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%% This BibTeX bibliography file was created using BibDesk.
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%% http://bibdesk.sourceforge.net/
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%% Created for Pierre-Francois Loos at 2020-08-03 21:30:38 +0200
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%% Created for Pierre-Francois Loos at 2020-08-07 13:35:24 +0200
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%% Saved with string encoding Unicode (UTF-8)
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@article{Burkatzki_2007,
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Author = {Burkatzki, M. and Filippi, C. and Dolg, M.},
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Date-Added = {2020-08-07 13:35:15 +0200},
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Date-Modified = {2020-08-07 13:35:15 +0200},
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Doi = {10.1063/1.2741534},
|
||||
Issn = {1089-7690},
|
||||
Journal = {J. Chem. Phys.},
|
||||
Month = {Jun},
|
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Number = {23},
|
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Pages = {234105},
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Publisher = {AIP Publishing},
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Title = {Energy-consistent pseudopotentials for quantum Monte Carlo calculations},
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Url = {http://dx.doi.org/10.1063/1.2741534},
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Volume = {126},
|
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Year = {2007},
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Bdsk-Url-1 = {http://dx.doi.org/10.1063/1.2741534}}
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@article{Burkatzki_2008,
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Author = {Burkatzki, M. and Filippi, Claudia and Dolg, M.},
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||||
Date-Added = {2020-08-07 13:35:15 +0200},
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Date-Modified = {2020-08-07 13:35:15 +0200},
|
||||
Doi = {10.1063/1.2987872},
|
||||
Issn = {1089-7690},
|
||||
Journal = {J. Chem. Phys.},
|
||||
Month = {Oct},
|
||||
Number = {16},
|
||||
Pages = {164115},
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||||
Publisher = {AIP Publishing},
|
||||
Title = {Energy-consistent small-core pseudopotentials for 3d-transition metals adapted to quantum Monte Carlo calculations},
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||||
Url = {http://dx.doi.org/10.1063/1.2987872},
|
||||
Volume = {129},
|
||||
Year = {2008},
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||||
Bdsk-Url-1 = {http://dx.doi.org/10.1063/1.2987872}}
|
||||
|
||||
@article{Schrodinger_1926,
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Author = {Schr\"odinger, Erwin},
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||||
Date-Added = {2020-08-03 21:29:29 +0200},
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@ -17,7 +51,8 @@
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Pages = {1049--1070},
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Title = {An Undulatory Theory of the Mechanics of Atoms and Molecules},
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Volume = {28},
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||||
Year = {1926}}
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Year = {1926},
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||||
Bdsk-Url-1 = {https://doi.org/10.1103/PhysRev.28.1049}}
|
||||
|
||||
@article{Evangelista_2014,
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Author = {Evangelista, Francesco A.},
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@ -367,21 +402,6 @@
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Title = {Gaussian˜16 {R}evision {C}.01},
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Year = {2016}}
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@article{Burkatzki_2008,
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||||
Author = {Burkatzki, M. and Filippi, Claudia and Dolg, M.},
|
||||
Doi = {10.1063/1.2987872},
|
||||
Issn = {1089-7690},
|
||||
Journal = {The Journal of Chemical Physics},
|
||||
Month = {Oct},
|
||||
Number = {16},
|
||||
Pages = {164115},
|
||||
Publisher = {AIP Publishing},
|
||||
Title = {Energy-consistent small-core pseudopotentials for 3d-transition metals adapted to quantum Monte Carlo calculations},
|
||||
Url = {http://dx.doi.org/10.1063/1.2987872},
|
||||
Volume = {129},
|
||||
Year = {2008},
|
||||
Bdsk-Url-1 = {http://dx.doi.org/10.1063/1.2987872}}
|
||||
|
||||
@article{Garniron_2019,
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||||
Author = {Garniron, Yann and Applencourt, Thomas and Gasperich, Kevin and Benali, Anouar and Fert{\'e}, Anthony and Paquier, Julien and Pradines, Barth{\'e}l{\'e}my and Assaraf, Roland and Reinhardt, Peter and Toulouse, Julien and Barbaresco, Pierrette and Renon, Nicolas and David, Gr{\'e}goire and Malrieu, Jean-Paul and V{\'e}ril, Micka{\"e}l and Caffarel, Michel and Loos, Pierre-Fran{\c c}ois and Giner, Emmanuel and Scemama, Anthony},
|
||||
Doi = {10.1021/acs.jctc.9b00176},
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@ -924,45 +944,42 @@
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Bdsk-Url-1 = {https://doi.org/10.13140/RG.2.1.3187.9766}}
