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123
Manuscript/Bib.bib
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@ -1459,32 +1459,11 @@ number = {5},
pages = {2304-2311},
year = {2018},
doi = {10.1021/acs.jctc.7b01196},
note ={PMID: 29614856},
URL = {
https://doi.org/10.1021/acs.jctc.7b01196
},
eprint = {
https://doi.org/10.1021/acs.jctc.7b01196
note ={PMID: 29614856},
URL = {https://doi.org/10.1021/acs.jctc.7b01196},
eprint = {https://doi.org/10.1021/acs.jctc.7b01196}
}
}
@article{Dubecky2014,
author = {Dubecky, Matus and Derian, Rene and Jurecka, Petr and Mitas, Lubos and Hobza, Pavel and Otyepka, Michal},
doi = {10.1039/C4CP02093F},
file = {:home/yeluo/.local/share/data/Mendeley Ltd./Mendeley Desktop/Downloaded/Dubecky et al. - 2014 - Quantum Monte Carlo for Noncovalent Interactions An Efficient Protocol Attaining Benchmark Accuracy.pdf:pdf},
OPTissn = {1463-9076},
journal = {Physical Chemistry Chemical Physics},
mendeley-groups = {Quantum espresso},
pages = {20915--20923},
volume = {16},
title = {{Quantum Monte Carlo for Noncovalent Interactions: An Efficient Protocol Attaining Benchmark Accuracy}},
OPTurl = {http://pubs.rsc.org/en/Content/ArticleLanding/2014/CP/C4CP02093F},
year = {2014}
}
@article{Petruzielo2012,
author = {Petruzielo, F R and Toulouse, Julien and Umrigar, C J},
@ -1513,15 +1492,9 @@ number = {6},
pages = {2583-2597},
year = {2016},
doi = {10.1021/acs.jctc.6b00160},
note ={PMID: 27175914},
URL = {
http://dx.doi.org/10.1021/acs.jctc.6b00160
},
eprint = {
http://dx.doi.org/10.1021/acs.jctc.6b00160
note ={PMID: 27175914},
URL = {http://dx.doi.org/10.1021/acs.jctc.6b00160},
eprint = {http://dx.doi.org/10.1021/acs.jctc.6b00160}
}
@ -1678,15 +1651,10 @@ number = {7},
pages = {5648-5652},
year = {1993},
doi = {10.1063/1.464913},
URL = {
https://doi.org/10.1063/1.464913
},
eprint = {
https://doi.org/10.1063/1.464913
URL = {https://doi.org/10.1063/1.464913},
eprint = {https://doi.org/10.1063/1.464913}
}
@article{B3LYP2,
title = {Development of the Colle-Salvetti correlation-energy formula into a functional of the electron density},
author = {Lee, Chengteh and Yang, Weitao and Parr, Robert G.},
@ -1803,29 +1771,6 @@ eprint = {
pages = "110201"
}% linear method optimization
@ARTICLE{QP2,
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},
title = "{Quantum Package 2.0: An Open-Source Determinant-Driven Suite of Programs}",
journal = {arXiv e-prints},
keywords = {Physics - Computational Physics, Condensed Matter - Strongly Correlated Electrons, Physics - Atomic Physics, Physics - Chemical Physics},
year = "2019",
month = "Feb",
eid = {arXiv:1902.08154},
pages = {arXiv:1902.08154},
archivePrefix = {arXiv},
eprint = {1902.08154},
primaryClass = {physics.comp-ph},
adsurl = {https://ui.adsabs.harvard.edu/\#abs/2019arXiv190208154G},
adsnote = {Provided by the SAO/NASA Astrophysics Data System}
}
@article{sharma2017,
author = {Sharma, Sandeep and Holmes, Adam A. and Jeanmairet, Guillaume and Alavi, Ali and Umrigar, C. J.},
title = {Semistochastic Heat-Bath Configuration Interaction Method: Selected Configuration Interaction with Semistochastic Perturbation Theory},
@ -1974,16 +1919,8 @@ number = {4},
pages = {1746-1749},
year = {1985},
doi = {10.1063/1.449362},
URL = {
https://doi.org/10.1063/1.449362
},
eprint = {
https://doi.org/10.1063/1.449362
}
URL = {https://doi.org/10.1063/1.449362},
eprint = {https://doi.org/10.1063/1.449362}
}
@ -1998,7 +1935,6 @@ pages = {894-905},
doi = {10.1002/jcc.540080616},
url = {https://onlinelibrary.wiley.com/doi/abs/10.1002/jcc.540080616},
