu[n]$ and $e^{-J}He^J$) contain similar physics. Considering the form of $\hat{H}^\mu[n]$ [see Eq.~\eqref{H_mu}], -one can notice that the differences with respect to the usual Hamiltonian come +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. 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 $\Psi^\mu$, while the effective one-body potential $\hat{\bar{V}}_{\text{Hxc}}^{\text{sr},\mu}[n_{\Psi^{\mu}}]$, -provided that it is exact, maintains the exact one-body density. -This is clearly what has been observed from +providing that it is exact, maintains the exact one-body density. +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{Ten-no2000Nov} +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. +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: they both deal with an effective non-divergent interaction but still produce a reasonable one-body density. -As a conclusion of the first part of this study, we can notice that: +%============================ +\subsection{Intermediate conclusion} +%============================ + +As a conclusion of the first part of this study, we can highlight the following observations: \begin{itemize} -\item with respect to the nodes of a KS determinant or a FCI wave function, +\item With respect to the nodes of a KS determinant or a FCI wave function, one can obtain a multi-determinant trial wave function $\Psi^\mu$ with a smaller - fixed-node error by properly choosing an optimal value of $\mu$ - in RS-DFT calculations, -\item the optimal value of $\mu$ depends on the system and the - basis set, and the larger the basis set, the larger the optimal value - of $\mu$, -\item numerical experiments (overlap $\braket*{\Psi^\mu}{\Psi^J}$, + fixed-node error by properly choosing an optimal value of $\mu$. +\item The optimal $\mu$ value is system- and basis-set-dependent, and it grows with basis set size. +\item Numerical experiments (overlap $\braket*{\Psi^\mu}{\Psi^J}$, one-body density, on-top pair density, and FN-DMC energy) indicate - that the RS-DFT scheme essentially plays the role of a simple Jastrow factor, - \ie, mimicking short-range correlation effects. The latter + that the RS-DFT scheme essentially plays the role of a simple Jastrow factor + by mimicking short-range correlation effects. This latter statement can be qualitatively understood by noticing that both RS-DFT and the trans-correlated approach deal with an effective non-divergent electron-electron interaction, while keeping the density constant.