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|
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@article{Tenno_2004,
|
||||
author = {Ten-no, Seiichiro},
|
||||
title = {{Explicitly correlated second order perturbation theory: Introduction of a
|
||||
rational generator and numerical quadratures}},
|
||||
journal = {J. Chem. Phys.},
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volume = {121},
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number = {1},
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pages = {117--129},
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||||
year = {2004},
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month = {Jul},
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issn = {0021-9606},
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publisher = {American Institute of Physics},
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||||
doi = {10.1063/1.1757439}
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}
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Author = {Ten-no, Seiichiro},
|
||||
Doi = {10.1063/1.1757439},
|
||||
Issn = {0021-9606},
|
||||
Journal = {J. Chem. Phys.},
|
||||
Month = {Jul},
|
||||
Number = {1},
|
||||
Pages = {117--129},
|
||||
Publisher = {American Institute of Physics},
|
||||
Title = {{Explicitly correlated second order perturbation theory: Introduction of a rational generator and numerical quadratures}},
|
||||
Volume = {121},
|
||||
Year = {2004},
|
||||
Bdsk-Url-1 = {https://doi.org/10.1063/1.1757439}}
|
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|
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@article{Ten-no2000Nov,
|
||||
author = {Ten-no, Seiichiro},
|
||||
title = {{A feasible transcorrelated method for treating electronic cusps using a frozen
|
||||
Gaussian geminal}},
|
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journal = {Chem. Phys. Lett.},
|
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volume = {330},
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number = {1},
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pages = {169--174},
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year = {2000},
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month = {Nov},
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issn = {0009-2614},
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publisher = {North-Holland},
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doi = {10.1016/S0009-2614(00)01066-6}
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}
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Author = {Ten-no, Seiichiro},
|
||||
Doi = {10.1016/S0009-2614(00)01066-6},
|
||||
Issn = {0009-2614},
|
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Journal = {Chem. Phys. Lett.},
|
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Month = {Nov},
|
||||
Number = {1},
|
||||
Pages = {169--174},
|
||||
Publisher = {North-Holland},
|
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Title = {{A feasible transcorrelated method for treating electronic cusps using a frozen Gaussian geminal}},
|
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Volume = {330},
|
||||
Year = {2000},
|
||||
Bdsk-Url-1 = {https://doi.org/10.1016/S0009-2614(00)01066-6}}
|
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|
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@article{Applencourt_2018,
|
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author = {Applencourt, Thomas and Gasperich, Kevin and
|
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Scemama, Anthony},
|
||||
title = {{Spin adaptation with determinant-based selected configuration interaction}},
|
||||
journal = {arXiv},
|
||||
year = {2018},
|
||||
month = {Dec},
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eprint = {1812.06902},
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url = {https://arxiv.org/abs/1812.06902v1}
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}
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Author = {Applencourt, Thomas and Gasperich, Kevin and Scemama, Anthony},
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Eprint = {1812.06902},
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Journal = {arXiv},
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||||
Month = {Dec},
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||||
Title = {{Spin adaptation with determinant-based selected configuration interaction}},
|
||||
Url = {https://arxiv.org/abs/1812.06902v1},
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||||
Year = {2018},
|
||||
Bdsk-Url-1 = {https://arxiv.org/abs/1812.06902v1}}
|
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|
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@software{qp2_2020,
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Author = {Anthony Scemama and Emmanuel Giner and Anouar Benali and Thomas Applencourt and Kevin Gasperich},
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@ -38,6 +38,7 @@
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\newcommand{\ANL}{Argonne Leadership Computing Facility, Argonne National Laboratory, Argonne, IL 60439, United States}
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\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}
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\DeclareMathOperator{\erfc}{erfc}
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\begin{document}