eprint = {https://onlinelibrary.wiley.com/doi/pdf/10.1002/jcc.540080616},
abstract = {Abstract A new algorithm for fitting atomic charges to molecular electrostatic potentials is presented. This method is non-iterative and rapid compared to previous work. Results from a variety of gaussian basis sets, including STO-3G, 3-21G and 6-31G*, are presented. Charges for a representative collection of molecules, comprising both first and second row atoms and anions are tabulated. The effects of using experimental and optimized geometries are explored. Charges derived from these fits are found to adequately reproduce SCF dipole moments. A small split valence representation, 3-21G, appears to yield consistently good results in a reasonable amount of time.},,
year = {1987}
}
@ -2097,28 +2033,21 @@ publisher={American Chemical Society},
address={[Washington, D.C.]}
}
@ARTICLE{QP2,
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},
title = "{Quantum Package 2.0: An Open-Source Determinant-Driven Suite of Programs}",
journal = {arXiv e-prints},
keywords = {Physics - Computational Physics, Condensed Matter - Strongly Correlated Electrons, Physics - Atomic Physics, Physics - Chemical Physics},
year = "2019",
month = "Feb",
eid = {arXiv:1902.08154},
pages = {arXiv:1902.08154},
archivePrefix = {arXiv},
eprint = {1902.08154},
primaryClass = {physics.comp-ph},
adsurl = {https://ui.adsabs.harvard.edu/\#abs/2019arXiv190208154G},
adsnote = {Provided by the SAO/NASA Astrophysics Data System}
}
@article{QP2,
Author = {Y. Garniron and K. Gasperich and T. Applencourt and A. Benali and A. Fert{\'e} and J. Paquier and B. Pradines and R. Assaraf and P. Reinhardt and J. Toulouse and P. Barbaresco and N. Renon and G. David and J. P. Malrieu and M. V{\'e}ril and M. Caffarel and P. F. Loos and E. Giner and A. Scemama},
Date-Added = {2019-04-07 13:54:16 +0200},
Date-Modified = {2019-06-12 14:59:52 +0200},
Doi = {10.1021/acs.jctc.9b00176},
Journal = {J. Chem. Theory Comput.},
Pages = {3591},
Title = {Quantum Package 2.0: A Open-Source Determinant-Driven Suite Of Programs},
Volume = {15},
Year = {2019},
Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.9b00176}}
@ARTICLE{2019JChPh.150h4103F,
author = {{Fert{\'e}}, Anthony and {Giner}, Emmanuel and {Toulouse}, Julien},
title = "{Range-separated multideterminant density-functional theory with a short-range correlation functional of the on-top pair density}",

85
Manuscript/main.tex
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@ -47,6 +47,8 @@ prb,
\usepackage{bm}% bold math
\usepackage{multirow}% Added by H. Shin to plot multi-row table
\usepackage{xcolor}
\usepackage{amstext,amsmath,amssymb,amsfonts,braket} % Package Maths
%\usepackage{hyperref}% add hypertext capabilities
%\usepackage[mathlines]{lineno}% Enable numbering of text and display math
%\linenumbers\relax % Commence numbering lines
@ -58,6 +60,21 @@ prb,
%%total={6.5in,8.75in}, top=1.2in, left=0.9in, includefoot,
%%height=10in,a5paper,hmargin={3cm,0.8in},
%]{geometry}
\newcommand{\beq}{\begin{equation}}
\newcommand{\eeq}{\end{equation}}
\newcommand{\beqq}{\begin{equation*}}
\newcommand{\eeqq}{\end{equation*}}
\newcommand{\bal}{\begin{align}}
\newcommand{\eal}{\end{align}}
\newcommand{\bcen}{\begin{center}}
\newcommand{\ecen}{\end{center}}
\newcommand{\tsp}{\textsuperscript}
\newcommand{\tsb}{\textsubscript}
\begin{document}
@ -133,6 +150,71 @@ The short-range correlation effects described by the Jastrow factor are partly t
In the ideal case, one would make the CIPSI determinant selection in the presence of the Jastrow factor to produce directly short CI expansions.
