diff --git a/Manuscript/rsdft-cipsi-qmc.tex b/Manuscript/rsdft-cipsi-qmc.tex index 1254a2f..b4832b0 100644 --- a/Manuscript/rsdft-cipsi-qmc.tex +++ b/Manuscript/rsdft-cipsi-qmc.tex @@ -251,7 +251,7 @@ estimate of the FCI energy, using a fixed value of the PT2 correction as a stopping criterion enforces a constant distance of all the calculations to the FCI energy. In this work, we target the chemical accuracy so all the CIPSI selections were made such that $\abs{\EPT} < -1$ millihartree. +1$ m\hartree{}. @@ -279,7 +279,7 @@ The main idea behind RS-DFT is to treat the short-range part of the interaction within KS-DFT, and the long-range part within a WFT method like FCI in the present case. The parameter $\mu$ controls the range of the separation, and allows to go continuously from the KS Hamiltonian ($\mu=0$) to -the FCI Hamiltonian ($\mu \to \infty$). +the FCI Hamiltonian ($\mu = \infty$). To rigorously connect WFT and DFT, the universal Levy-Lieb density functional \cite{Lev-PNAS-79,Lie-IJQC-83} is @@ -331,7 +331,7 @@ energy is obtained as Note that, for $\mu=0$, \titou{the long-range interaction vanishes}, \ie, $w_{\text{ee}}^{\text{lr},\mu=0}(r) = 0$, and thus RS-DFT reduces to standard KS-DFT and $\Psi^\mu$ -is the KS determinant. For $\mu\to\infty$, the long-range +is the KS determinant. For $\mu = \infty$, the long-range interaction becomes the standard Coulomb interaction, \ie, $w_{\text{ee}}^{\text{lr},\mu\to\infty}(r) = r^{-1}$, and thus RS-DFT reduces to standard WFT and $\Psi^\mu$ is @@ -369,12 +369,12 @@ In the outer (macro-iteration) loop (red), at the $k$th iteration, a CIPSI selec to obtain $\Psi^{\mu\,(k)}$ with the RS-DFT Hamiltonian $\hat{H}^{\mu\,(k)}$ parameterized using the current one-electron density $n^{(k)}$. At each iteration, the number of determinants in $\Psi^{\mu\,(k)}$ increases. -One exits the outer loop when the absolute energy difference between two successive macro-iterations $\Delta E^{(k)}$ is below a threshold $\tau_1$ that has been set to $10^{-3}$ hartree in the present study which is consistent with the CIPSI threshold (see Sec.~\ref{sec:comp-details}). +One exits the outer loop when the absolute energy difference between two successive macro-iterations $\Delta E^{(k)}$ is below a threshold $\tau_1$ that has been set to $10^{-3}$ \hartree{} in the present study which is consistent with the CIPSI threshold (see Sec.~\ref{sec:comp-details}). An inner (micro-iteration) loop (blue) is introduced to accelerate the convergence of the self-consistent calculation, in which the set of determinants in $\Psi^{\mu\,(k,l)}$ is kept fixed, and only the diagonalization of $\hat{H}^{\mu\,(k,l)}$ is performed iteratively with the updated density $n^{(k,l)}$. -The inner loop is exited when the absolute energy difference between two successive micro-iterations $\Delta E^{(k,l)}$ is below a threshold $\tau_2$ that has been here set to $10^{-2} \times \tau_1$ hartree. +The inner loop is exited when the absolute energy difference between two successive micro-iterations $\Delta E^{(k,l)}$ is below a threshold $\tau_2$ that has been here set to $10^{-2} \times \tau_1$ \hartree{}. The convergence of the algorithm was further improved by introducing a direct inversion in the iterative subspace (DIIS) step to extrapolate the density both in the outer and inner loops. \cite{Pulay_1980,Pulay_1982} @@ -407,7 +407,7 @@ and correlation functionals defined in Ref.~\onlinecite{GolWerStoLeiGorSav-CP-06} (see also Refs.~\onlinecite{TouColSav-JCP-05,GolWerSto-PCCP-05}). The convergence criterion for stopping the CIPSI calculations -has been set to $\EPT < 10^{-3}$ hartree or $ \Ndet > 10^7$. +has been set to $\EPT < 10^{-3}$ \hartree{} or $ \Ndet > 10^7$. All the wave functions are eigenfunctions of the $\Hat{S}^2$ spin operator, as described in Ref.~\onlinecite{Applencourt_2018}. @@ -431,7 +431,8 @@ with a time step of $5 \times 10^{-4}$ a.u. %%% TABLE I %%% \begin{table} - \caption{Fixed-node energies $\EDMC$ (in hartree) and number of determinants $\Ndet$ in \ce{H2O} with various trial wave functions $\Psi^{\mu}$.} + \caption{Fixed-node energy $\EDMC$ (in \hartree{}) and number of determinants $\Ndet$ in \ce{H2O} for various trial wave functions $\Psi^{\mu}$. + \titou{srPBE?}.} \label{tab:h2o-dmc} \centering \begin{ruledtabular} @@ -461,9 +462,9 @@ with a time step of $5 \times 10^{-4}$ a.u. \begin{figure} \centering \includegraphics[width=\columnwidth]{h2o-dmc.pdf} - \caption{Fixed-node energies of \ce{H2O} for different - values of $\mu$, using the srLDA or srPBE density - functionals to build the trial wave function.} + \caption{Fixed-node energy of \ce{H2O} as a function + of $\mu$ for various levels of theory to generate + the trial wave function.} \label{fig:h2o-dmc} \end{figure} %%% %%% %%% %%% @@ -481,31 +482,34 @@ parameters having an impact on the nodal surface fixed (\titou{such as ??