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@ -247,49 +248,52 @@ accuracy so all the CIPSI selections were made such that $|\EPT| <
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\label{sec:rsdft}
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Following the seminal work of Savin,\cite{Savin_1996,Toulouse_2004}
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the Coulomb electron-electron interaction is split into a short-range
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(sr) and a long range (lr) interaction as
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the Coulomb operator entering the interelectronic repulsion is split into two pieces:
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\begin{equation}
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\frac{1}{r_{ij}} = w_{\text{ee}}^{\text{lr}, \mu}(r_{ij}) + \qty(
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\frac{1}{r_{ij}} - w_{\text{ee}}^{\text{lr}, \mu}(r_{ij}) )
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\frac{1}{r}
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= w_{\text{ee}}^{\text{sr}, \mu}(r)
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+ w_{\text{ee}}^{\text{lr}, \mu}(r)
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\end{equation}
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where
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\begin{equation}
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w_{\text{ee}}^{\text{lr},\mu}(r_{ij}) = \frac{\erf \qty( \mu\, r_{ij})}{r_{ij}}
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\end{equation}
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The main idea is to treat the short-range electron-electron
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interaction with DFT, and the long range with wave function theory.
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The parameter $\mu$ controls the range of the separation, and allows
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to go continuously from the Kohn-Sham Hamiltonian ($\mu=0$) to
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the FCI Hamiltinoan ($\mu = \infty$).
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\begin{align}
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w_{\text{ee}}^{\text{sr},\mu}(r) & = \frac{\erfc \qty( \mu\, r)}{r},
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&
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w_{\text{ee}}^{\text{lr},\mu}(r) & = \frac{\erf \qty( \mu\, r)}{r}
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\end{align}
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are the singular short-range (sr) part and the non-singular long-range (lr) part, respectively, $\mu$ is the range-separation parameter which controls how rapidly the short-range part decays, $\erf(x)$ is the error function, and $\erfc(x) = 1 - \erf(x)$ is its complementary version.
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To rigorously connect wave function theory and DFT, the universal
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Levy-Lieb density functional\cite{Lev-PNAS-79,Lie-IJQC-83} is
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The main idea behind RS-DFT is to treat the short-range part of the
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interaction within KS-DFT, and the long range part within a WFT method like FCI in the present case.
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The parameter $\mu$ controls the range of the separation, and allows
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to go continuously from the KS Hamiltonian ($\mu=0$) to
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the FCI Hamiltonian ($\mu = \infty$).
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To rigorously connect WFT and DFT, the universal
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Levy-Lieb density functional \cite{Lev-PNAS-79,Lie-IJQC-83} is
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decomposed as
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\begin{equation}
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\mathcal{F}[n] = \mathcal{F}^{\mathrm{lr},\mu}[n] + \bar{E}_{\mathrm{Hxc}}^{\mathrm{sr,}\mu}[n],
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\mathcal{F}[n] = \mathcal{F}^{\text{lr},\mu}[n] + \bar{E}_{\text{Hxc}}^{\text{sr,}\mu}[n],
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\label{Fdecomp}
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\end{equation}
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where $n$ is a one-particle density,
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$\mathcal{F}^{\mathrm{lr},\mu}$ is a long-range universal density
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functional and $\bar{E}_{\mathrm{Hxc}}^{\mathrm{sr,}\mu}$ is the
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where $n$ is a one-electron density,
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$\mathcal{F}^{\text{lr},\mu}$ is a long-range universal density
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functional and $\bar{E}_{\text{Hxc}}^{\text{sr,}\mu}$ is the
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complementary short-range Hartree-exchange-correlation (Hxc) density
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functional\cite{Savin_1996,Toulouse_2004}.