\section{Range-separated DFT in a nutshell}
\subsection{Exact equations}
The exact ground-state energy of a $N$-electron system with nuclei-electron potential $v_\mathrm{ne}(\textbf{r})$ can be expressed by the following minimization over $N$-representable densities $n$~\cite{Lev-PNAS-79,Lie-IJQC-83}
\begin{equation}
E_0 = \min_n \left\{ \mathcal{F}[n] + \int v_\mathrm{ne}(\textbf{r}) n(\textbf{r}) \mathrm{d} \textbf{r} \right\},
\label{E0}
\end{equation}
with the standard constrained-search universal density functional
\begin{equation}
\mathcal{F}[n] = \min_{\Psi\rightarrow n} \langle\Psi|\hat{T}+\hat{W}_\mathrm{ee}|\Psi \rangle,
\label{Fn}
\end{equation}
where $\hat{T}$ and $\hat{W}_\mathrm{ee}$ are the kinetic-energy and Coulomb electron-electron interaction operators, respectively. The minimizing multideterminant wave function in Eq.~\eqref{Fn} will be denoted by $\Psi[n]$.
In RS-DFT, the universal density functional is decomposed as~\cite{Sav-INC-96,TouColSav-PRA-04}
\begin{equation}
\mathcal{F}[n] = \mathcal{F}^{\mathrm{lr},\mu}[n] + \bar{E}_{\mathrm{Hxc}}^{\mathrm{sr,}\mu}[n],
\label{Fdecomp}
\end{equation}
where $\mathcal{F}^{\mathrm{lr},\mu}[n]$ is a long-range (lr) universal density functional
\beq
\label{lr_univ_fonc}
\mathcal{F}^{\mathrm{lr},\mu}[n]= \min_{\Psi\rightarrow n} \langle\Psi|\hat{T}+\hat{W}_\mathrm{ee}^{\mathrm{lr},\mu}|\Psi \rangle,
\eeq
and $\bar{E}_{\mathrm{Hxc}}^{\,\mathrm{sr,}\mu}[n]$ is the complementary short-range (sr) Hartree-exchange-correlation (Hxc) density functional. In Eq.~(\ref{lr_univ_fonc}), $\hat{W}_\mathrm{ee}^{\mathrm{lr}}$ is the long-range electron-electron interaction defined as
\begin{equation}
\hat{W}_\mathrm{ee}^{\mathrm{lr},\mu} = \frac{1}{2} \iint w_{\mathrm{ee}}^{\mathrm{lr},\mu}(r_{12}) \hat{n}_2(\textbf{r}_1,\textbf{r}_2)
\mathrm{d} \textbf{r}_1 \mathrm{d} \textbf{r}_2,
\end{equation}
with the error-function potential $w_{\mathrm{ee}}^{\mathrm{lr},\mu}(r_{12})=\mathrm{erf}(\mu\, r_{12} )/r_{12}$ (expressed with the interelectronic distance $r_{12} = ||\textbf{r}_1-\textbf{r}_2||$) and the pair-density operator $\hat{n}_2(\textbf{r}_1, \textbf{r}_2)=\hat{n}(\textbf{r}_1) \hat{n}(\textbf{r}_2) - \delta(\textbf{r}_1-\textbf{r}_2) \hat{n}(\textbf{r}_1)$ where $\hat{n}(\textbf{r})$ is the density operator. The range-separation parameter $\mu$ corresponds to an inverse distance controlling the range of the separation and the strength of the interaction at $r_{12} = 0$. For a given density, we will denote by $\Psi^\mu[n]$ the minimizing multideterminant wave function in Eq. ~\eqref{lr_univ_fonc}. Inserting the decomposition of Eq.~\eqref{Fdecomp} into Eq.~\eqref{E0}, and recomposing the two-step minimization into a single one, leads to the following expression for the exact ground-state electronic energy
\begin{eqnarray}
\label{min_rsdft}
E_0= \min_{\Psi} \Big\{ \langle\Psi|\hat{T}+\hat{W}_\mathrm{{ee}}^{\mathrm{lr},\mu}+\hat{V}_{\mathrm{ne}}|\Psi\rangle + \bar{E}^{\mathrm{sr},\mu}_{\mathrm{Hxc}}[n_\Psi]\Big\},
\nonumber\\
\end{eqnarray}
where the minimization is done over normalized $N$-electron multideterminant wave functions, $\hat{V}_{\mathrm{ne}} = \int v_{\mathrm{ne}} (\textbf{r}) \hat{n}(\textbf{r}) \mathrm{d} \textbf{r}$, and $n_\Psi$ refers to the density of $\Psi$, i.e. $n_\Psi(\textbf{r})=\langle\Psi|\hat{n}(\textbf{r})|\Psi\rangle$.