}). \subsection{Fixed-node energy of $\Psi^\mu$} \label{sec:fndmc_mu} %====================================================== -From Table~\ref{tab:h2o-dmc} and Fig.~\ref{fig:h2o-dmc}, +From Table~\ref{tab:h2o-dmc} and Fig.~\ref{fig:h2o-dmc} where we report the fixed-node energy of \ce{H2O} as a function of $\mu$ for various short-range density functionals and basis sets, one can clearly observe that relying on FCI trial -wave functions ($\mu \to \infty$) give FN-DMC energies lower +wave functions ($\mu = \infty$) give FN-DMC energies lower than the energies obtained with a single KS 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. +a lowering 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. Coming now to the nodes of the trial wave functions $\Psi^{\mu}$ with intermediate values of $\mu$, Fig.~\ref{fig:h2o-dmc} shows that a smooth behaviour is obtained: -starting from $\mu=0$ (\textit{i.e.} the KS determinant), -the FN-DMC error is reduced continuously until it reaches a minimum for an optimal value of $\mu$, -and then the FN-DMC error raises until it reaches the $\mu=\infty$ limit (\textit{i.e.} the FCI wave function). -For instance, with respect to the FN-DMC energy of the FCI trial wave function in the double zeta basis set, -with the optimal value of $\mu$, one can obtain a lowering of the -FN-DMC energy of $2.6 \pm 0.7$~m\hartree{}, with an optimal value at -$\mu=1.75$~bohr$^{-1}$. +starting from $\mu=0$ (\ie, the KS determinant), +the FN-DMC error is reduced continuously until it reaches a minimum +for an optimal value of $\mu$ (which is obviously basis set and functional dependent), +and then the FN-DMC error raises until it reaches the $\mu=\infty$ limit (\ie, the FCI wave function). +For instance, with respect to the FN-DMC/VDZ-BFD energy at $\mu=\infty$, +one can obtain a lowering of the FN-DMC energy of $2.6 \pm 0.7$~m\hartree{} +with an optimal value of $\mu=1.75$~bohr$^{-1}$. When the basis set is increased, the gain in FN-DMC energy with respect to the FCI trial wave function is reduced, and the optimal value of $\mu$ is slightly shifted towards large $\mu$. Eventually, the nodes -of the wave functions $\Psi^\mu$ obtained with the short-range -LDA exchange-correlation functional give very similar FN-DMC energies with respect -to those obtained with the short-range PBE functional, even if the +of the wave functions $\Psi^\mu$ obtained with the srLDA +exchange-correlation functional give very similar FN-DMC energies with respect +to those obtained with the srPBE functional, even if the RS-DFT energies obtained with these two functionals differ by several tens of m\hartree{}. +\titou{The key fact here is that, at $\mu=1.75$~bohr$^{-1}$, $\Psi^{\mu}$ has \textit{only} $54\,540$ determinants at the srPBE/VDZ-BFD level, while the FCI wave function contains $200\,521$ determinants (see Table \ref{tab:h2o-dmc}). Even at the srPBE/VTZ-BFD level, we observe a reduction by a factor two in the number of determinants between the optimal $\mu$ value and $\mu = \infty$. +The take-home message here is that RS-DFT trial wave functions yield a lower fixed-node energy with more compact multideterminant expansion as compared to FCI.} + %====================================================== \subsection{Link between RS-DFT and Jastrow factors } \label{sec:rsdft-j} @@ -548,9 +552,9 @@ Then, within the same set of determinants we optimize the CI coefficients in the a simple one- and two-body Jastrow factor. This gives the CI expansion $\Psi^J$. Then, we can easily compare $\Psi^\mu$ and $\Psi^J$ as they are developed on the same Slater determinant basis. -In figure~\ref{fig:overlap}, we plot the overlaps +In Fig.~\ref{fig:overlap}, we plot the overlaps $\braket{\Psi^J}{\Psi^\mu}$ obtained for water and the fluorine dimer, -and in figure~\ref{dmc_small} the FN-DMC energy of the wave functions +and in Fig.~\ref{dmc_small} the FN-DMC energy of the wave functions $\Psi^\mu$ together with that of $\Psi^J$. %%% FIG 3 %%% @@ -663,7 +667,7 @@ $\mu=0.5$ and $\mu=1$~bohr$^{-1}$, constently with the FN-DMC energies and the overlap curve. 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 (\textit{i.e.} $H^\mu[n]$ and $e^{-J}He^J$) contain similar physics. +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 from the non-divergent two-body interaction $\hat{W}_{\text{ee}}^{\text{lr},\mu}$ @@ -694,7 +698,7 @@ As a conclusion of the first part of this study, we can notice that: \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, - \textit{i.e.} mimicking short-range correlation effects. The latter + \ie, mimicking short-range correlation effects. The latter statement can be qualitatively understood by noticing that both RS-DFT and transcorrelated approaches deal with an effective non-divergent electron-electron interaction, while keeping the density constant. @@ -764,7 +768,7 @@ 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