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functional. \cite{Savin_1996,Toulouse_2004}
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One obtains the following expression for the ground-state
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electronic energy
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\begin{equation}
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\label{min_rsdft} E_0= \min_{\Psi} \left\{
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\left
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\langle\Psi|\hat{T}+\hat{W}_\mathrm{{ee}}^{\mathrm{lr},\mu}+\hat{V}_{\mathrm{ne}}|\Psi\right
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\rangle
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+ \bar{E}^{\mathrm{sr},\mu}_{\mathrm{Hxc}}[n_\Psi]\right\}
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\label{min_rsdft} E_0= \min_{\Psi} \qty{
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\mel{\Psi}{\hat{T}+\hat{W}_\text{{ee}}^{\text{lr},\mu}+\hat{V}_{\text{ne}}}{\Psi}
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+ \bar{E}^{\text{sr},\mu}_{\text{Hxc}}[n_\Psi]
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},
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\end{equation}
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with $\hat{T}$ the kinetic energy operator,
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$\hat{W}_\mathrm{ee}^{\mathrm{lr}}$ the long-range
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$\hat{W}_\text{ee}^{\text{lr}}$ the long-range
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electron-electron interaction,
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$n_\Psi$ the one-particle density associated with $\Psi$,
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and $\hat{V}_{\mathrm{ne}}$ the electron-nucleus potential.
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The minimizing multi-determinant wave function $\Psi^\mu$
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$n_\Psi$ the one-electron density associated with $\Psi$,
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and $\hat{V}_{\text{ne}}$ the electron-nucleus potential.
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The minimizing multideterminant wave function $\Psi^\mu$
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can be determined by the self-consistent eigenvalue equation
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\begin{equation}
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\label{rs-dft-eigen-equation}
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@ -298,50 +302,51 @@ can be determined by the self-consistent eigenvalue equation
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with the long-range interacting Hamiltonian
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\begin{equation}
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\label{H_mu}
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\hat{H}^\mu[n_{\Psi^{\mu}}] = \hat{T}+\hat{W}_{\mathrm{ee}}^{\mathrm{lr},\mu}+\hat{V}_{\mathrm{ne}}+ \hat{\bar{V}}_{\mathrm{Hxc}}^{\mathrm{sr},\mu}[n_{\Psi^{\mu}}],
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\hat{H}^\mu[n_{\Psi^{\mu}}] = \hat{T}+\hat{W}_{\text{ee}}^{\text{lr},\mu}+\hat{V}_{\text{ne}}+ \hat{\bar{V}}_{\text{Hxc}}^{\text{sr},\mu}[n_{\Psi^{\mu}}],
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\end{equation}
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where
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$\hat{\bar{V}}_{\mathrm{Hxc}}^{\mathrm{sr},\mu}$
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$\hat{\bar{V}}_{\text{Hxc}}^{\text{sr},\mu}$
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is the complementary short-range Hartree-exchange-correlation
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potential operator.
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Once $\Psi^{\mu}$ has been calculated, the electronic ground-state
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energy is obtained by
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energy is obtained as
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\begin{equation}
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\label{E-rsdft}
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E_0= \mel{\Psi^{\mu}}{\hat{T}+\hat{W}_\mathrm{{ee}}^{\mathrm{lr},\mu}+\hat{V}_{\mathrm{ne}}}{\Psi^{\mu}}+\bar{E}^{\mathrm{sr},\mu}_{\mathrm{Hxc}}[n_{\Psi^\mu}].
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E_0= \mel{\Psi^{\mu}}{\hat{T}+\hat{W}_\text{{ee}}^{\text{lr},\mu}+\hat{V}_{\text{ne}}}{\Psi^{\mu}}+\bar{E}^{\text{sr},\mu}_{\text{Hxc}}[n_{\Psi^\mu}].