The minimizing multideterminant wave function $\Psi^\mu$ in Eq.~\eqref{min_rsdft} can be determined by the self-consistent eigenvalue equation
\beq
\label{rs-dft-eigen-equation}
\hat{H}^\mu[n_{\Psi^{\mu}}] \Ket{\Psi^{\mu}}= \mathcal{E}^{\mu} \Ket{\Psi^{\mu}},
\eeq
with the long-range interacting Hamiltonian
\beq
\label{H_mu}
\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}}],
\eeq
where $\hat{\bar{V}}_{\mathrm{Hxc}}^{\mathrm{sr},\mu}[n]=\int \delta \bar{E}^{\mathrm{sr},\mu}_{\mathrm{Hxc}}[n]/\delta n(\textbf{r}) \, \hat{n}(\textbf{r}) \mathrm{d} \textbf{r}$ is the complementary short-range Hartree-exchange-correlation potential operator. Note that $\Psi^{\mu}$ is not the exact physical ground-state wave function but only an effective wave function. However, its density $n_{\Psi^{\mu}}$ is the exact physical ground-state density. Once $\Psi^{\mu}$ has been calculated, the exact electronic ground-state energy is obtained by
\beq
\label{E-rsdft}
E_0= \braket{\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}].
\eeq
Note that, for $\mu=0$, the long-range interaction vanishes, $w_{\mathrm{ee}}^{\mathrm{lr},\mu=0}(r_{12}) = 0$, and thus RS-DFT reduces to standard KS-DFT. For $\mu\to\infty$, the long-range interaction becomes the standard Coulomb interaction, $w_{\mathrm{ee}}^{\mathrm{lr},\mu\to\infty}(r_{12}) = 1/r_{12}$, and thus RS-DFT reduces to standard wave-function theory (WFT).
\subsection{Approximations}
Provided that the exact short-range complementary functional is known and that the wave function $\Psi^{\mu}$ is developed in a complete basis set, the theory is exact and therefore provides the exact density and ground state energy whatever value of $\mu$ is chosen to split the universal Levy-Lieb functional (see Eqs. \eqref{Fdecomp} and \eqref{lr_univ_fonc}).
Of course in practice, two kind of approximations must be made: i) the approximation of the wave function by using a finite basis set, ii) the approximation on the exact complementary functionals.
As long as approximations are made, the theory is not exact anymore and might depend on $\mu$.
Despite its $\mu$ dependence, approximated RS-DFT schemes provide potentially appealing features as: i) the approximated wave function $\Psi^{\mu}$ is an eigenvector of a non-divergent operator $\hat{H}^\mu[n_{\Psi^{\mu}}]$ (see Eqs. \eqref{H_mu} and \eqref{rs-dft-eigen-equation}) and therefore converges more rapidly with respect to the basis set~\cite{FraMusLupTou-JCP-15}, and also produces more compact wave function~\cite{FerGinTou-JCP-18} as it will be illustrated here, ii) the semi-local approximations for RS-DFT complementary functionals are usually better suited to describe the remaining short-range part of the electron-electron correlation.
In this work, we use the short-range version of the Perdew-Burke-Ernzerhof (PBE)~\cite{PerBurErn-PRL-96} exchange and correlation functionals of Ref.~\onlinecite{GolWerStoLeiGorSav-CP-06} (see also Refs.~\onlinecite{TouColSav-JCP-05,GolWerSto-PCCP-05}) which takes the form
\begin{eqnarray}
\bar{E}^{\mathrm{sr},\mu,\textsc{pbe}}_{\mathrm{x/c}}[n] = \int \bar{e}_\mathrm{{x/c}}^\mathrm{sr,\mu,\textsc{pbe}}(n(\textbf{r}),\nabla n(\textbf{r})) \, \mathrm{d}\textbf{r}.
\end{eqnarray}
\section{\label{sec:Methods}Computational details}
All single determinant Hartree Fock (HF) and \textit{vanilla} density functional theory\cite{Hohenberg1964,kohn1965physrev} calculations were done using the PySCF code\cite{PYSCF}. To scan through multiple nodal surfaces, we used some of the most widely used XC functionals in the literature (LDA\cite{Kohn1965}, PBE\cite{PBE}, B3LYP\cite{B3LYP1,B3LYP2,B3LYP3,B3LYP4}, PBE0\cite{PBE,adamo1999jchemphys}, revPBE\cite{PhysRevLett.80.890}, and M06-L\cite{M06-L}).\\
@ -502,7 +584,8 @@ An award of computer time was provided by the Innovative and Novel Computational
% MDF citation - H. Shin 07/04
Input and output files for calculations are available on the Materials Data Facility, {\tt https://materialsdatafacility.org/}, DOI:To come.
\end{acknowledgments}
\bibliography{Bib}
\bibliography{Bib,biblio}
%\bibliography{Bib}
\end{document}
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