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\end{equation}
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Note that, for $\mu=0$, the long-range interaction vanishes,
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$w_{\mathrm{ee}}^{\mathrm{lr},\mu=0}(r_{12}) = 0$, and thus
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range-separated DFT (RS-DFT) reduces to standard KS-DFT and $\Psi^\mu$
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Note that, for $\mu=0$, \titou{the long-range interaction vanishes}, \ie,
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$w_{\text{ee}}^{\text{lr},\mu=0}(r) = 0$, and thus
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RS-DFT reduces to standard KS-DFT and $\Psi^\mu$
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is the KS determinant. For $\mu\to\infty$, the long-range
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interaction becomes the standard Coulomb interaction,
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$w_{\mathrm{ee}}^{\mathrm{lr},\mu\to\infty}(r_{12}) = 1/r_{12}$, and
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thus RS-DFT reduces to standard wave-function theory and $\Psi^\mu$ is
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interaction becomes the standard Coulomb interaction, \ie,
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$w_{\text{ee}}^{\text{lr},\mu\to\infty}(r) = r^{-1}$, and
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thus RS-DFT reduces to standard WFT and $\Psi^\mu$ is
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the FCI wave function.
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\begin{figure*}
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\centering
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\includegraphics[width=0.7\linewidth]{algorithm.pdf}
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\caption{Algorithm showing the generation of the RS-DFT wave
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function.}
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function $\Psi^{\mu}$ starting from the .}
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\label{fig:algo}
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\end{figure*}
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Hence we have a continuous path connecting the KS determinant to the
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FCI wave function, and as the KS nodes are of higher quality than the
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Hence, range separation creates a continuous path connecting smoothly the KS determinant to the
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FCI wave function. Because the KS nodes are of higher quality than the
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HF nodes, we expect that using wave functions built along this path
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will always provide reduced fixed-node errors compared to the path
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connecting HF to FCI using an increasing number of selected
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determinants.
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connecting HF to FCI which consists in increasing the number of determinants.
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We can follow this path by performing FCI calculations using the
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RS-DFT Hamiltonian with different values of $\mu$. In this work, we
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We follow the KS-to-FCI path by performing FCI calculations using the
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RS-DFT Hamiltonian with different values of $\mu$.
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Our algorithm, depicted in Fig.~\ref{fig:algo}, starts with a
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single- and multi-determinant wave function $\Psi^{(0)}$.
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One of the particularity of the present work is that we
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have used the CIPSI algorithm to perform approximate FCI calculations
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with the RS-DFT Hamiltonians,\cite{GinPraFerAssSavTou-JCP-18}
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$\hat{H}^\mu$ as shown in figure~\ref{fig:algo}. In the outer loop
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(red), a CIPSI selection is performed with a RS-Hamiltonian
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parameterized using the current density.
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with the RS-DFT Hamiltonians $\hat{H}^\mu$. \cite{GinPraFerAssSavTou-JCP-18}
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In the outer loop (red), a CIPSI selection is performed with a RS Hamiltonian
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parameterized using the current one-electron density.
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An inner loop (blue) is introduced to accelerate the
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convergence of the self-consistent calculation, in which the set of
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determinants is kept fixed, and only the diagonalization of the
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@ -349,31 +354,37 @@ RS-Hamiltonian is performed iteratively with the updated density.
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The convergence of the algorithm was further improved
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by introducing a direct inversion in the iterative subspace (DIIS)
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step to extrapolate the density both in the outer and inner loops.
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Note that any range-separated post-Hartree-Fock method can be
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Note that any range-separated post-HF method can be
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implemented using this scheme by just replacing the CIPSI step by the
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post-HF method of interest.
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\titou{T2: introduce $\tau_1$ and $\tau_2$. More description of the algorithm needed.}
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Note that, thanks to the self-consistent nature of the algorithm,
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the final trial wave function $\Psi^{\mu}$ is independent of the starting wave function $\Psi^{(0)}$.
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\section{Computational details}
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\label{sec:comp-details}
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\titou{The geometries for the G2 data set.}
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All the calculations were made using BFD
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pseudopotentials\cite{Burkatzki_2008} with the associated double-,
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All the calculations have been performed using Burkatzki-Filippi-Dolg (BFD)
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pseudopotentials \cite{Burkatzki_2007,Burkatzki_2008} with the associated double-,
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triple-, and quadruple-$\zeta$ basis sets (BFD-VXZ).
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CCSD(T) and KS-DFT calculations were made with
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\emph{Gaussian09},\cite{g16} using an unrestricted Hartree-Fock
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determinant as a reference for open-shell systems.
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The small-core BFD pseudopotentials include scalar relativistic effects.
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CCSD(T) and KS-DFT energies have been computed with
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\emph{Gaussian09},\cite{g16} using the unrestricted formalism for open-shell systems.
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All the CIPSI calculations were made with \emph{Quantum
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All the CIPSI calculations have been performed with \emph{Quantum
|
||||
Package}.\cite{Garniron_2019,qp2_2020} We used the short-range version
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of the Perdew-Burke-Ernzerhof (PBE)~\cite{PerBurErn-PRL-96} exchange
|
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and correlation functionals of
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of the Perdew-Burke-Ernzerhof (PBE) \cite{PerBurErn-PRL-96} exchange
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and correlation functionals defined in
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Ref.~\onlinecite{GolWerStoLeiGorSav-CP-06} (see also
|
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Refs.~\onlinecite{TouColSav-JCP-05,GolWerSto-PCCP-05}).
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The convergence criterion for stopping the CIPSI calculations
|
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was $\EPT < 1$~m\hartree{} $\vee \Ndet > 10^7$.
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All the wave functions are eigenfunctions of the $S^2$ operator, as
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described in ref~\onlinecite{Applencourt_2018}.
|
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has been set to $\EPT < 1$~m\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}.
|
||||
|
||||
Quantum Monte Carlo calculations were made with QMC=Chem,\cite{scemama_2013}
|
||||
in the determinant localization approximation (DLA),\cite{Zen_2019}
|
||||
@ -383,15 +394,18 @@ the pseudopotential is localized. Hence, in the DLA the fixed-node
|
||||
energy is independent of the Jastrow factor, as in all-electron
|
||||
calculations. Simple Jastrow factors were used to reduce the
|
||||
fluctuations of the local energy.
|
||||
|
||||
\titou{time step $5 \times 10^{-4}$.
|
||||
All-electron move DMC.}
|
||||
|
||||
|
||||
|
||||
\section{Influence of the range-separation parameter on the fixed-node
|
||||
error}
|
||||
\label{sec:mu-dmc}
|
||||
|
||||
%%% TABLE I %%%
|
||||
\begin{table}
|
||||
\caption{Fixed-node energies (in hartree) and number of determinants in \ce{H2O} and \ce{F2} with various trial wave functions.}
|
||||
\caption{Fixed-node energies $\EDMC$ (in hartree) and number of determinants $\Ndet$ in \ce{H2O} and \ce{F2} with various trial wave functions.}
|
||||
\label{tab:h2o-dmc}
|
||||
\centering
|
||||
\begin{ruledtabular}
|
||||
@ -425,21 +439,23 @@ System & $\mu$ & $\Ndet$ & $\EDMC$ & $\Ndet$ & $\ED
|
||||
\end{tabular}
|
||||
\end{ruledtabular}
|
||||
\end{table}
|
||||
%%% %%% %%% %%%
|
||||
|
||||
\begin{figure}
|
||||
\centering
|
||||
\includegraphics[width=\columnwidth]{h2o-dmc.pdf}
|
||||
\caption{Fixed-node energies of the water molecule for different
|
||||
values of $\mu$, using the sr-LDA or sr-PBE short-range density
|
||||
functionals to build the trial wave function.}
|
||||
\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.
|
||||
\titou{Toto: please remove hyphens in sr-LDA and sr-PBE.}}
|
||||
\label{fig:h2o-dmc}
|
||||
\end{figure}
|
||||
|
||||
\begin{figure}
|
||||
\centering
|
||||
\includegraphics[width=\columnwidth]{f2-dmc.pdf}
|
||||
\caption{Fixed-node energies of difluorine for different
|
||||
values of $\mu$.}
|
||||
\caption{Fixed-node energies of \ce{F2} for different
|
||||
values of $\mu$ using the srPBE density functional to build the trial wave function.}
|
||||
\label{fig:f2-dmc}
|
||||
\end{figure}
|
||||
The first question we would like to address is the quality of the
|
||||
@ -449,12 +465,12 @@ We generated trial wave functions $\Psi^\mu$ with multiple values of
|
||||
$\mu$, and computed the associated fixed node energy keeping all the
|
||||
parameters having an impact on the nodal surface fixed.
|
||||
We considered two weakly correlated molecular systems: the water
|
||||
molecule and the fluorine dimer, near their equilibrium
|
||||
geometry\cite{Caffarel_2016}.
|
||||
molecule and the fluorine dimer, near their equilibrium.
|
||||
geometry\cite{Caffarel_2016}
|
||||
|
||||
\subsection{Fixed node energy of $\Psi^\mu$}
|
||||
\subsection{Fixed-node energy of $\Psi^\mu$}
|
||||
\label{sec:fndmc_mu}
|
||||
From table~\ref{tab:h2o-dmc} and figures~\ref{fig:h2o-dmc}
|
||||
From Table~\ref{tab:h2o-dmc} and Figs.~\ref{fig:h2o-dmc}
|
||||
and~\ref{fig:f2-dmc}, one can clearly observe that using FCI trial
|
||||
wave functions ($\mu = \infty$) gives FN-DMC energies which are lower
|
||||
than the energies obtained with a single Kohn-Sham determinant ($\mu=0$):
|
||||
@ -476,7 +492,7 @@ LDA exchange-correlation functional give very similar FN-DMC energy with respect
|
||||
to those obtained with the short-range PBE functional, even if the RS-DFT energies obtained
|
||||
with these two functionals differ by several tens of m\hartree{}.
|
||||
|
||||
\subsection{Link between RS-DFT and jastrow factors }
|
||||
\subsection{Link between RS-DFT and Jastrow factors }
|
||||
\label{sec:rsdft-j}
|
||||
The data obtained in \ref{sec:fndmc_mu} show that RS-DFT can provide CI coefficients
|
||||
giving trial wave functions with better nodes than FCI wave functions.
|
||||
@ -571,12 +587,12 @@ These data suggest that the wave functions $\Psi^\mu$ and $\Psi^J$ are similar,
|
||||
and therefore that the operators that produced these wave functions (\textit{i.e.} $H^\mu[n]$ 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 $\hat{W}_{\mathrm{ee}}^{\mathrm{lr},\mu}$
|
||||
and the effective one-body potential $\hat{\bar{V}}_{\mathrm{Hxc}}^{\mathrm{sr},\mu}[n]$ which is the functional derivative of the Hartree-exchange-correlation functional.
|
||||
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 role 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 $\hat{\bar{V}}_{\mathrm{Hxc}}^{\mathrm{sr},\mu}[n_{\Psi^{\mu}}]$,
|
||||
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 the plots in figures ~\ref{fig:n1} and~\ref{fig:n2} in the case of the water molecule.
|
||||
Regarding now the transcorrelated Hamiltonian $e^{-J}He^J$, as pointed out by Ten-No\cite{Ten-no2000Nov},
|
||||
|
||||
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Reference in New Issue
Block a user